# Robust Fuzzy Adaptive Sliding Mode Stabilization for Fractional-Order Chaos
## 1 Introduction  
It is undeniable that fractional-calculus (computation of derivatives and integrals of any arbitrary real or complex order) and its applications focused much interest in the last three decades [1]. This growing popularity and importance is mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering [2; 3; 4].  
A great number of researchers focused their studies on fractional order systems presenting chaotic dynamics [5; 6; 7]. Recently, a class of fractional order piece-wise continuous systems was able to produce chaotic behavior with hidden attractors [8; 9]. Many control strategies have been proposed in the related literature for the stabilization and synchronization of chaos in nonlinear fractional-order systems such as fractional-order linear controllers [10], fractional-order \(\mathrm{PI}^{\mathrm{L}}\mathrm{D}^{\mu}\) control [11], adaptive control [12], adaptive \(\mathrm{H}^{\infty}\) control [13], fuzzy adaptive control [14], and adaptive backstepping control [15].  
Recently, sliding-mode control (SMC) has gained much attention for the control and synchronization of fractional-order chaotic systems [16; 17; 18]. This is mainly due to its robustness against parameter variations and external disturbances under matching conditions [19; 20]. In fact, robustness of a control system is very important because various uncertainties exist in practical applications; furthermore, noises in general are ubiquitous in natural and synthetic systems [21]. Based on the Lyapunov theory of stability, SMC is advantageous for this consideration, because it can switch the control law very fast to drive the states of the system from any initial states into a specifiedsliding surface, and to maintain the states on the surface for all subsequent time [22]. It is worth noticing that fuzzy sets have been extensively associated with SMC to deal with uncertainties and disturbing functions in both classical and fractional-order contexts [14; 23; 24].  
In this paper, a novel fuzzy adaptive sliding mode controller is introduced to stabilize a class of fractional-order chaotic systems. The control law is developed in two stages: First we consider the ideal case (free from uncertainties and disturbances) for which a sliding surface and a SMC control law is proposed that guaranties the stability of the system in the sense of [PERSON] theory. In the second stage, the control problem of this class of fractional-order nonlinear systems in presence of uncertainties and external disturbances is studied. A fuzzy adaptive SMC control law is proposed to stabilize asymptotically the closed-loop system. In addition, simulation results on fractional-order Chen system show that the recurrent SMC problem of chattering phenomena is eliminated by the proposed control law.  
In this design procedure, both the uncertainties in the fractional-order system model and the external disturbance signal are represented by the Takagi-Sugeno (TS) fuzzy-neural-network (FNN) model.  
The rest of the paper is organized as follows. Section 2 presents some basic definitions of fractional calculus and the general fractional-order chaotic systems. Section 3 presents an introduction to fuzzy systems theory. In Section 4, a fractional order sliding mode controller design is proposed for a class of three-dimensional fractional-order chaotic systems with numerical application results to the fractional chaotic Chen system. Section 5 investigates adaptive sliding mode control design for chaotic systems with uncertainties and external disturbances using fuzzy sets. The stability analysis is performed using Lyapunov theorem and the simulation example for the control of fractional Chen's system is clearly presented to show the viability and efficiency of the proposed fractional adaptive controllers. Finally, the paper is concluded in Section 6.