Networked Convergence of Fractional-Order Multiagent Systems with a Leader and Delay
Directory of Open Access Journals (Sweden)
Yuntao Shi
2015-01-01
Full Text Available This paper investigates the convergence of fractional-order discrete-time multiagent systems with a leader and sampling delay by using Hermite-Biehler theorem and the change of bilinearity. It is shown that such system can achieve convergence depending on the sampling interval h, the fractional-order α, and the sampling delay τ and its interconnection topology. Finally, some numerical simulations are given to illustrate the results.
Directory of Open Access Journals (Sweden)
Yuan Li
2017-03-01
Full Text Available In this study, an adaptive neural network synchronization (NNS approach, capable of guaranteeing prescribed performance (PP, is designed for non-identical fractional-order chaotic systems (FOCSs. For PP synchronization, we mean that the synchronization error converges to an arbitrary small region of the origin with convergence rate greater than some function given in advance. Neural networks are utilized to estimate unknown nonlinear functions in the closed-loop system. Based on the integer-order Lyapunov stability theorem, a fractional-order adaptive NNS controller is designed, and the PP can be guaranteed. Finally, simulation results are presented to confirm our results.
Li, Yuan; Lv, Hui; Jiao, Dongxiu
2017-03-01
In this study, an adaptive neural network synchronization (NNS) approach, capable of guaranteeing prescribed performance (PP), is designed for non-identical fractional-order chaotic systems (FOCSs). For PP synchronization, we mean that the synchronization error converges to an arbitrary small region of the origin with convergence rate greater than some function given in advance. Neural networks are utilized to estimate unknown nonlinear functions in the closed-loop system. Based on the integer-order Lyapunov stability theorem, a fractional-order adaptive NNS controller is designed, and the PP can be guaranteed. Finally, simulation results are presented to confirm our results.
DYNAMICS OF FRACTIONAL ORDER CHAOTIC SYSTEM
Directory of Open Access Journals (Sweden)
M. Jana
2017-02-01
Full Text Available This paper deals with the dynamics of chaos and synchronization for fractional order chaotic system. For fractional order derivative Captuo definition is used here and numerical simulations are done using Predictor-Correctors scheme by Diethlm based on the Adams-Baseforth-Moulton algorithm. Stability analysis is discussed here for non linear fractional order chaotic system and synchronization is achieved between two non identical fractional order chaotic systems: Finance chaotic system(driving systemand Lorenz system(response systemvia active control.Numerical simulations are performed to show the effectiveness of these approaches.
Control of Initialized Fractional-Order Systems
Hartly, Tom T.; Lorenzo, Carl F.
2002-01-01
Due to the importance of historical effects in fractional-order systems, this paper presents a general fractional-order control theory that includes the time-varying initialization response. Previous studies have not properly accounted for these historical effects. The initialization response, along with the forced response, for fractional-order systems is determined. Stability properties of fractional-order systems are presented in the complex Airplane, which is a transformation of the s-plane. Time responses are discussed with respect to pole positions in the complex Airplane and frequency response behavior is included. A fractional-order vector space representation, which is a generalization of the state space concept, is presented including the initialization response. Control methods for vector representations of initialized fractional-order systems are shown. Nyquist, root-locus, and other input-output control methods are adapted to the control of fractional-order systems. Finally, the fractional-order differintegral is generalized to continuous order-distributions that have the possibility of including a continuum of fractional orders in a system element.
Ping Zhou; Rui Ding
2012-01-01
The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
Directory of Open Access Journals (Sweden)
Ping Zhou
2012-01-01
Full Text Available The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
Characteristic ratio assignment in fractional order systems.
Tabatabaei, Mohammad; Haeri, Mohammad
2010-10-01
In this paper the characteristic ratios and generalized time constant are defined for all-pole commensurate fractional order systems. The sufficient condition for stability of these systems in terms of their characteristic ratios is obtained. Also an analytical approach for characteristic ratio assignment (CRA) to have a non-overshooting fast closed loop step response is introduced. The proposed CRA method is then employed to design a fractional order controller. Computer simulation results are presented to illustrate the performance of the CRA based designed fractional order controllers. Copyright © 2010 ISA. Published by Elsevier Ltd. All rights reserved.
Fractional-order adaptive fault estimation for a class of nonlinear fractional-order systems
N'Doye, Ibrahima
2015-07-01
This paper studies the problem of fractional-order adaptive fault estimation for a class of fractional-order Lipschitz nonlinear systems using fractional-order adaptive fault observer. Sufficient conditions for the asymptotical convergence of the fractional-order state estimation error, the conventional integer-order and the fractional-order faults estimation error are derived in terms of linear matrix inequalities (LMIs) formulation by introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach where the fractional-order α belongs to 0 < α < 1. A numerical example is given to demonstrate the validity of the proposed approach.
Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers
National Research Council Canada - National Science Library
Tianzeng Li; Yu Wang; Yong Yang
2014-01-01
In this paper, the synchronization of fractional-order chaotic systems is studied and a new fractional-order controller for hyperchaos synchronization is presented based on the Lyapunov stability theory...
Analogue Realization of Fractional-Order Dynamical Systems
Directory of Open Access Journals (Sweden)
Ladislav Pivka
2013-10-01
Full Text Available As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented.
Directory of Open Access Journals (Sweden)
Mohammadtaghi Hamidi Beheshti
2010-01-01
Full Text Available We propose a fractional-order controller to stabilize unstable fractional-order open-loop systems with interval uncertainty whereas one does not need to change the poles of the closed-loop system in the proposed method. For this, we will use the robust stability theory of Fractional-Order Linear Time Invariant (FO-LTI systems. To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of interval nonlinear systems and especially in fractional-order chaotic systems. Finally numerical simulations are presented to show the effectiveness of the proposed controller.
N U+02BC Doye, Ibrahima
2018-02-13
In this paper, we propose a robust fractional-order proportional-integral U+0028 FOPI U+0029 observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient conditions are derived in terms of linear matrix inequalities U+0028 LMIs U+0029 approach by using an indirect Lyapunov method. The proposed U+0028 FOPI U+0029 observer is robust against Lipschitz additive nonlinear uncertainty. It is also compared to the fractional-order proportional U+0028 FOP U+0029 observer and its performance is illustrated through simulations done on the fractional-order chaotic Lorenz system.
He, Shaobo; Sun, Kehui; Mei, Xiaoyong; Yan, Bo; Xu, Siwei
2017-01-01
In this paper, the numerical solutions of conformable fractional-order linear and nonlinear equations are obtained by employing the constructed conformable Adomian decomposition method (CADM). We found that CADM is an effective method for numerical solution of conformable fractional-order differential equations. Taking the conformable fractional-order simplified Lorenz system as an example, the numerical solution and chaotic behaviors of the conformable fractional-order simplified Lorenz system are investigated. It is found that rich dynamics exist in the conformable fractional-order simplified Lorenz system, and the minimum order for chaos is even less than 2. The results are validated by means of bifurcation diagram, Lyapunov characteristic exponents and phase portraits.
Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers
Directory of Open Access Journals (Sweden)
Tianzeng Li
2014-01-01
Full Text Available In this paper, the synchronization of fractional-order chaotic systems is studied and a new fractional-order controller for hyperchaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can be applied to an arbitrary four-dimensional fractional hyperchaotic system. And we give the optimal value of control parameters to achieve synchronization of fractional hyperchaotic system. This approach is universal, simple, and theoretically rigorous. Numerical simulations of several fractional-order hyperchaotic systems demonstrate the universality and the effectiveness of the proposed method.
Fractional order control and synchronization of chaotic systems
Vaidyanathan, Sundarapandian; Ouannas, Adel
2017-01-01
The book reports on the latest advances in and applications of fractional order control and synchronization of chaotic systems, explaining the concepts involved in a clear, matter-of-fact style. It consists of 30 original contributions written by eminent scientists and active researchers in the field that address theories, methods and applications in a number of research areas related to fractional order control and synchronization of chaotic systems, such as: fractional chaotic systems, hyperchaotic systems, complex systems, fractional order discrete chaotic systems, chaos control, chaos synchronization, jerk circuits, fractional chaotic systems with hidden attractors, neural network, fuzzy logic controllers, behavioral modeling, robust and adaptive control, sliding mode control, different types of synchronization, circuit realization of chaotic systems, etc. In addition to providing readers extensive information on chaos fundamentals, fractional calculus, fractional differential equations, fractional contro...
Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers
Tianzeng Li; Yu Wang; Yong Yang
2014-01-01
In this paper, the synchronization of fractional-order chaotic systems is studied and a new fractional-order controller for hyperchaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can be applied to an arbitrary four-dimensional fractional hyperchaotic system. And we give the optimal value of control parameters to achieve synchronization of fractional hyperchaotic system. This approach is universal, simple, and theoretically rigorous. Nu...
Directory of Open Access Journals (Sweden)
Sayyad Delshad Saleh
2010-01-01
Full Text Available Abstract We propose a fractional-order controller to stabilize unstable fractional-order open-loop systems with interval uncertainty whereas one does not need to change the poles of the closed-loop system in the proposed method. For this, we will use the robust stability theory of Fractional-Order Linear Time Invariant (FO-LTI systems. To determine the control parameters, one needs only a little knowledge about the plant and therefore, the proposed controller is a suitable choice in the control of interval nonlinear systems and especially in fractional-order chaotic systems. Finally numerical simulations are presented to show the effectiveness of the proposed controller.
Wang, L. M.
2017-09-01
A novel model-free adaptive sliding mode strategy is proposed for a generalized projective synchronization (GPS) between two entirely unknown fractional-order chaotic systems subject to the external disturbances. To solve the difficulties from the little knowledge about the master-slave system and to overcome the bad effects of the external disturbances on the generalized projective synchronization, the radial basis function neural networks are used to approach the packaged unknown master system and the packaged unknown slave system (including the external disturbances). Consequently, based on the slide mode technology and the neural network theory, a model-free adaptive sliding mode controller is designed to guarantee asymptotic stability of the generalized projective synchronization error. The main contribution of this paper is that a control strategy is provided for the generalized projective synchronization between two entirely unknown fractional-order chaotic systems subject to the unknown external disturbances, and the proposed control strategy only requires that the master system has the same fractional orders as the slave system. Moreover, the proposed method allows us to achieve all kinds of generalized projective chaos synchronizations by turning the user-defined parameters onto the desired values. Simulation results show the effectiveness of the proposed method and the robustness of the controlled system.
Zhou, Ping; Cheng, Yuan-Ming; Kuang, Fei
2010-09-01
Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.
Synchronization of Coupled Nonidentical Fractional-Order Hyperchaotic Systems
Directory of Open Access Journals (Sweden)
Zhouchao Wei
2011-01-01
Full Text Available Synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed by the active sliding mode method. By designing an active sliding mode controller and choosing proper control parameters, the master and slave systems are synchronized. Furthermore, synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is performed to show the effectiveness of the proposed controller.
Zhou, Ping; Bai, Rongji
2014-01-01
Based on a new stability result of equilibrium point in nonlinear fractional-order systems for fractional-order lying in 1 fractional-order Lorenz chaotic system with fractional-order 1 < q < 2 is considered. Numerical simulations show the validity and feasibility of the proposed scheme. PMID:25247207
Directory of Open Access Journals (Sweden)
Yimin Wu
2016-08-01
Full Text Available In this paper, we consider the control problem of a class of uncertain fractional-order chaotic systems preceded by unknown backlash-like hysteresis nonlinearities based on backstepping control algorithm. We model the hysteresis by using a differential equation. Based on the fractional Lyapunov stability criterion and the backstepping algorithm procedures, an adaptive neural network controller is driven. No knowledge of the upper bound of the disturbance and system uncertainty is required in our controller, and the asymptotical convergence of the tracking error can be guaranteed. Finally, we give two simulation examples to confirm our theoretical results.
Directory of Open Access Journals (Sweden)
Karthikeyan Rajagopal
2017-01-01
Full Text Available This paper introduces a fractional order memristor no equilibrium (FOMNE chaotic system and investigates its adaptive sliding mode synchronization. Firstly the dynamic properties of the integer order memristor no equilibrium system are analyzed. The fractional order memristor no equilibrium system is then derived from the integer order model. Lyapunov exponents and bifurcation with fractional order are investigated. An adaptive sliding mode control algorithm is derived to globally synchronize the identical fractional order memristor systems and genetically optimized fractional order PID controllers are designed and used to synchronize the FOMNE systems. Finally the fractional order memristor no equilibrium system is realized using FPGA.
Synchronization of chaotic fractional-order systems via linear control
Odibat, Zaid,; Corson, Nathalie; Aziz-Alaoui, Moulay; Bertelle, Cyrille
2010-01-01
International audience; The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived ...
Directory of Open Access Journals (Sweden)
Ping Zhou
2015-03-01
Full Text Available In this paper, a projective synchronization approach for a class of fractional-order chaotic systems with fractional-order 1 < q < 2 is demonstrated. The projective synchronization approach is established through precise theorization. To illustrate the effectiveness of the proposed scheme, we discuss two examples: (1 the fractional-order Lorenz chaotic system with fractional-order q = 1.1; (2 the fractional-order modified Chua’s chaotic system with fractional-order q = 1.02. The numerical simulations show the validity and feasibility of the proposed scheme.
Adaptive Synchronization for a Class of Uncertain Fractional-Order Neural Networks
Directory of Open Access Journals (Sweden)
Heng Liu
2015-10-01
Full Text Available In this paper, synchronization for a class of uncertain fractional-order neural networks subject to external disturbances and disturbed system parameters is studied. Based on the fractional-order extension of the Lyapunov stability criterion, an adaptive synchronization controller is designed, and fractional-order adaptation law is proposed to update the controller parameter online. The proposed controller can guarantee that the synchronization errors between two uncertain fractional-order neural networks converge to zero asymptotically. By using some proposed lemmas, the quadratic Lyapunov functions are employed in the stability analysis. Finally, numerical simulations are presented to confirm the effectiveness of the proposed method.
Synchronization in coupled nonidentical incommensurate fractional-order systems
Wang, Jun-Wei; Zhang, Yan-Bin
2009-12-01
Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.
Synchronization in coupled nonidentical incommensurate fractional-order systems
Energy Technology Data Exchange (ETDEWEB)
Wang Junwei, E-mail: wangjunweilj@yahoo.com.c [School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006 (China); Zhang Yanbin [School of Computer Science, Hangzhou Dianzi University, Hangzhou 310018 (China)
2009-12-28
Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.
Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems
Chen, Liping; Chai, Yi; Wu, Ranchao
2011-05-01
In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.
Synchronization of different fractional order chaotic systems using active control
Bhalekar, Sachin; Daftardar-Gejji, Varsha
2010-11-01
Synchronization of fractional order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this article we utilize active control technique to synchronize different fractional order chaotic dynamical systems. Further we investigate the interrelationship between the (fractional) order and synchronization in different chaotic dynamical systems. It is observed that synchronization is faster as the order tends to one.
Generalized Synchronization of Nonidentical Fractional-Order Chaotic Systems
Wang, Xing-Yuan; Hu, Zun-Wen; Luo, Chao
2013-10-01
In this paper, a chaotic synchronization scheme is proposed to achieve the generalized synchronization between two different fractional-order chaotic systems. Based on the stability theory of fractional-order systems and the pole placement technique, a controller is designed and theoretical proof is given. Two groups of examples are shown to verify the effectiveness of the proposed scheme, the first one is to realize the generalized synchronization between the fractional-order Chen system and the fractional-order Rössler system, the second one is between the fractional-order Lü system and the fractional-order hyperchaotic Lorenz system. The corresponding numerical simulations verify the effectiveness of the proposed scheme.
Stability Analysis of Fractional-Order Nonlinear Systems with Delay
Directory of Open Access Journals (Sweden)
Yu Wang
2014-01-01
Full Text Available Stability analysis of fractional-order nonlinear systems with delay is studied. We propose the definition of Mittag-Leffler stability of time-delay system and introduce the fractional Lyapunov direct method by using properties of Mittag-Leffler function and Laplace transform. Then some new sufficient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed firstly. And the application of Riemann-Liouville fractional-order systems is extended by the fractional comparison principle and the Caputo fractional-order systems. Numerical simulations of an example demonstrate the universality and the effectiveness of the proposed method.
LMI Conditions for Global Stability of Fractional-Order Neural Networks.
Zhang, Shuo; Yu, Yongguang; Yu, Junzhi
2017-10-01
Fractional-order neural networks play a vital role in modeling the information processing of neuronal interactions. It is still an open and necessary topic for fractional-order neural networks to investigate their global stability. This paper proposes some simplified linear matrix inequality (LMI) stability conditions for fractional-order linear and nonlinear systems. Then, the global stability analysis of fractional-order neural networks employs the results from the obtained LMI conditions. In the LMI form, the obtained results include the existence and uniqueness of equilibrium point and its global stability, which simplify and extend some previous work on the stability analysis of the fractional-order neural networks. Moreover, a generalized projective synchronization method between such neural systems is given, along with its corresponding LMI condition. Finally, two numerical examples are provided to illustrate the effectiveness of the established LMI conditions.
Chaotic Synchronization of Fractional-Order Spatiotemporal Coupled Lorenz System
Wang, Xing-Yuan; Zhang, Hao
2012-10-01
By utilizing the fractional calculus techniques and spatiotemporal chaos theory, this paper brings Lorenz system to fractional-order spatiotemporal coupled differential equation for the first time, and proposes the fractional-order spatiotemporal coupled Lorenz system. Based on that, we study the problem of chaotic synchronization of fractional-order spatiotemporal coupled Lorenz systems, design the linear controller and nonlinear controller by utilizing the Lyapunov stability theory and prove the correctness in theory. The numerical simulation results demonstrate the validity of controllers in high-dimension fractional-order spatiotemporal coupled Lorenz system.
Stability analysis of fractional-order Hopfield neural networks with time delays.
Wang, Hu; Yu, Yongguang; Wen, Guoguang
2014-07-01
This paper investigates the stability for fractional-order Hopfield neural networks with time delays. Firstly, the fractional-order Hopfield neural networks with hub structure and time delays are studied. Some sufficient conditions for stability of the systems are obtained. Next, two fractional-order Hopfield neural networks with different ring structures and time delays are developed. By studying the developed neural networks, the corresponding sufficient conditions for stability of the systems are also derived. It is shown that the stability conditions are independent of time delays. Finally, numerical simulations are given to illustrate the effectiveness of the theoretical results obtained in this paper. Copyright © 2014 Elsevier Ltd. All rights reserved.
Control of Initialized Fractional-Order Systems. Revised
Hartley, Tom T.; Lorenzo, Carl F.
2002-01-01
Due to the importance of historical effects in fractional-order systems, this paper presents a general fractional-order control theory that includes the time-varying initialization response. Previous studies have not properly accounted for these historical effects. The initialization response, along with the forced response, for fractional-order systems is determined. Stability properties of fractional-order systems are presented in the complex w-plane, which is a transformation of the s-plane. Time responses are discussed with respect to pole positions in the complex w-plane and frequency response behavior is included. A fractional-order vector space representation, which is a generalization of the state space concept, is presented including the initialization response. Control methods for vector representations of initialized fractional-order systems are shown. Nyquist, root-locus, and other input-output control methods are adapted to the control of fractional-order systems. Finally, the fractional-order differintegral is generalized to continuous order-distributions that have the possibility of including a continuum of fractional orders in a system element.
Qiong Huang; Chunyang Dong; Qianbin Chen
2014-01-01
A fractional-order scalar controller which involves only one state variable is proposed. By this fractional-order scalar controller, the unstable equilibrium points in the fractional-order Chen chaotic system can be asymptotically stable. The present control strategy is theoretically rigorous. Some circuits are designed to realize these control schemes. The outputs of circuit agree with the results of theoretical results.
Diverse Structure Synchronization of Fractional Order Hyper-Chaotic Systems
Wang, Xing-Yuan; Zhao, Guo-Bin; Yang, Yu-Hong
2013-04-01
This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor-corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient.
Parameter identification and synchronization of fractional-order chaotic systems
Yuan, Li-Guo; Yang, Qi-Gui
2012-01-01
The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.
Theory of fractional order elements based impedance matching networks
Radwan, Ahmed G.
2011-03-01
Fractional order circuit elements (inductors and capacitors) based impedance matching networks are introduced for the first time. In comparison to the conventional integer based L-type matching networks, fractional matching networks are much simpler and versatile. Any complex load can be matched utilizing a single series fractional element, which generally requires two elements for matching in the conventional approach. It is shown that all the Smith chart circles (resistance and reactance) are actually pairs of completely identical circles. They appear to be single for the conventional integer order case, where the identical circles completely overlap each other. The concept is supported by design equations and impedance matching examples. © 2010 IEEE.
Active disturbance rejection control for fractional-order system.
Li, Mingda; Li, Donghai; Wang, Jing; Zhao, Chunzhe
2013-05-01
Fractional-order proportional-integral (PI) and proportional-integral-derivative (PID) controllers are the most commonly used controllers in fractional-order systems. However, this paper proposes a simple integer-order control scheme for fractional-order system based on active disturbance rejection method. By treating the fractional-order dynamics as a common disturbance and actively rejecting it, active disturbance rejection control (ADRC) can achieve the desired response. External disturbance, sensor noise, and parameter disturbance are also estimated using extended state observer. The ADRC stability of rational-order model is analyzed. Simulation results on three typical fractional-order systems are provided to demonstrate the effectiveness of the proposed method. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Prescribed performance synchronization for fractional-order chaotic systems
Liu, Heng; Li, Sheng-Gang; Sun, Ye-Guo; Wang, Hong-Xing
2015-09-01
In this paper the synchronization for two different fractional-order chaotic systems, capable of guaranteeing synchronization error with prescribed performance, is investigated by means of the fractional-order control method. By prescribed performance synchronization we mean that the synchronization error converges to zero asymptotically, with convergence rate being no less than a certain prescribed function. A fractional-order synchronization controller and an adaptive fractional-order synchronization controller, which can guarantee the prescribed performance of the synchronization error, are proposed for fractional-order chaotic systems with and without disturbances, respectively. Finally, our simulation studies verify and clarify the proposed method. Project supported by the National Natural Science Foundation of China (Grant Nos. 11401243 and 61403157), the Fundamental Research Funds for the Central Universities of China (Grant No. GK201504002), and the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (Grant No. KJ2015A256).
Function projective lag synchronization of fractional-order chaotic systems
Wang, Sha; Yu, Yong-Guang; Wang, Hu; Ahmed, Rahmani
2014-04-01
Function projective lag synchronization of different structural fractional-order chaotic systems is investigated. It is shown that the slave system can be synchronized with the past states of the driver up to a scaling function matrix. According to the stability theorem of linear fractional-order systems, a nonlinear fractional-order controller is designed for the synchronization of systems with the same and different dimensions. Especially, for two different dimensional systems, the synchronization is achieved in both reduced and increased dimensions. Three kinds of numerical examples are presented to illustrate the effectiveness of the scheme.
Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks
Directory of Open Access Journals (Sweden)
Guanjun Li
2016-02-01
Full Text Available Stability of a class of fractional-order neural networks (FONNs is analyzed in this paper. First, two sufficient conditions for convergence of the solution for such systems are obtained by utilizing Gronwall–Bellman lemma and Laplace transform technique. Then, according to the fractional-order Lyapunov second method and linear feedback control, the synchronization problem between two fractional-order chaotic neural networks is investigated. Finally, several numerical examples are presented to justify the feasibility of the proposed methods.
Robust Synchronisation of Uncertain Fractional-Order Chaotic Unified Systems
Directory of Open Access Journals (Sweden)
Noghredani Naeimadeen
2017-04-01
Full Text Available Fractional-order chaotic unified systems include a variety of fractional-order chaotic systems such as Chen, Lorenz, Lu, Liu, and financial systems. This paper describes a sliding mode controller for synchronisation of fractional-order chaotic unified systems in the presence of uncertainties and external disturbances, and affirms the stability of the controller (which is composed of error dynamics. Moreover, the synchronisation of two separate fractional-order chaotic systems is studied. For this aim, fractional integral sliding surface is defined. Then the sliding mode control rule for stability of error dynamic is presented based on the Lyapunov stability theorem. Simulation results, obtained by using MATLAB, show that the proposed sliding mode has employed an appropriate approach against uncertainties and to reduce the chattering phenomenon that often occurs with sliding mode controllers.
Projective synchronization in fractional order chaotic systems and its control
Li, Chunguang
2006-01-01
The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.
Projective Synchronization in Fractional Order Chaotic Systems and Its Control
Chunguang, Li; Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China
2006-01-01
The chaotic dynamics of fractional (non-integer) order systems has begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.
Directory of Open Access Journals (Sweden)
Qiong Huang
2014-01-01
Full Text Available A fractional-order scalar controller which involves only one state variable is proposed. By this fractional-order scalar controller, the unstable equilibrium points in the fractional-order Chen chaotic system can be asymptotically stable. The present control strategy is theoretically rigorous. Some circuits are designed to realize these control schemes. The outputs of circuit agree with the results of theoretical results.
FRACTIONAL ORDER SYSTEM IDENTIFICATION BASED ON GENETIC ALGORITHMS
Directory of Open Access Journals (Sweden)
MAZIN Z. OTHMAN
2013-12-01
Full Text Available System identification deals with estimating the plant parameters under control using input-output measuring data. Most of practical plants have fractional order dynamic properties which are based on integration and differentiation of noninteger order. In this work the structure and the parameters of fractional order unknown transfer function are estimated using input-output data. Integer order Least Squares identification is used first to confirm the structure (order of the unknown transfer function. Then, Genetic Algorithms (GAs is followed to find the most accurate fractional order estimate that represents the system. Illustrative examples are presented in which fractional order transfer functions are identified in a way that faithfully estimates the dynamics of the unknown plants.
The Active Fractional Order Control for Maglev Suspension System
Directory of Open Access Journals (Sweden)
Peichang Yu
2015-01-01
Full Text Available Maglev suspension system is the core part of maglev train. In the practical application, the load uncertainties, inherent nonlinearity, and misalignment between sensors and actuators are the main issues that should be solved carefully. In order to design a suitable controller, the attention is paid to the fractional order controller. Firstly, the mathematical model of a single electromagnetic suspension unit is derived. Then, considering the limitation of the traditional PD controller adaptation, the fractional order controller is developed to obtain more excellent suspension specifications and robust performance. In reality, the nonlinearity affects the structure and the precision of the model after linearization, which will degrade the dynamic performance. So, a fractional order controller is addressed to eliminate the disturbance by adjusting the parameters which are added by the fractional order controller. Furthermore, the controller based on LQR is employed to compare with the fractional order controller. Finally, the performance of them is discussed by simulation. The results illustrated the validity of the fractional order controller.
Yang, Hongyong; Han, Fujun; Zhao, Mei; Zhang, Shuning; Yue, Jun
2017-08-01
Because many networked systems can only be characterized with fractional-order dynamics in complex environments, fractional-order calculus has been studied deeply recently. When diverse individual features are shown in different agents of networked systems, heterogeneous fractional-order dynamics will be used to describe the complex systems. Based on the distinguishing properties of agents, heterogeneous fractional-order multi-agent systems (FOMAS) are presented. With the supposition of multiple leader agents in FOMAS, distributed containment control of FOMAS is studied in directed weighted topologies. By applying Laplace transformation and frequency domain theory of the fractional-order operator, an upper bound of delays is obtained to ensure containment consensus of delayed heterogenous FOMAS. Consensus results of delayed FOMAS in this paper can be extended to systems with integer-order models. Finally, numerical examples are used to verify our results.
Yang, Li-xin; Jiang, Jun
2014-05-01
This paper investigates the adaptive synchronization in the drive-response fractional-order dynamical networks with uncertain parameters. By means of both the stability theory of fractional-order differential system and the adaptive control technique, a novel adaptive synchronization controller is developed with a more general and simpler analytical expression, which does not contain the parameters of the complex network, and effective adaptive laws of parameters. Furthermore, the very strong and conservative uniformly Lipschitz condition on the node dynamics of complex network is released. To demonstrate the validity of the proposed method, the examples for the synchronization of systems with the chaotic and hyper-chaotic node dynamics are presented.
Comparative study of synchronization methods of fractional order chaotic systems
Singh, Ajit K.; Yadav, Vijay K.; Das, S.
2016-09-01
In this article, the active control method and the backstepping method are used during the synchronization of fractional order chaotic systems. The salient feature of the article is the analysis of time of synchronization between fractional order Chen and Qi systems using both the methods. Numerical simulation and graphical results clearly exhibit that backstepping approach is better than active control method for synchronization of the considered pair of systems, as it takes less time to synchronize while using the first one compare to second one.
The synchronisation of fractional-order hyperchaos compound system
Indian Academy of Sciences (India)
Naeimadeen Noghredani
2018-01-24
Jan 24, 2018 ... security of secure communication. The corresponding theoretical analysis and results of simulation validated the effectiveness of the proposed synchronisation scheme using MATLAB. Keywords. Compound systems; memristor chaotic system; fractional-order chaotic systems; synchronisation. PACS Nos ...
Robust Stability Analysis of Fractional-Order Hopfield Neural Networks with Parameter Uncertainties
Directory of Open Access Journals (Sweden)
Shuo Zhang
2014-01-01
Full Text Available The issue of robust stability for fractional-order Hopfield neural networks with parameter uncertainties is investigated in this paper. For such neural system, its existence, uniqueness, and global Mittag-Leffler stability of the equilibrium point are analyzed by employing suitable Lyapunov functionals. Based on the fractional-order Lyapunov direct method, the sufficient conditions are proposed for the robust stability of the studied networks. Moreover, robust synchronization and quasi-synchronization between the class of neural networks are discussed. Furthermore, some numerical examples are given to show the effectiveness of our obtained theoretical results.
Stability and synchronization of memristor-based fractional-order delayed neural networks.
Chen, Liping; Wu, Ranchao; Cao, Jinde; Liu, Jia-Bao
2015-11-01
Global asymptotic stability and synchronization of a class of fractional-order memristor-based delayed neural networks are investigated. For such problems in integer-order systems, Lyapunov-Krasovskii functional is usually constructed, whereas similar method has not been well developed for fractional-order nonlinear delayed systems. By employing a comparison theorem for a class of fractional-order linear systems with time delay, sufficient condition for global asymptotic stability of fractional memristor-based delayed neural networks is derived. Then, based on linear error feedback control, the synchronization criterion for such neural networks is also presented. Numerical simulations are given to demonstrate the effectiveness of the theoretical results. Copyright © 2015 Elsevier Ltd. All rights reserved.
Active Control of a Chaotic Fractional Order Economic System
Directory of Open Access Journals (Sweden)
Haci Mehmet Baskonus
2015-08-01
Full Text Available In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.
Chaos synchronization between two different fractional-order hyperchaotic systems
Pan, Lin; Zhou, Wuneng; Zhou, Long; Sun, Kehui
2011-06-01
This work investigates chaos synchronization between two different fractional-order hyperchaotic system (FOHS)s. A novel FOHS is also proposed in this paper. The Chen FOHS is controlled to be a new FOHS and the Lü FOHS, respectively. The analytical conditions for the synchronization of these pairs of different FOHSs are derived by utilizing Laplace transform. Furthermore, synchronization between two different FOHSs is achieved by utilizing feedback control method in a quite short period and both remain in chaotic states. Numerical simulations are used to verify the theoretical analysis using different values of the fractional-order parameter.
Genetic Algorithm-Based Identification of Fractional-Order Systems
Directory of Open Access Journals (Sweden)
Shengxi Zhou
2013-05-01
Full Text Available Fractional calculus has become an increasingly popular tool for modeling the complex behaviors of physical systems from diverse domains. One of the key issues to apply fractional calculus to engineering problems is to achieve the parameter identification of fractional-order systems. A time-domain identification algorithm based on a genetic algorithm (GA is proposed in this paper. The multi-variable parameter identification is converted into a parameter optimization by applying GA to the identification of fractional-order systems. To evaluate the identification accuracy and stability, the time-domain output error considering the condition variation is designed as the fitness function for parameter optimization. The identification process is established under various noise levels and excitation levels. The effects of external excitation and the noise level on the identification accuracy are analyzed in detail. The simulation results show that the proposed method could identify the parameters of both commensurate rate and non-commensurate rate fractional-order systems from the data with noise. It is also observed that excitation signal is an important factor influencing the identification accuracy of fractional-order systems.
The adaptive synchronization of fractional-order Liu chaotic system ...
Indian Academy of Sciences (India)
In this paper, the chaos control and the synchronization of two fractional-order Liu chaotic systems with unknown parameters are studied. According to the Lyapunov stabilization theory and the adaptive control theorem, the adaptive control rule is obtained for the described error dynamic stabilization. Using the adaptive rule ...
Inverse synchronization of coupled fractional-order systems through ...
Indian Academy of Sciences (India)
A general explicit coupling via an open-plus-closed-loop control for inverse synchronization of two arbitrary unidirectionally or bidirectionally coupled fractional-order systems is proposed. The inverse synchronization is proved analytically based on the stability theorem of the fractional differential equations. A key feature of ...
Fractional-order Systems and Synchronous Generator Voltage Regulator
Directory of Open Access Journals (Sweden)
Wojciech Lubośny
2015-03-01
Full Text Available Modern regulators of synchronous generators, including voltage regulators, are digital systems, in their vast majority with standard structures contained in the IEEE standard. These are systems described with stationary differential equations of integral order. Differential equations of fractional order are not employed in regulators for synchronous generator control. This paper presents an analysis of the possibilities of using a system of fractional differential equations in the voltage regulator of an synchronous generator with a static excitation system.
Synchronization of a Class of Fractional-Order Chaotic Neural Networks
Directory of Open Access Journals (Sweden)
Yi Chai
2013-08-01
Full Text Available The synchronization problem is studied in this paper for a class of fractional-order chaotic neural networks. By using the Mittag-Leffler function, M-matrix and linear feedback control, a sufficient condition is developed ensuring the synchronization of such neural models with the Caputo fractional derivatives. The synchronization condition is easy to verify, implement and only relies on system structure. Furthermore, the theoretical results are applied to a typical fractional-order chaotic Hopfield neural network, and numerical simulation demonstrates the effectiveness and feasibility of the proposed method.
Active control technique of fractional-order chaotic complex systems
Mahmoud, Gamal M.; Ahmed, Mansour E.; Abed-Elhameed, Tarek M.
2016-06-01
Several kinds of synchronization of fractional-order chaotic complex systems are challenging research topics of current interest since they appear in many applications in applied sciences. Our main goal in this paper is to introduce the definition of modified projective combination-combination synchronization (MPCCS) of some fractional-order chaotic complex systems. We show that our systems are chaotic by calculating their Lyapunov exponents. The fractional Lyapunov dimension of the chaotic solutions of these systems is computed. A scheme is introduced to calculate MPCCS of four different (or identical) chaotic complex systems using the active control technique. Special cases of this type, which are projective and anti C-C synchronization, are discussed. Some figures are plotted to show that MPCCS is achieved and its errors approach zero.
Chaotic fractional-order Coullet system: Synchronization and control approach
Shahiri, M.; Ghaderi, R.; Ranjbar N., A.; Hosseinnia, S. H.; Momani, S.
2010-03-01
The main objective of this study is to investigate and control the chaotic behaviour of fractional-order Coullet system, including the necessary condition for appearance of chaos. An active control technique, in a master-slave structure for synchronization is suggested to control this chaotic system. The stability condition is obtained in both of theoretical analysis and simulation manner. The numerical simulation verifies the performance of the proposed controller.
Identification of fractional order systems using modulating functions method
Liu, Dayan
2013-06-01
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations.
The Active Fractional Order Control for Maglev Suspension System
Peichang Yu; Jie Li; Jinhui Li
2015-01-01
Maglev suspension system is the core part of maglev train. In the practical application, the load uncertainties, inherent nonlinearity, and misalignment between sensors and actuators are the main issues that should be solved carefully. In order to design a suitable controller, the attention is paid to the fractional order controller. Firstly, the mathematical model of a single electromagnetic suspension unit is derived. Then, considering the limitation of the traditional PD controller adaptat...
Electronic realization of the fractional-order systems
Directory of Open Access Journals (Sweden)
Františka Dorčáková
2007-10-01
Full Text Available This article is devoted to the electronic (analogue realization of the fractional-order systems – controllers or controlled objects whose we earlier used, identified, and analyzed as a mathematical models only ��� namely a fractional-order differential equation, and solved numerically using a method based on the truncated version of the Grunwald - Letnikov formula for fractional derivative. The electronic realization of the fractional derivative is based on the continued fraction expansion of the rational approximation of the fractional differentiator from which we obtained the values of the resistors and capacitors of the electronic circuit. Along with the mathematical description are presented also simulation and measurement results.
Projective Synchronization of Fractional Order Chaotic Systems Based on State Observer
Wang, Xing-Yuan; Hu, Zun-Wen
2012-12-01
Based on the stability theory of fractional order systems and the pole placement technique, this paper designs a synchronization scheme with the state observer method and achieves the projective synchronization of a class of fractional order chaotic systems. Taking an example for the fractional order unified system by using this observer controller, and numerical simulations of fractional order Lorenz-like system, fractional order Lü system and fractional order Chen system are provided to demonstrate the effectiveness of the proposed scheme.
Global asymptotical ω-periodicity of a fractional-order non-autonomous neural networks.
Chen, Boshan; Chen, Jiejie
2015-08-01
We study the global asymptotic ω-periodicity for a fractional-order non-autonomous neural networks. Firstly, based on the Caputo fractional-order derivative it is shown that ω-periodic or autonomous fractional-order neural networks cannot generate exactly ω-periodic signals. Next, by using the contraction mapping principle we discuss the existence and uniqueness of S-asymptotically ω-periodic solution for a class of fractional-order non-autonomous neural networks. Then by using a fractional-order differential and integral inequality technique, we study global Mittag-Leffler stability and global asymptotical periodicity of the fractional-order non-autonomous neural networks, which shows that all paths of the networks, starting from arbitrary points and responding to persistent, nonconstant ω-periodic external inputs, asymptotically converge to the same nonconstant ω-periodic function that may be not a solution. Copyright © 2015 Elsevier Ltd. All rights reserved.
