Abstract
In the paper (*), the realization problem of almost-linear systems which are special classes of pseudo-linear systems was presented with the following result.⌈For any invariant and affine input-response map A (equivalently, any input/output map with causality, time-invariance and affinety), there exist at least two canonical (quasi-reachable & observable) almost-linear systems which realize A And any two canonical almost-linear system with the same behavior (equivalently, input/output relation) are isomorphic each other.⌋
Moreover it is shown that the time-invariant and affine input-response map is completely characterized by a modified generalized impulse response map. And it is also shown that there exist“so-called”linear systems as examples of almost-linear system. Where“so-called”linear systems mean linear systems with non-zero initial state. Being based on the ubove results, we will investigate finite-dimensional almost-linear systems. Then we will obtain the following results.
1): a necessary and sufficient condition for finite-dimensional almost linear system to be canonical. 2): There exist uniquely a quasi-reachable standard system and an observable standard system as representatives of finite-dimensional canonical almost-linear systems. 3): a necessary and sufficient condition for the invariant and affine input-response map to be the behavior of finite dimensional almost-linear systems. 4): For a given invariant and affine input response map, there is given a realization procedure to obtain a canonical finite-dimensional almost system with the same behavior as it.
(*): Y. Hasegawa and T. Matsuo; Realization theory of Discrete-time Almost linear systems, Transaction, on SICE, 30-2 (1994)
