Abstract
The concept of state is studied in a new set-theoretic formalism for systems theory. Starting with the notion of a time system as a set of ordered pairs of abstract time functions, the concepts of (i) non-anticipation and (ii) causality are introduced. It is proved that the class of causal systems (those possessing a set of states and functional state transitions) is precisely the class of non-anticipatory systems. It is shown that every causal system has a series decomposition consisting of a transition system followed by a static system. It is proved that a state set for a causal system is always constructible using a class of “natural” partitions of the system input set. This latter construction generalizes the result known for certain functional discrete systems to a much more general situation.
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The author gratefully acknowledges the contribution of M. D. Mesarović to this paper. The research has been supported in part by the Office of Naval Research (Contract No. Nonr 1141(12)).
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Windeknecht, T.G. Mathematical systems theory: Causality. Math. Systems Theory 1, 279–288 (1967). https://doi.org/10.1007/BF01695163
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DOI: https://doi.org/10.1007/BF01695163