Algebraically Approximate and Noisy Realization of Almost Linear Systems

Abstract

Let the set of output’s values Y be a linear space over the real number field R. In the reference [Matsuo and Hasegawa, 2003], we introduced almost linear systems that are in a subclass of pseudo linear systems, which are very close to linear systems.

At first, their realization theory was stated. Namely, it was shown that any almost linear systems can be characterized by time-invariant, affine input response maps and any time-invariant, affine input response maps, that is, any input/output maps with causality, time-invariance and affinity can be completely characterized by two modified impulse responses, where the modified impulse response may be a slightly revised version of an impulse response in linear systems. An existence theorem and a uniqueness theorem were also proved.

Secondly, details of finite dimensional almost linear systems were investigated. A criterion for the canonical finite dimensional almost linear systems and representation theorems of isomorphic classes for canonical almost linear systems were given. Moreover, a criterion for the behavior of finite dimensional almost linear systems and a procedure to obtain the canonical almost linear systems were given. The criterion is the finite rank condition of an Input/output matrix, which is a natural extension of a finite rank of a Hankel matrix in linear systems.

Thirdly, their partial realization was discussed according to the above results. An algorithm to obtain an almost linear system from the given partial input response map was given.