A Local Parametrization of the State-Feedback Matrices in the Pole Assignment Problem

TL;DR

This paper develops a local parametrization of state-feedback matrices for pole assignment, revealing the manifold structure and providing a coordinate system via diffeomorphism, enhancing understanding of controllability and feedback design.

Contribution

It introduces a novel local parametrization of feedback matrices in the pole assignment problem, characterizing the set as a differentiable manifold and establishing a coordinate system through Lie group actions.

Findings

The set of feedback matrices forms a differentiable manifold.

A diffeomorphism links feedback matrices to orbit space of observability matrices.

The manifold's dimension is explicitly computed.

Abstract

Given a controllable system $(F,G)$, a local parametrization is obtained for the set of feedback gain matrices $K$ such that the state matrix, $F+GK$, of the closed loop system is in a prescribed similarity class. It is shown that this set can be endowed with the structure of a differentiable manifold whose dimension is also computed. Then a local parametrization and a local system of coordinates is provided using a diffeomorphism between this set of state feedback matrices and the orbit space of a set of truncated observability matrices via de action of a Lie group.