An efficient algorithm for checking the robust stability of a polytope of polynomials

Abstract

An efficient algorithm for checking the robust stability of a polytope of polynomials is proposed. This problem is equivalent to a zero exclusion condition at each frequency. It is shown that such a condition has to be checked at only afinite number of frequencies. We formulate this problem as aparametric linear program which can be solved by the Simplex procedure, with additional computations between steps consisting of polynomial evaluations and calculation of positive polynomial roots. Our algorithm requires a finite number of steps (corresponding to frequency checks) and in the important case when the polytope of parameters is a hypercube, this number is at most of orderO(m 3 n 2), wheren is the degree of the polynomials in the family andm is the number of parameters.