Synchronization of Two Different Chaotic Systems in the Presence of External Disturbances

All practical systems have disturbances present in them. Achieving synchronization of two non-identical non-linear chaotic systems subjected to external disturbances, while using the system dynamics as such, is often difficult for majority of systems. Thus, the synchronization controller design for different dynamical systems to get the two systems synchronized, is an important research area. This paper proposes an adaptive control law based scheme to synchronize two different systems, related by Lipschitz constant. The non-linear functions in the systems are mutually Lipschitz and unknown disturbances are norm bounded. Lyapunov stability conditions and Linear Matrix Inequality (LMI) theory are used for ensuring the asymptotic convergence of synchronization error trajectories. The efficacy of the proposed adaptive control mechanism is satisfactorily explored for two different sets of chaotic systems, taken as examples.