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Interconnection and Damping Assignment Passivity-Based Control using Sparse Neural ODEs

Nicolò Botteghi, Owen Brook, Urban Fasel, Federico Califano

TL;DR

IDA-PBC is limited by the need to solve matching PDEs exactly and by its stabilization-centric focus. The authors cast IDA-PBC as a multi-objective learning problem and use sparse neural ODEs with dictionary learning to learn J_d, R_d, H_d that shape a closed-loop port-Hamiltonian system while controlling the matching residuals. The approach yields nontrivial tasks such as regulation and discovery of periodic oscillations on a capacitor system, providing closed-form controller expressions and enabling stability analysis via the state-transition matrix. Residuals remain small along optimal trajectories, supporting practical deployment and interpretability. Overall, the method extends IDA-PBC to complex, task-driven control with interpretable, analyzable closed-loop models.

Abstract

Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a nonlinear control technique that assigns a port-Hamiltonian (pH) structure to a controlled system using a state-feedback law. While IDA-PBC has been extensively studied and applied to many systems, its practical implementation often remains confined to academic examples and, almost exclusively, to stabilization tasks. The main limitation of IDA-PBC stems from the complexity of analytically solving a set of partial differential equations (PDEs), referred to as the matching conditions, which enforce the pH structure of the closed-loop system. However, this is extremely challenging, especially for complex physical systems and tasks. In this work, we propose a novel numerical approach for designing IDA-PBC controllers without solving the matching PDEs exactly. We cast the IDA-PBC problem as the learning of a neural ordinary differential equation. In particular, we rely on sparse dictionary learning to parametrize the desired closed-loop system as a sparse linear combination of nonlinear state-dependent functions. Optimization of the controller parameters is achieved by solving a multi-objective optimization problem whose cost function is composed of a generic task-dependent cost and a matching condition-dependent cost. Our numerical results show that the proposed method enables (i) IDA-PBC to be applicable to complex tasks beyond stabilization, such as the discovery of periodic oscillatory behaviors, (ii) the derivation of closed-form expressions of the controlled system, including residual terms

Interconnection and Damping Assignment Passivity-Based Control using Sparse Neural ODEs

TL;DR

IDA-PBC is limited by the need to solve matching PDEs exactly and by its stabilization-centric focus. The authors cast IDA-PBC as a multi-objective learning problem and use sparse neural ODEs with dictionary learning to learn J_d, R_d, H_d that shape a closed-loop port-Hamiltonian system while controlling the matching residuals. The approach yields nontrivial tasks such as regulation and discovery of periodic oscillations on a capacitor system, providing closed-form controller expressions and enabling stability analysis via the state-transition matrix. Residuals remain small along optimal trajectories, supporting practical deployment and interpretability. Overall, the method extends IDA-PBC to complex, task-driven control with interpretable, analyzable closed-loop models.

Abstract

Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a nonlinear control technique that assigns a port-Hamiltonian (pH) structure to a controlled system using a state-feedback law. While IDA-PBC has been extensively studied and applied to many systems, its practical implementation often remains confined to academic examples and, almost exclusively, to stabilization tasks. The main limitation of IDA-PBC stems from the complexity of analytically solving a set of partial differential equations (PDEs), referred to as the matching conditions, which enforce the pH structure of the closed-loop system. However, this is extremely challenging, especially for complex physical systems and tasks. In this work, we propose a novel numerical approach for designing IDA-PBC controllers without solving the matching PDEs exactly. We cast the IDA-PBC problem as the learning of a neural ordinary differential equation. In particular, we rely on sparse dictionary learning to parametrize the desired closed-loop system as a sparse linear combination of nonlinear state-dependent functions. Optimization of the controller parameters is achieved by solving a multi-objective optimization problem whose cost function is composed of a generic task-dependent cost and a matching condition-dependent cost. Our numerical results show that the proposed method enables (i) IDA-PBC to be applicable to complex tasks beyond stabilization, such as the discovery of periodic oscillatory behaviors, (ii) the derivation of closed-form expressions of the controlled system, including residual terms

Paper Structure

This paper contains 14 sections, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Port-Hamiltonian systems and IDA-PBC
  4. Neural Ordinary Differential Equations
  5. Sparse Dictionary Learning
  6. Methodology
  7. A multi-objective perspective on IDA-PBC
  8. Sparse Neural ODE for IDA-PBC
  9. Numerical Results
  10. Regulation of the Capacitor's Moving Plate
  11. Discovery of an Oscillatory Behavior of the Capacitor's Moving Plate
  12. Stability of the Learned Periodic Oscillation
  13. Conclusion
  14. Differentiable L$_0$ Regularization

Figures (5)

  • Figure 1: State-feedback control law relying on a parametrization of $J_d(x)$, $R_d(x)$, and $H_d(x)$ using differentiable sparse dictionary models.
  • Figure 2: Evolution of the state variables.
  • Figure 3: Trend of the loss functions over training.
  • Figure 4: Evolution of the state variables for one period of oscillation $T$.
  • Figure 5: Evolution fo the state variables over multiple periods of oscillations.