A new LKF approach to stability analysis of linear systems with uncertain delays

Abstract

In this paper, we analyze the asymptotic stability of linear systems with multiple uncertain delays assuming the nominal delays to be constant with possible zero or nonzero values. We apply the simple Lyapunov–Krasovskii functional (LKF) with the derivative prescribed as a negative definite quadratic form of the “current” state of a system. We combine this functional with some integral bound for the derivative of the functional along the solutions of an uncertain system, what provides the novelty of approach. This bound consists of an essential integral part and some bounded expression. The fact is that the positiveness of the essential part provides itself the stability of a system. The study is based on recent works where the functional was shown to admit a quadratic lower bound on some special set of functions, though it is only known to have a local cubic one on the set of arbitrary functions. The derived stability condition is much simpler than most ones obtained by LKFs and outlined in the literature. It constitutes an algebraic inequality for delay perturbations based on the so-called Lyapunov matrix and converges to a necessary and sufficient condition in some sense.