Boundary energy shaping of linear distributed port-Hamiltonian systems

This paper deals with the energy-balancing passivity-based control of linear, lossless, distributed port-Hamiltonian systems. Once inputs and outputs have been chosen to obtain a well-defined boundary control system, the problem is tackled by determining, at first, the class of energy functions that can be employed in the energy-shaping procedure, together with the corresponding boundary state-feedback control actions. To verify the existence of solutions for the closed-loop system, the equivalence between energy-balancing and energy-Casimir methods is shown. For the latter approach, the conditions for having a particular set of Casimir functions in closed-loop are given, and then the existence of the associated semigroup is studied. Since both the methods provide the same control action, the existence result determined for the energy-Casimir method is valid also for the energy-balancing controller. Simple stability is obtained by shaping the open-loop Hamiltonian, while asymptotic stability is ensured if proper pervasive (boundary) damping is present. In this respect, a stability criterion is discussed. The methodology is illustrated with the help of a simple example, i.e. a Timoshenko beam with full-actuation on one side, and an inertia on the other side.