§We now show that the above
realization for F -1(s) indeed has neither uncontrollable modes nor
unobservable modes on the imaginary axis.
Assume
that jw0 is an
eigenvalue of H
but not a pole of F -1(s). Then jw0 must be
either an unobservable mode of ([R-1D*C R-1B*], H) or an uncontrollable mode of
(H, ). Suppose jw0 is an unobservable mode of
([R-1D*C
R-1B*], H). Then there exists
an such that
Hx0 = jw0 x0 , [R-1D*C R-1B*]x0= 0. Û
(jw0 I-A*)x1=0, (jw0 I+A*)x2=-C*Cx1,
D*Cx1+B*x2=0.
Since A has no imaginary axis
eigenvalues, we have x1 = 0 and x2 = 0. Contradiction!!!
Similarly,
a contradiction will also be arrived if jw0 is
assumed to be an uncontrollable mode of
(H, ).
