Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control

J Humberto Pérez-Cruz

doi:10.1155/2018/4989520

Abstract

Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems.

Highlights

Chaotic systems present interesting and peculiar features such as very high sensitivity to initial conditions, boundedness of solutions, and a rich dynamic behavior [1,2,3,4,5,6]
It is still possible to modify the behavior of such systems by means of a proper control input
The states of a chaotic system tend to an equilibrium point, generally the origin, by means of a proper control law

Summary

Introduction

Chaotic systems present interesting and peculiar features such as very high sensitivity to initial conditions, boundedness of solutions, and a rich dynamic behavior [1,2,3,4,5,6]. A chaotic system with control inputs known as slave must follow the dynamics of an autonomous chaotic system known as master Both systems should produce the same response in spite of the different initial conditions [39]. In practical situations, unmodeled dynamics and/or disturbance could be present Under this more realistic condition, the control laws designed in [66,67,68,69,70,71,72,73,74,75] are not effective anymore since the response is deteriorated, the estimations of the parameters start to grow unboundedly, and eventually the closed-loop system becomes unstable. The explained strategy can be applied to a broad class of chaotic systems

Zhang System and Problem Description

Adaptive Stabilization

Adaptive Synchronization

Conclusion