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

Yoshio Ebihara, Tomomichi Hagiwara

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9 Citations (Scopus)

Abstract

In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A (θ) {colon equals} A0 + θ A1 over θ ∈ [- 1, 1], where A0, A1 ∈ Rn × n with A0 being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form P (θ) {colon equals} ∑i = 0N θi Pi that ensures the robust Hurwitz stability of A (θ) via P (θ) > 0, P (θ) A (θ) + AT (θ) P (θ) < 0 (∀ θ ∈ [- 1, 1]). On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P (θ), then there always exists a first-degree PPDLM P (θ) = P0 + θ P1 that meets the desired conditions, regardless of the size or rank of A0 and A1. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A0, A1 ∈ R3 × 3 with A0 + θ A1 being Hurwitz stable for all θ ∈ [- 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 techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n ≥ 3 in general.

Original languageEnglish
Pages (from-to)1599-1603
Number of pages5
JournalAutomatica
Volume42
Issue number9
DOIs
Publication statusPublished - Sep 1 2006
Externally publishedYes

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Lyapunov functions
Polynomials
Robust stability

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

Cite this

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title = "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",
abstract = "In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A (θ) {colon equals} A0 + θ A1 over θ ∈ [- 1, 1], where A0, A1 ∈ Rn × n with A0 being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form P (θ) {colon equals} ∑i = 0N θi Pi that ensures the robust Hurwitz stability of A (θ) via P (θ) > 0, P (θ) A (θ) + AT (θ) P (θ) < 0 (∀ θ ∈ [- 1, 1]). On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P (θ), then there always exists a first-degree PPDLM P (θ) = P0 + θ P1 that meets the desired conditions, regardless of the size or rank of A0 and A1. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A0, A1 ∈ R3 × 3 with A0 + θ A1 being Hurwitz stable for all θ ∈ [- 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 techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n ≥ 3 in general.",
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AU - Hagiwara, Tomomichi

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AB - In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A (θ) {colon equals} A0 + θ A1 over θ ∈ [- 1, 1], where A0, A1 ∈ Rn × n with A0 being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form P (θ) {colon equals} ∑i = 0N θi Pi that ensures the robust Hurwitz stability of A (θ) via P (θ) > 0, P (θ) A (θ) + AT (θ) P (θ) < 0 (∀ θ ∈ [- 1, 1]). On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P (θ), then there always exists a first-degree PPDLM P (θ) = P0 + θ P1 that meets the desired conditions, regardless of the size or rank of A0 and A1. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A0, A1 ∈ R3 × 3 with A0 + θ A1 being Hurwitz stable for all θ ∈ [- 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 techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n ≥ 3 in general.

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