On the synthesis of robust strictly positive real discrete-time systems

The Strict Positive Real (SPR) property of discrete-time transfer functions plays a fundamental role in the analysis of the behaviour of several recursive schemes employed in identification and adaptive control. In the context of uncertain systems, this leads to the so called robust SPR problem (RSPR), i.e., given a set P of polynomials and a region Λ of the complex plane, determine, provided it exists, a polynomial or a rational filter F such that each rational function P/F, P ∈ P is strictly positive real over Λ. In this paper the discrete-time case of P being an l2 ball in coefficient space is considered. It will be shown in a constructive way that under the assumption that all the polynomials of such a family are Schur, the sought filter can be provided in closed form as a rational function of known degree. The proposed synthesis procedure is based on the solution of a polynomial factorization problem