Conservation Laws and Lumped System Dynamics

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, 4] 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 distributedparameter systems capturing their basic physical characteristics are needed more than ever. In this paper we are concerned with the development of a systematic framework for modeling multi-physics systems which is directly based on conservation laws. Modeling based on conservation laws is prevalent in a distributed-parameter context in areas such as fluid dynamics and hydraulic systems, chemical and thermodynamical systems [2], as well as electromagnetism, but is also underlying the basic structure of lumped-parameter systems such as electrical circuits. While the natural framework for formulating Kirchhoff’s laws for electrical circuits is the circuit graph we will show in this paper how distributed-parameter conservation laws can be discretized by using the proper generalization of the notion of graph to ’higherdimensional networks’, called k-complexes in algebraic topology. Furthermore, we show how these discretized conservation laws define a power-conserving intercon-