Interconnection of port-Hamiltonian systems and composition of Dirac structures

Abstract

Citation for published version (APA): Cervera, J., Schaft, A. J. V. D., & Baños, A. (2007). Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica, 43(2), 212-225. DOI: 10.1016/j.automatica.2006 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. AbstractPort-based network modeling of physical systems leads to a model class of nonlinear systems known as port-Hamiltonian systems. PortHamiltonian systems are defined with respect to a geometric structure on the state space, called a Dirac structure. Interconnection of portHamiltonian systems results in another port-Hamiltonian system with Dirac structure defined by the composition of the Dirac structures of the subsystems. In this paper the composition of Dirac structures is being studied, both in power variables and in wave variables (scattering) representation. This latter case is shown to correspond to the Redheffer star product of unitary mappings. An equational representation of the composed Dirac structure is derived. Furthermore, the regularity of the composition is being studied. Necessary and sufficient conditions are given for the achievability of a Dirac structure arising from the standard feedback interconnection of a plant port-Hamiltonian system and a controller port-Hamiltonian system, and an explicit description of the class of achievable Casimir functions is derived. ᭧

Structure-preserving model reduction of complex physical systems

“…Under general conditions [3] we can solve from these equations for e R , thus leaving a set of DAEs in the state variables x involving the external port-variables f P , e P . Furthermore, there always exists [6] a hybrid partitioning of the port-variables f P , e P into input variables u i = f P i , i ∈ K, u i = e P i , i / ∈ K, and complementary output variables…”

“…Indeed, it can be shown [3] that the equations LF x f x + LE x e x + LF R f R + LE R e R = 0 define the reduced Dirac structure…”

“…which results from interconnection of the original Dirac structure D with the Dirac structure on the space of external port variables F P ×E P defined by e P = 0 (see [3] for further information). The choice f P = 0 is similar, the difference being that L should now satisfy LE P = 0.…”

“…Let us now restrict attention to linear port-Hamiltonian systems without algebraic constraints given in input-state- 3 In the case of Dirac structures on manifolds these matrices will actually depend on the state variables x.…”

“…Now the reduced Dirac structure D ec red corresponding to the effort-constraint e 2 x = 0 is given by the explicit equations (see [3])…”