A Unified Framework for Deterministic and Probabilistic $\mathscr {D}$-Stability Analysis of Uncertain Polynomial Matrices

In control theory, we are often interested in robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D. Although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly D-stable, the problem of computing the probability of an uncertain matrix to be D-stable is still unexplored. The goal of this paper is to fill this gap in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set. This comprises many important cases in robust control theory. Second, the D-stability analysis problem is formulated in a probabilistic framework, by assuming that only few probabilistic information is available on the uncertain parameters, such as support and some moments. We will show how to compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments.