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Structural Controllability of Bilinear Systems on $\mathbb{SE(n)}$

A. Sanand Amita Dilip, Chirayu D. Athalye

TL;DR

This work addresses structural controllability of driftless bilinear systems on the special Euclidean group $SE(n)$ under uncertain interconnections. It develops a graph-theoretic framework using solid and broken edges and a transitive closure to characterize when a given sparsity pattern $\mathcal{B}_\Lambda$ yields controllability and accessibility, tying these properties to the Lie algebra generated by the pattern. The main contributions include graph-based necessary and sufficient conditions for structural controllability and accessibility, a sparsity-pattern characterization via spanning trees on $n+1$ vertices, and a minimum-cost greedy MST algorithm for optimal pattern selection. This framework enables scalable design of sparse interconnections in large-scale networked systems evolving on $SE(n)$, with potential applications in distributed coordination and rigid-body control.

Abstract

Structural controllability challenges arise from imprecise system modeling and system interconnections in large scale systems. In this paper, we study structural control of bilinear systems on the special Euclidean group. We employ graph theoretic methods to analyze the structural controllability problem for driftless bilinear systems and structural accessibility for bilinear systems with drift. This facilitates the identification of a sparsest pattern necessary for achieving structural controllability and discerning redundant connections. To obtain a graph theoretic characterization of structural controllability and accessibility on the special Euclidean group, we introduce a novel idea of solid and broken edges on graphs; subsequently, we use the notion of transitive closure of graphs.

Structural Controllability of Bilinear Systems on $\mathbb{SE(n)}$

TL;DR

This work addresses structural controllability of driftless bilinear systems on the special Euclidean group under uncertain interconnections. It develops a graph-theoretic framework using solid and broken edges and a transitive closure to characterize when a given sparsity pattern yields controllability and accessibility, tying these properties to the Lie algebra generated by the pattern. The main contributions include graph-based necessary and sufficient conditions for structural controllability and accessibility, a sparsity-pattern characterization via spanning trees on vertices, and a minimum-cost greedy MST algorithm for optimal pattern selection. This framework enables scalable design of sparse interconnections in large-scale networked systems evolving on , with potential applications in distributed coordination and rigid-body control.

Abstract

Structural controllability challenges arise from imprecise system modeling and system interconnections in large scale systems. In this paper, we study structural control of bilinear systems on the special Euclidean group. We employ graph theoretic methods to analyze the structural controllability problem for driftless bilinear systems and structural accessibility for bilinear systems with drift. This facilitates the identification of a sparsest pattern necessary for achieving structural controllability and discerning redundant connections. To obtain a graph theoretic characterization of structural controllability and accessibility on the special Euclidean group, we introduce a novel idea of solid and broken edges on graphs; subsequently, we use the notion of transitive closure of graphs.
Paper Structure (13 sections, 7 theorems, 21 equations, 2 figures, 1 algorithm)

This paper contains 13 sections, 7 theorems, 21 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and Contribution
  3. Organization
  4. Notation
  5. Problem Statement and Preliminaries
  6. Controllability and Accessibility of Bilinear Systems
  7. Structural Controllability and Accessibility
  8. Problem Formulation
  9. Preliminaries
  10. Main Results
  11. Sparsity and Structural Controllability
  12. Conclusion
  13. Appendix

Key Result

Theorem 2.1

The driftless bilinear system given by eq:driftless-bilinear is controllable if and only if $\mathbb{LG}$ is connected and the Lie subalgebra generated by $\{B_1,\ldots,B_m\}$ is equal to $\mathcal{LG}$.

Figures (2)

  • Figure 1: $l^{\rm th}$ transitive closure of $\mathcal{G}(\mathcal{B}_\Lambda)$ in Example \ref{['ex:s-controllable system']}.
  • Figure 2: $\mathcal{G}(\mathcal{B}_\Lambda)$ for the system in Example \ref{['ex:s-uncontrollable system']}.

Theorems & Definitions (17)

  • Theorem 2.1: JurdSuss
  • Proposition 2.1: JurdSuss1JurdSuss
  • Definition 2.1: Structural Controllability
  • Definition 2.2: Structural Accessibility
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Example 3.1
  • Example 3.2
  • ...and 7 more