Hyperchaos in fractional order nonlinear systems
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. [Electrical and Computer Engineering Department, University of Sharjah, P.O. Box 27272 Sharjah (United Arab Emirates)] e-mail: wajdi@sharjah.ac.ae
2005-12-01
We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7-10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + {epsilon}, where 1 > {epsilon} > 0.
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.
Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation
Petráš, Ivo
2011-01-01
"Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation" presents a study of fractional-order chaotic systems accompanied by Matlab programs for simulating their state space trajectories, which are shown in the illustrations in the book. Description of the chaotic systems is clearly presented and their analysis and numerical solution are done in an easy-to-follow manner. Simulink models for the selected fractional-order systems are also presented. The readers will understand the fundamentals of the fractional calculus, how real dynamical systems can be described using fractional derivatives and fractional differential equations, how such equations can be solved, and how to simulate and explore chaotic systems of fractional order. The book addresses to mathematicians, physicists, engineers, and other scientists interested in chaos phenomena or in fractional-order systems. It can be used in courses on dynamical systems, control theory, and applied mathematics at graduate or postgraduate level. ...
Chaos in fractional-order autonomous nonlinear systems
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M.; Sprott, J.C
2003-03-01
We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0,1], based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic 'jerk' model. In both models, numerical simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2+{epsilon}, 1>{epsilon}>0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated.
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 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.
Fuzzy fractional order sliding mode controller for nonlinear systems
Delavari, H.; Ghaderi, R.; Ranjbar, A.; Momani, S.
2010-04-01
In this paper, an intelligent robust fractional surface sliding mode control for a nonlinear system is studied. At first a sliding PD surface is designed and then, a fractional form of these networks PDα, is proposed. Fast reaching velocity into the switching hyperplane in the hitting phase and little chattering phenomena in the sliding phase is desired. To reduce the chattering phenomenon in sliding mode control (SMC), a fuzzy logic controller is used to replace the discontinuity in the signum function at the reaching phase in the sliding mode control. For the problem of determining and optimizing the parameters of fuzzy sliding mode controller (FSMC), genetic algorithm (GA) is used. Finally, the performance and the significance of the controlled system two case studies (robot manipulator and coupled tanks) are investigated under variation in system parameters and also in presence of an external disturbance. The simulation results signify performance of genetic-based fuzzy fractional sliding mode controller.
Institute of Scientific and Technical Information of China (English)
Jia Li-Xin; Dai Hao; Hui Meng
2010-01-01
This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems.Based on Lyapunov stability theory and numerical differentiation，a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems.Numerical simulation results are presented to illustrate the effectiveness of this method.
State-Feedback Control for Fractional-Order Nonlinear Systems Subject to Input Saturation
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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.
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.
2015-01-01
In this paper, the problem of robust control of nonlinear fractional-order systems in the presence of uncertainties and external disturbance is investigated. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed Lemmas, an adaptive fuzzy controller is designed. The proposed method can guarantee all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small ...
Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems
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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.
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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.
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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.
Lyapunov functions for fractional order systems
Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A.; Gallegos, Javier A.
2014-09-01
A new lemma for the Caputo fractional derivatives, when 0<α<1, is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
On nonlinear control design for autonomous chaotic systems of integer and fractional orders
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. E-mail: wajdi@sharjah.ac.ae; Harb, Ahmad M. E-mail: aharb@just.edu.jo
2003-11-01
In this paper, we address the problem of chaos control for autonomous nonlinear chaotic systems. We use the recursive 'backstepping' method of nonlinear control design to derive the nonlinear controllers. The controller effect is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. We study two nonlinear chaotic systems: an electronic chaotic oscillator model, and a mechanical chaotic 'jerk' model. We demonstrate the robustness of the derived controllers against system order reduction arising from the use of fractional integrators in the system models. Our results are validated via numerical simulations.
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.
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.
Pourmahmood Aghababa, Mohammad
2013-10-01
This paper investigates the problem of robust control of nonlinear fractional-order dynamical systems in the presence of uncertainties. First, a novel switching surface is proposed and its finite-time stability to the origin is proved. Subsequently, using the sliding mode theory, a robust fractional control law is proposed to ensure the existence of the sliding motion in finite time. We use a fractional Lyapunov stability theory to prove the stability of the system in a given finite time. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the fractional derivative of the control signal. The proposed chattering-free sliding mode technique is then applied for stabilisation of a broad class of three-dimensional fractional-order chaotic systems via a single variable driving control input. Simulation results reveal that the proposed fractional sliding mode controller works well for chaos control of fractional-order hyperchaotic Chen, chaotic Lorenz and chaotic Arneodo systems with no-chatter control inputs.
Fractional Order Nonlinear Feedback Controller Design for PMSM Drives
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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.
Stability of Fractional Order Switching Systems
HosseinNia, S Hassan; Vinagre, Blas M
2012-01-01
This paper addresses the stabilization issue for fractional order switching systems. Common Lyapunov method is generalized for fractional order systems and frequency domain stability equivalent to this method is proposed to prove the quadratic stability. Some examples are given to show the applicability and effectiveness of the proposed theory.
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.
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.
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.
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Xiaomin Tian
2014-02-01
Full Text Available In this paper, the problem of stabilizing a class of fractional-order chaotic systems with sector and dead-zone nonlinear inputs is investigated. The effects of model uncertainties and external disturbances are fully taken into account. Moreover, the bounds of both model uncertainties and external disturbances are assumed to be unknown in advance. To deal with the system’s nonlinear items and unknown bounded uncertainties, an adaptive fractional-order sliding mode (AFSM controller is designed. Then, Lyapunov’s stability theory is used to prove the stability of the designed control scheme. Finally, two simulation examples are given to verify the effectiveness and robustness of the proposed control approach.
The Active Fractional Order Control for Maglev Suspension System
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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.
Chaos in the fractional order nonlinear Bloch equation with delay
Baleanu, Dumitru; Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha
2015-08-01
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ = 0 , we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α = 0.8548 with subsequent period doubling that leads to chaos at α = 0.9436 . A periodic window is observed for the range 0.962 chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value α = 0.8532 , and the transition from two to four cycles at α = 0.9259 . With further increases in the fractional order, period doubling continues until at α = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α = 0.8441 , and α = 0.8635 , respectively. However, the system exhibits chaos at much lower values of α (α = 0.8635). A periodic window is observed in the interval 0.897 chaos again appearing for larger values of α . In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a
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.
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...
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.
Indian Academy of Sciences (India)
KARIMA RABAH; SAMIR LADACI; MOHAMED LASHAB
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.
Generalized Synchronization Between Different Fractional-Order Chaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHOU Ping; CHENG Xue-Feng; ZHANG Nian-Ying
2008-01-01
In this paper, using scalar feedback controller and stability theory of fractional-order systems, a gener-alized synchronization method for different fractional-order chaotic systems is established. Simulation results show the effectiveness of the theoretical results.
Frequency domain stability criteria for fractional-order control systems
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This paper concerns about the frequency domain stability criteria for fractional-order control systems. On the base of characteristics of the fractional-order equations solutions, we consider the Nyquist stability criterion in a wider sense and obtain a more common means to analyze the stability of fractional-order systems conveniently. Finally, this paper illustrates the generalized stability criteria with an example to show the effect of the parameters variation on the fractional-order control systems.
Realization of fractional-order Liu chaotic system by circuit
Institute of Scientific and Technical Information of China (English)
Lu Jun-Jie; Liu Chong-Xin
2007-01-01
In this paper, chaotic behaviours in the fractional-order Liu system are studied. Based on the approximation theory of fractional-order operator, circuits are designed to simulate the fractional- order Liu system with q = 0.1 - 0.9 in a step of 0.1, and an experiment has demonstrated the 2.7-order Liu system. The simulation results prove that the chaos exists indeed in the fractional-order Liu system with an order as low as 0.3. The experimental results prove that the fractional-order chaotic system can be realized by using hardware devices, which lays the foundation for its practical applications.
Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems
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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.
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.
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 ...
Formal modeling and verification of fractional order linear systems.
Zhao, Chunna; Shi, Likun; Guan, Yong; Li, Xiaojuan; Shi, Zhiping
2016-05-01
This paper presents a formalization of a fractional order linear system in a higher-order logic (HOL) theorem proving system. Based on the formalization of the Grünwald-Letnikov (GL) definition, we formally specify and verify the linear and superposition properties of fractional order systems. The proof provides a rigor and solid underpinnings for verifying concrete fractional order linear control systems. Our implementation in HOL demonstrates the effectiveness of our approach in practical applications.
Circuit realization of the fractional-order unified chaotic system
Institute of Scientific and Technical Information of China (English)
Chen Xiang-Rong; Liu Chong-Xin; Wang Fa-Qiang
2008-01-01
This paper studies the chaotic behaviours of the fractional-order unified chaotic system.Based on the approximation method in frequency domain,it proposes an electronic circuit model of tree shape to realize the fractional-order operator.According to the tree shape model,an electronic circuit is designed to realize the 2.7-order unified chaotic system.Numerical simulations and circuit experiments have verified the existence of chaos in the fraction-order unified system.
Preservation of Stability and Synchronization of a Class of Fractional-Order Systems
2012-01-01
We present sufficient conditions for the preservation of stability of fractional-order systems, and then we use this result to preserve the synchronization, in a master-slave scheme, of fractional-order systems. The systems treated herein are autonomous fractional differential linear and nonlinear systems with commensurate orders lying between 0 and 2, where the nonlinear ones can be described as a linear part plus a nonlinear part. These results are based on stability properties for equilibr...
Impulsive Control for Fractional-Order Chaotic Systems
Institute of Scientific and Technical Information of China (English)
ZHONG Qi-Shui; BAO Jing-Fu; YU Yong-Bin; LIAO Xiao-Feng
2008-01-01
@@ We propose an impulsive control scheme for fractional-order chaotic systems. Based on the Takagi-Sugeno (T-S) fuzzy model and linear matrix inequalities (LMIs), some sufficient conditions are given to stabilize the fractional-order chaotic system via impulsive control. Numerical simulation shows the effectiveness of this approach.
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.
Analogue Realization of Fractional-Order Dynamical Systems
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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.
Optimal design for fractional-order active isolation system
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Hao You
2015-12-01
Full Text Available The optimal control of fractional-order active isolation system is researched based on the optimal control theory, and the effect of fractional-order derivative on passive isolation system is also analyzed. The mechanical model is established where viscoelastic features of isolation materials are described by fractional-order derivative. The viscoelastic property of the fractional-order derivative in dynamical system is studied and the fractional-order derivative could be divided into linear stiffness and linear damping. It is found that both the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient but also the resonance frequency by the equivalent linear stiffness. Based on optimal control theory, the feedback gain of fractional-order active isolation system under harmonic excitation is obtained, which is changed with the excitation frequency. The statistical responses of the displacement and velocity for passive and active vibration isolation systems subjected to random excitation are also presented, which further verifies the excellent performance of fractional-order derivative in vibration control engineering.
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.
Dynamic behaviours and control of fractional-order memristor-based system
Indian Academy of Sciences (India)
Liping Chen; Yigang He; Xiao Lv; Ranchao Wu
2015-07-01
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 varying parameters. Furthermore, some conditions ensuring Hopf bifurcation with varying fractional orders and parameters are determined, respectively. By using a stabilization theorem proposed newly for a class of nonlinear systems, linear feedback controller is designed to stabilize the fractional-order system and the corresponding stabilization criterion is presented. Numerical simulations are given to illustrate and verify the effectiveness of our analysis results.
Intelligent Fractional Order Systems and Control An Introduction
Pan, Indranil
2013-01-01
Fractional order calculus is finding increasing interest in the control system community. Hardware realizations of fractional order controllers have sparked off a renewed zeal into the investigations of control system design in the light of fractional calculus. As such many notions of integer order LTI systems are being modified and extended to incorporate these new concepts. Computational Intelligence (CI) techniques have been applied to engineering problems to find solutions to many hitherto intractable conundrums and is a useful tool for dealing with problems of higher computational complexity. This book borders on the interface between CI techniques and fractional calculus, and looks at ways in which fractional order control systems may be designed or enhanced using CI based paradigms. To the best of the author’s knowledge this is the first book of its kind exclusively dedicated to the application of computational intelligence techniques in fractional order systems and control. The book tries to assimil...
Probabilistic robust stabilization of fractional order systems with interval uncertainty.
Alagoz, Baris Baykant; Yeroglu, Celaleddin; Senol, Bilal; Ates, Abdullah
2015-07-01
This study investigates effects of fractional order perturbation on the robust stability of linear time invariant systems with interval uncertainty. For this propose, a probabilistic stability analysis method based on characteristic root region accommodation in the first Riemann sheet is developed for interval systems. Stability probability distribution is calculated with respect to value of fractional order. Thus, we can figure out the fractional order interval, which makes the system robust stable. Moreover, the dependence of robust stability on the fractional order perturbation is analyzed by calculating the order sensitivity of characteristic polynomials. This probabilistic approach is also used to develop a robust stabilization algorithm based on parametric perturbation strategy. We present numerical examples demonstrating utilization of stability probability distribution in robust stabilization problems of interval uncertain systems.
Disturbance Rejection for Fractional-Order Time-Delay Systems
Hai-Peng Jiang; Yong-Qiang Liu
2016-01-01
This paper presents an equivalent-input-disturbance (EID-) based disturbance rejection method for fractional-order time-delay systems. First, a modified state observer is applied to reconstruct the state of the fractional-order time-delay plant. Then, a disturbance estimator is designed to actively compensate for the disturbances. Under such a construction of the system, by constructing a novel monochromatic Lyapunov function and using direct Lyapunov approach, the stability analysis and cont...
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.
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.
Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW
Zhu, Li; Fan, Qibin
2013-05-01
Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations.
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...
Horseshoe and entropy in a fractional-order unified system
Institute of Scientific and Technical Information of China (English)
Li Qing-Du; Chen Shu; Zhou Ping
2011-01-01
This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.
Observation of a class of disturbance in time series expansion for fractional order systems
Yiheng Wei; Hamid Reza Karimi; Jinwen Pan; Qing Gao; Yong Wang
2014-01-01
This paper is concerned with the problem of designing disturbance observer for fractional order systems, of which the disturbance is in time series expansion. The stability of a special observer with the selected nonlinear weighted function and transient dynamics function is rigorously analyzed for slowly varying disturbance. In addition, the result is also extended to estimate slope forms disturbance and higher order disturbance of fractional order systems. The efficacy of the proposed metho...
Static Output Feedback H-infinity Control for a Fractional-Order Glucose-Insulin System
2015-01-01
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 attemp...
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.
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.
Fractional-order Systems and Synchronous Generator Voltage Regulator
2015-01-01
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 volta...
Horseshoe and entropy in a fractional-order unified system
Li, Qing-Du; Chen, Shu; Zhou, Ping
2011-01-01
This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincaré section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.
Synchronization in a unified fractional-order chaotic system
Institute of Scientific and Technical Information of China (English)
Wu Zheng-Mao; Xie Jian-Ying
2007-01-01
In this paper, the synchronization in a unified fractional-order chaotic system is investigated by two methods. One is the frequency-domain method that is analysed by using the Laplace transform theory. The other is the time-domain method that is analysed by using the Lyapunov stability theory. Finally, the numerical simulations are used-to illustrate the effectiveness of the proposed synchronization methods.
Parrondo’s paradox for chaos control and anticontrol of fractional-order systems
Marius-F, Danca; Wallace, K. S. Tang
2016-01-01
We present the generalized forms of Parrondo’s paradox existing in fractional-order nonlinear systems. The generalization is implemented by applying a parameter switching (PS) algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N ≥ 2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words “winning” and “loosing” in the classical Parrondo’s paradox with “order” and “chaos", respectively, the PS algorithm leads to the generalized Parrondo’s paradox: chaos1 + chaos2 + ··· + chaosN = order and order1 + order2 + ··· + orderN = chaos. Finally, the concept is well demonstrated with the results based on the fractional-order Chen system.
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.
Analysing chaos in fractional-order systems with the harmonic balance method
Institute of Scientific and Technical Information of China (English)
Wu Zheng-Mao; Lu Jun-Guo; Xie Jian-Ying
2006-01-01
In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.
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. PMID:27783648
Circuit implementation of a new hyperchaos in fractional-order system
Institute of Scientific and Technical Information of China (English)
Liu Chong-Xin; Liu Ling
2008-01-01
This paper introduces a new four-dimensional (4D) hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincaré mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-ordersystem but also in the fractional-order system with an order as low as 3.6.
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.
Parameter identification of fractional order linear system based on Haar wavelet operational matrix.
Li, Yuanlu; Meng, Xiao; Zheng, Bochao; Ding, Yaqing
2015-11-01
Fractional order systems can be more adequate for the description of dynamical systems than integer order models, however, how to obtain fractional order models are still actively exploring. In this paper, an identification method for fractional order linear system was proposed. This is a method based on input-output data in time domain. The input and output signals are represented by Haar wavelet, and then fractional order systems described by fractional order differential equations are transformed into fractional order integral equations. Taking use of the Haar wavelet operational matrix of the fractional order integration, the fractional order linear system can easily be converted into a system of algebraic equation. Finally, the parameters of the fractional order system are determined by minimizing the errors between the output of the real system and that of the identified system. Numerical simulations, involving integral and fractional order systems, confirm the efficiency of the above methodology.
Static output feedback ℋ ∞ control for a fractional-order glucose-insulin system
N’Doye, Ibrahima
2015-05-23
This paper presents the ℋ
Series solutions of non-linear Riccati differential equations with fractional order
Energy Technology Data Exchange (ETDEWEB)
Cang Jie; Tan Yue; Xu Hang [School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030 (China); Liao Shijun [School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030 (China)], E-mail: sjliao@sjtu.edu.cn
2009-04-15
In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter h. Besides, it is proved that well-known Adomian's decomposition method is a special case of the homotopy analysis method when h = -1. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.
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.
Liping Chen; Shanbi Wei; Yi Chai; Ranchao Wu
2012-01-01
Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown par...
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 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that 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.
An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter
Ping Zhou; Rui Ding
2011-01-01
An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with di...
An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter
Directory of Open Access Journals (Sweden)
Ping Zhou
2011-01-01
Full Text Available An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with different fractional orders. Two examples are presented to demonstrate the effectiveness of the proposed method.
A novel adaptive-impulsive synchronization of fractional-order chaotic systems
Institute of Scientific and Technical Information of China (English)
Leung Y. T. Andrew; Li Xian-Feng; Chu Yan-Dong; Zhang Hui
2015-01-01
A novel adaptive–impulsive scheme is proposed for synchronizing fractional-order chaotic systems without the ne-cessity 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.
Institute of Scientific and Technical Information of China (English)
Qi Dong-Lian; Wang Qiao; Yang Jie
2011-01-01
Two different sliding mode controllers for a fractional order unified chaotic system are presented.The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system,and the fractional-order system can be made asymptotically stable by this controller.By proving the existence of a sliding manifold containing fractional integral,the controller for a fractional-order system is obtained,which can stabilize it.A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.
Chaos Control and Synchronization in Fractional-Order Lorenz-Like System
Directory of Open Access Journals (Sweden)
Sachin Bhalekar
2012-01-01
Full Text Available The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006. The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback.
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.
Institute of Scientific and Technical Information of China (English)
Ke Xiao; Shang-Bo Zhou; Wei-Wei Zhang
2008-01-01
For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional-order differential equations based on the definition of Grunwald-Letnikov is presented. The results of numerical solution by using the novel method and the frequency-domain method are compared, and the limitations of frequency-domain method arediscussed.
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.
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2010-01-01
Full Text Available We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.
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.
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.
Impulsive synchronisation of a class of fractional-order hyperchaotic systems
Institute of Scientific and Technical Information of China (English)
Wang Xing-Yuan; Zhang Yong-Lei; Lin Da; Zhang Na
2011-01-01
In this paper, an impulsive synchronisation scheme for a class of fractional-order hyperchaotic systems is proposed. The sufficient conditions of a class of integral-order hyperchaotic systems' impulsive synchronisation are illustrated. Furthermore, we apply the sufficient conditions to a class of fractional-order hyperchaotic systems and well achieve impulsive synchronisation of these fractional-order hyperchaotic systems, thereby extending the applicable scope of impulsive synchronisation. Numerical simulations further demonstrate the feasibility and effectiveness of the proposed scheme.
Research on the stability of control systems described by fractional-order transfer functions
Institute of Scientific and Technical Information of China (English)
Zeng Qingshan; Zhu Xinjian; Cao Guangyi
2005-01-01
The stability of control systems described by fractional-order transfer function form is mainly investigated. The stability analysis of integer-order linear systems was extended to the fractional-order control systems. The stability definition of fractional-order linear control systems is presented in terms of the Lyapunov's stability theory. Using the theorems of the Mittag-Leffler function in two parameters directly derives the stability conclusion. The illustrative examples are also given by simulation results.
One new fractional-order chaos system and its circuit simulation by electronic workbench
Institute of Scientific and Technical Information of China (English)
Zhou Ping; Cheng Xue-Feng; Zhang Nian-Ying
2008-01-01
This paper proposes a new chaotic system and its fractional-order chaotic system.The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained.It finds that this new fractional-order system is chaotic for q>0.783 if the system parameter m=6.The chaotic attractors for q=0.8,and q=0.9 are obtained.A circuit is designed to realize its fractional-order chaos system for q=0.9 by electronic workbench.
Q-S synchronization of the fractional-order unified system
Indian Academy of Sciences (India)
Yi Chai; Liping Chen; Rancho Wu; Juan Dai
2013-03-01
Concept of Q-S synchronization for fractional-order systems is introduced and Q-S synchronization of the fractional-order unified system is investigated in this paper. On the basis of the stability theory of the fractional-order system, two suitable control schemes are designed to achieve Q-S synchronization of the fractional-order unified systems under the given observable variables of drive system and the response system. Theoretical analysis and numerical simulations are shown to demonstrate the validity and feasibility of the proposed method.
Generalized Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions
Institute of Scientific and Technical Information of China (English)
WANG Sha; YU Yong-Guang
2012-01-01
The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated. According to the stability theory of linear fractional order systems, a sufficient condition to realize synchronization is obtained. The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions. The corresponding numerical results coincide with theoretical analysis.%The generalized projective synchronization of different dimensional fractional order chaotic systems is investigated.According to the stability theory of linear fractional order systems,a sufficient condition to realize synchronization is obtained.The fractional order chaotic and hyperchaotic systems are applied to achieve synchronization in both reduced and increased dimensions.The corresponding numerical results coincide with theoretical analysis.
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.
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems
Cuimei Jiang; Shutang Liu; Da Wang
2015-01-01
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, general...
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...
Regression model for tuning the PID controller with fractional order time delay system
S.P. Agnihotri; Laxman Madhavrao Waghmare
2014-01-01
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...
