Dynamic feedback linearization of control systems with symmetry (Q6588222)

Control systems of interest are usually invariant under Lie groups of transformations. This paper formulates the geometric framework based on Lie symmetry for such control systems, and obtains from this a sufficient condition for dynamical feedback linearizability. Furthermore, 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 the 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, the authors 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 \textit{DifferentialGeometry}.