10.3.Structured Robust Stability and Performance 203 since unitary operations do not change the singular values of a matrix.Note that uwt)I uwt)(d) P-diag(M-MEA22/MmPe ws(d> ws(I where P-and Pe are permutation matrices and where M;= uwt) uwt)(dl:) wsdl; ws Hence >u .3 uwt)(dl: ‘a(G1+)=/mac士， ws >u ·h ,3 maxT uwt) 1(dl)1 wsdl; 0 (wsdl∈lwt)9 mxs+td dl; Using Lemma 10.9,it is easy to show that the maximum is achieved at either I or 1 and that optimal dis given by de=tl△ ws(1 so the structured singular value is ‘a(G+》=wse+wt)Ie+Iws(llwt)Il(P)+d2 (10.27) Note that if ws and wt)are not too large,which are guaranteed if the nominal performance and robust stability conditions are satisfied,then the structured singular value is proportional to the square root of the plant condition number: P (G(1+)))ws(lwt)(P)2 (10.28) This confirms our intuition that an ill-conditioned plant with skewed specifications is hard to control
Structured Robust Stability and Performance since unitary operations do not change the singular values of a matrix Note that wt I wt d! wsd! wsI PdiagM MMmP where P and P are permutation matrices and where Mi wt wt di wsdi ws Hence Gj inf dR max i wt wt di wsdi ws inf dR max i wt wsdi h di i inf dR max i p jdi j jwsdi j jwt j inf dR max i s jwsj jwt j jwsdi j wt di Using Lemma it is easy to show that the maximum is achieved at either or and that optimal d is given by d jwt j jwsj so the structured singular value is Gj s jwsj jwt j jwsjjwt jP P Note that if jwsj and jwt j are not too large which are guaranteed if the nominal performance and robust stability conditions are satised then the structured singular value is proportional to the square root of the plant condition number Gj p jwsjjwt jP This conrms our intuition that an illconditioned plant with skewed specications is hard to control
/34 LAND U&SYN TH ESIS uex≤x→Approxim ation o{Multiple Full Blo ck1 The approximations given in thelast subsection can begeneralized to the mltiple block u problemby assuming that M is partitioned consistently with the structure of △，dig(△△△-△F so that M M M M, M M and D,diagd I△-drII) Now DMD-[ △dy:, And hence M=(M)≤(DMD-D, 撼raM r[st ≤ t ∑M,Fd An approximate D can be found by solving the following minimization problem d or,more conveniently,by minimizing ∑∑IM, with dF,u.Theoptimal di ninimizin ebvpobestisy,repectiy P dl, Mi1AI,4△-F-h (a./) <小Mb(中
AND SYNTHESIS Approximation of Multiple Full Block The approximations given in the last subsection can be generalized to the multiple block problem by assuming that M is partitioned consistently with the structure of diag F so that M M M MF M M MF MF MF MF F and D diagdI dF I I Now DMD Mij di dj dF And hence M inf DD DMD inf DD Mij di dj inf DD kMij k di dj inf DD v u u tX F i X F j kMij k d i d j inf DD v u u tX F i X F j kMij k F d i d j An approximate D can be found by solving the following minimization problem inf DDX F i X F j kMij k d i d j or more conveniently by minimizing inf DDX F i X F j kMij k F d i d j with dF The optimal di minimizing of the above two problems satisfy respectively d k P i k kMikk d i P j k kMkj k d j k F
099 Overview on u Synthesis (05 and P dp itk川Mi喔df 1#kM喔/a正 k△4(，，F-4 70,604 Using these relations,dk can be obtained by iterations, Example09 Consider a 6x 6 comp ler matrix [μ3j0-（i -(0j M△ 5j 63j-u36j -( Y-j] with structured=A diag7ou6-4 The largest singular value of M is 7M4A((..0.7 and the structured singular value of M comp uted using the ugAnalysis and Synthesis Toobor is equal to its up per bound; LmM4△7DMD]性△. with the op timal scaling Dept A diag0.6.55,0.>p,14 The optimal D minimizing FF EIa装 1++4 is Dsubt A diag70.6(u(,0.6,4which is solved from equation 7,(4 Using this Dsubt,we obtain anotheruper bound for the structured singular value h=7M4≤a7 DsubqtMD3qt4△μ.(56-. One may also use this Dsubot as an initial guess for the exact optimization, 10.4 Overview on-Synthesis This section briefly outlines various sy nthesis methods,The details are somewhat comg p licated and are treated in the other rarts of the boo,At thisp oint,we simp ly want to p oint out how the analysis theory discussed in the previous sections leads naturally to synthesis questions, from the analy sis results,we see that each case eventually leads to the evaluation of IMl.a△(，o,oru 70,6u4
Overview on Synthesis and d k P i k kMikk F d i P j k kMkj k F d j k F Using these relations dk can be obtained by iterations Example Consider a complex matrix M j j j j j j j j with structured diag The largest singular value of M is M and the structured singular value of M computed using the Analysis and Synthesis Toolbox is equal to its upper bound M inf DD DMD with the optimal scaling Dopt diag The optimal D minimizing inf DDX F i X F j kMij k d i d j is Dsubopt diag which is solved from equation Using this Dsubopt we obtain another upper bound for the structured singular value M DsuboptMD subopt One may also use this Dsubopt as an initial guess for the exact optimization Overview on Synthesis This section brie"y outlines various synthesis methods The details are somewhat com plicated and are treated in the other parts of the book At this point we simply want to point out how the analysis theory discussed in the previous sections leads naturally to synthesis questions From the analysis results we see that each case eventually leads to the evaluation of kMk or
