Arveson’s Distance Formulae and Robust Stabilization for Linear Time-Varying Systems

Abstract

Robustness properties for time invariant systems were first studied in the context of Youla parameterization by Vidyasagar and Kimura ([13], [12]). There, plants are given by co-prime factorizations and neighbourhoods of the plant are characterized in terms of perturbations of the numerator and denominator of a given co-prime factorization. In this paper we give a generalization of this result to the time-varying setting. Because the natural notions of numerator, denominator, poles, zeroes for time-invariant systems do not have an obvious (or in some cases any) meaning in the time-varying case, another formulation is useful. We use the framework described in [2], which allows us to give a purely operator-theoretic formulation of this problem, strongly related to that given for time-invariant systems in the fundamental paper of Georgiou and Smith ([1]). Our main result (Theorem 4.3) is strongly related to that of Shamma ([10]) on the necessary condition for the Small-Gain Theorem for time-varying systems and our proof can be adapted to give what seems to be a more transparent proof of that result.