Control of port-Hamiltonian systems with minimal energy supply

TL;DR

This paper studies optimal control of linear port-Hamiltonian systems, revealing how minimal energy control strategies relate to the system's conservative subspace and demonstrating the turnpike property in this context.

Contribution

It introduces a decomposition-based analysis of port-Hamiltonian systems, showing the boundedness of reachable states and the turnpike property with respect to the conservative subspace.

Findings

Reachable states are bounded in the dissipative subspace.

Optimal trajectories stay close to the conservative subspace over time.

All optimal steady states are located within the non-dissipative subspace.

Abstract

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.