Adaptive control and synchronization of a fractional-order chaotic system
Indian Academy of Sciences (India)
Chunlai Li; Yaonan Tong
2013-04-01
In this paper, the chaotic dynamics of a three-dimensional fractional-order chaotic system is investigated. The lowest order for exhibiting chaos in the fractional-order system is obtained. Adaptive schemes are proposed for control and synchronization of the fractional-order chaotic system based on the stability theory of fractional-order dynamic systems. The presented schemes, which contain only a single-state variable, are simple and flexible. Numerical simulations are used to demonstrate the feasibility of the presented methods.
Synchronization and an application of a novel fractional order King Cobra chaotic system.
Muthukumar, P; Balasubramaniam, P; Ratnavelu, K
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 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
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.
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.
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.
Realization of fractional-order Liu chaotic system by a new circuit unit
Institute of Scientific and Technical Information of China (English)
Xu Zhe; Liu Chong-Xin
2008-01-01
A new circuit unit for the analysis and the synthesis of the chaotic behaviours in a fractional-order Liu system is proposed in this paper.Based on the approximation theory of fractional-order operator,an electronic circuit is designed to describe the dynamic behaviours of the fractional-order Liu system with a=0.9.The results between simulation and experiment are in good agreement with each other,thereby proving that the chaos exists indeed in the fractional-order Liu system.
CONTROLLABILITY AND OBSERVABILITY FOR A CLASS OF FRACTIONAL-ORDER IMPULSIVE SYSTEMS
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
This paper studies the controllability and observability for a class of fractionalorder impulsive systems.First,the basic acknowledgement of fractional-order systems is presented.Then the solution of the fractional-order impulsive systems is given.Finally,necessary and sufficient criteria for controllability and observability are obtained.
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.
Chaotic synchronization for a class of fractional-order chaotic systems
Institute of Scientific and Technical Information of China (English)
Zhou Ping
2007-01-01
In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.
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.
Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems
Institute of Scientific and Technical Information of China (English)
Lu Jun-Guo
2005-01-01
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.
Energy Technology Data Exchange (ETDEWEB)
Zheng Yongai, E-mail: zhengyongai@163.co [Department of Computer, Yangzhou University, Yangzhou, 225009 (China); Nian Yibei [School of Energy and Power Engineering, Yangzhou University, Yangzhou, 225009 (China); Wang Dejin [Department of Computer, Yangzhou University, Yangzhou, 225009 (China)
2010-12-01
In this Letter, a kind of novel model, called the generalized Takagi-Sugeno (T-S) fuzzy model, is first developed by extending the conventional T-S fuzzy model. Then, a simple but efficient method to control fractional order chaotic systems is proposed using the generalized T-S fuzzy model and adaptive adjustment mechanism (AAM). Sufficient conditions are derived to guarantee chaos control from the stability criterion of linear fractional order systems. The proposed approach offers a systematic design procedure for stabilizing a large class of fractional order chaotic systems from the literature about chaos research. The effectiveness of the approach is tested on fractional order Roessler system and fractional order Lorenz system.
STABILITY CRITERIA FOR STOCHASTIC DISCRETE-TIME FRACTIONAL ORDER SYSTEMS
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Carmen BARBACIORU
2016-05-01
Full Text Available In this paper are discussed stability problems for a class of discrete-time fractional systems (DTFSs with independent random perturbations. Two notions of mean square stability (MSS and mean square asymptotic stability (MSAS are introduced for the DTFSs by using an approximating linear stochastic system. Necessary and sufficient conditions for MSS and MSA are then derived.
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.
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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.
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.
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.
Institute of Scientific and Technical Information of China (English)
SU XIN-WEI
2011-01-01
This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects. The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in [B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(2009), 251258].
Stability analysis of impulsive functional systems of fractional order
Stamova, Ivanka; Stamov, Gani
2014-03-01
In this paper, a class of impulsive fractional functional differential systems is investigated. Sufficient conditions for stability of the zero solution are proved, extending the corresponding theory of impulsive functional differential equations. The investigations are carried out by using the comparison principle, coupled with the Lyapunov function method. We apply our results to an impulsive single species model of Lotka-Volterra type.
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.
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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.
Atanasova-Pacemska, Tatjana; Jolevska-Tuneska, Biljana; Lazarova, Limonka
2016-01-01
Fractional-order systems have lately been attracting significant attention and gaining more acceptance as generalization to classical integer-order systems. Mathematical basics of fractional-order calculus were laid nearly 300 years ago and since that it has gained deeply rooted mathematical concepts. Today, it is known that many real dynamic systems cannot be described by a system of simple differential equation or of integer-order system. In practice we can encounter such systems in electro...
Chao, Luo
2015-11-01
In this paper, a novel digital secure communication scheme is firstly proposed. Different from the usual secure communication schemes based on chaotic synchronization, the proposed scheme employs asynchronous communication which avoids the weakness of synchronous systems and is susceptible to environmental interference. Moreover, as to the transmission errors and data loss in the process of communication, the proposed scheme has the ability to be error-checking and error-correcting in real time. In order to guarantee security, the fractional-order complex chaotic system with the shifting of order is utilized to modulate the transmitted signal, which has high nonlinearity and complexity in both frequency and time domains. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the scheme.
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.
Dynamical analysis of fractional-order Rössler and modified Lorenz systems
Energy Technology Data Exchange (ETDEWEB)
Letellier, Christophe, E-mail: Christophe.Letellier@coria.fr [Université de Rouen – CORIA, BP 12, F-76801 Saint-Etienne du Rouvray Cedex (France); Aguirre, Luis A. [Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, MG (Brazil)
2013-10-15
This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.
Dynamical analysis of fractional-order Rössler and modified Lorenz systems
Letellier, Christophe; Aguirre, Luis A.
2013-10-01
This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.
Institute of Scientific and Technical Information of China (English)
WANG Xing-Yuan; LIU Rong; ZHANG Na
2011-01-01
The purpose of this paper is to analyze the dynamic behavior of fractional-order four-order hyperchaotic Lii system, and use the Open-Plus-Closed-Looping (OPCL) coupling method to construct the system's corresponding response system, and then implement function projective synchronization (FPS) of fractional-order drive-response system with system parameters perturbation or not. Finally, the numerical simulations verify the effectiveness and robustness of this scheme.
A 3D Fractional-Order Chaotic System with Only One Stable Equilibrium and Controlling Chaos
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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.
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.
Institute of Scientific and Technical Information of China (English)
Zhang Ruo-Xun; Yang Shi-Ping
2012-01-01
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.
Inverse synchronization of coupled fractional-order systems through open-plus-closed-loop control
Indian Academy of Sciences (India)
Junwei Wang; Li Zeng; Qinghua Ma
2011-03-01
In this paper, the inverse synchronization problem of fractional-order dynamical systems is investigated. 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 this proposed scheme is that it can be applied not only to nonchaotic but also to chaotic fractional-order systems whenever they exhibit regular or irregular oscillations. Feasibility of the proposed inverse synchronization scheme is illustrated through numerical simulations.
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Chen Yi
2011-01-01
Full Text Available We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii's fixed point theorem are applied to establish the existence results.
Institute of Scientific and Technical Information of China (English)
Liu YANG; Zongmin QIAO
2012-01-01
In this paper,the existence and multiplicity of positive solutions for Robin type boundary value problem of differential equation involving the Riemann-Liouville fractional order derivative are established.
Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal
Energy Technology Data Exchange (ETDEWEB)
Lu Junguo [Department of Automation, Shanghai Jiaotong University, 1954 Hua Shan Road, Shanghai 200030 (China)] e-mail: jglu@sjtu.edu.cn
2006-01-01
In this paper, a drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.
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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
Jian Yuan; Bao Shi; Xiaoyun Zeng; Wenqiang Ji; Tetie Pan
2013-01-01
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 propo...
Collaboration Control of Fractional-Order Multiagent Systems with Sampling Delay
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Hong-yong Yang
2013-01-01
Full Text Available Because of the complexity of the practical environments, many distributed multiagent systems cannot be illustrated with the integer-order dynamics and can only be described with the fractional-order dynamics. In this paper, collaboration control problems of continuous-time networked fractional-order multiagent systems via sampled control and sampling delay are investigated. Firstly, the sampled-data control of multiagent systems with fractional-order derivative operator is analyzed in a directed weighted network ignoring sampling delay. Then, the collaborative control of fractional-order multiagent systems with sampled data and sampling delay is studied in a directed and symmetrical network. Many sufficient conditions for reaching consensus with sampled data and sampling delay are obtained. Some numerical simulations are presented to illustrate the utility of our theoretical results.
Research on the stability, controllability and observability for fractional order LTI systems
Institute of Scientific and Technical Information of China (English)
WANG Zhen-bin; CAO Guang-yi; ZHU Xin-jian
2006-01-01
The state space representations of fractional order linear time-invariant (LTI ) systems are introduced, and their solution formulas are deduced by means of Laplace transform. The stability condition of fractional order LTI systems is given, and its proof is deduced by means of using linear non-singularity transform and the derivative property of Mittag-Leffler function. The controllability condition of fractional order LTI systems is given, and its proof is deduced by means of using its characteristic polynomial and the Cayley-Hamilton theorem. The observability condition of fractional order LTI systems is given, and its proof is deduced by means of their solution formulas. Finally an example is given to prove the correctness of the stability, controllability, and observability conditions mentioned above. s are deduced by means of Laplace transform. Their stability, controllability and observability conditions are given as well as their proofs.
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.
Networked Convergence of Fractional-Order Multiagent Systems with a Leader and Delay
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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.
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.
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.
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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.
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.
El-Sayed, A. M. A.; Elsonbaty, A.; Elsadany, A. A.; Matouk, A. E.
2016-12-01
This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh-Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.
Stabilization of generalized fractional order chaotic systems using state feedback control
Energy Technology Data Exchange (ETDEWEB)
Ahmad, Wajdi M. E-mail: wajdi@sharjah.ac.ae; El-Khazali, Reyad E-mail: khazali@ece.ac.ae; Al-Assaf, Yousef E-mail: yassaf@aus.ac.ae
2004-10-01
In this paper, we address the problem of chaos control of three types of fractional order systems using simple state feedback gains. Electronic chaotic oscillators, mechanical 'jerk' systems, and the Chen system are investigated when they assume generalized fractional orders. We design the static gains to place the eigenvalues of the system Jacobian matrices in a stable region whose boundaries are determined by the orders of the fractional derivatives. We numerically demonstrate the effectiveness of the controller in eliminating the chaotic behavior from the state trajectories, and driving the states to the nearest equilibrium point in the basin of attraction. For the recently introduced Chen system, in particular, we demonstrate that with a proper choice of model parameters, chaotic behavior is preserved when the system order becomes fractional. Both state and output feedback controllers are then designed to stabilize a generalized fractional order Chen system.
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.
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.
Antisynchronization of Nonidentical Fractional-Order Chaotic Systems Using Active Control
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Sachin Bhalekar
2011-01-01
Full Text Available Antisynchronization phenomena are studied in nonidentical fractional-order differential systems. The characteristic feature of antisynchronization is that the sum of relevant state-variables vanishes for sufficiently large value of time variable. Active control method is used first time in the literature to achieve antisynchronization between fractional-order Lorenz and Financial systems, Financial and Chen systems, and Lü and Financial systems. The stability analysis is carried out using classical results. We also provide numerical results to verify the effectiveness of the proposed theory.
Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System
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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.
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.
Dynamic analysis of a fractional order delayed predator-prey system with harvesting.
Song, Ping; Zhao, Hongyong; Zhang, Xuebing
2016-06-01
In the study, we consider a fractional order delayed predator-prey system with harvesting terms. Our discussion is divided into two cases. Without harvesting, we investigate the stability of the model, as well as deriving some criteria by analyzing the associated characteristic equation. With harvesting, we investigate the dynamics of the system from the aspect of local stability and analyze the influence of harvesting to prey and predator. Finally, numerical simulations are presented to verify our theoretical results. In addition, using numerical simulations, we investigate the effects of fractional order and harvesting terms on dynamic behavior. Our numerical results show that fractional order can affect not only the stability of the system without harvesting terms, but also the switching times from stability to instability and to stability. The harvesting can convert the equilibrium point, the stability and the stability switching times.
Observer-Type Consensus Protocol for a Class of Fractional-Order Uncertain Multiagent Systems
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Hongjie Li
2012-01-01
Full Text Available This paper investigates the consensus problem for a class of fractional-order uncertain multiagent systems with general linear node dynamics. Firstly, an observer-type consensus protocol is proposed based on the relative observer states of neighboring agents. Secondly, based on property of the Kronecker product and stability theory of fractional-order system, some sufficient conditions are presented for robust asymptotical stability of the observer-based fractional-order control systems. Thirdly, robust stabilizing controllers are derived by using linear matrix inequality approach and matrix’s singular value decomposition. Our results are in the form of linear matrix inequalities which can easily be solved by LMI toolbox in MATLAB. Finally, a numerical simulation is performed to show the effectiveness of the theoretical results.
An innovative fixed-pole numerical approximation for fractional order systems.
Wei, Yiheng; Tse, Peter W; Du, Bin; Wang, Yong
2016-05-01
A novel numerical approximation scheme is proposed for fractional order systems by the concept of identification. An identical equation is derived firstly, from which one can obtain the exact state space model of fractional order systems. It reveals the nature of the approximation problem, and then provides an effective scheme to obtain the desired model. This research project also focuses on solving a knotty but crucial issue, i.e., the initial value problem of fractional order systems. The results generated by the study prove that it can reduce to the Caputo case by selecting some specific initial values. A careful simulation study is reported to illustrate the effectiveness of the proposed scheme. To exhibit the superiority clearly, the results are compared with that of the published fixed-pole finite model method.
Fractional-Order Fast Terminal Sliding Mode Control for a Class of Dynamical Systems
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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.
Multiobjective Optimization Design of a Fractional Order PID Controller for a Gun Control System
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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.
Robust Takagi-Sugeno fuzzy control for fractional order hydro-turbine governing system.
Wang, Bin; Xue, Jianyi; Wu, Fengjiao; Zhu, Delan
2016-11-01
A robust fuzzy control method for fractional order hydro-turbine governing system (FOHGS) in the presence of random disturbances is investigated in this paper. Firstly, the mathematical model of FOHGS is introduced, and based on Takagi-Sugeno (T-S) fuzzy rules, the generalized T-S fuzzy model of FOHGS is presented. Secondly, based on fractional order Lyapunov stability theory, a novel T-S fuzzy control method is designed for the stability control of FOHGS. Thirdly, the relatively loose sufficient stability condition is acquired, which could be transformed into a group of linear matrix inequalities (LMIs) via Schur complement as well as the strict mathematical derivation is given. Furthermore, the control method could resist random disturbances, which shows the good robustness. Simulation results indicate the designed fractional order T-S fuzzy control scheme works well compared with the existing method.
Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems
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Junhai Ma
2016-09-01
Full Text Available Based on the traditional IS–LM economic theory, which shows the relationship between interest rates and output in the goods and services market and the money market in macroeconomic. We established a four-dimensional IS–LM model involving four variables. With the Caputo fractional calculus theory, we improved it into a fractional order nonlinear model, analyzed the complexity and stability of the fractional order system. The existences conditions of attractors under different order conditions are compared, and obtain the orders when the system reaches a stable state. Have the detail analysis on the dynamic phenomena, such as the strange attractor, sensitivity to initial values through phase diagram and the power spectral. The order changes in two ways: orders changes synchronously or single order changes. The results show regardless of which the order situation is, the economic system will enter into multiple states, such as strong divergence, strange attractor and the convergence, finally, system will enter into the stable state under a certain order; parameter changes have similar effects on the economic system. Therefore, selecting an appropriate order is significant for an economic system, which guarantees a steady development. Furthermore, this paper construct the chaos control to IS–LM fractional-order macroeconomic model by means of linear feedback control method, by calculating and adjusting the feedback coefficient, we make the system return to the convergence state.
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.
Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems
Institute of Scientific and Technical Information of China (English)
Li Rui-Hong; Chen Wei-Sheng
2013-01-01
In this paper,the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated.The existence and uniqueness of solutions for this system are proved,and the stabilities of the equilibrium points are analyzed as one of the system parameters changes.The pitchfork bifurcation is discussed for the first time,and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived.The largest Lyapunov exponents and phase portraits are given to check the existence of chaos.Finally,the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable.Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.
Controlling Chaos for Fractional Order Loss Type of Coupled Dynamos Systems via Feedback
Hao, Jianhong; Xiong, Xueyan; Bin, Hong; Sun, Nayan
This paper studies the problem of chaos control for the fractional order modified coupled dynamos system that involves mechanical damping loss. Based on the Routh-Hurwitz criterion generalized to the fractional order stability theory, the stability conditions of the controlled system are discussed. We adopt a simple single-variable linear feedback method to suppress chaos to the unstable equilibrium point and limit cycle. Then, a modified feedback control method is developed in light of the sliding mode variable structure, namely exerting the controller only when the system trajectory is close to the target orbit. This method not only maintains the dynamics of the system, but provides the optimal control time and adjustable limit cycles radius. Numerical simulation proves the validity of this method.
A Color Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System
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Xia Huang
2014-12-01
Full Text Available In this paper, a new color image encryption algorithm based on a fractional-order hyperchaotic system is proposed. Firstly, four chaotic sequences are generated by a fractional-order hyperchaotic system. The parameters of such a system, together with the initial value, are regarded as the secret keys and the plain image is encrypted by performing the XOR and shufﬂing operations simultaneously. The proposed encryption scheme is described in detail with security analyses, including correlation analysis, histogram analysis, differential attacks, and key sensitivity analysis. Experimental results show that the proposed encryption scheme has big key space, and high sensitivity to keys properties, and resists statistical analysis and differential attacks, so it has high security and is suitable for color image encryption.
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.
Homotopy perturbation method for nonlinear partial differential equations of fractional order
Energy Technology Data Exchange (ETDEWEB)
Momani, Shaher [Department of Mathematics and Physics, Qatar University (Qatar)]. E-mail: shahermm@yahoo.com; Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa' Applied University, Salt (Jordan)]. E-mail: odibat@bau.edu.jo
2007-06-11
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.
Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems
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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.
The modified simple equation method for solving some fractional-order nonlinear equations
Indian Academy of Sciences (India)
KAPLAN MELIKE; BEKIR AHMET
2016-07-01
Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation method.
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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.
Institute of Scientific and Technical Information of China (English)
Wang Dong-Feng; Zhang Jin-Ying; Wang Xiao-Yan
2013-01-01
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.
The adaptive synchronization of fractional-order Liu chaotic system with unknown parameters
Indian Academy of Sciences (India)
NOURIAN ADELEH; BALOCHIAN SAEED
2016-06-01
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 and a proper Lyapunov candidate function, the unknown coefficients of the system are estimated and the stabilization of the synchronizer system is demonstrated. Finally, the numerical simulation illustrates the efficiency of the proposed method in synchronizing two chaotic systems.
Song, Jia; Wang, Lun; Cai, Guobiao; Qi, Xiaoqiang
2015-06-01
Near space hypersonic vehicle model is nonlinear, multivariable and couples in the reentry process, which are challenging for the controller design. In this paper, a nonlinear fractional order proportion integral derivative (NFOPIλDμ) active disturbance rejection control (ADRC) strategy based on a natural selection particle swarm (NSPSO) algorithm is proposed for the hypersonic vehicle flight control. The NFOPIλDμ ADRC method consists of a tracking-differentiator (TD), an NFOPIλDμ controller and an extended state observer (ESO). The NFOPIλDμ controller designed by combining an FOPIλDμ method and a nonlinear states error feedback control law (NLSEF) is to overcome concussion caused by the NLSEF and conversely compensate the insufficiency for relatively simple and rough signal processing caused by the FOPIλDμ method. The TD is applied to coordinate the contradiction between rapidity and overshoot. By attributing all uncertain factors to unknown disturbances, the ESO can achieve dynamic feedback compensation for these disturbances and thus reduce their effects. Simulation results show that the NFOPIλDμ ADRC method can make the hypersonic vehicle six-degree-of-freedom nonlinear model track desired nominal signals accurately and fast, has good stability, dynamic properties and strong robustness against external environmental disturbances.
Guaranteed Cost Finite-Time Control of Fractional-Order Positive Switched Systems
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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.
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
Institute of Scientific and Technical Information of China (English)
Si Gang-Quan; Sun Zhi-Yong; Zhang Yan-Bin
2011-01-01
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.
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Muhammad Asif Zahoor Raja
2011-01-01
Full Text Available A stochastic technique has been developed for the solution of fractional order system represented by Bagley-Torvik equation. The mathematical model of the equation was developed with the help of feed-forward artificial neural networks. The training of the networks was made with evolutionary computational intelligence based on genetic algorithm hybrid with pattern search technique. Designed scheme was successfully applied to different forms of the equation. Results are compared with standard approximate analytic, stochastic numerical solvers and exact solutions.
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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.
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.
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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.
Delayed feedback control of unstable steady states in fractional-order chaotic systems
Gjurchinovski, Aleksandar; Urumov, Viktor
2010-01-01
We study the possibility to stabilize unstable steady states in chaotic fractional-order dynamical systems by the time-delayed feedback method with both constant and time-varying delays. By performing a linear stability analysis in the constant delay case, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parametrizad by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. It is shown numerically that delayed feedback control with a variable time-delay significantly enlarges the stability region of the steady states in comparison to the classical time-delayed feedback scheme with a constant delay.
Criteria for Response Monotonicity Preserving in Approximation of Fractional Order Systems
Institute of Scientific and Technical Information of China (English)
Mohammad Saleh Tavazoei
2016-01-01
In approximation of fractional order systems,a significant objective is to preserve the important properties of the original system.The monotonicity of time/frequency responses is one of these properties whose preservation is of great importance in approximation process.Considering this importance,the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper.In these investigations,some conditions on approximating filters of fractional operators are found to guarantee the preservation of step/magnitude-frequency response monotonicity in approximation process.These conditions are also simplified in some special cases.In addition,numerical simulation results are presented to show the usefulness of the obtained conditions.
Energy Technology Data Exchange (ETDEWEB)
Liu, Xiaojun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China); School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001 (China); Hong, Ling, E-mail: hongling@mail.xjtu.edu.cn; Jiang, Jun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China)
2016-08-15
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Liu, Xiaojun; Hong, Ling; Jiang, Jun
2016-08-01
Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.
Zahra Yaghoubi; Hassan Zarabadipour
2012-01-01
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 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 effecti...
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.
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Xueliang Liu
2012-01-01
Full Text Available This paper is concerned with a containment problem of networked fractional-order system with multiple leaders under a fixed directed interaction graph. Based on the neighbor rule, a distributed protocol is proposed in delayed communication channels. By employing the algebraic graph theory, matrix theory, Nyquist stability theorem, and frequency domain method, it is analytically proved that the whole follower agents will flock to the convex hull which is formed by the leaders. Furthermore, a tight upper bound on the communication time-delay that can be tolerated in the dynamic network is obtained. As a special case, the interconnection topology under the undirected case is also discussed. Finally, some numerical examples with simulations are presented to demonstrate the effectiveness and correctness of the theoretical results.
