§**Characterization:** Let *g* > 0 and Then

where

and *R =**g** **2**I-D*D*.

§**Proof:** Let *F**(s) =**g** **2**I-G**~**(s)G(s).*

Then
||*G*||¥ < *g** *Û *F**(j**w**)>0, *" *w*Î **R **È {¥}Û det*F**(j**w**)**¹** 0, *" *w*Î **R **since** ***F**(*¥*)=R>0 *and *F**(j**w**) *is continuous. Û *F**(s) *has no imaginary axis zero.* *Û *F** **-1**(s) *has no imaginary axis pole.* *

Û *H* has no *j**w* axis eigenvalues if the above realization has neither uncontrollable modes nor
unobservable modes on the imaginary axis.