Control of Finite-Dimensional Port-Hamiltonian Systems

Abstract

We discuss in this chapter a number of approaches to exploit the model structure of port-Hamiltonian systems for control purposes. Actually, the formulation of physical control systems as port-Hamiltonian systems may lead in some cases to a re-thinking of standard control paradigms. Indeed, it opens up the way to formulate control problems in a way that is different and perhaps broader than usual. For example, formulating physical systems as port-Hamiltonian systems naturally leads to the consideration of ‘impedance’ control problems, where the behavior of the system at the interaction port is sought to be shaped by the addition of a controller system, and it suggests energy-transfer strategies, where the energy is sought to be transferred from one part the system to another. Furthermore, it naturally leads to the investigation of a particular type of dynamic controllers, namely those that can be also represented as port-Hamiltonian systems and that are attached to the given plant system in the same way as a physical system is interconnected to another physical system. As an application of this strategy of ‘control by interconnection’ within the port-Hamiltonian setting we consider the problem of (asymptotic) stabilization of a desired equilibrium by shaping the Hamiltonian into a Lyapunov function for this equilibrium. From a mathematical point of view we will show that the mathematical formalism of port-Hamiltonian systems provides various useful techniques, ranging from Casimir functions, Lyapunov function generation, shaping of the Dirac structure by composition, and the possibility to combine finitedimensional and infinite-dimensional systems.