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...
Indian Academy of Sciences (India)
Kumar Vishal; Saurabh K Agrawal; Subir Das
2016-01-01
In this paper, we have discussed the local stability of the Mathieu–van der Pol hyperchaotic system with the fractional-order derivative. The fractional Routh–Hurwitz stability conditions were provided and were used to discuss the stability. Feedback control method was used to control chaos in the Mathieu–van der Pol system with fractional-order derivative and after controlling the chaotic behaviour of the system the synchronization between the fractional-order hyperchaotic Mathieu–van der Pol system and controlled system was introduced. In this study, modified adaptive control methods with uncertain parameters at various equilibrium points were used. Also the analysis of control time with respect to different fractional-order derivatives is the key feature of this paper. Numerical simulation results achieved using Adams–Boshforth–Moulton method show that the method is effective and reliable.
Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach
Radek Matušů; Roman Prokop
2015-01-01
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 qua...
A new color image encryption scheme using CML and a fractional-order chaotic system.
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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.
A new color image encryption scheme using CML and a fractional-order chaotic system.
Wu, Xiangjun; Li, Yang; Kurths, Jürgen
2015-01-01
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.
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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.
Institute of Scientific and Technical Information of China (English)
曾庆山; 曹广益; 朱新坚
2004-01-01
The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov's stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.
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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.
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Yongjun Shen
2015-01-01
Full Text Available The single degree-of-freedom (SDOF system under the control of three semiactive methods is analytically studied in this paper, where a fractional-order derivative is used in the mathematical model. The three semiactive control methods are on-off control, limited relative displacement (LRD control, and relative control, respectively. The averaging method is adopted to provide an analytical study on the performance of the three different control methods. Based on the comparison between the analytical solutions with the numerical ones, it could be proved that the analytical solutions are accurate enough. The effects of the fractional-order parameters on the control performance, especially the relative and absolute displacement transmissibility, are analyzed. The research results indicate that the steady-state amplitudes of the three semiactive systems with fractional-order derivative in the model could be significantly reduced and the control performance can be greatly improved.
Maiti, Deepyaman; Konar, Amit
2008-01-01
This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The said simultaneous equations are generated and updated using particle swarm optimization (PSO) technique, the fitness function being the sum of squared deviations from the actual set of observations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.
Maiti, Deepyaman; Janarthanan, R; Konar, Amit
2008-01-01
This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.
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.
Luo, Chao; Wang, Xingyuan
2013-04-01
In this paper, a novel dynamic system, the fractional-order complex Chen system, is presented for the first time. Dynamic behaviors of system are studied analytically and numerically. Different routes to chaos are shown, and diverse kinds of motions are identified and exhibited by means of bifurcation diagram, portrait phase and the largest Lyapunov exponent. Secondly, an application to digital secure communication based on the novel system is proposed, in which security is enhanced by continually switching different orders of derivative in an irregular pattern. Furthermore, making full use of the advantage of high-capacity transmission of complex system, the improved digital secure communication scheme is achieved based on hybrid synchronization in coupled fractional-order complex Chen system, that means anti-synchronization in real part of state variables and projective synchronization in imaginary part, respectively. The corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed schemes.
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.
Energy Technology Data Exchange (ETDEWEB)
Wu, Yimin [School of Mathematics and Statistics, Suzhou University, Suzhou 234000 (China); Lv, Hui, E-mail: lvhui207@gmail.com [Department of Applied Mathematics, Huainan Normal University, Huainan 232038 (China)
2016-08-15
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.
Yu, Zhiyong; Jiang, Haijun; Hu, Cheng; Yu, Juan
2017-03-27
In this paper, the consensus of fractional-order multiagent systems (FOMASs) is considered via sampled-data control over directed communication topology with the order 0 system. Moreover, for the network with a dynamic leader, the sampling period, the coupling gain, and the spectrum of the Laplacian matrix are carefully devised, respectively. Finally, several simulation examples are employed to validate the effectiveness of the theoretical results.
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.
Energy Technology Data Exchange (ETDEWEB)
Melicio, R.; Catalao, J.P.S. [Department of Electromechanical Engineering, University of Beira Interior, R. Fonte do Lameiro, 6201-001 Covilha (Portugal); Mendes, V.M.F. [Department of Electrical Engineering and Automation, Instituto Superior de Engenharia de Lisboa, R. Conselheiro Emidio Navarro, 1950-062 Lisbon (Portugal)
2010-06-15
This paper presents a new integrated model for the simulation of wind energy systems. The proposed model is more realistic and accurate, considering a variable-speed wind turbine, two-mass rotor, permanent magnet synchronous generator (PMSG), different power converter topologies, and filters. Additionally, a new control strategy is proposed for the variable-speed operation of wind turbines with PMSG/full-power converter topology, based on fractional-order controllers. Comprehensive simulation studies are carried out with matrix and multilevel power converter topologies, in order to adequately assert the system performance in what regards the quality of the energy injected into the electric grid. Finally, conclusions are duly drawn. (author)
Dar, Mohammad Rafiq; Kant, Nasir Ali; Khanday, Farooq Ahmad
In this paper, electronic implementation of fractional-order Rössler system using operational transconductance amplifiers (OTAs) is presented which until now was only being investigated through numerical simulations. The realization offers the benefits of low-voltage implementation, integrability and electronic tunability. In addition, the proposed circuit is a MOS only design (as no BJTs have been used) which contains only grounded components and is therefore suitable for monolithic VLSI design. The chaotic behavior of the fractional-order Rössler system in consideration with the incommensurate orders has been demonstrated which finds many applications in several fields. The theoretical predictions of the proposed implementation have been verified through experimentation and HSPICE simulator using Austrian Micro System (AMS) 0.35μm CMOS process and the obtained results have been found in good agreement with the Matlab simulink theoretical results obtained using FOMCON simulink toolbox. Besides, a secure message communication system has been considered to demonstrate fully the usefulness of the chaotic system.
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.
Uniform Stability of a Class of Fractional-Order Nonautonomous Systems with Multiple Time Delays
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Tao Zou
2014-01-01
sufficient condition is established for the existence and uniqueness of solutions for such systems involving Caputo fractional derivative, and the uniform stability of solution is studied. At last, two examples are given to demonstrate the applicability of our results.
A numerical investigation for robust stability of fractional-order uncertain systems.
Senol, Bilal; Ates, Abdullah; Alagoz, B Baykant; Yeroglu, Celaleddin
2014-03-01
This study presents numerical methods for robust stability analysis of closed loop control systems with parameter uncertainty. Methods are based on scan sampling of interval characteristic polynomials from the hypercube of parameter space. Exposed-edge polynomial sampling is used to reduce the computational complexity of robust stability analysis. Computer experiments are used for demonstration of the proposed robust stability test procedures.
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.
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.
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Fei Gao
2013-01-01
Full Text Available In this paper, a non-Lyapunov novel approach is proposed to estimate the unknown parameters and orders together for noncommensurate and hyper fractional chaotic systems based on cuckoo search oriented statistically by the differential evolution (CSODE. Firstly, a novel Gaos’ mathematical model is proposed and analyzed in three submodels, not only for the unknown orders and parameters’ identification but also for systems’ reconstruction of fractional chaos systems with time delays or not. Then the problems of fractional-order chaos’ identification are converted into a multiple modal nonnegative functions’ minimization through a proper translation, which takes fractional-orders and parameters as its particular independent variables. And the objective is to find the best combinations of fractional-orders and systematic parameters of fractional order chaotic systems as special independent variables such that the objective function is minimized. Simulations are done to estimate a series of noncommensurate and hyper fractional chaotic systems with the new approaches based on CSODE, the cuckoo search, and Genetic Algorithm, respectively. The experiments’ results show that the proposed identification mechanism based on CSODE for fractional orders and parameters is a successful method for fractional-order chaotic systems, with the advantages of high precision and robustness.
Institute of Scientific and Technical Information of China (English)
ZHANG Dong-Li; TANG Ying-Gan; GUAN Xin-Ping
2014-01-01
Fractional order proportional-integral-derivative (FOPID) controller generalizes the standard PID controller. Compared to PID controller, FOPID controller has more pa-rameters and the tuning of parameters is more complex. In this paper, an improved artificial bee colony algorithm, which com-bines cyclic exchange neighborhood with chaos (CNC-ABC), is proposed for the sake of tuning the parameters of FOPID con-troller. The characteristic of the proposed CNC-ABC exists in two folds: one is that it enlarges the search scope of the solution by utilizing cyclic exchange neighborhood techniques, speeds up the convergence of artificial bee colony algorithm (ABC). The other is that it has potential to get out of local optima by exploit-ing the ergodicity of chaos. The proposed CNC-ABC algorithm is used to optimize the parameters of the FOPID controller for an automatic voltage regulator (AVR) system. Numerical sim-ulations show that the CNC-ABC FOPID controller has better performance than other FOPID and PID controllers.
Boundary Controllability of Nonlinear Fractional Integrodifferential Systems
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Ahmed HamdyM
2010-01-01
Full Text Available Sufficient conditions for boundary controllability of nonlinear fractional integrodifferential systems in Banach space are established. The results are obtained by using fixed point theorems. We also give an application for integropartial differential equations of fractional order.
Institute of Scientific and Technical Information of China (English)
Ke Xiao; Shang-Bo Zhou; Wei-Wei Zhang
2008-01-01
For a general nonlinear fractional-orderdifferential equation, the numerical solution is a goodway to approximate the trajectory of such systems. Inthis paper, a novel algorithm for numerical solution offractional-order differential equations based on thedefinition of Grunwald-Letnikov is presented. Theresults of numerical solution by using the novel methodand the frequency-domain method are compared, and the limitations of frequency-domain method arediscussed.
2013-01-01
Identification of the unknown parameters and orders of fractional chaotic systems is of vital significance in controlling and synchronization of fractional-order chaotic systems. In this paper, a non-Lyapunov novel approach is proposed to estimate the unknown parameters and orders together for non-commensurate and hyper fractional chaotic systems based on cuckoo search oriented statistically the differential evolution (CSODE). Firstly, a novel Gao's mathematical model is put and analysed in t...
Indian Academy of Sciences (India)
L M WANG
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.
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.
Laplace transform of fractional order differential equations
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Song Liang
2015-05-01
Full Text Available In this article, we show that Laplace transform can be applied to fractional system. To this end, solutions of linear fractional-order equations are first derived by a direct method, without using Laplace transform. Then the solutions of fractional-order differential equations are estimated by employing Gronwall and Holder inequalities. They are showed be to of exponential order, which are necessary to apply the Laplace transform. Based on the estimates of solutions, the fractional-order and the integer-order derivatives of solutions are all estimated to be exponential order. As a result, the Laplace transform is proved to be valid in fractional equations.
Fractional Order Models of Industrial Pneumatic Controllers
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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.
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Hammad Khalil
2016-06-01
Full Text Available In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear under $\\hat{m}$-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the boundary value problem to a system of easily solvable algebraic equations. We show the applicability of the scheme by solving some test problems. The scheme is computer oriented.
Digital implementation of fractional order PID controller and its application
Institute of Scientific and Technical Information of China (English)
Wang Zhenbin; Wang Zhenlei; Cao Guangyi; Zhu Xinjian
2005-01-01
A new discretization scheme is proposed for the design of a fractional order PID controller. In the design of a fractional order controller the interest is mainly focused on the s-domain, but there exists a difficult problem in the s-domain that needs to be solved, i.e. how to calculate fractional derivatives and integrals efficiently and quickly. Our scheme adopts the time domain that is well suited for Z-transform analysis and digital implementation. The main idea of the scheme is based on the definition of Grunwald-Letnicov fractional calculus. In this case some limited terms of the definition are taken so that it is much easier and faster to calculate fractional derivatives and integrals in the time domain or zdomain without loss much of the precision. Its effectiveness is illustrated by discretization of half-order fractional differential and integral operators compared with that of the analytical scheme. An example of designing fractional order digital controllers is included for illustration, in which different fractional order PID controllers are designed for the control of a nonlinear dynamic system containing one of the four different kinds of nonlinear blocks: saturation, deadzone, hysteresis, and relay.
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.
LMI Conditions for Global Stability of Fractional-Order Neural Networks.
Zhang, Shuo; Yu, Yongguang; Yu, Junzhi
2016-08-02
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.
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.
Research on Modeling of Hydropneumatic Suspension Based on Fractional Order
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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.
Development of a Novel Fractional Order Sliding Mode Controller for a Gun
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Qiang Gao
2013-09-01
Full Text Available To solve the nonlinearity phenomenon of a Gun Control System (GCS, a novel Fractional order Sliding Mode Control (FoSMC strategy is proposed in this study. By inducing the fractional order calculus, a Fractional Order PID (FOPID type sliding surface is especially designed and consequently an equivalent control discipline with fractional order dynamics is induced. The saturation function is employed as the switch function. By numerical simulation, the dynamic characteristics of the FoSMC based control system are analyzed and compared with Conventional Sliding Mode Control (CSMC system. The results demonstrate that the FoSMC system could reach up to the equilibrium state more smoothly, which shall significantly suppress the inherent chatter effects. Besides, the FoSMC based gun control system is of high response rate, better positioning accuracy and high robustness, which is suitable for fast, smooth and accurate adjustments of the gun.
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.
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.
Research on Modeling of Hydropneumatic Suspension Based on Fractional Order
Junwei Zhang; Sizhong Chen; Yuzhuang Zhao; Jianbo Feng; Chang Liu; Ying Fan
2015-01-01
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 considerin...
Synchronization of fractional order complex dynamical networks
Wang, Yu; Li, Tianzeng
2015-06-01
In this letter the synchronization of complex dynamical networks with fractional order chaotic nodes is studied. A fractional order controller for synchronization of complex network is presented. Some new sufficient synchronization criteria are proposed based on the Lyapunov stability theory and the LaSalle invariance principle. These synchronization criteria can apply to an arbitrary fractional order complex network in which the coupling-configuration matrix and the inner-coupling matrix are not assumed to be symmetric or irreducible. It means that this method is more general and effective. Numerical simulations of two fractional order complex networks demonstrate the universality and the effectiveness of the proposed method.
Seider, Warren D.; Ungar, Lyle H.
1987-01-01
Describes a course in nonlinear mathematics courses offered at the University of Pennsylvania which provides an opportunity for students to examine the complex solution spaces that chemical engineers encounter. Topics include modeling many chemical processes, especially those involving reaction and diffusion, auto catalytic reactions, phase…
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.
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.
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.
Institute of Scientific and Technical Information of China (English)
2013-01-01
In this paper, from the stability theory of fractional-order chaotic system, a kind of dislocated projective synchronization for fractional-order Chua’s system is successfully completed through a nonlinear controller. Meanwhile, the fractional-order unit circuit is designed, according to the series-parallel structure of resistor-capacitor and the approximate linear transfer function expression for the complex frequency domain. Thus, non-inductive modular circuit of dislocated projective synchronization of fractional-order Chua’s system is realized. The circuit simulation results prove the feasibility of the scheme. Furthermore, the method can be applied in secure communication through the improved chaotic masking. The information signal can be concealed and recovered. Numerical simulation results show the effectiveness of the proposed method.% 基于分数阶混沌系统稳定性理论，设计高效的非线性控制器，实现初始值不同的两个分数阶 Chua’s 系统错位投影同步。根据分数阶复频域近似方法，提出分数阶系统的等效电路，实现分数阶 Chua’s 系统错位投影同步的无感模块化电路。最后，利用改进的混沌掩盖通信原理，将以上同步方案应用于混沌保密通信中，在发送端使用分数阶混沌序列对有用信号加密传送，从接收端可以无失真地恢复出有用信号。数值仿真与电路仿真证实了提出方案的可行性。
Electronically Tunable Fractional Order All Pass Filter
Verma, Rakesh; Pandey, Neeta; Pandey, Rajeshwari
2017-08-01
In this paper, an electronically tunable fractional order all pass filter (FOAPF) based on operational transconductance amplifier (OTA) is presented. It uses two OTAs and single fractional order capacitor (FC) of non-integer order α to provide FOAPF of α order. Two different values of α, in particular 0.5 and 0.9, for FC are taken for investigation. The functionality of the proposal is verified through SPICE simulations using TSMC 0.18 μm Complementary Metal Oxide Semiconductor (CMOS) process parameters. Simulated and theoretical frequency and time domain responses are found to be in close agreement.
Mathematical modelling of fractional order circuits
Moreles, Miguel Angel
2016-01-01
In this work a classical derivation of fractional order circuits models is presented. Generalized constitutive equations in terms of fractional Riemann-Liouville derivatives are introduced in the Maxwell's equations. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. A fractional differential equation model is obtained with Caputo derivatives. Thus standard initial conditions apply.
Keshtkar, F.; Erjaee, G.; Boutefnouchet, M.
2014-01-01
In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts. NPRP . Grant Number: NP...
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...
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...
Adaptive Fractional Fuzzy Sliding Mode Control for Multivariable Nonlinear Systems
Directory of Open Access Journals (Sweden)
Junhai Luo
2014-01-01
Full Text Available This paper presents a robust adaptive fuzzy sliding mode control method for a class of uncertain nonlinear systems. The fractional order calculus is employed in the parameter updating stage. The underlying stability analysis as well as parameter update law design is carried out by Lyapunov based technique. In the simulation, two examples including a comparison with the traditional integer order counterpart are given to show the effectiveness of the proposed method. The main contribution of this paper consists in the control performance is better for the fractional order updating law than that of traditional integer order.
General Projective Synchronization and Fractional Order Chaotic Masking Scheme
Institute of Scientific and Technical Information of China (English)
Shi-Quan Shao
2008-01-01
In this paper, a fractional order chaotic masking scheme used for secure communication is introduced. Based on the general projective synchronization of two coupled fractional Chert systems, a popular masking scheme is designed. Numerical example is given to demonstrate the effectiveness of the proposed method.
General Projective Synchronization and Fractional Order Chaotic Masking Scheme
Institute of Scientific and Technical Information of China (English)
Shi-Quan Shao
2008-01-01
In this paper, a fractional order chaoticmasking scheme used for secure communication isintroduced. Based on the general projectivesynchronization of two coupled fractional Chen systems,a popular masking scheme is designed. Numericalexample is given to demonstrate the effectiveness of theproposed method.
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.
一类分数阶混沌金融系统的复杂性演化研究%Complexity evolvement of a chaotic fractional-order financial system
Institute of Scientific and Technical Information of China (English)
辛宝贵; 陈通; 刘艳芹
2011-01-01
A novel science branch, econophysics, is set up because various physical theories and methods are applied to economic and financial fields. More and more researchers are fascinated by the complex dynamical behavior of the fractional-order dynamical system. The paper analyzes the stability of the fractional-order financial system, and then simulates the generalized model complexity with Adams-Bashforth-Moulton predictor-corrector scheme by using bifurcation diagram, phase portrait and history time-series.%物理学理论与方法在经济与金融领域中的成功应用催生了一个新的科学分支--经济物理学(econophysics).分数阶微积分系统的复杂动力学现象受到了越来越多学者的关注.本文定性地分析一类分数阶混沌金融系统的均衡解的稳定性及Hopf分岔发生的条件,并运用亚当斯-巴什福斯-莫尔顿预估-校正的有限差分法,通过分岔图、相图和时间序列图对该系统的复杂性演化行为进行仿真研究.
Fractional-order integral and derivative controller for temperature proﬁle tracking
Indian Academy of Sciences (India)
Hyo-Sung Ahn; Varsha Bhambhani; YangQuan Chen
2009-10-01
This paper establishes a new strategy to tune a fractional order integral and derivative (ID) controller satisfying gain and phase margins based on Bode’s ideal transfer function as a reference model, for a temperature proﬁle tracking. A systematic analysis resulting in a non-linear equation relating user-deﬁned gain and phase margins to the fractional order controller is derived. The closed-loop system designed has a feature of robustness to gain variations with step responses exhibiting a nearly iso-damping property. This paper aims to apply the analytical tuning procedure to control the heat ﬂow systems at selected points in Quanser experimental platform. Thus, the main purpose of this paper is to examine performances of two different fractional order controllers in temperature proﬁle tracking. From experimental comparisons with the traditional PI/PID controller based on Ziegler–Nichols’ tuning method, it will be shown that the proposed mathodologies are speciﬁcally beneﬁcial in controlling temperature in time-delay heat ﬂow systems.
Energy Technology Data Exchange (ETDEWEB)
Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa' Applied University, Salt 19117 (Jordan)], E-mail: odibat@bau.edu.jo; Momani, Shaher [Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak (Jordan)], E-mail: shahermm@yahoo.com
2008-04-15
In this paper, a modification of He's homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p element of [0, 1] is constructed. The proposed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order.
Electroviscoelasticity of liquid/liquid interfaces: fractional-order model.
Spasic, Aleksandar M; Lazarevic, Mihailo P
2005-02-01
A number of theories that describe the behavior of liquid-liquid interfaces have been developed and applied to various dispersed systems, e.g., Stokes, Reiner-Rivelin, Ericksen, Einstein, Smoluchowski, and Kinch. A new theory of electroviscoelasticity describes the behavior of electrified liquid-liquid interfaces in fine dispersed systems and is based on a new constitutive model of liquids. According to this model liquid-liquid droplet or droplet-film structure (collective of particles) is considered as a macroscopic system with internal structure determined by the way the molecules (ions) are tuned (structured) into the primary components of a cluster configuration. How the tuning/structuring occurs depends on the physical fields involved, both potential (elastic forces) and nonpotential (resistance forces). All these microelements of the primary structure can be considered as electromechanical oscillators assembled into groups, so that excitation by an external physical field may cause oscillations at the resonant/characteristic frequency of the system itself (coupling at the characteristic frequency). Up to now, three possible mathematical formalisms have been discussed related to the theory of electroviscoelasticity. The first is the tension tensor model, where the normal and tangential forces are considered, only in mathematical formalism, regardless of their origin (mechanical and/or electrical). The second is the Van der Pol derivative model, presented by linear and nonlinear differential equations. Finally, the third model presents an effort to generalize the previous Van der Pol equation: the ordinary time derivative and integral are now replaced with the corresponding fractional-order time derivative and integral of order p<1.
Wang, Yaoyao; 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.
Controllability in nonlinear systems
Hirschorn, R. M.
1975-01-01
An explicit expression for the reachable set is obtained for a class of nonlinear systems. This class is described by a chain condition on the Lie algebra of vector fields associated with each nonlinear system. These ideas are used to obtain a generalization of a controllability result for linear systems in the case where multiplicative controls are present.
