Reduction of SISO H-infinity Output Feedback Control Problem

TL;DR

This paper presents a simplified and explicit solution to the scalar SISO H-infinity output feedback control problem by leveraging the non-uniqueness of perpendicular matrices and invariant zeros, unifying existing results.

Contribution

It introduces an explicit form of the optimal value for the scalar SISO H-infinity control problem, simplifying the LMI approach through null vectors and facial reduction techniques.

Findings

Explicit optimal value form derived for scalar SISO case

LMI problem simplified using null vectors of invariant zeros

Dual problem size reduced via facial reduction

Abstract

We consider the linear matrix inequality (LMI) problem of $H_\infty$ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of $H_\infty$ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.