Caspart, Sven

Hohmann, Soeren

Krebs, Stefan

Muller, Charles

Pfeifer, Martin

Pfeiffer, Silja

English

KITopen

Explicit Port-Hamiltonian Formulation of Bond GraphsM. Pfeifer∗, S. Pfeier, S. Caspart, C. Muller, S. Krebs, S. HohmannKarlsruhe Institute of Technology (KIT), Institute of Control Systems (IRS), Kaiserstr. 12,76131 KarlsruheKarlsruhe Institute of Technology (KIT), Institute of Algebra and Geometry (IAG), Kaiserstr.12, 76131 Karlsruhe∗Corresponding author (Tel: +49(0)721/608-3181, E-Mail: martin.pfeifer@kit.edu)Port-Hamiltonian theory is a well-known framework for control design in complex physicalsystems [2]. The majority of port-Hamiltonian controller and observer design methodsbase on an explicit input-state-output port-Hamiltonian model of the system under con-sideration. However, the derivation of a port-Hamiltonian system (PHS) is in general adicult task, especially for large-scale and networked systems. Remedy can be providedby systematic modeling procedures that are based on a graphical description of the sys-tem. The authors of [6] use open directed graphs to formulate port-Hamiltonian modelsfor various physical systems. In [3], schematics of analog circuits build the basis for an au-tomated generation of PHSs in form of dierential-algebraic equations (DAEs). Besidesdirected graphs and schematics, bond graphs are a particular promising starting pointfor the derivation of PHSs [2]. While the implicit port-Hamiltonian formulation of bondgraphs is well investigated in literature [4], approaches for an explicit formulation havebeen restriced to special case of bond graphs [1, 5]. In particular, algebraic conditions forthe existence of an explicit port-Hamiltonian formulation of a bond graph were missing.In this contribution, we present an automatable and mathematically rigorous method forthe derivation of explicit input-state-output PHSs from bond graphs. Fig. 1 illustratesthe approach which comprises the following four steps: (i) description of the set of power-continuous bond graph elements by a corresponding set of Dirac structures in implicitrepresentation; (ii) merging of the set of Dirac structures into one single Dirac structurein implicit form; (iii) conversion of the Dirac structure from an implicit into an explicitrepresentation; (iv) derivation of an explicit PHS by merging the Dirac structure obtainedfrom (iii) with the constitutive relations of storages and resistors from the bond graph.bond graph((i) set of Dirac structures (implicit)((ii) single Dirac structure (implicit)((iii) single Dirac structure (explicit)((iv) port-Hamiltoniansystem (explicit)Figure 1: Flowchart of the modeling approachAlong with the method of Fig. 1 we provide two algebraic conditions, one necessary andone sucient, for the existence of an explicit port-Hamiltonian formulation of a bondgraph. In the last part of this contribution, we interpret these algebraic conditions bymeans of bond graph causality which allows for some interesting results concerning theform of mathematical models (ODE vs. DAE) that can be obtained from bond graphs.The methods from this contribution are collected, integrated, and automated in the soft-ware tool Amoto which will be presented in the GMA special session on software toolsfor modeling, simulation, identication, and control design.References[1] Alejandro Donaire and Sergio Junco. Derivation of input-state-output port-Hamiltonian systems from bond graphs. Simulation Modelling Practice and Theory,17(1):137151, 2009.[2] Vincent Duindam, Alessandro Macchelli, Stefano Stramigioli, and Herman Bruyn-inckx, editors. Modeling and control of complex physical systems: the port-Hamiltonianapproach. Springer, Berlin Heidelberg, 2009.[3] Antoine Falaize and Thomas Hélie. Passive Guaranteed Simulation of Analog AudioCircuits: A Port-Hamiltonian Approach. Applied Sciences, 6(10), 2016.[4] Goran Golo, Arjan van der Schaft, Peter Breedveld, and Bernhard Maschke. Hamil-tonian formulation of bond graphs. In Nonlinear and hybrid systems in automotivecontrol, pages 351372. Springer-Verlag, London, 2003.[5] Nicolas Lopes. Approche passive pour la modélisation, la simulation et l'étude d'unbanc de test robotisé pour les instruments de type cuivre. PhD thesis, Université Pierreet Marie Curie, Paris, 2016.[6] Arjan van der Schaft and Bernhard Maschke. Port-Hamiltonian Systems on Graphs.SIAM Journal on Control and Optimization, 51(2):906937, 2013.

Explicit Port-Hamiltonian Formulation of Bond Graphs

https://publikationen.bibliothek.kit.edu/1000099899/48229676

Explicit Port-Hamiltonian Formulation of Bond Graphs

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