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Physics-informed Learning for Passivity-based Tracking Control

Thomas Beckers, Leonardo Colombo

TL;DR

This work tackles tracking control for systems with partially unknown dynamics within the port-Hamiltonian and IDA-PBC framework. It introduces a physics-informed learning approach, GP-PHS, to jointly learn the Hamiltonian while accounting for uncertainty in $J$, $R$, and $G$, and then applies a modified matching equation to design a tracking controller that guarantees probabilistic stability and semi-passivity. The method is validated by simulation on a mechanical-electrical example, showing bounded tracking error and a decreasing tracking-energy function $H_d$ despite model errors. The key contribution is enabling energy-based tracking guarantees under uncertainty, enabling robust tracking in multi-physics systems with data-driven learning.

Abstract

Passivity-based control ensures system stability by leveraging dissipative properties and is widely applied in electrical and mechanical systems. Port-Hamiltonian systems (PHS), in particular, are well-suited for interconnection and damping assignment passivity-based control (IDA-PBC) due to their structured, energy-centric modeling approach. However, current IDA-PBC faces two key challenges: (i) it requires precise system knowledge, which is often unavailable due to model uncertainties, and (ii) it is typically limited to set-point control. To address these limitations, we propose a data-driven tracking control approach based on a physics-informed model, namely Gaussian process Port-Hamiltonian systems, along with the modified matching equation. By leveraging the Bayesian nature of the model, we establish probabilistic stability and passivity guarantees. A simulation demonstrates the effectiveness of our approach.

Physics-informed Learning for Passivity-based Tracking Control

TL;DR

This work tackles tracking control for systems with partially unknown dynamics within the port-Hamiltonian and IDA-PBC framework. It introduces a physics-informed learning approach, GP-PHS, to jointly learn the Hamiltonian while accounting for uncertainty in , , and , and then applies a modified matching equation to design a tracking controller that guarantees probabilistic stability and semi-passivity. The method is validated by simulation on a mechanical-electrical example, showing bounded tracking error and a decreasing tracking-energy function despite model errors. The key contribution is enabling energy-based tracking guarantees under uncertainty, enabling robust tracking in multi-physics systems with data-driven learning.

Abstract

Passivity-based control ensures system stability by leveraging dissipative properties and is widely applied in electrical and mechanical systems. Port-Hamiltonian systems (PHS), in particular, are well-suited for interconnection and damping assignment passivity-based control (IDA-PBC) due to their structured, energy-centric modeling approach. However, current IDA-PBC faces two key challenges: (i) it requires precise system knowledge, which is often unavailable due to model uncertainties, and (ii) it is typically limited to set-point control. To address these limitations, we propose a data-driven tracking control approach based on a physics-informed model, namely Gaussian process Port-Hamiltonian systems, along with the modified matching equation. By leveraging the Bayesian nature of the model, we establish probabilistic stability and passivity guarantees. A simulation demonstrates the effectiveness of our approach.
Paper Structure (8 sections, 4 theorems, 25 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Port-Hamiltonian Systems
  4. Problem Setting
  5. Tracking Control of PHS
  6. Gaussian Process Port-Hamiltonian System
  7. Passivity-based Tracking control using GP-PHS
  8. Evaluation

Key Result

Theorem 1

Let for:gpphs be a GP-PHS model of the physical system eq:phs based on the dataset $\mathcal{D}$. Given the desired dynamics for:pchmodel with propy:1 that satisfyin the following, we omit the dependency on $\bm x,\bm x_d$ for brevity if obvious. where $H_d,R_d$ are designed such that for all $\{\bm\eta(\bm x)\in\mathbb{R}^n| |\eta_i(\bm x)|\leq \beta_i \mathop{\mathrm{var}}\nolimits\left(\dot{x

Figures (4)

  • Figure 1: Overview of the GP-PHS based tracking control approach. First, a GP-PHS model is used to learn the partially unknown dynamics of the physical systems. Then, a modified matching equations is used to design a tracking controller that is robustified against the modeling errors, enabling stability and passivity guarantees.
  • Figure 2: Electrostatic microactuator with unknown capacity $C(x_1)$.
  • Figure 3: Top 1 and 2: Closed-loop system with the proposed tracking control law. As there is uncertainty in the GP-PHS model, the closed-loop dynamics (solid) slightly deviates from the desired PHS (dashed) but remains in a neighborhood. Bottom: Computed control input over time.
  • Figure 4: The Hamiltonian function of the tracking error dynamics is decreasing as expected.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Corollary 2
  • proof