Synchronisation of fractional-order time delayed chaotic systems with ring connection
He, S.; Sun, K.; Wang, H.
2016-02-01
In this paper, synchronisation of fractional-order time delayed chaotic systems in ring networks is investigated. Based on Lyapunov stability theory, a new generic synchronisation criterion for N-coupled chaotic systems with time delay is proposed. The synchronisation scheme is applied to N-coupled fractional-order time delayed simplified Lorenz systems, and the Adomian decomposition method (ADM) is developed for solving these chaotic systems. Performance analysis of the synchronisation network is carried out. Numerical experiments demonstrate that synchronisation realises in both state variables and intermediate variables, which verifies the effectiveness of the proposed method.
Chaotic synchronization of a fractional-order system based on washout filter control
Zhou, Shangbo; Lin, Xiaoran; Li, Hua
2011-03-01
In this paper, chaos in a fractional-order neural network system with varying time delays is presented, and chaotic synchronization system with varying time delays is constructed. The stability of constructed synchronization system is analyzed by Laplace transformation theory. In addition, the bifurcation graph of the chaotic system is illustrated. The study results show that the chaos in such fractional-order neural networks with varying time delay can be synchronized, and Washout filter control can be used to reduce the range of coupled parameter.
Mata-Machuca, Juan L.; Aguilar-López, Ricardo
2018-01-01
This work deals with the adaptative synchronization of complex dynamical networks with fractional-order nodes and its application in secure communications employing chaotic parameter modulation. The complex network is composed of multiple fractional-order systems with mismatch parameters and the coupling functions are given to realize the network synchronization. We introduce a fractional algebraic synchronizability condition (FASC) and a fractional algebraic identifiability condition (FAIC) which are used to know if the synchronization and parameters estimation problems can be solved. To overcome these problems, an adaptative synchronization methodology is designed; the strategy consists in proposing multiple receiver systems which tend to follow asymptotically the uncertain transmitters systems. The coupling functions and parameters of the receiver systems are adjusted continually according to a convenient sigmoid-like adaptative controller (SLAC), until the measurable output errors converge to zero, hence, synchronization between transmitter and receivers is achieved and message signals are recovered. Indeed, the stability analysis of the synchronization error is based on the fractional Lyapunov direct method. Finally, numerical results corroborate the satisfactory performance of the proposed scheme by means of the synchronization of a complex network consisting of several fractional-order unified chaotic systems.
Robust outer synchronization between two complex networks with fractional order dynamics.
Asheghan, Mohammad Mostafa; Míguez, Joaquín; Hamidi-Beheshti, Mohammad T; Tavazoei, Mohammad Saleh
2011-09-01
Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics.
Stabilization of fractional-order singular uncertain systems.
Ji, Yude; Qiu, Jiqing
2015-05-01
This paper focuses on the state and static output feedback stabilization for fractional-order singular (FOS) uncertain linear systems with the fractional commensurate order 0controllers that guarantee the stability of resulting closed-loop control systems. First, the sufficient conditions for robust asymptotical stability of the closed-loop control systems are presented. Next, based on the matrix׳s singular value decomposition (SVD) and linear matrix inequality (LMI) technics, some new results in the form of LMI are developed to the state and static output feedback controller synthesis for the FOS systems. Finally, three numerical examples are given to illustrate the effectiveness of the proposed design methods. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Synchronization-based parameter estimation of fractional-order neural networks
Gu, Yajuan; Yu, Yongguang; Wang, Hu
2017-10-01
This paper focuses on the parameter estimation problem of fractional-order neural network. By combining the adaptive control and parameter update law, we generalize the synchronization-based identification method that has been reported in several literatures on identifying unknown parameters of integer-order systems. With this method, parameter identification and synchronization can be achieved simultaneously. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.
On chaos control and synchronization of the commensurate fractional order Liu system
Hegazi, A. S.; Ahmed, E.; Matouk, A. E.
2013-05-01
In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigated. We also obtain the necessary condition for the existence of chaotic attractors in the commensurate fractional order Liu system. The effect of fractional order on chaos control of this system is revealed by showing that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, we achieve chaos synchronization between the commensurate fractional order Liu system and its integer order counterpart via function projective synchronization. Numerical simulations are used to verify the analytical results.
Synchronization of fractional-order linear complex networks with directed coupling topology
Fang, Qingxiang; Peng, Jigen
2018-01-01
The synchronization of fractional-order complex networks with general linear dynamics under directed connected topology is investigated. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding independent subsystems by use of a pseudo-state transformation technique and real Jordan canonical form of matrix. Sufficient conditions in terms of linear matrix inequalities for synchronization are established according to stability theory of fractional-order differential equations. In a certain range of fractional order, the effects of the fractional order on synchronization is clearly revealed. Conclusions obtained in this paper generalize the existing results. Three numerical examples are provided to illustrate the validity of proposed conclusions.
Ding, Zhixia; Shen, Yi
2016-04-01
This paper investigates global projective synchronization of nonidentical fractional-order neural networks (FNNs) based on sliding mode control technique. We firstly construct a fractional-order integral sliding surface. Then, according to the sliding mode control theory, we design a sliding mode controller to guarantee the occurrence of the sliding motion. Based on fractional Lyapunov direct methods, system trajectories are driven to the proposed sliding surface and remain on it evermore, and some novel criteria are obtained to realize global projective synchronization of nonidentical FNNs. As the special cases, some sufficient conditions are given to ensure projective synchronization of identical FNNs, complete synchronization of nonidentical FNNs and anti-synchronization of nonidentical FNNs. Finally, one numerical example is given to demonstrate the effectiveness of the obtained results. Copyright © 2016 Elsevier Ltd. All rights reserved.
Fractional-order gradient descent learning of BP neural networks with Caputo derivative.
Wang, Jian; Wen, Yanqing; Gou, Yida; Ye, Zhenyun; Chen, Hua
2017-05-01
Fractional calculus has been found to be a promising area of research for information processing and modeling of some physical systems. In this paper, we propose a fractional gradient descent method for the backpropagation (BP) training of neural networks. In particular, the Caputo derivative is employed to evaluate the fractional-order gradient of the error defined as the traditional quadratic energy function. The monotonicity and weak (strong) convergence of the proposed approach are proved in detail. Two simulations have been implemented to illustrate the performance of presented fractional-order BP algorithm on three small datasets and one large dataset. The numerical simulations effectively verify the theoretical observations of this paper as well. Copyright © 2017 Elsevier Ltd. All rights reserved.
Rabah, Karima; Ladaci, Samir; Lashab, Mohamed
2017-09-01
In this paper, a new design of fractional-order sliding mode control scheme is proposed for the synchronization of a class of nonlinear fractional-order systems with chaotic behaviour. The considered design approach provides a set of fractional-order laws that guarantee asymptotic stability of fractional-order chaotic systems in the sense of the Lyapunov stability theorem. Two illustrative simulation examples on the fractional-order Genesio-Tesi chaotic systems and the fractional-order modified Jerk systems are provided. These examples show the effectiveness and robustness of this control solution.
A general fractional-order dynamical network: synchronization behavior and state tuning.
Wang, Junwei; Xiong, Xiaohua
2012-06-01
A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Rössler dynamics, respectively.
Directory of Open Access Journals (Sweden)
Liping Chen
2012-01-01
between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.
Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control
Wang, Xing-Yuan; Song, Jun-Mei
2009-08-01
Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems.
Inverse synchronization of coupled fractional-order systems through ...
Indian Academy of Sciences (India)
A general explicit coupling via an open-plus-closed-loop control for inverse synchronization of two arbitrary unidirectionally or bidirectionally coupled fractional-order sys- tems is proposed. The inverse synchronization is proved analytically based on the stability theorem of the fractional differential equations. A key feature of ...
Chen, Liping; Chai, Yi; Wu, Ranchao
2011-12-01
This paper is devoted to synchronization of uncertain fractional-order chaotic systems with fractional-order α: 0 fractional-order differential system and the observer-based robust control, two sufficient and necessary conditions for synchronizing uncertain fractional-order chaotic systems with parameter perturbations are presented in terms of linear matrix inequality, which is an efficient method and could be easily solved by the toolbox of MATLAB. Finally, fractional-order uncertain chaotic Lü system with fractional-order α = 0.95 and fractional-order uncertain chaotic Lorenz system with fractional-order α = 1.05 are taken as numerical examples to show the validity and feasibility of the proposed method.
Lag Full State Hybrid Projective Synchronization in Different Fractional-Order Chaotic Systems
Tang, Yang; Fang, Jian-An; Chen, Liang
In this paper, lag full state hybrid projective synchronization (LFSHPS) in fractional-order chaotic systems is first studied. We show that LFSHPS does exist in fractional-order chaotic systems. Based on active control theory, synchronization schemes for LFSHPS of the fractional-order chaotic systems are given. Numerical simulations are provided to illustrate and verify the effectiveness of the proposed methods.
Chaos Synchronization of Fractional Order Unified Chaotic System via Nonlinear Control
Chen, Xiang Rong; Liu, Chong Xin
Based on the stability theory of fractional order systems, an effective but theoretically rigorous nonlinear control method is proposed to synchronize the fractional order chaotic systems. Using this method, chaos synchronization between two identical fractional order unified systems is studied. Simulation results are shown to illustrate the effectiveness of this method.
Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems
Directory of Open Access Journals (Sweden)
Ping Zhou
2012-01-01
Full Text Available A hybrid projective synchronization scheme for two identical fractional-order chaotic systems is proposed in this paper. Based on the stability theory of fractional-order systems, a controller for the synchronization of two identical fractional-order chaotic systems is designed. This synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid projective synchronization for the fractional-order Chen chaotic system and hybrid projective synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the validity and feasibility of the proposed scheme.
Energy Technology Data Exchange (ETDEWEB)
Ge Zhengming [Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan (China)]. E-mail: zmg@cc.nctu.edu.tw; Jhuang Weiren [Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan (China)
2007-07-15
Chaos, its control and synchronization for a fractional order rotational mechanical system with a centrifugal governor are studied for both the autonomous and the nonautonomous cases. It is found that chaos exists in the fractional order systems with order less than and more than the number of states of the system. Controlling the chaotic motion of a fractional order system to its equilibrium point is obtained for both the autonomous and the nonautonomous cases. The rotational mechanical systems with the same fractional order and with the different fractional orders are synchronized by linear coupling for both the autonomous and the nonautonomous cases.
Modified adaptive controller for synchronization of incommensurate fractional-order chaotic systems
Zhang, Ruo-Xun; Yang, Shi-Ping
2012-03-01
We investigate the synchronization of a class of incommensurate fractional-order chaotic systems, and propose a modified adaptive controller for fractional-order chaos synchronization based on the Lyapunov stability theory, the fractional order differential inequality, and the adaptive strategy. This synchronization approach is simple, universal, and theoretically rigorous. It enables the synchronization of 0 fractional-order chaotic systems to be achieved in a systematic way. The simulation results for the fractional-order Qi chaotic system and the four-wing hyperchaotic system are provided to illustrate the effectiveness of the proposed scheme.
Hybrid Projective Synchronization of Fractional-Order Chaotic Systems with Time Delay
Directory of Open Access Journals (Sweden)
Li-xin Yang
2013-01-01
Full Text Available The hybrid projective synchronization for fractional-order chaotic systems with time delay is investigated in this paper. On the basis of stability analysis of fractional-order systems and pole placement technique, a novel and general approach is proposed. The hybrid projective synchronization of fractional-order chaotic and hyperchaotic systems with time delay is achieved via designing an appropriate controller. Corresponding numerical results are presented to demonstrate the effectiveness of the proposed synchronization scheme. Furthermore, the influence of the fractional order on the synchronization process is discussed. The result reveals that the fractional order has a significant effect on the synchronization speed.
Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems
Directory of Open Access Journals (Sweden)
Chunde Yang
2016-01-01
Full Text Available A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS, is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.
Compound synchronization for four chaotic systems of integer order and fractional order
Sun, Junwei; Yin, Quan; Shen, Yi
2014-05-01
By combining the tracking control and the stability theory of nonlinear fractional-order systems, a novel kind of compound synchronization among four chaotic systems of integer order and fractional order has been investigated, where the drive systems have been conceptually divided into two categories: the scaling drive system and the base drive system. A fractional-order chaotic system can realize compound synchronization with the novel system, which is a combination system among two integer-order chaotic systems and one different fractional-order chaotic system. Numerical simulation results are presented to demonstrate the effectiveness and correctness of the compound synchronization.
A New Fractional-Order Chaotic Complex System and Its Antisynchronization
Directory of Open Access Journals (Sweden)
Cuimei Jiang
2014-01-01
with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.
Parametric Control on Fractional-Order Response for Lü Chaotic System
Moaddy, K
2013-04-10
This paper discusses the influence of the fractional order parameter on conventional chaotic systems. These fractional-order parameters increase the system degree of freedom allowing it to enter new domains and thus it can be used as a control for such dynamical systems. This paper investigates the behaviour of the equally-fractional-order Lü chaotic system when changing the fractional-order parameter and determines the fractional-order ranges for chaotic behaviour. Five different parameter values and six fractional-order cases are discussed through this paper. Unlike the conventional parameters, as the fractional-order increases the system response begins with stability, passing by chaotic behaviour then reaches periodic response. As the system parameter α increases, a shift in the fractional order is required to maintain chaotic response.Therefore, the range of chaotic response can be expanded or minimized by controlling the fractional-order parameter. The non-standard finite difference method is used to solve the fractional-order Lü chaotic system numerically to validate these responses.
Das, S.; Yadav, V. K.
2016-10-01
We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems
Directory of Open Access Journals (Sweden)
Cuimei Jiang
2015-07-01
Full Text Available Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of the stability theory of fractional-order linear systems, we design a general controller via active control. Additionally, by virtue of two complex scaling matrices, generalized combination complex synchronization between fractional-order chaotic complex systems and real systems is investigated. Finally, three typical examples are given to demonstrate the effectiveness and feasibility of the schemes.
Niu, Yu-Jun; Wang, Xing-Yuan; Nian, Fu-Zhong; Wang, Ming-Jun
2010-12-01
Based on the stability theory of the fractional order system, the dynamic behaviours of a new fractional order system are investigated theoretically. The lowest order we found to have chaos in the new three-dimensional system is 2.46, and the period routes to chaos in the new fractional order system are also found. The effectiveness of our analysis results is further verified by numerical simulations and positive largest Lyapunov exponent. Furthermore, a nonlinear feedback controller is designed to achieve the generalized projective synchronization of the fractional order chaotic system, and its validity is proved by Laplace transformation theory.
Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2).
Zhang, Ruoxun; Tian, Gang; Yang, Shiping; Cao, Hefei
2015-05-01
This paper investigates the stability of n-dimensional fractional order nonlinear systems with commensurate order 0 fractional order nonlinear systems with order lying in (0, 2). According to this theory, stabilizing a class of fractional order nonlinear systems only need a linear state feedback controller. Simulation results demonstrate the effectiveness of the proposed theory. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Yong Xu; Hua Wang
2013-01-01
This paper is devoted to the problem of synchronization between fractional-order chaotic systems with Gaussian fluctuation by the method of fractional-order sliding mode control. A fractional integral (FI) sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove the proposed controller can ma...
Radwan, A.G.; Moaddy, K.; Salama, K. N.; S. Momani; Hashim, I.
2013-01-01
This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchroni...
Karthikeyan, Anitha; Rajagopal, Karthikeyan
2018-01-01
A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.
Robust synchronization for a class of fractional-order dynamical system via linear state variable
Li, C.; Xiong, J.; Li, W.; Tong, Y.; Zeng, Y.
2013-07-01
This paper reports the issue of robust synchronization for a class of fractional-order dynamical system with parameter mismatch and stochastic disturbance. Based on the fractional Lyapunov stability theorem, a linear controller is obtained to guarantee robust Mittag-Leffler synchronization of the fractional-order dynamical system. The gain matrix has been derived by solving the linear matrix inequality. As illustrative examples, the synchronization of fractional-order Newton-Leipnik chaotic system and fractional-order Chen hyperchaotic system are executed to verify the effectiveness of the proposed scheme.
Synchronization and an application of a novel fractional order King Cobra chaotic system
Energy Technology Data Exchange (ETDEWEB)
Muthukumar, P., E-mail: muthukumardgl@gmail.com; Balasubramaniam, P., E-mail: balugru@gmail.com [Department of Mathematics, Gandhigram Rural Institute‐Deemed University, Gandhigram 624 302, Tamilnadu (India); Ratnavelu, K., E-mail: kuru052001@gmail.com [Faculty of Science, Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur (Malaysia)
2014-09-01
In this paper, we design a new three dimensional King Cobra face shaped fractional order chaotic system. The multi-scale synchronization scheme of two fractional order chaotic systems is described. The necessary conditions for the multi-scale synchronization of two identical fractional order King Cobra chaotic systems are derived through feedback control. A new cryptosystem is proposed for an image encryption and decryption by using synchronized fractional order King Cobra chaotic systems with the supports of multiple cryptographic assumptions. The security of the proposed cryptosystem is analyzed by the well known algebraic attacks. Numerical simulations are given to show the effectiveness of the proposed theoretical results.
Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control
Jianeng Tang
2014-01-01
Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.
Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control
Directory of Open Access Journals (Sweden)
Jianeng Tang
2014-01-01
Full Text Available Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.
Diffusion control for a tempered anomalous diffusion system using fractional-order PI controllers.
Juan Chen; Zhuang, Bo; Chen, YangQuan; Cui, Baotong
2017-05-09
This paper is concerned with diffusion control problem of a tempered anomalous diffusion system based on fractional-order PI controllers. The contribution of this paper is to introduce fractional-order PI controllers into the tempered anomalous diffusion system for mobile actuators motion and spraying control. For the proposed control force, convergence analysis of the system described by mobile actuator dynamical equations is presented based on Lyapunov stability arguments. Moreover, a new Centroidal Voronoi Tessellation (CVT) algorithm based on fractional-order PI controllers, henceforth called FOPI-based CVT algorithm, is provided together with a modified simulation platform called Fractional-Order Diffusion Mobile Actuator-Sensor 2-Dimension Fractional-Order Proportional Integral (FO-Diff-MAS2D-FOPI). Finally, extensive numerical simulations for the tempered anomalous diffusion process are presented to verify the effectiveness of our proposed fractional-order PI controllers. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Fengjiao Wu
2016-01-01
Full Text Available The robust fuzzy control for fractional-order hydroturbine regulating system is studied in this paper. First, the more practical fractional-order hydroturbine regulating system with uncertain parameters and random disturbances is presented. Then, on the basis of interval matrix theory and fractional-order stability theorem, a fuzzy control method is proposed for fractional-order hydroturbine regulating system, and the stability condition is expressed as a group of linear matrix inequalities. Furthermore, the proposed method has good robustness which can process external random disturbances and uncertain parameters. Finally, the validity and superiority are proved by the numerical simulations.
Dual Synchronization of Fractional-Order Chaotic Systems via a Linear Controller
Jian Xiao; Zhen-zhen Ma; Ye-hong Yang
2013-01-01
The problem of the dual synchronization of two different fractional-order chaotic systems is studied. By a linear controller, we realize the dual synchronization of fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems. The numerical simulation shows the accuracy of the theory.
Dual synchronization of fractional-order chaotic systems via a linear controller.
Xiao, Jian; Ma, Zhen-zhen; Yang, Ye-Hong
2013-01-01
The problem of the dual synchronization of two different fractional-order chaotic systems is studied. By a linear controller, we realize the dual synchronization of fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems. The numerical simulation shows the accuracy of the theory.
Identification of fractional-order systems with unknown initial values and structure
Du, Wei; Miao, Qingying; Tong, Le; Tang, Yang
2017-06-01
In this paper, the identification problem of fractional-order chaotic systems is proposed and investigated via an evolutionary optimization approach. Different with other studies to date, this research focuses on the identification of fractional-order chaotic systems with not only unknown orders and parameters, but also unknown initial values and structure. A group of fractional-order chaotic systems, i.e., Lorenz, Lü, Chen, Rössler, Arneodo and Volta chaotic systems, are set as the system candidate pool. The identification problem of fractional-order chaotic systems in this research belongs to mixed integer nonlinear optimization in essence. A powerful evolutionary algorithm called composite differential evolution (CoDE) is introduced for the identification problem presented in this paper. Extensive experiments are carried out to show that the fractional-order chaotic systems with unknown initial values and structure can be successfully identified by means of CoDE.
Zheng, Mingwen; Li, Lixiang; Peng, Haipeng; Xiao, Jinghua; Yang, Yixian; Zhao, Hui
2016-09-01
In this paper, we study the finite-time stability and synchronization problem of a class of memristor-based fractional-order Cohen-Grossberg neural network (MFCGNN) with the fractional order α ∈ (0,1 ]. We utilize the set-valued map and Filippov differential inclusion to treat MFCGNN because it has discontinuous right-hand sides. By using the definition of Caputo fractional-order derivative, the definitions of finite-time stability and synchronization, Gronwall's inequality and linear feedback controller, two new sufficient conditions are derived to ensure the finite-time stability of our proposed MFCGNN and achieve the finite-time synchronization of drive-response systems which are constituted by MFCGNNs. Finally, two numerical simulations are presented to verify the rightness of our proposed theorems.
Phase and anti-phase synchronization of fractional order chaotic systems via active control
Taghvafard, Hadi; Erjaee, G. H.
2011-10-01
This paper is devoted to investigate the phase and anti-phase synchronization between two identical and non-identical fractional order chaotic systems using techniques from active control theory. The techniques are applied to fractional order chaotic Lü and Liu systems. Numerical results demonstrate the effectiveness and feasibility of the proposed control techniques.
Xin, Baogui; Chen, Tong; Liu, Yanqin
2011-11-01
A fractional-order energy resources demand-supply system is proposed. A projective synchronization scheme is proposed as an extension on the synchronization scheme of Odibat et al. (2010). The scheme is applied to achieve projective synchronization of the chaotic fractional-order energy resource demand-supply systems. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.
Directory of Open Access Journals (Sweden)
Hua Wang
2016-01-01
Full Text Available This paper proposes a new fractional-order approach for synchronization of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. A simple but practical method to synchronize many familiar fractional-order chaotic systems has been put forward. A new theorem is proposed for a class of cascade fractional-order systems and it is applied in chaos synchronization. Combined with the fact that the states of the fractional chaotic systems are bounded, many coupled items can be taken as zero items. Then, the whole system can be simplified greatly and a simpler controller can be derived. Finally, the validity of the presented scheme is illustrated by numerical simulations of the fractional-order unified system.
A comment on "α-stability and α-synchronization for fractional-order neural networks".
Kexue, Li; Jigen, Peng; Jinghuai, Gao
2013-12-01
In this paper, we point out that an inequality in the paper [J. Yu, C. Hu, H. Jiang, α-stability and α-synchronization for fractional-order neural networks, Neural Networks 35 (2012) 82-87] is not correct. The main theorems in this paper are not valid, since they are proved by this inequality. Copyright © 2013 Elsevier Ltd. All rights reserved.
Chaotic dynamics and synchronization of fractional-order Genesio Tesi systems
Lu, Jun-Guo
2005-08-01
In this paper, we investigate numerically the chaotic behaviours in the fractional-order Genesio-Tesi system. We find that chaos exists in the fractional-order Genesio-Tesi system with order less than 3. The lowest order we find to have chaos is 2.4 in this fractional-order Genesio-Tesi system. We propose a drive-response synchronization method for synchronizing the fractional-order chaotic Genesio-Tesi systems only using a scalar drive signal. This synchronization approach, based on stability theory of fractional-order systems, is simple and theoretically rigorous. It does not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method.
Alagoz, Baris Baykant
2017-09-01
With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Radwan, A.G.; Moaddy, K.; Salama, K.N.; Momani, S.; Hashim, I.
2013-01-01
This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters. PMID:25685479
Directory of Open Access Journals (Sweden)
A.G. Radwan
2014-01-01
Full Text Available This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters.
Radwan, A.G.
2013-03-13
This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters.
Radwan, A G; Moaddy, K; Salama, K N; Momani, S; Hashim, I
2014-01-01
This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters.
The adaptive synchronization of fractional-order Liu chaotic system ...
Indian Academy of Sciences (India)
in the world, such as circuits, mathematics, power systems, medicine, electrochemical biology, etc. [1,2]. Thus, chaos is one of the .... tronic circuit model of mixed shape unit. The components of the realization of chaotic .... Analysis of stability against noise having master system of eq. (4) and slave system are as follows: D q.
0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive
Directory of Open Access Journals (Sweden)
Baogui Xin
2013-01-01
weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.
On the stability of some fractional-order non-autonomous systems
Directory of Open Access Journals (Sweden)
Sheren Ahmed Abd El-Salam
2007-03-01
Full Text Available The fractional calculus (integration and differentiation of fractional-order is a one of the singular integral and integro-differential operators. In this work a class of fractional-order non-autonomous systems will be considered. The stability (and some other properties concerning the existence and uniqueness of the solution will be proved.
Synchronization of the Fractional-Order Brushless DC Motors Chaotic System
Directory of Open Access Journals (Sweden)
Shiyun Shen
2016-01-01
Full Text Available Based on the extension of Lyapunov direct method for nonlinear fractional-order systems, chaos synchronization for the fractional-order Brushless DC motors (BLDCM is discussed. A chaos synchronization scheme is suggested. By means of Lyapunov candidate function, the theoretical proof of chaos synchronization is addressed. The numerical results show that the chaos synchronization scheme is valid.
Synchronization of the Fractional-Order Brushless DC Motors Chaotic System
Shiyun Shen; Ping Zhou
2016-01-01
Based on the extension of Lyapunov direct method for nonlinear fractional-order systems, chaos synchronization for the fractional-order Brushless DC motors (BLDCM) is discussed. A chaos synchronization scheme is suggested. By means of Lyapunov candidate function, the theoretical proof of chaos synchronization is addressed. The numerical results show that the chaos synchronization scheme is valid.
Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control
Ding, Da-Wei; Yan, Jie; Wang, Nian; Liang, Dong
2017-09-01
In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fractional order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more practical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective. Supported by National Natural Science Foundation of China under Grant No. 61201227, National Natural Science Foundation of China Guangdong Joint Fund under Grant No. U1201255, the Natural Science Foundation of Anhui Province under Grant No. 1208085MF93, 211 Innovation Team of Anhui University under Grant Nos. KJTD007A and KJTD001B, and also supported by Chinese Scholarship Council
Directory of Open Access Journals (Sweden)
Yan-Ping Wu
2013-01-01
Full Text Available The synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems via sliding mode controller is investigated. By designing an active sliding mode controller and choosing proper control parameters, the drive and response systems are synchronized. Synchronization between the fractional-order Chen chaotic system and the integer-order Chen chaotic system and between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system is used to illustrate the effectiveness of the proposed synchronization approach. Numerical simulations coincide with the theoretical analysis.
Hybrid projective synchronization of chaotic fractional order systems with different dimensions
Wang, Sha; Yu, Yongguang; Diao, Miao
2010-11-01
The hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper. It is shown that the slave system can be synchronized with the projection of the master system generated through state transformation. Based on the stability theorem of linear fractional order systems, a suitable controller for achieving the synchronization is given. The hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical results verify the effectiveness of the proposed method.
The fractional-order modeling and synchronization of electrically coupled neuron systems
Moaddy, K.
2012-11-01
In this paper, we generalize the integer-order cable model of the neuron system into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the neuron response. Furthermore, the chaotic synchronization with a gap junction of two or multi-coupled-neurons of fractional-order are discussed. The circuit model, fractional-order state equations and the numerical technique are introduced in this paper for individual and multiple coupled neuron systems with different fractional-orders. Various examples are introduced with different fractional orders using the non-standard finite difference scheme together with the Grünwald-Letnikov discretization process which is easily implemented and reliably accurate. © 2011 Elsevier Ltd. All rights reserved.
QS synchronization of the fractional-order unified system
Indian Academy of Sciences (India)
State Key Laboratory of Power Transmission Equipment & System Security and New Technology, School of Automation, Chongqing University, Chongqing 400044, People's Republic of China; School of Mathematics, Anhui University, Hefei 230039, People's Republic of China; School of Automation, Beijing Institute of ...
The synchronisation of fractional-order hyperchaos compound system
Indian Academy of Sciences (India)
According to the suitability of compound synchronisation as a reliable solution for secure communication, we then examined the application of the proposed adaptive compound synchronisation scheme in the presence of noise for secure communication. In addition, the unpredictability and complexity of the drive systems ...
Dynamics of fractional-ordered Chen system with delay
Indian Academy of Sciences (India)
of fractional calculus to various branches of science and engineering have been real- ized only recently. It is becoming more and more clear that derivatives and integrals of non-integer orders are not mere mathematical curiosities but many processes, such as dielectric polarization [1], diffusion [2–5], viscoelastic systems [6] ...
Identification of fractional-order systems with unknown initial values and structure
Energy Technology Data Exchange (ETDEWEB)
Du, Wei, E-mail: duwei0203@gmail.com [Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237 (China); Miao, Qingying, E-mail: qymiao@sjtu.edu.cn [School of Continuing Education, Shanghai Jiao Tong University, Shanghai 200030 (China); Tong, Le, E-mail: tongle0328@gmail.com [Faculty of Applied Science and Textiles, The Hong Kong Polytechnic University, Hong Kong (China); Tang, Yang [Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237 (China)
2017-06-21
In this paper, the identification problem of fractional-order chaotic systems is proposed and investigated via an evolutionary optimization approach. Different with other studies to date, this research focuses on the identification of fractional-order chaotic systems with not only unknown orders and parameters, but also unknown initial values and structure. A group of fractional-order chaotic systems, i.e., Lorenz, Lü, Chen, Rössler, Arneodo and Volta chaotic systems, are set as the system candidate pool. The identification problem of fractional-order chaotic systems in this research belongs to mixed integer nonlinear optimization in essence. A powerful evolutionary algorithm called composite differential evolution (CoDE) is introduced for the identification problem presented in this paper. Extensive experiments are carried out to show that the fractional-order chaotic systems with unknown initial values and structure can be successfully identified by means of CoDE. - Highlights: • Unknown initial values and structure are introduced in the identification of fractional-order chaotic systems; • Only a series of output is utilized in the identification of fractional-order chaotic systems; • CoDE is used for the identification problem and the results are satisfactory when compared with other DE variants.
Directory of Open Access Journals (Sweden)
Junbiao Guan
2015-01-01
Full Text Available A new fractional-order chaotic system is addressed in this paper. By applying the continuous frequency distribution theory, the indirect Lyapunov stability of this system is investigated based on sliding mode control technique. The adaptive laws are designed to guarantee the stability of the system with the uncertainty and external disturbance. Moreover, the modified generalized projection synchronization (MGPS of the fractional-order chaotic systems is discussed based on the stability theory of fractional-order system, which may provide potential applications in secure communication. Finally, some numerical simulations are presented to show the effectiveness of the theoretical results.
Xu, Chang-Jin; Li, Pei-Luan; Pang, Yi-Cheng
2017-02-01
This paper is concerned with fractional-order bidirectional associative memory (BAM) neural networks with time delays. Applying Laplace transform, the generalized Gronwall inequality and estimates of Mittag-Leffler functions, some sufficient conditions which ensure the finite-time stability of fractional-order bidirectional associative memory neural networks with time delays are obtained. Two examples with their simulations are given to illustrate the theoretical findings. Our results are new and complement previously known results. Supported by National Natural Science Foundation of China under Grant Nos.~61673008, 11261010, 11101126, Project of High-Level Innovative Talents of Guizhou Province ([2016]5651), Natural Science and Technology Foundation of Guizhou Province (J[2015]2025 and J[2015]2026), 125 Special Major Science and Technology of Department of Education of Guizhou Province ([2012]011) and Natural Science Foundation of the Education Department of Guizhou Province (KY[2015]482)
Chaotic dynamics and synchronization of fractional order PMSM system
Directory of Open Access Journals (Sweden)
Vajiheh Vafaei
2015-12-01
Full Text Available In this paper, we investigate the chaotic behaviors of the fractional-order permanent magnet synchronous motor (PMSM system. The necessary condition for the existence of chaos in the fractional-order PMSM system is deduced and an active controller is developed based on the stability theory for fractional systems. The presented control scheme is simple and flexible, and it is suitable both for design and for implementation in practice. Simulation is carried out to verify that the obtained scheme is efficient and robust for controlling the fractional-order PMSM system.
Synchronization of fractional-order uncertain chaotic systems with input nonlinearity
Noghredani, Naeimadeen; Balochian, Saeed
2015-05-01
In this paper, sliding mode control has been designed for synchronization of fractional order uncertain chaotic systems with input nonlinearity. First, sliding mode control law has been taken from chaotic state error system which is asymptotically stable. Second, the shown sliding mode control ensures that fractional-order error system is asymptotically stable in the presence of uncertainty and nonlinear input. Simulation results using MATLAB software show that the designed controller is able to synchronize fractional-order chaotic systems in the presence of the mentioned factors.
Lag projective synchronization of two chaotic systems with different fractional orders
Sun, Zhenwu
2015-04-01
Lag projective synchronization (LPS) of chaotic systems with different fractional orders is investigated. A scheme of LPS is designed based on the stability of fractional nonlinear systems. LPS between two four-scroll hyperchaotic systems with different fractional orders is realized by using the scheme and is simulated by using a multi-step fractional differential transform method. All the theoretical analysis and simulation results show the effectiveness of the proposed controller. The work in this paper may accelerate the application of fractional-order chaotic systems in practice.
SOLVABILITY OF COUPLED SYSTEMS OF FRACTIONAL ORDER INTEGRO-DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
H.H.G. Hashem
2014-03-01
Full Text Available We present existence theorems for coupled systems of quadratic integral equations of fractional order. As applications we deduce existence theorems for twocoupled systems of Cauchy problems. Also, an example illustrating the existence theorem is given.
Ma, Yingdong; Lu, Junguo; Chen, Weidong
2014-03-01
This paper investigates the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observer-based controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Parameter identification of commensurate fractional-order chaotic system via differential evolution
Tang, Yinggan; Zhang, Xiangyang; Hua, Changchun; Li, Lixiang; Yang, Yixian
2012-01-01
Chaos can be observed in fractional-order nonlinear systems with appropriate orders. The knowledge about the parameters and orders are the basis of the control and synchronization of fractional-order chaotic systems. In this Letter, the problem of parameter identification of commensurate fractional-order chaotic systems is investigated. By treating the orders as additional parameters, the parameters and orders are identified together through minimizing an objective function. Differential evolution algorithm, a powerful and robust evolutionary algorithm, is applied to search the optimal solution of the objective function. Numerical simulations and comparisons with genetic algorithm (GA) demonstrate the effectiveness of the proposed method.
Simple Lmi-Based Synchronization of Fractional-Order Chaotic Systems
Kuntanapreeda, Suwat
This paper presents a simple scheme for synchronization of fractional-order chaotic systems. The scheme utilizes a recently developed LMI (Linear matrix inequality) stabilization theorem for fractional-order linear interval systems to design a linear controller. In contrast to existing schemes in the literature, the present scheme is straightforward and does not require that nonlinear parts of synchronization error dynamics are cancelled by the controller. The fractional-order Rössler, Lorenz, and hyperchaotic Chen systems are used as demonstrative examples. Numerical results illustrate the effectiveness of the present scheme.
Zhang, Ruo-Xun; Yang, Shi-Ping
2012-08-01
In this paper we investigate the synchronization of a class of three-dimensional fractional-order chaotic systems. Based on the Lyapunov stability theory and adaptive control technique, a single adaptive—feedback controller is developed to synchronize a class of fractional-order chaotic systems. The presented controller which only contains a single driving variable is simple both in design and in implementation. Numerical simulation and circuit experimental results for fractional-order chaotic system are provided to illustrate the effectiveness of the proposed scheme.
A 3D Fractional-Order Chaotic System with Only One Stable Equilibrium and Controlling Chaos
Directory of Open Access Journals (Sweden)
Shiyun Shen
2017-01-01
Full Text Available One 3D fractional-order chaotic system with only one locally asymptotically stable equilibrium is reported. To verify the chaoticity, the maximum Lyapunov exponent (MAXLE with respect to the fractional-order and chaotic attractors are obtained by numerical calculation for this system. Furthermore, by linear scalar controller consisting of a single state variable, one control scheme for stabilization of the 3D fractional-order chaotic system is suggested. The numerical simulations show the feasibility of the control scheme.
Dual Synchronization of Fractional-Order Chaotic Systems via a Linear Controller
Xiao, Jian; Ma, Zhen-zhen; Yang, Ye-hong
2013-01-01
The problem of the dual synchronization of two different fractional-order chaotic systems is studied. By a linear controller, we realize the dual synchronization of fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems. The numerical simulation shows the accuracy of the theory. PMID:24163612
Bhalekar, S.
2014-06-01
Chaos synchronization in fractional order chaotic systems is receiving increasing attention due to its applications in secure communications. In this article we use an active control technique to synchronize incommensurate non-identical fractional order chaotic dynamical systems. The relation between system order and the synchronization time is discussed. It is observed that the synchronization can be achieved faster by increasing the system order. Further we provide an application of the proposed theory in secure communication.
Robust Synchronization Controller Design for a Class of Uncertain Fractional Order Chaotic Systems
Directory of Open Access Journals (Sweden)
Lin Wang
2016-01-01
Full Text Available Synchronization problem for a class of uncertain fractional order chaotic systems is studied. Some fundamental lemmas are given to show the boundedness of a complicated infinite series which is produced by differentiating a quadratic Lyapunov function with fractional order. By using the fractional order extension of the Lyapunov stability criterion and the proposed lemma, stability of the closed-loop system is analyzed, and two sufficient conditions, which can enable the synchronization error to converge to zero asymptotically, are driven. Finally, an illustrative example is presented to confirm the proposed theoretical results.
