§Characterization: Let g > 0 and Then
•
• where
•
• and R =g 2I-D*D.
§Proof: Let F(s) =g 2I-G~(s)G(s).
§ Then
||G||¥ < g Û F(jw)>0, " wÎ R È {¥}Û detF(jw)¹ 0, " wÎ R since F(¥)=R>0 and F(jw) is continuous. Û F(s) has no imaginary axis zero. Û F -1(s) has no imaginary axis pole.
§
§
• Û H has no jw axis eigenvalues if the above realization has neither uncontrollable modes nor
unobservable modes on the imaginary axis.