On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems

Abstract

We consider infinite-dimensional port-Hamiltonian systems with respect to control issues. In contrast to the well-established representation relying on Stokes-Dirac structures that are based on skew-adjoint differential operators and the use of energy variables, we employ a different port-Hamiltonian framework. Based on this system representation conditions for Casimir functionals will be derived where in this context the variational derivative plays an extraordinary role. Furthermore the coupling of finite-and infinite-dimensional systems will be analyzed in the spirit of the control by interconnection problem.

Boundary L<inf>2</inf>-gain stabilisation of a distributed Port-Hamiltonian system with rectangular domain

“…3.1,Ċ has to be 0 for all the possible Hamiltonians. From (14), we obtain that δΨ δp is constant on D, and from the previous relation that it must be 0 on the boundary, which clearly implies (9). Due to the constraint (7), we obtain that…”

“…The generalization to the infinite dimensional scenario leads to the definition of distributed portHamiltonian systems [5]- [7], introduced about one decade ago as a particular case of the more general framework presented e.g. in [8], that deals with closed infinite dimensional Hamiltonian systems, and then extended in [9] to open systems. Distributed port-Hamiltonian systems have proved to represent a powerful framework for modeling, simulation and control of physical systems described by PDEs.…”

“…To properly shape the Hamiltonian function, the control by interconnection and Casimir generation developed for distributed parameter system with one-dimensional domain [9], [13]- [16] is here extended to the 2D case. Such control technique relies on the generation of a set of Casimir functions in closed-loop that independently from the Hamiltonian function relates the state of the plant with the state of the controller, a finite dimensional port-Hamiltonian system interconnected to the boundary of the distributed parameter one.…”

An Energy-Based Control Strategy for Battery Energy Storage Systems: A Case Study on Microgrid Applications

“…Recently the energy-based (EB) method employing the port-controlled Hamiltonian (PCH) principle has attracted more and more attention [17][18][19]. This method is a novel nonlinear robust control strategy based on passivity theory.…”

Asymptotic Stabilisation of Distributed Port-Hamiltonian Systems by Boundary Energy-Shaping Control

“…For example, in Rodriguez et al [2001], Melchiorri [2004, 2005], Pasumarthy and van der Schaft [2007], Siuka et al [2011], Schöberl and Siuka [2013], this task has been accomplished by looking at, or generating, a set of Casimir functions in closed-loop that robustly (i.e., independently from the Hamiltonian functions) relates the state of the infinite dimensional port-Hamiltonian system with the state of the controller, which is a finite dimensional portHamiltonian system interconnected to the boundary of the distributed parameter one. The shape of the closed-loop energy function is changed by acting on the Hamiltonian of the controller e.g.…”