Chapter

# Conservation Laws and Lumped System Dynamics

DOI: 10.1007/978-1-4419-0895-7_3 In book: Model-Based Control:, pp.31-48

ABSTRACT

Physical systems modeling, aimed at network modeling of complex multi-physics systems, has especially flourished in the fifties
and sixties of the 20-th century, see e.g. [11, 12] and references provided therein. With the reinforcement of the ’systems’
legacy in Systems & Control, the growing recognition that ’control’ is not confined to developing algorithms for processing
the measurements of the system into control signals (but instead is concerned with the design of the total controlled system),
and facing the complexity of modern technological and natural systems, systematic methods for physical systems modeling of
large-scale lumpedand distributed-parameter systems capturing their basic physical characteristics are needed more than ever.

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• "In this paragraph, we shall present finite-dimensional port Hamiltonian systems which may also be interpreted as systems of conservation laws but defined on some finite-dimensional spatial domain, represented as a graph [3]. The general case, really analogous to the systems of conservation laws presented in the paragraph II-A, defined on k-complexes may be found in [16]. A directed graph G consists of a finite set V of vertices and a finite set E of directed edges, together with a mapping from E to the set of ordered pairs of V, where no self-loops are allowed. "
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##### Article: Port-Hamiltonian Systems on Open Graphs
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ABSTRACT: In this talk we discuss how to define in an intrinsic manner port-Hamiltonian dynamics on open graphs. Open graphs are graphs where some of the vertices are boundary vertices (terminals), which allow interconnection with other systems. We show that a directed graph carries two natural Dirac structures, called the Kirchhoff-Dirac structure and the vertex-edge Dirac structure. The port-Hamiltonian dynamics corresponding to the Kirchhoff-Dirac structure is exemplified by the dynamics of an RLC-circuit. The port-Hamiltonian dynamics corresponding to the vertex-edge Dirac structure is illustrated by coordination control, in which case there is dynamics associated to every vertex and to every edge, and by standard consensus algorithms where there is dynamics associated to every vertex while every edge corresponds to a resistive relation.
Preview · Article · Jan 2010 · SIAM Journal on Control and Optimization
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ABSTRACT: The external behavior of linear resistive circuits with terminals is characterized as a linear input–output map given by a weighted Laplacian matrix. Conditions are derived for shaping the external behavior of the circuit by interconnection with an additional resistive circuit.
Preview · Article · Jul 2010 · Systems & Control Letters