A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization

In this paper, new control approaches for synchronization the master and the slave chaotic systems is established by means of novel coupled chaotic synchronous observers and coupled chaotic adaptive synchronous observers. The simultaneous estimation of the master and the slave systems’ states is accomplished, by means of the proposed observers for each of the master and the slave systems, to produce error signals between these estimated states. This estimated synchronization error signal and the state-estimation errors converge to the origin by means of a specific observers-based feedback control signal to ensure synchronization as well as state estimation. Using Lyapunov stability theory, nonadaptive and adaptive control laws and properties of nonlinearities, a convergence condition for the state-estimation errors and the estimated synchronization error is developed in the form of nonlinear matrix inequalities. Solution of the resulted inequality constraints using a two-step approach is presented, which provides the necessary and sufficient condition to obtain values of the observer gain and controller gain matrices. Further, a method requiring less computational efforts for solving the matrix inequalities for obtaining the observer and the controller gain matrices using decoupling technique is also proposed. Numerical simulation of the proposed synchronization technique for FitzHugh–Nagumo neuronal systems is illustrated to elaborate efficaciousness of the proposed observers-based control methodologies.