198 AND U-SYNTHESIS Maximum Singular Value and mu maximum singular value 1.5 mu bounds 0.5l 103 10 10~1 10° 10 102 103 frequency(rad/sec) Figure10.7:h×F(Gp(j3)）and(Gp(j3) -<Two Bloc 1<Robust p errormance Revisited Suppose that the uncertainty block is given by △= E RHoo △e with Allo<1 and that the interconnection model G is given by G(s)= Gufs)Gds) ∈RHo Gqls) Geds) Then the closed-loop sy stem is well-posed and internally stable iff sup,1(G(j3))<1. Let -[,小 d+ER< then D4G(3)D-4= G(3) d:Gjj3) 长Gei3) Gedj3) Hence by Theorem 10.4,at each frequency 3 Gulj3) dGij3) Ge(j3) (10.20) Gedj3)
AND SYNTHESIS 10−3 10−2 10−1 100 101 102 103 0.5 1 1.5 2 frequency (rad/sec) Maximum Singular Value and mu maximum singular value mu bounds Figure P Gpj and Gpj Two Block Robust Performance Revisited Suppose that the uncertainty block is given by RH with kk and that the interconnection model G is given by Gs Gs Gs Gs Gs RH Then the closedloop system is wellposed and internally stable i sup Gj Let D dI I d R then DGj D Gj dGj d Gj Gj Hence by Theorem at each frequency Gj inf dR Gj dGj d Gj Gj
083<Structured Robust Stability and Performance 1 Since the minimization is convex inlogd<see,Doyle 1.2(],the optimal d canbe found by a search;howev er,two approximations to d can be obtained easily by approximating the right,hand side of (10.(0): (1)Note that >u kG▣|3)k d<kG(3)k 11(G(3)≤ inf T 年kG2d|3)k kG22(|3)k dinf kG▣|3)k2+ekG(3)k2+五kG2|3)k2+kG2(3)k2 d Vkoc(l3)k+kGz(3)+(kG(13)kkG(3)k with the minimizing d<given by G2GI k 养VT ifGA0&G2口A0△ 0 ifG2D△0△ (10.(1) 比 (+ ifGE△0 (()Alternative approximation can be obtained by using the Frobenius norm 11(G(|3)≤ G▣|3) dG(3) dG2d3) G22(|3) inf kG3)好，+d长kG3)k+衣kG,d3)导+kGa(I3)号 Vk|3)k子+kG2z(I3)k子-+(kG色(I3)krkG2l3)kF with the minimizing d<given by 3 G2正 VTa☒F if GDA0&G2口A0△ 0 ifG20△0△ (10.() (+ ifGe△0 It can be shown that the approximations for the scalar d<obtained above are exact for a ((matrix G.For higher dimensional G,the approximations for d<are still reasonably good.Hence an approximation of u can be obtained as >4 .W G▣|3)G(|3) 11(G(|3)≤T G2dl3)G2(13) (10.(6)
Structured Robust Stability and Performance Since the minimization is convex in log d see Doyle the optimal d can be found by a search however two approximations to d can be obtained easily by approximating the right hand side of Note that Gj inf dR kGj k d kGj k d kGj k kGj k s inf dR kGj k d kGj k d kGj k kGj k q kGj k kGj k kGj k kGj k with the minimizing d given by d q kGjk kGjk if G G if G if G Alternative approximation can be obtained by using the Frobenius norm Gj inf dR Gj dGj d Gj Gj F s inf dR kGj k F d kGj k F d kGj k F kGj k F q kGj k F kGj k F kGj k F kGj k F with the minimizing d given by d q kG jkF kG jkF if G G if G if G It can be shown that the approximations for the scalar d obtained above are exact for a matrix G For higher dimensional G the approximations for d are still reasonably good Hence an approximation of can be obtained as Gj Gj dGj d Gj Gj
68 1 AND 1-SYNTHESIS or,alternatively,as 0. G-(j+) (G(j+)+ iG-dj+) 无Ge(j+) 8 (43./4) Gedj+) We can now see how these approximated 1 tests are compared with the sufficient conditions obt ained in Chapter.. Ex amp le 10.5 Consider again the robust performance problemof a systemwith out- put multiplicative uncertainty in Chapter.(see Figure.): P.,(I3W-△WP=Il△‖98 Then it is easy to show that the problem can be put in the general framework by selecting 1 0 WerW- .WeLWa G(s), W.SoW- WeSoWa and that the robust performance condition is sathsfied if and only if WEToW-oo+ (8./6) and Fu(G-)川o+u (43./>) for all A RHoo with lAll 9 u.But (uB./6)and (u./>are satisfied iff for each frequency 0 WeLW- 4(C+,盟 dWeToWa WeSo W- W.S.Wa +48 Note that,in contrast to the sufficient condition obtained in Chapter.,this condition is an exact test for robust performance.To compare the 1 test with the criteria obtained in Chapter.,some upper bounds for 1 can be derived.Let dw W.s.W-s WeTWal Then,using the first approximation for 1,we get (G(j+))+VlWeT.WIE3 IW.S.WallE3 /IWET.Wall w.S.W4 VIWeT.Wq3 w.s.Walle3 /(w=Wa)IlWer.w-lw.s.wall WeT.W43.(W二Wd)WeS.Wa‖
AND SYNTHESIS or alternatively as Gj Gj dGj d Gj Gj We can now see how these approximated tests are compared with the su cient conditions obtained in Chapter Example Consider again the robust performance problem of a system with out put multiplicative uncertainty in Chapter see Figure P I WWP kk Then it is easy to show that the problem can be put in the general framework by selecting Gs WToW WToWd WeSoW WeSoWd and that the robust performance condition is satised if and only if kWToWk and kFuG k for all RH with kk But and are satised i for each frequency Gj inf dR WToW dWToWd d WeSoW WeSoWd Note that in contrast to the su cient condition obtained in Chapter this condition is an exact test for robust performance To compare the test with the criteria obtained in Chapter some upper bounds for can be derived Let d s kWeSoWk kWToWdk Then using the rst approximation for we get Gj q kWToWk kWeSoWdk kWToWdk kWeSoWk q kWToWk kWeSoWdk W Wd kWToWk kWeSoWdk kWToWk W Wd kWeSoWdk
