qCorollary 12.5: Suppose (A,B) is stabilizable and (C,A) is detectable. Then
•A*X+XA-XBB*X+C*C=0
• has a unique positive semidefinite
solution. Moreover, it is stabilizing.
qCorollary 12.7: Suppose D has full column rank and denote R = D*D>0. Let H have the form
•
•
• Then H Î dom(Ric) iff (A,B) is stabilizable and has full column rank for all w. Furthermore, X=Ric(H)³0 if H Î dom(Ric) and Ker(X) = 0 if and only if (D^*C, A-BR-1D*C) has no stable unobservable modes.
qProof: This is because
has full column rank for all w
• Û ((I-DR-1D*)C, A-BR-1D*C) has no observable modes
on jw-axis.