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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