Chapter 7 Design of Robust Compensators based on u Theory There are two main limit ations in the use of Hoo theory for compensator design. First, only full complex perturb ations A(s)E Cnxm can be treated in a non-conservative way in an Hoo robust st ability test. Second, robust performance can only be handled in a conservative way even for full compler perturbations since st ability and performance can not be separated in the Hoo structure. The conservatism depends on the uncert ainty structure and on the condition number k of the sy stem. In this chapter, it will be demonstrated, that these limit ations can be avoided by using the structured singular value u First, the analysis problem will be considered, i.e. how given a compensator K(s) robust stability and robust performance is verified using 4. Then, the synthesis problem will be discussed, i.e. how to find a compensator, which is optimal with respect to u 7.1μ Analysis 7.1.1 Robust Stability In the sequel, control problems that can be represented in the block diagram structure shown in Figure 7.I will be considered. This structure will be refered to as the NAK structure The similarity between the NAK and the 2 x 2 block structure is obvious Here, however, A(s will not be restricted to be a full complex block. Instead, it is assumed that A(s has a cert ain block diagonal structure. Indeed, assume that A(s belongs to the following bounded B△={△(8)∈△|l(△(ju)<1} here△ is defined as: {diag(61L1,…,矶n,n,l,m,+x,…,bm。L, ∈R,∈C,△∈C Thus, both real and complex perturb ations which influen ces the nominal system via the NAk structure are considered. Very general robust st ability problems can be formulated via this structure, e.g. parametric uncert ainty, see Example 7. 1 on the following page. Obviously, the
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78 of △(s) N(8) K(8) Figure 7. 1: NAK formulation of the robust stability problem block diagonal structure of A(s) allow for a much more detailed uncert ainty description, than if A(s) simply consist s of one full complex block. Note, that a single full complex block of course is just a special case of the set A Example 7.1 Diagonal perturbation formulation D) This example is a slightly modified version of an example given in/Hol94 Ass ume that the system G(s) is given b g G(s)= (7.3) 68+1 where the dC gain a and the time constant B only are known with 10 9 uncertainty a=[27.0,33.0 B=[0.9,1.1 Erp ressing a and B by their nominal ual ues along with two perturbations Aa and As for which a, B< 1 can be obtained as (7.5) (7.6) △a∈[-1,+1 (77) Let ba denote the set[-1,+l. Then, the transfer function G(s) can be written as G(s 30(1+0.1△a) (78) (1+0.1△8)8+ with ,△a∈B△ (7.9) In block diagram form, G(s can be rep resented as shown in Figure 7.2 on
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Pher ammo il 0.1 F2gueg 7.2: Example 7.1: Block diagram rep res entation of G(s) o determine the Ns K formulation, thes block in Figure 7.2 is d, and the transfer functions from the three inputs ia(s), iB(s), and u(s) to the three Oa(s),og(s), and y(s) are determined. Standard block diagmm manipulation in matrix form gives 3(s)N (7.10) y(s) (8) The uncertainty blocks are given a (s) sB」Los(s) Nou, let w(s), 8, N(s), and s(s be given b w(s)N (s) (712) N (7.13) N(8)N oer er so er (7.14) The perturbed system can now be described as (8) y(s) u(8) (8)Ns(8)z(s) 717) and can immediately be put into the Ns K structure