(0) uAND U.SYNTHESIS for some transfer matrix M.Thus when the controller is put back into the problem,it in olves only a simple linear fractional transformation as shown in Figure 10.2 with M△F(GK)△G-+G-E(I u GeEK)]GE 1 0 where GGis choen,respectively,as GE-GEE 1 0 nominal performnce oly(A△0：G△PePe5 P&E P8s 1 0 P-P-6 ≤robust stability only:G△ P5-P6 「P-P-eP-i1 ≤robust performance G△P△ PeF∈Pe5 P6-PSEi P66 Figure 10.2:Synthesis Framework Each case then leads to the sy nthesis problem rkF1(G压kgfa6△(A+Aaru (10.6() which is sub ect to the internal stability of the nominal. The solutions of these problems for 6 A(and+are the focus of the rest of this book.The 6 A(case was already known in the 1.)0's,and the result presented in this book is simply a new interpretation.The two Riccati solutions for the 6 A+case were new products of the late 1.20's. The sy nthesis for the 6 Au case is not yet fully solved.Recall that u may be obtained by scaling and applying kk (for F<6 and S A0),a reasonable approach isto“solv e' DD D(G4K)D (10.66) by iteratively solv ing for K and D.This is the so-called D-K Iteration.The stable and minimum phase scaling matrix D(sl is chosen such that D(s)A(s)AA(s)D(s)(Note
AND SYNTHESIS for some transfer matrix M Thus when the controller is put back into the problem it involves only a simple linear fractional transformation as shown in Figure with M FG K G GKI GKG where G G G G G is chosen respectively as nominal performance only G P P P P robust stability only G P P P P robust performance G P P P P P P P P P P z w G K Figure Synthesis Framework Each case then leads to the synthesis problem min K kFG Kk for or which is sub ject to the internal stability of the nominal The solutions of these problems for and are the focus of the rest of this book The case was already known in the #s and the result presented in this book is simply a new interpretation The two Riccati solutions for the case were new products of the late #s The synthesis for the case is not yet fully solved Recall that may be obtained by scaling and applying kk for F and S a reasonable approach is to solve min K inf DDH DFG KD by iteratively solving for K and D This is the socalled DK Iteration The stable and minimum phase scaling matrix Ds is chosen such that Dss sDs Note
099 Overview on 1 Synthesis 13p that D(s)is not necessarily belong to D since D(s)is not necessarily Hermitian,see Remark ./)For a fixed scaling transfer matrix D,nink DF(GK)D is a standard H<optinization problem which will be solved in the later part of the book. For a given stabilizing controller K,infHDF(GK)D is a standard convex optimization problem and it can be solved pointwise in the frequency domain: SuP infDT [DF(GZK)(1+)D2 Indeed, GK)DD(GKXD This follows intuitively fromthe following arguments:the left hand side is always no smaller than the right hand side and,on the other hand,given the minimizing D from the right hand side across the frequency,thereis always a rational function D(s) uniformly approximating the magnitude frequency response D+. Note that when S,B(no scal ar blocks) D+,diag(djI22)-DA which is a block-diagonal scaling matrix applied pointwise across frequency to the fre quency responseF(GK)(I+). K Figure uB.<1-Sy nthesis via Scaling DK Iterations proceed by performing this two-parameter minimization in sequential fashion:first minimizing over K with D+fixed,then minimizing pointwise over D+ with K fixed,then again over K,and again over D,etc.Details of this process are summarized in the following steps: (i)Fix an initial estimate of the scaling matrix D+-D pointwise across frequency. (ii)Find scalar transfer functions d.(s)d(s)-RH<for i,F such that d(+4 d.This step can be done using the interpolation theory Youla and Saito,up;however,this will usually result in very high order transfer functions,which explains why this process is currently done mostly by graphical matching using lower order transfer functions
Overview on Synthesis that Ds is not necessarily belong to D since Ds is not necessarily Hermitian see Remark For a xed scaling transfer matrix D minK DFG KD is a standard H optimization problem which will be solved in the later part of the book For a given stabilizing controller K infDDH DFG KD is a standard convex optimization problem and it can be solved pointwise in the frequency domain sup inf DD DFG Kj D ! Indeed inf DDH DFG KD sup inf DD DFG Kj D ! This follows intuitively from the following arguments the left hand side is always no smaller than the right hand side and on the other hand given the minimizing D from the right hand side across the frequency there is always a rational function Ds uniformly approximating the magnitude frequency response D Note that when S no scalar blocks D diagd I d F I I D which is a blockdiagonal scaling matrix applied pointwise across frequency to the fre quency response FG Kj D D K G Figure Synthesis via Scaling DK Iterations proceed by performing this twoparameter minimization in sequential fashion rst minimizing over K with D xed then minimizing pointwise over D with K xed then again over K and again over D etc Details of this process are summarized in the following steps i Fix an initial estimate of the scaling matrix D D pointwise across frequency ii Find scalar transfer functions dis d i s RH for i F such that jdij j d i This step can be done using the interpolation theory Youla and Saito however this will usually result in very high order transfer functions which explains why this process is currently done mostly by graphical matching using lower order transfer functions