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Feedback linearization through the lens of data

C. De Persis, D. Gadginmath, F. Pasqualetti, P. Tesi

TL;DR

This work considers the feedback linearization problem of an unknown system when the solution must be found using experimental data and proposes a new method that learns the change of coordinates and the linearizing controller from a library of candidate functions with a simple algebraic procedure.

Abstract

Controlling nonlinear systems, especially when data are being used to offset uncertainties in the model, is hard. A natural approach when dealing with the challenges of nonlinear control is to reduce the system to a linear one via change of coordinates and feedback, an approach commonly known as feedback linearization. Here we consider the feedback linearization problem of an unknown system when the solution must be found using experimental data. We propose a new method that learns the change of coordinates and the linearizing controller from a library (a dictionary) of candidate functions with a simple algebraic procedure - the computation of the null space of a data-dependent matrix. Remarkably, we show that the solution is valid over the entire state space of interest and not just on the dataset used to determine the solution.

Feedback linearization through the lens of data

TL;DR

This work considers the feedback linearization problem of an unknown system when the solution must be found using experimental data and proposes a new method that learns the change of coordinates and the linearizing controller from a library of candidate functions with a simple algebraic procedure.

Abstract

Controlling nonlinear systems, especially when data are being used to offset uncertainties in the model, is hard. A natural approach when dealing with the challenges of nonlinear control is to reduce the system to a linear one via change of coordinates and feedback, an approach commonly known as feedback linearization. Here we consider the feedback linearization problem of an unknown system when the solution must be found using experimental data. We propose a new method that learns the change of coordinates and the linearizing controller from a library (a dictionary) of candidate functions with a simple algebraic procedure - the computation of the null space of a data-dependent matrix. Remarkably, we show that the solution is valid over the entire state space of interest and not just on the dataset used to determine the solution.
Paper Structure (20 sections, 8 theorems, 87 equations)

This paper contains 20 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Feedback linearization problem
  4. Objective of the paper and assumptions
  5. Main results
  6. Model-based solution
  7. Data-based solution and generalization from the sample space
  8. Discussion
  9. Basis functions
  10. The space of solutions of $\mathcal{F}(\mathbb{D})v=0$
  11. Concluding remarks
  12. appendix
  13. Proof of Proposition \ref{['lem:dimension-nu-space-V']}
  14. Proof of Proposition \ref{['prop3']}
  15. Further details on Example \ref{['empl:extended-Z']}
  16. ...and 5 more sections

Key Result

Lemma 1

The linearization problem is solvable if there exist a neighborhood $\mathcal{D}$ of $x^0$, functions $\gamma\colon \mathcal{D}\to \mathbb{R}^{m\times m}$, $\delta\colon \mathcal{D}\to \mathbb{R}^m$, with $\gamma(x)$ nonsingular for all $x\in \mathcal{D}$, and a coordinate transformation $\tau\col where $A_c=\text{diag}(A_1, \ldots, A_m)$, $B_c=\text{diag}(B_1, \ldots, B_m)$, with $r_1, \ldots,

Theorems & Definitions (14)

  • Lemma 1
  • Theorem 1
  • Example 1
  • Theorem 2
  • Proposition 1
  • Example 2
  • Remark 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • ...and 4 more