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The stability of a family of polynomials can be deduced from a finite number 0(k3) of frequency checks

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2 Author(s)
T. E. Djaferis ; Dept. of Electr. & Comput. Eng., Massachusetts, Univ., Amherst, MA, USA ; C. V. Hollot

Let φ(s,a)=φ0(s,a)+ a1φ1(s)+a2 φ2(s)+ . . .+akφ k(s)=φ0(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters ai which are confined in a k-dimensional hypercube Ωa . Let φ0(s) be stable of degree n and the φi(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ0(jω) lies outside the set of complex points -q(jω,Ωa) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ωa ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k3), and these critical frequencies correspond to the real nonnegative roots of some polynomials

Published in:

IEEE Transactions on Automatic Control  (Volume:34 ,  Issue: 9 )