The stability of a family of polynomials can be deduced from a finite number 0(k/sup 3/) of frequency checks

Let phi (s,a)= phi /sub 0/(s,a)+a/sub 1/ phi /sub 1/(s)+a/sub 2/ phi /sub 2/(s)+ . . .+a/sub k/ phi /sub k/(s)= phi /sub 0/(s)-q(s,a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a/sub i/ which are confined in a k-dimensional hypercube Omega /sub a/. Let phi /sub 0/(s) be stable of degree n and the phi /sub i/(s) polynomials (i>or=1) of degree less than n. A Nyquist argument shows that the family phi (s) is stable if and only if the complex number phi /sub 0/(j omega ) lies outside the set of complex points -q(j omega , Omega /sub a/) for every real omega . In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(j omega , Omega /sub a/), 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(k/sup 3/), and these critical frequencies correspond to the real nonnegative roots of some polynomials.<
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