Towards Polynomial Immersion of Port-Hamiltonian Systems

TL;DR

This paper presents a method to embed non-polynomial port-Hamiltonian systems into higher-dimensional polynomial systems, preserving key properties and enabling polynomial-based control design.

Contribution

It introduces a trajectory-preserving polynomial immersion technique for non-polynomial pH systems, facilitating stability analysis and control synthesis.

Findings

Preserves energy and interconnection structure during immersion

Enables stabilizing control design via sum-of-squares optimization

Validated through multiple illustrative examples

Abstract

Port-Hamiltonian (pH) systems offer a highly structured and energy-based modular framework for control systems. Many pH systems exhibit non-polynomial non-linearities. We consider the problem of immersing such systems into a higher-dimensional polynomial representation. We prove that, along system trajectories, important features of the non-polynomial pH system are preserved such as the internal interconnection geometry, the energy balance relation with passivity supply rate, as well as energy dissipation. We illustrate how the lifted system enables the design of stabilizing feedback laws by combining sum-of-squares optimization with concepts from passivity-based control. We draw upon several examples to illustrate our findings.