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Output-Feedback Controller Synthesis for Dissipativity and $H_2$ Performance of Autoregressive Systems from Noisy Input-Output Data

Pietro Kristović, Andrej Jokić, Mircea Lazar

Abstract

In this paper we propose a data-driven output-feedback controller synthesis method for discrete-time linear time-invariant systems in a specific autoregressive form. The synthesis goal is either to achieve dissipativity with respect to a given quadratic supply rate, or to achieve given $H_2$ performance level. It is assumed that the model of the plant is unknown, except for the disturbance term. To compensate for the lack of model knowledge, we have a recorded trajectory of the controlled input and the output available for control, which can be corrupted by an unknown but bounded disturbance. Derived controller synthesis method is in the form of linear matrix inequalities and is nonconservative within the considered problem setting.

Output-Feedback Controller Synthesis for Dissipativity and $H_2$ Performance of Autoregressive Systems from Noisy Input-Output Data

Abstract

In this paper we propose a data-driven output-feedback controller synthesis method for discrete-time linear time-invariant systems in a specific autoregressive form. The synthesis goal is either to achieve dissipativity with respect to a given quadratic supply rate, or to achieve given performance level. It is assumed that the model of the plant is unknown, except for the disturbance term. To compensate for the lack of model knowledge, we have a recorded trajectory of the controlled input and the output available for control, which can be corrupted by an unknown but bounded disturbance. Derived controller synthesis method is in the form of linear matrix inequalities and is nonconservative within the considered problem setting.

Paper Structure

This paper contains 17 sections, 5 theorems, 56 equations, 1 figure, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Dissipativity
  4. $H_2$ performance
  5. Matrix S-lemma
  6. Dualization lemma
  7. Problem definition
  8. The plant, the controller and the state-space realization of the closed-loop system
  9. The unknown matrices and the known data
  10. Assumptions regarding the known data
  11. The disturbance model
  12. The data-driven realization of the closed-loop system
  13. The set of plants consistent with data
  14. The control problem
  15. Controller synthesis
  16. ...and 2 more sections

Key Result

Theorem B.2

Let $M, H \in \mathbb{R}^{(p+r)\times (p+r)}$ be symmetric matrices and consider the partition where $H_{11}\in \mathbb{R}^{p \times p}$. Let the set $S_{H}$ be defined as AssumeNote that in c21 it is in fact required that $H_{11}-H_{12}H_{22}^\dagger H_{12}^\top \succeq 0$ (i.e., the generalized Schur complement of $H$ with respect to $H_{22}$ is positive semidefinite). This condition is equiva

Figures (1)

  • Figure E1: Frequency responses (solid lines) of closed-loop system with data-based controller, and associated $H_\infty$ performance bounds (dashed lines) for $\sigma \in (0,0.01, 0.05, 0.1, 0.2)$ in red, green, magenta, black and blue color, respectively.

Theorems & Definitions (9)

  • Remark B.1
  • Theorem B.2
  • Lemma B.3
  • Remark C.1
  • Lemma C.5
  • proof
  • Theorem D.1
  • Theorem D.2
  • Remark D.3