Stability analysis of linear time-invariant systems in the presence of polytopic uncertainty and a time delay state

Abstract

This paper deals with the robust stability analysis problem of linear time-invariant systems with a state delay and polytopic uncertainty. The delay parameter is constant which does not limited by any upper bound. The system’s matrices linearly depend on uncertain parameters which belong to the unit simplex known as the uncertain space. An extended uncertain system is introduced based on the original delay model that contains the uncertain parameters and two slack parameters. The slack parameters belong to the boundary of the unit circle in two dimensional real space. Then, it has been proven that the delay system is robust stable if and only if the extended uncertain system is robust stable over the uncertain space and the unit circle’s boundary. Since the unit circle’s boundary is not convex, a containing polygonal space is developed that consists of a group of polygonal subspaces. This paper proposes a novel algorithm to check system’s robust stability via linear dependent Lyapunov functions that correspondingly defined for each polygonal subspace. The algorithm is able to establish robust stability of the model for all positive values of the delay parameter. There simulation examples are provided to clarify the notations and facts of this paper. First example presents the notation and results in an instance uncertain delay system. Two extensive examples compare feasibility performances of the proposed algorithm to some existing methods that reveal the superiority of the prosed algorithm.