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290 CONTROLLER REDUCTION Lemma 15.4 The family of all admissible controllers such that T<y can also be uritten as K(s)=F(M,Q)=(日11Q+日12)（Θ2Q+日22）1：=UV-1 三 (Q612+622)(Q61+621):=立-10 where QE RH,Ql<7,and UV-1 and -i are respectively right and left coprime factorizations over RHo and A-BDa C2 Ba-B Da D22 BiDat 11 日12 D12-1 02 日22 C1-D11D2C2 Du Dar D22 Du1Da1 -D5C2 -D2D2 D A- 611 012 B2Di2C B1 -B2 Di2 Du1 -B2D2 白21 62 C2-D22D2C1 D21-D22D2D1-D22D D2C1 D2Du D A-B2DiC B2Di2 B1-B2D2D11 0-1= D2C1 D Di2D1 C2-D22D3C1 D22D2 Da-D22Di Di1 A-BI DaC2 -BID5 B2-B1D2D22 6-1= Da C2 Da D5D22 C1-D11D5C2 -Du Dan D12-D11D5D22 Proof. The results follow immediately from Lemma 9.2 Theorem15.5 Let Ko =01202 be the central Hoo controller such that lF(G,Ko)川o<and iet,∥∈RHo with det V(oo)≠0 be such that [o-(-[]L 1/w2. (15.2) Then K=UV-1 is also a stabilizing controller such that F(G,K)<. Proof.Note that by Lemma 15.4,K is an admissible controller such that Tll< if and only if there exists a QE RHo withll<y such that []-[88+8a- (15.3) CONTROLLER REDUCTION Lemma The family of al l admissible control lers such that kTzwk can also be written as Ks FM Q Q Q U V Q Q V U where Q RH kQk and U V and V U are respectively right and left coprime factorizations over RH and A B D C B B D D B D C D D C D D D D D D D C D D D A B D C B B D D B D C D D C D D D D D D D C D D D A B D C B D B B D D D C D D D C D D C D D D D D D A B D C B D B B D D D C D D D C D D C D D D D D D Proof The results follow immediately from Lemma Theorem Let K be the central H control ler such that kFG Kk and let U V RH with det V be such that I I U V p Then K U V is also a stabilizing control ler such that kFG K k Proof Note that by Lemma K is an admissible controller such that kTzwk if and only if there exists a Q RH with kQk such that U V Q Q Q I