Observer-Based Robust Controller Design for Nonlinear Fractional-Order Uncertain Systems via LMI
Directory of Open Access Journals (Sweden)
Jing Qiu
2017-01-01
Full Text Available We discuss the observer-based robust controller design problem for a class of nonlinear fractional-order uncertain systems with admissible time-variant uncertainty in the case of the fractional-order satisfying 0<α<1. Based on direct Lyapunov approach, a sufficient condition for the robust asymptotic stability of the observer-based nonlinear fractional-order uncertain systems is presented. Employing Finsler’s Lemma, the systematic robust stabilization design algorithm is then proposed in terms of linear matrix inequalities (LMIs. The efficiency and advantage of the proposed algorithm are finally illustrated by two numerical simulations.
Directory of Open Access Journals (Sweden)
Sajewski Łukasz
2017-03-01
Full Text Available Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass–Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.
Sliding Mode Control of the Fractional-Order Unified Chaotic System
Directory of Open Access Journals (Sweden)
Jian Yuan
2013-01-01
Full Text Available This paper deals with robust synchronization of the fractional-order unified chaotic systems. Firstly, control design for synchronization of nominal systems is proposed via fractional sliding mode technique. Then, systematic uncertainties and external disturbances are considered in the fractional-order unified chaotic systems, and adaptive sliding mode control is designed for the synchronization issue. Finally, numerical simulations are carried out to verify the effectiveness of the two proposed control techniques.
Regression model for tuning the PID controller with fractional order time delay system
Directory of Open Access Journals (Sweden)
S.P. Agnihotri
2014-12-01
Full Text Available In this paper a regression model based for tuning proportional integral derivative (PID controller with fractional order time delay system is proposed. The novelty of this paper is that tuning parameters of the fractional order time delay system are optimally predicted using the regression model. In the proposed method, the output parameters of the fractional order system are used to derive the regression function. Here, the regression model depends on the weights of the exponential function. By using the iterative algorithm, the best weight of the regression model is evaluated. Using the regression technique, fractional order time delay systems are tuned and the stability parameters of the system are maintained. The effectiveness and feasibility of the proposed technique is demonstrated through the MATLAB/Simulink platform, as well as testing and comparison using the classical PID controller, Ziegler–Nichols tuning method, Wang tuning method and curve fitting technique base tuning method.
State-dependent switching control of switched positive fractional-order systems.
Zhao, Xudong; Yin, Yunfei; Zheng, Xiaolong
2016-05-01
In this paper, the problem of switching stabilization for a class of continuous-time switched positive fractional-order systems is studied by using state-dependent switching. First, the asymptotic stability condition of switched positive fractional-order systems with state-dependent switching is given, which is based on the fractional co-positive Lyapunov method. Moreover, by the sliding sector method, the stability condition of switched positive fractional-order systems whose subsystems are possibly all unstable is obtained. A variable structure (VS) switching law with sliding sector is also proposed to guarantee the switched positive fractional-order system to be asymptotically stable. Finally, two numerical examples are given to demonstrate the advantages and effectiveness of our developed results. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Parameter identification of commensurate fractional-order chaotic system via differential evolution
Energy Technology Data Exchange (ETDEWEB)
Tang, Yinggan, E-mail: ygtang@yahoo.cn [Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004 (China); National Engineering Research Center for Equipment and Technology of Cold Strip Rolling (China); Zhang, Xiangyang; Hua, Changchun [Institute of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei 066004 (China); Li, Lixiang; Yang, Yixian [Research Center on Fictitious Economy and Data Science, Chinese Academy of Sciences, Beijing 100190 (China)
2012-01-09
Chaos can be observed in fractional-order nonlinear systems with appropriate orders. The knowledge about the parameters and orders are the basis of the control and synchronization of fractional-order chaotic systems. In this Letter, the problem of parameter identification of commensurate fractional-order chaotic systems is investigated. By treating the orders as additional parameters, the parameters and orders are identified together through minimizing an objective function. Differential evolution algorithm, a powerful and robust evolutionary algorithm, is applied to search the optimal solution of the objective function. Numerical simulations and comparisons with genetic algorithm (GA) demonstrate the effectiveness of the proposed method. -- Highlights: ► We study the parameter identification problem of fractional order chaotic system. ► The orders and the parameters can be identified simultaneously. ► Numerical results verify the effectiveness of the proposed method.
Consensus of Fractional-Order Multiagent Systems with Double Integrator under Switching Topologies
Directory of Open Access Journals (Sweden)
Shiyun Shen
2017-01-01
Full Text Available Due to the complexity of the practical environments, many systems can only be described with the fractional-order dynamics. In this paper, the consensus of fractional-order multiagent systems with double integrator under switching topologies is investigated. By applying Mittag-Leffler function, Laplace transform, and dwell time technique, a sufficient condition on consensus is obtained. Finally, a numerical simulation is presented to illustrate the effectiveness of the theoretical result.
State-Feedback Control for Fractional-Order Nonlinear Systems Subject to Input Saturation
Directory of Open Access Journals (Sweden)
Junhai Luo
2014-01-01
Full Text Available We give a state-feedback control method for fractional-order nonlinear systems subject to input saturation. First, a sufficient condition is derived for the asymptotical stability of a class of fractional-order nonlinear systems. Then based on Gronwall-Bellman lemma and a sector bounded condition of the saturation function, a linear state-feed back controller is designed. Finally, two simulation examples are presented to show the validity of the proposed method.
Universal block diagram based modeling and simulation schemes for fractional-order control systems.
Bai, Lu; Xue, Dingyü
2017-05-08
Universal block diagram based schemes are proposed for modeling and simulating the fractional-order control systems in this paper. A fractional operator block in Simulink is designed to evaluate the fractional-order derivative and integral. Based on the block, the fractional-order control systems with zero initial conditions can be modeled conveniently. For modeling the system with nonzero initial conditions, the auxiliary signal is constructed in the compensation scheme. Since the compensation scheme is very complicated, therefore the integrator chain scheme is further proposed to simplify the modeling procedures. The accuracy and effectiveness of the schemes are assessed in the examples, the computation results testify the block diagram scheme is efficient for all Caputo fractional-order ordinary differential equations (FODEs) of any complexity, including the implicit Caputo FODEs. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Yu Huang
Full Text Available Parameter estimation for fractional-order chaotic systems is an important issue in fractional-order chaotic control and synchronization and could be essentially formulated as a multidimensional optimization problem. A novel algorithm called quantum parallel particle swarm optimization (QPPSO is proposed to solve the parameter estimation for fractional-order chaotic systems. The parallel characteristic of quantum computing is used in QPPSO. This characteristic increases the calculation of each generation exponentially. The behavior of particles in quantum space is restrained by the quantum evolution equation, which consists of the current rotation angle, individual optimal quantum rotation angle, and global optimal quantum rotation angle. Numerical simulation based on several typical fractional-order systems and comparisons with some typical existing algorithms show the effectiveness and efficiency of the proposed algorithm.
Huang, Yu; Guo, Feng; Li, Yongling; Liu, Yufeng
2015-01-01
Parameter estimation for fractional-order chaotic systems is an important issue in fractional-order chaotic control and synchronization and could be essentially formulated as a multidimensional optimization problem. A novel algorithm called quantum parallel particle swarm optimization (QPPSO) is proposed to solve the parameter estimation for fractional-order chaotic systems. The parallel characteristic of quantum computing is used in QPPSO. This characteristic increases the calculation of each generation exponentially. The behavior of particles in quantum space is restrained by the quantum evolution equation, which consists of the current rotation angle, individual optimal quantum rotation angle, and global optimal quantum rotation angle. Numerical simulation based on several typical fractional-order systems and comparisons with some typical existing algorithms show the effectiveness and efficiency of the proposed algorithm.
Tabatabaei, Mohammad
2017-07-01
In this paper, a new method for determination of the desired characteristic equation and zero location of commensurate fractional order systems is presented. The concept of the characteristic ratio is extended for zero-including commensurate fractional order systems. The generalized version of characteristic ratios is defined such that the time-scaling property of characteristic ratios is also preserved. The monotonicity of the magnitude frequency response is employed to assign the generalized characteristic ratios for commensurate fractional order transfer functions with one zero. A simple pattern for characteristic ratios is proposed to reach a non-overshooting step response. Then, the proposed pattern is revisited to reach a low overshoot (say for example 2%) step response. Finally, zero-including controllers such as fractional order PI or lag (lead) controllers are designed using generalized characteristic ratios assignment method. Numerical simulations are provided to show the efficiency of the so designed controllers. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Xue, Wei; Xu, Jin-Kang; Cang, Shi-Jian; Jia, Hong-Yan
2014-06-01
In this paper, the synchronization of the fractional-order generalized augmented Lü system is investigated. Based on the predictor—corrector method, we obtain phase portraits, bifurcation diagrams, Lyapunov exponent spectra, and Poincaré maps of the fractional-order system and find that a four-wing chaotic attractor exists in the system when the system parameters change within certain ranges. Further, by varying the system parameters, rich dynamical behaviors occur in the 2.7-order system. According to the stability theory of a fractional-order linear system, and adopting the linearization by feedback method, we have designed a nonlinear feedback controller in our theoretical analysis to implement the synchronization of the drive system with the response system. In addition, the synchronization is also shown by an electronic circuit implementation for the 2.7-order system. The obtained experiment results accord with the theoretical analyses, which further demonstrate the feasibility and effectiveness of the proposed synchronization scheme.
Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings
Zhang, Limin; Sun, Kehui; He, Shaobo; Wang, Huihai; Xu, Yixin
2017-01-01
Based on the Adomian decomposition method (ADM), the numerical solution of a fractional-order 5-D hyperchaotic system with four wings is investigated. Dynamics of the system are analyzed by means of phase diagram, bifurcation diagram, Lyapunov exponents spectrum and chaos diagram. The method of one-dimensional linear path through the multidimensional parameter space is proposed to observe the evolution law of the system dynamics with parameters varying. The results illustrate that the system has abundant dynamical behaviors. Both the system order and parameters can be taken as bifurcation parameters. The phenomenon of multiple attractors is found, which means that some attractors are generated simultaneously from different initial values. The spectral entropy (SE) algorithm is applied to estimate the fractional-order system complexity, and we found that the complexity decreases with the increasing of system order. In order to verify the reliability of numerical solution, the fractional-order 5-D system with four wings is implemented on a DSP platform. The phase portraits of fractional-order system generated on DSP agree well with those obtained by computer simulations. It is shown that the fractional-order hyperchaotic system is a potential model for application in the field of chaotic secure communication.
Velmurugan, G; Rakkiyappan, R; Vembarasan, V; Cao, Jinde; Alsaedi, Ahmed
2017-02-01
As we know, the notion of dissipativity is an important dynamical property of neural networks. Thus, the analysis of dissipativity of neural networks with time delay is becoming more and more important in the research field. In this paper, the authors establish a class of fractional-order complex-valued neural networks (FCVNNs) with time delay, and intensively study the problem of dissipativity, as well as global asymptotic stability of the considered FCVNNs with time delay. Based on the fractional Halanay inequality and suitable Lyapunov functions, some new sufficient conditions are obtained that guarantee the dissipativity of FCVNNs with time delay. Moreover, some sufficient conditions are derived in order to ensure the global asymptotic stability of the addressed FCVNNs with time delay. Finally, two numerical simulations are posed to ensure that the attention of our main results are valuable. Copyright © 2016 Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Xiaona Song
2017-01-01
Full Text Available The synchronization of two fractional-order complex chaotic systems is discussed in this paper. The parameter uncertainty and external disturbance are included in the system model, and the synchronization of the considered chaotic systems is implemented based on the finite-time concept. First, a novel fractional-order nonsingular terminal sliding surface which is suitable for the considered fractional-order systems is proposed. It is proven that once the state trajectories of the system reach the proposed sliding surface they will converge to the origin within a given finite time. Second, in terms of the established nonsingular terminal sliding surface, combining the fuzzy control and the sliding mode control schemes, a novel robust single fuzzy sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface over a finite time. Finally, using the fractional Lyapunov stability theorem, the stability of the proposed method is proven. The proposed method is implemented for synchronization of two fractional-order Genesio-Tesi chaotic systems with uncertain parameters and external disturbances to verify the effectiveness of the proposed fractional-order nonsingular terminal fuzzy sliding mode controller.
Geng, Lingling; Yu, Yongguang; Zhang, Shuo
2016-09-01
In this paper, the function projective synchronization between integer-order and stochastic fractional-order nonlinear systems is investigated. Firstly, according to the stability theory of fractional-order systems and tracking control, a controller is designed. At the same time, based on the orthogonal polynomial approximation, the method of transforming stochastic error system into an equivalent deterministic system is given. Thus, the stability of the stochastic error system can be analyzed through its equivalent deterministic one. Finally, to demonstrate the effectiveness of the proposed scheme, the function projective synchronization between integer-order Lorenz system and stochastic fractional-order Chen system is studied. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Chaos in fractional-order Genesio-Tesi system and its synchronization
Faieghi, Mohammad Reza; Delavari, Hadi
2012-02-01
In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.
Directory of Open Access Journals (Sweden)
Yong Xu
2013-01-01
Full Text Available Chaotic systems are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems with Gaussian fluctuation. A fractional integral (FI sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove that the proposed controller can make the system achieve synchronization. As a case study, the presented method is applied to the fractional-order Chen-Lü system, and simulation results show that the proposed control approach is capable to go against Gaussian noise well.
Hybrid Stability Checking Method for Synchronization of Chaotic Fractional-Order Systems
Directory of Open Access Journals (Sweden)
Seng-Kin Lao
2014-01-01
Full Text Available A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits.
Wu, Ailong; Zeng, Zhigang
2016-02-01
We show that the ω-periodic fractional-order fuzzy neural networks cannot generate non-constant ω-periodic signals. In addition, several sufficient conditions are obtained to ascertain the boundedness and global Mittag-Leffler stability of fractional-order fuzzy neural networks. Furthermore, S-asymptotical ω-periodicity and global asymptotical ω-periodicity of fractional-order fuzzy neural networks is also characterized. The obtained criteria improve and extend the existing related results. To illustrate and compare the theoretical criteria, some numerical examples with simulation results are discussed in detail. Crown Copyright © 2015. Published by Elsevier Ltd. All rights reserved.
Dynamical properties and complexity in fractional-order diffusionless Lorenz system
He, Shaobo; Sun, Kehui; Banerjee, Santo
2016-08-01
In this paper, dynamics and complexity of the fractional-order diffusionless Lorenz system which is solved by the developed discrete Adomian decomposition method are investigated numerically. Dynamical properties of the fractional-order diffusionless Lorenz system with the control parameter and derivative order varying is analyzed by using bifurcation diagrams, and period-doubling route to chaos in different cases is observed. The complexity of the system is investigated by means of Lyapunov characteristic exponents, multi-scale spectral entropy algorithm and multiscale Renyi permutation entropy algorithm. It can be observed that the three methods illustrate consistent results and the system has rich complex dynamics. Interestingly, complexity decreases with the increase of derivative order. It shows that the fractional-order diffusionless Lorenz system is a good model for real applications such as information encryption and secure communication.
Directory of Open Access Journals (Sweden)
Xiaomin Tian
2014-01-01
Full Text Available The complex modified projective synchronization (CMPS between fractional-order chaotic real and complex systems is investigated for the first time. The parameters of both master and slave systems are assumed to be unknown in advance; moreover, the slave system is perturbed by unknown but bounded external disturbances. The master and slave systems that achieved CMPS can be synchronized up to a complex constant matrix. On the basis of frequency distributed model of fractional integrator and Lyapunov stability theory, a robust adaptive control law is designed to realize the CMPS for two different types of fractional-order chaotic systems. Meanwhile, to deal with these unknown parameters, some fractional-order type parametric update laws are provided. An example is given to demonstrate the effectiveness and feasibility of the proposed synchronization scheme.
Taming stochastic bifurcations in fractional-order systems via noise and delayed feedback
Sun, Zhongkui; Zhang, Jintian; Yang, Xiaoli; Xu, Wei
2017-08-01
The dynamics in fractional-order systems have been widely studied during the past decade due to the potential applications in new materials and anomalous diffusions, but the investigations have been so far restricted to a fractional-order system without time delay(s). In this paper, we report the first study of random responses of fractional-order system coupled with noise and delayed feedback. Stochastic averaging method has been utilized to determine the stationary probability density functions (PDFs) by means of the principle of minimum mean-square error, based on which stochastic bifurcations could be identified through recognizing the shape of the PDFs. It has been found that by changing the fractional order the shape of the PDFs can switch from unimodal distribution to bimodal one, or from bimodal distribution to unimodal one, thus announcing the onset of stochastic bifurcation. Further, we have demonstrated that by merely modulating the time delay, the feedback strengths, or the noise intensity, the shapes of PDFs can transit between a single peak and a double peak. Therefore, it provides an efficient candidate to control, say, induce or suppress, the stochastic bifurcations in fractional-order systems.
Adaptive - Synchronization of Fractional-Order Chaotic Systems with Nonidentical Structures
Directory of Open Access Journals (Sweden)
Li-xin Yang
2013-01-01
Full Text Available This paper investigates the adaptive - synchronization of the fractional-order chaotic systems with nonidentical structures. Based on the stability of fractional-order systems and adaptive control technique, a general formula for designing the controller and parameters update law is proposed to achieve adaptive - synchronization between two different chaotic systems with different structures. The effective scheme parameters identification and - synchronization of chaotic systems can be realized simultaneously. Furthermore, two typical illustrative numerical simulations are given to demonstrate the effectiveness of the proposed scheme, for each case, we design the controller and parameter update laws in detail. The numerical simulations are performed to verify the effectiveness of the theoretical results.
Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System
Directory of Open Access Journals (Sweden)
Xiaojun Liu
2014-01-01
Full Text Available The Hopf bifurcation of a fractional-order Van der Pol (VDP for short system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.
Chaotic incommensurate fractional order Rössler system: active control and synchronization
Directory of Open Access Journals (Sweden)
Baleanu Dumitru
2011-01-01
Full Text Available Abstract In this article, we present an active control methodology for controlling the chaotic behavior of a fractional order version of Rössler system. The main feature of the designed controller is its simplicity for practical implementation. Although in controlling such complex system several inputs are used in general to actuate the states, in the proposed design, all states of the system are controlled via one input. Active synchronization of two chaotic fractional order Rössler systems is also investigated via a feedback linearization method. In both control and synchronization, numerical simulations show the efficiency of the proposed methods.
Directory of Open Access Journals (Sweden)
Darui Zhu
2013-01-01
Full Text Available A novel fractional-order hyperchaotic system is proposed; the theoretical analysis and numerical simulation of this system are studied. Based on the stability theory of fractional calculus, we propose a novel drive-response synchronization scheme. In order to achieve this synchronization control, the Adams-Bashforth-Moulton algorithm is studied. And then, a drive-response synchronization controller is designed to realize the synchronization of the drive and response system, and the simulation results are given. At last, the fractional oscillator circuit of the new fractional-order hyperchaotic system is designed based on the EWB software, and it is verified that the simulation results of the fractional-order oscillator circuit are consistent with the numerical simulation results through circuit simulation.
Generation and control of multi-scroll chaotic attractors in fractional order systems
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. [Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Sharjah (United Arab Emirates)] e-mail: wajdi@sharjah.ac.ae
2005-08-01
The objective of this paper is twofold: on one hand we demonstrate the generation of multi-scroll attractors in fractional order chaotic systems. Then, we design state feedback controllers to eliminate chaos from the system trajectories. It is demonstrated that modifying the underlying nonlinearity of the fractional chaotic system results in the birth of multiple chaotic attractors, thus forming the so called multi-scroll attractors. The presence of chaotic behavior is evidenced by a positive largest Lyapunov exponent computed for the output time series. We investigate generation and control of multi-scroll attractors in two different models, both of which are fractional order and chaotic: an electronic oscillator, and a mechanical 'jerk' model. The current findings extend previously reported results on generation of n-scroll attractors from the domain of integer order to the domain of fractional order chaotic systems, and addresses the issue of controlling such chaotic behaviors. Our investigations are validated through numerical simulations.
Chaos Suppression in Fractional order Permanent Magnet Synchronous Generator in Wind Turbine Systems
Rajagopal, Karthikeyan; Karthikeyan, Anitha; Duraisamy, Prakash
2017-06-01
In this paper we investigate the control of three-dimensional non-autonomous fractional-order uncertain model of a permanent magnet synchronous generator (PMSG) via a adaptive control technique. We derive a dimensionless fractional order model of the PMSM from the integer order presented in the literatures. Various dynamic properties of the fractional order model like eigen values, Lyapunov exponents, bifurcation and bicoherence are investigated. The system chaotic behavior for various orders of fractional calculus are presented. An adaptive controller is derived to suppress the chaotic oscillations of the fractional order model. As the direct Lyapunov stability analysis of the robust controller is difficult for a fractional order first derivative, we have derived a new lemma to analyze the stability of the system. Numerical simulations of the proposed chaos suppression methodology are given to prove the analytical results derived through which we show that for the derived adaptive controller and the parameter update law, the origin of the system for any bounded initial conditions is asymptotically stable.
Wang, Chenhui
2016-01-01
In this paper, control of uncertain fractional-order financial chaotic system with input saturation and external disturbance is investigated. The unknown part of the input saturation as well as the system's unknown nonlinear function is approximated by a fuzzy logic system. To handle the fuzzy approximation error and the estimation error of the unknown upper bound of the external disturbance, fractional-order adaptation laws are constructed. Based on fractional Lyapunov stability theorem, an adaptive fuzzy controller is designed, and the asymptotical stability can be guaranteed. Finally, simulation studies are given to indicate the effectiveness of the proposed method.
Directory of Open Access Journals (Sweden)
Heng Liu
2017-01-01
Full Text Available We investigate the synchronization problem of fractional-order chaotic systems with input saturation and unknown external disturbance by means of adaptive fuzzy control. An adaptive controller, accompanied with fractional adaptation law, is established, fuzzy logic systems are used to approximate the unknown nonlinear functions, and the fractional Lyapunov stability theorem is used to analyze the stability. This control method can realize the synchronization of two fractional-order chaotic or hyperchaotic systems and the synchronization error tends to zero asymptotically. Finally, we show the effectiveness of the proposed method by two simulation examples.
Directory of Open Access Journals (Sweden)
Chenhui Wang
Full Text Available In this paper, control of uncertain fractional-order financial chaotic system with input saturation and external disturbance is investigated. The unknown part of the input saturation as well as the system's unknown nonlinear function is approximated by a fuzzy logic system. To handle the fuzzy approximation error and the estimation error of the unknown upper bound of the external disturbance, fractional-order adaptation laws are constructed. Based on fractional Lyapunov stability theorem, an adaptive fuzzy controller is designed, and the asymptotical stability can be guaranteed. Finally, simulation studies are given to indicate the effectiveness of the proposed method.
Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations
Directory of Open Access Journals (Sweden)
Yongjin Li
2017-01-01
Full Text Available We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs. We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.
Synchronization for a Class of Fractional-Order Hyperchaotic System and Its Application
Directory of Open Access Journals (Sweden)
Wen Tan
2012-01-01
Full Text Available A new controller design method is proposed to synchronize the fractional-order hyperchaotic system through the stability theory of fractional calculus; the synchronization between two identical fractional-order Chen hyperchaotic systems is realized by designing only two suitable controllers in the response system. Furthermore, this control scheme can be used in secure communication via the technology of chaotic masking using the complex nonperiodic information as trial message, and the useful information can be recovered at the receiver. Numerical simulations coincide with the theoretical analysis.
Observer-based adaptive backstepping control for fractional order systems with input saturation.
Sheng, Dian; Wei, Yiheng; Cheng, Songsong; Wang, Yong
2017-07-03
An observer-based fractional order anti-saturation adaptive backstepping control scheme is proposed for incommensurate fractional order systems with input saturation and partial measurable state in this paper. On the basis of stability analysis, a novel state observer is established first since the only information we could acquire is the system output. In order to compensate the saturation, a series of virtual signals are generated via the construction of fractional order auxiliary system. Afterwards, the controller design is carried out in accordance with the adaptive backstepping control method by introduction of the indirect Lyapunov method. To highlight the effectiveness of the proposed control scheme, simulation examples are demonstrated at last. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Mittag-Leffler synchronization of fractional-order uncertain chaotic systems
Wang, Qiao; Ding, Dong-Sheng; Qi, Dong-Lian
2015-06-01
This paper deals with the synchronization of fractional-order chaotic systems with unknown parameters and unknown disturbances. An adaptive control scheme combined with fractional-order update laws is proposed. The asymptotic stability of the error system is proved in the sense of generalized Mittag-Leffler stability. The two fractional-order chaotic systems can be synchronized in the presence of model uncertainties and additive disturbances. Finally these new developments are illustrated in examples and numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme. Project supported by the National Natural Science Foundation of China (Grant No. 61171034) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. R1110443).
Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization
Wang, Yan; Sun, Kehui; He, Shaobo; Wang, Huihai
2014-06-01
In this paper, dynamical behaviors of the fractional-order sinusoidally forced simplified Lorenz are investigated by employing the time-domain solution algorithm of fractional-order calculus. The system parameters and the fractional derivative orders q are treated as bifurcation parameters. The range of the bifurcation parameters in which the system generates chaos is determined by bifurcation, phase portrait, and Poincaré section, and different bifurcation motions are visualized by virtue of a systematic numerical analysis. We find that the lowest order of this system to yield chaos is 3.903. Based on fractional-order stability theory, synchronization is achieved by using nonlinear feedback control method. Simulation results show the scheme is effective and a chaotic secure communication scheme is present based on this synchronization.
New Methods of Finite-Time Synchronization for a Class of Fractional-Order Delayed Neural Networks
Directory of Open Access Journals (Sweden)
Weiwei Zhang
2017-01-01
Full Text Available Finite-time synchronization for a class of fractional-order delayed neural networks with fractional order α, 0<α≤1/2 and 1/2<α<1, is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequality skills, two sufficient conditions are considered to ensure synchronization of fractional-order delayed neural networks in a finite-time interval. Numerical example is given to verify the feasibility of the theoretical results.
On stable integral manifolds for impulsive Kolmogorov systems of fractional order
Stamov, Gani; Stamova, Ivanka
2017-05-01
In this paper, an impulsive Kolmogorov-type system using the Caputo fractional-order derivative is developed. The fractional-order system displays many interesting dynamic behaviors and fractional integrals can be used to describe the fractal media. The existence and stability of integral manifolds for the impulsive fractional model are considered. The main results are proved by means of piecewise continuous Lyapunov functions and the new fractional comparison principle. The impulses are realized at variable impulsive moments of time and can be considered as a control. Finally, an example is given to illustrate our results.
Fractional-Order Fast Terminal Sliding Mode Control for a Class of Dynamical Systems
Directory of Open Access Journals (Sweden)
Guoliang Zhao
2013-01-01
Full Text Available This paper introduces a novel fractional fast terminal sliding mode control strategy for a class of dynamical systems with uncertainty. In this strategy, a fractional-order sliding surface is proposed, the corresponding control law is derived based on Lyapunov stability theory to guarantee the sliding condition, and the finite time stability of the closeloop system is also ensured. Further, to achieve the equivalence between convergence rate and singularity avoidance, a fractional-order nonsingular fast terminal sliding mode controller is studied and the stability is presented. Finally, numerical simulation results are presented to illustrate the effectiveness of the proposed method.
A New Scheme to Projective Synchronization of Fractional-Order Chaotic Systems
Wang, Jun-Wei; Chen, Ai-Min
2010-11-01
We demonstrate that the projective synchronization can be observed in coupled fractional-order chaotic systems. A new systematic and powerful coupling scheme is developed to investigate the projective synchronization via the open-plus-closed-loop control, which allows us to arbitrarily manipulate the scaling factor of projective synchronization. The proposed scheme is proved analytically on the basis of the stability theorem of the fractional differential equations. Numerical simulations on the fraction-order chaotic Chen system are presented to justify the theoretical analysis.
Ding, Zhixia; Shen, Yi; Wang, Leimin
2016-01-01
This paper is concerned with the global Mittag-Leffler synchronization for a class of fractional-order neural networks with discontinuous activations (FNNDAs). We give the concept of Filippov solution for FNNDAs in the sense of Caputo's fractional derivation. By using a singular Gronwall inequality and the properties of fractional calculus, the existence of global solution under the framework of Filippov for FNNDAs is proved. Based on the nonsmooth analysis and control theory, some sufficient criteria for the global Mittag-Leffler synchronization of FNNDAs are derived by designing a suitable controller. The proposed results enrich and enhance the previous reports. Finally, one numerical example is given to demonstrate the effectiveness of the theoretical results. Copyright © 2015 Elsevier Ltd. All rights reserved.
Si, Gang-Quan; Sun, Zhi-Yong; Zhang, Yan-Bin
2011-08-01
This paper investigates the synchronization between integer-order and fractional-order chaotic systems. By introducing fractional-order operators into the controllers, the addressed problem is transformed into a synchronization one among integer-order systems. A novel general method is presented in the paper with rigorous proof. Based on this method, effective controllers are designed for the synchronization between Lorenz systems with an integer order and a fractional order, and for the synchronization between an integer-order Chen system and a fractional-order Liu system. Numerical results, which agree well with the theoretical analyses, are also given to show the effectiveness of this method.
Directory of Open Access Journals (Sweden)
Bin Wang
2016-01-01
Full Text Available This paper studies the application of frequency distributed model for finite time control of a fractional order nonlinear hydroturbine governing system (HGS. Firstly, the mathematical model of HGS with external random disturbances is introduced. Secondly, a novel terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite time stability and sliding mode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional order HGS. Finally, simulation results show the effectiveness and robustness of the proposed scheme.
Directory of Open Access Journals (Sweden)
Yu-xin Zhao
2014-01-01
Full Text Available This paper presents a novel wavelet kernel neural network (WKNN with wavelet kernel function. It is applicable in online learning with adaptive parameters and is applied on parameters tuning of fractional-order PID (FOPID controller, which could handle time delay problem of the complex control system. Combining the wavelet function and the kernel function, the wavelet kernel function is adopted and validated the availability for neural network. Compared to the conservative wavelet neural network, the most innovative character of the WKNN is its rapid convergence and high precision in parameters updating process. Furthermore, the integrated pressurized water reactor (IPWR system is established by RELAP5, and a novel control strategy combining WKNN and fuzzy logic rule is proposed for shortening controlling time and utilizing the experiential knowledge sufficiently. Finally, experiment results verify that the control strategy and controller proposed have the practicability and reliability in actual complicated system.
Static output feedback ℋ ∞ control for a fractional-order glucose-insulin system
N’Doye, Ibrahima
2015-05-23
This paper presents the ℋ∞ static output feedback control of nonlinear fractional-order systems. Based on the extended bounded real lemma, the ℋ∞ control is formulated and sufficient conditions are derived in terms of linear matrix inequalities (LMIs) formulation by using the fractional Lyapunov direct method where the fractional-order α belongs to 0 < α < 1. The control approach is finally applied to the regulation of the glucose level in diabetes type 1 treatment. Therefore, it is attempted to incorporate fractional-order into the mathematical minimal model of glucose-insulin system dynamics and it is still an interesting challenge to show, how the order of a fractional differential system affects the dynamics of the system in the presence of meal disturbance. Numerical simulations are carried out to illustrate our proposed results and show that the nonlinear fractional-order glucose-insulin systems are, at least, as stable as their integer-order counterpart in the presence of exogenous glucose infusion or meal disturbance. © 2015 Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg
Fractional order fuzzy control of hybrid power system with renewable generation using chaotic PSO.
Pan, Indranil; Das, Saptarshi
2016-05-01
This paper investigates the operation of a hybrid power system through a novel fuzzy control scheme. The hybrid power system employs various autonomous generation systems like wind turbine, solar photovoltaic, diesel engine, fuel-cell, aqua electrolyzer etc. Other energy storage devices like the battery, flywheel and ultra-capacitor are also present in the network. A novel fractional order (FO) fuzzy control scheme is employed and its parameters are tuned with a particle swarm optimization (PSO) algorithm augmented with two chaotic maps for achieving an improved performance. This FO fuzzy controller shows better performance over the classical PID, and the integer order fuzzy PID controller in both linear and nonlinear operating regimes. The FO fuzzy controller also shows stronger robustness properties against system parameter variation and rate constraint nonlinearity, than that with the other controller structures. The robustness is a highly desirable property in such a scenario since many components of the hybrid power system may be switched on/off or may run at lower/higher power output, at different time instants. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Eigenvalues for Iterative Systems of (n,p-Type Fractional Order Boundary Value Problems
Directory of Open Access Journals (Sweden)
K. R. Prasad
2014-04-01
Full Text Available In this paper, we determine the eigenvalue intervals of λ1, λ2, ..., λn for which the iterative system of (n,p-type fractional order two-point boundary value problem has a positive solution by an application of Guo-Krasnosel’skii fixed point theorem on a cone.
Adaptive function projective synchronization between different fractional-order chaotic systems
Zhou, Ping; Ding, Rui
2012-06-01
An adaptive function projective synchronization scheme between two entirely different fractional-order chaotic systems with uncertain parameters has been addressed. The adaptive controller has been designed, and the parameter update law is also gained. One example has been presented to demonstrate the effectiveness of the proposed method.
Chen, Diyi; Zhang, Runfan; Sprott, J C; Chen, Haitao; Ma, Xiaoyi
2012-06-01
In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.
Chen, Diyi; Zhang, Runfan; Sprott, J. C.; Chen, Haitao; Ma, Xiaoyi
2012-06-01
In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.
Sliding-Mode Synchronization Control for Uncertain Fractional-Order Chaotic Systems with Time Delay
Directory of Open Access Journals (Sweden)
Haorui Liu
2015-06-01
Full Text Available Specifically setting a time delay fractional financial system as the study object, this paper proposes a single controller method to eliminate the impact of model uncertainty and external disturbances on the system. The proposed method is based on the stability theory of Lyapunov sliding-mode adaptive control and fractional-order linear systems. The controller can fit the system state within the sliding-mode surface so as to realize synchronization of fractional-order chaotic systems. Analysis results demonstrate that the proposed single integral, sliding-mode control method can control the time delay fractional power system to realize chaotic synchronization, with strong robustness to external disturbance. The controller is simple in structure. The proposed method was also validated by numerical simulation.
Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case
Zeng, Cheng; Liang, Shan; Su, Yingying
2013-01-01
Remarkable improvements in the asymptotic properties of discrete system zeros may be achieved by properly adjusted fractional-order hold (FROH) circuit. This paper analyzes asymptotic properties of the limiting zeros, as the sampling period $T$ tends to zero, of the sampled-data models on the basis of the normal form representation of the continuous-time systems with FROH. Moreover, when the relative degree of the continuous-time system is equal to one or two, an approximate expression of the...
Dynamic Analysis and Circuit Design of a Novel Hyperchaotic System with Fractional-Order Terms
Directory of Open Access Journals (Sweden)
Abir Lassoued
2017-01-01
Full Text Available A novel hyperchaotic system with fractional-order (FO terms is designed. Its highly complex dynamics are investigated in terms of equilibrium points, Lyapunov spectrum, and attractor forms. It will be shown that the proposed system exhibits larger Lyapunov exponents than related hyperchaotic systems. Finally, to enhance its potential application, a related circuit is designed by using the MultiSIM Software. Simulation results verify the effectiveness of the suggested circuit.
Dai, Hao; Si, Gangquan; Jia, Lixin; Zhang, Yanbin
2013-11-01
This paper investigates generalized function matrix projective lag synchronization between fractional-order and integer-order complex networks with delayed coupling, non-identical topological structures and different dimensions. Based on Lyapunov stability theory, generalized function matrix projective lag synchronization criteria are derived by using the adaptive control method. In addition, the three-dimensional fractional-order chaotic system and the four-dimensional integer-order hyperchaotic system as the nodes of the drive and the response networks, respectively, are analyzed in detail, and numerical simulation results are presented to illustrate the effectiveness of the theoretical results.
Qiang Gao; Liang Zheng; Jilin Chen; Li Wang; Yuanlong Hou
2014-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipment (GCE) with excellent performances. In this paper, a novel disturbance observer (DOB) based fractional order PD (FOPD) control strategy is proposed for the GCE. By adopting the DOB, the control system behaves as if it were the nominal closed-loop system in the absence of disturbances and uncertainties. The optimal control parameters of the FOPD are determined from the loop-shaping perspective, and th...
Ding, Da-Wei; Liu, Fang-Fang; Chen, Hui; Wang, Nian; Liang, Dong
2017-12-01
In this paper, a simplest fractional-order delayed memristive chaotic system is proposed in order to control the chaos behaviors via sliding mode control strategy. Firstly, we design a sliding mode control strategy for the fractional-order system with time delay to make the states of the system asymptotically stable. Then, we obtain theoretical analysis results of the control method using Lyapunov stability theorem which guarantees the asymptotic stability of the non-commensurate order and commensurate order system with and without uncertainty and an external disturbance. Finally, numerical simulations are given to verify that the proposed sliding mode control method can eliminate chaos and stabilize the fractional-order delayed memristive system in a finite time. Supported by the National Nature Science Foundation of China under Grant No. 61201227, Funding of China Scholarship Council, the Natural Science Foundation of Anhui Province under Grant No. 1208085M F93, 211 Innovation Team of Anhui University under Grant Nos. KJTD007A and KJTD001B
Directory of Open Access Journals (Sweden)
Baogui Xin
2012-01-01
Full Text Available Based on linear feedback control technique, a projective synchronization scheme of N-dimensional chaotic fractional-order systems is proposed, which consists of master and slave fractional-order financial systems coupled by linear state error variables. It is shown that the slave system can be projectively synchronized with the master system constructed by state transformation. Based on the stability theory of linear fractional order systems, a suitable controller for achieving synchronization is designed. The given scheme is applied to achieve projective synchronization of chaotic fractional-order financial systems. Numerical simulations are given to verify the effectiveness of the proposed projective synchronization scheme.
