Synchronization of uncertain chaotic systems with minimal parametric information

Abstract

The chaotic synchronization is mainly hampered by uncertain system dynamics in terms of underlying parameters. To obtain accurate parameter estimates for the proper synchronization of uncertain chaotic systems (UCSs) through adaptive control, it is necessary to satisfy the persistence of excitation (PE) condition. Furthermore, the challenges imposed by the explosion of complexity in sequential stabilization procedures, slow performances, and lack of information in UCSs hinder the synchronization process through existing control techniques. Therefore, the contribution of this research is two-fold: First, a systematic stabilization approach with minimal calculations is proposed to achieve proper synchronization between chaotic drive and response systems. The key concept of the proposed theory is to obtain an invariant manifold by immersing the error dynamics (resulting from the mismatch between the drive and response systems) into lower-order target dynamics. From there, the control law for synchronizing these chaotic systems can be derived by defining passive outputs and associated storage functions. The proposed framework, which combines the concepts of immersion and passivity, is referred to as the Passivity and Immersion (P&I) approach. Second, unlike adaptive control, the synchronization of UCSs is made possible by selecting the appropriate target dynamics without any parameter estimation (or with minimal parametric information in certain cases). The simplicity and superiority of the proposed approach over existing techniques are preserved and demonstrated by application to a number of well-known chaotic systems.