Abstract
It is shown that fractional-order (FO) nonlinear systems can also show higher nonlinearity and complex dynamics. FO chaotic systems have wider applications in secure communication, signal processing, financial field due to FO chaos has larger key space and more complex random sequences than integer-order chaos. Thanks to the lack of the effective analytical methods and controller design methods of integer-order chaotic systems can not be applied directly to FO chaos systems, to control chaos of FO chaotic systems is a very interesting and difficult problem, especially for FO chaotic system with order α:1<α<2. Based on the stability theory of FO systems and the linear state feedback control, an LMI criterion for controlling a class of fractional-order chaotic systems with fractional-order α:1<α<2 is addressed in this paper. The proposed method can be easily verified and resolved by using the Matlab LMI toolbox. Moreover, the proposed controller is linear, easy to implement and overcome some defects in the recent literature, which have improved the existing results. The method employed in this letter can effectively avoid control cost and inaccuracy in the literatures, and can be be applied to FO hyperchaos systems and synchronization controller design of FO chaotic system. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed methods.
Highlights
The main property of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time
Some methods have been applied to deal with this problem, such as OGY method [5], the feedback control method [6], impulsive control [7], backstepping method [8], adaptive control [9], bang-bang control [10], sliding mode control [11], nonlinear control [12], active control [13], and many others
On the other hand, during the past several years, with the development of theory of fractional-order calculus, it has been surprisingly found that many FO differential systems display complex bifurcation and chaos phenomena, for instance, FO Duffing’s oscillators [14], FO Chua’s circuit system [15], FO Rössler system [16], FO Chen system [17], FO Lorenz system [18], FO Lüsystem [19], FO Liu system [20], FO financial system [21], FO Volta’s system [22], FO hyperchaotic Chen system [23], FO hyperchaotic Lorenz system [24], FO hyperchaotic Rössler system [25], FO love model [26] and so on
Summary
The main property of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time This feature would lead systems to unstable, performance-degraded, and even catastrophic situations. Note that the proposed schemes in most of the previous works about chaos control, are too complex in design to implement in applications, some of these control schemes are not applicable to fractional-order chaotic systems. Simplicity in configuration and implementation, the linear feedback control is especially attractive and has been commonly adopted for practical implementations It possesses a high value in application. Based on the stability theory of fractional-order differential system and the linear feedback control, chaos control of a class of chaotic fractional-order systems is considered, a LMI conditions is established, feedback gains could be found directly from the LMI formulation. Notations: Throughout this paper, Rn and Rn×m, denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices; MT denotes transpose of matrix M; The notation M > 0(M < 0) means that the matrix M is positive (negative) definite; Sym*M+ is used to denote the expression M + MT, and ⋆ is used to denote a block matrix element that is induced by transposition
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
