離散時間線形表現系の部分実現問題
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概要
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A partial realization problem of discrete time linear representation systems (abb. L.R.S.) is discussed. In reference 7), we have already established the realization theory of an arbitrary black-box with causality condition (equivalently, an input response map a∈F(U*, Y)) by L.R.S.. It contains the uniqueness theorem and existence theorem. We have also investigated finite-dimensional L.R.S. in reference 9). One of the main theorems is realization condition of finite-dimensional L.R.S..Based upon the preceding results, we investigate in this paper a partial realization problem for finite input-output data of length N of a non-linear black-box (or equivalenty, a partial input response map a∈F(U*(≤N), Y) of length N). The partial realization problem is stated as follows:"For an arbitrary partial input response map a∈F(U*(≤N), Y), seek a minimum L.R.S. which partially realizes a and want to prove that minimum partial realizations are unique up to isomorphisms." Existence of minimum partial realizations are trivially presented. It is shown that minimum partial realizations are not unique even up to isomorphisms. To solve the uniqueness problem, we introduce the notion of natural partial realizations. The main contents are following:1). A necessary and sufficient condition for the existence of the natural partial realizations of a∈F(U*(≤N), Y) is given by the rank condition of finite Hankel matrix of a.2). The existence condition of natural partial realizations of a∈F(U*(≤N), Y) is equivalent to the uniqueness condition of minimum partial realizations of a modulo isomorphisms.3). An algorithm to obtain a natural partial realization from a partial input response map a∈F(U*(≤N), Y).
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