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and can immediately be put into the NP K structure. Note, that in this case, the block structure for P asg contains repeated scalar blocks 03 In general, for an arbitrary uncertain system, several equivalent Np k formulations will be possible, which can contain different P asg structures. It might be difficult to determine a r Inlr sI formulation, where the size of p asg is the smallest possible As Illustust)d by to sbn) tCn )xsr pI)s,(Ig(ly stuuctuud unc)uts Inty r nd)ls osn b)up us)nt)d In t( NP K stuuctuu). Unfnutunst )ly, xt us ctIng to unc)utsInty blncks cs n Invnlv) snr)t)dInus slg)bus. In tO sTLs BTE u tnnlb nx t(u))xlsts, (nC))u s v)uy(sndy e functIon unif ore, C(Ic( fs dlltstssn sutnr Izs tInn nf t(ls punc)ss Dy r Ics l unc)uts Inty cs n s lsn b)Includ)d vIs cnr plx bIncks nf sppunpust) dlr nslnns NnC, It Flalasg, Kasgg= Pasg d)nnt)to tus nsfu functIon nbtsIn)d by cInslng t( InC)u Innp In Flguu)7.1 nn ps g)78. Pasg Is to generlized clos ed loop transfer function snd Is Pasg= FiaNasg, Kas (731) 1asg (732) EOn, glv)n s stuuctuu)d unc)utsInty P asg B BA, unbust sts blllty Is d)t)ur In)d t(unug( O fnllnCIng t(nur, C(Ic( Is s g)n)usllzs tInn nf to B oo unbust stsblllty t()r(s)) 2 Theore 7-1 Assume that the system Pasg is stable, and that the perturbation P asg is of such nature, that the closed loop s ystem is stable f and only if the Nyquist curve for det al PasgP asgg does not encircle the origin. Then the closed loop system in Figure 7. 1 on page 78 is stable for all perturbations P asg BBa sf and only if PajwgP ajwgg≠0 BB△ paPajwgP aywgg <o awgBE△ (734) (735) Proof of Theorem 7.1 The proof follows immediately from the proof for the r oo robust stability theorem( Theorem5.2 on page52)with△(s)oB△ Nnt), t(st(7.35)Is nnly s sufficl )nt cnndltInn fnu unbust sts blllty. N)c)sslty nf t( cmu) spnndlng CnndltInn fnu unst)d unc)utsIntI)s fnlInCs fur t( fs ct, st t( unstuuctuu)d s)t cnts Ins all P asg CIt( 8aP ajwgg <o. NnC, (nC)v)u to p)utuubs tInn s)t Is u) stu ct ) d tn P asg B Ba snd, t(us, t( cnndltlnn (7.35)r lg(t In g)n)usl b) subltus uly cnns)uvs tlv RstOut(sn s unbust sts blllty CnndltInn bs s)d nn slngulsu vslu)s, s cnndltlnn Is u))d C(Ic( tsk)st( stuuctuu) nf to p)utuubs tInn Intn Cnnsl d)us tinn. 2(Is ls pu)cls)ly to vlutu) nf tO stuuctuud slngulsu vsu)A Glv)n sny r stulx PbCnx to pnsltlv)u)sl functIon u Is d)find by △a △ min 8aP g: P BA, detaI - PPg=0f
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O)K812 xcept if no 4, g makes I s PA singular(det(Is PA)=O; in this case, by definition Ae(P)=0. Hence, 1/A(P)is the 'magnitude'of the smallest perturb ation A measured by its singular Hlue 8(4) making Is P singular. If P(s)is a transfer matrix, 1/4(P(24 ))can be interpreted as the magnitude of the smallest perturb ation which mofes the characteristic loci of P(s) into the Ny Aist point(s 1, 0)at the angular fre Auency 4 From the definition of A and Pheorem 7 1 on the page before the following theorem for determining robust stability can be formulated(see also [DP87, PD931 Theorem 7. 2(Robust stability with 4 Assume that the system P(s) is stable, and that the perturbation A(s) is such that the closed loop system is stable if and only if the Nyquist curve for det(i s P(sA(s) does not encircle the origin. Then, the closed loop system in Figure 7. 