On the existence and approximation of a dissipating feedback

TL;DR

This paper investigates conditions for and methods to compute feedback matrices that ensure a system's stability by making the pair (A, B) dissipative, focusing on existence, parameterization, and approximation strategies.

Contribution

It reviews classical results, introduces new matrix properties, and discusses computational approaches for approximating minimal Frobenius norm dissipating feedback matrices.

Findings

Classical results on dissipating matrices are expanded.

New matrix properties related to the problem are identified.

Strategies for approximating minimal Frobenius norm feedback are discussed.

Abstract

Given a matrix $A\in \R^{n\times n}$ and a tall rectangular matrix $B \in \R^{n\times q}$, $q < n$, we consider the problem of making the pair $(A,B)$ dissipative, that is the determination of a {\it feedback} matrix $K \in \R^{q\times n}$ such that the field of values of $A-B K$ lies in the left half open complex plane. We review and expand classical results available in the literature on the existence and parameterization of the class of dissipating matrices, and we explore new matrix properties associated with the problem. In addition, we discuss various computational strategies for approximating the minimal Frobenius norm dissipating $K$.