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

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.

Synchronization of a New Chaotic System Using Adaptive Control: Design and Experimental Implementation

“…By means of the Lyapunov theory, Wang et al [76] proposed a nonlinear adaptive system to ensure the synchronization of two Hindmarsh-Rose neuron models and its simulation results verifed the feasibility and efectiveness of the designed controller. Also, Pérez-Cruz [77] added a robustifying term to the adaptive control law for the stabilization and synchronization of an uncertain Zhang system; the performance of this robust approach was verifed through numerical simulations. In [78], Khennaoui et al proposed a onedimensional adaptive control strategy that forces the states of discrete-time chaotic systems to tend asymptotically to zero; numerical results were presented to confrm the success of these synchronization schemes.…”

“…Based on Pecora and Carroll's contribution, the research related to the design of controllers for synchronization of chaotic systems has been intensively studied during the last four decades . Te proposed control strategies reported in those works can be classifed as active control [36][37][38][39][40][41][42], nonlinear control [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], linear feedback control [58][59][60][61][62][63][64][65], sliding modes [66][67][68][69][70][71][72], and adaptive control [73][74][75][76][77][78][79][80][81][82][83][...…”

Synchronization of a Non-Equilibrium Four-Dimensional Chaotic System Using a Disturbance-Observer-Based Adaptive Terminal Sliding Mode Control Method

“…For instance, Pérez-Cruz et al proposed a novel linear feedback controller for synchronization of chaotic master and slave systems [39]. In another study, Pérez-Cruz also proposed an adaptive control scheme for synchronization of uncertain systems [40].…”

Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems

“…What can we say about the asymptotic behavior (as t → ∞) of solutions of perturbed system (1)? This question represents one of the fundamental problems in the area of robust stability and robustness of the systems in general and so the effect of (known or unknown) perturbations on the solutions of nominal system as a potential source of instability attracts the attention and interest of scientific community for a long time in the various contexts, recently for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. A comprehensive overview of the most significant results on robust control theory as a stand-alone subfield of control theory and its history is presented in [17,18].…”