Energy Technology Data Exchange (ETDEWEB)
Lin, Tsung-Chih, E-mail: tclin@fcu.edu.tw [Department of Electronic Engineering, Feng-Chia University, Taichung, Taiwan (China); Lee, Tun-Yuan [Department of Electronic Engineering, Feng-Chia University, Taichung, Taiwan (China); Balas, Valentina E. [Aurel Vlaicu University of Arad, B-dul Revolutiei 77, 310130 Arad (Romania)
2011-10-15
Highlights: > We study uncertain fractional order chaotic systems synchronization. > Lyapunov synthesis is used to derive control law and adaptive laws. > Based on sliding mode control, chattering phenomena in the control effort can be reduced. - Abstract: This paper deals with chaos synchronization between two different uncertain fractional order chaotic systems based on adaptive fuzzy sliding mode control (AFSMC). With the definition of fractional derivatives and integrals, a fuzzy Lyapunov synthesis approach is proposed to tune free parameters of the adaptive fuzzy controller on line by output feedback control law and adaptive law. Moreover, chattering phenomena in the control efforts can be reduced. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but also the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and synchronization performance of the advocated design methodology.
Directory of Open Access Journals (Sweden)
Teh-Lu Liao
2014-01-01
Full Text Available This paper investigates the robust synchronization problem for a class of fractional-order hyperchaotic systems subjected to unmatched uncertainties and input nonlinearity. Based on the sliding mode control (SMC technique, this approach only uses a single controller to achieve chaos synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero or into predictable bounds for matched and unmatched perturbations, respectively, even with input nonlinearity.
Directory of Open Access Journals (Sweden)
Hua Chen
2013-01-01
Full Text Available A second-order sliding mode (SOSM controller is proposed to synchronize a class of incommensurate fractional-order chaotic systems with model uncertainties and external disturbances. Based on the chattering free SOSM control scheme, it can be rigorously proved that the dynamics of the synchronization error is globally asymptotically stable by using the Lyapunov stability theorem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed controller design approach.
Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems
Directory of Open Access Journals (Sweden)
Hai Zhang
2014-01-01
Full Text Available We investigate the delay-independently asymptotic stability of fractional-order linear singular delay differential systems. Based on the algebraic approach, the sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria, one can avoid solving the roots of transcendental equations. An example is also provided to illustrate the effectiveness and applicability of the theoretical results.
Multiobjective Optimization Design of a Fractional Order PID Controller for a Gun Control System
Qiang Gao; Jilin Chen; Li Wang; Shiqing Xu; Yuanlong Hou
2013-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipments possessing excellent performances. In this paper, a typical fractional order PID control strategy is employed for the gun control system. To obtain optimal parameters of the controller, a multiobjective optimization scheme is developed from the loop-shaping perspective. To solve the specified nonlinear optimization problem, a novel Pareto optimal solution based multiobjective differential evolution...
Zhang, Xinxin; Niu, Peifeng; Ma, Yunpeng; Wei, Yanqiao; Li, Guoqiang
2017-10-01
This paper is concerned with the stability analysis issue of fractional-order impulsive neural networks. Under the one-side Lipschitz condition or the linear growth condition of activation function, the existence of solution is analyzed respectively. In addition, the existence, uniqueness and global Mittag-Leffler stability of equilibrium point of the fractional-order impulsive neural networks with one-side Lipschitz condition are investigated by the means of contraction mapping principle and Lyapunov direct method. Finally, an example with numerical simulation is given to illustrate the validity and feasibility of the proposed results. Copyright © 2017 Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Adel Ouannas
2017-01-01
Full Text Available We present new approaches to synchronize different dimensional master and slave systems described by integer order and fractional order differential equations. Based on fractional order Lyapunov approach and integer order Lyapunov stability method, effective control schemes to rigorously study the coexistence of some synchronization types between integer order and fractional order chaotic systems with different dimensions are introduced. Numerical examples are used to validate the theoretical results and to verify the effectiveness of the proposed schemes.
Aghababa, Mohammad Pourmahmood
2012-06-01
This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.
Wang, Dong-Feng; Zhang, Jin-Ying; Wang, Xiao-Yan
2013-04-01
This paper provides a novel method to synchronize uncertain fractional-order chaotic systems with external disturbance via fractional terminal sliding mode control. Based on Lyapunov stability theory, a new fractional-order switching manifold is proposed, and in order to ensure the occurrence of sliding motion in finite time, a corresponding sliding mode control law is designed. The proposed control scheme is applied to synchronize the fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with uncertainty and external disturbance parameters. The simulation results show the applicability and efficiency of the proposed scheme.
Directory of Open Access Journals (Sweden)
Jin-E Zhang
2017-01-01
Full Text Available In this paper, the global O(t-α synchronization problem is investigated for a class of fractional-order neural networks with time delays. Taking into account both better control performance and energy saving, we make the first attempt to introduce centralized data-sampling approach to characterize the O(t-α synchronization design strategy. A sufficient criterion is given under which the drive-response-based coupled neural networks can achieve global O(t-α synchronization. It is worth noting that, by using centralized data-sampling principle, fractional-order Lyapunov-like technique, and fractional-order Leibniz rule, the designed controller performs very well. Two numerical examples are presented to illustrate the efficiency of the proposed centralized data-sampling scheme.
Dynamics of a fractional-order simplified unified system based on the Adomian decomposition method
Xu, Yixin; Sun, Kehui; He, Shaobo; Zhang, Limin
2016-06-01
In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order simplified unified system. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, power spectrum and maximum Lyapunov exponent diagram. Route to chaos by period doubling is observed. Chaos is found to exist in the system with order as low as 1.371. In addition, chaotic behaviors are studied with different dimensional order, and the results show that this system is chaotic when only the first or third dimension is fractional.
Directory of Open Access Journals (Sweden)
Ping Zhou
2011-01-01
Full Text Available An adaptive hybrid function projective synchronization (AHFPS scheme between different fractional-order chaotic systems with uncertain system parameter is addressed in this paper. In this proposed scheme, the drive and response system could be synchronized up to a vector function factor. This proposed scheme is different with the function projective synchronization (FPS scheme, in which the drive and response system could be synchronized up to a scaling function factor. The adaptive controller and the parameter update law are gained. Two examples are presented to demonstrate the effectiveness of the proposed scheme.
Finite-Time Synchronizing Fractional-Order Chaotic Volta System with Nonidentical Orders
Directory of Open Access Journals (Sweden)
Jian-Bing Hu
2013-01-01
Full Text Available We investigate synchronizing fractional-order Volta chaotic systems with nonidentical orders in finite time. Firstly, the fractional chaotic system with the same structure and different orders is changed to the chaotic systems with identical orders and different structure according to the property of fractional differentiation. Secondly, based on the lemmas of fractional calculus, a controller is designed according to the changed fractional chaotic system to synchronize fractional chaotic with nonidentical order in finite time. Numerical simulations are performed to demonstrate the effectiveness of the method.
Directory of Open Access Journals (Sweden)
Wafaa S. Sayed
2017-01-01
Full Text Available This paper proposes a novel generalized switched synchronization scheme among n fractional-order chaotic systems with various operating modes. Digital dynamic switches and dynamic scaling factors are employed, which offer many new capabilities. Dynamic switches determine the role of each system as a master or a slave. A system can either have a fixed role throughout the simulation time (static switching or switch its role one or more times (dynamic switching. Dynamic scaling factors are used for each state variable of the master system. Such scaling factors control whether the master is a single system or a combination of several systems. In addition, these factors determine the generalized relation between the original systems from which the master system is built as well as the slave system(s. Moreover, they can be utilized to achieve different kinds of generalized synchronization relations for the purpose of generating new attractor diagrams. The paper presents a mathematical formulation and analysis of the proposed synchronization scheme. Furthermore, many numerical simulations are provided to demonstrate the successful generalized switched synchronization of several fractional-order chaotic systems. The proposed scheme provides various functions suitable for applications such as different master-slave communication models and secure communication systems.
Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection
Delshad, Saleh Sayyad; Asheghan, Mohammad Mostafa; Beheshti, Mohammadtaghi Hamidi
2011-09-01
In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to synchronize N-coupled incommensurate fractional-order chaotic systems with ring connection. Chaos synchronization is studied by utilizing the coupling method that includes unidirectional and bidirectional coupling. The main contribution of this work is to design the coupling parameters in which it is shown that our proposed controller does stabilize error dynamics around all of equilibriums of master system. Finally, the claims are justified through a set of simulations and results coincide with the theoretical analysis.
Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems
Directory of Open Access Journals (Sweden)
Huiling Xi
2014-11-01
Full Text Available In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems.
Stabilization and control of fractional order systems a sliding mode approach
Bandyopadhyay, Bijnan
2015-01-01
In the last two decades fractional differential equations have been used more frequently in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electro chemistry and many others. It opens a new and more realistic way to capture memory dependent phenomena and irregularities inside the systems by using more sophisticated mathematical analysis.This monograph is based on the authors' work on stabilization and control design for continuous and discrete fractional order systems. The initial two chapters and some parts of the third chapter are written in tutorial fashi
Guaranteed Cost Finite-Time Control of Fractional-Order Positive Switched Systems
Directory of Open Access Journals (Sweden)
Leipo Liu
2017-01-01
Full Text Available The problem of guaranteed cost finite-time control of fractional-order positive switched systems (FOPSS is considered in this paper. Firstly, a new cost function is defined. Then, by constructing linear copositive Lyapunov functions and using the average dwell time (ADT approach, a state feedback controller and a static output feedback controller are constructed, respectively, and sufficient conditions are derived to guarantee that the corresponding closed-loop systems are guaranteed cost finite-time stable (GCFTS. Such conditions can be easily solved by linear programming. Finally, two examples are given to illustrate the effectiveness of the proposed method.
Directory of Open Access Journals (Sweden)
Zhao Yige
2011-01-01
Full Text Available Abstract In this article, we study the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders where 2 < α ≤ 3, 3 < β ≤ 4, , are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] × [0, +∞ → [0, +∞ are given continuous functions, f(t, 0 ≡ 0, g(t, 0 ≡ 0. Our analysis relies on fixed point theorems on cones. Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. As an application, examples are presented to illustrate the main results.
Directory of Open Access Journals (Sweden)
C. Ünlü
2013-01-01
Full Text Available A modification of the variational iteration method (VIM for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.
Merrikh-Bayat, Farshad
2011-04-01
One main approach for time-domain simulation of the linear output-feedback systems containing fractional-order controllers is to approximate the transfer function of the controller with an integer-order transfer function and then perform the simulation. In general, this approach suffers from two main disadvantages: first, the internal stability of the resulting feedback system is not guaranteed, and second, the amount of error caused by this approximation is not exactly known. The aim of this paper is to propose an efficient method for time-domain simulation of such systems without facing the above mentioned drawbacks. For this purpose, the fractional-order controller is approximated with an integer-order transfer function (possibly in combination with the delay term) such that the internal stability of the closed-loop system is guaranteed, and then the simulation is performed. It is also shown that the resulting approximate controller can effectively be realized by using the proposed method. Some formulas for estimating and correcting the simulation error, when the feedback system under consideration is subjected to the unit step command or the unit step disturbance, are also presented. Finally, three numerical examples are studied and the results are compared with the Oustaloup continuous approximation method. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Hai Zhang
2017-01-01
Full Text Available This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the U-layer and V-layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.
Gao, Qiang; Zheng, Liang; Chen, Jilin; Wang, Li; Hou, Yuanlong
2014-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipment (GCE) with excellent performances. In this paper, a novel disturbance observer (DOB) based fractional order PD (FOPD) control strategy is proposed for the GCE. By adopting the DOB, the control system behaves as if it were the nominal closed-loop system in the absence of disturbances and uncertainties. The optimal control parameters of the FOPD are determined from the loop-shaping perspective, and the Q-filter of the DOB is deliberately designed with consideration of system robustness. The linear frame of the proposed control system will enable the analysis process more convenient. The disturbance rejection properties and the tracking performances of the control system are investigated by both numerical and experimental tests, the results demonstrate that the proposed DOB based FOPD control system is of more robustness, and it is much more suitable for the gun control system with strong nonlinearity and disturbance.
Analysis of tristable energy harvesting system having fractional order viscoelastic material
Energy Technology Data Exchange (ETDEWEB)
Oumbé Tékam, G. T.; Woafo, P. [Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé (Cameroon); Kitio Kwuimy, C. A. [Center for Nonlinear Dynamics and Control, Department of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, Pennsylvania 19085 (United States)
2015-01-15
A particular attention is devoted to analyze the dynamics of a strongly nonlinear energy harvester having fractional order viscoelastic flexible material. The strong nonlinearity is obtained from the magnetic interaction between the end free of the flexible material and three equally spaced magnets. Periodic responses are computed using the KrylovBogoliubov averaging method, and the effects of fractional order damping on the output electric energy are analyzed. It is obtained that the harvested energy is enhanced for small order of the fractional derivative. Considering the order and strength of the fractional viscoelastic property as control parameter, the complexity of the system response is investigated through the Melnikov criteria for horseshoes chaos, which allows us to derive the mathematical expression of the boundary between intra-well motion and bifurcations appearance domain. We observe that the order and strength of the fractional viscoelastic property can be effectively used to control chaos in the system. The results are confirmed by the smooth and fractal shape of the basin of attraction as the order of derivative decreases. The bifurcation diagrams and the corresponding Lyapunov exponents are plotted to get insight into the nonlinear response of the system.
A novel adaptive-impulsive synchronization of fractional-order chaotic systems
Leung, Y. T. Andrew; Li, Xian-Feng; Chu, Yan-Dong; Zhang, Hui
2015-10-01
A novel adaptive-impulsive scheme is proposed for synchronizing fractional-order chaotic systems without the necessity of knowing the attractors’ bounds in priori. The nonlinear functions in these systems are supposed to satisfy local Lipschitz conditions but which are estimated with adaptive laws. The novelty is that the combination of adaptive control and impulsive control offers a control strategy gathering the advantages of both. In order to guarantee the convergence is no less than an expected exponential rate, a combined feedback strength design is created such that the symmetric axis can shift freely according to the updated transient feedback strength. All of the unknown Lipschitz constants are also updated exponentially in the meantime of achieving synchronization. Two different fractional-order chaotic systems are employed to demonstrate the effectiveness of the novel adaptive-impulsive control scheme. Project supported by the National Natural Science Foundations of China (Grant Nos. 11161027 and 11262009), the Key Natural Science Foundation of Gansu Province, China (Grant No. 1104WCGA195), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20136204110001).
Estimation of Multiple Point Sources for Linear Fractional Order Systems Using Modulating Functions
Belkhatir, Zehor
2017-06-28
This paper proposes an estimation algorithm for the characterization of multiple point inputs for linear fractional order systems. First, using polynomial modulating functions method and a suitable change of variables the problem of estimating the locations and the amplitudes of a multi-pointwise input is decoupled into two algebraic systems of equations. The first system is nonlinear and solves for the time locations iteratively, whereas the second system is linear and solves for the input’s amplitudes. Second, closed form formulas for both the time location and the amplitude are provided in the particular case of single point input. Finally, numerical examples are given to illustrate the performance of the proposed technique in both noise-free and noisy cases. The joint estimation of pointwise input and fractional differentiation orders is also presented. Furthermore, a discussion on the performance of the proposed algorithm is provided.
Wang, Li-Ming; Tang, Yong-Guang; Chai, Yong-Quan; Wu, Feng
2014-10-01
An adaptive fuzzy sliding mode strategy is developed for the generalized projective synchronization of a fractional-order chaotic system, where the slave system is not necessarily known in advance. Based on the designed adaptive update laws and the linear feedback method, the adaptive fuzzy sliding controllers are proposed via the fuzzy design, and the strength of the designed controllers can be adaptively adjusted according to the external disturbances. Based on the Lyapunov stability theorem, the stability and the robustness of the controlled system are proved theoretically. Numerical simulations further support the theoretical results of the paper and demonstrate the efficiency of the proposed method. Moreover, it is revealed that the proposed method allows us to manipulate arbitrarily the response dynamics of the slave system by adjusting the desired scaling factor λi and the desired translating factor ηi, which may be used in a channel-independent chaotic secure communication.
Ding, Zhixia; Zeng, Zhigang; Wang, Leimin
2017-03-10
This paper is concerned with robust finite-time stabilization for a class of fractional-order neural networks (FNNs) with two types of activation functions (i.e., discontinuous and continuous activation function) under uncertainty. It is worth noting that there exist few results about FNNs with discontinuous activation functions, which is mainly because classical solutions and theories of differential equations cannot be applied in this case. Especially, there is no relevant finite-time stabilization research for such system, and this paper makes up for the gap. The existence of global solution under the framework of Filippov for such system is guaranteed by limiting discontinuous activation functions. According to set-valued analysis and Kakutani's fixed point theorem, we obtain the existence of equilibrium point. In particular, based on differential inclusion theory and fractional Lyapunov stability theory, several new sufficient conditions are given to ensure finite-time stabilization via a novel discontinuous controller, and the upper bound of the settling time for stabilization is estimated. In addition, we analyze the finite-time stabilization of FNNs with Lipschitz-continuous activation functions under uncertainty. The results of this paper improve corresponding ones of integer-order neural networks with discontinuous and continuous activation functions. Finally, three numerical examples are given to show the effectiveness of the theoretical results.
Fractional order phase shaper design with Bode's integral for iso-damped control system.
Saha, Suman; Das, Saptarshi; Ghosh, Ratna; Goswami, Bhaswati; Balasubramanian, R; Chandra, A K; Das, Shantanu; Gupta, Amitava
2010-04-01
The phase curve of an open loop system is flat in nature if the derivative of its phase with respect to frequency is zero. With a flat-phase curve, the corresponding closed loop system exhibits an iso-damped property i.e. maintains constant overshoot with the change of gain. This implies enhanced parametric robustness e.g. to variation in system gain. In the recent past, fractional order (FO) phase shapers have been proposed by contemporary researchers to achieve enhanced parametric robustness. In this paper, a simple methodology is proposed to design an appropriate FO phase shaper to achieve phase flattening in a control loop, comprising a plant controlled by a classical Proportional Integral Derivative (PID) controller. The methodology is demonstrated with MATLAB simulation of representative plants and accompanying PID controllers. Copyright 2009 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Seyed Abbas Taher
2014-03-01
Full Text Available In this paper, fractional order PID (FOPID controller was proposed for load frequency control (LFC in an interconnected power system. This controller had five parameters to be tuned; thus, it provided two more degrees of freedom in comparison with the conventional PID. For proper tuning of the controller parameters, imperialist competitive algorithm (ICA was used. ICA is a new evolutionary algorithm with proved efficiency. In this study, simulation investigations were carried out on a three-area power system with different generating units. These results showed that FOPID controller was robust to the parameter changes in the power system. Also, the simulation results certified much better performance of FOPID controller for LFC in comparison with conventional PID controllers.
Chen, Di-Yi; Zhang, Run-Fan; Ma, Xiao-Yi; Wang, Juan
2012-12-01
In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integer-order chaotic system, in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method. Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus. Moreover, three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results. Finally, results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.
Model-free adaptive fractional order control of stable linear time-varying systems.
Yakoub, Z; Amairi, M; Chetoui, M; Saidi, B; Aoun, M
2017-03-01
This paper presents a new model-free adaptive fractional order control approach for linear time-varying systems. An online algorithm is proposed to determine some frequency characteristics using a selective filtering and to design a fractional PID controller based on the numerical optimization of the frequency-domain criterion. When the system parameters are time-varying, the controller is updated to keep the same desired performances. The main advantage of the proposed approach is that the controller design depends only on the measured input and output signals of the process. The effectiveness of the proposed method is assessed through a numerical example. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Multiobjective optimization design of a fractional order PID controller for a gun control system.
Gao, Qiang; Chen, Jilin; Wang, Li; Xu, Shiqing; Hou, Yuanlong
2013-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipments possessing excellent performances. In this paper, a typical fractional order PID control strategy is employed for the gun control system. To obtain optimal parameters of the controller, a multiobjective optimization scheme is developed from the loop-shaping perspective. To solve the specified nonlinear optimization problem, a novel Pareto optimal solution based multiobjective differential evolution algorithm is proposed. To enhance the convergent rate of the optimization process, an opposition based learning method is embedded in the chaotic population initialization process. To enhance the robustness of the algorithm for different problems, an adapting scheme of the mutation operation is further employed. With assistance of the evolutionary algorithm, the optimal solution for the specified problem is selected. The numerical simulation results show that the control system can rapidly follow the demand signal with high accuracy and high robustness, demonstrating the efficiency of the proposed controller parameter tuning method.
H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach.
Lin, Tsung-Chih; Kuo, Chia-Hao
2011-10-01
This paper presents a novel adaptive fuzzy logic controller (FLC) equipped with an adaptive algorithm to achieve H(∞) synchronization performance for uncertain fractional order chaotic systems. In order to handle the high level of uncertainties and noisy training data, a desired synchronization error can be attenuated to a prescribed level by incorporating fuzzy control design and H(∞) tracking approach. Based on a Lyapunov stability criterion, not only the performance of the proposed method is satisfying with an acceptable synchronization error level, but also a rather simple stability analysis is performed. The simulation results signify the effectiveness of the proposed control scheme. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Jianyu Lin
2015-01-01
uncertain systems with a commensurate order 1<α<2 under controller gain uncertainties. The uncertainties are of norm-bounded type. Based on the stability criterion of fractional order system, sufficient conditions on the decentralized stabilization of fractional order large-scale uncertain systems in both cases of additive and multiplicative gain perturbations are established by using the complex Lyapunov inequality. Moreover, the decentralized nonfragile controllers are designed. Finally, some numerical examples are given to validate the proposed method.
Lu Liu; Feng Pan; Dingyu Xue
2015-01-01
Fractional-order time-delay system is thought to be a kind of oscillatory complex system which could not be controlled efficaciously so far because it does not have an analytical solution when using inverse Laplace transform. In this paper, a type of fractional-order controller based on numerical inverse Laplace transform algorithm INVLAP was proposed for the mentioned systems by searching for the optimal controller parameters with the objective function of ITAE index due to the verified natu...
Biswas, Karabi; Caponetto, Riccardo; Mendes Lopes, António; Tenreiro Machado, José António
2017-01-01
This book focuses on two specific areas related to fractional order systems – the realization of physical devices characterized by non-integer order impedance, usually called fractional-order elements (FOEs); and the characterization of vegetable tissues via electrical impedance spectroscopy (EIS) – and provides readers with new tools for designing new types of integrated circuits. The majority of the book addresses FOEs. The interest in these topics is related to the need to produce “analogue” electronic devices characterized by non-integer order impedance, and to the characterization of natural phenomena, which are systems with memory or aftereffects and for which the fractional-order calculus tool is the ideal choice for analysis. FOEs represent the building blocks for designing and realizing analogue integrated electronic circuits, which the authors believe hold the potential for a wealth of mass-market applications. The freedom to choose either an integer- or non-integer-order analogue integrator...
Srivastava, Mayank; Agrawal, Saurabh; Das, Subir
2013-10-01
The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.
Directory of Open Access Journals (Sweden)
Yan Yan
2016-01-01
Full Text Available This paper deals with the synchronization of a class of fractional order chaotic systems with unknown parameters and external disturbance. Based on the Lyapunov stability theory, a fractional order sliding mode is constructed and a controller is proposed to realize chaos synchronization. The presented method not only realizes the synchronization of the considered chaotic systems but also enhances the robustness of sliding mode synchronization. Finally, some simulation results demonstrate the effectiveness and robustness of the proposed method.
Cao, Hefei; Zhang, Ruoxun; Yan, Fengli
2013-02-01
A one-driving-variable adaptive controller for synchronization of a kind of fractional order chaotic system is designed. Based on the theory of spread spectrum communication and the synchronization of one-driving-variable fractional order chaotic system, we propose a new scheme for general spread spectrum communication. Numerical simulation and circuit experiment results are provided to illustrate the effectiveness of the proposed scheme.
Directory of Open Access Journals (Sweden)
Qiang Gao
2014-01-01
Full Text Available Motion control of gun barrels is an ongoing topic for the development of gun control equipment (GCE with excellent performances. In this paper, a novel disturbance observer (DOB based fractional order PD (FOPD control strategy is proposed for the GCE. By adopting the DOB, the control system behaves as if it were the nominal closed-loop system in the absence of disturbances and uncertainties. The optimal control parameters of the FOPD are determined from the loop-shaping perspective, and the Q-filter of the DOB is deliberately designed with consideration of system robustness. The linear frame of the proposed control system will enable the analysis process more convenient. The disturbance rejection properties and the tracking performances of the control system are investigated by both numerical and experimental tests, the results demonstrate that the proposed DOB based FOPD control system is of more robustness, and it is much more suitable for the gun control system with strong nonlinearity and disturbance.
Zheng, Liang; Chen, Jilin; Wang, Li; Hou, Yuanlong
2014-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipment (GCE) with excellent performances. In this paper, a novel disturbance observer (DOB) based fractional order PD (FOPD) control strategy is proposed for the GCE. By adopting the DOB, the control system behaves as if it were the nominal closed-loop system in the absence of disturbances and uncertainties. The optimal control parameters of the FOPD are determined from the loop-shaping perspective, and the Q-filter of the DOB is deliberately designed with consideration of system robustness. The linear frame of the proposed control system will enable the analysis process more convenient. The disturbance rejection properties and the tracking performances of the control system are investigated by both numerical and experimental tests, the results demonstrate that the proposed DOB based FOPD control system is of more robustness, and it is much more suitable for the gun control system with strong nonlinearity and disturbance. PMID:24616616
Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach
Directory of Open Access Journals (Sweden)
Radek Matušů
2015-01-01
Full Text Available The paper deals with a graphical approach to investigation of robust stability for a feedback control loop with an uncertain fractional order time-delay plant and integer order or fractional order controller. Robust stability analysis is based on plotting the value sets for a suitable range of frequencies and subsequent verification of the zero exclusion condition fulfillment. The computational examples present the typical shapes of the value sets of a family of closed-loop characteristic quasipolynomials for a fractional order plant with uncertain gain, time constant, or time-delay term, respectively, and also for combined cases. Moreover, the practically oriented example focused on robust stability analysis of main irrigation canal pool controlled by either classical integer order PID or fractional order PI controller is included as well.
Yeroğlu, Celaleddin; Kavuran, Gürkan
2015-01-01
This paper presents a design method for a sliding mode controller with the contribution of a fractional order differential operator. The conventional sliding mode controller has been widely studied in different control applications. This paper proposes that the fractional order differential operator enlarges the output span of the classical sliding mode controller to obtain a better-fitting control signal for enhanced control performance. The sliding surface and the equivalent control ...
Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case
Directory of Open Access Journals (Sweden)
Cheng Zeng
2013-01-01
Full Text Available Remarkable improvements in the asymptotic properties of discrete system zeros may be achieved by properly adjusted fractional-order hold (FROH circuit. This paper analyzes asymptotic properties of the limiting zeros, as the sampling period T tends to zero, of the sampled-data models on the basis of the normal form representation of the continuous-time systems with FROH. Moreover, when the relative degree of the continuous-time system is equal to one or two, an approximate expression of the limiting zeros for the sampled-data system with FROH is also given as power series with respect to a sampling period up to the third-order term. And, further, the corresponding stability conditions of the sampling zeros are discussed for fast sampling rates. The ideas of the paper here provide a more accurate approximation for asymptotic zeros, and certain known achievements on asymptotic behavior of limiting zeros are shown to be particular cases of the results presented.
Directory of Open Access Journals (Sweden)
Li Junyi
2015-01-01
Full Text Available A fractional order PID (FOPID controller, which is suitable for control system designing for being insensitive to the variation in system parameter, is proposed for hydroturbine governing system in the paper. The simultaneous optimization for several parameters of controller, that is, Ki, Kd, Kp, λ, and μ, is done by a recently developed metaheuristic nature-inspired algorithm, namely, the firefly algorithm (FA, for the first time, where the selecting, moving, attractiveness behavior between fireflies and updating of brightness, and decision range are studied in detail to simulate the optimization process. Investigation clearly reveals the advantages of the FOPID controller over the integer controllers in terms of reduced oscillations and settling time. The present work also explores the superiority of FA based optimization technique in finding optimal parameters of the controller. Further, convergence characteristics of the FA are compared with optimum integer order PID (IOPID controller to justify its efficiency. What is more, analysis confirms the robustness of FOPID controller under isolated load operation conditions.
Multiobjective Optimization Design of a Fractional Order PID Controller for a Gun Control System
Chen, Jilin; Wang, Li; Xu, Shiqing; Hou, Yuanlong
2013-01-01
Motion control of gun barrels is an ongoing topic for the development of gun control equipments possessing excellent performances. In this paper, a typical fractional order PID control strategy is employed for the gun control system. To obtain optimal parameters of the controller, a multiobjective optimization scheme is developed from the loop-shaping perspective. To solve the specified nonlinear optimization problem, a novel Pareto optimal solution based multiobjective differential evolution algorithm is proposed. To enhance the convergent rate of the optimization process, an opposition based learning method is embedded in the chaotic population initialization process. To enhance the robustness of the algorithm for different problems, an adapting scheme of the mutation operation is further employed. With assistance of the evolutionary algorithm, the optimal solution for the specified problem is selected. The numerical simulation results show that the control system can rapidly follow the demand signal with high accuracy and high robustness, demonstrating the efficiency of the proposed controller parameter tuning method. PMID:23766721
Multiobjective Optimization Design of a Fractional Order PID Controller for a Gun Control System
Directory of Open Access Journals (Sweden)
Qiang Gao
2013-01-01
Full Text Available Motion control of gun barrels is an ongoing topic for the development of gun control equipments possessing excellent performances. In this paper, a typical fractional order PID control strategy is employed for the gun control system. To obtain optimal parameters of the controller, a multiobjective optimization scheme is developed from the loop-shaping perspective. To solve the specified nonlinear optimization problem, a novel Pareto optimal solution based multiobjective differential evolution algorithm is proposed. To enhance the convergent rate of the optimization process, an opposition based learning method is embedded in the chaotic population initialization process. To enhance the robustness of the algorithm for different problems, an adapting scheme of the mutation operation is further employed. With assistance of the evolutionary algorithm, the optimal solution for the specified problem is selected. The numerical simulation results show that the control system can rapidly follow the demand signal with high accuracy and high robustness, demonstrating the efficiency of the proposed controller parameter tuning method.
A new color image encryption scheme using CML and a fractional-order chaotic system.
Directory of Open Access Journals (Sweden)
Xiangjun Wu
Full Text Available The chaos-based image cryptosystems have been widely investigated in recent years to provide real-time encryption and transmission. In this paper, a novel color image encryption algorithm by using coupled-map lattices (CML and a fractional-order chaotic system is proposed to enhance the security and robustness of the encryption algorithms with a permutation-diffusion structure. To make the encryption procedure more confusing and complex, an image division-shuffling process is put forward, where the plain-image is first divided into four sub-images, and then the position of the pixels in the whole image is shuffled. In order to generate initial conditions and parameters of two chaotic systems, a 280-bit long external secret key is employed. The key space analysis, various statistical analysis, information entropy analysis, differential analysis and key sensitivity analysis are introduced to test the security of the new image encryption algorithm. The cryptosystem speed is analyzed and tested as well. Experimental results confirm that, in comparison to other image encryption schemes, the new algorithm has higher security and is fast for practical image encryption. Moreover, an extensive tolerance analysis of some common image processing operations such as noise adding, cropping, JPEG compression, rotation, brightening and darkening, has been performed on the proposed image encryption technique. Corresponding results reveal that the proposed image encryption method has good robustness against some image processing operations and geometric attacks.
Rajagopal, Karthikeyan; Karthikeyan, Anitha
2016-09-01
Most of the Real systems shows chaotic behavior when they approach complex states. Especially in physical and chemical systems these behaviors define the character of the system. The control of these chaotic behaviors is of very high practical importance and hence mathematical models of these chaotic systems proves vital in deciding the control structures. One such model of chemical reactors is the Willamowski-Rössler system (WR). In this paper we derive a fractional order sliding mode control scheme where the states of the WR system are driven back to the defined equilibrium points. We have also synchronized master and slave fractional order WR system using sliding mode control. As the entire control law is defined in fractional order, we derived a new methodology to prove the stability of the controller. The numerical simulation and analysis are achieved with LabVIEW.
Directory of Open Access Journals (Sweden)
Lu Liu
2015-01-01
Full Text Available Fractional-order time-delay system is thought to be a kind of oscillatory complex system which could not be controlled efficaciously so far because it does not have an analytical solution when using inverse Laplace transform. In this paper, a type of fractional-order controller based on numerical inverse Laplace transform algorithm INVLAP was proposed for the mentioned systems by searching for the optimal controller parameters with the objective function of ITAE index due to the verified nature that fractional-order controllers were the best means of controlling fractional-order systems. Simulations of step unit tracking and load-disturbance responses of the proposed fractional-order optimal PIλDμ controller (FOPID and corresponding conventional optimal PID (OPID controller have been done on three typical kinds of fractional time-delay system with different ratio between time delay (L and time constant (T and a complex high-order fractional time delay system to verify the availability of the presented control method.
Directory of Open Access Journals (Sweden)
Qiang Gao
2016-03-01
Full Text Available To achieve perfect behavior of the unbalanced barrel of a gun control system, a novel control strategy for simultaneous balancing and positioning of the system is proposed, physically being on the basis of a novel hydraulic cylinder with three cavities. The fuzzy fractional order proportional–integral–derivative controller is developed, and the particle swarm optimization algorithm is adopted for optimal selection of the control parameters for the gun control system. The results demonstrate that the fuzzy fractional order proportional–integral–derivative control strategy can finely improve dynamic performance of the control system, and the nonlinear characteristics of system can be effectively suppressed.
Wang, Huihai; Sun, Kehui; He, Shaobo
2015-06-01
By adopting Adomian decomposition method, the fractional-order simplified Lorenz system is solved and implemented on a digital signal processor (DSP). The Lyapunov exponent (LE) spectra of the system is calculated based on QR-factorization, and it accords well with the corresponding bifurcation diagrams. We analyze the influence of the parameter and the fractional derivative order on the system characteristics by color maximum LE (LEmax) and chaos diagrams. It is found that the smaller the order is, the larger the LEmax is. The iteration step size also affects the lowest order at which the chaos exists. Further, we implement the fractional-order simplified Lorenz system on a DSP platform. The phase portraits generated on DSP are consistent with the results that were obtained by computer simulations. It lays a good foundation for applications of the fractional-order chaotic systems.
Pan, Lin; Zhou, Wuneng; Fang, Jian'an; Li, Dequan
2010-12-01
This paper discusses the synchronization and anti-synchronization of new uncertain fractional-order unified chaotic systems (UFOUCS). Based on the idea of active control, a novel active pinning control strategy is presented, which only needs a state of new UFOUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UFOUCS. Numerical simulations of new UFOUCS show that the controller can make fractional-order unified chaotic systems (FOUCS) achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.
Directory of Open Access Journals (Sweden)
Zahra Yaghoubi
2012-01-01
Full Text Available Synchronization of fractional-order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this paper, a drive-response synchronization method is studied for “phase and antiphase synchronization” of a class of fractional-order chaotic systems via active control method, using the 3-cell and Volta systems as an example. These examples are used to illustrate the effectiveness of the synchronization method.
Zhang, Bitao; Pi, YouGuo
2013-07-01
The traditional integer order proportional-integral-differential (IO-PID) controller is sensitive to the parameter variation or/and external load disturbance of permanent magnet synchronous motor (PMSM). And the fractional order proportional-integral-differential (FO-PID) control scheme based on robustness tuning method is proposed to enhance the robustness. But the robustness focuses on the open-loop gain variation of controlled plant. In this paper, an enhanced robust fractional order proportional-plus-integral (ERFOPI) controller based on neural network is proposed. The control law of the ERFOPI controller is acted on a fractional order implement function (FOIF) of tracking error but not tracking error directly, which, according to theory analysis, can enhance the robust performance of system. Tuning rules and approaches, based on phase margin, crossover frequency specification and robustness rejecting gain variation, are introduced to obtain the parameters of ERFOPI controller. And the neural network algorithm is used to adjust the parameter of FOIF. Simulation and experimental results show that the method proposed in this paper not only achieve favorable tracking performance, but also is robust with regard to external load disturbance and parameter variation. Crown Copyright © 2013. Published by Elsevier Ltd. All rights reserved.
Ding, Dawei; Qian, Xin; Hu, Wei; Wang, Nian; Liang, Dong
2017-11-01
In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay. The associated characteristic equation is established by regarding the time delay as a bifurcation parameter. A set of conditions which ensure the existence of the Hopf bifurcation are gained by analyzing the corresponding characteristic equation. Then, we discuss the influence of feedback gain on the critical value of fractional order and time delay in the controlled system. Theoretical analysis shows that the controller is effective in delaying the Hopf bifurcation critical value via decreasing the feedback gain. Finally, some numerical simulations are presented to prove the validity of our theoretical analysis and confirm that the time-delayed feedback controller is valid in controlling chaos and Hopf bifurcation in the fractional-order memristor-based system.
Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System
Ma, Junhai; Ren, Wenbo
On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.
(王会海, H. H. Wang; (孙克辉, K. H. Sun; (贺少波, S. B. He
2015-01-01
By adopting the Adomian decomposition method, the fractional-order Lorenz-Stenflo (LS) system is solved and implemented in a digital signal processor (DSP). The discrete iterative formula of the system is deduced, and a Lyapunov exponent spectrum algorithm is designed. The dynamics of the fractional-order LS system with sets of parameters are analyzed by means of Lyapunov exponent spectra, bifurcation diagrams and 0-1 test. The results illustrate that the fractional-order LS system has rich dynamic behaviors, and both the system parameter and the fractional order can be taken as bifurcation parameters. We implement the fractional-order LS system on a DSP platform. Phase portraits of the fractional-order LS system generated in the DSP agree well with those obtained by computer simulations. This lays a good foundation for the application of the fractional-order LS system.
