Realization Theory of Discrete-Time Finite-Dimensional Pseudo-Linear Systems
Yasumichi Hasegawa, Tsuyoshi Matsuo · Transactions of the Society of Instrument and Control Engineers · 1993
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 σ1, σ2 be canonical pseudo-linear systems with same behavior, then there exists an unique isomorphic system morphism T: σ1→σ2.]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)