A convex optimization procedure to compute ℋ︁2 and ℋ︁∞ norms for uncertain linear systems in polytopic domains

Abstract

AbstractIn this paper, a convergent numerical procedure to compute ℋ︁2 and ℋ︁∞ norms of uncertain time‐invariant linear systems in polytopic domains is proposed. The norms are characterized by means of homogeneous polynomially parameter‐dependent Lyapunov functions of arbitrary degree g solving parameter‐dependent linear matrix inequalities. Using an extension of Pólya's Theorem to the case of matrix‐valued polynomials, a sequence of linear matrix inequalities is constructed in terms of an integer d providing a Lyapunov solution for a given degree g and guaranteed ℋ︁2 and ℋ︁∞ costs whenever such a solution exists. As the degree of the homogeneous polynomial matrices increases, the guaranteed costs tend to the worst‐case norm evaluations in the polytope. Both continuous‐ and discrete‐time uncertain systems are investigated, as illustrated by numerical examples that include comparisons with other techniques from the literature. Copyright © 2007 John Wiley & Sons, Ltd.