§We now show that the above
realization for F-1(s) indeed has neither uncontrollable modes nor
unobservable modes on the imaginary axis.
eigenvalue of H
but not a pole of F-1(s). Then jw0must be
either an unobservable mode of ([R-1D*CR-1B*], H) or an uncontrollable mode of
(H,). Suppose jw0is an unobservable mode of
•([R-1D*CR-1B*], H). Then there exists
•Hx0 = jw0x0 , [R-1D*CR-1B*]x0= 0.Û
•Since A has no imaginary axis
eigenvalues, we have x1 = 0 and x2 = 0. Contradiction!!!
a contradiction will also be arrived if jw0is
assumed to be an uncontrollable mode of(H,).