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

Abstract

ForewordThis report is an extended version of the paper A unified framework for deterministic and probabilistic D-stability analysis of uncertain polynomial matrices submitted by the authors to the IEEE Transactions on Automatic Control. AbstractIn 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 of the complex plane. 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 by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main 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 efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.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 of the complex plane. 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 by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main 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 1 arXiv:1604.02031v3 [math.OC] 17 Jun 2018 is available on the uncertain parameters, such as support and some moments. We will show how to efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.To this end, we develop a unified framework for deterministic (robust) and probabilistic D-stability analysis. A semi-infinite linear progr...

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