Towards Polynomial Immersion of Port-Hamiltonian Systems

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.