Observer-based synchronization scheme for a class of chaotic systems using contraction theory

Abstract

In this paper, an adaptive synchronization scheme is proposed for a class of nonlinear systems. The design utilizes an adaptive observer, which is quite useful in establishing a transmitter-receiver kind of synchronization scheme. The proposed approach is based on contraction theory and provides a very simple way of establishing exponential convergence of observer states to actual system states. The class of systems addressed here has uncertain parameters, associated with the part of system dynamics that is a function of measurable output only. The explicit conditions for the stability of the observer are derived in terms of gain selection of the observer. Initially, the case without uncertainty is considered and then the results are extended to the case with uncertainty in parameters of the system. An application of the proposed approach is presented to synchronize the family of N chaotic systems which are coupled through the output variable only. The numerical results are presented for designing an adaptive observer for the chaotic Chua system to verify the efficacy of the proposed approach. Explicit bounds on observer gains are derived by exploiting the properties of the chaotic attractor exhibited by Chua's system. Convergence of uncertain parameters is also analyzed for this case and numerical simulations depict the convergence of parameter estimates to their true value.

Simple Observer Based Synchronization of Lorenz System with Parametric Uncertainty

“…The following Lemma is used to analyze the convergence of observer states to actual system states and taken from [16]. Lemma 2.If a virtual system defined for actual system and observer system along with parameter adaptationlaw, then incremental behaviour in terms of virtual displacement could be represented in matrix form as…”

“…And the output of the virtual system is given by (16) Now using the definition of Φ and yy in virtual system (15) and defining the virtual displacement Φ as…”

Synchronization of Chaotic Systems via Active Disturbance Rejection Control

“…Adaptive feedback control design method, where controller parameters are assumed to be unknown and are evolved using adaptation laws so as to achieve synchronization was presented in [4]. The contraction theory for the adaptive synchronization was used in [5]. The adaptive synchronization and parameters identification of uncertain Rössler hyperchaotic system were presented in [6].…”

Single input sliding mode control for hyperchaotic Lu system with parameter uncertainty

“…These chaotic and/or hyperchaotic systems show different dynamical properties. So far, various types of synchronization phenomenon have been found such as functional synchronization [2], complete synchronization [3], observer based synchronization [4], phase synchronization [5], generalized synchronization [6], anti-synchronization [7][8][9], distributed synchronization [10], robust synchronization [11], and projective synchronization [12,13]. Transition of synchronization [14][15][16][17][18] induced in the chaotic systems has been explored extensively.…”