A Nonlinear Adaptive Controller for the Synchronization of Unknown Identical Chaotic Systems

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This paper proposes a novel nonlinear adaptive controller for the synchronization of two identical unknown chaotic systems. The proposed controller accomplishes fast convergence of the steady-state error to the origin and reduces the amplitude of the oscillations in the error signals during the transient. The rapid convergence increases the disturbance rejection attributes of the controller. The reduction in the oscillations’ magnitude decreases energy consumption and lowers the actuators’ degradation, reducing the probability of failure. This controller consists of linear adaptive and nonlinear control components, and they jointly synthesize control signals to penalize the synchronization error. The direct linear adaptive term keeps the closed-loop system stable; it ensures that the synchronization error converges to zero. The nonlinear terms heavily penalize the large state error vector in the transient, and its effect becomes minimal in the steady-state. The linear adaptive control is active in the steady-state, which synthesizes a smooth control signal in the vicinity of zero; this behavior causes oscillation free convergence of the error. Analysis based on the Lyapunov direct theorem proves the global asymptotic stabilization of the closed-loop at the origin. The paper also introduces parameters update laws that estimate the controller and unknown parameters of the master system and assure the convergence. This work also presents a comparative study of a numerical example of two identical Lorenz-Like unknown chaotic systems.