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Mathematical Structures in Network Modelling of Physical Systems : From Analysis to Control

Key concept in modern modelling approaches is "modularity" : (complex) physical systems are modelled as the interconnection of simpler sub-systems. From the system-theoretic point of view this naturally emphasizes the need for models of systems with external (interconnection) variables, which however do not possess an a priori imposed input-output structure. Modular modelling generally leads to systems described by mixed sets of (nonlinear) differential and algebraic equations, that is, not of the standard type as usually considered in control theory. For electro-mechanical systems there is a lot of additional structure, which should be exploited for simulation and control purposes. Indeed, disregarding dissipative elements, all sub-systems admit a standard Hamiltonian representation, while the interconnections are power-conserving. This suggests that the interconnected system described by differential and algebraic equations is also Hamiltonian in some generalized sense. Crucial concept in this formulation of implicit Hamiltonian systems is a mathematical geometric structure, called Dirac structure, which can be regarded as an abstract formulation of Tellegen's theorem. The Dirac structure of an interconnected system is determined by the Dirac structures of the sub-systems, together with the interconnection constraints. Dissipation is included by terminating some of the system ports by dissipative elements. The proposed Hamiltonian modelling of physical systems emphasizes modularity, yields proper tools for structural and stability analysis, and forms a natural starting point for the design of passive controllers with inherent robustness properties. Some applications towards the control of electro-mechanical systems and power-converters will be indicated

Type of Seminar:
Special Series on Nonlinear Systems
Prof. Arjan van der Schaft
University of Twente, Enschede, The Netherlands
Apr 07, 1999   14:15

ETH Zurich, Building VAW, Room B1, Gloriastrasse 37-39, 8006 Zurich
Contact Person:

Prof. M.Morari
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Biographical Sketch:
Arjan van der Schaft obtained his PhD degree in 1983 with a thesis entitled "System theoretic descriptions of physical systems" (thesis advisor: Jan C. Willems). Since then he has worked in the broad area of (nonlinear) systems and control theory and the mathematical modelling of open physical systems. He is the author of the books "L-2 Gain and Passivity Techniques in Nonlinear Control" (1996, revised edition 1999), "Nonlinear Dynamical Control Systems" (with H. Nijmeijer, 1990), "Variational and Hamiltonian control systems" (with P.E. Crouch, 1987), and "An Introduction to Hybrid Dynamical Systems" (with J.M. Schumacher, to appear 1999).