• Hence
•
• has no zeros on the imaginary axis
since
•
•
• has no poles on the imaginary
axis. Next, note that
•-X(jwI-A)-(jwI-A)*X+XBB*X/g2I +C*C=0
• Multiply B*{(jwI-A)*}-1 on the left and (jwI-A)-1B on the right of the above equation to get
• -B*{(jwI-A)*}-1XB-B*X(jwI-A)-1B+B*{(jwI-A)*}-1C*C (jwI-A)-1B
•+ B*{(jwI-A)*}-1XBB*X(jwI-A)-1B/g2 =0
• Completing square, we
have
•G*(jw)G(jw)= g2I-W*(jw)W(jw)