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Dynamic Feedback Linearization of Control Systems with Symmetry

Jeanne N. Clelland, Taylor J. Klotz, Peter J. Vassiliou

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.

Dynamic Feedback Linearization of Control Systems with Symmetry

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.

Paper Structure

This paper contains 20 sections, 18 theorems, 170 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on explicit integrability, dynamic feedback linearizability, and flatness
  3. Geometry of control systems
  4. Brunovský normal forms
  5. Goursat bundles
  6. Control admissible symmetries
  7. Relative Goursat bundles
  8. The contact sub-connection
  9. Cascade feedback linearization
  10. How hard is it to implement cascade linearization?
  11. Relation between cascade linearization & differential flatness
  12. Framework for dynamic feedback linearization of invariant control systems
  13. Signature of an explicit solution
  14. Dynamic linearization via symmetry reduction
  15. Explicit solution via the contact sub-connection
  16. ...and 5 more sections

Key Result

Proposition 2.8

A regular control system of the form controlSystem is explicitly integrable if and only if it is regularly dynamic feedback linearizable.

Figures (5)

  • Figure 1: Principal bundle equivalence where $\boldsymbol{\gamma}^G=\operatorname{ann}{\mathcal{H}}_G$ and $c\colon\mathbb{R}\to J^\kappa$ is an integral curve of $\boldsymbol{\beta}^\kappa$.
  • Figure 2: Items (1) and (2) in Definition \ref{['cfl-def']} are labelled by (1) and (2) in this diagram.
  • Figure 3: Here $\varphi'$ is the static feedback map that linearizes the prolonged contact sub-connection.
  • Figure 4: Deformation of $\widetilde{\widetilde{\varphi}}$ to a static feedback transformation $\vartheta$.
  • Figure 5: In this diagram the signature $\kappa$ decomposes as $\kappa=\nu+\nu^\perp$. Additionally, the maps $\varphi$, $\bar{\varphi}$, $\widetilde{\varphi}$, and $\vartheta$ are all SFTs.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 3.1
  • ...and 56 more