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Data-driven control of nonlinear systems from input-output data

Xiaoyan Dai, Claudio De Persis, Nima Monshizadeh, Pietro Tesi

TL;DR

The paper develops a data-driven method to design dynamic output-feedback controllers for nonlinear discrete-time systems using input/output data under a uniform observability assumption. By introducing a dynamic extension with a chain of integrators and leveraging a data-dependent, linear-in-parameters representation, it derives a solvable SDP that yields a stabilizing output-feedback gain from IO data. The approach provides local stability guarantees for the extended system and includes a procedure to estimate the region of attraction from data, demonstrated on a pendulum example. This work enables IO-based controller synthesis for nonlinear systems with no direct state measurements, advancing practical data-driven control when only IO measurements are available.

Abstract

The design of controllers from data for nonlinear systems is a challenging problem. In a recent paper, De Persis, Rotulo and Tesi, "Learning controllers from data via approximate nonlinearity cancellation," IEEE Transactions on Automatic Control, 2023, a method to learn controllers that make the closed-loop system stable and dominantly linear was proposed. The approach leads to a simple solution based on data-dependent semidefinite programs. The method uses input-state measurements as data, while in a realistic setup it is more likely that only input-output measurements are available. In this note we report how the design principle of the above mentioned paper can be adjusted to deal with input-output data and obtain dynamic output feedback controllers in a favourable setting.

Data-driven control of nonlinear systems from input-output data

TL;DR

The paper develops a data-driven method to design dynamic output-feedback controllers for nonlinear discrete-time systems using input/output data under a uniform observability assumption. By introducing a dynamic extension with a chain of integrators and leveraging a data-dependent, linear-in-parameters representation, it derives a solvable SDP that yields a stabilizing output-feedback gain from IO data. The approach provides local stability guarantees for the extended system and includes a procedure to estimate the region of attraction from data, demonstrated on a pendulum example. This work enables IO-based controller synthesis for nonlinear systems with no direct state measurements, advancing practical data-driven control when only IO measurements are available.

Abstract

The design of controllers from data for nonlinear systems is a challenging problem. In a recent paper, De Persis, Rotulo and Tesi, "Learning controllers from data via approximate nonlinearity cancellation," IEEE Transactions on Automatic Control, 2023, a method to learn controllers that make the closed-loop system stable and dominantly linear was proposed. The approach leads to a simple solution based on data-dependent semidefinite programs. The method uses input-state measurements as data, while in a realistic setup it is more likely that only input-output measurements are available. In this note we report how the design principle of the above mentioned paper can be adjusted to deal with input-output data and obtain dynamic output feedback controllers in a favourable setting.
Paper Structure (14 sections, 5 theorems, 62 equations, 1 figure)

This paper contains 14 sections, 5 theorems, 62 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Dataset
  4. Uniform Observability
  5. An auxiliary system
  6. Design of an output feedback controller from data
  7. A dynamic extension
  8. Control input design
  9. Data-dependent representation of the closed-loop system
  10. Region of Attraction
  11. Main result
  12. Numerical example
  13. CONCLUSIONS
  14. Proof of Lemma \ref{['lem-equiv']}

Key Result

Lemma 1

Let system nonl satisfy Assumption asspt-uo-on-set. Consider arbitrary $k_0\in \mathbb{Z}$, $x(k-N)\in \mathcal{X}$ and $u_{[k-N, k-1]}\in \mathcal{U}^N$ for all $k\in \mathbb{Z}_{\ge k_0}$. Consider the system with $\tilde{f}, \tilde{h}$ defined in mappings. If the input $v(k)$ applied to nonl-eq satisfies $v(k)=u_{[k-N,k-1]}$ for all $k\in \mathbb{Z}_{\ge k_0}$ and the initial condition of nonl

Figures (1)

  • Figure 1: The blue area represents the estimate of the ROA of system \ref{['eq:sys_example_pendulum']} in closed-loop with the controller \ref{['chain-integrators-with-u']}, \ref{['chain-integrators-with-y']}, \ref{['u-tv']}, where $v_{k-k_0}=0$ for $k_0\le k\le k_0+N-1$ and $\kappa$ is given in \ref{['kappa.example']}.

Theorems & Definitions (11)

  • Lemma 1
  • Example 1
  • Remark 1
  • Lemma 2
  • Proposition 1
  • Definition 1
  • Definition 2
  • Corollary 1
  • Remark 2
  • Proposition 2
  • ...and 1 more