By Tsuyoshi Matsuo

This monograph extends cognizance concept to the discrete-time area. It contains new effects and constructs a brand new and intensely large inclusion relation for numerous non-linear dynamical structures. After developing a few gains of discrete-time dynamical structures it provides effects touching on structures that are proposed by means of the authors for the 1st time. They introduce basic Dynamical platforms, Linear illustration structures, Affine Dynamical structures, Pseudo Linear platforms, virtually Linear structures and So-called Linear structures for discrete-time and display the connection among them and the opposite dynamical structures. This ebook is meant for graduate scholars and researchers who examine keep watch over concept.

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One is the rank condition of infinite Hankel matrix, and the other is the application of Kleene’s Theorem obtained in automata theory. Lastly, we give a procedure to obtain the quasi-reachable standard system which realizes a given input response map. B. 6) Corollary Let T be a Linear Representation System morphism T : σ1 → σ2 , then aσ1 = aσ2 holds. [proof] This is a direct calculation by the definition of the behavior and Linear Representation System morphism. There is a fact about finite dimensional linear spaces that an n-dimensional linear space over the field K is isomorphic to K n and L(K n , K m ) is iso- 42 4 Linear Representation Systems morphic to K m×n (See Halmos [1958]).

A). This means that a state transition equation is described by a difference equation. These are the same as the automata’s theory of Eilenberg. 4), we gave two canonical realizations for an arbitrary input response map. The quotient set of them may be derived from Nerode equivalence [1958], and the set may be closed to right congruence in Eilenberg [1974]. Note that Nerode proposed the so-called nerode equivalence for automata with linear input/output maps. This concept was used in the finite dimensional linear systems by Kalman [1965] and in the infinite dimensional linear systems by Balarkrishnan [1965], [1967] and Matsuo [1968], [1969].

24) Proposition For any given a ∈ F (ΩN , Y ), there always exists a minimal partial realization of it. [proof] For any ω ∈ ΩN , set a(ω) = 0. 18) implies that there exists a finite dimensional partial realization of a. Therefore, there exists a minimal partial realization. Minimal partial realizations are, in general, not unique modulo isomorphism. Therefore, we introduce a natural partial realization, and we show that natural partial realizations exist if and only if they are isomorphic. 25) Definition For a Linear Representation System σ = ((X, F ), x0 , h) and some p ∈ N , if X = {φF (ω)x0 ; ω ∈ Ωp } , then σ is said to be p-quasi-reachable.