Energy Matching in Reduced Passive and Port-Hamiltonian Systems

It is well known that any port-Hamiltonian (pH) system is passive, and, conversely, any minimal and stable passive system has a pH representation. Nevertheless, this equivalence is only concerned with the input-output mapping but not with the Hamiltonian itself. Thus, we propose to view a pH system either as an enlarged dynamical system with the Hamiltonian as additional output or as two dynamical systems with the input-output and the Hamiltonian dynamic. Our first main result is a structure-preserving Kalman-like decomposition of the enlarged pH system that separates the controllable and zero-state observable parts. Moreover, for further approximations in the context of structure-preserving model-order reduction (MOR), we propose to search for a Hamiltonian in the reduced pH system that minimizes the $H_2$-distance to the full-order Hamiltonian without altering the input-output dynamic, thus discussing a particular aspect of the corresponding multiobjective minimization problem corresponding to $H_2$-optimal MOR for pH systems. We show that this optimization problem is uniquely solvable and can be recast as a standard semidefinite program, and we present two numerical approaches for solving it. The results are illustrated with three academic examples.

SIAM Journal on Control and Optimization