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
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
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
15.1.Hoo Controller Reductions 291 and K=UV-1 Hence,to show that K=UV-1 with U and V satisfying equation (15.2)is also a stabilizing controller such that F(G,K)<,we need to show that there is another coprime factorization for K=UV-1 and a QE RHoo with lQlle<y such that equation(15.3)is satisfied. Define -[w。8][]) and partition△as a-] Then [-[8]-[a-a] and ]-e[aar"门 Define U:=U(I-△v)-1,V:=V(I-△v)-and Q:=-yAw(I-△v)l.Then UV-1 is another coprime factorization forK.To show that=UV=V-isa stabilizing controller such that (GK)l,we need to show thatu(I-Av)7, or equivalently‖△v(I-△v)厂1‖o<1.Now AI-A)1=「I0A(I-[0I1△) and by Lemma 15.1△u(I-△v)1‖<1 since l is a contraction and2△‖e<l. I Similarly,we have the following theorem
H Controller Reductions and K U V Hence to show that K U V with U and V satisfying equation is also a stabilizing controller such that kFG K k we need to show that there is another coprime factorization for K U V and a Q RH with kQk such that equation is satis ed De ne I I U V and partition as U V Then U V I I U I V and U I V V I V U I V I De ne U U I V V V I V and Q U I V Then U V is another coprime factorization for K To show that K U V U V is a stabilizing controller such that kFG K k we need to show that U I V or equivalently U I V Now U I V h I i I h I i F h I i I p h I p i p A and by Lemma U I V since h I i I p h I p i is a contraction and p Similarly we have the following theorem
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
15.3.Prob lems 293 1,<Problems Prob lem 15.1 Find the lowest order controller for the system in Erample 1.,(when ·=2, Prob lem 15.2 Find the lowest order controller for Prollem 1(,)when.1.1.opt where.opt is the optimal norm, Prob lem 15.3 Find the lowest order controller for the HIMAT control problem in Prob' lem 1(,11 when.=1.1.pt where.opt is the optimal norm,Compare the controller reduction methods presented in this chapter with other available methods, Prob lem 15.4 Let G be a generalized plant and K be a stalilizing controller,Let △=diag-△p-△k）be a suitably dimensioned perturbation and let T be the transfer 2 matrix from to乏= in the following diagram -K)W Let W-W- e Hoo be a given transfer matriz,Show that the following statements are equivalent 51 ,FoG-K)51andW-Fu王-Ap)‖51 for all-Ap)≤1； 51 for all-△k)≤1； Prob lem 15.3 In the part fi of Problem =(if we let Ag=K-K)W-then T the system and satis es
Problems Problems Problem Find the lowest order control ler for the system in Example when Problem Find the lowest order control ler for Problem when opt where opt is the optimal norm Problem Find the lowest order control ler for the HIMAT control problem in Prob lem when opt where opt is the optimal norm Compare the control ler reduction methods presented in this chapter with other available methods Problem Let G be a generalized plant and K be a stabilizing control ler Let diagp k be a suitably dimensioned perturbation and let Tzw be the transfer matrix from w w w to z z z in the fol lowing diagram f G K K KW W zw z w z w Let W W H be a given transfer matrix Show that the fol lowing statements are equivalent I W Tzw kFG Kk and WFuTzw p for al l p WTzw and F I W Tzw k for al l k Problem In the part of Problem if we let k K KW then Tzw GI KG and F I W Tzw k FG K Thus K wil l stabilizes the system and satises FG K if kkk K KW and the
294 CON TROLLER RED UCTION art of Prolilem-=-is satis/ed by a controller K,Hence to reduce the order of the cntroller KEit is sufficient to solve a frequency weighted model reduction roblem if W can be calculated,In the single in ut and single out ut caseEa "smallestoweighting function WS)can be calculated using the art of Prollem-=-as follows IW4w川6spFu4Tm4w),·p儿 Re eated Problem-=-and Problem-)using the above method,(Hint"W can be com uted frequency ly frequency using u software and then Atted by a stable and minimum Vhase transfer function,1 Problem -,2 One way to generalize the method in Prollem-==to MIMO case is to tae a diagonal W W=diag4W,Wr,Wm） and let Wi be com uted from IW4w)川6sup |eFum4w),·p)引 )s where e is the i th unit vector. Nert tet a)be com uted from 1 a4iw)6 sup |W-1FuT雪w),·p)川 71 P)S where W=diag④V,W, ,...Wm),Then a sutable W can be tallen as W=aW. A ly this method to Problem-3, Problem -fi Generalize the rocedures in Problem-==and Problem-=6 to rob" lems with additional structured uhcertainty cases,(A more general case can be fourd in Yang and Pacard 9fff-il
CONTROLLER REDUCTION part of Problem is satised by a control ler K Hence to reduce the order of the control ler K it is su cient to solve a frequency weighted model reduction problem if W can be calculated In the single input and single output case a smal lest weighting function Ws can be calculated using the part of Problem as fol lows jWjj sup P jFuTzw j pj Repeated Problem and Problem using the above method Hint W can be com puted frequency by frequency using software and then tted by a stable and minimum phase transfer function Problem One way to generalize the method in Problem to MIMO case is to take a diagonal W W diagW WWm and let W i be computed from jW ijj sup P je T i FuTzw j pj where ei is the i th unit vector Next let s be computed from j jj sup P jW FuTzw j pj where W diagW W W m Then a suitable W can be taken as W W Apply this method to Problem Problem Generalize the procedures in Problem and Problem to prob lems with additional structured uncertainty cases A more general case can be found in Yang and Packard
