State Space Realizations

Given *G(s),* find (*A,B,C,D*) such that

which is a state space realization of *G(s).*

A state space realization (*A,B,C,D*) of *G(s)* is *minimal* if and only if (*A,B*) is controllable and (*C,A*) is observable.

Let (*A**1**,B**1**,C**1**,D*) and (*A**2**,B**2**,C**2**,D*) be two minimal realizations of *G(s).* Then there exists a unique
nonsingular *T *such that

Furthermore, *T* can be specified as

where *C**1**, C**2**, O**1**,* and *O**2* are the
corresponding controllability and observability matrices respectively.