Problem
Research questions and friction points this paper is trying to address.
symmetry
control systems
weak invariance
cascade decomposition
group affine systems
A Weak Notion of Symmetry for Control Systems
🤖 AI Summary
This study addresses the limitations of classical symmetries, which are often overly restrictive and susceptible to disruption by external forces, by introducing the concept of "weak invariance" that leverages control systems on Lie groups to capture asymmetric residuals. Methodologically, this work integrates Lie group theory, differential geometry, and cascade system decomposition techniques. It proves that weak invariance supports cascade decomposition and can be factored out from error dynamics to achieve further dimensionality reduction, thereby generalizing classical symmetry to group-affine systems. In application, a nine-dimensional weakly symmetric model is constructed for aerial vehicles subject to gravity, establishing a theoretical foundation for flexible symmetry-based control.
📝 Abstract
Symmetry (or invariance) is a powerful structural property that enables efficient, effective solutions for estimation and control. However, the constraints imposed on a system's dynamics by classical invariance make symmetry a very rigid property, which may be broken by external forces or confined to only a portion of the overall system. Seeking greater flexibility, this work introduces a novel relaxed notion of symmetry, termed ``weak invariance'', in which the non-symmetric part of the dynamics (the ``residual'') can be captured entirely by another control system evolving on the symmetry group. Weakly invariant systems are strictly more general than classical invariant systems, but they nonetheless enjoy many similar favorable properties. In particular, we prove that any weakly invariant system admits a cascade decomposition in which the driven subsystem is group affine, showing that weak symmetry generalizes not only classical symmetry, but also the (thus far distinct) class of group affine systems. We also show that a weak symmetry with autonomous residual can be factored out of the system's error dynamics, enabling yet a greater reduction of dimensionality as compared to classical symmetries. Finally, we study the example of an aerial vehicle under the influence of gravity, for which we propose a nine-dimensional weak symmetry (strictly containing the system's familiar four-dimensional classical symmetry). Weak invariance thus generalizes classical symmetry while also preserving key structural properties, thereby laying a foundation for more flexible methods of symmetry-informed control.
Innovation
Methods, ideas, or system contributions that make the work stand out.
weak invariance
symmetry
control systems
cascade decomposition
group affine
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