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