Sliding mode control for a class of variable-order fractional chaotic systems

Abstract This paper concerns the stability analysis and controller design for a class of variable order fractional chaotic systems. By use of Fractional Comparison Principle, the Mittag-Leffler stability criterion is proposed and proved for variable order fractional systems. For the unperturbed system, two different kinds of sliding mode controllers are proposed and the stability of the controlled systems is proved based on the proposed stability criterion. The first control law is designed on the constructing of a variable order fractional integral sliding mode surface and results in a free of chattering signal. While, the second one introduces a variable order fractional derivative sliding mode surface, which is also adapted for the variable order fractional system with uncertainty and external disturbance. And the sign  ·  function in the switching control law is transferred into the fractional derivative of the control input such that it avoids the undesirable chattering. In addition, for the systems with uncertainty and external disturbance, an adaptive sliding mode control law is designed for the variable order fractional chaotic system. And the unknown bounds of the uncertainty and external disturbance are estimated by the variable order fractional derivative adaptive laws. Based on the fractional order Barbalat’s Lemma which is extended from the integer order Barbalat’s Lemma, the asymptotical stability is proved for the controlled uncertain system. At last, numerical simulations are presented to verify the validity and efficiency of the proposed fractional order controllers.