Institute of Scientific and Technical Information of China (English)
李志民; 孙其振; 孙勇; 韩绪鹏
2012-01-01
在总结分析基于永磁同步发电机PMSG(permanent magnet synchronous generator)的风力发电系统数学模型的基础上,以PMSG风力发电系统的最大风能追踪为控制目标,引入分数阶PIλ控制理论,设计了基于分数阶pIλ的PMSG风力发电系统控制方案,采用遗传算法对分数阶PIλ控制器的参数进行优化整定.在Matlab/Simulink软件平台上建立了PMSG风力发电系统的仿真模型,并与传统整数阶PI控制进行了对比.采用分数阶PIλ控制方案使风力机在额定风速以下能够有效地保持最佳叶尖速比运行,系统控制性能得到有效提高,有较强的鲁棒性.仿真结果证明了该方案的可行性与正确性.%On the basis of analyzing the mathematical model of PMSG (permanent magnet synchronous generator) wind power generation system, to realize the control target of the maximum wind energy tracking for PMSG wind power system, a novel control theory of fractional order PI1 control is proposed and the PMSG wind power system control scheme is designed based on the fractional order PP. The genetic algorithm is applied to optimize and set the parameters of the design. A simulation model of PMSG wind power control system is established based on the Matlab/Simulink software platform. The novel fractional order PP control method is compared with the classical integer order PI control method. The proposed PP control method can effectively keep optimum tip-speed ratio under rated wind speed and the system control performance is improved with good robustness. The simulation results verify the validity and feasibility of the proposed control method.
Fractional Order AGC for Distributed Energy Resources Using Robust Optimization
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.
Design and implementation of grid multi-scroll fractional-order chaotic attractors.
Chen, Liping; Pan, Wei; Wu, Ranchao; Tenreiro Machado, J A; Lopes, António M
2016-08-01
This paper proposes a novel approach for generating multi-scroll chaotic attractors in multi-directions for fractional-order (FO) systems. The stair nonlinear function series and the saturated nonlinear function are combined to extend equilibrium points with index 2 in a new FO linear system. With the help of stability theory of FO systems, stability of its equilibrium points is analyzed, and the chaotic behaviors are validated through phase portraits, Lyapunov exponents, and Poincaré section. Choosing the order 0.96 as an example, a circuit for generating 2-D grid multiscroll chaotic attractors is designed, and 2-D 9 × 9 grid FO attractors are observed at most. Numerical simulations and circuit experimental results show that the method is feasible and the designed circuit is correct.
Design and implementation of grid multi-scroll fractional-order chaotic attractors
Energy Technology Data Exchange (ETDEWEB)
Chen, Liping, E-mail: lip-chenhut@126.com; Pan, Wei [School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009 (China); Wu, Ranchao [School of Mathematics, Anhui University, Hefei 230039 (China); Tenreiro Machado, J. A. [Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto (Portugal); Lopes, António M. [UISPA–LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto (Portugal)
2016-08-15
This paper proposes a novel approach for generating multi-scroll chaotic attractors in multi-directions for fractional-order (FO) systems. The stair nonlinear function series and the saturated nonlinear function are combined to extend equilibrium points with index 2 in a new FO linear system. With the help of stability theory of FO systems, stability of its equilibrium points is analyzed, and the chaotic behaviors are validated through phase portraits, Lyapunov exponents, and Poincaré section. Choosing the order 0.96 as an example, a circuit for generating 2-D grid multiscroll chaotic attractors is designed, and 2-D 9 × 9 grid FO attractors are observed at most. Numerical simulations and circuit experimental results show that the method is feasible and the designed circuit is correct.
Lugiato, Luigi; Brambilla, Massimo
2015-01-01
Guiding graduate students and researchers through the complex world of laser physics and nonlinear optics, this book provides an in-depth exploration of the dynamics of lasers and other relevant optical systems, under the umbrella of a unitary spatio-temporal vision. Adopting a balanced approach, the book covers traditional as well as special topics in laser physics, quantum electronics and nonlinear optics, treating them from the viewpoint of nonlinear dynamical systems. These include laser emission, frequency generation, solitons, optically bistable systems, pulsations and chaos and optical pattern formation. It also provides a coherent and up-to-date treatment of the hierarchy of nonlinear optical models and of the rich variety of phenomena they describe, helping readers to understand the limits of validity of each model and the connections among the phenomena. It is ideal for graduate students and researchers in nonlinear optics, quantum electronics, laser physics and photonics.
Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay
Directory of Open Access Journals (Sweden)
S. J. Sadati
2010-01-01
Full Text Available Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputo's derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.
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.
Nonlinear systems in medicine.
Higgins, John P
2002-01-01
Many achievements in medicine have come from applying linear theory to problems. Most current methods of data analysis use linear models, which are based on proportionality between two variables and/or relationships described by linear differential equations. However, nonlinear behavior commonly occurs within human systems due to their complex dynamic nature; this cannot be described adequately by linear models. Nonlinear thinking has grown among physiologists and physicians over the past century, and non-linear system theories are beginning to be applied to assist in interpreting, explaining, and predicting biological phenomena. Chaos theory describes elements manifesting behavior that is extremely sensitive to initial conditions, does not repeat itself and yet is deterministic. Complexity theory goes one step beyond chaos and is attempting to explain complex behavior that emerges within dynamic nonlinear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in health and in disease states.
Adaptive Fractional Fuzzy Sliding Mode Control for Multivariable Nonlinear Systems
Junhai Luo; Heng Liu
2014-01-01
This paper presents a robust adaptive fuzzy sliding mode control method for a class of uncertain nonlinear systems. The fractional order calculus is employed in the parameter updating stage. The underlying stability analysis as well as parameter update law design is carried out by Lyapunov based technique. In the simulation, two examples including a comparison with the traditional integer order counterpart are given to show the effectiveness of the proposed method. The main contribution of th...
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.
DEFF Research Database (Denmark)
Jørgensen, Michael Finn
1995-01-01
It is generally very difficult to solve nonlinear systems, and such systems often possess chaotic solutions. In the rare event that a system is completely solvable, it is said to integrable. Such systems never have chaotic solutions. Using the Inverse Scattering Transform Method (ISTM) two...
Chaos control via a simple fractional-order controller
Energy Technology Data Exchange (ETDEWEB)
Tavazoei, Mohammad Saleh [Advanced Control System Lab., Electrical Engineering Department, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Haeri, Mohammad [Advanced Control System Lab., Electrical Engineering Department, Sharif University of Technology, Tehran (Iran, Islamic Republic of)], E-mail: haeri@sina.sharif.edu
2008-02-04
In this Letter, we propose a fractional-order controller to stabilize the unstable fixed points of an unstable open-loop system. Also, we show that this controller has strong ability to eliminate chaotic oscillations or reduce them to regular oscillations in the chaotic systems. This controller has simple structure and is designed very easily. 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 uncertain chaotic systems.
On Fractional Order Hybrid Differential Equations
Directory of Open Access Journals (Sweden)
Mohamed A. E. Herzallah
2014-01-01
Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Synchronization of fractional-order complex-valued neural networks with time delay.
Bao, Haibo; Park, Ju H; Cao, Jinde
2016-09-01
This paper deals with the problem of synchronization of fractional-order complex-valued neural networks with time delays. By means of linear delay feedback control and a fractional-order inequality, sufficient conditions are obtained to guarantee the synchronization of the drive-response systems. Numerical simulations are provided to show the effectiveness of the obtained results.
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.
Study on the Nonsingular Problem of Fractional-Order Terminal Sliding Mode Control
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Kening Li
2013-01-01
Full Text Available An improved type of control strategy combining the fractional calculus with nonsingular terminal sliding mode control named non-singular fractional terminal sliding mode control (NFOTSM is proposed for the nonlinear tire-road friction control system of vehicle in this paper. A fractional-order switching manifold is proposed, and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed controller ensures the finite time stability of the closed-loop system. Then, a terminal attractor is introduced to solve the singularity problem of fractional terminal sliding mode control (FOTSM. Finally, the performance of the NFOTSM is fully investigated compared with other related algorithms to find the effectiveness for the tire-road friction system. The results show that the NFOTSM has better performance than other related algorithms.
Institute of Scientific and Technical Information of China (English)
杨红; 王瑞
2011-01-01
According to the stability of fractional order linear systems theory, the system is decomposed into stable linear parts and the corresponding nonlinear parts. The active controller is designed to compensate the nonlinear parts, and the fractional order chaotic system is suppressed to an equilibrium point. In order to improve the compensation ability of active controller, a multiple least square support vector machine （M-LS-SVM） regression model is presented based on feedback. The subtractive clustering is adopted to divide the input space into several sub-spaces, and sub-models are built by a LS- SVM in each sub-space. In order to minimize the severe correlation among sub-models and to improve the accuracy and the robustness of the model, the sub-models are combined by principal component regression （PCR）. The experiment result shows that by using the method the compensation accuracy and the system response indices can be improved.%根据分数阶线性系统的稳定理论，将混沌系统分成稳定的线性部分和相应的非线性部分．设计主动控制器，对非线性部分进行补偿，从而将分数阶混沌系统控制到平衡点．为了提高主动控制器的补偿能力，提出基于反馈的多最小二乘支持向量机（M-LS-SVM）拟合模型．通过减聚类方法将输入空间划分为一些小的局部空间，在每个局部空间中用LS-SVM建立子模型．为解决子模型相互之间的严重相关问题，提高模型的精度和鲁棒性，各个子模型的预测输出通过主元递归（PCR）方法连接．仿真实验表明该方法有助于提高补偿精度和系统响应指标．
Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment
Liu, Da Yan
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.
A New Model of the Fractional Order Dynamics of the Planetary Gears
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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.
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
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Ammar SOUKKOU
2016-07-01
Full Text Available This paper will establish the importance and significance of studying the fractional-order control of nonlinear dynamical systems. The foundation and the sources related to this research scope is going to be set. Then, the paper incorporates a brief overview on how this study is performed and present the organization of this study. The present work investigates the effectiveness of the physical-fractional and biological-genetic operators to develop an Optimal Form of Fractional-order PID Controller (O2Fo-PIDC. The newly developed Fo-PIDC with optimal structure and parameters can, also, improve the performances required in the modeling and control of modern manufacturing-industrial process (MIP. The synthesis methodology of the proposed O2Fo-PIDC can be viewed as a multi-level design approach. The hierarchical Multiobjective genetic algorithm (MGA, adopted in this work, can be visualized as a combination of structural and parametric genes of a controller orchestrated in a hierarchical fashion. Then, it is applied to select an optimal structure and knowledge base of the developed fractional controller to satisfy the various design specification contradictories (simplicity, accuracy, stability and robustness.
SOLVING FRACTIONAL-ORDER COMPETITIVE LOTKA-VOLTERRA MODEL BY NSFD SCHEMES
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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.
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.
Projective synchronization of a complex network with different fractional order chaos nodes
Institute of Scientific and Technical Information of China (English)
wang Ming-Jun; wang Xing-Yuan; Niu Yu-Jun
2011-01-01
Based on the stability theory of the linear fractional order system, projective synchronization of a complex network is studied in the paper, and the coupling functions of the connected nodes are identified. With this method, the projective synchronization of the network with different fractional order chaos nodes can be achieved, besides, the number of the system, Liu system and Coullet system are chosen as examples to show the effectiveness of the scheme.
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.
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.
On fractional order composite model reference adaptive control
Wei, Yiheng; Sun, Zhenyuan; Hu, Yangsheng; Wang, Yong
2016-08-01
This paper presents a novel composite model reference adaptive control approach for a class of fractional order linear systems with unknown constant parameters. The method is extended from the model reference adaptive control. The parameter estimation error of our method depends on both the tracking error and the prediction error, whereas the existing method only depends on the tracking error, which makes our method has better transient performance in the sense of generating smooth system output. By the aid of the continuous frequency distributed model, stability of the proposed approach is established in the Lyapunov sense. Furthermore, the convergence property of the model parameters estimation is presented, on the premise that the closed-loop control system is stable. Finally, numerical simulation examples are given to demonstrate the effectiveness of the proposed schemes.
Controllability of nonlinear systems.
Sussmann, H. J.; Jurdjevic, V.
1972-01-01
Discussion of the controllability of nonlinear systems described by the equation dx/dt - F(x,u). Concepts formulated by Chow (1939) and Lobry (1970) are applied to establish criteria for F and its derivatives to obtain qualitative information on sets which can be obtained from x which denotes a variable of state in an arbitrary, real, analytical manifold. It is shown that controllability implies strong accessibility for a large class of manifolds including Euclidean spaces.-
2007-03-01
IEEE Transactions on Automatic Control , AC- 48, pp. 1712-1723, (2003). [14] C.I. Byrnes, A. Isidori...Nonlinear internal models for output regulation,” IEEE Transactions on Automatic Control , AC-49, pp. 2244-2247, (2004). [15] C.I. Byrnes, F. Celani, A...approach,” IEEE Transactions on Automatic Control , 48 (Dec. 2003), 2172–2190. 2. C. I. Byrnes, “Differential Forms and Dynamical Systems,” to appear
Fault Detection for Nonlinear Systems
DEFF Research Database (Denmark)
Stoustrup, Jakob; Niemann, H.H.
1998-01-01
The paper describes a general method for designing fault detection and isolation (FDI) systems for nonlinear processes. For a rich class of nonlinear systems, a nonlinear FDI system can be designed using convex optimization procedures. The proposed method is a natural extension of methods based...
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.
Directory of Open Access Journals (Sweden)
DJAIRO G. DEFIGUEIREDO
2000-12-01
Full Text Available In this paper we treat the question of the existence of solutions of boundary value problems for systems of nonlinear elliptic equations of the form - deltau = f (x, u, v,Ñu,Ñv, - deltav = g(x, u, v, Ñu, Ñv, in omega, We discuss several classes of such systems using both variational and topological methods. The notion of criticality takes into consideration the coupling, which plays important roles in both a priori estimates for the solutions and Palais-Smale conditions for the associated functional in the variational case.
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.
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.
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.
Novel Fractional Order Calculus Extended PN for Maneuvering Targets
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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.
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.
Global Mittag-Leffler Stabilization of Fractional-Order Memristive Neural Networks.
Wu, Ailong; Zeng, Zhigang
2015-12-22
According to conventional memristive neural network theories, neurodynamic properties are powerful tools for solving many problems in the areas of brain-like associative learning, dynamic information storage or retrieval, etc. However, as have often been noted in most fractional-order systems, system analysis approaches for integral-order systems could not be directly extended and applied to deal with fractional-order systems, and consequently, it raises difficult issues in analyzing and controlling the fractional-order memristive neural networks. By using the set-valued maps and fractional-order differential inclusions, then aided by a newly proposed fractional derivative inequality, this paper investigates the global Mittag--Leffler stabilization for a class of fractional-order memristive neural networks. Two types of control rules (i.e., state feedback stabilizing control and output feedback stabilizing control) are designed for the stabilization of fractional-order memristive neural networks, while a list of stabilization criteria is established. Finally, two numerical examples are given to show the effectiveness and characteristics of the obtained theoretical results.
Balancing for unstable nonlinear systems
Scherpen, J.M.A.
1993-01-01
A previously obtained method of balancing for stable nonlinear systems is extended to unstable nonlinear systems. The similarity invariants obtained by the concept of LQG balancing for an unstable linear system can also be obtained by considering a past and future energy function of the system. By c
INITIAL VALUE PROBLEM FOR FRACTIONAL ORDER EQUATION WITH CONSTANT COEFFICIENTS
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Bogatyreva F. T.
2016-12-01
Full Text Available In this paper we construct an explicit representation of the solution of the Cauchy problem for ordinary differential equation of fractional order with Dzhrbashyan-Nersesyan operators.
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.
Tuning algorithms for fractional order internal model controllers for time delay processes
Muresan, Cristina I.; Dutta, Abhishek; Dulf, Eva H.; Pinar, Zehra; Maxim, Anca; Ionescu, Clara M.
2016-03-01
This paper presents two tuning algorithms for fractional-order internal model control (IMC) controllers for time delay processes. The two tuning algorithms are based on two specific closed-loop control configurations: the IMC control structure and the Smith predictor structure. In the latter, the equivalency between IMC and Smith predictor control structures is used to tune a fractional-order IMC controller as the primary controller of the Smith predictor structure. Fractional-order IMC controllers are designed in both cases in order to enhance the closed-loop performance and robustness of classical integer order IMC controllers. The tuning procedures are exemplified for both single-input-single-output as well as multivariable processes, described by first-order and second-order transfer functions with time delays. Different numerical examples are provided, including a general multivariable time delay process. Integer order IMC controllers are designed in each case, as well as fractional-order IMC controllers. The simulation results show that the proposed fractional-order IMC controller ensures an increased robustness to modelling uncertainties. Experimental results are also provided, for the design of a multivariable fractional-order IMC controller in a Smith predictor structure for a quadruple-tank system.
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.
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.
Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
Directory of Open Access Journals (Sweden)
Johnston S. J.
2016-01-01
Full Text Available The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.
Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
Johnston, S. J.; Jafari, H.; Moshokoa, S. P.; Ariyan, V. M.; Baleanu, D.
2016-07-01
The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.
Meyer, George
1997-01-01
The paper describes a method for guiding a dynamic system through a given set of points. The paradigm is a fully automatic aircraft subject to air traffic control (ATC). The ATC provides a sequence of way points through which the aircraft trajectory must pass. The way points typically specify time, position, and velocity. The guidance problem is to synthesize a system state trajectory which satisfies both the ATC and aircraft constraints. Complications arise because the controlled process is multi-dimensional, multi-axis, nonlinear, highly coupled, and the state space is not flat. In addition, there is a multitude of possible operating modes, which may number in the hundreds. Each such mode defines a distinct state space model of the process by specifying the state space coordinatization, the partition of the controls into active controls and configuration controls, and the output map. Furthermore, mode transitions must be smooth. The guidance algorithm is based on the inversion of the pure feedback approximations, which is followed by iterative corrections for the effects of zero dynamics. The paper describes the structure and modules of the algorithm, and the performance is illustrated by several example aircraft maneuvers.
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.
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.
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.
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.
Lyapunov functions for a class of nonlinear systems using Caputo derivative
Fernandez-Anaya, G.; Nava-Antonio, G.; Jamous-Galante, J.; Muñoz-Vega, R.; Hernández-Martínez, E. G.
2017-02-01
This paper presents an extension of recent results that allow proving the stability of Caputo nonlinear and time-varying systems, by means of the fractional order Lyapunov direct method, using quadratic Lyapunov functions. This article introduces a new way of building polynomial Lyapunov functions of any positive integer order as a way of determining the stability of a greater variety of systems when the order of the derivative is 0 < α < 1. Some examples are given to validate these results.
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.
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.
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.
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.
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.
Approximation of Analytic Functions by Bessel's Functions of Fractional Order
Directory of Open Access Journals (Sweden)
Soon-Mo Jung
2011-01-01
Full Text Available We will solve the inhomogeneous Bessel's differential equation x2y″(x+xy′(x+(x2-ν2y(x=∑m=0∞amxm, where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.
Nonlinear Waves in Complex Systems
DEFF Research Database (Denmark)
2007-01-01
The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations, it is the ......The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations......, it is the universality and robustness of the main models with respect to perturbations that developped the field. This is true for both continuous and discrete equations. In this volume we keep this broad view and draw new perspectives for nonlinear waves in complex systems. In particular we address energy flow...
Arshad, Muhammad; Lu, Dianchen; Wang, Jun
2017-07-01
In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order partial differential equations (PDEs) can be resolved effectively. The most distinct aspect of this method is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. We utilize the proposed scheme on some initial value problems and approximate numerical solutions of linear and nonlinear time fractional PDEs are obtained, which shows that the method is highly accurate and simple to apply. The proposed technique is thus an influential technique for solving the fractional PDEs and fractional order problems occurring in the field of engineering, physics etc. Numerical results are obtained for verification and demonstration purpose by using Mathematica software.
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.
Implementation of fractional-order electromagnetic potential through a genetic algorithm
Jesus, Isabel S.; Machado, J. A. Tenreiro
2009-05-01
Several phenomena present in electrical systems motivated the development of comprehensive models based on the theory of fractional calculus (FC). Bearing these ideas in mind, in this work are applied the FC concepts to define, and to evaluate, the electrical potential of fractional order, based in a genetic algorithm optimization scheme. The feasibility and the convergence of the proposed method are evaluated.
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.
Fractional Calculus: Integral and Differential Equations of Fractional Order
Gorenflo, Rudolf
2008-01-01
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.
2009-11-18
analytic semigroup T(t) ~ eAl is exponentially stable (Notice that it is also a contraction semigroup ). 3. Be 3(U, Z) and P e £(W, 2) are bounded. 4. Ce...quite often in practice, .4 is self-adjoint. We also note that, since we assume (—A) is sectorial, we work with the semigroup exp(.4f) rather than...Uniform Output Regulation of Nonlinear Sys- tems: A convergent Dynamics Approach, Birkhauser, Boston, 2006. 23 135] A. Pazy, Semigroups of Linear
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.
Fractional-order theory of heat transport in rigid bodies
Zingales, Massimiliano
2014-11-01
The non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of α-order Marchaud fractional-order derivative (0⩽α⩽1). Thermodynamical consistency of the model is provided in the context of Clausius-Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.
Fractional order sliding mode control for tethered satellite deployment with disturbances
Kang, Junjie; Zhu, Zheng H.; Wang, Wei; Li, Aijun; Wang, Changqing
2017-01-01
This paper proposes a fractional order sliding mode control for the deployment of tethered space systems with the consideration of uncertainty of external disturbances and unmodeled system dynamics. The proposed fractional order sliding mode control consists of two sub-sliding manifolds that are defined separately for the actuated and unactuated states. This, in turn, generates a control scheme to make all states move toward to the desired states. The stability analysis of the proposed control law indicates not only all states converge to the desired states at equilibrium but also disturbances caused by the uncertainty can be suppressed satisfactorily. Parametric studies are conducted to investigate the influences of fractional order and sub-sliding manifold of unactuated states on the performance of the proposed control law. The performance is compared with the sliding mode, PD and fractional order PD control laws for a baseline scenario of tether deployment. The proposed control law performs better than others in the settling time and the maximum pitch angle control in the presence of unwanted disturbances. Effectiveness and robustness of the proposed control law are demonstrated by computer simulations.
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.
Nonlinear robust hierarchical control for nonlinear uncertain systems
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Leonessa Alexander
1999-01-01
Full Text Available A nonlinear robust control-system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving nominal system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions, a hierarchical nonlinear robust control strategy is developed that robustly stabilizes a given nonlinear system over a prescribed range of system uncertainty by robustly stabilizing a collection of nonlinear controlled uncertain subsystems. The robust switching nonlinear controller architecture is designed based on a generalized (lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized nominal system equilibria. The proposed framework robustly stabilizes a compact positively invariant set of a given nonlinear uncertain dynamical system with structured parametric uncertainty. Finally, the efficacy of the proposed approach is demonstrated on a jet engine propulsion control problem with uncertain pressure-flow map data.
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.
