Shaping Conditions and the Stability of Systems with Parameter Uncertainty

Abstract

Let ϕ (s, a) = ϕ₀ (s) + a₁ϕ₁ (s) + a₂ϕ₂ (s) +...+ ϕ_k (s) = ϕ₀ (s) + q (s, a) be a family of real polynomials in s, with coefficients that depend linearly on a. which are confined in a k dimensional hypercube Ω_a. Let ϕ₀ (s) be stable of degree n and the ϕ₁ (s) polynomials (i ≥ 1) of degree less than n. It follows from the Nyquist Theorem that the family ϕ (s, a) is stable if and only if the complex number ϕ₀ (jω) lies outside the set of complex numbers -q(jω, Ω_a) for every real ω. In this paper we show that -q(jω, Ω_a), the so called “-q locus”, is in general a 2k convex parpolygon — a convex polygon with an even number of sides (2k) in which opposite sides are equal and parallel. We then exploit this special structure and show that to test for stability only a finite number of frequency checks need to be done. The number of critical frequencies is shown to be polynomial in k, 0(k³), and these frequencies correspond to the real nonnegative roots of some polynomials.