Control of finite-dimensional port-Hamiltonian systems

We discuss in this chapter a number of approaches to exploit the modelstructure of port-Hamiltonian systems for control purposes. Actually, the formulationof physical control systems as port-Hamiltonian systems may lead in somecases to a re-thinking of standard control paradigms. Indeed, it opens up the way toformulate control problems in a way that is different and perhaps broader than usual.For example, formulating physical systems as port-Hamiltonian systems naturallyleads to the consideration of impedance control problems, where the behavior ofthe system at the interaction port is sought to be shaped by the addition of a controllersystem, and it suggests energy-transfer strategies, where the energy is soughtto be transferred from one part the system to another. Furthermore, it naturally leadsto the investigation of a particular type of dynamic controllers, namely those thatcan be also represented as port-Hamiltonian systems and that are attached to thegiven plant system in the same way as a physical system is interconnected to anotherphysical system. As an application of this strategy of control by interconnectionwithin the port-Hamiltonian setting we consider the problem of (asymptotic)stabilization of a desired equilibrium by shaping the Hamiltonian into a Lyapunovfunction for this equilibrium. From a mathematical point of view we will show thatthe mathematical formalism of port-Hamiltonian systems provides various usefultechniques, ranging from Casimir functions, Lyapunov function generation, shapingof the Dirac structure by composition, and the possibility to combine finitedimensionaland infinite-dimensional systems