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Data-Driven Boundary Control of Distributed Port-Hamiltonian Systems

Thomas Beckers, Leonardo Colombo

Abstract

Distributed Port-Hamiltonian (dPHS) theory provides a powerful framework for modeling physical systems governed by partial differential equations and has enabled a broad class of boundary control methodologies. Their effectiveness, however, relies heavily on the availability of accurate system models, which may be difficult to obtain in the presence of nonlinear and partially unknown dynamics. To address this challenge, we combine Gaussian Process distributed Port-Hamiltonian system (GP-dPHS) learning with boundary control by interconnection. The GP-dPHS model is used to infer the unknown Hamiltonian structure from data, while its posterior uncertainty is incorporated into an energy-based robustness analysis. This yields probabilistic conditions under which the closed-loop trajectories remain bounded despite model mismatch. The method is illustrated on a simulated shallow water system.

Data-Driven Boundary Control of Distributed Port-Hamiltonian Systems

Abstract

Distributed Port-Hamiltonian (dPHS) theory provides a powerful framework for modeling physical systems governed by partial differential equations and has enabled a broad class of boundary control methodologies. Their effectiveness, however, relies heavily on the availability of accurate system models, which may be difficult to obtain in the presence of nonlinear and partially unknown dynamics. To address this challenge, we combine Gaussian Process distributed Port-Hamiltonian system (GP-dPHS) learning with boundary control by interconnection. The GP-dPHS model is used to infer the unknown Hamiltonian structure from data, while its posterior uncertainty is incorporated into an energy-based robustness analysis. This yields probabilistic conditions under which the closed-loop trajectories remain bounded despite model mismatch. The method is illustrated on a simulated shallow water system.

Paper Structure

This paper contains 10 sections, 3 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian Process Regression
  4. Distributed Port-Hamiltonian systems
  5. Problem Setting
  6. Control Design based on GP-dPHS
  7. Data Collection
  8. Training of the Gaussian Process distributed Port-Hamiltonian model
  9. Boundary Control by Interconnection
  10. Evaluation

Key Result

Proposition 1

Let $C \colon X \times \mathbb{R}^{n_c} \to \mathbb{R}$ be a Casimir function of the autonomous closed-loop system obtained from the controller eq:controller and modeldPHSform, connected via eq:interconnection. Then, $C$ is also a Casimir function for the autonomous closed-loop system obtained from

Figures (3)

  • Figure A1: Overview of the control-by-interconnection approach. A GP-dPHS model is used to learn the partially unknown dynamics of the PDE systems. Then, the model enables the design of a boundary controller if the system dissipation dominates the model's uncertainty.
  • Figure D1: Controlled water level. At $t=0$, the water level is constant over the spatial domain (gray dashed). Then, the controller lets water flow into the system from the left boundary, reaching the desired equilibrium (black dashed) after 10 seconds.
  • Figure D2: Comparison of the evolution of the Hamiltonian with a controller designed based on a model (a) and the actual system dynamics (b). As stated in \ref{['prop:3']}, the model error leads to a Hamiltonian that converges to a set around the steady state.

Theorems & Definitions (8)

  • Remark 1
  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2: adapted from macchelli2015control
  • proof
  • Proposition 3
  • proof