Ebihara, Yoshio (Kyoto Univ.), Hagiwara, Tomomichi (Kyoto Univ.)

On the Degree of Polynomial Parameter-Dependent Lyapunov Functions for Robust Stability of Single Parameter-Dependent LTI Systems: A Counter-Example to Barmish's Conjecture

Scheduled for presentation during the Regular Session "Convex Optimization II" (FrP09), Friday, July 28, 2006, 15:20−15:45, Room J

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 19, 2024

Abstract

In this brief paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix $A(theta):=A_0+theta A_1$ over $theta in [-1,1]$, where $A_0,A_1 in R^{ntimes n}$ with $A_0$ being Hurwitz stable. In particular, we are interested in the degree $N$ of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form $P(theta):=sum_{i=0}^N theta^i P_i$ that ensures the robust Hurwitz stability of $A(theta)$ via $P(theta)>0, P(theta)A(theta)+A^T(theta)P(theta)<0 (forall theta in [-1,1])$. On the degree of PPDLMs, Barmish conjectured in early 90's that if there exists such $P(theta)$, then there always exists a first-degree PPDLM $P(theta)=P_0+theta P_1$ that meets the desired conditions, regardless of the size or rank of $A_0$ and $A_1$. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices $A_0,A_1 in R^{3 times 3}$ with $A_0+ theta A_1$ being Hurwitz stable for all $theta in [-1,1]$ and prove rigorously that the desired first-degree PPDLM does not exist for this particular pair. The proof is based on the recently developed LMI techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture by Barmish is not valid when $n ge 3$ in general.