On symmetries of nonlinear systems in state representation and application of invariant feedback design

TL;DR

This paper explores the geometric symmetries of nonlinear control systems and develops invariant feedback laws that preserve these symmetries, with applications demonstrated in a predator-prey bioreactor model.

Contribution

It introduces a geometric approach to symmetry analysis and invariant feedback design for nonlinear systems, including a method for reduced-order system derivation.

Findings

Invariant feedback laws can be constructed using group invariants.

Normalization procedures help determine invariant tracking errors.

Application to a predator-prey bioreactor demonstrates controlled symmetry implementation.

Abstract

Symmetries of nonlinear control systems in state representation are considered. To this end, a geometric approach to ordinary differential equations is advocated. Invariant feedback laws for systems with Lie symmetries, i.e. feedback laws that preserve the symmetry group of a considered plant, can be constructed based on invariants of the considered group action. Under minor technical assumptions suitable invariant tracking errors can be determined by following a normalization procedure. The underlying local geometric meaning of this procedure is discussed and it is shown how it can also be applied in order to derive a local, reduced-order system representation. Further, the idea of controlled symmetries, i.e. imposing desired symmetry properties on a given control system by state feedback, is discussed by outlining an exemplary control design for a predator-prey bioreactor.