# Mathematical structures in the network representation of energy-conserving physical systems

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Mathematical Structures in the Network Representation of Energy-Conserving Physical Systems - Decision and Control, 1996., Proceedings of the 35th IEEE WA09 10:40 Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996 Mathematical Structures in the Network Representation of Energy-Conserving Physical Systems A.J. van der Schaftl, M. Dalsmo2, B.M. Maschke3 Abstract It is shown that network modelling of energy- conserving physical systems naturally leads to the con- sideration of (nonlinear) implicit generalized Hamilto- nian systems. Behavioral systems theory may be in- voked to formulate and analyze the system-theoretic properties of these systems. 1 Generalized Hamiltonian modelling Most of the current modelling approaches of physical systems (e.g. multi-body systems) are based on some sort of network representation, where the physical sys- tem under consideration is seen as the interconnection of a possible large number of simple sub-systems (the elementary building blocks). This way of modelling has several advantages. The knowledge about sub-systems can be stored in libraries, and is re-usable for later occa- sions. Because of the modularity the modelling process can be performed in a “recursive” manner, first neglect- ing certain effects and gradually refining the model by adding other sub-systems. Further, the approach is suited to general control design where the overall be- havior of the system is sought to be improved by the addition of other sub-systems (controllers). In our previous work [2], [3], [4], (51, [6], [7], [8], [9] we have mainly concentrated on energy-conserving physi- cal systems, where we have argued that the basic dy- namic building blocks are of the form Here z = (21,. . . , 2,) denotes the vector of (indepen- dent) energy variables, coordinatizing the state space manifold X , H(z1 , . . . , z,) is the total stored energy of the sub-system, wit

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