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

I. Baragaña, F. Puerta, I. Zaballa

TL;DR

The paper tackles the local parametrization of the set $\mathcal{H}_{(F,G)}=\{K\in\mathbb{R}^{m\times n}: F+GK\in \mathcal{O}(\underline{\alpha})\}$ of state-feedback matrices that force the closed-loop matrix $F+GK$ to lie in a prescribed similarity class $\mathcal{O}(\underline{\alpha})$ for a controllable system $(F,G)$. It proves that $\mathcal{H}_{(F,G)}$ is a differentiable (immersed) submanifold of $\mathbb{R}^{m\times n}$ and computes its dimension. A local parametrization is obtained via a diffeomorphism between $\mathcal{H}_{(F,G)}$ and the orbit space of truncated observability matrices under a Lie group action, yielding a reduced form and local coordinates. These results enable perturbation and optimization analyses of pole placement by preserving the prescribed invariant class, with potential applications to robust and efficient controller design.

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.

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

TL;DR

The paper tackles the local parametrization of the set of state-feedback matrices that force the closed-loop matrix to lie in a prescribed similarity class for a controllable system . It proves that is a differentiable (immersed) submanifold of and computes its dimension. A local parametrization is obtained via a diffeomorphism between and the orbit space of truncated observability matrices under a Lie group action, yielding a reduced form and local coordinates. These results enable perturbation and optimization analyses of pole placement by preserving the prescribed invariant class, with potential applications to robust and efficient controller design.

Abstract

Given a controllable system , a local parametrization is obtained for the set of feedback gain matrices such that the state matrix, , 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.

Paper Structure

This paper contains 4 sections, 1 theorem, 3 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminary results
  3. Partitions
  4. Real Jordan and Weyr Canonical Forms

Key Result

Proposition 2.1

If $\underline{a}$ and $\underline{b}$ are partitions then where $\underline{a}^\ast$, $\underline{b}^\ast$ and $(\underline{a}+\underline{b})^\ast$ are the conjugate partitions of $\underline{a}^\ast$, $\underline{b}^\ast$ and $(\underline{a}+\underline{b})^\ast$, respectively.

Theorems & Definitions (1)

  • Proposition 2.1