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16.1.Robust Stabilization of Coprime Factors 99 where Q is the solution to the following Lyapunov equation Q(A-YCC)+(A-YCC)Q+CC=0. Moreover&if the above conditions hold then for any y>min a controller achieving is given by A-BB.Xo-YC.C-YC. K(s)= -B.Xoo 0 where x=Q(- Proof. f.Note t at te Ham lton an matrx asscaesg g ven by -YC.C -BB+寸 C.CY H= 0 Ha 0 Hg=A-Yb.c -CC -(A-YC.C) poa e x,(0g）=ImQ and tre stable nvar ant subs ace of Hoo s given by -YO X(t)= 0 X (Ha)=Im y2, Hence t ere Q>0 Robust Stabilization of Coprime Factors where Q is the solution to the fol lowing Lyapunov equation QA Y CCA Y CCQ CC Moreover if the above conditions hold then for any min a control ler achieving K I I P KM is given by Ks A BBX Y CC Y C BX where X Q I Y Q Proof Note that the Hamiltonian matrix associated with X is given by H A Y CC BB Y CCY CC A Y CC Straightforward calculation shows that H I Y I Hq I Y I where Hq A Y CC CC A Y CC It is clear that the stable invariant subspace of Hq is given by XHq Im I Q and the stable invariant subspace of H is given by XH I Y I XHq Im I Y Q Q Hence there is a nonnegative de nite stabilizing solution to the algebraic Riccati equa tion of X if and only if I YQ