Qiang Gao; Kang Li; Yuanlong Hou; Runmin Hou; Chao Wang
2016-01-01
To achieve perfect behavior of the unbalanced barrel of a gun control system, a novel control strategy for simultaneous balancing and positioning of the system is proposed, physically being on the basis of a novel hydraulic cylinder with three cavities. The fuzzy fractional order proportional–integral–derivative controller is developed, and the particle swarm optimization algorithm is adopted for optimal selection of the control parameters for the gun control system. The results demonstrate t...
IMC-PID-fractional-order-filter controllers design for integer order systems.
Maâmar, Bettayeb; Rachid, Mansouri
2014-09-01
One of the reasons of the great success of standard PID controllers is the presence of simple tuning rules, of the automatic tuning feature and of tables that simplify significantly their design. For the fractional order case, some tuning rules have been proposed in the literature. However, they are not general because they are valid only for some model cases. In this paper, a new approach is investigated. The fractional property is not especially imposed by the controller structure but by the closed loop reference model. The resulting controller is fractional but it has a very interesting structure for its implementation. Indeed, the controller can be decomposed into two transfer functions: an integer transfer function which is generally an integer PID controller and a simple fractional filter. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Song, Shuai; Song, Xiaona; Balsera, Ines Tejado
2017-05-01
This paper investigates the mixed H∞ and passive projective synchronization problem for fractional-order (FO) memristor-based neural networks with time delays. Our aim is to design a controller such that, though the unavoidable phenomena of time delay and external disturbances is fully considered, the resulting closed-loop system is stable with a mixed H∞ and passive performance level. By combining sliding mode control and adaptive control methods, a novel adaptive sliding mode control strategy is designed for the synchronization of time-delayed FO dynamic networks. Via the application of FO system stability theory, the projective synchronization conditions are addressed in terms of linear matrix inequalities. Based on the conditions, a desired controller which can guarantee the stability of the closed-loop system and also ensure a mixed H∞ and passive performance level is designed. Finally, two simulation examples are given to illustrate the effectiveness of the proposed method.
Moafi, Milad; Marzband, Mousa; Savaghebi, Mehdi; Guerrero, Josep M.
2016-01-01
The need to reduce greenhouse effect using distributed energy resources (DER) has significantly increased in recent years, particularly with the advent of deregulatedmarket. Climate changes causes large swings in output power of renewable resources and the resulting fluctuations in frequency in the islanded Microgrid (MG).To increase performance for a wide range of power system operating conditions, an energy management systems (EMS) is proposed based on a fuzzy fractional orderPID (FFOPID) c...
DEFF Research Database (Denmark)
Moafi, Milad; Marzband, Mousa; Savaghebi, Mehdi
2016-01-01
in the islanded Microgrid (MG). To increase performance for a wide range of power system operating conditions, an energy management systems (EMS) is proposed based on a fuzzy fractional order PID (FFOPID) controller. It is able to analyze and simulate the dynamic behavior in grid connected MGs. This controller...... storage (ES) is used to improve the system dynamic response, reduce the distortion and provide damping for frequency oscillations caused by renewable resources. ES overload capacity is utilized for rapid initial control of frequency in MG. To achieve this goal, EMS based on fuzzy decision mechanism...... combined with a PID-controller (termed as FLPID) and Fuzzy fractional order PID (termed as FFOPID) are implemented according to the characteristics and limitations of overloading and state of charge (SOC). The obtained results show good performance of FFOPID controllers by improving the transient stability...
Zhong, Jianpeng; Li, Lichuan
2014-07-01
This paper presents the application of fractional-order system identification (FOSI) and proportional-derivative (PD(µ)) control to a solid-core magnetic bearing (MB). A practical strategy for closed-loop incommensurate FOSI along with a modified error criterion is utilized to model the MB system and a corresponding, verification experiment is carried out. Based on the identified model, integer-order (IO) PD and fractional-order (FO) PD(µ) controllers are designed and compared with the same specifications. Besides, the relation between the two categories of controllers is discussed by their feasible control zones. Final simulation and experimental results show that the FO PD(µ) controller can significantly improve the transient and steady-state performance of the MB system comparing with the IO PD controller. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Al-Sawalha M. Mossa
2016-01-01
Full Text Available The objective of this article is to implement and extend applications of adaptive control to anti-synchronize different fractional order chaotic and hyperchaotic dynamical systems. The sufficient conditions for achieving anti–synchronization are derived by using the Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. Theoretical analysis and numerical simulations are shown to verify the results.
Al-Sawalha, M. Mossa; Al-Sawalha, Ayman
2016-01-01
The objective of this article is to implement and extend applications of adaptive control to anti-synchronize different fractional order chaotic and hyperchaotic dynamical systems. The sufficient conditions for achieving anti-synchronization are derived by using the Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. Theoretical analysis and numerical simulations are shown to verify the results.
Peng, Xiao; Wu, Huaiqin; Song, Ka; Shi, Jiaxin
2017-10-01
This paper is concerned with the global Mittag-Leffler synchronization and the synchronization in finite time for fractional-order neural networks (FNNs) with discontinuous activations and time delays. Firstly, the properties with respect to Mittag-Leffler convergence and convergence in finite time, which play a critical role in the investigation of the global synchronization of FNNs, are developed, respectively. Secondly, the novel state-feedback controller, which includes time delays and discontinuous factors, is designed to realize the synchronization goal. By applying the fractional differential inclusion theory, inequality analysis technique and the proposed convergence properties, the sufficient conditions to achieve the global Mittag-Leffler synchronization and the synchronization in finite time are addressed in terms of linear matrix inequalities (LMIs). In addition, the upper bound of the setting time of the global synchronization in finite time is explicitly evaluated. Finally, two examples are given to demonstrate the validity of the proposed design method and theoretical results. Copyright © 2017 Elsevier Ltd. All rights reserved.
Balochian, Saeed; Sedigh, Ali Khaki
2012-01-01
This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1control with a sliding sector. First, a sufficient condition for the stability of an LTI-FO switched system with order 1control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Hongjuan Liu
2014-01-01
Full Text Available A new general and systematic coupling scheme is developed to achieve the modified projective synchronization (MPS of different fractional-order systems under parameter mismatch via the Open-Plus-Closed-Loop (OPCL control. Based on the stability theorem of linear fractional-order systems, some sufficient conditions for MPS are proposed. Two groups of numerical simulations on the incommensurate fraction-order system and commensurate fraction-order system are presented to justify the theoretical analysis. Due to the unpredictability of the scale factors and the use of fractional-order systems, the chaotic data from the MPS is selected to encrypt a plain image to obtain higher security. Simulation results show that our method is efficient with a large key space, high sensitivity to encryption keys, resistance to attack of differential attacks, and statistical analysis.
Chaos and Synchronization in Complex Fractional-Order Chua’s System
Lin, Xiaoran; Zhou, Shangbo; Li, Hua
In this paper, chaos in complex-order Chua’s system and its chaotic synchronization for secure communication are studied based on the fractional derivative investigated. Chaos in complex-order Chua’s system is illustrated by presenting its waveform graphs, states diagrams and bifurcation graphs. The dynamic behaviors in the complex-order Chua’s system and time-delayed Chua’s system are compared. In addition, the largest Lyapunov exponents of normal and time-delayed Chua’s systems are calculated and verified by numerical simulations. The experiment indicates that the chaos in such Chua’s system can achieve two channel communications.
Directory of Open Access Journals (Sweden)
Chaojun Wu
2015-01-01
Full Text Available An efficient approach of inverse optimal control and adaptive control is developed for global asymptotic stabilization of a novel fractional-order four-wing hyperchaotic system with uncertain parameter. Based on the inverse optimal control methodology and fractional-order stability theory, a control Lyapunov function (CLF is constructed and an adaptive state feedback controller is designed to achieve inverse optimal control of a novel fractional-order hyperchaotic system with four-wing attractor. Then, an electronic oscillation circuit is designed to implement the dynamical behaviors of the fractional-order four-wing hyperchaotic system and verify the satisfactory performance of the controller. Comparing with other fractional-order chaos control methods which may have more than one nonlinear state feedback controller, the inverse optimal controller has the advantages of simple structure, high reliability, and less control effort that is required and can be implemented by electronic oscillation circuit.
Synchronization of Fractional Chaotic Systems via Fractional-Order Adaptive Controller
Hosseinnia, S. H.; Ghaderi, R.; N., A. Ranjbar; Sadati, J.; Momani, S.
2012-01-01
In this paper, an adaptive fractional controller has been designed to control chaotic systems. In fact, this controller is a fractional PID controller, which the coefficients will be tuned according to a proper adaptation mechanism. The adaptation law will be constructed from a sliding surface via gradient method. The adaptive fractional controller is implemented on a gyro system to signify the performance of the proposed technique.
ℋ∞ Adaptive observer for nonlinear fractional-order systems
Ndoye, Ibrahima
2016-06-23
In this paper, an adaptive observer is proposed for the joint estimation of states and parameters of a fractional nonlinear system with external perturbations. The convergence of the proposed observer is derived in terms of linear matrix inequalities (LMIs) by using an indirect Lyapunov method.The proposed ℋ∞ adaptive observer is also robust against Lipschitz additive nonlinear uncertainty. The performance of the observer is illustrated through some examples including the chaotic Lorenz and Lü\\'s systems. © 2016 John Wiley & Sons, Ltd.
Fractional Order Element Based Impedance Matching
Radwan, Ahmed Gomaa
2014-06-24
Disclosed are various embodiments of methods and systems related to fractional order element based impedance matching. In one embodiment, a method includes aligning a traditional Smith chart (|.alpha.|=1) with a fractional order Smith chart (|.alpha.|.noteq.1). A load impedance is located on the traditional Smith chart and projected onto the fractional order Smith chart. A fractional order matching element is determined by transitioning along a matching circle of the fractional order Smith chart based at least in part upon characteristic line impedance. In another embodiment, a system includes a fractional order impedance matching application executed in a computing device. The fractional order impedance matching application includes logic that obtains a first set of Smith chart coordinates at a first order, determines a second set of Smith chart coordinates at a second order, and determines a fractional order matching element from the second set of Smith chart coordinates.
Q-S synchronization of the fractional-order unified system
Indian Academy of Sciences (India)
2013-03-03
Mar 3, 2013 ... engineering application and information science, in particular in secure communication. [11–14]. Since Pecora and Carroll [15] introduced a method to realize synchroniza- tion between two identical chaotic systems with different initial values in 1990, many important and fundamental results have been ...
Fractional order differentiation by integration: An application to fractional linear systems
Liu, Dayan
2013-02-04
In this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with time-varying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative of a truncated Jacobi polynomial series expansion. We then approximate the FDE by applying to each fractional derivative this formal algebraic integral estimator. Consequently, the fractional derivatives of the solution are applied on the used Jacobi polynomials and then we need to identify the unknown coefficients of the truncated series expansion of the solution. Modulating functions method is used to estimate these coefficients by solving a linear system issued from the approximated FDE and some initial conditions. A numerical result is given to confirm the reliability of the proposed method. © 2013 IFAC.
Directory of Open Access Journals (Sweden)
Meysam Gheisarnezhad
2015-01-01
Full Text Available Fractional-order PID (FOPID controller is a generalization of standard PID controller using fractional calculus. Compared with the Standard PID controller, two adjustable variables “differential order” and “integral order” are added to the PID controller.Three tank system is a nonlinear multivariable process that is a good prototype of chemical industrial processes. Cuckoo Optimization Algorithm (COA, that was recently introduced has shown its good performance in optimization problems. In this study, Improved Cuckoo Optimization Algorithm (ICOA has been presented. The aim of the paper is to compare different controllers tuned with a Improved Cuckoo Optimization Algorithm (ICOA for Three Tank System. In order to compare the performance of the optimized FOPID controller with other controllers, Genetic Algorithm(GA, Particle swarm optimization (PSO, Cuckoo Optimization Algorithm (COA and Imperialist Competitive Algorithm (ICA.
Directory of Open Access Journals (Sweden)
Qiang Gao
2015-12-01
Full Text Available Aiming at balancing and positioning of a new electro-hydraulic servo system with iso-actuation configuration, an extended state observer–based fractional order proportional–integral–derivative controller is proposed in this study. To meet the lightweight requirements of heavy barrel weapons with large diameters, an electro-hydraulic servo system with a three-chamber hydraulic cylinder is especially designed. In the electro-hydraulic servo system, the balance chamber of the hydraulic cylinder is used to realize active balancing of the unbalanced forces, while the driving chambers consisting of the upper and lower chambers are adopted for barrel positioning and dynamic compensation of external disturbances. Compared with conventional proportional–integral–derivative controllers, the fractional order proportional–integral–derivative possesses another two adjustable parameters by expanding integer order to arbitrary order calculus, resulting in more flexibility and stronger robustness of the control system. To better compensate for strong external disturbances and system nonlinearities, the extended state observer strategy is further introduced to the fractional order proportional–integral–derivative control system. Numerical simulation and bench test indicate that the extended state observer–based fractional order proportional–integral–derivative significantly outperforms proportional–integral–derivative and fractional order proportional–integral–derivative control systems with better control accuracy and higher system robustness, well demonstrating the feasibility and effectiveness of the proposed extended state observer–based fractional order proportional–integral–derivative control strategy.
Directory of Open Access Journals (Sweden)
Lakhdar Chaib
2017-06-01
Full Text Available This paper proposes a novel robust power system stabilizer (PSS, based on hybridization of fractional order PID controller (PIλDμ and PSS for optimal stabilizer (FOPID-PSS for the first time, using a new metaheuristic optimization Bat algorithm (BA inspired by the echolocation behavior to improve power system stability. The problem of FOPID-PSS design is transformed as an optimization problem based on performance indices (PI, including Integral Absolute Error (IAE, Integral Squared Error (ISE, Integral of the Time-Weighted Absolute Error (ITAE and Integral of Time multiplied by the Squared Error (ITSE, where, BA is employed to obtain the optimal stabilizer parameters. In order to examine the robustness of FOPID-PSS, it has been tested on a Single Machine Infinite Bus (SMIB power system under different disturbances and operating conditions. The performance of the system with FOPID-PSS controller is compared with a PID-PSS and PSS. Further, the simulation results obtained with the proposed BA based FOPID-PSS are compared with those obtained with FireFly algorithm (FFA based FOPID-PSS. Simulation results show the effectiveness of BA for FOPID-PSS design, and superior robust performance for enhancement power system stability compared to other with different cases.
Song, Xiao-Na; Song, Shuai; Tejado Balsera, Inés; Liu, Lei-Po
2017-10-01
This paper investigates the mixed H ∞ and passive projective synchronization problem for fractional-order (FO) memristor-based neural networks. Our aim is to design a controller such that, though the unavoidable phenomena of time-delay and parameter uncertainty are fully considered, the resulting closed-loop system is asymptotically stable with a mixed H ∞ and passive performance level. By combining active and adaptive control methods, a novel hybrid control strategy is designed, which can guarantee the robust stability of the closed-loop system and also ensure a mixed H ∞ and passive performance level. Via the application of FO Lyapunov stability theory, the projective synchronization conditions are addressed in terms of linear matrix inequality techniques. Finally, two simulation examples are given to illustrate the effectiveness of the proposed method. Supported by National Natural Science Foundation of China under Grant Nos. U1604146, U1404610, 61473115, 61203047, Science and Technology Research Project in Henan Province under Grant Nos. 152102210273, 162102410024, and Foundation for the University Technological Innovative Talents of Henan Province under Grant No. 18HASTIT019
Pashaei, Shabnam; Badamchizadeh, Mohammadali
2016-07-01
This paper investigates the stabilization and disturbance rejection for a class of fractional-order nonlinear dynamical systems with mismatched disturbances. To fulfill this purpose a new fractional-order sliding mode control (FOSMC) based on a nonlinear disturbance observer is proposed. In order to design the suitable fractional-order sliding mode controller, a proper switching surface is introduced. Afterward, by using the sliding mode theory and Lyapunov stability theory, a robust fractional-order control law via a nonlinear disturbance observer is proposed to assure the existence of the sliding motion in finite time. The proposed fractional-order sliding mode controller exposes better control performance, ensures fast and robust stability of the closed-loop system, eliminates the disturbances and diminishes the chattering problem. Finally, the effectiveness of the proposed fractional-order controller is depicted via numerical simulation results of practical example and is compared with some other controllers. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
K. R. Prasad
2014-01-01
Full Text Available This paper establishes the existence of at least three positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, D0+β1(ϕp(D0+α1u(t=f1(t,u(t,v(t, t∈(0,1, D0+β2(ϕp(D0+α2v(t=f2(t,u(t,v(t, t∈(0,1, u(0=D0+q1u(0=0, γu(1+δD0+q2u(1=0, D0+α1u(0=D0+α1u(1=0, v(0=D0+q1v(0=0, γv(1+δD0+q2v(1=0, D0+α2v(0=D0+α2v(1=0, by applying five functionals fixed point theorem.
Directory of Open Access Journals (Sweden)
Adel Ouannas
2015-01-01
Full Text Available A robust control approach is presented to study the problem of Q-S synchronization between Integer-order and fractional-order chaotic systems with different dimensions. Based on Laplace transformation and stability theory of linear integer-order dynamical systems, a new control law is proposed to guarantee the Q-S synchronization between n-dimensional integer-order master system and m-dimensional fractional-order slave system. This paper provides further contribution to the topic of Q-S chaos synchronization between integer-order and fractional-order systems and introduces a general control scheme that can be applied to wide classes of chaotic and hyperchaotic systems. Illustrative example and numerical simulations are used to show the effectiveness of the proposed method.
Kingni, S. T.; Jafari, S.; Simo, H.; Woafo, P.
2014-05-01
This paper proposes a three-dimensional chaotic autonomous system with only one stable equilibrium. This system belongs to a newly introduced category of chaotic systems with hidden attractors. The nonlinear dynamics of the proposed chaotic system is described through numerical simulations which include phase portraits, bifurcation diagrams and new cost function for parameter estimation of chaotic flows. The coexistence of a stable equilibrium point with a strange attractor is found in the proposed system for specific parameters values. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcard-PSpice software. A good qualitative agreement is shown between the simulations and the experimental results. Based on the Routh-Hurwitz conditions and for a specific choice of linear controllers, it is shown that the proposed chaotic system is controlled to its equilibrium point. Chaos synchronization of an identical proposed system is achieved by using the unidirectional linear and nonlinear error feedback coupling. Finally, the fractional-order form of the proposed system is studied by using the stability theory of fractional-order systems and numerical simulations. A necessary condition for the commensurate fractional order of this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three.
Wang, Fei; Yang, Yongqing
2017-09-01
In this paper, we study the leader-following exponential consensus of multi-agent system. Each agent in the system is described by nonlinear fractional order differential equation. Both the internal delay and coupling delay are taken into consideration. The heterogeneous impulsive control is used for ensuring the consensus of all agents. Based on Lyapunov function method and matrix analysis, some sufficient conditions for exponential consensus are obtained. Finally, some illustrative examples are given to show the effectiveness of the obtained results.
Zhang, Jun-Feng; Pei, Qiu-Yu; Zhang, Xiao-Li
2011-08-01
In this paper, we have found a kind of interesting nonlinear phenomenon—hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are antiphase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams-Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.
Fractional Order Models of Industrial Pneumatic Controllers
Directory of Open Access Journals (Sweden)
Abolhassan Razminia
2014-01-01
Full Text Available This paper addresses a new approach for modeling of versatile controllers in industrial automation and process control systems such as pneumatic controllers. Some fractional order dynamical models are developed for pressure and pneumatic systems with bellows-nozzle-flapper configuration. In the light of fractional calculus, a fractional order derivative-derivative (FrDD controller and integral-derivative (FrID are remodeled. Numerical simulations illustrate the application of the obtained theoretical results in simple examples.
Muthukumar, P; Balasubramaniam, P; Ratnavelu, K
2017-07-26
This paper proposes a generalized robust synchronization method for different dimensional fractional order dynamical systems with mismatched fractional derivatives in the presence of function uncertainty and external disturbance by a designing sliding mode controller. Based on the proposed theory of generalized robust synchronization criterion, a novel audio cryptosystem is proposed for sending or sharing voice messages secretly via insecure channel. Numerical examples are given to verify the potency of the proposed theories. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Ferroelectric Fractional-Order Capacitors
Agambayev, Agamyrat
2017-07-25
Poly(vinylidene fluoride)-based polymers and their blends are used to fabricate electrostatic fractional-order capacitors. This simple but effective method allows us to precisely tune the constant phase angle of the resulting fractional-order capacitor by changing the blend composition. Additionally, we have derived an empirical relation between the ratio of the blend constituents and the constant phase angle to facilitate the design of a fractional order capacitor with a desired constant phase angle. The structural composition of the fabricated blends is investigated using Fourier transform infrared spectroscopy and X-ray diffraction techniques.
Hyperchaotic Chameleon: Fractional Order FPGA Implementation
Directory of Open Access Journals (Sweden)
Karthikeyan Rajagopal
2017-01-01
Full Text Available There are many recent investigations on chaotic hidden attractors although hyperchaotic hidden attractor systems and their relationships have been less investigated. In this paper, we introduce a hyperchaotic system which can change between hidden attractor and self-excited attractor depending on the values of parameters. Dynamic properties of these systems are investigated. Fractional order models of these systems are derived and their bifurcation with fractional orders is discussed. Field programmable gate array (FPGA implementations of the systems with their power and resource utilization are presented.
The fractional - order controllers: Methods for their synthesis and application
Petras, I.
2000-01-01
This paper deals with fractional-order controllers. We outline mathematical description of fractional controllers and methods of their synthesis and application. Synthesis method is a modified root locus method for fractional-order systems and fractional-order controllers. In the next section we describe how to apply the fractional controller on control systems.
Directory of Open Access Journals (Sweden)
S. Alonso-Quesada
2010-01-01
Full Text Available This paper presents a strategy for designing a robust discrete-time adaptive controller for stabilizing linear time-invariant (LTI continuous-time dynamic systems. Such systems may be unstable and noninversely stable in the worst case. A reduced-order model is considered to design the adaptive controller. The control design is based on the discretization of the system with the use of a multirate sampling device with fast-sampled control signal. A suitable on-line adaptation of the multirate gains guarantees the stability of the inverse of the discretized estimated model, which is used to parameterize the adaptive controller. A dead zone is included in the parameters estimation algorithm for robustness purposes under the presence of unmodeled dynamics in the controlled dynamic system. The adaptive controller guarantees the boundedness of the system measured signal for all time. Some examples illustrate the efficacy of this control strategy.
Ndoye, Ibrahima
2014-12-01
In this paper, an adaptive observer design with parameter identification for a nonlinear system with external perturbations and unknown parameters is proposed. The states of the nonlinear system are estimated by a nonlinear observer and the unknown parameters are also adapted to their values. Sufficient conditions for the stability of the adaptive observer error dynamics are derived in terms of linear matrix inequalities. Simulation results for chaotic Lorenz systems with unknown parameters in the presence of external perturbations are given to illustrate the effectiveness of our proposed approach. © 2014 IEEE.
Directory of Open Access Journals (Sweden)
S. J. Sadati
2012-01-01
Full Text Available A fractional-order controller will be proposed to regulate the inlet oxygen into the heart-lung machine. An analytical approach will be explained to satisfy some requirements together with practical implementation of some restrictions for the first time. Primarily a nonlinear single-input single-output (SISO time-delay model which was obtained previously in the literature is introduced for the oxygen generation process in the heart-lung machine system and we will complete it by adding some new states to control it. Thereafter, the system is linearized using the state feedback linearization approach to find a third-order time-delay dynamics. Consequently classical PID and fractional order controllers are gained to assess the quality of the proposed technique. A set of optimal parameters of those controllers are achieved through the genetic algorithm optimization procedure through minimizing a cost function. Our design method focuses on minimizing some famous performance criterions such as IAE, ISE, and ITSE. In the genetic algorithm, the controller parameters are chosen as a random population. The best relevant values are achieved by reducing the cost function. A time-domain simulation signifies the performance of controller with respect to a traditional optimized PID controller.
Directory of Open Access Journals (Sweden)
Khalid Hilal
2016-01-01
Full Text Available This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order 1<α≤2 is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.
Directory of Open Access Journals (Sweden)
H. Khalil
2015-06-01
Full Text Available We study shifted Legendre polynomials and develop some operational matrices of integrations. We use these operational matrices and develop new sophisticated technique for numerical solutions to the following coupled system of fredholm integro differential equations DU(x = f(x + 11 Z 1 0 K11(x, tU(tdt + 12 Z 1 0 K12(x, tV (tdt, DV (x = g(x + 21 Z 1 0 K21(x, tU(tdt + 22 Z 1 0 K22(x, tV (tdt, U(0 = C1, V (0 = C2, where D is fractional derivative of order with respect to x, 0 < 6 1, 11, 12, 21, 22 are real constants, f, g 2 C([0, 1] and K11, K12, K21, K22 2 C([0, 1]×[0, 1]. We develop simple procedure to reduce the coupled system of equations to a system of algebraic equations without discretizing the system. We provide examples and numerical simulations to show the applicability and simplicity of our results and to demonstrate that the results obtained using the new technique matches well with the exact solutions of the problem. We also provide error analysis.
Directory of Open Access Journals (Sweden)
Javad Morsali
2017-02-01
Full Text Available In this paper, fractional order proportional-integral-differential (FOPID controller is employed in the design of thyristor controlled series capacitor (TCSC-based damping controller in coordination with the secondary integral controller as automatic generation control (AGC loop. In doing so, the contribution of the TCSC in tie-line power exchange is extracted mathematically for small load disturbance. Adjustable parameters of the proposed FOPID-based TCSC damping controller and the AGC loop are optimized concurrently via an improved particle swarm optimization (IPSO algorithm which is reinforced by chaotic parameter and crossover operator to obtain a globally optimal solution. The powerful FOMCON toolbox is used along with MATLAB for handling fractional order modeling and control. An interconnected multi-source power system is simulated regarding the physical constraints of generation rate constraint (GRC nonlinearity and governor dead band (GDB effect. Simulation results using FOMCON toolbox demonstrate that the proposed FOPID-based TCSC damping controller achieves the greatest dynamic performance under different load perturbation patterns in comparison with phase lead-lag and classical PID-based TCSC damping controllers, all in coordination with the integral AGC. Moreover, sensitivity analyses are performed to show the robustness of the proposed controller under various uncertainty scenarios.
Belkhatir, Zehor
2017-05-31
This paper proposes a two-stage estimation algorithm to solve the problem of joint estimation of the parameters and the fractional differentiation orders of a linear continuous-time fractional system with non-commensurate orders. The proposed algorithm combines the modulating functions and the first-order Newton methods. Sufficient conditions ensuring the convergence of the method are provided. An error analysis in the discrete case is performed. Moreover, the method is extended to the joint estimation of smooth unknown input and fractional differentiation orders. The performance of the proposed approach is illustrated with different numerical examples. Furthermore, a potential application of the algorithm is proposed which consists in the estimation of the differentiation orders of a fractional neurovascular model along with the neural activity considered as input for this model.
Directory of Open Access Journals (Sweden)
Sadalla Talar
2017-12-01
Full Text Available The paper aims at presenting the influence of an open-loop time delay on the stability and tracking performance of a second-order open-loop system and continuoustime fractional-order PI controller. The tuning method of this controller is based on Hermite- Biehler and Pontryagin theorems, and the tracking performance is evaluated on the basis of two integral performance indices, namely IAE and ISE. The paper extends the results and methodology presented in previous work of the authors to analysis of the influence of time delay on the closed-loop system taking its destabilizing properties into account, as well as concerning possible application of the presented results and used models.
National Research Council Canada - National Science Library
Chen, Guan-Chun; Lin, Chia-Hung; Li, Chien-Ming; Hsieh, Kai-Sheng; Du, Yi-Chun; Chen, Tainsong
2015-01-01
...) using fractional-order vascular access tracker and haptic force producer. An endoscope or a catheter for diagnosis and surgery of cardiovascular disease has been commonly used in minimally invasive surgery...
Directory of Open Access Journals (Sweden)
Muhammad Iqbal
2017-01-01
Full Text Available We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: D cαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],D cαyt-ft,yt=gt,xt,Iαxt, t∈J=0,1, D cpx0=ψxη1, x′0=0,…,xn-20=0, D cpx1=ψxη2, D cpy0=ψyη1, y′0=0,…,yn-20=0, D cpy1=ψyη2, where p,η1,η2∈0,1, ψ is linear, D cα is Caputo fractional derivative of order α, with n-1<α≤n, n∈N, and Iα is fractional integral of order α. The nonlinear functions f, g are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.
Dynamical models of happiness with fractional order
Song, Lei; Xu, Shiyun; Yang, Jianying
2010-03-01
This present study focuses on a dynamical model of happiness described through fractional-order differential equations. By categorizing people of different personality and different impact factor of memory (IFM) with different set of model parameters, it is demonstrated via numerical simulations that such fractional-order models could exhibit various behaviors with and without external circumstance. Moreover, control and synchronization problems of this model are discussed, which correspond to the control of emotion as well as emotion synchronization in real life. This study is an endeavor to combine the psychological knowledge with control problems and system theories, and some implications for psychotherapy as well as hints of a personal approach to life are both proposed.
Chaos and Adaptive Control of the Fractional-Order Magnetic-Field Electromechanical Transducer
Luo, Shaohua; Li, Shaobo; Tajaddodianfar, Farid
2017-12-01
In this paper, we investigate chaos and adaptive control of the magnetic-field electromechanical transducer wherein the electric characteristics of the capacitor contain the fractional-order derivative. The phase diagrams for different values of the fractional-order exhibit chaotic characteristics of the magnetic-field electromechanical transducer. In the process of controller design, a continuous frequency distributed model is utilized to construct the indirect Lyapunov stability criterion and a Chebyshev neural network with weight, and fractional-order adaptive law is introduced to approximate the complicated unknown function. To suppress chaotic oscillation, an adaptive control scheme by fusing Chebyshev neural network and backstepping is presented to guarantee that the closed-loop system is globally asymptotically stable. To illustrate the feasibility of the proposed approach, simulation studies are done in the end.
Adaptive control and synchronization of a fractional-order chaotic ...
Indian Academy of Sciences (India)
order chaotic system based on the stability theory of fractional-order dynamic ... of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China; School of Information and Communication Engineering, Hunan ...
Fractional Order Nonlinear Feedback Controller Design for PMSM Drives
Directory of Open Access Journals (Sweden)
Jian-Ping Wen
2013-01-01
Full Text Available Fractional order integral is introduced into active disturbance rejection controller (ADRC to establish the structure of fractional order proportional integral controller (FPI. Fractional order ADRC (FADRC is designed by replacing the nonlinear state error feedback control law using nonlinear function combination in ADRC with FPI, which can combine the high performance of ADRC estimating disturbances with the characteristics of fractional order calculus more really describing the physical object and spreading the stable region of the system parameters. The proposed FADRC is applied to permanent magnet synchronous motor (PMSM speed servo system in order to improve robustness of system against the disturbances. Compared with ADRC, simulation results verify that the proposed control method has given very good robust results and fast speed tracking performance.
Fractional-order in a macroeconomic dynamic model
David, S. A.; Quintino, D. D.; Soliani, J.
2013-10-01
In this paper, we applied the Riemann-Liouville approach in order to realize the numerical simulations to a set of equations that represent a fractional-order macroeconomic dynamic model. It is a generalization of a dynamic model recently reported in the literature. The aforementioned equations have been simulated for several cases involving integer and non-integer order analysis, with some different values to fractional order. The time histories and the phase diagrams have been plotted to visualize the effect of fractional order approach. The new contribution of this work arises from the fact that the macroeconomic dynamic model proposed here involves the public sector deficit equation, which renders the model more realistic and complete when compared with the ones encountered in the literature. The results reveal that the fractional-order macroeconomic model can exhibit a real reasonable behavior to macroeconomics systems and might offer greater insights towards the understanding of these complex dynamic systems.
Fractional-Order Control of Pneumatic Position Servosystems
Directory of Open Access Journals (Sweden)
Cao Junyi
2011-01-01
Full Text Available A fractional-order control strategy for pneumatic position servosystem is presented in this paper. The idea of the fractional calculus application to control theory was introduced in many works, and its advantages were proved. However, the realization of fractional-order controllers for pneumatic position servosystems has not been investigated. Based on the relationship between the pressure in cylinder and the rate of mass flow into the cylinder, the dynamic model of pneumatic position servo system is established. The fractional-order controller for pneumatic position servo and its implementation in industrial computer is designed. The experiments with fractional-order controller are carried out under various conditions, which include sine position signal with different frequency and amplitude, step position signal, and variety inertial load. The results show the effectiveness of the proposed scheme and verify their fine control performance for pneumatic position servo system.
Classical controller design techniques for fractional order case.
Yeroglu, Celaleddin; Tan, Nusret
2011-07-01
This paper presents some classical controller design techniques for the fractional order case. New robust lag, lag-lead, PI controller design methods for control systems with a fractional order interval transfer function (FOITF) are proposed using classical design methods with the Bode envelopes of the FOITF. These controllers satisfy the robust performance specifications of the fractional order interval plant. In order to design a classical PID controller, an optimization technique based on fractional order reference model is used. PID controller parameters are obtained using the least squares optimization method. Different PID controller parameters that satisfy stability have been obtained for the same plant. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
Asheghan, Mohammad Mostafa; Beheshti, Mohammad Taghi Hamidi; Tavazoei, Mohammad Saleh
2011-06-01
In this note some points to paper [L. Pan, W. Zhou, J. Fang, D. Li, Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control, Commun Nonlinear Sci Numer Simulat 2010;15:3754-3762] are presented. Hereby, we illustrate that the way that authors in [1] treat with fractional version of Lyapunov stability theorem suffers lack of a correct justification.
Fractional Order Signal Processing Introductory Concepts and Applications
Das, Saptarshi
2012-01-01
The book tries to briefly introduce the diverse literatures in the field of fractional order signal processing which is becoming an emerging topic among an interdisciplinary community of researchers. This book is aimed at postgraduate and beginning level research scholars who would like to work in the field of Fractional Order Signal processing (FOSP). The readers should have preliminary knowledge about basic signal processing techniques. Prerequisite knowledge of fractional calculus is not essential and is exposited at relevant places in connection to the appropriate signal processing topics. Basic signal processing techniques like filtering, estimation, system identification, etc. in the light of fractional order calculus are presented along with relevant application areas. The readers can easily extend these concepts to varied disciplines like image or speech processing, pattern recognition, time series forecasting, financial data analysis and modeling, traffic modeling in communication channels, optics, b...
Adaptive control and synchronization of a fractional-order chaotic ...
Indian Academy of Sciences (India)
journal of. April 2013 physics pp. 583–592. Adaptive control and synchronization of a fractional-order chaotic system. CHUNLAI LI1,∗ and YAONAN TONG2. 1College of Physics and Electronics; 2School of Information and Communication Engineering,. Hunan Institute of Science and Technology, Yueyang 414006, China.
FPGA implementation of fractional-order discrete memristor chaotic ...
Indian Academy of Sciences (India)
Home; Journals; Pramana – Journal of Physics; Volume 90; Issue 1. FPGA implementation of fractional-order discrete memristor chaotic system and its commensurate and incommensurate synchronisations. ANITHA KARTHIKEYAN KARTHIKEYAN RAJAGOPAL. Research Article Volume 90 Issue 1 January 2018 Article ID ...
Pitchfork bifurcation and vibrational resonance in a fractional-order ...
Indian Academy of Sciences (India)
Duffing oscillator with delayed feedback and excited by two harmonic signals. Using an approxi- ... phenomenon in the fractional-order Duffing oscillator with time delay feedback. The equation of motion of the system we ..... where T = 2π/ω and r is a positive integer which should be chosen big enough. For convenience, we ...
Martínez-Guerra, Rafael; Gómez-Cortés, Gian Carlo
2015-01-01
This book provides a general overview of several concepts of synchronization and brings together related approaches to secure communication in chaotic systems. This is achieved using a combination of analytic, algebraic, geometrical and asymptotical methods to tackle the dynamical feedback stabilization problem. In particular, differential-geometric and algebraic differential concepts reveal important structural properties of chaotic systems and serve as guide for the construction of design procedures for a wide variety of chaotic systems. The basic differential algebraic and geometric concepts are presented in the first few chapters in a novel way as design tools, together with selected experimental studies demonstrating their importance. The subsequent chapters treat recent applications. Written for graduate students in applied physical sciences, systems engineers, and applied mathematicians interested in synchronization of chaotic systems and in secure communications, this self-contained text requires only...
Fractional-order RC and RL circuits
Radwan, Ahmed Gomaa
2012-05-30
This paper is a step forward to generalize the fundamentals of the conventional RC and RL circuits in fractional-order sense. The effect of fractional orders is the key factor for extra freedom, more flexibility, and novelty. The conditions for RC and RL circuits to act as pure imaginary impedances are derived, which are unrealizable in the conventional case. In addition, the sensitivity analyses of the magnitude and phase response with respect to all parameters showing the locations of these critical values are discussed. A qualitative revision for the fractional RC and RL circuits in the frequency domain is provided. Numerical and PSpice simulations are included to validate this study. © Springer Science+Business Media, LLC 2012.
Modeling and analysis of fractional order DC-DC converter.
