A Polynomial time Algorithm for Checking the Robust Stability of a Polytope of Polynomials

A efficient algorithm to check the robust stability of a polytope of polynomials is proposed. This problem is equivalent with a zero exclusion condition at each frequency. It is shown that such a condition has to be checked at only a finite number of frequencies. We formulate this problem as a parametric 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 of the polytope of parameters being a hypercube, this number is at most of order O(m3n), where n is the degee of the polynomials in the family and m is the number of parameters.