### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 058, 49 pages      arXiv:2103.05078      https://doi.org/10.3842/SIGMA.2024.058
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

### Dynamic Feedback Linearization of Control Systems with Symmetry

Jeanne N. Clelland a, Taylor J. Klotz b and Peter J. Vassiliou c
a) Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA
b) Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall (Keller Hall 401A), Honolulu, Hawaii 96822, USA
c) Mathematical Sciences Institute, Australian National University, Canberra, ACT, 2601 Australia

Received July 29, 2023, in final form May 30, 2024; Published online July 01, 2024

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.

Key words: Lie symmetry reduction; contact geometry; static feedback linearization; explicit integrability; flat outputs; principal bundle.

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