1 on page 78 is stable for all perturbations A(s),ag f and only if gAe(P(s))g 7 1 (737) gA(P(s)gs=s△(P(24) (738) 7.1.2 Robust Performance In order to analyze a sy stem with respect to robust performance, the normalized exogenous disturbances d'(s) and the normalized error signals() are included in the NAK formulation Now, a general framework for the analy sis and synthesis of linear systems can be formulat ee see Figure 7. 4. Any linear combination of control inputs u, measured outputs y, dist urbances d', error signals e, perturbations w and compensator K can be described Fia this ' generic' Generel formulering Figure 7. 4: A general framework for the analysis and for compensator design for linear sys
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Pher tu wo ni W28(2 82 aiN 1+Ik, NMG2 N..34, an b)m), Ny 81+3BocNiaN3 D)) F2gurr7 4 BFa.2 g B/8). L4 8e)e back BIbi[, goMe gy N. 1)2(82 gy(nb) NbgH+1b ).2 8+8 aIN 3)KI NaIX o aeg gud 8(N8 Beg aBeg n. P aeg M, +F N2). 8++) 2,+.N8),8(1824 aeg 2y B/18).28+x1 batckg aHt, gi28),81a(8().2),yh+z 8081 +2 Buay b N.: PN. +:80)81++u8Bu8y a N. uPi ss aN b)2),8f). N 8(anYe)K 78Ix4aeg28()+[7.2+8()2b)aⅠ L,8( NNGa+2 Hbuy8 B)I4+F. No), 8( aIN y2)K1 N81% 2F-: Beg 8+aBeg 2y yBu 2. s(2y 8IN y2)K N8l% 2y g2),bC dBeg79o34 aegpP egg马g (739) 7 422aegf 42saegP agam[ 4ssaegP aeggas4 s2aeg: Beg (7.40) I,(7.40),422aeg28(1)2(8).,+2M6)+ENo)8Ny)28(2+1)xN)8()+ 9), 484 28GN.,(,a), 9oa aegpP aegg 2 80 1)2(8. B)18ulb). B)E+F No 8IN 3)Kl sO Hbu8 B)I+I NolNulraN, H b)2+F u 8. u 2 g(7.40)Np 904 aegpP egg!.7904a即a8c0paug(B△(.41) N+), 8(188 aHt; 28 +, a+KHbu8 B)I+F No)24+A ue. N Ng2 gu kv Ma b hu., jug N NFbug8 38/ O+. 284+ 2+Ku, 38FioBur. u o)18N2 8G s(ugh 28 al b)at, ou). 8(18 the robust penformance condition(7. 41)is satisfied f and only if the sys tem 9oay aegpP egg is robustly stable in the face of a norm bounded perturbation P peg with 8aP naud gg +o, OL. H),o), bG Mg[), 82 g 8)B)bUlb Nect y8FioBu l bG NauzuoH Bix'B)E+F N o batck' reeg, Hbu 8 B)E+F No al b)v)]fi). v2NNFbug8 g&; 284+. Fu180F. +r, 8(2 Ng[ ) 8N8C+ +z 80, o)18N2 8G g 8FioBuF al b)aNl2. +u8 2 Nqu28), NuFM1 NG ug2 g n N80 N[ gab 2) 38FioBu F) &+Kn 2y Bro2 )G Nbatck. ig+ M+) 8(2y 2N228M8)y 8)a++1 2g 80+ a+KNg)yg2 g Hbuy8B)I4+H. No, y)N+DP87, PD93 Theorem 7.3(Robust performance with A) Assume that penformance specfications have been given as an A. specification f the transfer matric from: Beg to bEg(typically a weighted sens itivity specification )of the form 99oa4 aegpP aega a 7 agony aul gpP aul gBgc o (742) Then the perturbed closed loop system goad aegpP aegg and the per formance specification 9 90a4 aegpP aeggg c o, OP aeg( Ba if and only if chere the perturbation structure has been augmented by a full compler per formance block △71.2 aP P(△pPn( r r 8()+732y+)+8()IN+y1(G8()△,,+26+5KNM)~+2)EFNo) d)8N282yb+.).bG8()23892guNN),a82y/+kb) 2+KFbug 8 y8Nb228G and robust penformance 2 N, ++, y) INn)1 N Iz 80 u a 182 8G 2y +:2. 2 NBFo2 )1 NG 2). 2 NugG8[ y i Iaeg( g aHfuu 2BB)NK2 BIN8lo), 80, 80 7 C, 28C+, 4+KHbug8 B)E+" No 2y b+8(, oyyNIGN. yuffio2,& s(ugh 8()+fy
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provide less conservative conditions for robust performance comp ared to the corresponding ility and performance can be separated, far more det ailed uncert ainty descriptions can be formulated due to the block diagonal structure of A, and non-conservative conditions for robust performance are obtained, even for ill-conditioned systems(k( G(w))> (7. 43)on the page before provides a simple test for robust performance. If pA(P(w) plotted against frequency, it is easy to check whether the condition(7.43)in Theorem 7. 