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An accessibility condition for discrete-time linear systems on Lie groups

Thiago Matheus Cavalheiro, Alexandre José Santana, Eduardo Celso Viscovini

Abstract

In this paper, we characterize the accessibility of discrete-time linear control systems on Lie groups. Using an exceptional notion of derivative, we construct a subalgebra $\mathfrak{h}$ based on the infinitesimal automorphism of the system such that if its dimension is maximal, the system is accessible. Our criteria provide simple conditions in a general context for the discrete-time case. Additionally, we prove a sufficient condition for local controllability at the identity using the infinitesimal automorphism, akin to the ad-rank condition in the continuous case.

An accessibility condition for discrete-time linear systems on Lie groups

Abstract

In this paper, we characterize the accessibility of discrete-time linear control systems on Lie groups. Using an exceptional notion of derivative, we construct a subalgebra based on the infinitesimal automorphism of the system such that if its dimension is maximal, the system is accessible. Our criteria provide simple conditions in a general context for the discrete-time case. Additionally, we prove a sufficient condition for local controllability at the identity using the infinitesimal automorphism, akin to the ad-rank condition in the continuous case.

Paper Structure

This paper contains 6 sections, 18 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conditions for Accessibility
  4. Main results
  5. Examples
  6. Conclusion

Key Result

Proposition \oldthetheorem

Consider the above discrete-time linear control system defined on a Lie group $G$. Then for all $g \in G$ and $u = (u_i)_{i \in \mathbb{Z}} \in \mathcal{U}$

Theorems & Definitions (45)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 35 more