Robust Strict Positive Realness: Characterization and Construction

Let 9 be a convex set of real polynomials. This paper considers 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 ) E 9. Necessary and sufficient conditions are found for the transfer function b(s ) case, and when the degree of the polynomials in 9 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.