qTheorem 12.4: Suppose H has the form
•
•
• Then H Î dom(Ric) iff (A,B) is stabilizable and (C,A) has no unobservable modes on the
imaginary axis. Furthermore, X=Ric(H)³0. And X > 0 if and only if (C,A) has no stable
unobservable modes.
qProof: Only need to show that, assuming (A,B) is stabilizable, H has no jw eigenvalues iff (C,A) has no unobservable
modes on the imaginary axis.
q Suppose that jw is an eigenvalue and is a corresponding
eigenvector. Then
•Ax-BB*z=jw x, -C*Cx-A*z=jw z
• Re-arrange: (A- jwI)x=BB*z, -(A-jwI)*z=C*Cx