Some Properties of Past Determined Systems

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In modern systems theory dynamic systems are almost always treated by their state space representations. However, a state space, in general, cannot be uniquely determined by the external behavior (i.e., input and output) of a given system. This ambiguity of the state space may be one of the biggest difficulties of the theoretical aspect of the state space approach. In order to avoid this difficulty Mesarovic introduced the concept of a past determined system where the future output of the system is uniquely determined by the future input and by the past input-output pair. It was shown that a past determined system has a "natural state" which is uniquely determined from its external quantities.In this paper we considered what kind of class of systems is past determined and gave necessary conditions and sufficient conditions for a general linear system to be past determined. In particular, we demonstrated that the classes of linear constant coefficient ordinary differential and difference equation systems are past determined. We also clarified the relation between the class of finite memory machines of finite automata and that of past determined systems, which may open a passage between the control theory and the automata theory.
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