Probability-guaranteed robust H∞ performance analysis and state-feedback design

Abstract

This paper addresses the common engineering practice of specifying a required probability of attaining some performance level. The problem setup is that of a robust H ∞ performance analysis/state-feedback synthesis of an affinely parameter-dependent linear system, except that the parameter hyper-rectangle (box) is allowed to shrink—representing a probability less than one—in order to accommodate a polytopic performance goal that is better than the one attainable for the original parameter box. A new version of the bounded real lemma (BRL), which assigns a different Lyapunov function to each of the vertices of the uncertainty polytope and includes a scalar-free parameter, seems to reduce the overdesign associated with the polytopic problem to the bare minimum. The shrinking of the parameter box leads to bi-linear matrix inequalities (BLMIs), since the final vertices are also unknown. These BLMIs are solved iteratively; three steps have sufficed, both in the analysis and in the state-feedback design examples. A uniform distribution is assumed for all the system parameters, following the uniformity principle. The probability requirement is expressed by a set of linear matrix inequalities (LMIs) that is derived by extending an existing second-order cone method; these LMIs are concurrently solved with the BLMIs of the BRL. The features of the proposed method are demonstrated via two examples.