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Neural Distributed Controllers with Port-Hamiltonian Structures

Muhammad Zakwan, Giancarlo Ferrari-Trecate

TL;DR

This work addresses the challenge of designing distributed controllers for large-scale nonlinear dissipative networks with limited local information while ensuring safety and stability. It introduces a free-parametrization of neural distributed controllers based on port-Hamiltonian structure, guaranteeing a finite $L_2$ gain and closed-loop stability for all trainable parameters, thus enabling unconstrained SGD-based training. The approach supports arbitrary nonlinear storage functions and sparsity patterns, outperforming previous methods that restricted the storage function to a quadratic form or required weight constraints. The authors validate the framework on Kuramoto oscillator synchronization, demonstrating robust convergence across multiple network topologies and illustrating practical potential for scalable, safe learning-enabled control in dissipation-dominated systems.

Abstract

Controlling large-scale cyber-physical systems necessitates optimal distributed policies, relying solely on local real-time data and limited communication with neighboring agents. However, finding optimal controllers remains challenging, even in seemingly simple scenarios. Parameterizing these policies using Neural Networks (NNs) can deliver good performance, but their sensitivity to small input changes can destabilize the closed-loop system. This paper addresses this issue for a network of nonlinear dissipative systems. Specifically, we leverage well-established port-Hamiltonian structures to characterize deep distributed control policies with closed-loop stability guarantees and a finite $\mathcal{L}_2$ gain, regardless of specific NN parameters. This eliminates the need to constrain the parameters during optimization and enables training with standard methods like stochastic gradient descent. A numerical study on the consensus control of Kuramoto oscillators demonstrates the effectiveness of the proposed controllers.

Neural Distributed Controllers with Port-Hamiltonian Structures

TL;DR

This work addresses the challenge of designing distributed controllers for large-scale nonlinear dissipative networks with limited local information while ensuring safety and stability. It introduces a free-parametrization of neural distributed controllers based on port-Hamiltonian structure, guaranteeing a finite gain and closed-loop stability for all trainable parameters, thus enabling unconstrained SGD-based training. The approach supports arbitrary nonlinear storage functions and sparsity patterns, outperforming previous methods that restricted the storage function to a quadratic form or required weight constraints. The authors validate the framework on Kuramoto oscillator synchronization, demonstrating robust convergence across multiple network topologies and illustrating practical potential for scalable, safe learning-enabled control in dissipation-dominated systems.

Abstract

Controlling large-scale cyber-physical systems necessitates optimal distributed policies, relying solely on local real-time data and limited communication with neighboring agents. However, finding optimal controllers remains challenging, even in seemingly simple scenarios. Parameterizing these policies using Neural Networks (NNs) can deliver good performance, but their sensitivity to small input changes can destabilize the closed-loop system. This paper addresses this issue for a network of nonlinear dissipative systems. Specifically, we leverage well-established port-Hamiltonian structures to characterize deep distributed control policies with closed-loop stability guarantees and a finite gain, regardless of specific NN parameters. This eliminates the need to constrain the parameters during optimization and enables training with standard methods like stochastic gradient descent. A numerical study on the consensus control of Kuramoto oscillators demonstrates the effectiveness of the proposed controllers.
Paper Structure (6 sections, 2 theorems, 17 equations, 4 figures)

This paper contains 6 sections, 2 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Neural $\mathcal{L}_2$-stable Hamiltonian Controllers
  4. Synchronization in Kuramoto Oscillators
  5. Conclusion
  6. Proof of Theorem \ref{['thm:finite_l2_gain']}

Key Result

Theorem 1

Consider the closed-loop system $\Sigma_1 \Vert_f \Sigma_2$ given in Fig. fig:small_gain_theorem.

Figures (4)

  • Figure 1: An example of a large-scale system $\Sigma_s$ and a distributed controller $\Sigma_c$ for $N = 4$. The solid lines represent interactions between the subsystems of $\Sigma_s$, and the dashed lines represent the flow of information between the system $\Sigma_s$ and the controller $\Sigma_c$.
  • Figure 2: Standard feedback interconnection $\Sigma_1 \Vert_f \Sigma_2$.
  • Figure 3: The consensus metric $r(t)$ for an uncontrolled network with a fully connected communication topology demonstrating unsynchronized behavior of the oscillators.
  • Figure 4: The consensus metric $r(t)$ for the closed-loop with different communication topologies exhibiting consensus.

Theorems & Definitions (7)

  • Definition 1: Dissipativity, vanderSchaft2017
  • Theorem 1: vanderSchaft2017
  • Remark 1: Passivity by design
  • Theorem 2
  • Remark 2: Selection of Hamiltonian
  • Remark 3: Comparison with RENs
  • Remark 4: Communication among sub-controllers