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Approximate robust output regulation of boundary control systems

Jukka-Pekka Humaloja, Mikael Kurula, Lassi Paunonen

Abstract

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.

Approximate robust output regulation of boundary control systems

Abstract

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.

Paper Structure

This paper contains 12 sections, 12 theorems, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The wave equation
  3. The wave equation as a formal boundary control system
  4. Exponential stabilization and admissible observation
  5. The plant, the controller, and the exosystem
  6. Output regulation
  7. A regulating controller
  8. The Internal Model Principle
  9. An approximate robust controller
  10. A robust controller
  11. Approximate robust regulation of the wave equation
  12. Conclusions

Key Result

Theorem 2.2

The triple $(\widetilde{\mathfrak B},\widetilde{\mathfrak A},\widetilde{\mathfrak C})$ is a passive BCS on $(\mathcal{U},X_\mathcal{H},\mathcal{U})$.

Figures (4)

  • Figure 1: The output profile $y$ of the controlled wave equation and the reference profile $y_{ref}$ for $t \in [0, 10]$ and in the same scales.
  • Figure 2: The regulation error $\int\limits_t^{t+1}\|y(s) - y_{ref}(s)\|^2ds$ for $t \in [0, 20]$.
  • Figure 3: The wave profile of the controlled system at $t = 9$.
  • Figure 4: The disturbance signal $d$ for $t \in [0, 6]$.

Theorems & Definitions (30)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • Definition 4.1
  • ...and 20 more