3 on the preceding p age is satisfied Since△1=diag{△,0}and△2=diag{0,△} are special cases of the general structure△∈△ it is obvious that △(P()≥max{p△(B1(元),△n(P2(ju)=(P2(j) (745) which means that a necessary condition for robust performance is that the closed loop syster must be robustly st able, and that the nominal system satisfies the performance specifications 7.1.3 Computing p As illustrated above, u is a very useful tool for determining robust st ability and robust per formance in the face of structured as well as unstructed uncertainties. Unfortunately, the comput ation of A it self is a complicated problem, which does not allow a general mat hematical solution. The trouble is, that(7. 36)on page 82 can not be used directly for computing u since the optimization problem involved in general will have several local optima DP87, FTD9] Upper and lower bounds for u, however, can be computed both for purely complex pertur bation sets(mr =0 in(7. 2)on page 77) and for mixed real and complex perturb ation sets Algorithms for computing these bounds were the subject of intense research activities in the beginning of the 1990s, see e. g. DP87, YND91. In the sequel some of the bounds will be presented. To avoid making the not ation any more complicated than required, it will be assumed that the generalized system P(s is square, P(s)ECn 7.1.3.1 A for complex perturbations First, the comput ation of u will be considered in the case, where the perturb ation structure consists entirely of complex blocks, i. e. when m =0 in(7. 2)on page 77. It is not difficult to show that ua(P) can be computed by st andard functions, when A is one of the following two sets, see e.g. ZDG96 ·If△={8hn|∈C}(mr=0,me=1,me=0in(7.2), then u△(P)=p(P) the spectral radius of P(the largest absolute value of any eigenvalue of P, P(P ax;3(P)) ·If△={△|△∈Cnxn}(mr=0,me=0,mc=1in(7.2),then△(P)=0(P),the largest singular value of P For a general complex perturb ation A the following holds {6Ln1°∈C}c△c{△△∈cnxn} (746)
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P(P)f ae(P)f 8(P) (747) △(])u-190t],(11)v)2=2)y)808]982,8)tde=du)8l)0p(P)=t8(P d0u)=h2822).△9],8()u90t](747)(x)8+u)2)f0)t.△(7d=0u)t+0) 8(219g(d2=780s2=870]gP1(ddN△Myas5yae(P)980)v)28(3],[+thf p(P)=0t8(P).A+8(=8)0t,t)f0)8()gH7g]9u])8 N, f 8 N1(sisiNI tg(D2(o(D(41+(m Di a e (D*ND1>,(am(l>,}(7.49) I8d00u)](400)g.8)+2g0=3e=e)240auyD+2[D82)8(=82=y8aa (9+2 1(id( mr N,),Q a l =0t d a e 80gtS+1 10g(+it] Q*aI( Q8 aa(sQa a(8(QsN88Q)N8(8( (7.51) F2(7.50)=t(7.51),8)H7g8()-2)[d0u)t)2)t Theorem 7.4 (Upper and lower bounds for A) FN 2AyQ a I aDAe yhs iNe ig h ae(PQ) Nae(QP)Nae(P)Nae(DPD-) (752) u2△s(△(7.47)5△sf△ Asd esd gen p(QP)f ae(P)f DE(DPD) (7.53) A0341)2 u+90t g vxQen p(QP)i 70 98 =0 it )oara(g vxQen p(QP)Nae(P)),u9890g12p 890=8)3,8)90d740p(QP)0-8d0v)s,=t70g)0)237817(x)])v)23:3[=7 H)0d,=09[)2d3])ad2([7048g9=208))t8+f0tau9828(2j9]8=:H1)2 u+90t.O08(+8)2(=0t,8()9ee)2u+90t7=d0v)se2+u[,=0t89,8(g:-=3[mp 49[FDgf8(DPD-)d07e205]u)t)820)t.U028908)3,8(9ee)2 u1907]1)8)]8287).8)g2Hu=37f[9[[(8048u))q9=38+a.18d0u)](40 8(=82)e)d]eae)282=8740]82d892),7).92mrN,0tmn+mcfu8( H41)v)2s2]9k82)]1(mc+me>v=0ts2[+8[=82J]P,a1u)]8zo ]8=0HIDf品(DPD-1).008()+80)2(=0t,09[)2zd13)se)2)0d0td=8)8(=8)v)0s2 预nc+mc>8)9ee)2u-490t9]9=08((3d0])2=87) W78( 80) MATLAB M a AO=]a =ot Sy08(] 4++3u1-ts [BDG+93), dH 04x=2u2)s2de98g8()u+90t]Δ)+2)[74.F42e2=Bd3de)0=842t)吧0 (=83)+892e92)3de3)se)282=870]),8()[=8([=8d3e2+u3[]v+3)tm8(dp e98=8740a])[]8+u)羽)]]r0id0d
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