Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs

Abstract

We consider port-Hamiltonian systems with dissipation (PHSD) whose underlying geometric structure is represented as the composition of a Dirac and a resistive structure. We show how the choice of a new passive input-output pair for a PHSD is reflected in a new Dirac structure. We define a general class of new passive inputoutput pairs for a PHSD and subsequently compute (in a constructive manner) the resulting new Dirac structure and examine the achievable Casimirs for this new Dirac structure. We focus on the special case where only the passive output is changed (while retaining the original input) and subsequently define a general class of new passive outputs for the PHSD. We then identify (on the basis of the achievable Casimirs) the precise form of the so-called dissipation obstacle, and how this obstacle may be removed by changing the passive output.We also review the “swapping the damping” procedure for computing a new passive output, and show how this can be obtained as a special case within our approach. We finally consider the examples of the RLC-circuit and MEMS optical switch to investigate the role played by the new class of passive outputs in shaping the system's energy.