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Feedback Linearizable Discretizations of Second Order Mechanical Systems using Retraction Maps

Shreyas N. B., David Martin Diego, Ravi Banavar

Abstract

Mechanical systems are most often described by a set of continuous-time, nonlinear, second-order differential equations (SODEs) of a particular structure governed by the covariant derivative. The digital implementation of controllers for such systems requires a discrete model of the system and hence requires numerical discretization schemes. Feedback linearizability of such sampled systems, however, depends on the discretization scheme employed. In this article, we utilize retraction maps and their lifts to construct feedback linearizable discretizations for SODEs which can be applied to many mechanical systems.

Feedback Linearizable Discretizations of Second Order Mechanical Systems using Retraction Maps

Abstract

Mechanical systems are most often described by a set of continuous-time, nonlinear, second-order differential equations (SODEs) of a particular structure governed by the covariant derivative. The digital implementation of controllers for such systems requires a discrete model of the system and hence requires numerical discretization schemes. Feedback linearizability of such sampled systems, however, depends on the discretization scheme employed. In this article, we utilize retraction maps and their lifts to construct feedback linearizable discretizations for SODEs which can be applied to many mechanical systems.
Paper Structure (27 sections, 6 theorems, 64 equations, 7 figures)

This paper contains 27 sections, 6 theorems, 64 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Retraction and Discretization Maps
  4. Feedback Linearization
  5. Constructing feedback linearizable discretization maps
  6. Mechanical Control Systems
  7. Mechanical Feedback Linearization
  8. Tangent Lifts of Discretization Maps
  9. Commutator diagram
  10. Feedback Linearizable Discretizations for SODEs
  11. Example: Symmetric Discretization
  12. MF-Linearizable discretizations
  13. Example
  14. Example 1
  15. MF-Linearization
  16. ...and 12 more sections

Key Result

Proposition I.1

Let $\phi: M \longrightarrow N:= \mathbb{R}^{n}$ be the linearizing coordinate change, and $\psi: M \times U \longrightarrow \mathbb{R}^{m}$ be the linearizing feedback, where $U \subset \mathbb{R}^{m}$ is the control space. Let $R_d$ be a discretization map on $N$ that discretizes the continuous-ti is a discretization map on $M$, which discretizes the continuous-time system (CTS) to a feedback li

Figures (7)

  • Figure 1.1: $R_d$ and $R_{d,\phi}$ commute as shown
  • Figure 3.1: $R_d^T$ and $R^T_{d,\phi}$ commute as shown. Description in \ref{['sec:double-comm']}.
  • Figure 6.1: The inertia wheel pendulum
  • Figure 6.2: System states $x_k$ for symmetric discretization plotted against exact discretization (ODE45) $x(t_k)$ for $t_k \in [0, 1]$
  • Figure 6.3: Magnitude of error norm for $\theta_1$ and $\dot{\theta}_1$
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition I.1
  • Remark I.1
  • Definition I.2
  • Proposition I.1
  • Definition II.1
  • Theorem III.1
  • Proposition III.2
  • Definition III.1
  • Proposition IV.1
  • proof
  • ...and 6 more