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Dynamic Output-Feedback Controller Synthesis for Dissipativity and $H_2$ Performance from Noisy Input-State Data

Pietro Kristović, Andrej Jokić, Mircea Lazar

Abstract

In this paper we propose dynamic output-feedback controller synthesis methods for discrete-time linear time-invariant systems. The synthesis goal is to achieve dissipativity with respect to a given quadratic supply rate or a given $H_2$ performance level. It is assumed that the model of system dynamics is unknown, expect for the disturbance term. Instead, we have a recorded trajectory of the control input and the state, which can be corrupted by an unknown but bounded disturbance. The state data is used only for the purpose of controller synthesis, while the designed controller is output feedback controller, i.e., the full state is not used for control in real time. The presented synthesis method is formulated in terms of linear matrix inequalities parametrized by a scalar variable. Within the considered setting, the synthesis procedure is non-conservative.

Dynamic Output-Feedback Controller Synthesis for Dissipativity and $H_2$ Performance from Noisy Input-State Data

Abstract

In this paper we propose dynamic output-feedback controller synthesis methods for discrete-time linear time-invariant systems. The synthesis goal is to achieve dissipativity with respect to a given quadratic supply rate or a given performance level. It is assumed that the model of system dynamics is unknown, expect for the disturbance term. Instead, we have a recorded trajectory of the control input and the state, which can be corrupted by an unknown but bounded disturbance. The state data is used only for the purpose of controller synthesis, while the designed controller is output feedback controller, i.e., the full state is not used for control in real time. The presented synthesis method is formulated in terms of linear matrix inequalities parametrized by a scalar variable. Within the considered setting, the synthesis procedure is non-conservative.

Paper Structure

This paper contains 15 sections, 5 theorems, 54 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Dissipativity
  4. $H_2$ performance
  5. Matrix S-lemma
  6. Problem definition
  7. The plant
  8. Problem and data recording settings
  9. The disturbance model
  10. The set of plants consistent with the data
  11. The control problem
  12. Controller synthesis
  13. EXAMPLE
  14. CONCLUSIONS
  15. Performance certificates and controller realizations

Key Result

Theorem II.2

Let $M, H \in \mathbb{R}^{(n+r)\times (n+r)}$ be symmetric matrices with partitions where $H_{11}\in \mathbb{R}^{n \times n}$, and let the set $S_{H}$ be defined as Assume that $H_{22}\prec 0$ and $S_{H} \neq \emptyset$. Then, we have that if and only if there exists scalar $\alpha \geq 0$ such that

Figures (2)

  • Figure 1: $H_\infty$ performance $\gamma$ as a function of the parameter $\alpha$ using dynamic state (dashed) and output (solid) feedback controller synthesis with $\sigma \in (0, 0.05, 0.1, 0.2)$ in red, green, black and blue color, respectively.
  • Figure 2: Frequency response of the closed-loop system using dynamic state (dashed) and output (solid) feedback controller with $\sigma \in (0, 0.05, 0.1, 0.2)$ in red, green, black and blue color, respectively.

Theorems & Definitions (12)

  • Remark II.1
  • Theorem II.2
  • Lemma III.2
  • proof
  • Definition III.4
  • Theorem IV.1
  • proof
  • Theorem IV.2
  • proof
  • Remark IV.3
  • ...and 2 more