Radwan, Ahmed G; Emira, Ahmed A; AbdelAty, Amr M; Azar, Ahmed Taher
2017-07-11
Due to the non-idealities of commercial inductors, the demand for a better model that accurately describe their dynamic response is elevated. So, the fractional order models of Buck, Boost and Buck-Boost DC-DC converters are presented in this paper. The detailed analysis is made for the two most common modes of converter operation: Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM). Closed form time domain expressions are derived for inductor currents, voltage gain, average current, conduction time and power efficiency where the effect of the fractional order inductor is found to be strongly present. For example, the peak inductor current at steady state increases with decreasing the inductor order. Advanced Design Systems (ADS) circuit simulations are used to verify the derived formulas, where the fractional order inductor is simulated using Valsa Constant Phase Element (CPE) approximation and Generalized Impedance Converter (GIC). Different simulation results are introduced with good matching to the theoretical formulas for the three DC-DC converter topologies under different fractional orders. A comprehensive comparison with the recently published literature is presented to show the advantages and disadvantages of each approach. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Research on Modeling of Hydropneumatic Suspension Based on Fractional Order
Directory of Open Access Journals (Sweden)
Junwei Zhang
2015-01-01
Full Text Available With such excellent performance as nonlinear stiffness, adjustable vehicle height, and good vibration resistance, hydropneumatic suspension (HS has been more and more applied to heavy vehicle and engineering vehicle. Traditional modeling methods are still confined to simple models without taking many factors into consideration. A hydropneumatic suspension model based on fractional order (HSM-FO is built with the advantage of fractional order (FO in viscoelastic material modeling considering the mechanics property of multiphase medium of HS. Then, the detailed calculation method is proposed based on Oustaloup filtering approximation algorithm. The HSM-FO is implemented in Matlab/Simulink, and the results of comparison among the simulation curve of fractional order, integral order, and the curve of real experiment prove the feasibility and validity of HSM-FO. The damping force property of the suspension system under different fractional orders is also studied. In the end of this paper, several conclusions concerning HSM-FO are drawn according to analysis of simulation.
Fractional Order AGC for Distributed Energy Resources Using Robust Optimization
Pan, Indranil; Das, Saptarshi
2016-01-01
The applicability of fractional order (FO) automatic generation control (AGC) for power system frequency oscillation damping is investigated in this paper, employing distributed energy generation. The hybrid power system employs various autonomous generation systems like wind turbine, solar photovoltaic, diesel engine, fuel-cell and aqua electrolyzer along with other energy storage devices like the battery and flywheel. The controller is placed in a remote location while receiving and sending...
Fractional Processes and Fractional-Order Signal Processing Techniques and Applications
Sheng, Hu; Qiu, TianShuang
2012-01-01
Fractional processes are widely found in science, technology and engineering systems. In Fractional Processes and Fractional-order Signal Processing, some complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations (local and long memory), are introduced and examined from the ‘fractional’ perspective using simulation, fractional-order modeling and filtering and realization of fractional-order systems. These fractional-order signal processing (FOSP) techniques are based on fractional calculus, the fractional Fourier transform and fractional lower-order moments. Fractional Processes and Fractional-order Signal Processing: • presents fractional processes of fixed, variable and distributed order studied as the output of fractional-order differential systems; • introduces FOSP techniques and the fractional signals and fractional systems point of view; • details real-world-application examples of FOSP techniques to demonstr...
A General Method for Designing Fractional Order PID Controller
Directory of Open Access Journals (Sweden)
Marzieh Safaei
2013-01-01
Full Text Available The idea of using fractional order calculus in control became apparent when this kind of calculus was accepted as a powerful tool in many applications. This resulted in a new generation of PID controller called fractional order PID Controller, named as Controller. controller is more flexible and provides a better response with larger stability region as compared with standard PID controller. This paper presents a simple and reliable method for finding the family of controllers. The required calculations are done in frequency domain based on frequency response of the system and the stability region is specified in the parameters space. This method can be used for time-delay systems and, more generally, for any system with no transfer function.
Numerical Analysis of Fractional Order Epidemic Model of Childhood Diseases
Directory of Open Access Journals (Sweden)
Fazal Haq
2017-01-01
Full Text Available The fractional order Susceptible-Infected-Recovered (SIR epidemic model of childhood disease is considered. Laplace–Adomian Decomposition Method is used to compute an approximate solution of the system of nonlinear fractional differential equations. We obtain the solutions of fractional differential equations in the form of infinite series. The series solution of the proposed model converges rapidly to its exact value. The obtained results are compared with the classical case.
Dynamic behaviours and control of fractional-order memristor-based ...
Indian Academy of Sciences (India)
Dynamics of fractional-order memristor circuit system and its control are investigated in this paper. With the help of stability theory of fractional-order systems, stability of its equilibrium points is analysed. Then, the chaotic behaviours are validated using phase portraits, the Lyapunov exponents and bifurcation diagrams with ...
Liu, Ling; Liang, Deliang; Liu, Chongxin; Zhang, Qun
2012-12-01
In this paper, a nonlinear state observer control strategy is developed for projective self-synchronization of the fractional-order chaotic attractors of a permanent magnet synchronous motor (PMSM) system. The mathematical model of PMSM system in a smooth fractional-order form is derived by using the fractional derivative theory. A state observer control design can achieve the full-state projective synchronization of the fractional-order PMSM (FO-PMSM) system without the limitation of partial-linearity. Global stability and asymptotic synchronization between the outputs of drive system and response system can be obtained. Simulation results are provided to demonstrate the effectiveness of the approach.
Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment
Liu, Dayan
2015-03-31
The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters\\' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann-Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.
Implementation of fractional order integrator/differentiator on field programmable gate array
Directory of Open Access Journals (Sweden)
K.P.S. Rana
2016-06-01
Full Text Available Concept of fractional order calculus is as old as the regular calculus. With the advent of high speed and cost effective computing power, now it is possible to model the real world control and signal processing problems using fractional order calculus. For the past two decades, applications of fractional order calculus, in system modeling, control and signal processing, have grown rapidly. This paper presents a systematic procedure for hardware implementation of the basic operators of fractional calculus i.e. fractional integrator and derivative, using Grünwald–Letnikov definition, on field programmable gate array (FPGA in LabVIEW environment. The simulation and hardware implementation results for fractional order integrator and derivative of sinusoid and square waveform signals for some selected fractional orders have been presented. A close agreement between the simulated and the experimental results demonstrated the suitability of FPGA device in fractional order control and signal processing applications. LabVIEW being one of the finest tools for measurement and control, and signal processing applications the fractional order operator implementation is expected to further enhance the capability of the tool to cater to the needs of advanced experimental research employing fractional order operators.
Fractional-order PI based STATCOM and UPFC controller to diminish subsynchronous resonance.
Koteswara Raju, D; Umre, Bhimrao S; Junghare, Anjali S; Thakre, Mohan P; Motamarri, Rambabu; Somu, Chaitanya
2016-01-01
This research article proposes a powerful fractional-order PI controller to mitigate the subsynchronous oscillations in turbine-generator shaft due to subsynchronous resonance (SSR) with flexible AC transmission system devices such as static synchronous compensator (STATCOM) and unified power flow controller (UPFC). The diminution of SSR is achieved by the raising of network damping at those frequencies which are proximate to the torsional mode frequency of the turbine-generator shaft. The increase of network damping is obtained with the injection of subsynchronous frequency component of current and both current and voltage into the line. The subsynchronous component of current and voltage are derived from the measured signal of the system and further the same amount of shunt current is injected with STATCOM and simultaneous injection of current and voltage with UPFC into the transmission line to make the subsynchronous current to zero which is the prime source of turbine shaft oscillations. The insertion and proper tuning of Fractional-order PI controller in the control scheme, the subsynchronous oscillations are reduced to 92 % in case of STATCOM and 98 % in case of UPFC as compared to without controller and 14 % as compared with the results of conventional PI controller. The IEEE first benchmark model has adopted for analyze the effectiveness and speed of the proposed control scheme using MATLAB-Simulink and the corresponding results illustrates the precision and robustness of the proposed controller.
Ultracapacitor Modelling and Control Using Discrete Fractional Order State-Space Model
Directory of Open Access Journals (Sweden)
Andrzej Dzielinski
2008-03-01
Full Text Available In this paper the modelling of ultracapacitor system using discrete fractional order state-space system is presented. The obtained model is used for design and testing of state feedback controller with observer.
Finite-time synchronization of fractional-order simplest two-component chaotic oscillators
Kengne, Romanic; Tchitnga, Robert; Mezatio, Anicet; Fomethe, Anaclet; Litak, Grzegorz
2017-05-01
The problem of finite-time synchronization of fractional-order simplest two-component chaotic oscillators operating at high frequency and application to digital cryptography is addressed. After the investigation of numerical chaotic behavior in the system, an adaptive feedback controller is designed to achieve the finite-time synchronization of two oscillators, based on the Lyapunov function. This controller could find application in many other fractional-order chaotic circuits. Applying synchronized fractional-order systems in digital cryptography, a well secured key system is obtained. Numerical simulations are given to illustrate and verify the analytic results.
SOLVING FRACTIONAL-ORDER COMPETITIVE LOTKA-VOLTERRA MODEL BY NSFD SCHEMES
Directory of Open Access Journals (Sweden)
S.ZIBAEI
2016-12-01
Full Text Available In this paper, we introduce fractional-order into a model competitive Lotka- Volterra prey-predator system. We will discuss the stability analysis of this fractional system. The non-standard nite difference (NSFD scheme is implemented to study the dynamic behaviors in the fractional-order Lotka-Volterra system. Proposed non-standard numerical scheme is compared with the forward Euler and fourth order Runge-Kutta methods. Numerical results show that the NSFD approach is easy and accurate for implementing when applied to fractional-order Lotka-Volterra model.
Study on the Business Cycle Model with Fractional-Order Time Delay under Random Excitation
Directory of Open Access Journals (Sweden)
Zifei Lin
2017-07-01
Full Text Available Time delay of economic policy and memory property in a real economy system is omnipresent and inevitable. In this paper, a business cycle model with fractional-order time delay which describes the delay and memory property of economic control is investigated. Stochastic averaging method is applied to obtain the approximate analytical solution. Numerical simulations are done to verify the method. The effects of the fractional order, time delay, economic control and random excitation on the amplitude of the economy system are investigated. The results show that time delay, fractional order and intensity of random excitation can all magnify the amplitude and increase the volatility of the economy system.
Xue, Dingyü; Li, Tingxue
2017-04-27
The parameter optimization method for multivariable systems is extended to the controller design problems for multiple input multiple output (MIMO) square fractional-order plants. The algorithm can be applied to search for the optimal parameters of integer-order controllers for fractional-order plants with or without time delays. Two examples are given to present the controller design procedures for MIMO fractional-order systems. Simulation studies show that the integer-order controllers designed are robust to plant gain variations. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Fractional-order permanent magnet synchronous motor and its adaptive chaotic control
Li, Chun-Lai; Yu, Si-Min; Luo, Xiao-Shu
2012-10-01
In this paper we investigate the chaotic behaviors of the fractional-order permanent magnet synchronous motor (PMSM). The necessary condition for the existence of chaos in the fractional-order PMSM is deduced. And an adaptive-feedback controller is developed based on the stability theory for fractional systems. The presented control scheme, which contains only one single state variable, is simple and flexible, and it is suitable both for design and for implementation in practice. Simulation is carried out to verify that the obtained scheme is efficient and robust against external interference for controlling the fractional-order PMSM system.
Electronically Tunable Fully Integrated Fractional-Order Resonator
Tsirimokou, Georgia
2017-03-20
A fully integrated implementation of a parallel fractional-order resonator which employs together a fractional order capacitor and a fractional-order inductor is proposed in this paper. The design utilizes current-controlled Operational Transconductance Amplifiers as building blocks, designed and fabricated in AMS 0:35m CMOS process, and based on a second-order approximation of a fractional-order differentiator/ integrator magnitude optimized in the range 10Hz–700Hz. An attractive benefit of the proposed scheme is its electronic tuning capability.
Position control of an industrial robot using fractional order controller
Clitan, Iulia; Muresan, Vlad; Abrudean, Mihail; Clitan, Andrei; Miron, Radu
2017-02-01
This paper presents the design of a control structure that ensures no overshoot for the movement of an industrial robot, used for the evacuation of round steel blocks from inside a rotary hearth furnace. First, a mathematical model for the positioning system is derived from a set of experimental data, and further, the paper focuses on obtaining a PID type controller, using the relay method as tuning method in order to obtain a stable closed loop system. The controller parameters are further tuned in order to achieve the imposed set of performances for the positioning of the industrial robot through computer simulation, using trial and error method. Further, a fractional - order PID controller is obtained in order to improve the control signal variation, so as to fit within the range of unified current's variation, 4 to 20 mA.
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan
2012-12-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Butterworth passive filter in the fractional-order
Sołtan, Ahmed
2011-12-01
In this paper, the generalized analysis of the first Butterworth filter based on two passive elements is introduced in the fractional-order sense. The fractional-order condition of the Butterworth circuit is presented for the first time where it will lead us back to the known condition of the integer-order circuit when the two fractional-orders equal one. Therefore, the conventional behavior of the integer-order circuit is a narrow subset of the fractional-order ones. The circuit is studied under same and different order cases, and verified through their numerical simulations. Stability analysis is also introduced showing the poles location in the fractional-order versus integer order cases. © 2011 IEEE.
Characteristic Analysis of Fractional-Order 4D Hyperchaotic Memristive Circuit
Directory of Open Access Journals (Sweden)
Jun Mou
2017-01-01
Full Text Available Dynamical behaviors of the 4D hyperchaotic memristive circuit are analyzed with the system parameter. Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical solution of fractional-order 4D hyperchaotic memristive circuit is investigated. The distribution of stable and unstable regions of the fractional-order 4D hyperchaotic memristive circuit is determined, and dynamical characteristics are studied by phase portraits, Lyapunov exponents spectrum, and bifurcation diagram. Complexities are calculated by employing the spectral entropy (SE algorithm and C0 algorithm. Complexity results are consistent with that of the bifurcation diagrams, and this means that complexity can also reflect the dynamic characteristics of a chaotic system. Results of this paper provide a theoretical and experimental basis for the application of fractional-order 4D hyperchaotic memristive circuit in the field of encryption and secure communication.
Novel Fractional Order Calculus Extended PN for Maneuvering Targets
Directory of Open Access Journals (Sweden)
Jikun Ye
2017-01-01
Full Text Available Based on the theory of fractional order calculus (FOC, a novel extended proportional guidance (EPN law for intercepting the maneuvering target is proposed. In the first part, considering the memory function and filter characteristic of FOC, the novel extended PN guidance algorithm is developed based on the conventional PN after introducing the properties and operation rules of FOC. Further, with the help of FOC theory, the average load and ballistics characteristics of proposed guidance law are analyzed. Then, using the small offset kinematic model, the robustness of the new guidance law against autopilot parameters is studied theoretically by analyzing the sensitivity of the closed loop guidance system. At last, representative numerical results show that the designed guidance law obtains a better performance than the traditional PN for maneuvering target.
Dynamical analysis of fractional order model of immunogenic tumors
Directory of Open Access Journals (Sweden)
Sadia Arshad
2016-07-01
Full Text Available In this article, we examine the fractional order model of the cytotoxic T lymphocyte response to a growing tumor cell population. We investigate the long-term behavior of tumor growth and explore the conditions of tumor elimination analytically. We establish the conditions for the tumor-free equilibrium and tumor-infection equilibrium to be asymptotically stable and provide the expression of the basic reproduction number. Existence of physical significant tumor-infection equilibrium points is investigated analytically. We show that tumor growth rate, source rate of immune cells, and death rate of immune cells play vital role in tumor dynamics and system undergoes saddle-node and transcritical bifurcation based on these parameters. Furthermore, the effect of cancer treatment is discussed by varying the values of relevant parameters. Numerical simulations are presented to illustrate the analytical results.
Generalized Lacunary Statistical Difference Sequence Spaces of Fractional Order
Directory of Open Access Journals (Sweden)
Ugur Kadak
2015-01-01
Full Text Available We generalize the lacunary statistical convergence by introducing the generalized difference operator Δνα of fractional order, where α is a proper fraction and ν=(νk is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator and investigate the topological structures of related sequence spaces. Furthermore, we introduce some properties of the strongly Cesaro difference sequence spaces of fractional order involving lacunary sequences and examine various inclusion relations of these spaces. We also determine the relationship between lacunary statistical and strong Cesaro difference sequence spaces of fractional order.
Atmospheric Turbulence Modeling for Aerospace Vehicles: Fractional Order Fit
Kopasakis, George (Inventor)
2015-01-01
An improved model for simulating atmospheric disturbances is disclosed. A scale Kolmogorov spectral may be scaled to convert the Kolmogorov spectral into a finite energy von Karman spectral and a fractional order pole-zero transfer function (TF) may be derived from the von Karman spectral. Fractional order atmospheric turbulence may be approximated with an integer order pole-zero TF fit, and the approximation may be stored in memory.
Rajagopal, Karthikeyan; Karthikeyan, Anitha; Duraisamy, Prakash
2016-12-01
In this paper we investigate the control of three-dimensional non-autonomous fractional-order model of a permanent magnet synchronous motor (PMSM) and PI controlled fractional order Induction motor via recursive extended back stepping control technique. A robust generalized weighted controllers are derived to suppress the chaotic oscillations of the fractional order model. As the direct Lyapunov stability analysis of the controller is difficult for a fractional order first derivative, we have derived a new lemma to analyze the stability of the system. Numerical simulations of the proposed chaos suppression methodology are given to prove the analytical results.
Comlekoglu, T.; Weinberg, S. H.
2017-09-01
Cardiac memory is the dependence of electrical activity on the prior history of one or more system state variables, including transmembrane potential (Vm), ionic current gating, and ion concentrations. While prior work has represented memory either phenomenologically or with biophysical detail, in this study, we consider an intermediate approach of a minimal three-variable cardiomyocyte model, modified with fractional-order dynamics, i.e., a differential equation of order between 0 and 1, to account for history-dependence. Memory is represented via both capacitive memory, due to fractional-order Vm dynamics, that arises due to non-ideal behavior of membrane capacitance; and ionic current gating memory, due to fractional-order gating variable dynamics, that arises due to gating history-dependence. We perform simulations for varying Vm and gating variable fractional-orders and pacing cycle length and measure action potential duration (APD) and incidence of alternans, loss of capture, and spontaneous activity. In the absence of ionic current gating memory, we find that capacitive memory, i.e., decreased Vm fractional-order, typically shortens APD, suppresses alternans, and decreases the minimum cycle length (MCL) for loss of capture. However, in the presence of ionic current gating memory, capacitive memory can prolong APD, promote alternans, and increase MCL. Further, we find that reduced Vm fractional order (typically less than 0.75) can drive phase 4 depolarizations that promote spontaneous activity. Collectively, our results demonstrate that memory reproduced by a fractional-order model can play a role in alternans formation and pacemaking, and in general, can greatly increase the range of electrophysiological characteristics exhibited by a minimal model.
Comlekoglu, T; Weinberg, S H
2017-09-01
Cardiac memory is the dependence of electrical activity on the prior history of one or more system state variables, including transmembrane potential (Vm), ionic current gating, and ion concentrations. While prior work has represented memory either phenomenologically or with biophysical detail, in this study, we consider an intermediate approach of a minimal three-variable cardiomyocyte model, modified with fractional-order dynamics, i.e., a differential equation of order between 0 and 1, to account for history-dependence. Memory is represented via both capacitive memory, due to fractional-order Vm dynamics, that arises due to non-ideal behavior of membrane capacitance; and ionic current gating memory, due to fractional-order gating variable dynamics, that arises due to gating history-dependence. We perform simulations for varying Vm and gating variable fractional-orders and pacing cycle length and measure action potential duration (APD) and incidence of alternans, loss of capture, and spontaneous activity. In the absence of ionic current gating memory, we find that capacitive memory, i.e., decreased Vm fractional-order, typically shortens APD, suppresses alternans, and decreases the minimum cycle length (MCL) for loss of capture. However, in the presence of ionic current gating memory, capacitive memory can prolong APD, promote alternans, and increase MCL. Further, we find that reduced Vm fractional order (typically less than 0.75) can drive phase 4 depolarizations that promote spontaneous activity. Collectively, our results demonstrate that memory reproduced by a fractional-order model can play a role in alternans formation and pacemaking, and in general, can greatly increase the range of electrophysiological characteristics exhibited by a minimal model.
Fractional order PID controller for improvement of PMSM speed control in aerospace applications
Saraji, Ali Motalebi; Ghanbari, Mahmood
2014-12-01
Because of the benefits reduced size, cost and maintenance, noise, CO2 emissions and increased control flexibility and precision, to meet these expectations, electrical equipment increasingly utilize in modern aircraft systems and aerospace industry rather than conventional mechanic, hydraulic, and pneumatic power systems. Electric motor drives are capable of converting electrical power to drive actuators, pumps, compressors, and other subsystems at variable speeds. In the past decades, permanent magnet synchronous motor (PMSM) and brushless dc (BLDC) motor were investigated for aerospace applications such as aircraft actuators. In this paper, the fractional-order PID controller is used in the design of speed loop of PMSM speed control system. Having more parameters for tuning fractional order PID controller lead to good performance ratio to integer order. This good performance is shown by comparison fractional order PID controller with the conventional PI and tuned PID controller by Genetic algorithm in MATLAB soft wear.
Fractional order PID controller for improvement of PMSM speed control in aerospace applications
Energy Technology Data Exchange (ETDEWEB)
Saraji, Ali Motalebi [Young Researchers and Elite Club, AliAbad Katoul Branch, Islamic Azad University, AliAbad Katoul (Iran, Islamic Republic of); Ghanbari, Mahmood [Department of Electrical Engineering, AliAbad Katoul Branch, Islamic Azad University, AliAbad Katoul (Iran, Islamic Republic of)
2014-12-10
Because of the benefits reduced size, cost and maintenance, noise, CO2 emissions and increased control flexibility and precision, to meet these expectations, electrical equipment increasingly utilize in modern aircraft systems and aerospace industry rather than conventional mechanic, hydraulic, and pneumatic power systems. Electric motor drives are capable of converting electrical power to drive actuators, pumps, compressors, and other subsystems at variable speeds. In the past decades, permanent magnet synchronous motor (PMSM) and brushless dc (BLDC) motor were investigated for aerospace applications such as aircraft actuators. In this paper, the fractional-order PID controller is used in the design of speed loop of PMSM speed control system. Having more parameters for tuning fractional order PID controller lead to good performance ratio to integer order. This good performance is shown by comparison fractional order PID controller with the conventional PI and tuned PID controller by Genetic algorithm in MATLAB soft wear.
Robust fractional order differentiators using generalized modulating functions method
Liu, Dayan
2015-02-01
This paper aims at designing a fractional order differentiator for a class of signals satisfying a linear differential equation with unknown parameters. A generalized modulating functions method is proposed first to estimate the unknown parameters, then to derive accurate integral formulae for the left-sided Riemann-Liouville fractional derivatives of the studied signal. Unlike the improper integral in the definition of the left-sided Riemann-Liouville fractional derivative, the integrals in the proposed formulae can be proper and be considered as a low-pass filter by choosing appropriate modulating functions. Hence, digital fractional order differentiators applicable for on-line applications are deduced using a numerical integration method in discrete noisy case. Moreover, some error analysis are given for noise error contributions due to a class of stochastic processes. Finally, numerical examples are given to show the accuracy and robustness of the proposed fractional order differentiators.
Calculating area of fractional-order memristor pinched hysteresis loop
Directory of Open Access Journals (Sweden)
Ya-Juan Yu
2015-11-01
Full Text Available A fractional-order current-controlled memristor pinched hysteresis loop area is calculated in this study. The area is divided into two parts: one equals to the half of instantaneous power and the other is the part memory of the memristor. Moreover, two parts of the area are affected not only by the cosine components, but also by the sine components. The voltage of the fractional-order current-controlled memristor is no longer an odd function with respect to time and the coefficient of cos(ωt in its Fourier series is zero. In a closed loop, the average power and the memory rely only on sine harmonics of the voltage. Meanwhile, the power and the memory are related to the order of the fractional-order derivative.
Fractional-order Fourier analysis for ultrashort pulse characterization.
Brunel, Marc; Coetmellec, Sébastien; Lelek, Mickael; Louradour, Frédéric
2007-06-01
We report what we believe to be the first experimental demonstration of ultrashort pulse characterization using fractional-order Fourier analysis. The analysis is applied to the interpretation of spectral interferometry resolved in time (SPIRIT) traces [which are spectral phase interferometry for direct electric field reconstruction (SPIDER)-like interferograms]. First, the fractional-order Fourier transformation is shown to naturally allow the determination of the cubic spectral phase coefficient of pulses to be analyzed. A simultaneous determination of both cubic and quadratic spectral phase coefficients of the pulses using the fractional-order Fourier series expansion is further demonstrated. This latter technique consists of localizing relative maxima in a 2D cartography representing decomposition coefficients. It is further used to reconstruct or filter SPIRIT traces.
Front dynamics in fractional-order epidemic models.
Hanert, Emmanuel; Schumacher, Eva; Deleersnijder, Eric
2011-06-21
A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic's front and a power-law decay of the front's leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics. Copyright © 2011 Elsevier Ltd. All rights reserved.
Onset of fractional-order thermal convection in porous media
Karani, Hamid; Rashtbehesht, Majid; Huber, Christian; Magin, Richard L.
2017-12-01
The macroscopic description of buoyancy-driven thermal convection in porous media is governed by advection-diffusion processes, which in the presence of thermophysical heterogeneities fail to predict the onset of thermal convection and the average rate of heat transfer. This work extends the classical model of heat transfer in porous media by including a fractional-order advective-dispersive term to account for the role of thermophysical heterogeneities in shifting the thermal instability point. The proposed fractional-order model overcomes limitations of the common closure approaches for the thermal dispersion term by replacing the diffusive assumption with a fractional-order model. Through a linear stability analysis and Galerkin procedure, we derive an analytical formula for the critical Rayleigh number as a function of the fractional model parameters. The resulting critical Rayleigh number reduces to the classical value in the absence of thermophysical heterogeneities when solid and fluid phases have similar thermal conductivities. Numerical simulations of the coupled flow equation with the fractional-order energy model near the primary bifurcation point confirm our analytical results. Moreover, data from pore-scale simulations are used to examine the potential of the proposed fractional-order model in predicting the amount of heat transfer across the porous enclosure. The linear stability and numerical results show that, unlike the classical thermal advection-dispersion models, the fractional-order model captures the advance and delay in the onset of convection in porous media and provides correct scalings for the average heat transfer in a thermophysically heterogeneous medium.
Directory of Open Access Journals (Sweden)
Jian Liu
2014-11-01
Full Text Available This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively.
Two-Dimensional Fractional Order Generalized Thermoelastic Porous Material
Directory of Open Access Journals (Sweden)
Ibrahim A. Abbas
Full Text Available AbstractIn the work, a two-dimensional problem of a porous material is considered within the context of the fractional order generalized thermoelasticity theory with one relaxation time. The medium is assumed initially quiescent for a thermoelastic half space whose surface is traction free and has a constant heat flux. The normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled equations. The effect of the fractional order of the temperature, displacement components, the stress components, changes in volume fraction field and temperature distribution have been depicted graphically.
An extended integrable fractional-order KP soliton hierarchy
Energy Technology Data Exchange (ETDEWEB)
Li Li, E-mail: li07099@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2011-01-17
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.
Wiener-Hopf optimal control of a hydraulic canal prototype with fractional order dynamics.
Feliu-Batlle, Vicente; Feliu-Talegón, Daniel; San-Millan, Andres; Rivas-Pérez, Raúl
2017-06-26
This article addresses the control of a laboratory hydraulic canal prototype that has fractional order dynamics and a time delay. Controlling this prototype is relevant since its dynamics closely resembles the dynamics of real main irrigation canals. Moreover, the dynamics of hydraulic canals vary largely when the operation regime changes since they are strongly nonlinear systems. All this makes difficult to design adequate controllers. The controller proposed in this article looks for a good time response to step commands. The design criterium for this controller is minimizing the integral performance index ISE. Then a new methodology to control fractional order processes with a time delay, based on the Wiener-Hopf control and the Padé approximation of the time delay, is developed. Moreover, in order to improve the robustness of the control system, a gain scheduling fractional order controller is proposed. Experiments show the adequate performance of the proposed controller. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Fractional Order Digital Differentiator Design Based on Power Function and Least squares
Kumar, Manjeet; Rawat, Tarun Kumar
2016-10-01
In this article, we propose the use of power function and least squares method for designing of a fractional order digital differentiator. The input signal is transformed into a power function by using Taylor series expansion, and its fractional derivative is computed using the Grunwald-Letnikov (G-L) definition. Next, the fractional order digital differentiator is modelled as a finite impulse response (FIR) system that yields fractional order derivative of the G-L type for a power function. The FIR system coefficients are obtained by using the least squares method. Two examples are used to demonstrate that the fractional derivative of the digital signals is computed by using the proposed technique. The results of the third and fourth examples reveal that the proposed technique gives superior performance in comparison with the existing techniques.
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis
Directory of Open Access Journals (Sweden)
Carson Ingo
2014-11-01
Full Text Available Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.
A novel auto-tuning method for fractional order PI/PD controllers.
De Keyser, Robin; Muresan, Cristina I; Ionescu, Clara M
2016-05-01
Fractional order PID controllers benefit from an increasing amount of interest from the research community due to their proven advantages. The classical tuning approach for these controllers is based on specifying a certain gain crossover frequency, a phase margin and a robustness to gain variations. To tune the fractional order controllers, the modulus, phase and phase slope of the process at the imposed gain crossover frequency are required. Usually these values are obtained from a mathematical model of the process, e.g. a transfer function. In the absence of such model, an auto-tuning method that is able to estimate these values is a valuable alternative. Auto-tuning methods are among the least discussed design methods for fractional order PID controllers. This paper proposes a novel approach for the auto-tuning of fractional order controllers. The method is based on a simple experiment that is able to determine the modulus, phase and phase slope of the process required in the computation of the controller parameters. The proposed design technique is simple and efficient in ensuring the robustness of the closed loop system. Several simulation examples are presented, including the control of processes exhibiting integer and fractional order dynamics. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.
Ingo, Carson; Magin, Richard L; Parrish, Todd B
2014-11-01
Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag-Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.
Optimal fractional order PID design via Tabu Search based algorithm.
Ateş, Abdullah; Yeroglu, Celaleddin
2016-01-01
This paper presents an optimization method based on the Tabu Search Algorithm (TSA) to design a Fractional-Order Proportional-Integral-Derivative (FOPID) controller. All parameter computations of the FOPID employ random initial conditions, using the proposed optimization method. Illustrative examples demonstrate the performance of the proposed FOPID controller design method. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Fractional order differential inclusions on the half-line
Directory of Open Access Journals (Sweden)
Mouffak Benchohra
2010-06-01
Full Text Available We are concerned with the existence of bounded solutions of a boundary value problem on an unbounded domain for fractional order differential inclusions involving the Caputo fractional derivative. Our results are based on the fixed point theorem of Bohnnenblust-Karlin combined with the diagonalization method.
Spherical angular spectrum and the fractional order Fourier transform.
Pellat-Finet, Pierre; Durand, Pierre-Emmanuel; Fogret, Eric
2006-12-01
The notion of a spherical angular spectrum leads to the decomposition of the field amplitude on a spherical emitter into a sum of spherical waves that converge onto the Fourier sphere of the emitter. Unlike the usual angular spectrum, the spherical angular spectrum is propagated as the field amplitude, in a way that can be expressed by a fractional order Fourier transform.
Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation.
Wen, Shao-Fang; Shen, Yong-Jun; Wang, Xiao-Na; Yang, Shao-Pu; Xing, Hai-Jun
2016-08-01
In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.
Tian, Yan; Zhong, Lin-Feng; He, Gui-Tian; Yu, Tao; Luo, Mao-Kang; Stanley, H. Eugene
2018-01-01
We study stochastic resonance (SR) in an oscillator with nonlinear noise, fractional-order external damping, and fractional-order intrinsic damping. Using a moment equation, we derive the exact analytical expression of the output amplitude and find that fluctuations in the output amplitude are non-monotonic. Using numerical simulations we verify the accuracy of this analytical result. We find (i) that nonlinear noise plays a key role in system behavior and that the resonance of the output amplitude is diverse when there is nonlinear noise, (ii) that the order of the fractional-order damping strongly impacts resonant intensity and that the impact on resonant intensity of fractional-order external damping is opposite that of fractional-order intrinsic damping, and (iii) that the evolution of the output amplitude versus the frequency of the external periodic force exhibits three behaviors: a resonance with one peak, a resonance with one peak and one valley, and a resonance with one valley.
Deniz, Furkan Nur; Alagoz, Baris Baykant; Tan, Nusret; Atherton, Derek P
2016-05-01
This paper introduces an integer order approximation method for numerical implementation of fractional order derivative/integrator operators in control systems. The proposed method is based on fitting the stability boundary locus (SBL) of fractional order derivative/integrator operators and SBL of integer order transfer functions. SBL defines a boundary in the parametric design plane of controller, which separates stable and unstable regions of a feedback control system and SBL analysis is mainly employed to graphically indicate the choice of controller parameters which result in stable operation of the feedback systems. This study reveals that the SBL curves of fractional order operators can be matched with integer order models in a limited frequency range. SBL fitting method provides straightforward solutions to obtain an integer order model approximation of fractional order operators and systems according to matching points from SBL of fractional order systems in desired frequency ranges. Thus, the proposed method can effectively deal with stability preservation problems of approximate models. Illustrative examples are given to show performance of the proposed method and results are compared with the well-known approximation methods developed for fractional order systems. The integer-order approximate modeling of fractional order PID controllers is also illustrated for control applications. Copyright © 2016 ISA. All rights reserved.
Li, Mingjie; Zhou, Ping; Zhao, Zhicheng; Zhang, Jinggang
2016-03-01
Recently, fractional order (FO) processes with dead-time have attracted more and more attention of many researchers in control field, but FO-PID controllers design techniques available for the FO processes with dead-time suffer from lack of direct systematic approaches. In this paper, a simple design and parameters tuning approach of two-degree-of-freedom (2-DOF) FO-PID controller based on internal model control (IMC) is proposed for FO processes with dead-time, conventional one-degree-of-freedom control exhibited the shortcoming of coupling of robustness and dynamic response performance. 2-DOF control can overcome the above weakness which means it realizes decoupling of robustness and dynamic performance from each other. The adjustable parameter η2 of FO-PID controller is directly related to the robustness of closed-loop system, and the analytical expression is given between the maximum sensitivity specification Ms and parameters η2. In addition, according to the dynamic performance requirement of the practical system, the parameters η1 can also be selected easily. By approximating the dead-time term of the process model with the first-order Padé or Taylor series, the expressions for 2-DOF FO-PID controller parameters are derived for three classes of FO processes with dead-time. Moreover, compared with other methods, the proposed method is simple and easy to implement. Finally, the simulation results are given to illustrate the effectiveness of this method. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Saïd Abbas
2013-01-01
Full Text Available In the present paper we investigate the existence of solutions for a system of integral inclusions of fractional order with multiple delay. Our results are obtained upon suitable fixed point theorems, namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler for the nonconvex case.
On the solutions of fractional order of evolution equations
Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.
2017-01-01
In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.
Mathematical modelling of fractional order circuit elements and bioimpedance applications
Moreles, Miguel Angel; Lainez, Rafael
2017-05-01
In this work a classical derivation of fractional order circuits models is presented. Generalised constitutive equations in terms of fractional Riemann-Liouville derivatives are introduced in the Maxwell's equations for each circuit element. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. It is shown that from basic properties of Fractional Calculus, a fractional differential equation model with Caputo derivatives is obtained. Thus standard initial conditions apply. Finally, models for bioimpedance are revisited.
A new operational matrix of fractional order integration for the ...
Indian Academy of Sciences (India)
49
integration in the Riemann-Liouville sense for the CWs is derived. Hat functions ... matrices of fractional order integration for the Haar wavelets [27], Legendre wavelets [28–33] and the. 1. 1. 2. 3. 4. 5. 6. 7. 8. 9 ..... and |u (x)| ≤ M2, can be expanded as an infinite sum of CWs, and the series converges uniformly to u(x), that is:.
A Fractional Order Recovery SIR Model from a Stochastic Process.
Angstmann, C N; Henry, B I; McGann, A V
2016-03-01
Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.
Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative
Directory of Open Access Journals (Sweden)
Yongjun Shen
2014-01-01
Full Text Available The subharmonic resonance of van der Pol (VDP oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC and equivalent linear stiffness coefficient (ELSC for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.
High-performance fractional order terminal sliding mode control strategy for DC-DC Buck converter.
Wang, Jianlin; Xu, Dan; Zhou, Huan; Bai, Anning; Lu, Wei
2017-01-01
This paper presents an adaption of the fractional order terminal sliding mode control (AFTSMC) strategy for DC-DC Buck converter. The following strategy aims to design a novel nonlinear sliding surface function, with a double closed-loop structure of voltage and current. This strategy is a fusion of two characteristics: terminal sliding mode control (TSMC) and fractional order calculation (FOC). In addition, the influence of "the controller parameters" on the "performance of double closed-loop system" is investigated. It is observed that the value of terminal power has to be chosen to make a compromise between start-up and transient response of the converter. Therefore the AFTSMC strategy chooses the value of the terminal power adaptively, and this strategy can lead to the appropriate number of fractional order as well. Furthermore, through the fractional order analysis, the system can reach the sliding mode surface in a finite time. And the theoretical considerations are verified by numerical simulation. The performance of the AFTSMC and TSMC strategies is tested by computer simulations. And the comparison simulation results show that the AFTSMC exhibits a considerable improvement in terms of a faster output voltage response during load changes. Moreover, AFTSMC obtains a faster dynamical response, smaller steady-state error rate and lower overshoot.
High-performance fractional order terminal sliding mode control strategy for DC-DC Buck converter.
