The Largest Stability Hypercube for Families of Polynomials with Linear Uncertainty
T.E. Djaferis · 1989
Let ϕ(s) = ϕ0(s) = ϕ0(s) + a1ϕ1(s) + a2ϕ2(s) +...+ akϕk(s) be a polynomial with coefficients that depend linearly on real parameters ai, 10(s) be stable of degree n and ϕi, 1iare allowed to take values in the k dimensional hypercube Ωλa= {(a1,...,ak)εRk: a-i≤' ai≤ λ a+i}, where a-1i+i> 0, 1 ≤ i ≤ k are fixed and λ ≤ 0. In this paper we consider the problem of how to compute the largest value λ* such that the family is stable in Ωλafor 0 ≤ λ < λ*. Considering the problem in the frequency domain, a function of frequency can be constructed whose infimum is λ*. In this paper we show that to compute λ* one need only consider the values of this function at a finite number of frequencies. The number of frequencies is polynomial in k, and the frequencies themselves are explicitly determined from the given data.