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288 CONTROLLER REDUCTION where Ko may be interpreted as a nominal,higher order controller,A is a stable per- turbation,with stable,minimum phase,and invertible weighting functions Wi and W2. Suppose that (G,Ko)l<.A natural question is whether it is possible to obtain a reduced order cont roller K in this class such that the Htoo performance bound remains valid when K is in place of Ko.Note that this is somewhat a special case of the above general problem;the specific form of K restricts that K and Ko must possess the same right half plane poles,thus to a certain degree limiting the set of attainable reduced order controllers. Suppose K is a suboptimal Hoo controller,i.e.,there is a QERHoo with Qlo< such that K=F(Moo,Q).It follows from simple algebra that Q=F(r。',K) where -[[ Furt hermore,it follows from straightforward mani pulations that llQllo <7 → F(,<y → F(K。l,Kg+W2△W)‖o< → F(△<1 where [l][we] W and R is given by the star product ]小 It is easy to see that Ri2 and R2 are both minimum phase and invertible,and hence have full column and full row rank,respectively for all wE RUoo.Consequently, by invoking Lemma 15.1,we conclude that if R is a contraction and lAl<1 then FR)<1.This guarantees the existence of a Q such thato qilntly,theseofasch that.This obervation leads to the following theorem. Theorem 15.2 Suppose Wi and W2 are stable,minimum phase and invertible transfer matrices such that R is a contraction.Let Ko be a stabilizing controller such that CONTROLLER REDUCTION where K may be interpreted as a nominal higher order controller is a stable per turbation with stable minimum phase and invertible weighting functions W and W Suppose that kFG Kk A natural question is whether it is possible to obtain a reduced order controller K in this class such that the H performance bound remains valid when K is in place of K Note that this is somewhat a special case of the above general problem the speci c form of K restricts that K and K must possess the same right half plane poles thus to a certain degree limiting the set of attainable reduced order controllers Suppose K is a suboptimal H controller ie there is a Q RH with kQk such that K FM Q It follows from simple algebra that Q FK a K where K a I I M I I Furthermore it follows from straightforward manipulations that kQk FK a K FK a K WW FR where R I W R R R R I W and R is given by the star product R R R R SK a Ko I I It is easy to see that R and R are both minimum phase and invertible and hence have full column and full row rank respectively for all R Consequently by invoking Lemma we conclude that if R is a contraction and kk then FR This guarantees the existence of a Q such that kQk or equivalently the existence of a K such that FG K This observation leads to the following theorem Theorem Suppose W and W are stable minimum phase and invertible transfer matrices such that R is a contraction Let K be a stabilizing control ler such that