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A Weak Notion of Symmetry for Dynamical Systems

Jake Welde, Pieter van Goor

Abstract

Many nonlinear dynamical systems exhibit symmetry, affording substantial benefits for control design, observer architecture, and data-driven control. While the classical notion of group invariance enables a cascade decomposition of the system into highly structured subsystems, it demands very rigid structure in the original system. Conversely, much more general notions (e.g., partial symmetry) have been shown to be sufficient for obtaining less-structured decompositions. In this work, we propose a middle ground termed "weak invariance", studying diffeomorphisms (resp., vector fields) that are group invariant up to a diffeomorphism of (resp., vector field on) the symmetry group. Remarkably, we prove that weak invariance implies that this diffeomorphism of (resp., vector field on) the symmetry group must be an automorphism (resp., group linear). Additionally, we demonstrate that a vector field is weakly invariant if and only if its flow is weakly invariant, where the associated group linear vector field generates the associated automorphisms. Finally, we show that weakly invariant systems admit a cascade decomposition in which the dynamics are group affine along the orbits. Weak invariance thus generalizes both classical invariance and the important class of group affine dynamical systems on Lie groups, laying a foundation for new methods of symmetry-informed control and observer design.

A Weak Notion of Symmetry for Dynamical Systems

Abstract

Many nonlinear dynamical systems exhibit symmetry, affording substantial benefits for control design, observer architecture, and data-driven control. While the classical notion of group invariance enables a cascade decomposition of the system into highly structured subsystems, it demands very rigid structure in the original system. Conversely, much more general notions (e.g., partial symmetry) have been shown to be sufficient for obtaining less-structured decompositions. In this work, we propose a middle ground termed "weak invariance", studying diffeomorphisms (resp., vector fields) that are group invariant up to a diffeomorphism of (resp., vector field on) the symmetry group. Remarkably, we prove that weak invariance implies that this diffeomorphism of (resp., vector field on) the symmetry group must be an automorphism (resp., group linear). Additionally, we demonstrate that a vector field is weakly invariant if and only if its flow is weakly invariant, where the associated group linear vector field generates the associated automorphisms. Finally, we show that weakly invariant systems admit a cascade decomposition in which the dynamics are group affine along the orbits. Weak invariance thus generalizes both classical invariance and the important class of group affine dynamical systems on Lie groups, laying a foundation for new methods of symmetry-informed control and observer design.

Paper Structure

This paper contains 13 sections, 8 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Smooth Manifolds
  4. Lie Groups and Automorphisms
  5. Group Actions
  6. Group Invariance
  7. Notions of Weak Invariance
  8. Weak Invariance of Diffeomorphisms
  9. Weak Invariance of Flows
  10. Weak Invariance of Vector Fields
  11. Cascade Decompositions of Weakly Invariant Dynamical Systems
  12. Group Affine Dynamical Systems
  13. Conclusion

Key Result

Proposition 1

Let ${\Phi : G \times M \to M}$ be an effective group action and ${f : M \to M}$ be a diffeomorphism that is weakly $\Phi$-invariant with respect to $\sigma : G \to G$. Then, $\sigma \in \mathop{\mathrm{Aut}}\nolimits(G)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Lemma 1
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 3
  • proof
  • ...and 11 more