Model-free adaptive sliding mode controller design for generalized projective synchronization of the fractional-order chaotic system via radial basis function neural networks

Abstract

A novel model-free adaptive sliding mode strategy is proposed for a generalized projective synchronization (GPS) between two entirely unknown fractional-order chaotic systems subject to the external disturbances. To solve the difficulties from the little knowledge about the master–slave system and to overcome the bad effects of the external disturbances on the generalized projective synchronization, the radial basis function neural networks are used to approach the packaged unknown master system and the packaged unknown slave system (including the external disturbances). Consequently, based on the slide mode technology and the neural network theory, a model-free adaptive sliding mode controller is designed to guarantee asymptotic stability of the generalized projective synchronization error. The main contribution of this paper is that a control strategy is provided for the generalized projective synchronization between two entirely unknown fractional-order chaotic systems subject to the unknown external disturbances, and the proposed control strategy only requires that the master system has the same fractional orders as the slave system. Moreover, the proposed method allows us to achieve all kinds of generalized projective chaos synchronizations by turning the user-defined parameters onto the desired values. Simulation results show the effectiveness of the proposed method and the robustness of the controlled system.

Appendix: Dynamic equations of the fractional-order chaotic systems

The fractional-order chaotic Lorenz system [28] is described by the following equation:

$$\begin{aligned} \left\{ {\begin{array}{l} D_t ^{q_1 }x_1 =10(x_2 -x_1 ), \\ D_t ^{q_2 }x_2 =4x_1 +5x_2 -x_1 x_3 \\ D_t ^{q_3 }x_3 =x_2 x_3 -8/3x_3, \\ \end{array}}, \right. \quad (q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97). \end{aligned}$$

(A.1)

Through numerical simulations, one can find that system (A.1) will exhibit chaotic behaviours when $(q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97)$, which is shown in figure 6.

Fig. 6

Chaotic attractor and projections of the fractional-order Lorenz chaotic system when $(q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97)$.

The fractional-order chaotic Chen system [29] is described by the following equation:

$$\begin{aligned} \left\{ {\begin{array}{l} D_t ^{q_1 }x_1 =10(x_2 -x_1 ), \\ D_t ^{q_2 }x_2 =40x_1 -10x_1 x_3 \\ D_t ^{q_3 }x_3 =4x_1 ^{2}-2.5x_3, \\ \end{array}}, \right. \quad (q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97). \end{aligned}$$

(A.2)

Through numerical simulations, one can find that system (A.2) will exhibit chaotic behaviours when $(q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97)$, which is shown in figure 7.

Fig. 7

Chaotic attractor and projections of the fractional-order Chen chaotic system when $(q_1 ,q_2 ,q_3 )=(0.98,0.95,0.97)$.