On a discretization process of fractional-order Logistic differential equation
National Research Council Canada - National Science Library
El Raheem, Z.F; Salman, S.M
2014-01-01
.... First of all, we consider the fractional-order Logistic differential equation then, we consider the corresponding fractional-order Logistic differential equation with piecewise constant arguments...
Nonlinear Waves in Complex Systems
DEFF Research Database (Denmark)
2007-01-01
The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations......, it is the universality and robustness of the main models with respect to perturbations that developped the field. This is true for both continuous and discrete equations. In this volume we keep this broad view and draw new perspectives for nonlinear waves in complex systems. In particular we address energy flow...... in Fourier space and equipartition, the role of inhomogeneities and complex geometry and the importance of coupled systems....
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.
Systems of nonlinear Volterra integro-differential equations of arbitrary order
Directory of Open Access Journals (Sweden)
Kourosh Parand
2018-10-01
Full Text Available In this paper, a new approximate method for solving of systems of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs. Also, we construct the fractional derivative operational matrix of order $\\alpha$ in the Caputo's definition for GFCFs. This method reduced a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.
Nonlinear input-output systems
Hunt, L. R.; Luksic, Mladen; Su, Renjeng
1987-01-01
Necessary and sufficient conditions that the nonlinear system dot-x = f(x) + ug(x) and y = h(x) be locally feedback equivalent to the controllable linear system dot-xi = A xi + bv and y = C xi having linear output are found. Only the single input and single output case is considered, however, the results generalize to multi-input and multi-output systems.
Practical stability of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatolii Andreevich
1990-01-01
This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
Stability analysis of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatoly A
2015-01-01
The book investigates stability theory in terms of two different measure, exhibiting the advantage of employing families of Lyapunov functions and treats the theory of a variety of inequalities, clearly bringing out the underlying theme. It also demonstrates manifestations of the general Lyapunov method, showing how this technique can be adapted to various apparently diverse nonlinear problems. Furthermore it discusses the application of theoretical results to several different models chosen from real world phenomena, furnishing data that is particularly relevant for practitioners. Stability Analysis of Nonlinear Systems is an invaluable single-sourse reference for industrial and applied mathematicians, statisticians, engineers, researchers in the applied sciences, and graduate students studying differential equations.
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.
PBH tests for nonlinear systems
Kawano, Yu; Ohtsuka, Toshiyuki
2017-01-01
Recently, concepts of nonlinear eigenvalues and eigenvectors are introduced. In this paper, we establish connections between the nonlinear eigenvalues and nonlinear accessibility/observability. In particular, we provide a generalization of Popov- Belevitch-Hautus (PBH) test to nonlinear accessibilit
Nonlinear dynamics in biological systems
Carballido-Landeira, Jorge
2016-01-01
This book presents recent research results relating to applications of nonlinear dynamics, focusing specifically on four topics of wide interest: heart dynamics, DNA/RNA, cell mobility, and proteins. The book derives from the First BCAM Workshop on Nonlinear Dynamics in Biological Systems, held in June 2014 at the Basque Center of Applied Mathematics (BCAM). At this international meeting, researchers from different but complementary backgrounds, including molecular dynamics, physical chemistry, bio-informatics and biophysics, presented their most recent results and discussed the future direction of their studies using theoretical, mathematical modeling and experimental approaches. Such was the level of interest stimulated that the decision was taken to produce this publication, with the organizers of the event acting as editors. All of the contributing authors are researchers working on diverse biological problems that can be approached using nonlinear dynamics. The book will appeal especially to applied math...
Nonlinear Response of Strong Nonlinear System Arisen in Polymer Cushion
Directory of Open Access Journals (Sweden)
Jun Wang
2013-01-01
Full Text Available A dynamic model is proposed for a polymer foam-based nonlinear cushioning system. An accurate analytical solution for the nonlinear free vibration of the system is derived by applying He's variational iteration method, and conditions for resonance are obtained, which should be avoided in the cushioning design.
Nonlinear elliptic systems with exponential nonlinearities
Directory of Open Access Journals (Sweden)
Said El Manouni
2002-12-01
Full Text Available In this paper we investigate the existence of solutions for {gather*} -mathop{m div}( a(| abla u | ^N| abla u |^{N-2}u = f(x,u,v quad mbox{in } Omega -mathop{m div}(a(| abla v| ^N| abla v |^{N-2}v = g(x,u,v quad mbox{in } Omega u(x = v(x = 0 quad mbox{on }partial Omega. end{gather*} Where $Omega$ is a bounded domain in ${mathbb{R}}^N$, $Ngeq 2$, $f$ and $g$ are nonlinearities having an exponential growth on $Omega$ and $a$ is a continuous function satisfying some conditions which ensure the existence of solutions.
On balanced truncation for symmetric nonlinear systems
Fujimoto, K.; Scherpen, Jacqueline M.A.
2014-01-01
This paper is concerned with model order reduction based on balanced realization for symmetric nonlinear systems. A new notion of symmetry for nonlinear systems was characterized recently. It plays an important role in linear systems theory and is expected to provide new insights to nonlinear system
A fractional order model for lead-acid battery crankability estimation
Sabatier, J.; Cugnet, M.; Laruelle, S.; Grugeon, S.; Sahut, B.; Oustaloup, A.; Tarascon, J. M.
2010-05-01
With EV and HEV developments, battery monitoring systems have to meet the new requirements of car industry. This paper deals with one of them, the battery ability to start a vehicle, also called battery crankability. A fractional order model obtained by system identification is used to estimate the crankability of lead-acid batteries. Fractional order modelling permits an accurate simulation of the battery electrical behaviour with a low number of parameters. It is demonstrated that battery available power is correlated to the battery crankability and its resistance. Moreover, the high-frequency gain of the fractional model can be used to evaluate the battery resistance. Then, a battery crankability estimator using the battery resistance is proposed. Finally, this technique is validated with various battery experimental data measured on test rigs and vehicles.
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.
The Restoration of Textured Images Using Fractional-Order Regularization
Directory of Open Access Journals (Sweden)
Ying Fu
2014-01-01
Full Text Available Image restoration problem is ill-posed, so most image restoration algorithms exploit sparse prior in gradient domain to regularize it to yield high-quality results, reconstructing an image with piecewise smooth characteristics. While sparse gradient prior has good performance in noise removal and edge preservation, it also tends to remove midfrequency component such as texture. In this paper, we introduce the sparse prior in fractional-order gradient domain as texture-preserving strategy to restore textured images degraded by blur and/or noise. And we solve the unknown variables in the proposed model using method based on half-quadratic splitting by minimizing the nonconvex energy functional. Numerical experiments show our algorithm's robust outperformance.
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.
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.
Complex motions and chaos in nonlinear systems
Machado, José; Zhang, Jiazhong
2016-01-01
This book brings together 10 chapters on a new stream of research examining complex phenomena in nonlinear systems—including engineering, physics, and social science. Complex Motions and Chaos in Nonlinear Systems provides readers a particular vantage of the nature and nonlinear phenomena in nonlinear dynamics that can develop the corresponding mathematical theory and apply nonlinear design to practical engineering as well as the study of other complex phenomena including those investigated within social science.
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.
An Adaptive Nonlinear Filter for System Identification
Directory of Open Access Journals (Sweden)
Tokunbo Ogunfunmi
2009-01-01
Full Text Available The primary difficulty in the identification of Hammerstein nonlinear systems (a static memoryless nonlinear system in series with a dynamic linear system is that the output of the nonlinear system (input to the linear system is unknown. By employing the theory of affine projection, we propose a gradient-based adaptive Hammerstein algorithm with variable step-size which estimates the Hammerstein nonlinear system parameters. The adaptive Hammerstein nonlinear system parameter estimation algorithm proposed is accomplished without linearizing the systems nonlinearity. To reduce the effects of eigenvalue spread as a result of the Hammerstein system nonlinearity, a new criterion that provides a measure of how close the Hammerstein filter is to optimum performance was used to update the step-size. Experimental results are presented to validate our proposed variable step-size adaptive Hammerstein algorithm given a real life system and a hypothetical case.
Institute of Scientific and Technical Information of China (English)
温少芳; 申永军; 杨绍普
2016-01-01
With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffng oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coeffcients are defined by the feedback coeffcient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coeffcient, fractional order and nonlinear stiffness coeffcient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coeffcient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coeffcient determines the stabilities and the bending
Nonlinearity of colloid systems oxyhydrate systems
Sucharev, Yuri I
2008-01-01
The present monograph is the first systematic study of the non-linear characteristic of gel oxy-hydrate systems involving d- and f- elements. These are the oxyhydrates of rare-earth elements and oxides - hydroxides of d- elements (zirconium, niobium, titanium, etc.) The non-linearity of these gel systems introduces fundamental peculiarities into their structure and, consequently, their properties. The polymer-conformational diversity of energetically congenial gel fragments, which continu-ously transform under the effect of, for instance, system dissipation heat, is central to the au-thor's hy
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.
Institute of Scientific and Technical Information of China (English)
路永坤
2015-01-01
针对含参数不确定的整数阶统一混沌系统，提出一种鲁棒分数阶比例-微分(PDµ)控制。通过变换将受控统一混沌系统转换成等效被控对象及其等效控制器。针对等效被控对象，基于一种改进Monje-Vinagre方法并考虑到求解性能约束方程组的复杂度，设计了鲁棒PDµ控制器。通过基于最小相角边界传递函数和最大增益边界传递函数的设计约束来保证受控统一混沌系统对参数不确定性的鲁棒性能。数值仿真验证了所提出方法的有效性。%In this paper, a robust fractional-order proportional-derivative (PDµ) control is designed for controling in integer-order unified chaotic systems with parametric uncertainties. Equivalent plant is obtained by transforming the controlled dynamic system, and then the PDµ controller as an equivalent controller is applied to the equivalent plant. In the uncertain controlled unified chaotic systems, one equation is certain, and the other two equations are uncertain . The equivalent controller for the certain system is then designed based on a fractional-order proportional-derivative controller, in which three specifications for phase margin, gain crossover frequency, and robustness should be met. On the other hand, the robustness of uncertain systems is achieved by an improved Monje-Vinagre tuning method, however, the pre-specified frequency band should be replaced by the gain crossover frequency in order to reduce the complexity in determining the controllers for the uncertain systems. Specifications related to phase margin for the lower bound of the phase, gain crossover frequency for the upper bound of the gain, and robustness for the lower bound of the phase constraints are satisfied by the uncertain system. Parameters of the equivalent controller are determined based on a graphical method. Origins of the unstable equilibrium can be asymptotically stabilized by the proposed strategy for the integer
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.
Nonlinear cross Gramians and gradient systems
Ionescu, T. C.; Scherpen, J.M.A.
2007-01-01
We study the notion of cross Gramians for nonlinear gradient systems, using the characterization in terms of prolongation and gradient extension associated to the system. The cross Gramian is given for the variational system associated to the original nonlinear gradient system. We obtain linearization results that precisely correspond to the notion of a cross Gramian for symmetric linear systems. Furthermore, first steps towards relations with the singular value functions of the nonlinear Han...
Sharma, Richa; Gaur, Prerna; Mittal, A P
2015-09-01
The robotic manipulators are multi-input multi-output (MIMO), coupled and highly nonlinear systems. The presence of external disturbances and time-varying parameters adversely affects the performance of these systems. Therefore, the controller designed for these systems should effectively deal with such complexities, and it is an intriguing task for control engineers. This paper presents two-degree of freedom fractional order proportional-integral-derivative (2-DOF FOPID) controller scheme for a two-link planar rigid robotic manipulator with payload for trajectory tracking task. The tuning of all controller parameters is done using cuckoo search algorithm (CSA). The performance of proposed 2-DOF FOPID controllers is compared with those of their integer order designs, i.e., 2-DOF PID controllers, and with the traditional PID controllers. In order to show effectiveness of proposed scheme, the robustness testing is carried out for model uncertainties, payload variations with time, external disturbance and random noise. Numerical simulation results indicate that the 2-DOF FOPID controllers are superior to their integer order counterparts and the traditional PID controllers.
Saha, Suman; Das, Shantanu; Gupta, Amitava
2012-01-01
A novel conformal mapping based Fractional Order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI{\\lambda}D{\\mu}) controller have been approximated in this paper vis-\\`a-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI{\\lambda}D{\\mu} controller pushes the open loop zeros of the equivalent PID cont...
Observability and Controllability for Smooth Nonlinear Systems
Schaft, A.J. van der
1982-01-01
The definition of a smooth nonlinear system as proposed recently, is elaborated as a natural generalization of the more common definitions of a smooth nonlinear input-output system. Minimality for such systems can be defined in a very direct geometric way, and already implies a usual notion of observability, namely, local weak observability. As an application of this theory, it is shown that observable nonlinear Hamiltonian systems are necessarily controllable, and vice versa.
Computing abstractions of nonlinear systems
Reißig, Gunther
2009-01-01
We present an efficient algorithm for computing discrete abstractions of arbitrary memory span for nonlinear discrete-time and sampled systems, in which, apart from possibly numerically integrating ordinary differential equations, the only nontrivial operation to be performed repeatedly is to distinguish empty from non-empty convex polyhedra. We also provide sufficient conditions for the convexity of attainable sets, which is an important requirement for the correctness of the method we propose. It turns out that requirement can be met under rather mild conditions, which essentially reduce to sufficient smoothness in the case of sampled systems. Practicability of our approach in the design of discrete controllers for continuous plants is demonstrated by an example.
Nonlinear cross Gramians and gradient systems
Ionescu, T. C.; Scherpen, J. M. A.
2007-01-01
We study the notion of cross Gramians for nonlinear gradient systems, using the characterization in terms of prolongation and gradient extension associated to the system. The cross Gramian is given for the variational system associated to the original nonlinear gradient system. We obtain
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.
Computational Models for Nonlinear Aeroelastic Systems Project
National Aeronautics and Space Administration — Clear Science Corp. and Duke University propose to develop and demonstrate new and efficient computational methods of modeling nonlinear aeroelastic systems. The...
Model Updating Nonlinear System Identification Toolbox Project
National Aeronautics and Space Administration — ZONA Technology (ZONA) proposes to develop an enhanced model updating nonlinear system identification (MUNSID) methodology that utilizes flight data with...
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.
Discontinuity and complexity in nonlinear physical systems
Baleanu, Dumitru; Luo, Albert
2014-01-01
This unique book explores recent developments in experimental research in this broad field, organized in four distinct sections. Part I introduces the reader to the fractional dynamics and Lie group analysis for nonlinear partial differential equations. Part II covers chaos and complexity in nonlinear Hamiltonian systems, important to understand the resonance interactions in nonlinear dynamical systems, such as Tsunami waves and wildfire propagations; as well as Lev flights in chaotic trajectories, dynamical system synchronization and DNA information complexity analysis. Part III examines chaos and periodic motions in discontinuous dynamical systems, extensively present in a range of systems, including piecewise linear systems, vibro-impact systems and drilling systems in engineering. And in Part IV, engineering and financial nonlinearity are discussed. The mechanism of shock wave with saddle-node bifurcation and rotating disk stability will be presented, and the financial nonlinear models will be discussed....
Research on Nonlinear Dynamical Systems.
1983-01-10
investigated fundamental aspects of functional differential equations, including qualitative questions (stability, nonlinear oscillations ), in 142,45,47,52...Bifurcation in the Duffing equation with several parameters, II. Proc. of the Royal Society of Edinburgh, Series A, 79A (1977), pp.317-326. 1I.J (with ;Ibtoas...Lecture Notes in Mathematics, Vol. 730 (1979). [54] Nonlinear oscillations in equations with delays. Proc. at A.M.S. 10th Summer Seminar on Nonlinear
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.
Stability of fractional positive nonlinear systems
Directory of Open Access Journals (Sweden)
Kaczorek Tadeusz
2015-12-01
Full Text Available The conditions for positivity and stability of a class of fractional nonlinear continuous-time systems are established. It is assumed that the nonlinear vector function is continuous, satisfies the Lipschitz condition and the linear part is described by a Metzler matrix. The stability conditions are established by the use of an extension of the Lyapunov method to fractional positive nonlinear systems.
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.
Maximized Gust Loads of a Closed-Loop, Nonlinear Aeroelastic System Using Nonlinear Systems Theory
Silva, Walter A.
1999-01-01
The problem of computing the maximized gust load for a nonlinear, closed-loop aeroelastic aircraft is discusses. The Volterra theory of nonlinear systems is applied in order to define a linearized system that provides a bounds on the response of the nonlinear system of interest. The method is applied to a simplified model of an Airbus A310.
Indian Academy of Sciences (India)
Sana P Ansari; Saurabh K Agrawal; Subir Das
2015-01-01
This paper presents the synchronization between a pair of identical susceptible–infected–recovered (SIR) epidemic chaotic systems and fractional-order time derivative using active control method. The fractional derivative is described in Caputo sense. Numerical simulation results show that the method is effective and reliable for synchronizing the fractional-order chaotic systems while it allows the system to remain in chaotic state. The striking features of this paper are: the successful presentation of the stability of the equilibrium state and the revelation that time for synchronization varies with the variation in fractional-order derivatives close to the standard one for different specified values of the parameters of the system.
Stability analysis of nonlinear systems with slope restricted nonlinearities.
Liu, Xian; Du, Jiajia; Gao, Qing
2014-01-01
The problem of absolute stability of Lur'e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.
Stability Analysis of Nonlinear Systems with Slope Restricted Nonlinearities
Directory of Open Access Journals (Sweden)
Xian Liu
2014-01-01
Full Text Available The problem of absolute stability of Lur’e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.
DISTURBANCE ATTENUATION FOR UNCERTAIN NONLINEAR CASCADED SYSTEMS
Institute of Scientific and Technical Information of China (English)
BI Weiping; MU Xiaowu; SUN Yuqiang
2004-01-01
In present paper, the disturbance attenuation problem of uncertain nonlinear cascaded systems is studied. Based on the adding one power integrator technique and recursive design, a feedback controller that solves the disturbance attenuation problem is constructed for uncertain nonlinear cascaded systems with internal stability.
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.
Quantum Dynamics of Nonlinear Cavity Systems
Nation, Paul D.
2010-01-01
We investigate the quantum dynamics of three different configurations of nonlinear cavity systems. To begin, we carry out a quantum analysis of a dc superconducting quantum interference device (SQUID) mechanical displacement detector comprised of a SQUID with a mechanically compliant loop segment. The SQUID is approximated by a nonlinear current-dependent inductor, inducing a flux tunable nonlinear Duffing term in the cavity equation of motion. Expressions are derived for the detector signal ...
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.
Raju, D. Koteswara; Umre, Bhimrao S.; Junghare, A. S.; Chitti Babu, B.
2016-12-01
This paper explores a robust Fractional-order PI (FOPI) controller to diminish Subsynchronous Resonance (SSR) using Static Synchronous series compensator (SSSC). The diminution of SSR is accomplished by increasing the network damping with the injection of voltage of subsynchronous component into the line at those frequencies which are proximate to the torsional mode frequency of the turbine-generator shaft. The voltage of subsynchronous frequency component is extracted from the transmission line and further the similar quantity of series voltage is injected by SSSC into the line to make the current of subsynchronous frequency component to zero which is the major source of oscillations in the turbine-generator shaft. The insertion and fine tuning of Fractional-order PI controller in the control scheme of SSSC the subsynchronous oscillations are reduced to 4 % as compared to conventional PI controller. The studied system is modelled and simulated using MATLAB-Simulink and the results are analysed to show the precision and robustness of the proposed control strategy.
Projective synchronization of fractional-order memristor-based neural networks.
Bao, Hai-Bo; Cao, Jin-De
2015-03-01
This paper investigates the projective synchronization of fractional-order memristor-based neural networks. Sufficient conditions are derived in the sense of Caputo's fractional derivation and by combining a fractional-order differential inequality. Two numerical examples are given to show the effectiveness of the main results. The results in this paper extend and improve some previous works on the synchronization of fractional-order neural networks.
Hopf Bifurcation in a Nonlinear Wave System
Institute of Scientific and Technical Information of China (English)
HE Kai-Fen
2004-01-01
@@ Bifurcation behaviour of a nonlinear wave system is studied by utilizing the data in solving the nonlinear wave equation. By shifting to the steady wave frame and taking into account the Doppler effect, the nonlinear wave can be transformed into a set of coupled oscillators with its (stable or unstable) steady wave as the fixed point.It is found that in the chosen parameter regime, both mode amplitudes and phases of the wave can bifurcate to limit cycles attributed to the Hopf instability. It is emphasized that the investigation is carried out in a pure nonlinear wave framework, and the method can be used for the further exploring routes to turbulence.
FORCED OSCILLATIONS IN NONLINEAR FEEDBACK CONTROL SYSTEM
Since a nonlinear feedback control system may possess more than one type of forced oscillations, it is highly desirable to investigate the type of...method for finding the existence of forced oscillations and response curve characteristics of a nonlinear feedback control system by means of finding the...second order feedback control system are investigated; the fundamental frequency forced oscillation for a higher order system and the jump resonance
Nonlinear identification of power electronic systems
Chau, KT; Chan, CC
1995-01-01
This paper presents a new approach to modelling power electronic systems using nonlinear system identification. By employing the nonlinear autoregressive moving average with exogenous input (NARMAX) technique, the parametric model of power electronic systems can be derived from the time-domain data. This approach possesses some advantages over available circuit-oriented modelling approaches, such as no small-signal approximation, no circuit idealization and no detailed knowledge of system ope...
Quadratic stabilization of switched nonlinear systems
Institute of Scientific and Technical Information of China (English)
DONG YaLi; FAN JiaoJiao; MEI ShengWei
2009-01-01
In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems.
Advances and applications in nonlinear control systems
Volos, Christos
2016-01-01
The book reports on the latest advances and applications of nonlinear control systems. It consists of 30 contributed chapters by subject experts who are specialized in the various topics addressed in this book. The special chapters have been brought out in the broad areas of nonlinear control systems such as robotics, nonlinear circuits, power systems, memristors, underwater vehicles, chemical processes, observer design, output regulation, backstepping control, sliding mode control, time-delayed control, variables structure control, robust adaptive control, fuzzy logic control, chaos, hyperchaos, jerk systems, hyperjerk systems, chaos control, chaos synchronization, etc. Special importance was given to chapters offering practical solutions, modeling and novel control methods for the recent research problems in nonlinear control systems. This book will serve as a reference book for graduate students and researchers with a basic knowledge of electrical and control systems engineering. The resulting design proce...
Yang, Yongge; Xu, Wei; Sun, Yahui; Xiao, Yanwen
2017-01-01
This paper aims to investigate the stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation. Firstly, the original stochastic vibroimpact system with fractional derivative is transformed into equivalent stochastic vibroimpact system without fractional derivative. Then, the non-smooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system. At last, in order to verify the effectiveness of the above mentioned approach, the van der Pol vibroimpact system with fractional derivative is worked out in detail. A very satisfactory agreement can be found between the analytical results and the numerical results. An interesting phenomenon we found in this paper is that the fractional order and fractional coefficient of the stochastic van der Pol vibroimpact system can induce the occurrence of stochastic P-bifurcation. To the best of authors' knowledge, the stochastic P-bifurcation phenomena induced by fractional order and fractional coefficient have not been found in the present available literature which studies the dynamical behaviors of stochastic system with fractional derivative under Gaussian white noise excitation.