10.3.Structured Robust Stability and Performance 2 where W_is assumed to be invertible in the last two inequalities.The last term is exactly the sufficient robust perormance criteria obtained in Chapter E It is dlear that any term precedin}the last forms a ti]hter test since.(W.Wd 3 u Yet another alternative sufficient test can be obtained from the above sequence ofinequalities: 11(G(I+))0 v.(W:Wd(kwTow k+kWesods)2 Note that this sufficient condition is not easy to }et from the approach taken in Chapter E and is potentially less conservative than the bounds derived there. Next we consider the skewed specification problem,but first the followin}lemma is needed in the sequel. Lemma10.9 Suppose士×1-317322231m×1）≤them d{+}士+÷： Proof.Consider a function y xx+ux,then y is a convex function and the maxi- mization over a closed interval is achieved at the boundary ofthe interval.Hence for any fixed d (dr+ (dr) ×(1+ (d) which }ives dx=.The result then ollows from substitutin}d Ex amp le 10.6 As another example,consider alain the skewed specification problem from Chapter E Then the correspondin}G matrix is }iven by 1w TW 1 W KSod G× 2 WesoPwL weSowd So the robust performance specification is satisfied iff 1 W TW
Structured Robust Stability and Performance where W is assumed to be invertible in the last two inequalities The last term is exactly the su cient robust performance criteria obtained in Chapter It is clear that any term preceding the last forms a tighter test since W Wd Yet another alternative su cient test can be obtained from the above sequence of inequalities Gj q W WdkWToWk kWeSoWdk Note that this su cient condition is not easy to get from the approach taken in Chapter and is potentially less conservative than the bounds derived there Next we consider the skewed specication problem but rst the following lemma is needed in the sequel Lemma Suppose m then inf dR max i di di Proof Consider a function y x x then y is a convex function and the maxi mization over a closed interval is achieved at the boundary of the interval Hence for any xed d max i di di max d d d d Then the minimization over d is obtained i d d d d which gives d The result then follows from substituting d Example As another example consider again the skewed specication problem from Chapter Then the corresponding G matrix is given by G WTiW WKSoWd WeSoPW WeSoWd So the robust performance specication is satised i Gj inf dR WTiW dWKSoWd d WeSoPW WeSoWd
13 1 AND 12SYNTHESIS for all +6 B.As in the last example,an upper bound can be obt ained by taking d≤， WeSoP W1业8 WOKSoWall Then (G(j+))0 V.(WaPW1(IIWoR Will 3 IWesowall)8 In particular,this suggests that therobust performance margin isinversely proportional to the square root of the plant condition number if Wd,I and W1,I.This can be further illustrated by considering a plant-inverting control system To simplify the exposition,we shall make the following assumptions: We,wsI=Wd,I=W1,I=W0,wtI= and P is stable and has a stableinverse(i.e,minimmphase)(P can be strictly proper). Furt hermore,we shall assume that the controller has the form K(s),P-s)1(s) where l(s)is a scalar loop transfer function which makes K(s)proper and stabilizes the closed-loop.This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements,namely So,S,31( μ一I=T0,,21I8 Denote 1(s) (s),31可))，3i阿 and substitute these expressions into G;we get uwt)I uwt)P-1] ws(P ws(I The structured singular value for G at frequency can be computed by uwt)I △(G(j+), ws(dP Let the singular value decomposition of P(j+)at frequency be P(j+),UV=∑，diag(1=0888=m) with 1,and'm,where m is the dimension of P.Then △(G(i+), uwt)I uwt)(ds)-1 ws(d> ws(I
AND SYNTHESIS for all As in the last example an upper bound can be obtained by taking d s kWeSoPWk kWKSoWdk Then Gj q W d PWkWTiWk kWeSoWdk In particular this suggests that the robust performance margin is inversely proportional to the square root of the plant condition number if Wd I and W I This can be further illustrated by considering a plantinverting control system To simplify the exposition we shall make the following assumptions We wsI Wd I W I W wtI and P is stable and has a stable inverse ie minimum phase P can be strictly proper Furthermore we shall assume that the controller has the form Ks P sls where ls is a scalar loop transfer function which makes Ks proper and stabilizes the closedloop This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements namely So Si ls I To Ti ls ls I Denote s ls s ls ls and substitute these expressions into G we get G wt I wt P wsP wsI The structured singular value for G at frequency can be computed by Gj inf dR wt I wt dP wsdP wsI Let the singular value decomposition of P j at frequency be P j U!V ! diag m with and m where m is the dimension of P Then Gj inf dR wt I wt d! wsd! wsI