Passivity-based Control of Implicit Port-Hamiltonian systems

The main contribution of this paper is the generalisation of well-known energy-based control techniques (i.e., energy-balancing passivity-based control, passivity-based control with state- modulated source, and interconnection and damping assignment passivity-based control), to the case in which the plant is a port-Hamiltonian system in implicit form. A typical situation is when (part of) the system is obtained from the spatial discretisation of an infinite dimensional port-Hamiltonian system: in this case, the dynamics is not given in standard input-state-output form, but as a set of DAEs. Consequently, the most popular control by energy-shaping techniques have to be extended to deal with dynamical systems with constraints, and geometrical conditions for the existence of the regulator based on the properties of Dirac and resistive structures of the plant are also necessary. The methodological results are discussed with the help of simple but illustrative examples, in which the port-Hamiltonian systems in implicit form are the finite-element models of a lossless transmission line, and of a Timoshenko beam.