Application of Facial Reduction to $H_\infty$ State Feedback Control Problem

TL;DR

This paper applies facial reduction techniques to address numerical difficulties in solving $H_ abla$ control LMIs, linking feasibility issues to invariant zeros and improving numerical stability in control design.

Contribution

It introduces a facial reduction approach for $H_ abla$ control LMIs, providing conditions related to invariant zeros and demonstrating improved numerical stability.

Findings

Facial reduction clarifies feasibility conditions linked to invariant zeros.

The method improves numerical stability in $H_ abla$ control problems.

Null vectors associated with invariant zeros are key to addressing numerical issues.

Abstract

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss the reason from the viewpoint of optimization, and provide necessary and sufficient conditions for LMI problem and its dual not to be strongly feasible. Moreover, we interpret them in terms of control system. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that a necessary and sufficient condition closely related to the existence of invariant zeros in the closed left-half plane in the system, and present a way to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.