Song, Junqiang; Leng, Hongze; Lu, Fengshun
2014-01-01
We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303
Linearization of Systems of Nonlinear Diffusion Equations
Institute of Scientific and Technical Information of China (English)
KANG Jing; QU Chang-Zheng
2007-01-01
We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.
Model Updating Nonlinear System Identification Toolbox Project
National Aeronautics and Space Administration — ZONA Technology proposes to develop an enhanced model updating nonlinear system identification (MUNSID) methodology by adopting the flight data with state-of-the-art...
Computational Models for Nonlinear Aeroelastic Systems Project
National Aeronautics and Space Administration — Clear Science Corp. and Duke University propose to develop and demonstrate a new and efficient computational method of modeling nonlinear aeroelastic systems. The...
Interactive optomechanical coupling with nonlinear polaritonic systems
Bobrovska, N; Liew, T C H; Kyriienko, O
2016-01-01
We study a system of interacting matter quasiparticles strongly coupled to photons inside an optomechanical cavity. The resulting normal modes of the system are represented by hybrid polaritonic quasiparticles, which acquire effective nonlinearity. Its strength is influenced by the presence of the mechanical mode and depends on the resonance frequency of the cavity. This leads to an interactive type of optomechanical coupling, being distinct from the previously studied dispersive and dissipative couplings in optomechanical systems. The emergent interactive coupling is shown to generate effective optical nonlinearity terms of high order, being quartic in the polariton number. We consider particular systems of exciton-polaritons and dipolaritons, and show that the induced effective optical nonlinearity due to the interactive coupling can exceed in magnitude the strength of Kerr nonlinear terms, such as those arising from polariton-polariton interactions. As applications, we show that the higher order terms give...
Pitchfork bifurcation and vibrational resonance in a fractional-order Duffing oscillator
Indian Academy of Sciences (India)
J H Yang; M A F Sanjuán; W Xiang; H Zhu
2013-12-01
The pitchfork bifurcation and vibrational resonance are studied in a fractional-order Duffing oscillator with delayed feedback and excited by two harmonic signals. Using an approximation method, the bifurcation behaviours and resonance patterns are predicted. Supercritical and subcritical pitchfork bifurcations can be induced by the fractional-order damping, the exciting highfrequency signal and the delayed time. The fractional-order damping mainly determines the pattern of the vibrational resonance. There is a bifurcation point of the fractional order which, in the case of double-well potential, transforms vibrational resonance pattern from a single resonance to a double resonance, while in the case of single-well potential, transforms vibrational resonance from no resonance to a single resonance. The delayed time influences the location of the vibrational resonance and the bifurcation point of the fractional order. Pitchfork bifurcation is the necessary condition for the double resonance. The theoretical predictions are in good agreement with the numerical simulations.
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.
Chaotification for a class of nonlinear systems
Institute of Scientific and Technical Information of China (English)
Liu Na; Guan Zhi-Hong
2009-01-01
More and more attention has been focused on effectively generating chaos via simple physical devices. The problem of creating chaotic attractors is considered for a class of nonlinear systems with backlash function in this paper. By utilizing the Silnikov heteroclinic and homoclinic theorems, some sufficient conditions are established to guarantee that the nonlinear system has horseshoe-type chaos. Examples and simulations are given to verify the effectiveness of the theoretical results.
APPROXIMATE OUTPUT REGULATION FOR AFFINE NONLINEAR SYSTEMS
Institute of Scientific and Technical Information of China (English)
Yali DONG; Daizhan CHENG; Huashu QIN
2003-01-01
Output regulation for affine nonlinear systems driven by an exogenous signal is investigated in this paper. In the absence of the standard exosystem hypothesis, we assume availability of the instantaneous values of the exogenous signal and its first time-derivative for use in the control law.For affine nonlinear systems, the necessary and sufficient conditions of the solvability of approximate output regulation problem are obtained. The precise form of the control law is presented under some suitable assumptions.
Qualitative stability of nonlinear networked systems
Angulo, Marco Tulio; Slotine, Jean-Jacques
2016-01-01
In many large systems, such as those encountered in biology or economics, the dynamics are nonlinear and are only known very coarsely. It is often the case, however, that the signs (excitation or inhibition) of individual interactions are known. This paper extends to nonlinear systems the classical criteria of linear sign stability introduced in the 70's, yielding simple sufficient conditions to determine stability using only the sign patterns of the interactions.
Fractional order nonsingular terminal sliding mode control for flexible spacecraft attitude tracking
Institute of Scientific and Technical Information of China (English)
GAO; Junshan; DENG; Liwei; SONG; Shenmin
2016-01-01
This paper investigates a fractional terminal sliding mode control for flexible spacecraft attitude tracking in the presence of inertia uncertainties and external disturbances. The controller is based on the fractional calculus and nonsingular terminal sliding mode control technique,and it guarantees the convergence of attitude tracking error in finite time rather than in the asymptotic sense. With respect to the controller,a fractional order sliding surface is given,the corresponding control scheme is proposed based on Lyapunov stability theory to guarantee the sliding condition,and the finite time stability of the whole close loop system is also proven. Finally,numerical simulations are presented to illustrate the performance of the proposed scheme.
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.
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.
Nonlinear Differential Systems with Prescribed Invariant Sets
DEFF Research Database (Denmark)
Sandqvist, Allan
1999-01-01
We present a class of nonlinear differential systems for which invariant sets can be prescribed.Moreover,we show that a system in this class can be explicitly solved if a certain associated linear homogeneous system can be solved.As a simple application we construct a plane autonomous system having...
Nonlinear characteristics of an autoparametric vibration system
Yan, Zhimiao; Taha, Haithem E.; Tan, Ting
2017-03-01
The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.
Nonlinear vibrating system identification via Hilbert decomposition
Feldman, Michael; Braun, Simon
2017-02-01
This paper deals with the identification of nonlinear vibration systems, based on measured signals for free and forced vibration regimes. Two categories of time domain signal are analyzed, one of a fast inter-modulation signal and a second as composed of several mono-components. To some extent, this attempts to imitate analytic studies of such systems, with its two major analysis groups - the perturbation and the harmonic balance methods. Two appropriate signal processing methods are then investigated, one based on demodulation and the other on signal decomposition. The Hilbert Transform (HT) has been shown to enable effective and simple methods of analysis. We show that precise identification of the nonlinear parameters can be obtained, contrary to other average HT based methods where only approximation parameters are obtained. The effectiveness of the proposed methods is demonstrated for the precise nonlinear system identification, using both the signal demodulation and the signal decomposition methods. Following the exposition of the tools used, both the signal demodulation as well as decomposition are applied to classical examples of nonlinear systems. Cases of nonlinear stiffness and damping forces are analyzed. These include, among other, an asymmetric Helmholtz oscillator, a backlash with nonlinear turbulent square friction, and a Duffing oscillator with dry friction.
Modal analysis of nonlinear mechanical systems
2014-01-01
The book first introduces the concept of nonlinear normal modes (NNMs) and their two main definitions. The fundamental differences between classical linear normal modes (LNMs) and NNMs are explained and illustrated using simple examples. Different methods for computing NNMs from a mathematical model are presented. Both advanced analytical and numerical methods are described. Particular attention is devoted to the invariant manifold and normal form theories. The book also discusses nonlinear system identification.
NONLINEAR DYNAMIC ANALYSIS OF FLEXIBLE MULTIBODY SYSTEM
Institute of Scientific and Technical Information of China (English)
A.Y.T.Leung; WuGuorong; ZhongWeifang
2004-01-01
The nonlinear dynamic equations of a multibody system composed of flexible beams are derived by using the Lagrange multiplier method. The nonlinear Euler beam theory with inclusion of axial deformation effect is employed and its deformation field is described by exact vibration modes. A numerical procedure for solving the dynamic equations is presented based on the Newmark direct integration method combined with Newton-Raphson iterative method. The results of numerical examples prove the correctness and efficiency of the method proposed.
Chaos and chaotic control in a fractional-order electronic oscillator
Gao, Xin; Yu, Jue-Bang
2005-05-01
In this paper, we study the chaotic behaviours in a fractional-order chaotic electronic oscillator. We find that chaos exists in the fractional-order electronic oscillator with an order being less than 3. In addition, we numerically simulate the continuance of the chaotic behaviours in the electronic oscillator with orders ranging from 2.8 to 3.2. Finally, we further investigate the method of controlling a fractional-order electronic oscillator based on adaptive backstepping. Numerical simulations show the effectiveness and feasibility of this approach.
The Oscillation of a Class of the Fractional-Order Delay Differential Equations
Directory of Open Access Journals (Sweden)
Qianli Lu
2014-01-01
Full Text Available Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.
Chaos and chaotic control in a fractional-order electronic oscillator
Institute of Scientific and Technical Information of China (English)
Gao Xin; Yu Jue-Bang
2005-01-01
In this paper, we study the chaotic behaviours in a fractional-order chaotic electronic oscillator. We find that chaos exists in the fractional-order electronic oscillator with an order being less than 3. In addition, we numerically simulate the continuance of the chaotic behaviours in the electronic oscillator with orders ranging from 2.8 to 3.2. Finally, we further investigate the method of controlling a fractional-order electronic oscillator based on adaptive backstepping.Numerical simulations show the effectiveness and feasibility of this approach.
Gradient realization of nonlinear control systems
Cortes monforte, J.; Cortés, J.; Crouch, P.E.; Astolfi, A.; van der Schaft, Arjan; Gordillo, F.
2003-01-01
We investigate necessary and su?cient conditions under which a nonlinear afine control system with outputs can be written as a gradient control system corresponding to some pseudo-Riemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with
Damage detection in initially nonlinear systems
Energy Technology Data Exchange (ETDEWEB)
Bornn, Luke [Los Alamos National Laboratory; Farrar, Charles [Los Alamos National Laboratory; Park, Gyuhae [Los Alamos National Laboratory
2009-01-01
The primary goal of Structural Health Monitoring (SHM) is to detect structural anomalies before they reach a critical level. Because of the potential life-safety and economic benefits, SHM has been widely studied over the past decade. In recent years there has been an effort to provide solid mathematical and physical underpinnings for these methods; however, most focus on systems that behave linearly in their undamaged state - a condition that often does not hold in complex 'real world' systems and systems for which monitoring begins mid-lifecycle. In this work, we highlight the inadequacy of linear-based methodology in handling initially nonlinear systems. We then show how the recently developed autoregressive support vector machine (AR-SVM) approach to time series modeling can be used for detecting damage in a system that exhibits initially nonlinear response. This process is applied to data acquired from a structure with induced nonlinearity tested in a laboratory environment.
Controller Design of Complex System Based on Nonlinear Strength
Directory of Open Access Journals (Sweden)
Rongjun Mu
2015-01-01
Full Text Available This paper presents a new idea of controller design for complex systems. The nonlinearity index method was first developed for error propagation of nonlinear system. The nonlinearity indices access the boundary between the strong and the weak nonlinearities of the system model. The algorithm of nonlinearity index according to engineering application is first proposed in this paper. Applying this method on nonlinear systems is an effective way to measure the nonlinear strength of dynamics model over the full flight envelope. The nonlinearity indices access the boundary between the strong and the weak nonlinearities of system model. According to the different nonlinear strength of dynamical model, the control system is designed. The simulation time of dynamical complex system is selected by the maximum value of dynamic nonlinearity indices. Take a missile as example; dynamical system and control characteristic of missile are simulated. The simulation results show that the method is correct and appropriate.
Nonlinear System Identification and Behavioral Modeling
Huq, Kazi Mohammed Saidul; Kabir, A F M Sultanul
2010-01-01
The problem of determining a mathematical model for an unknown system by observing its input-output data pair is generally referred to as system identification. A behavioral model reproduces the required behavior of the original analyzed system, such as there is a one-to-one correspondence between the behavior of the original system and the simulated system. This paper presents nonlinear system identification and behavioral modeling using a work assignment.
Discrete time learning control in nonlinear systems
Longman, Richard W.; Chang, Chi-Kuang; Phan, Minh
1992-01-01
In this paper digital learning control methods are developed primarily for use in single-input, single-output nonlinear dynamic systems. Conditions for convergence of the basic form of learning control based on integral control concepts are given, and shown to be satisfied by a large class of nonlinear problems. It is shown that it is not the gross nonlinearities of the differential equations that matter in the convergence, but rather the much smaller nonlinearities that can manifest themselves during the short time interval of one sample time. New algorithms are developed that eliminate restrictions on the size of the learning gain, and on knowledge of the appropriate sign of the learning gain, for convergence to zero error in tracking a feasible desired output trajectory. It is shown that one of the new algorithms can give guaranteed convergence in the presence of actuator saturation constraints, and indicate when the requested trajectory is beyond the actuator capabilities.
Theoretical aspects of nonlinear echo image system
Institute of Scientific and Technical Information of China (English)
ZHANG Ruiquan; FENG Shaosong
2003-01-01
In order to develop the nonlinear echo image system to diagnose pathological changes in biological tissue , a simple physical model to analyse the character of nonlinear reflected wave in biological medium is postulated. The propagation of large amplitude plane sound wave in layered biological media is analysed for the one dimensional case by the method of successive approximation and the expression for the second order wave reflected from any interface of layered biological media is obtained. The relations between the second order reflection coefficients and the nonlinear parameters of medium below the interface are studied in three layers interfaces. Finally, the second order reflection coefficients of four layered media are calculated numerically. The results indicate that the nonlinear parameter B/A of each layer of biological media can be determined by the reflection method.
Nonlinear system identification in offshore structural reliability
Energy Technology Data Exchange (ETDEWEB)
Spanos, P.D. [Rice Univ., Houston, TX (United States); Lu, R. [Hudson Engineering Corporation, Houston, TX (United States)
1995-08-01
Nonlinear forces acting on offshore structures are examined from a system identification perspective. The nonlinearities are induced by ocean waves and may become significant in many situations. They are not necessarily in the form of Morison`s equation. Various wave force models are examined. The force function is either decomposed into a set of base functions or it is expanded in terms of the wave and structural kinematics. The resulting nonlinear system is decomposed into a number of parallel no-memory nonlinear systems, each followed by a finite-memory linear system. A conditioning procedure is applied to decouple these linear sub-systems; a frequency domain technique involving autospectra and cross-spectra is employed to identify the linear transfer functions. The structural properties and those force transfer parameters are determine with the aid of the coherence functions. The method is verified using simulated data. It provides a versatile and noniterative approach for dealing with nonlinear interaction problems encountered in offshore structural analysis and design.
BINARY NONLINEARIZATION FOR THE DIRAC SYSTEMS
Institute of Scientific and Technical Information of China (English)
MAWENXIU
1997-01-01
A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the Dirac systems. It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Linuville integrable Hamiltonian system and that under the control of the spatial part, the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are interpreted as a hierarchy of commutative, finite dimensional Linuville integrable Hamiltoian systems whose Hamiltonian functions consist of a series of integrals of motion for the spatial part. Moreover an invaiutive representation of solutions of the Dirac systems exhibits their integrability by quadratures. This kind of symmetry constraint procedure involving thespectral problem and the adjoint spectral problem is referred to as a binary nonlinearization technique like a binary Darhoux transformation.
Saha, Suman; Das, Saptarshi; Das, Shantanu; Gupta, Amitava
2012-09-01
A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PIλDμ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PIλDμ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as "M-curve". This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller's effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.
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.
EXISTENCE OF MILD SOLUTIONS TO SEMILINEAR FRACTIONAL ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we use the analytic semigroup theory of linear operators and fixed point method to prove the existence of mild solutions to a semilinear fractional order functional differential equations in a Banach space.
Ontology of Earth's nonlinear dynamic complex systems
Babaie, Hassan; Davarpanah, Armita
2017-04-01
As a complex system, Earth and its major integrated and dynamically interacting subsystems (e.g., hydrosphere, atmosphere) display nonlinear behavior in response to internal and external influences. The Earth Nonlinear Dynamic Complex Systems (ENDCS) ontology formally represents the semantics of the knowledge about the nonlinear system element (agent) behavior, function, and structure, inter-agent and agent-environment feedback loops, and the emergent collective properties of the whole complex system as the result of interaction of the agents with other agents and their environment. It also models nonlinear concepts such as aperiodic, random chaotic behavior, sensitivity to initial conditions, bifurcation of dynamic processes, levels of organization, self-organization, aggregated and isolated functionality, and emergence of collective complex behavior at the system level. By incorporating several existing ontologies, the ENDCS ontology represents the dynamic system variables and the rules of transformation of their state, emergent state, and other features of complex systems such as the trajectories in state (phase) space (attractor and strange attractor), basins of attractions, basin divide (separatrix), fractal dimension, and system's interface to its environment. The ontology also defines different object properties that change the system behavior, function, and structure and trigger instability. ENDCS will help to integrate the data and knowledge related to the five complex subsystems of Earth by annotating common data types, unifying the semantics of shared terminology, and facilitating interoperability among different fields of Earth science.
Emulating “Chaos + Chaos = Order” in Chen’s Circuit of Fractional Order by Parameter Switching
Tang, Wallace K. S.; Danca, Marius-F.
2016-06-01
In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos + Chaos = Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.
Design of a Fractional Order PID Controller Using Particle Swarm Optimization Technique
Maiti, Deepyaman; Konar, Amit
2008-01-01
Particle Swarm Optimization technique offers optimal or suboptimal solution to multidimensional rough objective functions. In this paper, this optimization technique is used for designing fractional order PID controllers that give better performance than their integer order counterparts. Controller synthesis is based on required peak overshoot and rise time specifications. The characteristic equation is minimized to obtain an optimum set of controller parameters. Results show that this design method can effectively tune the parameters of the fractional order controller.
Robustness analysis for a class of nonlinear descriptor systems
Institute of Scientific and Technical Information of China (English)
吴敏; 张凌波; 何勇
2004-01-01
The robustness analysis problem of a class of nonlinear descriptor systems is studied. Nonlinear matrix inequality which has the good computation property of convex feasibility is employed to derive some sufficient conditions to guarantee that the nonlinear descriptor systems have robust disturbance attenuation performance, which avoids the computational difficulties in conversing nonlinear matrix and Hamilton-Jacobi inequality. The computation property of convex feasibility of nonlinear matrix inequality makes it possible to apply the results of nonlinear robust control to practice.
Nonlinear Control of Delay and PDE Systems
Bekiaris-Liberis, Nikolaos
In this dissertation we develop systematic procedures for the control and analysis of general nonlinear systems with delays and of nonlinear PDE systems. We design predictor feedback laws (i.e., feedback laws that use the future, rather than the current state of the system) for the compensation of delays (i.e., after the control signal reaches the system for the first time, the system behaves as there were no delay at all) that can be time-varying or state-dependent, on the input and on the state of nonlinear systems. We also provide designs of predic- tor feedback laws for linear systems with constant distributed delays and known or unknown plant parameters, and for linear systems with simultaneous known or unknown constant delays on the input and the state. Moreover, we intro- duce infinite-dimensional backstepping transformations for each particular prob-lem, which enables us to construct Lyapunov-Krasovskii functionals. With the available Lyapunov-Krasovskii functionals we study stability, as well as, robust- ness of our control laws to plant uncertainties. We deal with coupled PDE-ODE systems. We consider nonlinear systems with wave actuator dynamics, for which we design a predictor inspired feedback law. We study stability of the closed-loop system either by constructing Lyapunov functionals, or using arguments of explicit solutions. We also consider linear sys- tems with distributed actuator and sensor dynamics governed by diffusion or wave PDEs, for which we design stabilizing feedback laws. We study stability of the closed-loop systems using Lyapunov functionals that we construct with the intro- duction of infinite-dimensional transformations of forwarding type. Finally, we develop a control design methodology for coupled nonlinear first-order hyperbolic PDEs through an application to automotive catalysts.
Controller reconfiguration for non-linear systems
Kanev, S.; Verhaegen, M.
2000-01-01
This paper outlines an algorithm for controller reconfiguration for non-linear systems, based on a combination of a multiple model estimator and a generalized predictive controller. A set of models is constructed, each corresponding to a different operating condition of the system. The interacting m
Dynamic disturbance decoupling for nonlinear systems
Huijberts, H.J.C.; Nijmeijer, H.; Wegen, van der L.L.M.
1992-01-01
In analogy with the dynamic input-output decoupling problem the dynamic disturbance decoupling problem for nonlinear systems is introduced. A local solution of this problem is obtained in the case that the system under consideration is invertible. The solution is given in algebraic as well as in geo
Fault detection for nonlinear systems - A standard problem approach
DEFF Research Database (Denmark)
Stoustrup, Jakob; Niemann, Hans Henrik
1998-01-01
The paper describes a general method for designing (nonlinear) fault detection and isolation (FDI) systems for nonlinear processes. For a rich class of nonlinear systems, a nonlinear FDI system can be designed using convex optimization procedures. The proposed method is a natural extension...
Mitigation of Subsynchronous Resonance with Fractional-order PI based UPFC controller
Raju, D. Koteswara; Umre, Bhimrao S.; Junghare, Anjali S.; Babu, B. Chitti
2017-02-01
Due to incorporation of series capacitor compensation in transmission line for stability improvement, subsynchronous oscillations are generated at turbine-generator shaft. These oscillations can damage the shaft system if these are not well suppressed. In order to damp out these oscillations, usually power system network should have sufficient damping and the increase of network damping is obtained by the injection of subsynchronous component of voltage and current into the line, which are extracted from the measured signal of the system. However, the effectiveness of damp out of these subsynchronous oscillations is possibly by incorporating UPFC in the transmission line network is of high interest and it should be further investigated. This research article proposes the mitigation of subsynchronous resonance (SSR) using fractional-order PI (FOPI) based unified power flow controller (UPFC). The robustness of the proposed controller is tested for 25%, 55% and 70% series compensation with a symmetrical fault (L-L-L fault). Further, Eigenvalue analysis and Fast Fourier Transform (FFT) analysis against operating point variations and uncertainties in the system are also examined. The IEEE first benchmark model is adopted for this study and the superiority of the FOPI based UPFC controller over PI based UPFC controller is discussed by comparing the results with various performance indices.
Network science, nonlinear science and infrastructure systems
2007-01-01
Network Science, Nonlinear Science and Infrastructure Systems has been written by leading scholars in these areas. Its express purpose is to develop common theoretical underpinnings to better solve modern infrastructural problems. It is felt by many who work in these fields that many modern communication problems, ranging from transportation networks to telecommunications, Internet, supply chains, etc., are fundamentally infrastructure problems. Moreover, these infrastructure problems would benefit greatly from a confluence of theoretical and methodological work done with the areas of Network Science, Dynamical Systems and Nonlinear Science. This book is dedicated to the formulation of infrastructural tools that will better solve these types of infrastructural problems. .
