12502 - Interconnection structures in physical systems a...

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Unformatted text preview: Interconnection structures in physical systems: a mathematical formulation Goran Golo, Orest V. Iftime, Abraham J. van der Schaft Cornelis J. Drebbel Institute for Mechatronics Department of Applied Mathematics, University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands Abstract The power-conserving structure of a physical system is known as interconnection structure. This paper presents a mathematical formulation of the interconnection struc- ture in Hilbert spaces. Some properties of interconnection structures are pointed out and their three natural representations are treated. The developed theory is illustrated on two examples: electrical circuit and one-dimensional transmission line. 1 In troduction Most of the current modelling and simulation approaches to (complex) physical systems are based on some sort of network representation . The physical system under consideration is seen as the interconnection of a number of subsystems possibly from different domains (mechanical, electrical, and so on). This way of modelling has several advantages. One of them is that the knowledge about subsystems can be stored in libraries, and is reusable for later occasions. Also, due to this modularity, the modelling process can be performed in an iterative way, gradually refining the model -if necessary- by adding other subsystems. Further, the approach is suited to general control design where the overall behaviour of the system is sought to be improved by the addition of other subsystems or controlling devices. In this paper we concentrate on the mathematical description of power-conserving part of a network representation of a physical system called interconnection structure . The relevance of interconnection structures in analysis of network models is enormous. It is used for the structural analysis of networks models [1, 2] and for the derivation of simulation model [3]. The proper treatment of interconnection structure is essential for the spatial discretisation of a class of physical systems described by partial differential equations [4]. Our starting assumption is that an interconnection structure is a Dirac structure 1 . This approach was initiated in [9, 10]. In these papers the authors show the relevance of Dirac structures in descriptions of LC-circuits with dependent storage elements [9] and how Dirac structures can be used in the description of kinematic structures of mechanisms [10]. These ideal are further elaborated in [11, 12, 13, 14]. The concept of Dirac structures (slightly 1 The notation of Dirac strictures was introduced by Courant and Weinstein [5] and furthermore investigate by Courant in [6] as a generalisation of Poisson and (pre)-symplectic structures. Dorfman [7, 8] developed an algebraic theory of Dirac structures in the context of the study of completely integrable systems of partial differential equations....
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12502 - Interconnection structures in physical systems a...

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