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292 CONTROLLER REDUCTION Theorem-5.6 Let Ko =62262 be the central H2 controller such that lF-GKo)l2<yand let立，立，Rt2 with det 0 be such that eee 1/2. Then K=is also a stabilizing coutroller such thatK)l2<5 The above two theorems show that the sufficient conditions for H2 controller re- ductionobemto frequencyweighted modeleduction obem H2 Controller Reduction Procedures (i)Let Ko=122)be a subo timal H2 central controller-=0)such that T:wll2 <7. (ii)ind a reduced order controllerK=vor)such that the following frequency weighted H2 error 。。(][]儿 <1/W taa1Ie1a[gl儿 <1/W2 Then the closed-loo system with the reduced order controller K is stable and the rmance is mainained with the reduced order controller I:ll2 =F-G,K)2<7 15.2 Notes and References The main results resented in this cha ter are based on the wor of Goddard and Glover [1993,1994].VOther controller reduction methods include the stability oriented controller reduction criterion ro osed by Enns [1964],the weighted and unweighted co rime factor controller reducio met hods studied by Liu and Anderson [1966,1990], Li,Anderson,and Ly 1990,Anderson and Liu [199,and Anderson [1993,the nor- malized H2 controller reduction by Mustafa and Glover 1991],the normalized co rime factor method by McFarlane and Glover 1990 in the H2 loo sha ing set-u y and the controller reduction in the v-ga metric setu by Vinnicombe [1993.Lenz,Khar- gonear and Doyle 1967 have also Vro osed another H2 controller reduction method with guaranteed erformance for a dass of H2 roblems. CONTROLLER REDUCTION Theorem Let K be the central H control ler such that kFG Kk and let U V RH with det V be such that h i h U V i I I p Then K V U is also a stabilizing control ler such that kFG K k The above two theorems show that the sucient conditions for H controller re duction problem are equivalent to frequency weighted H model reduction problems H Controller Reduction Procedures i Let K be a suboptimal H central controller Q such that kTzwk ii Find a reduced order controller K U V or V U such that the following frequency weighted H error I I U V p or h i h U V i I I p Then the closedloop system with the reduced order controller K is stable and the performance is maintained with the reduced order controller ie kTzwk FG K Notes and References The main results presented in this chapter are based on the work of Goddard and Glover Other controller reduction methods include the stability oriented controller reduction criterion proposed by Enns the weighted and unweighted coprime factor controller reduction methods studied by Liu and Anderson Liu Anderson and Ly Anderson and Liu and Anderson the nor malized H controller reduction by Mustafa and Glover the normalized coprime factor method by McFarlane and Glover in the H loop shaping setup and the controller reduction in the gap metric setup by Vinnicombe Lenz Khar gonekar and Doyle have also proposed another H controller reduction method with guaranteed performance for a class of H problems