Random Walks for Probabilistic Robustness

In this paper, we explore the use of Markov chain sampling techniques for applications in probabilistic robustness of control systems. First, we analyze the general hit-and-run (HR) method for uniform sampling in convex bodies, and discuss several key issues related to the so called mixing rate of this process and to Hoeffding-type inequalities for dependent samples. Then, we apply the HR method for uniform sampling in the interior of a generic LMI feasible set. Two specific applications of this latter problem which are relevant in probabilistic robust control are studied: the uniform generation of stable transfer functions bounded in the H∞ norm, and uniform sampling in matrix spectral (maximum singular value) norm balls.