The transformation matrix W leading from (A, b) to the
BF form can be computed as W = (Q)(QBF)-l, where Q B F
is the controllability matrix of (AB,, bBF)O. f course, the
controllability matrices Q and QBF can be replaced with
the generalised controllability ones T and TB, , because,
as can be seen from eqns. 4 and 5, the state transformation
W modifies T according to the relation TBF =
W-'T. Therefore we have