%0 Journal Article
%T Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems
%A Jian Yuan
%A Bao Shi
%A Wenqiang Ji
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/576709
%X Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented. 1. Introduction Fractional calculus is an old and yet novel topic whose infancy dates back to the end of the 17th century, the time when Newton and Leibniz established the foundations of classical calculus. For three centuries, fractional calculus developed mainly as a pure theoretical mathematical field without applications. However, in the last two decades it has attracted the interest of researchers in several areas including mathematics, physics, chemistry, material, engineering, finance, and even social science. The stability of fractional differential equations (FDEs) and fractional control have both gained rapid development very recently [1¨C3]. One of the most important areas of application is the chaos theory. In recent years, fractional chaotic systems have intensively attracted a great deal of attention due to the ease of their electronics implementations and the rapid development of the stability of FDEs. More and more fractional dynamics described in the pseudo state space exhibiting chaos have been found, such as the fractional Chua circuit [4], the fractional Van der Pol oscillator [5¨C7], the fractional Lorenz system [8, 9], the fractional Chen system [10¨C12], the fractional L¨¹ system [13], the fractional Liu system [14], the fractional R£¿ssler system [15, 16], the fractional Arneodo system [17], the fractional Newton-Leipnik system [18¨C20], the fractional Lotka-Volterra system [21, 22], the fractional finance system [20, 23], and the fractional Rucklidge system [24]. Most of the above papers have used numerical methods to present chaotic behaviors. In particular, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. Moreover, several control and synchronization methods have been proposed based on the stability of fractional differential equations in the pseudo state space [22]. The linear state feedback control algorithm based upon the stability criterion of linear
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