In reference (*), we have proposed pseudo-linear systems as a special system of affine dynamocal systems which are general non-linear systems. And we have established the realization theory of discrete-time pseudo-linear systems. The main theorem is the following. [For any time-invariant input response map (that is equivalent to any input/output map with causality and time-invariance), there exist at least two canonical (quasi-reachable & observable) pseudo-linear systems which realize it. And let σ
1, σ
2 be canonical pseudo-linear systems with same behavior, then there exists an unique isomorphic system morphism
T: σ
1→σ
2.]
In this paper, we investigate details of finite dimensional pseudo-linear systems. In order to obtain practical results, we assume that the set
U of input values is finite and show that this assumption is not so special. A necessary and sufficient condition for a finite dimensional pseudo-linear system to be canonical is given. In an isomorphic class of canonical pseudo-linear systems, there exist two standard systems. One is the quasi-reachable standard system, the other is the observable standard system.
A condition for a time-invariant input response to be the behavior of a finite dimensional pseudo-linear system is equivalent to a condition that the rank of an infinite Hankel matrix is finite.
A procedure to obtain the quasi-reachable standard system from a time-invariant input response map is given.
These theories signify an extension of systematizationin linear system theory established by R. E. Kalman et al.
*) Realization Theory of Discrete-Time Pseudo-Linear Systems, Trasns, of SICE, 28-2, 199/207 (1992)
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