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

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 ∈ R^{n × 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 = 0}^N θⁱ P_i that ensures the robust Hurwitz stability of A (θ) via P (θ) > 0, P (θ) A (θ) + A^T (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 ∈ R^{3 × 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.