CEP course ”New Trends in Nonlinear Control”, IIT Bombay Theory of Port-Hamiltonian systems Chapter 2: Control of Port-Hamiltonian systems

Port-based network modeling of physical systems directly leads to their representation as portHamiltonian systems. Key feature of port-Hamiltonian systems is that the power-conserving interconnection of port-Hamiltonian systems results in another port-Hamiltonian system, with total state space the product of the state spaces of the components, total Hamiltonian being the sum of the Hamiltonian functions, and with Dirac structure defined by the composition of the Dirac structures of the subsystems. In this way port-Hamiltonian systems serve as a unified framework for the modeling and analysis of complex physical systems consisting of many components from different physical domains, finiteor infinite-dimensional. 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 portHamiltonian 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 transferred from one part the system to another. Furthermore, it 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, ∗Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, [email protected]