Nonlinear system compound inverse control method
Institute of Scientific and Technical Information of China (English)
Yan ZHANG; Zengqiang CHEN; Peng YANG; Zhuzhi YUAN
2005-01-01
A compound neural network is utilized to identify the dynamic nonlinear system.This network is composed of two parts: one is a linear neural network,and the other is a recurrent neural network.Based on the inverse theory a compound inverse control method is proposed.The controller has also two parts:a linear controller and a nonlinear neural network controller.The stability condition of the closed-loop neural network-based compound inverse control system is demonstrated based on the Lyapunov theory.Simulation studies have shown that this scheme is simple and has good control accuracy and robustness.
Explicit solutions of nonlinear wave equation systems
Institute of Scientific and Technical Information of China (English)
Ahmet Bekir; Burcu Ayhan; M.Naci (O)zer
2013-01-01
We apply the (G'/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions,trigonometric functions,and rational functions with arbitrary parameters.We highlight the power of the (G'/G)-expansion method in providing generalized solitary wave solutions of different physical structures.It is shown that the (G'/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
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.
Evolutionary quantitative genetics of nonlinear developmental systems.
Morrissey, Michael B
2015-08-01
In quantitative genetics, the effects of developmental relationships among traits on microevolution are generally represented by the contribution of pleiotropy to additive genetic covariances. Pleiotropic additive genetic covariances arise only from the average effects of alleles on multiple traits, and therefore the evolutionary importance of nonlinearities in development is generally neglected in quantitative genetic views on evolution. However, nonlinearities in relationships among traits at the level of whole organisms are undeniably important to biology in general, and therefore critical to understanding evolution. I outline a system for characterizing key quantitative parameters in nonlinear developmental systems, which yields expressions for quantities such as trait means and phenotypic and genetic covariance matrices. I then develop a system for quantitative prediction of evolution in nonlinear developmental systems. I apply the system to generating a new hypothesis for why direct stabilizing selection is rarely observed. Other uses will include separation of purely correlative from direct and indirect causal effects in studying mechanisms of selection, generation of predictions of medium-term evolutionary trajectories rather than immediate predictions of evolutionary change over single generation time-steps, and the development of efficient and biologically motivated models for separating additive from epistatic genetic variances and covariances.
Workshop on Nonlinear Phenomena in Complex Systems
1989-01-01
This book contains a thorough treatment of neural networks, cellular-automata and synergetics, in an attempt to provide three different approaches to nonlinear phenomena in complex systems. These topics are of major interest to physicists active in the fields of statistical mechanics and dynamical systems. They have been developed with a high degree of sophistication and include the refinements necessary to work with the complexity of real systems as well as the more recent research developments in these areas.
New results in global stabilization for stochastic nonlinear systems
Institute of Scientific and Technical Information of China (English)
Tao BIAN; Zhong-Ping JIANG
2016-01-01
This paper presents new results on the robust global stabilization and the gain assignment problems for stochastic nonlinear systems. Three stochastic nonlinear control design schemes are developed. Furthermore, a new stochastic gain assignment method is developed for a class of uncertain interconnected stochastic nonlinear systems. This method can be combined with the nonlinear small-gain theorem to design partial-state feedback controllers for stochastic nonlinear systems. Two numerical examples are given to illustrate the effectiveness of the proposed methodology.
Nonlinear distortion in wireless systems modeling and simulation with Matlab
Gharaibeh, Khaled M
2011-01-01
This book covers the principles of modeling and simulation of nonlinear distortion in wireless communication systems with MATLAB simulations and techniques In this book, the author describes the principles of modeling and simulation of nonlinear distortion in single and multichannel wireless communication systems using both deterministic and stochastic signals. Models and simulation methods of nonlinear amplifiers explain in detail how to analyze and evaluate the performance of data communication links under nonlinear amplification. The book addresses the analysis of nonlinear systems
Exploring Nonlinearities in Financial Systemic Risk
Wolski, M.
2013-01-01
We propose a new methodology of assessing the effects of individual institution's risk on the others and on the system as a whole. We build upon the Conditional Value-at-Risk approach, however, we introduce the explicit Granger causal linkages and we account for possible nonlinearities in the
Oscillatority Conditions for Nonlinear Systems with Delay
Directory of Open Access Journals (Sweden)
Denis V. Efimov
2007-01-01
Full Text Available Sufficient conditions for oscillatority in the sense of Yakubovich for a class of time delay nonlinear systems are proposed. Under proposed conditions, upper and lower bounds for oscillation amplitude are given. Examples illustrating analytical results by computer simulation are presented.
A polynomial approach to nonlinear system controllability
Zheng, YF; Willems, JC; Zhang, CH
2001-01-01
This note uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input-single-output (SISO) nonlinear systems. The condition is presented in terms of common factors of a noncommutative polynomial expression. This result exposes controllability
Periodic Solutions for Highly Nonlinear Oscillation Systems
DEFF Research Database (Denmark)
Ghadimi, M; Barari, Amin; Kaliji, H.D
2012-01-01
In this paper, Frequency-Amplitude Formulation is used to analyze the periodic behavior of tapered beam as well as two complex nonlinear systems. Many engineering structures, such as offshore foundations, oil platform supports, tower structures and moving arms, are modeled as tapered beams...
Optimized spectral estimation for nonlinear synchronizing systems.
Sommerlade, Linda; Mader, Malenka; Mader, Wolfgang; Timmer, Jens; Thiel, Marco; Grebogi, Celso; Schelter, Björn
2014-03-01
In many fields of research nonlinear dynamical systems are investigated. When more than one process is measured, besides the distinct properties of the individual processes, their interactions are of interest. Often linear methods such as coherence are used for the analysis. The estimation of coherence can lead to false conclusions when applied without fulfilling several key assumptions. We introduce a data driven method to optimize the choice of the parameters for spectral estimation. Its applicability is demonstrated based on analytical calculations and exemplified in a simulation study. We complete our investigation with an application to nonlinear tremor signals in Parkinson's disease. In particular, we analyze electroencephalogram and electromyogram data.
Statistical mechanics of a discrete nonlinear system
Rasmussen; Cretegny; Kevrekidis; Gronbech-Jensen
2000-04-24
Statistical mechanics of the discrete nonlinear Schrodinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of T = infinity, we identify a phase transition through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breatherlike localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.
Nonlinear dynamics in distributed systems
Adjali, I; Gell-Mann, Murray; Iqbal Adjali; Jose-Luis Fernandez-Villacanas; Michael Gell
1994-01-01
formulate it in a way that the deterministic and stochastic processes within the system are clearly separable. We show how internal fluctuations can be analysed in a systematic way using Van Kanpen's expansion method for Markov processes. We present some results for both stationary and time-dependent states. Our approach allows the effect of fluctuations to be explored, particularly in finite systems where such processes assume increasing importance.
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...
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
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.
Roy, Prasanta; Roy, Binoy Krishna
2016-07-01
The Quadruple Tank Process (QTP) is a well-known benchmark of a nonlinear coupled complex MIMO process having both minimum and nonminimum phase characteristics. This paper presents a novel self tuning type Dual Mode Adaptive Fractional Order PI controller along with an Adaptive Feedforward controller for the QTP. The controllers are designed based on a novel Variable Parameter Transfer Function model. The effectiveness of the proposed model and controllers is tested through numerical simulation and experimentation. Results reveal that the proposed controllers work successfully to track the reference signals in all ranges of output. A brief comparison with some of the earlier reported similar works is presented to show that the proposed control scheme has some advantages and better performances than several other similar works.
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.
Energy Technology Data Exchange (ETDEWEB)
Abbas, Ibrahim A., E-mail: aabbas5@kau.edu.sa [Department of Mathematics, Faculty of Science and Arts-Khulais, King Abdulaziz University, Jeddah (Saudi Arabia); Department of mathematics, Faculty of Science, Sohag University, Sohag (Egypt)
2015-03-01
In the present work, we consider the problem of fractional order thermoelastic interaction in a material placed in a magnetic field and subjected to a moving plane of heat source. The basic equations have been written in the form of a vector–matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. The inverse Laplace transforms are computed numerically and some comparisons have been shown in figures to estimate the effect of each of the fractional order, heat source velocity, time and the magnetic field and parameters. - Highlights: • The problem of fractional order thermoelastic interaction in a material placed in a magnetic field and subjected to a moving plane of heat source. • The eigenvalue approach gives exact solution in the Laplace domain without any assumed restrictions on the actual physical quantities. • Numerical results for the temperature, displacement and the stress distributions are represented graphically.
A Fractional-Order Phase-Locked Loop with Time-Delay and Its Hopf Bifurcation
Yu, Ya-Juan; Wang, Zai-Hua
2013-11-01
A fractional-order phase-locked loop (PLL) with a time-delay is firstly proposed on the basis of the fact that a capacitor has memory. The existence of Hopf bifurcation of the fractional-order PLL with a time-delay is investigated by studying the root location of the characteristic equation, and the bifurcated periodic solution and its stability are studied simply by using “pseudo-oscillator analysis". The results are checked by numerical simulation. It is found that the fractional-order PLL with a time-delay reduces the locking time, and it minimizes the amplitude of the bifurcated periodic solution when the order is properly chosen.
Institute of Scientific and Technical Information of China (English)
Ibrahim A. Abbas
2015-01-01
The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials (FGM) (i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach. The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.
Chakraborty, Mithun; Konar, Amit
2008-01-01
The Proportional-Integral-Derivative Controller is widely used in industries for process control applications. Fractional-order PID controllers are known to outperform their integer-order counterparts. In this paper, we propose a new technique of fractional-order PID controller synthesis based on peak overshoot and rise-time specifications. Our approach is to construct an objective function, the optimization of which yields a possible solution to the design problem. This objective function is optimized using two popular bio-inspired stochastic search algorithms, namely Particle Swarm Optimization and Differential Evolution. With the help of a suitable example, the superiority of the designed fractional-order PID controller to an integer-order PID controller is affirmed and a comparative study of the efficacy of the two above algorithms in solving the optimization problem is also presented.
Rakkiyappan, R; Velmurugan, G; Cao, Jinde
2015-04-01
In this paper, the problem of the existence, uniqueness and uniform stability of memristor-based fractional-order neural networks (MFNNs) with two different types of memductance functions is extensively investigated. Moreover, we formulate the complex-valued memristor-based fractional-order neural networks (CVMFNNs) with two different types of memductance functions and analyze the existence, uniqueness and uniform stability of such networks. By using Banach contraction principle and analysis technique, some sufficient conditions are obtained to ensure the existence, uniqueness and uniform stability of the considered MFNNs and CVMFNNs with two different types of memductance functions. The analysis results establish from the theory of fractional-order differential equations with discontinuous right-hand sides. Finally, four numerical examples are presented to show the effectiveness of our theoretical results.
Variable Separation Approach to Solve Nonlinear Systems
Institute of Scientific and Technical Information of China (English)
SHEN Shou-Feng; PAN Zu-Liang; ZHANG Jun
2004-01-01
The variable separation approach method is very useful to solving (2+ 1 )-dimensional integrable systems. But the (1+1)-dimensional and (3+ 1 )-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1+1) dimensions by taking the Redekopp system as a simple example and (3+1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3+ 1 )-dimensional universal formula obtained from many (2+ 1 )-dimensional systems is extended.
Variable Separation Approach to Solve Nonlinear Systems
Institute of Scientific and Technical Information of China (English)
SHENShou-Feng; PANZu-Liang; ZHANGJun
2004-01-01
The variable separation approach method is very useful to solving (2+1)-dimensional integrable systems.But the (1+1)-dimensional and (3+1)-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1+1) dimensions by taking the Redekopp system as a simp!e example and (3+1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3+1)-dimensional universal formula obtained from many (2+1)-dimensional systems is extended.
Spectral decomposition of nonlinear systems with memory.
Svenkeson, Adam; Glaz, Bryan; Stanton, Samuel; West, Bruce J
2016-02-01
We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.
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.
Conditions on Structural Controllability of Nonlinear Systems: Polynomial Method
Directory of Open Access Journals (Sweden)
Qiang Ma
2011-03-01
Full Text Available In this paper the structural controllability of a class of a nonlinear system is investigated. The transfer function (matrix of nonlinear systems is obtained by putting the nonlinear system model on non-commutative ring. Conditions of structural controllability of nonlinear systems are presented according to the criterion of linear systems structural controllability in frequency domain. An example is used to testify the presented conditions finally.
Adaptive explicit Magnus numerical method for nonlinear dynamical systems
Institute of Scientific and Technical Information of China (English)
LI Wen-cheng; DENG Zi-chen
2008-01-01
Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group,an efficient numerical method is proposed for nonlinear dynamical systems.To improve computational efficiency,the integration step size can be adaptively controlled.Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system,the van der Pol system with strong stiffness,and the nonlinear Hamiltonian pendulum system.
Nonlinear System Control Using Neural Networks
Directory of Open Access Journals (Sweden)
Jaroslava Žilková
2006-10-01
Full Text Available The paper is focused especially on presenting possibilities of applying off-linetrained artificial neural networks at creating the system inverse models that are used atdesigning control algorithm for non-linear dynamic system. The ability of cascadefeedforward neural networks to model arbitrary non-linear functions and their inverses isexploited. This paper presents a quasi-inverse neural model, which works as a speedcontroller of an induction motor. The neural speed controller consists of two cascadefeedforward neural networks subsystems. The first subsystem provides desired statorcurrent components for control algorithm and the second subsystem providescorresponding voltage components for PWM converter. The availability of the proposedcontroller is verified through the MATLAB simulation. The effectiveness of the controller isdemonstrated for different operating conditions of the drive system.
Control of nonlinear systems with applications
Pan, Haizhou
In practical applications of feedback control, most actuators exhibit physical constraints that limit the control amplitude and/or rate. The principal challenge of control design problem for linear systems with input constraints is to ensure closed-loop stability and yield a good transient performance in the presence of amplitude and/or rate-limited control. Since actuator saturation manifests itself as a nonlinear behavior in an otherwise linear system, the development of a nonconservative saturation control design methodology poses a significant challenge. In particular, it is well known that unstable linear systems can be stabilized using smooth controllers only in a local sense in the presence of actuator saturation. Thus, it is of paramount importance to develop a saturation control design methodology that yields a nonconservative estimate of the stability domain for closed-loop system. The first part of this research focuses on a numerically tractable formulation of the control synthesis problem for linear systems with actuator amplitude and rate saturation nonlinearity using a linear-matrix-inequality (LMI) framework. Following the recent trend in the actuator saturation control research, we (i) utilize absolute stability theory to ensure closed-loop stability and (ii) minimize a quadratic cost to account for the closed-loop system performance degradation. In order to reduce the inherent conservatism of the absolute stability based saturation control framework, we exploit stability multipliers (of, e.g., weighted circle criterion, Popov criterion, etc.). For the control of linear systems with simultaneous actuator amplitude and rate saturation nonlinearities, by virtue of a rate limiter that is predicated on designing the control amplitude and then computing the control rates, we directly account for rate constraints. Both continuous- and discrete-time systems with actuator saturation are considered. A number of design examples are presented to demonstrate
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.
Stability Analysis of a Class of Fractional-order Neural Networks
Directory of Open Access Journals (Sweden)
Tao Zou
2013-09-01
Full Text Available In this paper, the problems of the existence and uniqueness of solutions and stability for a class of fractional-order neural networks are studied by using Banach fixed point principle and analysis technique, respectively. A sufficient condition is given to ensure the existence and uniqueness of solutions and uniform stability of solutions for fractional-order neural networks with variable coefficients and multiple time delays. The obtained results improve and extend some previous works to some extent, and they are easy to check in practice. An illustrative example is presented to show the validity and application of the proposed results.
Constructing conservation laws for fractional-order integro-differential equations
Lukashchuk, S. Yu.
2015-08-01
In a class of functions depending on linear integro-differential fractional-order variables, we prove an analogue of the fundamental operator identity relating the infinitesimal operator of a point transformation group, the Euler-Lagrange differential operator, and Noether operators. Using this identity, we prove fractional-differential analogues of the Noether theorem and its generalizations applicable to equations with fractional-order integrals and derivatives of various types that are Euler-Lagrange equations. In explicit form, we give fractional-differential generalizations of Noether operators that gives an efficient way to construct conservation laws, which we illustrate with three examples.
Finite-time synchronization of fractional-order memristor-based neural networks with time delays.
Velmurugan, G; Rakkiyappan, R; Cao, Jinde
2016-01-01
In this paper, we consider the problem of finite-time synchronization of a class of fractional-order memristor-based neural networks (FMNNs) with time delays and investigated it potentially. By using Laplace transform, the generalized Gronwall's inequality, Mittag-Leffler functions and linear feedback control technique, some new sufficient conditions are derived to ensure the finite-time synchronization of addressing FMNNs with fractional order α:1neural networks. Finally, three numerical examples are presented to show the effectiveness of our proposed theoretical results.
Consensus tracking for multiagent systems with nonlinear dynamics.
Dong, Runsha
2014-01-01
This paper concerns the problem of consensus tracking for multiagent systems with a dynamical leader. In particular, it proposes the corresponding explicit control laws for multiple first-order nonlinear systems, second-order nonlinear systems, and quite general nonlinear systems based on the leader-follower and the tree shaped network topologies. Several numerical simulations are given to verify the theoretical results.
Nonlinear dynamic macromodeling techniques for audio systems
Ogrodzki, Jan; Bieńkowski, Piotr
2015-09-01
This paper develops a modelling method and a models identification technique for the nonlinear dynamic audio systems. Identification is performed by means of a behavioral approach based on a polynomial approximation. This approach makes use of Discrete Fourier Transform and Harmonic Balance Method. A model of an audio system is first created and identified and then it is simulated in real time using an algorithm of low computational complexity. The algorithm consists in real time emulation of the system response rather than in simulation of the system itself. The proposed software is written in Python language using object oriented programming techniques. The code is optimized for a multithreads environment.
Merrikh-Bayat, Farshad
2017-03-15
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+kd2s(μ) 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.
Model reduction of systems with localized nonlinearities.
Energy Technology Data Exchange (ETDEWEB)
Segalman, Daniel Joseph
2006-03-01
An LDRD funded approach to development of reduced order models for systems with local nonlinearities is presented. This method is particularly useful for problems of structural dynamics, but has potential application in other fields. The key elements of this approach are (1) employment of eigen modes of a reference linear system, (2) incorporation of basis functions with an appropriate discontinuity at the location of the nonlinearity. Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system with a small basis set. For problems involving small amplitude dynamics, the addition of discontinuous (joint) modes appears to capture the nonlinear mechanics correctly while preserving the modal form of the predictions. For problems involving large amplitude dynamics of realistic joint models (macro-slip), the use of appropriate joint modes along with sufficient basis eigen modes to capture the frequencies of the system greatly enhances convergence, though the modal nature the result is lost. Also observed is that when joint modes are used in conjunction with a small number of elastic eigen modes in problems of macro-slip of realistic joint models, the resulting predictions are very similar to those of the full solution when seen through a low pass filter. This has significance both in terms of greatly reducing the number of degrees of freedom of the problem and in terms of facilitating the use of much larger time steps.
Nonlinear Filtering Preserves Chaotic Synchronization via Master-Slave System
Directory of Open Access Journals (Sweden)
J. S. González-Salas
2013-01-01
Full Text Available We present a study on a class of interconnected nonlinear systems and give some criteria for them to behave like a filter. Some chaotic systems present this kind of interconnected nonlinear structure, which enables the synchronization of a master-slave system. Interconnected nonlinear filters have been defined in terms of interconnected nonlinear systems. Furthermore, their behaviors have been studied numerically and theoretically on different input signals.
Coordinated formation control of multiple nonlinear systems
Institute of Scientific and Technical Information of China (English)
Wei KANG; Ning XI; Jindong TAN; Yiwen ZHAO; Yuechao WANG
2005-01-01
A general method of controller design is developed for the purpose of formation keeping and reconfiguration of nonlinear systems with multiple subsystems,such as the formation of multiple aircraft,ground vehicles,or robot arms.The model consists of multiple nonlinear systems.Controllers are designed to keep the subsystems in a required formation and to coordinate the subsystems in the presence of environmental changes.A step-by-step algorithm of controller design is developed.Sufficient conditions for the stability of formation tracking are proved.Simulations and experiments are conducted to demonstrate some useful coordination strategies such as movement with a leader,simultaneous movement,series connection of formations,and human-machine interaction.
Nonlinear Energy Collimation System for Linear Colliders
Resta-Lopez, Javier
2011-01-01
The post-linac energy collimation system of multi-TeV linear colliders is designed to fulfil an important function of protection of the Beam Delivery System (BDS) against miss-steered beams likely generated by failure modes in the main linac. For the case of the Compact Linear Collider (CLIC), the energy collimators are required to withstand the impact of a full bunch train in case of failure. This is a very challenging task, assuming the nominal CLIC beam parameters at 1.5 TeV beam energy. The increase of the transverse spot size at the collimators using nonlinear magnets is a potential solution to guarantee the survival of the collimators. In this paper we present an alternative nonlinear optics based on a skew sextupole pair for energy collimation. Performance simulation results are also presented.
Adaptive stabilization for cascade nonlinear systems
Institute of Scientific and Technical Information of China (English)
陈岚萍; 王洪元; 吴波
2004-01-01
An adaptive controller of full state feedback for certain cascade nonlinear systems achieving input-to-state stability with respect to unknown bounded disturbance is designed using backstepping and control Lyapunov function (CLF)techniques. We show that unknown bounded disturbance can be estimated by update laws, which requires less information on unknown disturbance, as a part of stabilizing control. The design method achieves the desired property: global robust stability. Our contribution is illustrated with the example of a disturbed pendulum.
Inverse Problems for Nonlinear Delay Systems
2011-03-15
Ba82]. For nonlinear delay systems such as those discussed here, approximation in the context of a linear semigroup framework as presented [BBu1, BBu2...linear part generates a linear semigroup as in [BBu1, BBu2, BKap]. One then uses the linear semigroup in a vari- ation of parameters implicit...BBu2, BKap] (for the linear semigroup ) plus a Gronwall inequality. An alternative (and more general) approach given in [Ba82] eschews use of the Trotter
Adaptive Control of Nonlinear Flexible Systems
1994-05-26
Proceedings of the American Control Conference , pp. 547-551, San Francisco, June 1993. 3 2 1.3 Personnel Dr. Robert Kosut and Dr. M. Giintekin Kabuli worked on...Control of Nonlinear Systems Under Matching Conditions," Proceedings of the American Control Conference , pp. 549-555, San Diego, CA, May 1990. [10] I...Poolla, P. Khargonekar, A. Tikku, J. Krause and K. Nagpal, "A time-domain ap- proach to model validation," Proceedings
Controllability of nonlinear degenerate parabolic cascade systems
Directory of Open Access Journals (Sweden)
Mamadou Birba
2016-08-01
Full Text Available This article studies of null controllability property of nonlinear coupled one dimensional degenerate parabolic equations. These equations form a cascade system, that is, the solution of the first equation acts as a control in the second equation and the control function acts only directly on the first equation. We prove positive null controllability results when the control and a coupling set have nonempty intersection.
