On Robustness in the Gap Metric and Coprime Factor Uncertainty for LTV Systems

TL;DR

This paper extends operator theoretic methods to analyze robust stabilization of linear time-varying systems with coprime factor uncertainty, providing bounds, conditions for compactness, and links to the Operator Corona Theorem.

Contribution

It generalizes LTI results to LTV systems, deriving bounds for stability margins and conditions for Hankel operator compactness, enabling computation of controllers.

Findings

Computed upper bounds for stability margins.

Identified conditions for Hankel operator compactness.

Linked robust stabilization to the Operator Corona Theorem.

Abstract

In this paper, we study the problem of robust stabilization for linear time-varying (LTV) systems subject to time-varying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, we compute an upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the time-varying stability margin and TV controller. A connection between robust stabilization for LTV systems and an Operator Corona Theorem is also pointed out.