Dynamic Feedback Linearization of Control Systems with Symmetry

TL;DR

This paper develops a geometric framework using Lie symmetry for dynamic feedback linearization of control systems, allowing for fully nonlinear, time-varying systems without restrictive assumptions, and demonstrates practical applications including aerospace systems.

Contribution

It introduces a symmetry-based geometric approach for dynamic feedback linearization applicable to fully nonlinear, time-varying control systems without additional local form constraints.

Findings

Provides a sufficient condition for dynamic feedback linearizability.

Systematic procedure for obtaining all smooth, generic trajectories.

Automated the constructions in Maple package DifferentialGeometry.

Abstract

Control systems of interest are often invariant under Lie groups of transformations. For such control systems, a geometric framework based on Lie symmetry is formulated, and from this a sufficient condition for dynamic feedback linearizability obtained. Additionally, a systematic procedure for obtaining all the smooth, generic system trajectories is shown to follow from the theory. Besides smoothness and the existence of symmetry, no further assumption is made on the local form of a control system, which is therefore permitted to be fully nonlinear and time varying. Likewise, no constraints are imposed on the local form of the dynamic compensator. Particular attention is given to the consideration of geometric (coordinate independent) structures associated to control systems with symmetry. To show how the theory is applied in practice we work through illustrative examples of control systems, including the vertical take-off and landing system, demonstrating the significant role that Lie symmetry plays in dynamic feedback linearization. Besides these, a number of more elementary pedagogical examples are discussed as an aid to reading the paper. The constructions have been automated in the Maple package DifferentialGeometry.