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Observability on the classes of non-nilpotent solvable three-dimensional Lie groups

Thiago Matheus Cavalheiro, Alexandre José Santana, Victor Ayala

TL;DR

The paper tackles the problem of observability for linear control systems on solvable, non-nilpotent three‑dimensional Lie groups by linking group structure, derivations, and projections to homogeneous spaces. It develops a drift–kernel analysis, using simply connected subgroups and homomorphisms to characterize when different outputs yield distinguishability of states. The main contributions include a classification of local and global observability conditions across five 3D affine Lie group classes, and a detailed treatment of how the drift’s derivation forms (e.g., diagonal vs. Jordan-type) influence fixed points and kernel intersections. The results offer a principled framework for understanding observability in nonlinear settings with Lie-group symmetry and point to computational avenues for higher-dimensional generalizations and practical control applications.

Abstract

In control theory, researchers need to understand a system's local and global behaviors in relation to its initial conditions. When discussing observability, the main focus is on the ability to analyze the system using an output space defined by an output map. In this study, our objective was to establish conditions for characterizing the observability properties of linear control systems on Lie groups. We will focus on five classes of solvable, non-nilpotent three-dimensional Lie groups, examining local and global perspectives. This analysis explores the kernels of homomorphisms between the state space and its simply connected subgroups, where the output is projected onto the quotient space.

Observability on the classes of non-nilpotent solvable three-dimensional Lie groups

TL;DR

The paper tackles the problem of observability for linear control systems on solvable, non-nilpotent three‑dimensional Lie groups by linking group structure, derivations, and projections to homogeneous spaces. It develops a drift–kernel analysis, using simply connected subgroups and homomorphisms to characterize when different outputs yield distinguishability of states. The main contributions include a classification of local and global observability conditions across five 3D affine Lie group classes, and a detailed treatment of how the drift’s derivation forms (e.g., diagonal vs. Jordan-type) influence fixed points and kernel intersections. The results offer a principled framework for understanding observability in nonlinear settings with Lie-group symmetry and point to computational avenues for higher-dimensional generalizations and practical control applications.

Abstract

In control theory, researchers need to understand a system's local and global behaviors in relation to its initial conditions. When discussing observability, the main focus is on the ability to analyze the system using an output space defined by an output map. In this study, our objective was to establish conditions for characterizing the observability properties of linear control systems on Lie groups. We will focus on five classes of solvable, non-nilpotent three-dimensional Lie groups, examining local and global perspectives. This analysis explores the kernels of homomorphisms between the state space and its simply connected subgroups, where the output is projected onto the quotient space.

Paper Structure

This paper contains 26 sections, 11 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Initial Results
  3. Affine 3-dimensional Lie group
  4. Linear Vector Fields
  5. Fixed points of $\mathcal{X}$
  6. Subgroups of $R_2$
  7. The Subgroup $G_1 = \mathbb{R} \times (\mathbb{R} \times \{0\})$
  8. Case 1: $\det B \neq 0.$
  9. Case 2: $\det B = 0.$
  10. Subgroup $G_2 = \mathbb{R} \times_{\rho} (\{0\} \times \mathbb{R})$
  11. Subgroup $G_3 = \{0\} \times_{\rho} \mathbb{R}^2$
  12. Other subgroups
  13. Subgroups of $R_3$
  14. The Subgroup $G_1$
  15. The Subgroup $G_2$
  16. ...and 11 more sections

Key Result

Proposition 1.3

Theorems & Definitions (29)

  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 19 more