Gilbert’s Realization

Let G(s) be a p×m
transfer matrix *G*(s) =*N*(s)/*d*(s)

with *d(s)* a scalar polynomial. For simplicity, we shall
assume that *d(s)* has only real and distinct roots li ¹ lj if *i** *¹ *j* and

Then *G(s)* has the following partial fractional expansion:

Suppose
rank *W**i** = k**i* and let
B*i*Î **R**ki×m and C*i *Î **R**p×ki be two constant matrices such that *W**i** = C**i**B**i*.

Then a realization is given by

This realization is controllable and observable (minimal)
by PBH tests.