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

In this paper, a convergent numerical procedure to compute ℋ2 and ℋ∞ 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 Polya'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 ℋ2 and ℋ∞ 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.