Stability of Lurie-type systems with asynchronous and synchronous switching and constant delays

Abstract

The problems of analysis for systems with synchronous and asynchronous switching have been actively studied for the linear case. In this paper, a switched system of difference-differential equations, in which the right-hand side consists of a linear term and an essentially nonlinear part containing sector-type components is considered. This kind of systems belongs to the class of Lurie indirect control systems. Sufficient conditions on the system parameters and the switching law are investigated for asymptotic stability to be guaranteed both in the case of synchronous switching between subsystems and in asynchronous one. In the latter case it is supposed that the nonlinear delayed part switches with a lag equal to the corresponding delay. It is required that stability should be preserved for any constant positive delays. The problem is solved using the Lyapunov — Krasovsky approach. The functional is chosen that includes a quadratic form and integrals of nonlinearities. Restrictions that ensure asymptotic stability for an arbitrary switching law are found. With such an approach for the asynchronous case these conditions turn out to be less restrictive. By using multiple functionals the restrictions on the lengths of intervals between switchings are also obtained. This type of conditions are similar for both cases of synchronous and asynchronous switching. Theoretical results are demonstrated by a specially selected example.