By Tsuyoshi Matsuo
This monograph extends consciousness conception to the discrete-time area. It contains new effects and constructs a brand new and extremely vast inclusion relation for varied non-linear dynamical structures. After constructing a few gains of discrete-time dynamical platforms it provides effects touching on platforms that are proposed by means of the authors for the 1st time. They introduce normal Dynamical structures, Linear illustration structures, Affine Dynamical platforms, Pseudo Linear structures, nearly Linear structures and So-called Linear structures for discrete-time and display the connection among them and the opposite dynamical platforms. This booklet is meant for graduate scholars and researchers who examine regulate concept.
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Additional info for Realization Theory of Discrete-Time Dynamical Systems
Sample text
Namely, the behavior of A is the same as one of σ. By the above discussion, a K − U -automaton can be considered as a finite dimensional Linear Representation System. 11) Theorem A Linear Representation System σ = ((K n , F ), x0 , h) is canonical if and only if the following conditions 1) and 2) hold. 1) rank [x0 , F (u1 )x0 , · · · , F (um )x0 , · · · , F (u1 )2 x0 , F (u1 )F (u2 )x0 , · · · , F (u1 )F (um )x0 , · · · , F (um )2 x0 , · · · , F (u1 )n−1 x0 , F (u2 )F (u1 )n−2 x0 , · · · , F (um )n−1 x0 ] = n.
1) Linearly independent vectors in a set Sl (Ω)a are obviously a1 := a and a2 := Sl (u1 )a. 2) Let a state set X be R2 and an initial state x0 be e1 . 3) Let an output map hs : R2 → Y be hs = 0 b . 4) Since Sl (u1 )a1 = a2 and Sl (u1 )a2 = a2 , Fs (u1 ) can be given by the following. Since Sl (u2 )a1 = a1 and Sl (u2 )a2 = a2 , Fs (u2 ) can be given by 48 4 Linear Representation Systems the following. And since Sl (u3 )a1 = a1 and Sl (u3 )a2 = a1 , Fs (u3 ) can be given by the following. 0 1 1 0 1 1 Fs (u1 ) = , Fs (u2 ) = .
7) Proposition Let X be a linear space over the field K. For any map F : U → L(X), a map φ : Ω → L(X) obtained by the following formula *) is a monoid morphism. Moreover, this correspondence is bijective. ∗) : φ(ω) = F (ω(|ω|) · F (ω(|ω| − 1) · F (ω(1) φ(1 ) = I for any ω ∈ Ω. 54 4 Linear Representation Systems Therefore, a linear U -action (X, F ) injectively corresponds to a linear Ωmodule (X, φ) by the formula *). Let (X, F ) be a linear U -action corresponding to a linear Ω-module (X, φ), then (X, F ) may be written as (X, φF ).