Directory of Open Access Journals (Sweden)
Jianlin Wang
Full Text Available This paper presents an adaption of the fractional order terminal sliding mode control (AFTSMC strategy for DC-DC Buck converter. The following strategy aims to design a novel nonlinear sliding surface function, with a double closed-loop structure of voltage and current. This strategy is a fusion of two characteristics: terminal sliding mode control (TSMC and fractional order calculation (FOC. In addition, the influence of "the controller parameters" on the "performance of double closed-loop system" is investigated. It is observed that the value of terminal power has to be chosen to make a compromise between start-up and transient response of the converter. Therefore the AFTSMC strategy chooses the value of the terminal power adaptively, and this strategy can lead to the appropriate number of fractional order as well. Furthermore, through the fractional order analysis, the system can reach the sliding mode surface in a finite time. And the theoretical considerations are verified by numerical simulation. The performance of the AFTSMC and TSMC strategies is tested by computer simulations. And the comparison simulation results show that the AFTSMC exhibits a considerable improvement in terms of a faster output voltage response during load changes. Moreover, AFTSMC obtains a faster dynamical response, smaller steady-state error rate and lower overshoot.
Fractional-order modeling and State-of-Charge estimation for ultracapacitors
Zhang, Lei; Hu, Xiaosong; Wang, Zhenpo; Sun, Fengchun; Dorrell, David G.
2016-05-01
Ultracapacitors (UCs) have been widely recognized as an enabling energy storage technology in various industrial applications. They hold several advantages including high power density and exceptionally long lifespan over the well-adopted battery technology. Accurate modeling and State-of-Charge (SOC) estimation of UCs are essential for reliability, resilience, and safety in UC-powered system operations. In this paper, a novel fractional-order model composed of a series resistor, a constant-phase-element (CPE), and a Walburg-like element, is proposed to emulate the UC dynamics. The Grünald-Letnikov derivative (GLD) is then employed to discretize the continuous-time fractional-order model. The model parameters are optimally extracted using genetic algorithm (GA), based on the time-domain data acquired through the Federal Urban Driving Schedule (FUDS) test. By means of this fractional-order model, a fractional Kalman filter is synthesized to recursively estimate the UC SOC. Validation results prove that the proposed fractional-order modeling and state estimation scheme is accurate and outperforms current practice based on integer-order techniques.
Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials
Directory of Open Access Journals (Sweden)
Riccardo Caponetto
2016-11-01
Full Text Available The paper presents a fractional order model of a heating process and a comparison of fractional and standard PI controllers in its closed loop system. Preliminarily, an enhanced fractional order model for the heating process on non-continuous materials has been identified through a fitting algorithm on experimental data. Experimentation has been carried out on a finite length beam filled with three non-continuous materials (air, styrofoam, metal buckshots in order to identify a model in the frequency domain and to obtain a relationship between the fractional order of the heating process and the different materials’ properties. A comparison between the experimental model and the theoretical one has been performed, proving a significant enhancement of the fitting performances. Moreover the obtained modelling results confirm the fractional nature of the heating processes when diffusion occurs in non-continuous composite materials, and they show how the model’s fractional order can be used as a characteristic parameter for non-continuous materials with different composition and structure. Finally, three different kinds of controllers have been applied and compared in order to keep constant the beam temperature constant at a fixed length.
Das, Saptarshi; Pan, Indranil; Das, Shantanu
2013-07-01
Fuzzy logic based PID controllers have been studied in this paper, considering several combinations of hybrid controllers by grouping the proportional, integral and derivative actions with fuzzy inferencing in different forms. Fractional order (FO) rate of error signal and FO integral of control signal have been used in the design of a family of decomposed hybrid FO fuzzy PID controllers. The input and output scaling factors (SF) along with the integro-differential operators are tuned with real coded genetic algorithm (GA) to produce optimum closed loop performance by simultaneous consideration of the control loop error index and the control signal. Three different classes of fractional order oscillatory processes with various levels of relative dominance between time constant and time delay have been used to test the comparative merits of the proposed family of hybrid fractional order fuzzy PID controllers. Performance comparison of the different FO fuzzy PID controller structures has been done in terms of optimal set-point tracking, load disturbance rejection and minimal variation of manipulated variable or smaller actuator requirement etc. In addition, multi-objective Non-dominated Sorting Genetic Algorithm (NSGA-II) has been used to study the Pareto optimal trade-offs between the set point tracking and control signal, and the set point tracking and load disturbance performance for each of the controller structure to handle the three different types of processes. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Rasouli, H; Fatehi, A; Zamanian, H
2015-03-01
In this paper, a routine algorithm is presented to design a fractional order controller for tracking the reference model. Using this algorithm, a pole placement controller can be designed by assigning the desired integer and fractional order closed loop transfer functions. Considering the desired time response and using the generalized characteristic ratio assignment (CRA) method for fractional order systems and coefficient diagram method (CDM) for integer order systems, the desired closed loop system can be achieved. For various practical experiments, having the desired time responses is vital for magnetic flux in Damavand tokamak. To approach this, at first, the desired reference models are obtained based on CRA and CDM methods. After that, a fractional order pole placement controller is designed and simulated by this algorithm. At last, this controller is implemented on a digital signal processor to control the vertical magnetic flux of Damavand tokamak plant. The practical results show the satisfactory performance of the controller.
Robust stability of fractional order polynomials with complicated uncertainty structure
Şenol, Bilal; Pekař, Libor
2017-01-01
The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition. PMID:28662173
A Solution to the Fundamental Linear Fractional Order Differential Equation
Hartley, Tom T.; Lorenzo, Carl F.
1998-01-01
This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.
Atmospheric Turbulence Modeling for Aero Vehicles: Fractional Order Fits
Kopasakis, George
2015-01-01
Atmospheric turbulence models are necessary for the design of both inlet/engine and flight controls, as well as for studying coupling between the propulsion and the vehicle structural dynamics for supersonic vehicles. Models based on the Kolmogorov spectrum have been previously utilized to model atmospheric turbulence. In this paper, a more accurate model is developed in its representative fractional order form, typical of atmospheric disturbances. This is accomplished by first scaling the Kolmogorov spectral to convert them into finite energy von Karman forms and then by deriving an explicit fractional circuit-filter type analog for this model. This circuit model is utilized to develop a generalized formulation in frequency domain to approximate the fractional order with the products of first order transfer functions, which enables accurate time domain simulations. The objective of this work is as follows. Given the parameters describing the conditions of atmospheric disturbances, and utilizing the derived formulations, directly compute the transfer function poles and zeros describing these disturbances for acoustic velocity, temperature, pressure, and density. Time domain simulations of representative atmospheric turbulence can then be developed by utilizing these computed transfer functions together with the disturbance frequencies of interest.
New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties
Directory of Open Access Journals (Sweden)
Ali Karcı
2016-05-01
Full Text Available The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes.
Directory of Open Access Journals (Sweden)
Chunde Yang
2016-01-01
Full Text Available A modified fractional-order Chua chaotic circuit is proposed in this paper, and the chaotic attractor is obtained for q=0.98. Based on the Mittag-Leffler function in two parameters and Gronwall’s Lemma, two control schemes are proposed to stabilize the modified fractional-order Chua chaotic system via the Caputo derivative of a single input. The numerical simulation shows the validity and feasibility of the control scheme.
Directory of Open Access Journals (Sweden)
Mohammad Reza Rahmani Mehdiabadi
2014-05-01
Full Text Available This paper addresses synchronizing two coupled chaotic FitzHugh–Nagumo (FHN neurons with weakly gap junction under external electrical stimulation (EES. To transmit information among coupled neurons, by generalization of the integer-order FHN equations of the coupled system into the fractional-order in frequency domain using Crone approach, the behavior of each coupled neuron relies on its past behavior and the memorized system can be a better fit for the neuron response. An adaptive fractional-order controller Based on the Lyaponuv stability theory has been designed to synchronize two neurons electrically coupled with gap junction in EES. The proposed controller is also robust to the inevitable random noise such as disturbances of ionic channels. The simulation results demonstrate the effectiveness of the control scheme.
Mehdiabadi, M R Rahmani; Rouhani, E; Mashhadi, S K Mousavi; Jalali, A A
2014-01-01
This paper addresses synchronizing two coupled chaotic FitzHugh-Nagumo (FHN) neurons with weakly gap junction under external electrical stimulation (EES). To transmit information among coupled neurons, by generalization of the integer-order FHN equations of the coupled system into the fractional-order in frequency domain using Crone approach, the behavior of each coupled neuron relies on its past behavior and the memorized system can be a better fit for the neuron response. An adaptive fractional-order controller based on the Lyaponuv stability theory was designed to synchronize two neurons electrically coupled with gap junction in EES. The proposed controller is also robust to the inevitable random noise such as disturbances of ionic channels. The simulation results demonstrate the effectiveness of the control scheme.
Mehdiabadi, M. R. Rahmani; Rouhani, E.; Mashhadi, S. K. Mousavi; Jalali, A. A.
2014-01-01
This paper addresses synchronizing two coupled chaotic FitzHugh–Nagumo (FHN) neurons with weakly gap junction under external electrical stimulation (EES). To transmit information among coupled neurons, by generalization of the integer-order FHN equations of the coupled system into the fractional-order in frequency domain using Crone approach, the behavior of each coupled neuron relies on its past behavior and the memorized system can be a better fit for the neuron response. An adaptive fractional-order controller based on the Lyaponuv stability theory was designed to synchronize two neurons electrically coupled with gap junction in EES. The proposed controller is also robust to the inevitable random noise such as disturbances of ionic channels. The simulation results demonstrate the effectiveness of the control scheme. PMID:25337373
Zheng, WeiJia; Luo, Ying; Wang, XiaoHong; Pi, YouGuo; Chen, YangQuan
2017-05-01
In order to achieve a desired control performance characterized by satisfying specifications in both frequency-domain and time-domain simultaneously, an optimal fractional order proportional integral derivative (PI λ D μ ) controller design strategy is proposed based on analytical calculation and Differential Evolution algorithm for a permanent magnet synchronous motor (PMSM) servo system in this paper. In this controller design, the frequency-domain specifications can guarantee the system stability with both gain margin and phase margin, and also the system robustness to loop gain variations. The time-domain specifications can ensure the desired step response performance with rapid rising curve, constrained overshoot, and proper power consuming. Compared with the PI λ controller and the traditional PID controller, PI λ D μ controller can get obvious benefits from two more degrees of freedom of the fractional orders λ and μ on satisfying multiple constraints simultaneously and achieving better servo tracking performance for the PMSM servo system. PMSM speed tracking simulations and experiments are demonstrated to show the significant advantages of using the proposed optimal PI λ D μ controller over the optimal fractional order PI λ controller and traditional integer order PID controller. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Subordination Principle for a Class of Fractional Order Differential Equations
Directory of Open Access Journals (Sweden)
Emilia Bazhlekova
2015-05-01
Full Text Available The fractional order differential equation \\(u'(t=Au(t+\\gamma D_t^{\\alpha} Au(t+f(t, \\ t>0\\, \\(u(0=a\\in X\\ is studied, where \\(A\\ is an operator generating a strongly continuous one-parameter semigroup on a Banach space \\(X\\, \\(D_t^{\\alpha}\\ is the Riemann–Liouville fractional derivative of order \\(\\alpha \\in (0,1\\, \\(\\gamma>0\\ and \\(f\\ is an \\(X\\-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \\(C_{0}\\-semigroup, generated by the operator \\(A\\. As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered.
Tunable fractional-order capacitor using layered ferroelectric polymers
Directory of Open Access Journals (Sweden)
Agamyrat Agambayev
2017-09-01
Full Text Available Pairs of various Polyvinylidene fluoride P(VDF-based polymers are used for fabricating bilayer fractional order capacitors (FOCs. The polymer layers are constructed using a simple drop casting approach. The resulting FOC has two advantages: It can be easily integrated with printed circuit boards, and its constant phase angle (CPA can be tuned by changing the thickness ratio of the layers. Indeed, our experiments show that the CPA of the fabricated FOCs can be tuned within the range from -83° to -65° in the frequency band changing from 150 kHz to 10 MHz. Additionally, we provide an empirical formula describing the relationship between the thickness ratio and the CPA, which is highly useful for designing FOCs with the desired CPA.
Fractional order viscoelasticity in characterization for atrial tissue
Shen, Jing Jin; Li, Cheng Gang; Wu, Hong Tao; Kalantari, Masoud
2013-05-01
Atrial tissue due to its solid-like and fluid-like constituents shows highly viscoelastic properties. Up to now, the distribution pattern of muscle fiber in heart is not well established, and it is hard to establish the constitutive model for atrial tissue completely based on the microstructure level. Consider the equivalence between the fractional viscoelasticity and the fractal spring-dashpot model, a generalized fractional order Maxwell model is proposed to model the porcine atrial tissue in the phenomenological sense. This model has a simple expression and intuitively physical meanings. The constitutive parameters in the model are estimated in the complex domain by a genetic algorithm. Final results illustrate the proposed model gets a well agreement with the experimental data.
Tunable fractional-order capacitor using layered ferroelectric polymers
Agambayev, Agamyrat; Patole, Shashikant; Bagci, Hakan; Salama, Khaled N.
2017-09-01
Pairs of various Polyvinylidene fluoride P(VDF)-based polymers are used for fabricating bilayer fractional order capacitors (FOCs). The polymer layers are constructed using a simple drop casting approach. The resulting FOC has two advantages: It can be easily integrated with printed circuit boards, and its constant phase angle (CPA) can be tuned by changing the thickness ratio of the layers. Indeed, our experiments show that the CPA of the fabricated FOCs can be tuned within the range from -83° to -65° in the frequency band changing from 150 kHz to 10 MHz. Additionally, we provide an empirical formula describing the relationship between the thickness ratio and the CPA, which is highly useful for designing FOCs with the desired CPA.
Tunable fractional-order capacitor using layered ferroelectric polymers
Agambayev, Agamyrat
2017-09-05
Pairs of various Polyvinylidene fluoride P(VDF)-based polymers are used for fabricating bilayer fractional order capacitors (FOCs). The polymer layers are constructed using a simple drop casting approach. The resulting FOC has two advantages: It can be easily integrated with printed circuit boards, and its constant phase angle (CPA) can be tuned by changing the thickness ratio of the layers. Indeed, our experiments show that the CPA of the fabricated FOCs can be tuned within the range from -83° to -65° in the frequency band changing from 150 kHz to 10 MHz. Additionally, we provide an empirical formula describing the relationship between the thickness ratio and the CPA, which is highly useful for designing FOCs with the desired CPA.
Zhu, Zhiwei; Zhou, Xiaoqin
2012-01-01
The main contribution of this paper is the development of a linearized model for describing the dynamic hysteresis behaviors of piezoelectrically actuated fast tool servo (FTS). A linearized hysteresis force model is proposed and mathematically described by a fractional order differential equation. Combining the dynamic modeling of the FTS mechanism, a linearized fractional order dynamic hysteresis (LFDH) model for the piezoelectrically actuated FTS is established. The unique features of the LFDH model could be summarized as follows: (a) It could well describe the rate-dependent hysteresis due to its intrinsic characteristics of frequency-dependent nonlinear phase shifts and amplitude modulations; (b) The linearization scheme of the LFDH model would make it easier to implement the inverse dynamic control on piezoelectrically actuated micro-systems. To verify the effectiveness of the proposed model, a series of experiments are conducted. The toolpaths of the FTS for creating two typical micro-functional surfaces involving various harmonic components with different frequencies and amplitudes are scaled and employed as command signals for the piezoelectric actuator. The modeling errors in the steady state are less than ±2.5% within the full span range which is much smaller than certain state-of-the-art modeling methods, demonstrating the efficiency and superiority of the proposed model for modeling dynamic hysteresis effects. Moreover, it indicates that the piezoelectrically actuated micro systems would be more suitably described as a fractional order dynamic system.
DEFF Research Database (Denmark)
Xie, Chuan; Zhao, Xin; Savaghebi, Mehdi
2017-01-01
This paper presents a multi-rate fractional-order repetitive control (MRFORC) scheme for three-phase shunt active power filter (APF). The proposed APF control scheme includes an inner proportional-integral (PI) control loop with a sampling rate identical to switching frequency and an external plug......-in RC loop with a reduced sampling rate. The MRFORC loop is implemented in dq-frame with interleaving for the reduction of computational burden in each control cycle. Moreover, in order to deal with the harmonics in the presence of wide grid frequency variations, FORC, which replaces the fractional-order...... elements with the Lagrange-interpolation based finite impulse response (FIR) filter, is adopted instead of the conventional RC. The synthesis, stability analysis and parameters tuning criteria of the MRFORC system are derived in detail. A step by step design example of the proposed controller is also given...
Tavakoli-Kakhki, Mahsan; Haeri, Mohammad
2011-07-01
Fractional order PI and PID controllers are the most common fractional order controllers used in practice. In this paper, a simple analytical method is proposed for tuning the parameters of these controllers. The proposed method is useful in designing fractional order PI and PID controllers for control of complicated fractional order systems. To achieve the goal, at first a reduction technique is presented for approximating complicated fractional order models. Then, based on the obtained reduced models some analytical rules are suggested to determine the parameters of fractional order PI and PID controllers. Finally, numerical results are given to show the efficiency of the proposed tuning algorithm. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
Beaulieu, Alexandre; Bossé, Dominick; Micheau, Philippe; Avoine, Olivier; Praud, Jean-Paul; Walti, Hervé
2012-02-01
This study presents a methodology for applying the forced-oscillation technique in total liquid ventilation. It mainly consists of applying sinusoidal volumetric excitation to the respiratory system, and determining the transfer function between the delivered flow rate and resulting airway pressure. The investigated frequency range was f ∈ [0.05, 4] Hz at a constant flow amplitude of 7.5 mL/s. The five parameters of a fractional order lung model, the existing "5-parameter constant-phase model," were identified based on measured impedance spectra. The identification method was validated in silico on computer-generated datasets and the overall process was validated in vitro on a simplified single-compartment mechanical lung model. In vivo data on ten newborn lambs suggested the appropriateness of a fractional-order compliance term to the mechanical impedance to describe the low-frequency behavior of the lung, but did not demonstrate the relevance of a fractional-order inertance term. Typical respiratory system frequency response is presented together with statistical data of the measured in vivo impedance model parameters. This information will be useful for both the design of a robust pressure controller for total liquid ventilators and the monitoring of the patient's respiratory parameters during total liquid ventilation treatment. © 2011 IEEE
Chen, Jiawang; Gu, Linyi
2014-01-01
For the 4-DOF (degrees of freedom) trajectory tracking control problem of underwater remotely operated vehicles (ROVs) in the presence of model uncertainties and external disturbances, a novel output feedback fractional-order nonsingular terminal sliding mode control (FO-NTSMC) technique is introduced in light of the equivalent output injection sliding mode observer (SMO) and TSMC principle and fractional calculus technology. The equivalent output injection SMO is applied to reconstruct the full states in finite time. Meanwhile, the FO-NTSMC algorithm, based on a new proposed fractional-order switching manifold, is designed to stabilize the tracking error to equilibrium points in finite time. The corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. Comparative numerical simulation results are presented and analyzed to demonstrate the effectiveness of the proposed method. Finally, it is noteworthy that the proposed output feedback FO-NTSMC technique can be used to control a broad range of nonlinear second-order dynamical systems in finite time. PMID:24983004
Fractional-order mathematical model of an irrigation main canal pool
Directory of Open Access Journals (Sweden)
Shlomi N. Calderon-Valdez
2015-09-01
Full Text Available In this paper a fractional order model for an irrigation main canal is proposed. It is based on the experiments developed in a laboratory prototype of a hydraulic canal and the application of a direct system identification methodology. The hydraulic processes that take place in this canal are equivalent to those that occur in real main irrigation canals and the results obtained here can therefore be easily extended to real canals. The accuracy of the proposed fractional order model is compared by deriving two other integer-order models of the canal of a complexity similar to that proposed here. The parameters of these three mathematical models have been identified by minimizing the Integral Square Error (ISE performance index existing between the models and the real-time experimental data obtained from the canal prototype. A comparison of the performances of these three models shows that the fractional-order model has the lowest error and therefore the higher accuracy. Experiments showed that our model outperformed the accuracy of the integer-order models by about 25%, which is a significant improvement as regards to capturing the canal dynamics.
Matouk, A. E.
2011-02-01
In this work, stability analysis of the fractional-order modified Autonomous Van der Pol-Duffing (MAVPD) circuit is studied using the fractional Routh-Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.
On fractional Langevin equation involving two fractional orders
Baghani, Omid
2017-01-01
In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.
Magneto-thermoelasticity with two fractional order heat transfer
Directory of Open Access Journals (Sweden)
Magdy A. Ezzat
2016-02-01
Full Text Available A new mathematical model of the equations of two-temperature magneto-thermoelasticity theory for a perfect conducting solid has been constructed in the context of a new consideration of heat conduction with a time-fractional derivative of order α (0 < α ⩽ 1 and a time-fractional integral of order υ (0 < υ ⩽ 2. This model is applied to one-dimensional problem for a perfect conducting half-space of elastic solid with heat source distribution in the presence of a constant magnetic field. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and their graphs, conclusions about the new theory are given. Some comparisons are shown in figures to estimate the effects of the fractional order parameters and the temperature discrepancy on all the studied fields.
Incorporation of fractional-order dynamics into an existing PI/PID DC motor control loop.
Tepljakov, Aleksei; Gonzalez, Emmanuel A; Petlenkov, Eduard; Belikov, Juri; Monje, Concepción A; Petráš, Ivo
2016-01-01
The problem of changing the dynamics of an existing DC motor control system without the need of making internal changes is considered in the paper. In particular, this paper presents a method for incorporating fractional-order dynamics in an existing DC motor control system with internal PI or PID controller, through the addition of an external controller into the system and by tapping its original input and output signals. Experimental results based on the control of a real test plant from MATLAB/Simulink environment are presented, indicating the validity of the proposed approach. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Stabilization and Tracking Control of Inverted Pendulum Using Fractional Order PID Controllers
Directory of Open Access Journals (Sweden)
Sunil Kumar Mishra
2014-01-01
Full Text Available This work focuses on the use of fractional calculus to design robust fractional-order PID (PIλDμ controller for stabilization and tracking control of inverted pendulum (IP system. A particle swarm optimisation (PSO based direct tuning technique is used to design two PIλDμ controllers for IP system without linearizing the actual nonlinear model. The fitness function is minimized by running the SIMULINK model of IP system according to the PSO program in MATLAB. The performance of proposed PIλDμ controllers is compared with two PID controllers. Simulation results are also obtained by adding disturbances to the model to show the robustness of the proposed controllers.
Directory of Open Access Journals (Sweden)
Teerawat Sangpet
2014-01-01
Full Text Available Noncollocated control of flexible structures results in nonminimum-phase systems because the separation between the actuator and the sensor creates an input-output delay. The delay can deteriorate stability of closed-loop systems. This paper presents a simple approach to improve the delay-margin of the noncollocated vibration control of piezo-actuated flexible beams using a fractional-order controller. Results of real life experiments illustrate efficiency of the controller and show that the fractional-order controller has better stability robustness than the integer-order controller.
Complex systems and networks dynamics, controls and applications
Yu, Xinghuo; Chen, Guanrong; Yu, Wenwu
2016-01-01
This elementary book provides some state-of-the-art research results on broad disciplinary sciences on complex networks. It presents an in-depth study with detailed description of dynamics, controls and applications of complex networks. The contents of this book can be summarized as follows. First, the dynamics of complex networks, for example, the cluster dynamic analysis by using kernel spectral methods, community detection algorithms in bipartite networks, epidemiological modeling with demographics and epidemic spreading on multi-layer networks, are studied. Second, the controls of complex networks are investigated including topics like distributed finite-time cooperative control of multi-agent systems by applying homogenous-degree and Lyapunov methods, composite finite-time containment control for disturbed second-order multi-agent systems, fractional-order observer design of multi-agent systems, chaos control and anticontrol of complex systems via Parrondos game and many more. Third, the applications of ...
Yang, Ningning; Han, Yuchao; Wu, Chaojun; Jia, Rong; Liu, Chongxin
2017-08-01
Not Available Project supported by the National Natural Science Foundation of China (Grant No. 51507134) and the Science Fund from the Education Department of Shaanxi Province, China (Grant No. 15JK1537).
On the fragility of fractional-order PID controllers for FOPDT processes.
Padula, Fabrizio; Visioli, Antonio
2016-01-01
This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the controller parameters on the achieved control system robustness and performance are investigated. Results show that this kind of controllers is more fragile with respect to the standard proportional-integral-derivative controllers and therefore a significant attention should be paid by the user in their tuning. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Wang, Zhen; Wang, Xiaohong; Li, Yuxia; Huang, Xia
2017-12-01
In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results.
Directory of Open Access Journals (Sweden)
Bahatdin Daşbaşı
2017-06-01
Full Text Available In this study, it is described the general forms of fractional-order differential equations and asymtotic stability of their system’s equilibria. In addition that, the stability analysis of equilibrium points of the local bacterial infection model which is fractional-order differential equation system, is made. Results of this analysis are supported via numerical simulations drawn by datas obtained from literature for mycobacterium tuberculosis and the antibiotics isoniazid (INH, rifampicin (RIF, streptomycin (SRT and pyrazinamide (PRZ used against this bacterial infection.
Improved Fractional Order VSS Inc-Cond MPPT Algorithm for Photovoltaic Scheme
Directory of Open Access Journals (Sweden)
R. Arulmurugan
2014-01-01
Full Text Available Nowadays a hot topic among the research community is the harnessing energy from the free sunlight which is abundant and pollution-free. The availability of cheap solar photovoltaic (PV modules has to harvest solar energy with better efficiency. The nature of solar modules is nonlinear and therefore the proper impedance matching is essential. The proper impedance matching ensures the extraction of the maximum power from solar PV module. Maximum power point tracking (MPPT algorithm is acting as a significant part in solar power generating system because it varies in the output power from a PV generating set for various climatic conditions. This paper suggested a new improved work for MPPT of PV energy system by using the optimized novel improved fractional order variable step size (FOVSS incremental conductance (Inc-Cond algorithm. The new proposed controller combines the merits of both improved fractional order (FO and variable step size (VSS Inc-Cond which is well suitable for design control and execution. The suggested controller results in attaining the desired transient reaction under changing operating points. MATLAB simulation effort shows MPPT controller and a DC to DC Luo converter feeding a battery load is achieved. The laboratory experimental results demonstrate that the new proposed MPPT controller in the photovoltaic generating system is valid.
Fractional Order PID Control of Rotor Suspension by Active Magnetic Bearings
Directory of Open Access Journals (Sweden)
Parinya Anantachaisilp
2017-01-01
Full Text Available One of the key issues in control design for Active Magnetic Bearing (AMB systems is the tradeoff between the simplicity of the controller structure and the performance of the closed-loop system. To achieve this tradeoff, this paper proposes the design of a fractional order Proportional-Integral-Derivative (FOPID controller. The FOPID controller consists of only two additional parameters in comparison with a conventional PID controller. The feasibility of FOPID for AMB systems is investigated for rotor suspension in both the radial and axial directions. Tuning methods are developed based on the evolutionary algorithms for searching the optimal values of the controller parameters. The resulting FOPID controllers are then tested and compared with a conventional PID controller, as well as with some advanced controllers such as Linear Quadratic Gausian (LQG and H ∞ controllers. The comparison is made in terms of various stability and robustness specifications, as well as the dimensions of the controllers as implemented. Lastly, to validate the proposed method, experimental testing is carried out on a single-stage centrifugal compressor test rig equipped with magnetic bearings. The results show that, with a proper selection of gains and fractional orders, the performance of the resulting FOPID is similar to those of the advanced controllers.
A New Model of the Fractional Order Dynamics of the Planetary Gears
Directory of Open Access Journals (Sweden)
Vera Nikolic-Stanojevic
2013-01-01
Full Text Available A theoretical model of planetary gears dynamics is presented. Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. In the paper, it has been indicated that even the small disturbance in design realizations of this gear cause nonlinear properties of dynamics which are the source of vibrations and noise in the gear transmission. Dynamic model of the planetary gears with four degrees of freedom is used. Applying the basic principles of analytical mechanics and taking the initial and boundary conditions into consideration, it is possible to obtain the system of equations representing physical meshing process between the two or more gears. This investigation was focused to a new model of the fractional order dynamics of the planetary gear. For this model analytical expressions for the corresponding fractional order modes like one frequency eigen vibrational modes are obtained. For one planetary gear, eigen fractional modes are obtained, and a visualization is presented. By using MathCAD the solution is obtained.
Wu, L.; Liu, S.; Yang, Yingjie
2016-01-01
Traditional integer order buffer operator is extended to fractional order buffer operator, the corresponding relationship between the weakening buffer operator and the strengthening buffer operator is revealed. Fractional order buffer operator not only can generalize the weakening buffer operator and the strengthening buffer operator, but also realize tiny adjustment of buffer effect. The effectiveness of GM(1,1) with the fractional order buffer operator is validated by six cases.
A fractional-order controller for vibration suppression of uncertain structures.
Aghababa, Mohammad Pourmahmood
2013-11-01
The problem of active control of vibration structures has attracted much attention over the past decades. A general description of the control problem of vibration systems is to design an active controller to suppress the vibrations of the system induced by external disturbances such as an earthquake. In this paper, a novel fractional-order sliding mode control is introduced to attenuate the vibrations of structures with uncertainties and disturbances. After establishing a stable fractional sliding surface, a sliding mode control law is proposed. Then, the global asymptotic stability of the closed-loop system is analytically proved using fractional Lyapunov stability theorem. Finally, the robustness and applicability of the technique are verified using two examples, including a three degree of freedom structure and a two-story shear building. Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Computer network defense system
Urias, Vincent; Stout, William M. S.; Loverro, Caleb
2017-08-22
A method and apparatus for protecting virtual machines. A computer system creates a copy of a group of the virtual machines in an operating network in a deception network to form a group of cloned virtual machines in the deception network when the group of the virtual machines is accessed by an adversary. The computer system creates an emulation of components from the operating network in the deception network. The components are accessible by the group of the cloned virtual machines as if the group of the cloned virtual machines was in the operating network. The computer system moves network connections for the group of the virtual machines in the operating network used by the adversary from the group of the virtual machines in the operating network to the group of the cloned virtual machines, enabling protecting the group of the virtual machines from actions performed by the adversary.
Amplitude Modulation and Synchronization of Fractional-Order Memristor-Based Chua's Circuit
Directory of Open Access Journals (Sweden)
A. G. Radwan
2013-01-01
Full Text Available This paper presents a general synchronization technique and an amplitude modulation of chaotic generators. Conventional synchronization and antisynchronization are considered a very narrow subset from the proposed technique where the scale between the output response and the input response can be controlled via control functions and this scale may be either constant (positive, negative or time dependent. The concept of the proposed technique is based on the nonlinear control theory and Lyapunov stability theory. The nonlinear controller is designed to ensure the stability and convergence of the proposed synchronization scheme. This technique is applied on the synchronization of two identical fractional-order Chua's circuit systems with memristor. Different examples are studied numerically with different system parameters, different orders, and with five alternative cases where the scaling functions are chosen to be positive/negative and constant/dynamic which covers all possible cases from conventional synchronization to the amplitude modulation cases to validate the proposed concept.
Fractional-order integral and derivative controller for temperature ...
Indian Academy of Sciences (India)
), 1 Oryong-dong, Buk-gu, Gwangju 500 712, Korea; Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical and Computer Engineering, 4160 Old Main Hill, Utah State University, Logan, UT 84322-4160, USA ...
Fractional-order integral and derivative controller for temperature ...
Indian Academy of Sciences (India)
. Nonlinear Dynamics. (Guest Editor) 29: 1–385. Malki Heidar A, Dave Misir, Denny Feigenspan, Guanrong Chen 1997 Fuzzy PID control of a flexible- joint robot arm with uncertainties from time-varying loads. IEEE Trans. on Control Systems ...
Pitchfork bifurcation and vibrational resonance in a fractional-order ...
Indian Academy of Sciences (India)
School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, People's Republic of China; Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain; Department of ...
Experimental demonstration of fractional-order oscillators of orders 2.6 and 2.7
Elwakil, A.S.
2017-02-07
The purpose of this work is to provide an experimental demonstration for the development of sinusoidal oscillations in a fractional-order Hartley-like oscillator. Solid-state fractional-order electric double-layer capacitors were first fabricated using graphene-percolated P(VDF-TrFE-CFE) composite structure, and then characterized by using electrochemical impedance spectroscopy. The devices exhibit the fractional orders of 0.6 and 0.74 respectively (using the model Zc=Rs+1/(jω)αCα), with the corresponding pseudocapacitances of approximately 93nFsec−0.4 and 1.5nFsec−0.26 over the frequency range 200kHz–6MHz (Rs < 15Ω). Then, we verified using these fractional-order devices integrated in a Hartley-like circuit that the fractional-order oscillatory behaviors are of orders 2.6 and 2.74.
Kumar, Anupam; Kumar, Vijay
2017-05-01
In this paper, a novel concept of an interval type-2 fractional order fuzzy PID (IT2FO-FPID) controller, which requires fractional order integrator and fractional order differentiator, is proposed. The incorporation of Takagi-Sugeno-Kang (TSK) type interval type-2 fuzzy logic controller (IT2FLC) with fractional controller of PID-type is investigated for time response measure due to both unit step response and unit load disturbance. The resulting IT2FO-FPID controller is examined on different delayed linear and nonlinear benchmark plants followed by robustness analysis. In order to design this controller, fractional order integrator-differentiator operators are considered as design variables including input-output scaling factors. A new hybridized algorithm named as artificial bee colony-genetic algorithm (ABC-GA) is used to optimize the parameters of the controller while minimizing weighted sum of integral of time absolute error (ITAE) and integral of square of control output (ISCO). To assess the comparative performance of the IT2FO-FPID, authors compared it against existing controllers, i.e., interval type-2 fuzzy PID (IT2-FPID), type-1 fractional order fuzzy PID (T1FO-FPID), type-1 fuzzy PID (T1-FPID), and conventional PID controllers. Furthermore, to show the effectiveness of the proposed controller, the perturbed processes along with the larger dead time are tested. Moreover, the proposed controllers are also implemented on multi input multi output (MIMO), coupled, and highly complex nonlinear two-link robot manipulator system in presence of un-modeled dynamics. Finally, the simulation results explicitly indicate that the performance of the proposed IT2FO-FPID controller is superior to its conventional counterparts in most of the cases. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Owolabi, Kolade M.
2017-03-01
In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 super-diffusive (1 computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.
Comparison of the methods for discrete approximation of the fractional-order operator
Directory of Open Access Journals (Sweden)
Zborovjan Martin
2003-12-01
Full Text Available In this paper we will present some alternative types of discretization methods (discrete approximation for the fractional-order (FO differentiator and their application to the FO dynamical system described by the FO differential equation (FDE. With analytical solution and numerical solution by power series expansion (PSE method are compared two effective methods - the Muir expansion of the Tustin operator and continued fraction expansion method (CFE with the Tustin operator and the Al-Alaoui operator. Except detailed mathematical description presented are also simulation results. From the Bode plots of the FO differentiator and FDE and from the solution in the time domain we can see, that the CFE is a more effective method according to the PSE method, but there are some restrictions for the choice of the time step. The Muir expansion is almost unusable.
Directory of Open Access Journals (Sweden)
Najeeb Alam Khan
2012-01-01
Full Text Available This paper suggests two component homotopy method to solve nonlinear fractional integrodifferential equations, namely, Volterra's population model. Padé approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order α are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems.
New Approaches to Minimum-Energy Design of Integer- and Fractional-Order Perfect Control Algorithms
Hunek, Wojciech P.; Wach, Łukasz
2017-10-01
In this paper the new methods concerning the energy-based minimization of the perfect control inputs is presented. For that reason the multivariable integer- and fractional-order models are applied which can be used for describing a various real world processes. Up to now, the classical approaches have been used in forms of minimum-norm/least squares inverses. Notwithstanding, the above-mentioned tool do not guarantee the optimal control corresponding to optimal input energy. Therefore the new class of inversebased methods has been introduced, in particular the new σ- and H-inverse of nonsquare parameter and polynomial matrices. Thus a proposed solution remarkably outperforms the typical ones in systems where the control runs can be understood in terms of different physical quantities, for example heat and mass transfer, electricity etc. A simulation study performed in Matlab/Simulink environment confirms the big potential of the new energy-based approaches.
An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems
Directory of Open Access Journals (Sweden)
Mohammad Maleki
2012-01-01
Full Text Available An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE. By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.
1985-01-01
Long-term and short-term objectives for the development of a network operating system for the Space Station are stated. The short-term objective is to develop a prototype network operating system for a 100 megabit/second fiber optic data bus. The long-term objective is to establish guidelines for writing a detailed specification for a Space Station network operating system. Major milestones are noted. Information is given in outline form.
Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE
Directory of Open Access Journals (Sweden)
Chunlong Sun
2017-01-01
Full Text Available The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.
Ebrahimkhani, Sadegh
2016-07-01
Wind power plants have nonlinear dynamics and contain many uncertainties such as unknown nonlinear disturbances and parameter uncertainties. Thus, it is a difficult task to design a robust reliable controller for this system. This paper proposes a novel robust fractional-order sliding mode (FOSM) controller for maximum power point tracking (MPPT) control of doubly fed induction generator (DFIG)-based wind energy conversion system. In order to enhance the robustness of the control system, uncertainties and disturbances are estimated using a fractional order uncertainty estimator. In the proposed method a continuous control strategy is developed to achieve the chattering free fractional order sliding-mode control, and also no knowledge of the uncertainties and disturbances or their bound is assumed. The boundedness and convergence properties of the closed-loop signals are proven using Lyapunov׳s stability theory. Simulation results in the presence of various uncertainties were carried out to evaluate the effectiveness and robustness of the proposed control scheme. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations
Directory of Open Access Journals (Sweden)
Said Abbas
2011-11-01
Full Text Available In this article we investigate the existence and uniqueness of solutions for the initial value problems, for a class of hyperbolic impulsive fractional order differential equations by using some fixed point theorems.
