Robust Stabilization of a Class of Uncertain Fractional-order Chaotic Systems via a Novel Sliding Mode Control Scheme

Abstract

This paper proposes a novel sliding mode control (SMC) scheme to stabilize a class of fractional-order chaotic systems. Through constructing two sliding mode variables, the control problem of n-dimensional system can be transformed to the equivalent stabilizing problem of a reduced-order system. Subsequently, on the basis of second-order sliding mode (SOSM) technique, a robust control law is designed, which strongly attenuates the chattering phenomenon inherent in traditional sliding mode controller, and guarantees the existence of sliding motion in a finite time. The stability of two sliding mode variables to the origin is proved by conventional and fractional Lyapunov theories, respectively. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach. 246Copyright ⓒ 2014 SERSC to realize the control/synchronization of fractional-order chaotic systems in the past decades [13][14][15][16][17].However, the traditional SMC is of the first order, and there exists an inevitable drawback in applying such SMC, that is the so-called chattering phenomenon cannot be avoided. In order to improve the control accuracy and avoid, or at least strongly attenuate the undesired chattering effect, the second-order sliding mode (SOSM) method is proposed. The SOSM control approach was developed starting from the mid eighties, and its main aim is enhancing the control precision and eliminating the chattering phenomenon by removing the control discontinuity. To the best of our knowledge, so far, there are few works available in stabilizing/synchronizing fractional-order chaotic systems by SOSM control approach.Take the above discussions into account, this article considers the roust stabilization problem of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. The main contribution of this article is that using the novel SMC approach, the control task of n-dimensional system is transformed to the equivalent problem of stabilizing a reduced-order system. In other words, under the effect of the proposed method, if the reduced-order system is stable, then the n-dimensional system also is stable, so we only need to ensure the asymptotic stability of the reduced-order system. We use the traditional and fractional Lyapunov functions to demonstrate that the reaching phases of two sliding surfaces are stable. Besides, on the basis of SOSM control theory, a robust control law is designed, which can strongly attenuates the chattering phenomenon inherent in conventional sliding mode controller.The structure of this paper is organized as follows. In Section 2, relevant definitions and lemmas are presented. Section 3 gives the main results. In Section 4, two simulation examples are provided to verify the effectiveness of the proposed method. Conclusions are drawn in Section 5.

Linear matrix inequalities design approach for robust stabilization of uncertain nonlinear systems with perturbation based on optimally-tuned global sliding mode control

“…In Delavari (2015), based on the Lyapunov stability theory and adaptive switching gain, robust synchronization control of a class of uncertain integer and fractional-order chaotic systems is investigated. In Tian et al (2014), a novel chattering-free SMC structure based on traditional and fractional Lyapunov stability theories is offered to stabilize uncertain fractional-order chaotic systems which proven that both reaching phases of two switching surfaces are stable. In Chen and Yang (2015), based on the fractional-order Lyapunov function and generalized Gronwall inequality, the robust finite-time stabilization of fractional-order linear and time-varying impulsive systems is considered.…”