q
qA*X+XA+Q = 0
qSuppose X is the solution of the
Lyapunov equation, then
• Reli(A)£0 if X > 0 and Q³0.
• A is stable if X > 0 and Q > 0.
• A is stable if X ³0, Q³0 and (Q, A) is detectable.
qProof: Let l be an eigenvalue and v be a corresponding eigenvector of A, I.e., Av=lv. Then
w 0=v*(A*X+XA)v+v*Qv=(l*+l)v*Xv+v*Qv=2 Re l v*Xv+v*Qv
w ³ 2 Re l v*Xv since Q³0. Thus
Re l £0.
w If Re l =0, then v*Qv=0. Thus Qv=0 and Av=lv. By PBH test, (Q, A) is not detectable.