Dynamic Feedback Linearization of Control Systems with Symmetry

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Clelland, Jeanne N., Klotz, Taylor J., Vassiliou, Peter J.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212249
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Dynamic Feedback Linearization of Control Systems with Symmetry. Jeanne N. Clelland, Taylor J. Klotz and Peter J. Vassiliou. SIGMA 20 (2024), 058, 49 pages

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 is 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 with 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, many more elementary pedagogical examples are discussed as an aid to reading the paper. The constructions have been automated in the Maple package DifferentialGeometry.