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12.1.Hoo Controller Reductions -9 KF(G5..ThenKsiing ontroller such that F(GIK)5 k△k。=W1(K(K)Ww 512 Since R can always be made contractive for sufficiertly small w and W,there are infinite many w and W.that satisfy the conditions in the theorer.It is obvious that to makeW,(本(K)W, 5 1 for some K,one would like to select the "largest" w and w. such that Ris contraction. Lemma 1 2.3 Assume kR koo 5 and define 0（R 0 1.7I)2 Then R is a contra ction if w and W t机s 小 Proof. See Goddard and Glover 1993 ◇ An algorithm that maximizes det(W<W )det(WW<)has been developed by God- dard and Glover [1993.The procedure below,devised drectly from the above theorem, can be used to generate a required reduced order controller which will preserve the dlosed-loop Hee performance boundF(GK)5.. 1.Let K be a full order controller such that kF(GiK )k 5. Compute W and W..so that Ris a contraction; 3.Using model reduction method to find a K so that 15.1.2 Coprime Factor Reduction The Hoo controller reduction problem can also be considered in the coprime factor framework.For that purpose,we need the following alternative representation of all admissible Hoo controllers. H Controller Reductions kFG Kk Then K is also a stabilizing control ler such that FG K if kk W K KW Since R can always be made contractive for suciently small W and W there are in nite many W and W that satisfy the conditions in the theorem It is obvious that to make W K KW for some K one would like to select the largest W and W such that R is a contraction Lemma Assume kRk and dene L L L L L F R R R R R R R R I Then R is a contraction if W and W satisfy W W WW L L L L Proof See Goddard and Glover An algorithm that maximizes detW W detWW has been developed by God dard and Glover The procedure below devised directly from the above theorem can be used to generate a required reduced order controller which will preserve the closedloop H performance bound FG K Let K be a full order controller such that kFG Kk Compute W and W so that R is a contraction Using model reduction method to nd a K so that W K KW Coprime Factor Reduction The H controller reduction problem can also be considered in the coprime factor framework For that purpose we need the following alternative representation of all admissible H controllers