Algebraically Approximate and Noisy Realization of Linear Systems

Abstract

Let the set Y of output’s values be a linear space over the real number field R. It is well known that Linear System Theory was established in the algebraic sense [Kalman, 1969]. The main theorem says that for any causal linear input/output map, there exist at least two canonical (controllable and observable) Linear Systems which realize (faithfully describe) it and any two canonical Linear Systems with the same behavior are isomorphic.

Details of finite dimensional Linear Systems were investigated. The criterion for the canonical finite dimensional Linear Systems and various standard canonical Linear Systems were given.

Their partial realization was also discussed according to the above results. We will state an algorithm to obtain a canonical partial realization from a given partial input/output map.

Based on fundamentally established results, an approximate realization problem and a noisy realization problem were discussed through solving partial differential equations of rational polynomial in multi-variables [Hasegawa, 2008].

In this chapter, approximate realization problem and noisy realization problems will be discussed by executing only algebraic operations and comparing with the results of the reference [Hasegawa, 2008].