Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory

In this paper stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance.     

Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach

In this paper, a stability test procedure is proposed for linear nonhomogeneous fractional order systems with a pure time delay. Some basic results from the area of finite time and practical stability are extended to linear, continuous, fractional order time-delay systems given in state-space form. Sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems. A numerical example is given to illustrate the validity of the proposed procedure.     

分数阶Duffing混沌系统的动力性态及其由单一主动控制的混沌同步

随着物理与技术的深入研究,分数阶非线性系统的动力性态及其分数阶混沌系统的同步成为研究的焦点.研究了分数阶Duffing系统的动力性态包括混沌性质,并且由分数阶非线性稳定性准则得到了分数阶非自治系统的混沌同步.特别地,研究了由单一主动控制的分数阶Duffing系统的同步.相应的数值结果演示了方法的有效性.     

An indirect Lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty

The paper is concerned with the problem of the robust stabilization for a class of fractional order linear systems with positive real uncertainty under Riemann–Liouville (RL) derivatives. Firstly, by utilizing the continuous frequency distributed model of the fractional integrator, the fractional order system is expressed as an infinite dimensional integral order system. And via using indirect Lyapunov approach and linear matrix inequality techniques, sufficient condition for robust asymptotic stability of the fractional order systems and design methods of the state feedback controller are presented. Secondly, by using matrixs singular value decomposition technique the static output feedback controller and observer-based controller for asymptotically stabilizing the fractional order systems are derived. Finally, the validity of the proposed methods are demonstrated by numerical examples.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

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.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

Chaos synchronization of a fractional‐order modified Van der Pol–Duffing system via new linear control,backstepping control and Takagi–Sugeno fuzzy approaches

In this article, based on the stability theory of fractional‐order systems, chaos synchronization is achieved in the fractional‐order modified Van der Pol–Duffing system via a new linear control approach. A fractional backstepping controller is also designed to achieve chaos synchronization in the proposed system. Takagi‐Sugeno fuzzy models‐based are also presented to achieve chaos synchronization in the fractional‐order modified Van der Pol–Duffing system via linear control technique. Numerical simulations are used to verify the effectiveness of the synchronization schemes. © 2015 Wiley Periodicals, Inc. Complexity 21: 116–124, 2016     

Chaos synchronization of the fractional-order Chen’s system

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora–Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

A robust method for new fractional hybrid chaos synchronization

In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer‐order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used. Copyright © 2016 John Wiley & Sons, Ltd.     

On the stability of fractional neutron point kinetics (FNPK)

The aim of this work is investigate the stability of fractional neutron point kinetics (FNPK). The method applied in this work considers the stability of FNPK as a linear fractional differential equation by transforming the s − plane to the W − plane. The FNPK equations is an approximation of the dynamics of the reactor that includes three new terms related to fractional derivatives, which are explored in this work with an aim to understand their effect in the system stability. Theoretical study of reactor dynamical systems plays a significant role in understanding the behavior of neutron density, which is important in the analysis of reactor safety. The fractional relaxation time (τ^α) for values of fractional-order derivative (α) were analyzed, and the minimum absolute phase was obtained in order to establish the stability of the system. The results show that nuclear reactor stability with FNPK is a function of the fractional relaxation time.     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

On chaos control and synchronization of the commensurate fractional order Liu system

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.     

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.     

Switching adaptive controllers to control fractional‐order complex systems with unknown structure and input nonlinearities

This article investigates the chaos control problem for the fractional‐order chaotic systems containing unknown structure and input nonlinearities. Two types of nonlinearity in the control input are considered. In the first case, a general continuous nonlinearity input is supposed in the controller, and in the second case, the unknown dead‐zone input is included. In each case, a proper switching adaptive controller is introduced to stabilize the fractional‐order chaotic system in the presence of unknown parameters and uncertainties. The control methods are designed based on the boundedness property of the chaotic system's states, where, in the proposed methods the nonlinear/linear dynamic terms of the fractional‐order chaotic systems are assumed to be fully unknown. The analytical results of the mentioned techniques are proved by the stability analysis theorem of fractional‐order systems and the adaptive control method. In addition, as an application of the proposed methods, single input adaptive controllers are adopted for control of a class of three‐dimensional nonlinear fractional‐order chaotic systems. And finally, some numerical examples illustrate the correctness of the analytical results. © 2014 Wiley Periodicals, Inc. Complexity 21: 211–223, 2015     

Modified projective synchronization of uncertain fractional order hyperchaotic systems

Base on the stability theory of fractional order system, this work mainly investigates modified projective synchronization of two fractional order hyperchaotic systems with unknown parameters. A controller is designed for synchronization of two different fractional order hyperchaotic systems. The method is successfully applied to modified projective synchronization between fractional order Rössler hyperchaotic system and fractional order Chen hyperchaotic system, and numerical simulations illustrate the effectiveness of the obtained results.     

Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control

Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems.     

分数阶时变广义系统稳定性分析

基于Caputo分数阶导数,研究了分数阶时变广义线性系统和分数阶时变广义非线性系统的稳定性问题.首先利用相关不等式,给出了一个时变广义线性系统无脉冲且稳定的充分条件.然后,通过慢子系统来判断快子系统的变化,并利用Riccati方程,建立了分数阶时变广义非线性系统是渐近稳定的判定准则.最后,给出了算例和Simulink仿真结果,以说明结论的正确性.     

Fractional order adaptive high-gain controllers for a class of linear systems

In this paper we show that a fractional adaptive controller based on high gain output feedback can always be found to stabilize any given linear, time-invariant, minimum phase, siso systems of relative degree one. We generalize the stability theorem of integer order controllers to the fractional order case, and we introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm.     

Variable structure control of linear time invariant fractional order systems using a finite number of state feedback law

In this paper, an approach based on the variable structure control is proposed for stabilization of linear time invariant fractional order systems (LTI-FOS) using a finite number of available state feedback controls, none of which is capable of stabilizing the LTI-FOS by itself. First, a system with integer order derivatives is defined and its existence is proved, which has stability equivalent properties with respect to the fractional system. This makes it possible to use Lyapunov function and convex analysis in order to define the sliding sector and develop a variable structure control which enables the switching between available control gains and stabilizing the fractional order system.