Dynamic Feedback Linearization of Control Systems with Symmetry

Jeanne Clelland, Taylor Klotz, Peter Vassiliou

Published 2021-03-08Version 1

Control systems of interest are often invariant under Lie groups of transformations. Given such a control system, assumed to not be static feedback linearizable, a verifiable geometric condition is described and proven to guarantee its dynamic feedback linearizability. Additionally, a systematic procedure for obtaining all the system trajectories is shown to follow from this condition. 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 those systems requiring non-trivial dynamic extensions; that is, beyond augmentation by chains of integrators. Nevertheless, the results are illustrated by an example of each type. Firstly, a control system that can be dynamically linearized by a chain of integrators, and secondly, one which does not possess any linearizing chains of integrators and for which a dynamic feedback linearization is nevertheless derived. These systems are discussed in some detail. The constructions have been automated in the package DifferentialGeometry running on Maple.