Neural Port-Hamiltonian Models for Nonlinear Distributed Control: An Unconstrained Parametrization Approach

TL;DR

The framework of port-Hamiltonian systems is used to design continuous-time distributed control policies for nonlinear systems that guarantee closed-loop stability and finite $\mathcal{L}_2$ or incremental $\mathcal{L}_2$ gains, independent of the optimzation parameters of the controllers.

Abstract

The control of large-scale cyber-physical systems requires optimal distributed policies relying solely on limited communication with neighboring agents. However, computing stabilizing controllers for nonlinear systems while optimizing complex costs remains a significant challenge. Neural Networks (NNs), known for their expressivity, can be leveraged to parametrize control policies that yield good performance. However, NNs' sensitivity to small input changes poses a risk of destabilizing the closed-loop system. Many existing approaches enforce constraints on the controllers' parameter space to guarantee closed-loop stability, leading to computationally expensive optimization procedures. To address these problems, we leverage the framework of port-Hamiltonian systems to design continuous-time distributed control policies for nonlinear systems that guarantee closed-loop stability and finite $\mathcal{L}_2$ or incremental $\mathcal{L}_2$ gains, independent of the optimzation parameters of the controllers. This eliminates the need to constrain parameters during optimization, allowing the use of standard techniques such as gradient-based methods. Additionally, we discuss discretization schemes that preserve the dissipation properties of these controllers for implementation on embedded systems. The effectiveness of the proposed distributed controllers is demonstrated through consensus control of non-holonomic mobile robots subject to collision avoidance and averaged voltage regulation with weighted power sharing in DC microgrids.

Figures (13)

• Figure 1: An example of an interconnected 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 mobile wheeled robot used in the experiment. The triple $(q_x, q_y, q_\theta)$ denotes the position and the orientation of the robot, and $d$ is the distance from the center to the wheels.
• Figure 4: Forward velocities of the swarm demonstrating consensus. The translucent blue color lines represent the evolution of trajectories starting from the initial conditions sampled from a ball with radius $0.05$ centered at the nominal condition for each agent.
• Figure 5: Relative distance $r_{ij}$ among the agents with collision avoidance. Note that $r_{ij} < 0.5$ indicates a collision. The translucent blue color lines represent the evolution of trajectories starting from the initial conditions sampled from a ball with radius $0.05$ centered at the nominal condition for each agent.

Theorems & Definitions (17)

• Definition 1: Dissipativity, vanderSchaft2017
• Theorem 1: vanderSchaft2017
• Definition 2: Incremental Dissipativity, verhoek2023convex
• Theorem 2: vanderSchaft2017
• Definition 3: Differential Dissipativity, verhoek2023convex
• Remark 1: Time-varying parameters
• Remark 2: Selection of Hamiltonian
• Theorem 3
• Remark 3: Comparison with RENs
• Remark 4: Learning models with finite $\mathcal{L}_2$ gains