Abstract
We have established in (1) the realization theory of discrete-time linear representation systems (l.r.s.). The main theorems say that for any input response map a∈F(Ω, Y), there exist at least two canonical (quasi-reachable and distinguishable) l.r.s. with the behavior a, and if σ1 and σ2 are two canonical realizations of a, there exists a unique isomorphic linear representation system morphism T: σ1→σ2, where Ω is the concatenation monoid of a set U of input values and Y is a set of output values which is a linear space over a field K.
In this paper, we investigate details of finite-dimensional l.r.s. (f.l.r.s.). A necessary and sufficient condition of the quasi-reachability [distinguishability] of a finite-dimensional linear U-action with an initial state [a linear readout map] is given. There are given representation theorems of isomorphic classes of canonical f.l.r.s.. In an isomorphic classes of canonical f.l.r.s., there exists at least one strongly quasi-reachable [strongly distinguishable] f.l.r.s.. A necessary and sufficient condition for the input response map a∈F(Ω, Y) to be the behavior of a finite-dimensional l.r.s. is given.
When the set U of input values is finite, the preceding results are rewritten in a more practical form as follows.
In an isomorphic class of canonical f.l.r.s., there exists a unique quasi-reachable [distinguishable] standard canonical system.
The following conditions are equivalent: (1) an input response map a∈F(Ω, K) is the behavior of a f.l.r.s., (2) the rank of the infinite Hankel matrix Ha is finite and (3) a is rational. A procedure to obtain the quasi-reachable standard canonical system from an input response map a∈F(Ω, K) is given.
