Optimal quadratic Lyapunov functions for robust stability problem

Abstract

The robust stability problem for nominally linear system with nonlinear, time-varying structured perturbations is considered. The system is of the form $$\dot x = A_N x + \sum\limits_{j = 1}^q { p_j A_j x.} $$ The Lyapunov direct method has been often utilized to determine the bounds for nonlinear, time-dependent functions pj, which can be tolerated by a stable nominal system. In most cases, quadratic forms are used either as components of vector Lyapunov function or as a function itself. The resulting estimates are usually conservative. Optimizing the Lyapunov function reduces the conservatism of the bounds. The main result of this article is the theorem which makes possible a recursive procedure of a design of optimal Lyapunov function. Examples demonstrate application of the proposed method.