Fractional sliding mode surface controller for projective synchronization of fractional hyperchaotic systems

Abstract

In this paper, a sliding mode controller with fractional sliding surface is designed for projective synchronization (PS) of fractional hyperchaotic systems via active control theory and sliding mode theory. The existence and the stability of PS between two fractional hyperchaotic systems are studied based on Lyapunov theory, fractional stability theory and the theory of fractional calculus. The criterions for the practical stability of the PS error system are presented also. Numerical simulation of PS between fractional Chen system and a new hyperchaotic system shows the effectiveness of the proposed controller law.

Robust sliding mode control for fractional-order chaotic economical system with parameter uncertainty and external disturbance

“…The following theorem will give sufficient conditions of finite time convergence to sliding surface s(t) for system (27) with parameter uncertainty and external disturbance. Theorem 3 If system ( 27) is controlled with the following controller:…”

“…Once the states of system (27) reach the sliding surface (7), the desired convergence characteristic (state y(t) converges to zero in finite time) will be subsequently achieved, and then the asymptotical stability of states x(t) and z(t) can be obtained by the designed sliding surface (7).…”

“…Moreover, the control and synchronization of fractionalorder dynamical systems has attracted a great deal of interest. [16][17][18][19][20][21][22] The most common control methods include the active control method, [23][24][25] sliding mode control method, [26][27][28][29] adaptive control method, [30][31][32] observer-based control method, [33][34][35] impulsive control method, [36][37][38] etc. Among these methods, sliding mode control (SMC) is an efficient method to deal with the robust control scheme.…”

Modified impulsive synchronization of fractional order hyperchaotic systems

“…From inequalties (15) and ( 16) it follows that V (e(t)) → 0 and e(t) → 0 as t → 0 (or k → 0). Therefore, error system ( 6) is globally asymptotically stable, which implies that the impulsively controlled response system (5) asymptotically synchronizes with drive system (1).…”

Backstepping-based lag synchronization of a complex permanent magnet synchronous motor system

“…After that, synchronizations have been widely studied. [5][6][7] Currently, the study of chaos has developed from lowdimension and integral order to high-dimension and fractional order. People have proposed many different kinds of synchronization schemes according to various systems.…”