C h a p ter 1 5 H。Loop Shap:ng This chapter introduces a design technique which incorporates loop shaping methods to obtain performance/robust stability tradeoffs,and a particular H2 optimization problem to guarantee closed loop stability and a level of robust stability at all frequen1 cies.The proposed technique uses only the basic concept of loop shaping met hods and then a robust stabilization controller for the normalized coprime factor perturbed sys1 tem is used to construct the final controller.This chapter is arranged as follows:The H2 theory is applied to solve the stabilization problem of a normalized coprime factor perturbed system in Section 16.1.The loop shaping design procedure is described in Section 16.2.The theoretical justification for the loop shaping design procedure is given in Section 16.3.Some further loop shaping guidelines are given in Section 16.4. R obust Stab,Lzat on of Come Factors n this section,we use the H2 control theory developed in the previous chapters to solve the robust stabilization of a left coprime factor perturbed plant given by P△=(M+△M)1(N+△w) with M,N,△w,△w7 RH2 and<se gure 16.1.The transfer matrices(M,N)are assumed to be a left coprime factorization of P(i.e.,P=MEIN). and K internally stalilizes the nominal sy stem. It has been shown in Chapter-that the system is robustly stable iff [] I+PK)EIME 1/e. 2 Finding a controller such that the above norm condition holds is an 72 norm minI imization problem which can be solved using H2 theory developed in the previous chapters. 295
Chapter H Loop Shaping This chapter introduces a design technique which incorporates loop shaping methods to obtain performance robust stability tradeo s and a particular H optimization problem to guarantee closedloop stability and a level of robust stability at all frequen cies The proposed technique uses only the basic concept of loop shaping methods and then a robust stabilization controller for the normalized coprime factor perturbed sys tem is used to construct the nal controller This chapter is arranged as follows The H theory is applied to solve the stabilization problem of a normalized coprime factor perturbed system in Section The loop shaping design procedure is described in Section The theoretical justi cation for the loop shaping design procedure is given in Section Some further loop shaping guidelines are given in Section Robust Stabilization of Coprime Factors In this section we use the H control theory developed in the previous chapters to solve the robust stabilization of a left coprime factor perturbed plant given by P M M N N with M N M N RH and h N M i see Figure The transfer matrices M N are assumed to be a left coprime factorization of P ie P M N and K internally stabilizes the nominal system It has been shown in Chapter that the system is robustly stable i K I I P KM Finding a controller such that the above norm condition holds is an H norm min imization problem which can be solved using H theory developed in the previous chapters
296 H LOOP SHAPING 22 Figure 16.1:Left Coprime Factor Perturbed Systems Suppose P has a stabilizable and detectable state space realization given by B and let L be a matrix such that A+LC is stable then a left coprime factorization of P=MI-N is given by [=[ B+LD L ZC ZD Z where Z can be any nonsingular matrix. In particular,we shall choose Z =(I+ DD*)1-2if P=MI-Nis chosen to be a normalized left coprime factorization.Denote K=/K then the system diagram can be put in an LFT form as in Figure 16.2 with the gener- alized plant 以，aT可 ty A D_D-2 C2 D2-D22 To apply the Hoo control formulae in Chapter 14,we need to normalize the "D_2" matrix first.Note that []-[r+ro明o-[4o (I+DD)1÷ EDI+DD)1÷
H LOOP SHAPING f f f y w z z r M M N N K Figure Left Coprime Factor Perturbed Systems Suppose P has a stabilizable and detectable state space realization given by P A B C D and let L be a matrix such that A LC is stable then a left coprime factorization of P M N is given by h N M i A LC B LD L ZC ZD Z where Z can be any nonsingular matrix In particular we shall choose Z I DD if P M N is chosen to be a normalized left coprime factorization Denote K K then the system diagram can be put in an LFT form as in Figure with the gener alized plant Gs M I P M P A LZ B C Z I D C Z D A B B C D D C D D To apply the H control formulae in Chapter we need to normalize the D matrix rst Note that I D U I I DD where U D I DD I DD I DD DI DD
6%y Robust Stabilization of Coprime Factors 297 Figure 16.2:LFT Diagram for Coprime Factor Stabilization and Uis a unitary matrix.Let r=(1+DD)5衣Z Then Tllo =U*Tllo =Twllo and the problem becomes one of finding a controller K so that lo 5.with the following generalized plant 0 (I+DD) A -LZ1- 0 (I+D*D)D*ZI- I ZC I ZD(I+D*D) Now the formulae in Chapter 14 can be applied to G to obtain a controller K and then the K can be obtained from K=-(I+D*D)1KZ.We shall leave the detail to the reader.In the sequel,we shall consider the case D=0 and Z=I.In this case,we have 1 and Xo4-)+(4-9rx-Xan-x。+C-0 Yo(4+LC)*+(4+LC)Yo -Yo C*CYo =0
Robust Stabilization of Coprime Factors M i K N y u z z w Figure LFT Diagram for Coprime Factor Stabilization and U is a unitary matrix Let K I DD KZ z z U z z Then kTzwk kUTzwk kTzw k and the problem becomes one of nding a controller K so that kTzw k with the following generalized plant G U Z G I I DD A LZ B I DD C I DD DC I DD Z I DD DZ I ZC I ZDI DD Now the formulae in Chapter can be applied to G to obtain a controller K and then the K can be obtained from K I DD KZ We shall leave the detail to the reader In the sequel we shall consider the case D and Z I In this case we have and XA LC A LC X XBB LL X CC YA LC A LCY YCCY