Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems

Abstract

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Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

“…Let us consider firstly the CT case. This supposition implies that Re(λ) < −0.5η * for all λ ∈ spc(A(p)) for all p ∈ P. In fact, from (6)- (8), (18) and Lemma 3 in [3], the first two constraints in (12) imply…”

“…The following theorem is given in [3,18,12] and investigates robust stability of (1) via a HPD-QLF of degree m, i.e. via a Lyapunov function of the form x ′ P (p, β)x.…”

“…Indeed, observe that the constraint tr(S(β)) = 1 is not restrictive since S(β), R(β), U(α and T (β) are linear functions, and it is introduced in order to normalized the solution of (12). Let us consider the instability statement.…”

“…Theorem 2 requires to solve (12), which is a generalized eigenvalue problem and hence a quasi-convex optimization problem. The pairs (u, y) in R q 0 × R n 0 satisfying (15) can be found with the technique in [19,12] which amounts to finding the roots of a polynomial obtained via pivoting.…”

“…The pairs (u, y) in R q 0 × R n 0 satisfying (15) can be found with the technique in [19,12] which amounts to finding the roots of a polynomial obtained via pivoting. The vectors c 1 , .…”