On the design of linear dynamic controllers for the Master–Slave synchronization of chaotic oscillators

In this study, we focus on the synchronization of unidirectionally coupled chaotic oscillators that can be written in the Lure form. As a first step, we evaluate the performance of the classic master–slave scheme based on static output feedback, and also, the performance of a recently presented synchronization strategy based on a first-order linear dynamic controller. Then, we highlight some of the limitations of these controllers, in particular, for both cases, it is shown that there exist single-input and single-output combinations, for which the controllers fail to induce the desired synchronous behavior in the coupled oscillators. In order to overcome these limitations, we introduce a novel control law that combines static output feedback and dynamic control. The stability of the closed-loop system is investigated using the master stability function formalism and also Lyapunov stability theory for perturbed linear systems. In the resulting synchronization error dynamics, the difference of the nonlinear terms of the oscillators is considered as a bounded vanishing perturbation. Using the Gershgorin Circle Theorem, we show that synchronization is possible with the proposed controller because it increases the perturbation bound that the error system can tolerate. Through the analysis, the Rössler system is considered as a benchmark example and numerical simulations are provided to illustrate the results. Ultimately, it is demonstrated that the proposed controller induces synchronization even in the cases where classic static and dynamic controllers fail.