Nonlinear dynamics analysis of a new autonomous chaotic system
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, a new nonlinear autonomous system introduced by Chlouverakis and Sprott is studied further, to present very rich and complex nonlinear dynamical behaviors. Some basic dynamical properties are studied either analytically or nuchaotic system with very high Lyapunov dimensions is constructed and investigated. Two new nonlinear autonomous systems can be changed into one another by adding or omitting some constant coefficients.
Solving linear fractional-order differential equations via the enhanced homotopy perturbation method
Energy Technology Data Exchange (ETDEWEB)
Naseri, E; Ghaderi, R; Sadati, J; Mahmoudian, M; Hosseinnia, S H [Intelligent System Research Group, Babol, Noushirvani University of Technology, Faculty of Electrical and Computer Engineering, PO Box 47135-484, Babol (Iran, Islamic Republic of); Ranjbar N, A [Golestan University, Gorgan (Iran, Islamic Republic of); Momani, S [Mutah University, PO Box 7, Al-Karak (Jordan)], E-mail: h.hoseinnia@stu.nit.ac.ir, E-mail: a.ranjbar@nit.ac.ir, E-mail: shahermm@yahoo.com
2009-10-15
The linear fractional differential equation is solved using the enhanced homotopy perturbation method (EHPM). In this method, the convergence has been provided by selecting a stabilizing linear part. The most significant features of this method are its simplicity and its excellent accuracy and convergence for the whole range of fractional-order differential equations.
Solving linear fractional-order differential equations via the enhanced homotopy perturbation method
Naseri, E.; Ghaderi, R.; Ranjbar N, A.; Sadati, J.; Mahmoudian, M.; Hosseinnia, S. H.; Momani, S.
2009-10-01
The linear fractional differential equation is solved using the enhanced homotopy perturbation method (EHPM). In this method, the convergence has been provided by selecting a stabilizing linear part. The most significant features of this method are its simplicity and its excellent accuracy and convergence for the whole range of fractional-order differential equations.
Directory of Open Access Journals (Sweden)
Archana Chauhan
2011-08-01
Full Text Available In this work we consider a class of impulsive fractional-order semilinear evolution equations with a nonlocal initial condition. By means of solution operator and application of fixed point theorems we established the existence and uniqueness of a mild solution.
Indian Academy of Sciences (India)
Syed Abbas; V Kavitha; R Murugesu
2015-08-01
In this article, we study the concept of Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. We establish the results with Lipschitz condition and without Lipschitz condition on the forcing term. An interesting example is presented to illustrate the main findings. The results proven are new and complement the existing ones.
Multi-Rate Fractional-Order Repetitive Control of Shunt Active Power Filter
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...
Identification of Nonlinear Systems Using Neurofuzzy Networks
Institute of Scientific and Technical Information of China (English)
LI Ying; JIAO Licheng
2001-01-01
This paper presents a compound neu-ral network model, I.e., adaptive neurofuzzy network(ANFN), which can be used for identifying the com-plicated nonlinear system. The proposed ANFN has asimple structure and exploits a hybrid algorithm com-bining supervised learning and unsupervised learning.In addition, ANFN is capable of overcoming the errorof system identification due to the existence of somechanging points and improving the accuracy of identi-fication of the whole system. The effectiveness of themodel and its algorithm are tested on the identifica-tion results of missile attacking area.
Das, Saptarshi; Maharatna, Koushik
2016-01-01
In this paper, an efficient control strategy for physiological interaction based anaesthetic drug infusion model is explored using the fractional order (FO) proportional integral derivative (PID) controllers. The dynamic model is composed of several human organs by considering the brain response to the anaesthetic drug as output and the drug infusion rate as the control input. Particle Swarm Optimisation (PSO) is employed to obtain the optimal set of parameters for PID/FOPID controller structures. With the proposed FOPID control scheme much less amount of drug-infusion system can be designed to attain a specific anaesthetic target and also shows high robustness for +/-50% parametric uncertainty in the patient's brain model.
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.
Indian Academy of Sciences (India)
AYYAZ ALI; MUHAMMAD ASAD IQBAL; SYED TAUSEEF MOHYUD-DIN
2016-11-01
In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp$(−\\psi(\\eta)$)-expansion method is implemented tofind exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effectiveand can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method providesadditional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order andcould be protracted to other physical phenomena.
Adaptive Neuro Fuzzy Inference Controller for Full Vehicle Nonlinear Active Suspension Systems
Directory of Open Access Journals (Sweden)
A. Aldair
2010-12-01
Full Text Available The main objective of designed the controller for a vehicle suspension system is to reduce the discomfort sensed by passengers which arises from road roughness and to increase the ride handling associated with the pitching and rolling movements. This necessitates a very fast and accurate controller to meet as much control objectives, as possible. Therefore, this paper deals with an artificial intelligence Neuro-Fuzzy (NF technique to design a robust controller to meet the control objectives. The advantage of this controller is that it can handle the nonlinearities faster than other conventional controllers. The approach of the proposed controller is to minimize the vibrations on each corner of vehicle by supplying control forces to suspension system when travelling on rough road. The other purpose for using the NF controller for vehicle model is to reduce the body inclinations that are made during intensive manoeuvres including braking and cornering. A full vehicle nonlinear active suspension system is introduced and tested. The robustness of the proposed controller is being assessed by comparing with an optimal Fractional Order PIλ Dμ (FOPID controller. The results show that the intelligent NF controller has improved the dynamic response measured by decreasing the cost function.
Tracking Control for Switched Cascade Nonlinear Systems
Directory of Open Access Journals (Sweden)
Xiaoxiao Dong
2015-01-01
Full Text Available The issue of H∞ output tracking for switched cascade nonlinear systems is discussed in this paper, where not all the linear parts of subsystems are stabilizable. The conditions of the solvability for the issue are given by virtue of the structural characteristics of the systems and the average dwell time method, in which the total activation time for stabilizable subsystems is longer than that for the unstabilizable subsystems. At last, a simulation example is used to demonstrate the validity and advantages of the proposed approach.
Dynamics of Nonlinear Time-Delay Systems
Lakshmanan, Muthusamy
2010-01-01
Synchronization of chaotic systems, a patently nonlinear phenomenon, has emerged as a highly active interdisciplinary research topic at the interface of physics, biology, applied mathematics and engineering sciences. In this connection, time-delay systems described by delay differential equations have developed as particularly suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant. This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics. Special attention is devoted to scalar chaotic/hyperchaotic time-delay systems, and some higher order models, occurring in different bran...
Control of self-organizing nonlinear systems
Klapp, Sabine; Hövel, Philipp
2016-01-01
The book summarizes the state-of-the-art of research on control of self-organizing nonlinear systems with contributions from leading international experts in the field. The first focus concerns recent methodological developments including control of networks and of noisy and time-delayed systems. As a second focus, the book features emerging concepts of application including control of quantum systems, soft condensed matter, and biological systems. Special topics reflecting the active research in the field are the analysis and control of chimera states in classical networks and in quantum systems, the mathematical treatment of multiscale systems, the control of colloidal and quantum transport, the control of epidemics and of neural network dynamics.
On stability of randomly switched nonlinear systems
Chatterjee, Debasish
2007-01-01
This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure global asymptotic stability using Lyapunov-based methods when individual subsystems are stable and a certain ``slow switching'' condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markov jump systems in particular. For systems with control inputs we provide explicit control s...
Synchronization between two different chaotic systems with nonlinear feedback control
Institute of Scientific and Technical Information of China (English)
Lü Ling; Guo Zhi-An; Zhang Chao
2007-01-01
This paper presents chaos synchronization between two different chaotic systems by using a nonlinear controller, in which the nonlinear functions of the system are used as a nonlinear feedback term. The feedback controller is designed on the basis of stability theory, and the area of feedback gain is determined. The artificial simulation results show that this control method is commendably effective and feasible.
Model Reduction for Nonlinear Systems by Incremental Balanced Truncation
Besselink, Bart; van de Wouw, Nathan; Scherpen, Jacquelien M. A.; Nijmeijer, Henk
2014-01-01
In this paper, the method of incremental balanced truncation is introduced as a tool for model reduction of nonlinear systems. Incremental balanced truncation provides an extension of balanced truncation for linear systems towards the nonlinear case and differs from existing nonlinear balancing tech
Model Reduction for Nonlinear Systems by Incremental Balanced Truncation
Besselink, Bart; van de Wouw, Nathan; Scherpen, Jacquelien M. A.; Nijmeijer, Henk
2014-01-01
In this paper, the method of incremental balanced truncation is introduced as a tool for model reduction of nonlinear systems. Incremental balanced truncation provides an extension of balanced truncation for linear systems towards the nonlinear case and differs from existing nonlinear balancing tech
Nonlinear control for dual quaternion systems
Price, William D.
The motion of rigid bodies includes three degrees of freedom (DOF) for rotation, generally referred to as roll, pitch and yaw, and 3 DOF for translation, generally described as motion along the x, y and z axis, for a total of 6 DOF. Many complex mechanical systems exhibit this type of motion, with constraints, such as complex humanoid robotic systems, multiple ground vehicles, unmanned aerial vehicles (UAVs), multiple spacecraft vehicles, and even quantum mechanical systems. These motions historically have been analyzed independently, with separate control algorithms being developed for rotation and translation. The goal of this research is to study the full 6 DOF of rigid body motion together, developing control algorithms that will affect both rotation and translation simultaneously. This will prove especially beneficial in complex systems in the aerospace and robotics area where translational motion and rotational motion are highly coupled, such as when spacecraft have body fixed thrusters. A novel mathematical system known as dual quaternions provide an efficient method for mathematically modeling rigid body transformations, expressing both rotation and translation. Dual quaternions can be viewed as a representation of the special Euclidean group SE(3). An eight dimensional representation of screw theory (combining dual numbers with traditional quaternions), dual quaternions allow for the development of control techniques for 6 DOF motion simultaneously. In this work variable structure nonlinear control methods are developed for dual quaternion systems. These techniques include use of sliding mode control. In particular, sliding mode methods are developed for use in dual quaternion systems with unknown control direction. This method, referred to as self-reconfigurable control, is based on the creation of multiple equilibrium surfaces for the system in the extended state space. Also in this work, the control problem for a class of driftless nonlinear systems is
Nonlinear and Variable Structure Excitation Controller for Power System Stability
Institute of Scientific and Technical Information of China (English)
Wang Ben; Ronnie Belmans
2006-01-01
A new nonlinear variable structure excitation controller is proposed. Its design combines the differential geometry theory and the variable structure controlling theory. The mathematical model in the form of "an affine nonlinear system" is set up for the control of a large-scale power system. The static and dynamic performances of the nonlinear variable structure controller are simulated. The response of system with the controller proposed is compared to that of the nonlinear optimal controller when the system is subjected to a variety of disturbances. Simulation results show that the nonlinear variable structure excitation controller gives more satisfactorily static and dynamic performance and better robustness.
μ Synthesis Method for Robust Control of Uncertain Nonlinear Systems
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
μ synthesis method for robust control of uncertain nonlinear systems is propored, which is based on feedback linearization. First, nonlinear systems are linearized as controllable linear systems by I/O linearization,such that uncertain nonlinear systems are expressed as the linear fractional transformations (LFTs) on the generalized linearized plants and uncertainty.Then,linear robust controllers are obtained for the LFTs usingμsynthesis method based on H∞ optimization.Finally,the nonlinear robust controllers are constructed by combining the linear robust controllers and the nonlinear feedback.An example is given to illustrate the design.
Sliding mode identifier for parameter uncertain nonlinear dynamic systems with nonlinear input
Institute of Scientific and Technical Information of China (English)
张克勤; 庄开宇; 苏宏业; 褚健; 高红
2002-01-01
This paper presents a sliding mode(SM) based identifier to deal with the parameter idenfification problem for a class of parameter uncertain nonlinear dynamic systems with input nonlinearity. A sliding mode controller (SMC) is used to ensure the global reaching condition of the sliding mode for the nonlinear system;an identifier is designed to identify the uncertain parameter of the nonlinear system. A numerical example is studied to show the feasibility of the SM controller and the asymptotical convergence of the identifier.
Institute of Scientific and Technical Information of China (English)
SUN WeiJie; HUANG Jie
2009-01-01
In this paper,we consider the global robust output regulation problem for a class of uncertain nonlinear systems with nonlinear exosystems.By employing the internal model approach,we show that this problem boils down to a global robust stabilization problem of a time-varying nonlinear system in lower triangular form,the solution of which will lead to the solution of the global robust output regulation problem.An example shows the effectiveness of the proposed approach.
Fractional order Buck-Boost converter in CCM: modelling, analysis and simulations
Wang, Faqiang; Ma, Xikui
2014-12-01
In this paper, the modelling, analysis and the power electronics simulator (PSIM) simulations of the fractional order Buck-Boost converter operating in continuous conduction mode (CCM) operation are investigated. Based on the three-terminal switch device method, the average circuit model of the fractional order Buck-Boost converter is established, and the corresponding DC equivalent circuit model and AC small signal equivalent circuit model are presented. And then, the equilibrium point and the transfer functions are derived. It is found that the equilibrium point is not influenced by the inductor's or the capacitor's order, but both these orders are included in the derived transfer functions. Finally, the comparisons between the theoretical analysis and the PSIM simulations are given for confirmation.
An Approach to Differential Geometry of Fractional Order via Modified Riemann-Liouville Derivative
Institute of Scientific and Technical Information of China (English)
Guy JUMARIE
2012-01-01
In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative,one (Jumarie)has proposed recently an alternative referred to as (local) modified Riemann-Liouville definition,which directly,provides a Taylor's series of fractional order for non differentiable functions.We examine here in which way this calculus can be used as a framework for a differential geometry of fractional order.One will examine successively implicit function,manifold,length of curves,radius of curvature,Christoffel coefficients,velocity,acceleration.One outlines the application of this framework to Lagrange optimization in mechanics,and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.
Xiao, Min; Zheng, Wei Xing; Jiang, Guoping; Cao, Jinde
2015-12-01
In this paper, a fractional-order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated, such as the stability, Hopf bifurcations, and undamped oscillations. The stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate-order neural network. Based on the stability analysis, the critical values of the fractional order are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state. Then, the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate-order networks. Meanwhile, it is shown that the incommensurate-order neural network can also exhibit a Hopf bifurcation as the network parameter passes through a critical value which can be determined exactly. The frequency and amplitude of bifurcated oscillations are determined.
A proposed fractional-order Gompertz model and its application to tumour growth data.
Bolton, Larisse; Cloot, Alain H J J; Schoombie, Schalk W; Slabbert, Jacobus P
2015-06-01
A fractional-order Gompertz model of orders between 0 and 2 is proposed. The main purpose of this investigation is to determine whether the ordinary or proposed fractional Gompertz model would best fit our experimental dataset. The solutions for the proposed model are obtained using fundamental concepts from fractional calculus. The closed-form equations of both the proposed model and the ordinary Gompertz model are calibrated using an experimental dataset containing tumour growth volumes of a Rhabdomyosarcoma tumour in a mouse. With regard to the proposed model, the order, within the interval mentioned, that resulted in the best fit to the data was used in a further investigation into the prediction capability of the model. This was compared to the prediction capability of the ordinary Gompertz model. The result of the investigation was that a fractional-order Gompertz model of order 0.68 produced a better fit to our experimental dataset than the well-known ordinary Gompertz model.
Observability and Information Structure of Nonlinear Systems,
1985-10-01
defined by Shannon and used as a measure of mut.:al infor-mation between event x. and y4. If p(x.l IY.) I I(x., y.) xil -in (1/p(x.)) =- JInp (x.) (2...entropy H(x,y) in a similar way as H(x,y) = - fx,yp(xiy)lnp(x,y)cdlY, = -E[ JInp (x,y)]. (3-13) With the above definitions, mutual information between x...Observabiity of Nonlinear Systems, Eng. Cybernetics, Volume 1, pp 338-345, 1972. 18. Sen , P., Chidambara, M.R., Observability of a Class of Nonli-.ear
Identification methods for nonlinear stochastic systems.
Fullana, Jose-Maria; Rossi, Maurice
2002-03-01
Model identifications based on orbit tracking methods are here extended to stochastic differential equations. In the present approach, deterministic and statistical features are introduced via the time evolution of ensemble averages and variances. The aforementioned quantities are shown to follow deterministic equations, which are explicitly written within a linear as well as a weakly nonlinear approximation. Based on such equations and the observed time series, a cost function is defined. Its minimization by simulated annealing or backpropagation algorithms then yields a set of best-fit parameters. This procedure is successfully applied for various sampling time intervals, on a stochastic Lorenz system.
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
Energy Technology Data Exchange (ETDEWEB)
Li Xicheng; Xu Mingyu [Institute of Applied Mathematics, School of Mathematics and System Science, Shandong University, Jinan 250100 (China); Wang Shaowei [Department of Mechanics and Engineering Science, Peking University, Beijing 100871 (China)], E-mail: xichengli@yahoo.com.cn
2008-04-18
In this paper, we give similarity solutions of partial differential equations of fractional order with a moving boundary condition. The solutions are given in terms of a generalized Wright function. The time-fractional Caputo derivative and two types of space-fractional derivatives are considered. The scale-invariant variable and the form of the solution of the moving boundary are obtained by the Lie group analysis. A comparison between the solutions corresponding to two types of fractional derivative is also given.
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.
Boundary control of long waves in nonlinear dispersive systems
DEFF Research Database (Denmark)
Hasan, Agus; Foss, Bjarne; Aamo, Ole Morten
2011-01-01
Unidirectional propagation of long waves in nonlinear dispersive systems may be modeled by the Benjamin-Bona-Mahony-Burgers equation, a third order partial differential equation incorporating linear dissipative and dispersive terms, as well as a term covering nonlinear wave phenomena. For higher...... orders of the nonlinearity, the equation may have unstable solitary wave solutions. Although it is a one dimensional problem, achieving a global result for this equation is not trivial due to the nonlinearity and the mixed partial derivative. In this paper, two sets of nonlinear boundary control laws...... that achieve global exponential stability and semi-global exponential stability are derived for both linear and nonlinear cases....
Comparative Study of Controllers for a Variable Area MIMO Interacting NonLinear System
Directory of Open Access Journals (Sweden)
Priya Chandrasekar
2014-03-01
Full Text Available Most of the industrial processes are basically Multi Input Multi Output (MIMO system. In this paper a new combination of Spherical Conical Interacting Tank System (SCITS which is a variable area nonlinear MIMO system is considered for study and various control algorithms based on Ziegler Nichol’s tuning method, Hagglund Astrom Robust tuning method, Fractional Order (FO control and Passivity Based Control (PBC are used and compared for the level control of spherical tank system and conical tank system connected with interaction. Transfer function matrix of the system is obtained experimentally from the open loop response of the system. The designed controllers are tested for servo and regulatory operations. The controllers are compared in terms of time domain specification and performance index criterion. From the analysis of the simulation results, it is seen that FO controller gives improved performance when compared to conventional Integer Order (IO controller and overall Passivity Based Controller (PBCr gives improved performance comparatively for spherical conical interacting MIMO system.
Shahnazi, Reza
2015-01-01
An adaptive fuzzy output feedback controller is proposed for a class of uncertain MIMO nonlinear systems with unknown input nonlinearities. The input nonlinearities can be backlash-like hysteresis or dead-zone. Besides, the gains of unknown input nonlinearities are unknown nonlinear functions. Based on universal approximation theorem, the unknown nonlinear functions are approximated by fuzzy systems. The proposed method does not need the availability of the states and an observer based on strictly positive real (SPR) theory is designed to estimate the states. An adaptive robust structure is used to cope with fuzzy approximation error and external disturbances. The semi-global asymptotic stability of the closed-loop system is guaranteed via Lyapunov approach. The applicability of the proposed method is also shown via simulations.
Nonlinear Systems of Second-Order ODEs
Directory of Open Access Journals (Sweden)
Patricio Cerda
2008-02-01
Full Text Available We study existence of positive solutions of the nonlinear system Ã¢ÂˆÂ’(p1(t,u,vuÃ¢Â€Â²Ã¢Â€Â²=Ã¢Â€Â…h1(tf1(t,u,v in (0,1; Ã¢ÂˆÂ’(p2(t,u,vvÃ¢Â€Â²Ã¢Â€Â²=h2(tf2(t,u,v in (0,1; u(0=u(1=v(0=v(1=0, where p1(t,u,v=1/(a1(t+c1g1(u,v and p2(t,u,v=1/(a2(t+c2g2(u,v. Here, it is assumed that g1, g2 are nonnegative continuous functions, a1(t, a2(t are positive continuous functions, c1,c2Ã¢Â‰Â¥0, h1,h2Ã¢ÂˆÂˆL1(0,1, and that the nonlinearities f1,Ã¢Â€Â…f2 satisfy superlinear hypotheses at zero and +Ã¢ÂˆÂž. The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of u and v.
Impulse position control algorithms for nonlinear systems
Energy Technology Data Exchange (ETDEWEB)
Sesekin, A. N., E-mail: sesekin@list.ru [Ural Federal University, 19 S. Mira, Ekaterinburg, 620002 (Russian Federation); Institute of Mathematics and Mechanics, Ural Division of Russian Academy of Sciences, 16 S. Kovalevskaya, Ekaterinburg, 620990 (Russian Federation); Nepp, A. N., E-mail: anepp@urfu.ru [Ural Federal University, 19 S. Mira, Ekaterinburg, 620002 (Russian Federation)
2015-11-30
The article is devoted to the formalization and description of impulse-sliding regime in nonlinear dynamical systems that arise in the application of impulse position controls of a special kind. The concept of trajectory impulse-sliding regime formalized as some limiting network element Euler polygons generated by a discrete approximation of the impulse position control This paper differs from the previously published papers in that it uses a definition of solutions of systems with impulse controls, it based on the closure of the set of smooth solutions in the space of functions of bounded variation. The need for the study of such regimes is the fact that they often arise when parry disturbances acting on technical or economic control system.
Impulse position control algorithms for nonlinear systems
Sesekin, A. N.; Nepp, A. N.
2015-11-01
The article is devoted to the formalization and description of impulse-sliding regime in nonlinear dynamical systems that arise in the application of impulse position controls of a special kind. The concept of trajectory impulse-sliding regime formalized as some limiting network element Euler polygons generated by a discrete approximation of the impulse position control This paper differs from the previously published papers in that it uses a definition of solutions of systems with impulse controls, it based on the closure of the set of smooth solutions in the space of functions of bounded variation. The need for the study of such regimes is the fact that they often arise when parry disturbances acting on technical or economic control system.