離散時間系の実現理論

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  • Realization Theory of Discrete Time Systems
  • リサン ジカンケイ ノ ジツゲン リロン

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In this paper, the general realization problem of discrete time systems including nonlinear, time-varying, and infinite dimensional systems is discussed.<br>The general realization problem is stated as follows.“Given a black-box whose input values and output values belong to arbitrary different sets U and Y respectively, find a“Mathematical Model”of the black-box which fits to the input-output data.”<br>Our“Mathematical Model”in this paper is a dynamical system. The definition of the general dynamical system was proposed in 1960, while the dynamical system theory was established in 1969 by R. E. Kalman only for linear, time-invariant, finite dimensional systems.3) Recently, T. Matsuo has established the realization theory of general dynamical syetems for continuous-time case.1)<br>We present the realization theorems of general discrete-time systems by rewriting Matsuo's theory. The special features of discrete time systems are as follows. 1) There exists only one concatenation monoid FS(N+, U), namely, a free monoid U* over the set U of input values. Note that several kinds of concatenation monoids exist in continuous-time case. 2) An Ω-module (X, Ψ) can be described by a U-action (X, F). 3) An. Ω-morphism is equivalent to a U-morphism.<br>By products of this realization theory, we have that: 1) An automaton is a special discrete-time dynamical system. 2) The condition of minimal control system which includes the minimal automaton is equivalent to the condition of multiple experimentablility.

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