Directory of Open Access Journals (Sweden)
Syuan-Yi Chen
2016-11-01
Full Text Available The object of this study is to develop a self-tuning fractional order proportional-integral-derivative (SFOPID controller for controlling the mover position of a direct drive linear voice coil motor (VCM accurately under different operational conditions. The fractional order proportional-integral-derivative (FOPID controller can improve the control performances of the conventional integer order PID controller with respect to the additional fractional differential and integral orders; however, choosing five interdependent control parameters including proportional, integral, and derivative gains, as well as fractional differential and integral orders appropriately is arduous in practical applications. In this regard, the SFOPID controller is proposed in which the five control parameters are optimized dynamically and concurrently according to an adaptive differential evolution algorithm with a high efficiency adaptive selection mechanism. Experimental results reveal that the SFOPID controller outperforms PID and FOPID controllers with regard to the nonlinear VCM control system under both nominal and payload conditions.
Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order
Directory of Open Access Journals (Sweden)
Yang Liu
2012-01-01
Full Text Available We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo’s fractional order derivative. The existence results are obtained by means of the Avery-Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative.
A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting.
Zhang, Yi; Pu, Yi-Fei; Hu, Jin-Rong; Liu, Yan; Zhou, Ji-Liu
2011-01-01
In this paper, we propose a new metal artifacts reduction algorithm based on fractional-order total-variation sinogram inpainting model for X-ray computed tomography (CT). The numerical algorithm for our fractional-order framework is also analyzed. Simulations show that, both quantitatively and qualitatively, our method is superior to conditional interpolation methods and the classic integral-order total variation model.
A Novel Sigma-Delta Modulator with Fractional-Order Digital Loop Integrator
Directory of Open Access Journals (Sweden)
Chi Xu
2017-01-01
Full Text Available This paper proposes using a fractional-order digital loop integrator to improve the robust stability of Sigma-Delta modulator, thus extending the integer-order Sigma-Delta modulator to a non-integer-order (fractional-order one in the Sigma-Delta ADC design field. The proposed fractional-order Sigma-Delta modulator has reasonable noise characteristics, dynamic range, and bandwidth; moreover the signal-to-noise ratio (SNR is improved remarkably. In particular, a 2nd-order digital loop integrator and a digital PIλDμ controller are combined to work as the fractional-order digital loop integrator, which is realized using FPGA; this will reduce the ASIC analog circuit layout design and chip testing difficulties. The parameters of the proposed fractional-order Sigma-Delta modulator are tuned by using swarm intelligent algorithm, which offers opportunity to simplify the process of tuning parameters and further improve the noise performance. Simulation results are given and they demonstrate the efficiency of the proposed fractional-order Sigma-Delta modulator.
Parameter Sensitivity Analysis for Fractional-Order Modeling of Lithium-Ion Batteries
Directory of Open Access Journals (Sweden)
Daming Zhou
2016-02-01
Full Text Available This paper presents a novel-fractional-order lithium-ion battery model that is suitable for use in embedded applications. The proposed model uses fractional calculus with an improved Oustaloup approximation method to describe all the internal battery dynamic behaviors. The fractional-order model parameters, such as equivalent circuit component coefficients and fractional-order values, are identified by a genetic algorithm. A modeling parameters sensitivity study using the statistical Multi-Parameter Sensitivity Analysis (MPSA method is then performed and discussed in detail. Through the analysis, the dynamic effects of parameters on the model output performance are obtained. It has been found out from the analysis that the fractional-order values and their corresponding internal dynamics have different degrees of impact on model outputs. Thus, they are considered as crucial parameters to accurately describe a battery’s dynamic voltage responses. To experimentally verify the accuracy of developed fractional-order model and evaluate its performance, the experimental tests are conducted with a hybrid pulse test and a dynamic stress test (DST on two different types of lithium-ion batteries. The results demonstrate the accuracy and usefulness of the proposed fractional-order model on battery dynamic behavior prediction.
Dominant pole placement with fractional order PID controllers: D-decomposition approach.
Mandić, Petar D; Šekara, Tomislav B; Lazarević, Mihailo P; Bošković, Marko
2017-03-01
Dominant pole placement is a useful technique designed to deal with the problem of controlling a high order or time-delay systems with low order controller such as the PID controller. This paper tries to solve this problem by using D-decomposition method. Straightforward analytic procedure makes this method extremely powerful and easy to apply. This technique is applicable to a wide range of transfer functions: with or without time-delay, rational and non-rational ones, and those describing distributed parameter systems. In order to control as many different processes as possible, a fractional order PID controller is introduced, as a generalization of classical PID controller. As a consequence, it provides additional parameters for better adjusting system performances. The design method presented in this paper tunes the parameters of PID and fractional PID controller in order to obtain good load disturbance response with a constraint on the maximum sensitivity and sensitivity to noise measurement. Good set point response is also one of the design goals of this technique. Numerous examples taken from the process industry are given, and D-decomposition approach is compared with other PID optimization methods to show its effectiveness. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Triangulation positioning system network
Directory of Open Access Journals (Sweden)
Sfendourakis Marios
2017-01-01
Full Text Available This paper presents ongoing work on localization and positioning through triangulation procedure for a Fixed Sensors Network - FSN.The FSN has to work as a system.As the triangulation problem becomes high complicated in a case with large numbers of sensors and transmitters, an adequate grid topology is needed in order to tackle the detection complexity.For that reason a Network grid topology is presented and areas that are problematic and need further analysis are analyzed.The Network System in order to deal with problems of saturation and False Triangulations - FTRNs will have to find adequate methods in every sub-area of the Area Of Interest - AOI.Also, concepts like Sensor blindness and overall Network blindness, are presented. All these concepts affect the Network detection rate and its performance and ought to be considered in a way that the network overall performance won’t be degraded.Network performance should be monitored contentiously, with right algorithms and methods.It is also shown that as the number of TRNs and FTRNs is increased Detection Complexity - DC is increased.It is hoped that with further research all the characteristics of a triangulation system network for positioning will be gained and the system will be able to perform autonomously with a high detection rate.
Design of CMOS analog integrated fractional-order circuits applications in medicine and biology
Tsirimokou, Georgia; Elwakil, Ahmed
2017-01-01
This book describes the design and realization of analog fractional-order circuits, which are suitable for on-chip implementation, capable of low-voltage operation and electronic adjustment of their characteristics. The authors provide a brief introduction to fractional-order calculus, followed by design issues for fractional-order circuits of various orders and types. The benefits of this approach are demonstrated with current-mode and voltage-mode filter designs. Electronically tunable emulators of fractional-order capacitors and inductors are presented, where the behavior of the corresponding chips fabricated using the AMS 0.35um CMOS process has been experimentally verified. Applications of fractional-order circuits are demonstrated, including a pre-processing stage suitable for the implementation of the Pan-Tompkins algorithm for detecting the QRS complexes of an electrocardiogram (ECG), a fully tunable implementation of the Cole-Cole model used for the modeling of biological tissues, and a simple, non-i...
Luo, Ying; Chen, Yangquan; Pi, Youguo
2011-01-01
In a reference of the previous work, a new systematic design method for fractional order proportional and derivative (FOPD) controller is proposed for a class of typical second-order plants. Simulation and experimental results in the reference show that, the dynamic performance and robustness with the designed FOPD controller outperforms that with the optimized traditional integer order proportional integral (IOPI) controller at normal speed. Furthermore, it is found that, for the ultra low-speed position tracking with a significant friction effect, the tracking performance using the designed FOPD controller is much better than that using the optimized IOPI controller. However, the reason of this advantage is unclear. In this paper, using the describing function method and Bode plots analysis, the observed advantage of the designed FOPD controller over the optimized IOPI controller, for the nonlinear low-speed position tracking system with friction effect, is explained with the theoretical analysis. This explanation for the priority of the designed FOPD controller is consistently demonstrated by the extended experimental results in this paper. Copyright © 2010 ISA. Published by Elsevier Ltd. All rights reserved.
Performance Analysis of Fractional-Order PID Controller for a Parabolic Distributed Solar Collector
Elmetennani, Shahrazed
2017-09-01
This paper studies the performance of a fractional-order proportional integral derivative (FOPID) controller designed for parabolic distributed solar collectors. The control problem addressed in concentrated solar collectors aims at forcing the produced heat to follow a desired reference despite the unevenly varying solar irradiance. In addition to the unpredictable variations of the energy source, the parabolic solar collectors are subject to inhomogeneous distributed efficiency parameters affecting the heat production. The FOPID controller is well known for its simplicity with better tuning flexibility along with robustness with respect to disturbances. Thus, we propose a control strategy based on FOPID to achieve the control objectives. First, the FOPID controller is designed based on a linear approximate model describing the system dynamics under nominal working conditions. Then, the FOPID gains and differentiation orders are optimally tuned in order to fulfill the robustness design specifications by solving a nonlinear optimization problem. Numerical simulations are carried out to evaluate the performance of the proposed FOPID controller. A comparison to the robust integer order PID is also provided. Robustness tests are performed for the nominal model to show the effectiveness of the FOPID. Furthermore, the proposed FOPID is numerically tested to control the distributed solar collector under real working conditions.
Yang, Qingxia; Xu, Jun; Cao, Binggang; Li, Xiuqing
2017-01-01
Identification of internal parameters of lithium-ion batteries is a useful tool to evaluate battery performance, and requires an effective model and algorithm. Based on the least square genetic algorithm, a simplified fractional order impedance model for lithium-ion batteries and the corresponding parameter identification method were developed. The simplified model was derived from the analysis of the electrochemical impedance spectroscopy data and the transient response of lithium-ion batteries with different states of charge. In order to identify the parameters of the model, an equivalent tracking system was established, and the method of least square genetic algorithm was applied using the time-domain test data. Experiments and computer simulations were carried out to verify the effectiveness and accuracy of the proposed model and parameter identification method. Compared with a second-order resistance-capacitance (2-RC) model and recursive least squares method, small tracing voltage fluctuations were observed. The maximum battery voltage tracing error for the proposed model and parameter identification method is within 0.5%; this demonstrates the good performance of the model and the efficiency of the least square genetic algorithm to estimate the internal parameters of lithium-ion batteries. PMID:28212405
Haze image enhancement based on space fractional-order partial differential equation
Xue, Wendan; Zhao, Fengqun
2017-07-01
Based on good amplitude frequency characteristics and the spatial global correlation of fractional-order differential, an energy functional of haze image enhancement is established by taking fractional derivative on both sides of the atmospheric physics scattering model, and a haze image enhancement model based on space fractional-order partial differential equation is obtained by using steepest descent method. Based on fast wavelet transform, the low-frequency part of patch transmission and the high-frequency part of point transmission are fused to estimate the transmission. Finally, the numerical solution of the fractional-order partial differential equation is obtained by the finite difference method. The experimental results show that the algorithm can improve the contrast, brightness and clarity of the image, and it is an effective image enhancement method for haze images.
Directory of Open Access Journals (Sweden)
Samaneh Jenab
2014-01-01
Full Text Available Application of fractional order proportional integral (FOPI controller to improve transient performance of wind turbine (WT with Doubly fed induction generator (DFIG is presented and studied in this paper. By small signal analysis, it is found that the dynamic behavior of the DFIG based WT, during the variation of operating conditions, is strongly affected by the stator dynamics. Since the DFIG electrical dynamics are nonlinear, the linear control (PI scheme cannot work properly under change in wind speed and stator modes are not damped appropriately. The proposed fractional order controller generalizes the conventional integer order PI controller whose integral order are fractional number rather than integer. This expansion can provide more flexibility in achieving control objectives. By time domain simulations, a comparative analysis is made with respect to the standard PI controller to demonstrate effectiveness of the fractional order PI controller during wind speed perturbation.
Liu, Da-Yan
2015-04-30
This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.
Kumar, Rajneesh; Singh, Kulwinder; Pathania, Devinder Singh
2017-07-01
The purpose of this paper is to study the variations in temperature, radial and normal displacement, normal stress, shear stress and couple stress in a micropolar thermoelastic solid in the context of fractional order theory of thermoelasticity. Eigen value approach together with Laplace and Hankel transforms are employed to obtain the general solution of the problem. The field variables corresponding to different fractional order theories of thermoelasticity have been obtained in the transformed domain. The general solution is applied to an infinite space subjected to a concentrated load at the origin. To obtained solution in the physical domain numerical inversion technique has been applied and numerically computed results are depicted graphically to analyze the effects of fractional order parameter on the field variables.
Demonstrative fractional order - PID controller based DC motor drive on digital platform.
Khubalkar, Swapnil W; Junghare, Anjali S; Aware, Mohan V; Chopade, Amit S; Das, Shantanu
2017-09-21
In industrial drives applications, fractional order controllers can exhibit phenomenal impact due to realization through digital implementation. Digital fractional order controllers have created wide scope as it possess the inherent advantages like robustness against the plant parameter variation. This paper provides brief design procedure of fractional order proportional-integral-derivative (FO-PID) controller through the indirect approach of approximation using constant phase technique. The new modified dynamic particle swarm optimization (IdPSO) technique is proposed to find controller parameters. The FO-PID controller is implemented using floating point digital signal processor. The building blocks are designed and assembled with all peripheral components for the 1.5kW industrial DC motor drive. The robust operation for parametric variation is ascertained by testing the controller with two separately excited DC motors with the same rating but different parameters. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Jian-feng Zhao
2017-01-01
Full Text Available This paper presents a three-dimensional autonomous chaotic system with high fraction dimension. It is noted that the nonlinear characteristic of the improper fractional-order chaos is interesting. Based on the continuous chaos and the discrete wavelet function map, an image encryption algorithm is put forward. The key space is formed by the initial state variables, parameters, and orders of the system. Every pixel value is included in secret key, so as to improve antiattack capability of the algorithm. The obtained simulation results and extensive security analyses demonstrate the high level of security of the algorithm and show its robustness against various types of attacks.
Zeng Cheng; Liang Shan; Zhang Yuzhe; Zhong Jiaqi; Su Yingying
2014-01-01
Remarkable improvements in the stability properties of discrete system zeros may be achieved by using a new design of the fractional-order hold (FROH) circuit. This paper first analyzes asymptotic behaviors of the limiting zeros, as the sampling period T tends to zero, of the sampled-data models on the basis of the normal form representation for continuous-time systems with a new hold proposed. Further, we also give the approximate expression of limiting zeros of the resulting sampled-data sy...
Directory of Open Access Journals (Sweden)
Fukang Yin
2013-01-01
Full Text Available A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs. The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs. The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations
Chakraverty, S.; Smita, Tapaswini
2014-12-01
The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.
Kopasakis, George
2014-01-01
The presentation covers a recently developed methodology to model atmospheric turbulence as disturbances for aero vehicle gust loads and for controls development like flutter and inlet shock position. The approach models atmospheric turbulence in their natural fractional order form, which provides for more accuracy compared to traditional methods like the Dryden model, especially for high speed vehicle. The presentation provides a historical background on atmospheric turbulence modeling and the approaches utilized for air vehicles. This is followed by the motivation and the methodology utilized to develop the atmospheric turbulence fractional order modeling approach. Some examples covering the application of this method are also provided, followed by concluding remarks.
Non-asymptotic fractional order differentiators via an algebraic parametric method
Liu, Dayan
2012-08-01
Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie\\'s modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations. © 2012 IEEE.
Network systems security analysis
Yilmaz, Ä.°smail
2015-05-01
Network Systems Security Analysis has utmost importance in today's world. Many companies, like banks which give priority to data management, test their own data security systems with "Penetration Tests" by time to time. In this context, companies must also test their own network/server systems and take precautions, as the data security draws attention. Based on this idea, the study cyber-attacks are researched throughoutly and Penetration Test technics are examined. With these information on, classification is made for the cyber-attacks and later network systems' security is tested systematically. After the testing period, all data is reported and filed for future reference. Consequently, it is found out that human beings are the weakest circle of the chain and simple mistakes may unintentionally cause huge problems. Thus, it is clear that some precautions must be taken to avoid such threats like updating the security software.
Merrikh-Bayat, Farshad
2017-05-01
In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to [Formula: see text] , where kj and αj are unknown and known real parameters respectively, is introduced. Without any loss of generality, a special form of MFOPID with transfer function kp+ki/s+kd1s+kd2sμ where kp, ki, kd1, and kd2 are unknown real and μ is a known positive real parameter, is considered. Similar to PID and TID, MFOPID is also linear in its parameters which makes it possible to study all of them in a same framework. Tuning the parameters of PID, TID, and MFOPID based on loop shaping using Linear Matrix Inequalities (LMIs) is discussed. For this purpose separate LMIs for closed-loop stability (of sufficient type) and adjusting different aspects of the open-loop frequency response are developed. The proposed LMIs for stability are obtained based on the Nyquist stability theorem and can be applied to both integer and fractional-order (not necessarily commensurate) processes which are either stable or have one unstable pole. Numerical simulations show that the performance of the four-variable MFOPID can compete the trivial five-variable FOPID and often excels PID and TID. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Directory of Open Access Journals (Sweden)
Jianxin Han
2017-01-01
Full Text Available This paper focuses on chaos suppression strategy of a microresonator actuated by two symmetrical electrodes. Dynamic behavior of this system under the case where the origin is the only stable equilibrium is investigated first. Numerical simulations reveal that system may exhibit chaotic motion under certain excitation conditions. Then, bifurcation diagrams versus amplitude or frequency of AC excitation are drawn to grasp system dynamics nearby its natural frequency. Results show that the vibration is complex and may exhibit period-doubling bifurcation, chaotic motion, or dynamic pull-in instability. For the suppression of chaos, a novel control algorithm, based on an integer-order nonsingular fast terminal sliding mode and a fractional-order switching law, is proposed. Fractional Lyapunov Stability Theorem is used to guarantee the asymptotic stability of the system. Finally, numerical results with both fractional-order and integer-order control laws show that our proposed control law is effective in controlling chaos with system uncertainties and external disturbances.
Numerical solution of two dimensional time fractional-order biological population model
Directory of Open Access Journals (Sweden)
Prakash Amit
2016-01-01
Full Text Available In this work, we provide an approximate solution of a parabolic fractional degenerate problem emerging in a spatial diffusion of biological population model using a fractional variational iteration method (FVIM. Four test illustrations are used to show the proficiency and accuracy of the projected scheme. Comparisons between exact solutions and numerical solutions are presented for different values of fractional order α.
Ateş, I.; Zegeling, P. A.|info:eu-repo/dai/nl/073634433
2017-01-01
In this paper we describe the application of the homotopy perturbation method (HPM) to two-point boundary-value problems with fractional-order derivatives of Caputo-type. We show that HPM is equivalent to the semi-analytical Adomian decomposition method when applied to a class of nonlinear
A fractional-order model of HIV infection with drug therapy effect
Directory of Open Access Journals (Sweden)
A.A.M. Arafa
2014-10-01
Full Text Available In this paper, the fractional-order model that describes HIV infection of CD4+ T cells with therapy effect is given. Generalized Euler Method (GEM is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.
Directory of Open Access Journals (Sweden)
Ahmad Neirameh
2018-07-01
Full Text Available In this study, we propose a new algorithm to find exact solitary wave solutions of nonlinear time- fractional order of extended biological population model. The new algorithm basically illustrates how two powerful algorithms, conformable fractional derivative and the homogeneous balance method can be combined and used to get exact solutions of fractional partial differential equations.
Magin, Richard L; Abdullah, Osama; Baleanu, Dumitru; Zhou, Xiaohong Joe
2008-02-01
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.
L1-Solutions of Boundary Value Problems for Implicit Fractional Order Differential Equations
Directory of Open Access Journals (Sweden)
Mouffak Benchohra
2015-12-01
Full Text Available The aim of this paper is to present new results on the existence of solutions for a class of boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem.
ECG artifact cancellation in surface EMG signals by fractional order calculus application.
Miljković, Nadica; Popović, Nenad; Djordjević, Olivera; Konstantinović, Ljubica; Šekara, Tomislav B
2017-03-01
New aspects for automatic electrocardiography artifact removal from surface electromyography signals by application of fractional order calculus in combination with linear and nonlinear moving window filters are explored. Surface electromyography recordings of skeletal trunk muscles are commonly contaminated with spike shaped artifacts. This artifact originates from electrical heart activity, recorded by electrocardiography, commonly present in the surface electromyography signals recorded in heart proximity. For appropriate assessment of neuromuscular changes by means of surface electromyography, application of a proper filtering technique of electrocardiography artifact is crucial. A novel method for automatic artifact cancellation in surface electromyography signals by applying fractional order calculus and nonlinear median filter is introduced. The proposed method is compared with the linear moving average filter, with and without prior application of fractional order calculus. 3D graphs for assessment of window lengths of the filters, crest factors, root mean square differences, and fractional calculus orders (called WFC and WRC graphs) have been introduced. For an appropriate quantitative filtering evaluation, the synthetic electrocardiography signal and analogous semi-synthetic dataset have been generated. The examples of noise removal in 10 able-bodied subjects and in one patient with muscle dystrophy are presented for qualitative analysis. The crest factors, correlation coefficients, and root mean square differences of the recorded and semi-synthetic electromyography datasets showed that the most successful method was the median filter in combination with fractional order calculus of the order 0.9. Statistically more significant (p < 0.001) ECG peak reduction was obtained by the median filter application compared to the moving average filter in the cases of low level amplitude of muscle contraction compared to ECG spikes. The presented results suggest
Magin, Richard L.; Li, Weiguo; Velasco, M. Pilar; Trujillo, Juan; Reiter, David A.; Morgenstern, Ashley; Spencer, Richard G.
2011-01-01
We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T1 and T2). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T1 and T2 relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T2 relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T1 was observed in BNC. In the single-component gels, for T2 measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for microstructural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T2 NMR relaxation processes in biological tissues. PMID:21498095
Visiting Power Laws in Cyber-Physical Networking Systems
Directory of Open Access Journals (Sweden)
Ming Li
2012-01-01
Full Text Available Cyber-physical networking systems (CPNSs are made up of various physical systems that are heterogeneous in nature. Therefore, exploring universalities in CPNSs for either data or systems is desired in its fundamental theory. This paper is in the aspect of data, aiming at addressing that power laws may yet be a universality of data in CPNSs. The contributions of this paper are in triple folds. First, we provide a short tutorial about power laws. Then, we address the power laws related to some physical systems. Finally, we discuss that power-law-type data may be governed by stochastically differential equations of fractional order. As a side product, we present the point of view that the upper bound of data flow at large-time scaling and the small one also follows power laws.
Directory of Open Access Journals (Sweden)
Temirkhan Aleroev
2009-04-01
Full Text Available We carry out spectral analysis of one class of integral operators associated with fractional order differential equations that arises in mechanics. We establish a connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness and completeness the systems of the eigenvalues.
Revisiting the approximate analytical solution of fractional-order gas dynamics equation
Directory of Open Access Journals (Sweden)
Mohammad Tamsir
2016-06-01
Full Text Available In this paper, an approximate analytical solution of the time fractional gas dynamics equation arising in the shock fronts, is obtained using a recent semi-analytical method referred as fractional reduced differential transform method. The fractional derivatives are considered in the Caputo sense. To validate the efficiency and reliability of the method, four numerical examples of the linear and nonlinear gas dynamics equations are considered. Computed results are compared with results available in the literature. It is found that obtained results agree excellently with DTM, and FHATM. The solutions behavior and its effects for different values of the fractional order are shown graphically. The main advantage of the method is easiness to implement and requires small size of computation. Hence, it is a very effective and efficient semi-analytical method for solving the fractional order gas dynamics equation.
Adaptive fractional-order total variation image restoration with split Bregman iteration.
Li, Dazi; Tian, Xiangyi; Jin, Qibing; Hirasawa, Kotaro
2017-09-08
Alleviating the staircase artifacts for variation method and adjusting the regularization parameters adaptively with the characteristics of different regions are two main issues in image restoration regularization process. An adaptive fractional-order total variation l1 regularization (AFOTV-l1) model is proposed, which is resolved by using split Bregman iteration algorithm (SBI) for image estimation. An improved fractional-order differential kernel mask (IFODKM) with an extended degree of freedom (DOF) is proposed, which can preserve more image details and effectively avoid the staircase artifact. With the SBI algorithm adopted in this paper, fast convergence and small errors are achieved. Moreover, a novel regularization parameters adaptive strategy is given. Experimental results, by using the standard image library (SIL), the lung imaging database consortium and image database resource initiative (LIDC-IDRI), demonstrate that the proposed methods have better approximation, robustness and fast convergence performances for image restoration. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate
Directory of Open Access Journals (Sweden)
Adnane Boukhouima
2017-01-01
Full Text Available We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results.
Monotonic solutions of functional integral and differential equations of fractional order
Directory of Open Access Journals (Sweden)
Ahmed El-Sayed
2009-02-01
Full Text Available The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equations have been studied by J. Banas. Here we are concerned with a singular quadratic functional integral equations. The existence of positive monotonic solutions $x \\in L_1[0,1]$ will be proved. The fractional order nonlinear functional differential equation will be given as a special case.
Directory of Open Access Journals (Sweden)
Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
Directory of Open Access Journals (Sweden)
Asma Ali Elbeleze
2014-01-01
Full Text Available We are concerned here with singular partial differential equations of fractional order (FSPDEs. The variational iteration method (VIM is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.
Couceiro, Micael
2015-01-01
This book examines the bottom-up applicability of swarm intelligence to solving multiple problems, such as curve fitting, image segmentation, and swarm robotics. It compares the capabilities of some of the better-known bio-inspired optimization approaches, especially Particle Swarm Optimization (PSO), Darwinian Particle Swarm Optimization (DPSO) and the recently proposed Fractional Order Darwinian Particle Swarm Optimization (FODPSO), and comprehensively discusses their advantages and disadvantages. Further, it demonstrates the superiority and key advantages of using the FODPSO algorithm, suc
Directory of Open Access Journals (Sweden)
A. Patra
2016-03-01
Full Text Available Radiating extended surfaces are usually utilized to enhance the heat transfer between primary surface and the environment. In this paper, temperature distribution, fin efficiency, efficacy of convective straight fins with constant and temperature-dependent thermal conductivity are solved by implementing homotopy perturbation sumudu transform method (HPSTM. The proposed method is very useful and practical for solving the fractional order nonlinear diffusion equation, which is associated with variable thermal conductivity condition. A dimensionless analytical expression has been developed for fin effectiveness. The fin efficiency and the fin effectiveness have been attained as a function of thermo-geometric fin parameter. It can be noticed that the thermal conductivity parameter has a strong influence over the fin efficiency. The analytical solutions acquired by the present method illustrate the approach is easy to implement and computationally very interesting. The obtained results are compared with previously found classical order results using variational iteration method (VIM, Adomian decomposition method, and the results from Galerkin method in order to show the competence of this present method. HPSTM is a simple and effective method for rapid assessment of physical systems although the fractional order energy balance equations comprise with strong nonlinear terms. The subsequent correlation equations can benefit thermal design engineers for designing of innovative straight fins with both constant and temperature-dependent thermal conductivity.
Directory of Open Access Journals (Sweden)
Duan Jun-Sheng
2017-12-01
Full Text Available We conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the order α of the fractional derivative and the parameter γ parameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameter γ, not exactly like the case of single fractional-order derivative for the order α. The case of the distributed-order derivative provides us more options for the weight function and parameters.
Directory of Open Access Journals (Sweden)
Sayed Hamed Hosseini
2017-06-01
Full Text Available This paper presents a two-loop approach for velocity and stator currents control of an Interior-type Permanent Magnet Synchronous Motor (IPMSM. In the outer loop, the reference torque obtained from a conventional PI controller gives two-axis stator reference currents based on Maximum-Torque per Ampere (MTPA strategy. In the inner loop, an adaptive fractional order sliding mode controller is designed to reach the two-axis stator currents to their reference values obtained from the MTPA method. To achieve this idea, fractional order sliding surfaces and an adaptive controller with adjustable parameters are employed. The adaptive controller is designed to increase the robustness of the proposed method against the uncertainties in stator resistance and inductances. A Lyapunov based adaptation mechanism is proposed for adjustment of the controller parameters. The optimal value of the fractional orders are obtained by optimization of an integral time absolute error performance index. The simulation results show the robustness of the proposed method against the uncertainties in stator resistance and stator inductances.
Khater method for nonlinear Sharma Tasso-Olever (STO equation of fractional order
Directory of Open Access Journals (Sweden)
Sadaf Bibi
Full Text Available In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering. Keywords: Khater method, Modified Riemann-Liouville derivative, Exact solutions, Fractional complex transform, Space-time fractional Sharma Tasso-Olever (STO equation
Directory of Open Access Journals (Sweden)
Cheng Liu
2017-05-01
Full Text Available In this paper, a novel nonlinear robust damping controller is proposed to suppress power oscillation in interconnected power systems. The proposed power oscillation damping controller exhibits good nonlinearity and robustness. It can consider the strong nonlinearity of power oscillation and uncertainty of its model. First, through differential homeomorphic mapping, a mathematical model of the system can be transformed into the Brunovsky standard. Next, an extended state observer (ESO estimated and compensated for model errors and external disturbances as well as uncertain factors to achieve dynamic linearization of the nonlinear model. A power oscillation damping controller for interconnected power systems was designed on a backstepping-fractional order sliding mode variable structure control theory (BFSMC. Compared with traditional methods, the controller exhibits good dynamic performance and strong robustness. Simulations involving a four-generator two-area and partial test system of Northeast China were conducted under various disturbances to prove the effectiveness and robustness of the proposed damping control method.
Adaptive fuzzy synchronization for a class of fractional-order neural networks
Liu, Heng; Li, Sheng-Gang; Wang, Hong-Xing; Li, Guan-Jun
2017-03-01
Not Available Project supported by the National Natural Science Foundation of China (Grant Nos. 11401243 and 61403157), the Foundation for Distinguished Young Talents in Higher Education of Anhui Province, China (Grant No. GXYQZD2016257), the Fundamental Research Funds for the Central Universities of China (Grant No. GK201504002), the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (Grant Nos. KJ2015A256 and KJ2016A665), the Natural Science Foundation of Anhui Province, China (Grant No. 1508085QA16), and the Innovation Funds of Graduate Programs of Shaanxi Normal University, China (Grant No. 2015CXB008).
Dynamic behaviours and control of fractional-order memristor-based ...
Indian Academy of Sciences (India)
attention, because of its potential applications in programmable logic, signal processing, neural networks, control systems, ... On the other hand, fractional calculus, an old mathematical topic, is now attracting intensive research in nearly all ... with a variation of system parameters are studied. Besides, two new stability and ...
Design of a Fractional Order Phase Shaper for Iso-Damped Control of a PHWR Under Step-Back Condition
Saha, Suman; Das, Saptarshi; Ghosh, Ratna; Goswami, Bhaswati; Balasubramanian, R.; Chandra, A. K.; Das, Shantanu; Gupta, Amitava
2010-06-01
Phase shaping using fractional order (FO) phase shapers has been proposed by many contemporary researchers as a means of producing systems with iso-damped closed loop response due to a stepped variation in input. Such systems, with the closed loop damping remaining invariant to gain changes can be used to produce dead-beat step response with only rise time varying with gain. This technique is used to achieve an active step-back in a Pressurized Heavy Water Reactor (PHWR) where it is desired to change the reactor power to a pre-determined value within a short interval keeping the power undershoot as low as possible. This paper puts forward an approach as an alternative for the present day practice of a passive step-back mechanism where the control rods are allowed to drop during a step-back action by gravity, with release of electromagnetic clutches. The reactor under a step-back condition is identified as a system using practical test data and a suitable Proportional plus Integral plus Derivative (PID) controller is designed for it. Then the combined plant is augmented with a phase shaper to achieve a dead-beat response in terms of power drop. The fact that the identified static gain of the system depends on the initial power level at which a step-back is initiated, makes this application particularly suited for using a FO phase shaper. In this paper, a model of a nuclear reactor is developed for a control rod drop scenario involving rapid power reduction in a 500 MWe Canadian Deuterium Uranium (CANDU) reactor using AutoRegressive Exogenous (ARX) algorithm. The system identification and reduced order modeling are developed from practical test data. For closed loop active control of the identified reactor model, the fractional order phase shaper along with a PID controller is shown to perform better than the present Reactor Regulating System (RRS) due to its iso-damped nature.
Perotto, E.
1987-08-01
The Network Operating System is an addition to CMS designed to allow multitasking operation, while conserving all the facilities of CMS: file system, interactivity, high level language environment. Multitasking is useful for server virtual machines, e.g. Network Transport Managers, File Managers, Disk space Managers, Tape Unit Managers, where the execution of a task involves long waits due to I/O completion, VCMF communication delays or human responses, during which the task status stays as a control block in memory, while the virtual machine serves other users executing the same lines of code. Multitasking is not only for multi-user service: a big data reduction program may run as a main task, while a side task, connected to the virtual console, gives reports on the ongoing work of the main task in response to user commands and steers the main task through common data. All the service routines (Wait, Create and Delete Task, Get and Release Buffer, VMCF Open and Close Link, Send and Receive, I/O and Console Routines) are FORTRAN callable, and may be used from any language environment consistent with the same parameter passing conventions. The outstanding feature of this system is efficiency, no user defined SVC are used, and the use of other privileged instructions as LPSW or SSM is the bare necessary, so that CP (with the associated overhead) is not too involved. System code and read-only data are write-protected with a different storage key from CMS and user program.
Das, Saptarshi; Pan, Indranil; Das, Shantanu
2015-09-01
An optimal trade-off design for fractional order (FO)-PID controller is proposed with a Linear Quadratic Regulator (LQR) based technique using two conflicting time domain objectives. A class of delayed FO systems with single non-integer order element, exhibiting both sluggish and oscillatory open loop responses, have been controlled here. The FO time delay processes are handled within a multi-objective optimization (MOO) formalism of LQR based FOPID design. A comparison is made between two contemporary approaches of stabilizing time-delay systems withinLQR. The MOO control design methodology yields the Pareto optimal trade-off solutions between the tracking performance and total variation (TV) of the control signal. Tuning rules are formed for the optimal LQR-FOPID controller parameters, using median of the non-dominated Pareto solutions to handle delayed FO processes. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Freed, Alan; Diethelm, Kai; Luchko, Yury
2002-01-01
This is the first annual report to the U.S. Army Medical Research and Material Command for the three year project "Advanced Soft Tissue Modeling for Telemedicine and Surgical Simulation" supported by grant No. DAMD17-01-1-0673 to The Cleveland Clinic Foundation, to which the NASA Glenn Research Center is a subcontractor through Space Act Agreement SAA 3-445. The objective of this report is to extend popular one-dimensional (1D) fractional-order viscoelastic (FOV) materials models into their three-dimensional (3D) equivalents for finitely deforming continua, and to provide numerical algorithms for their solution.
Regarding on the prototype solutions for the nonlinear fractional-order biological population model
Energy Technology Data Exchange (ETDEWEB)
Baskonus, Haci Mehmet, E-mail: hmbaskonus@gmail.com [Department of Computer Engineering, Tunceli University, Tunceli (Turkey); Bulut, Hasan [Department of Mathematics, Fzrat University, Elazig, Turkey, email: hbulut@firat.edu.tr (Turkey)
2016-06-08
In this study, we have submitted to literature a method newly extended which is called as Improved Bernoulli sub-equation function method based on the Bernoulli Sub-ODE method. The proposed analytical scheme has been expressed with steps. We have obtained some new analytical solutions to the nonlinear fractional-order biological population model by using this technique. Two and three dimensional surfaces of analytical solutions have been drawn by wolfram Mathematica 9. Finally, a conclusion has been submitted by mentioning important acquisitions founded in this study.
Directory of Open Access Journals (Sweden)
Hussein A. H. Salem
2013-01-01
Full Text Available The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.
Spatiotemporal chaos of fractional order logistic equation in nonlinear coupled lattices
Zhang, Ying-Qian; Wang, Xing-Yuan; Liu, Li-Yan; He, Yi; Liu, Jia
2017-11-01
We investigate a new spatiotemporal dynamics with fractional order differential logistic map and spatial nonlinear coupling. The spatial nonlinear coupling features such as the higher percentage of lattices in chaotic behaviors for most of parameters and none periodic windows in bifurcation diagrams are held, which are more suitable for encryptions than the former adjacent coupled map lattices. Besides, the proposed model has new features such as the wider parameter range and wider range of state amplitude for ergodicity, which contributes a wider range of key space when applied in encryptions. The simulations and theoretical analyses are developed in this paper.
Directory of Open Access Journals (Sweden)
Ibrahim A. Abbas
2016-06-01
Full Text Available In this article, we consider the problem of a thermoelastic infinite body with a spherical cavity in the context of the theory of fractional order thermoelasticity. The inner surface of the cavity is taken traction free and subjected to a thermal shock. The form of a vector–matrix differential equation has been considered for the governing equations in the Laplace transform domain. The analytical solutions are given by the eigenvalue approach. The graphical results indicate that the fractional parameter effect plays a significant role on all the physical quantities.
Surfaces with Prescribed Nodes and Minimum Energy Integral of Fractional Order
Directory of Open Access Journals (Sweden)
H. Gunawan
2011-11-01
Full Text Available his paper presents a method of finding a continuous, real-valued, function of two variables z = u(x,y defined on the square S := [0,1]^2, which minimizes an energy integral of fractional order, subject to the condition u(0,y = u(1,y = u(x,0 = u(x,1 = 0 and u(x_i,y_j = c_{ij}, where 0 < x_1 < ... < x_M < 1, 0 < y_1 < ... є ℝ are given. The function is expressed as a double Fourier sine series, and an iterative procedure to obtain the function will be presented.
Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order
Directory of Open Access Journals (Sweden)
Ming Li
2013-01-01
Full Text Available This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.