Port-Hamiltonian Systems with Dissipation Potential: Modelling and Trajectory Tracking Control

TL;DR

This paper introduces a port-Hamiltonian system with dissipation potential (PHS-DP) that simplifies trajectory tracking control by replacing the damping matrix with scalar convex dissipation potentials, maintaining physical interpretability.

Contribution

It proposes a novel PHS-DP framework that enables PDE-free energy shaping for trajectory tracking, preserving the system's energy structure and simplifying the control design process.

Findings

Validation on magnetic levitation system shows effective trajectory tracking.

The method reduces design complexity compared to traditional PDE-based approaches.

The approach maintains the physical interpretability of the port-Hamiltonian structure.

Abstract

Port-Hamiltonian systems (PHS) and interconnection and damping assignment passivity-based control (IDA-PBC) have achieved broad success in modelling and stabilisation of physical systems. However, the absence of a dedicated scalar potential for the momentum channel forces any modification of the momentum-dependent dynamics to proceed indirectly through the interconnection and damping matrices, rendering the matching partial differential equation (PDE) difficult to solve and complicating extensions to trajectory tracking. This paper proposes a port-Hamiltonian system with dissipation potential (PHS-DP), in which the damping matrix is replaced by scalar convex dissipation potentials, providing independent scalar objects for the momentum and auxiliary state channels and restoring the variational symmetry between stored and dissipated energy. Building on this framework, Dual Potential Shaping Control (DPSC) achieves trajectory tracking by sequentially shaping the potential energy and dissipation potentials without modifying the interconnection structure. Contraction of the closed-loop cascade is established via a hierarchical contraction argument, and the matching condition is satisfied automatically for any admissible choice of shaped potentials, requiring no PDE to be solved. In contrast to existing PDE-free energy shaping approaches, which achieve this by abandoning the port-Hamiltonian closed-loop structure and sacrificing physical interpretability, the proposed framework preserves the interconnection structure and retains a transparent energy-based interpretation at every stage of the design. Validation on a magnetic levitation system demonstrates tracking performance comparable to timed IDA-PBC with substantially reduced design complexity.