Robust strict positive realness: characterization and construction

Let P be a convex set of real polynomials. Consideration is given to the question of when there exists a real polynomial b (s), or more generally, a real transfer function b(s), such that p(s)/b(s) is strictly positive real for all p(s)∈P. Necessary and sufficient conditions are found for the transfer function b(s) case, and when the degree of the polynomials in P is restricted, such conditions are also found for the polynomial b(s) case. Closely related results are also obtained for a z-transform version of the problem. The results have application in adaptive systems