Chapter 7 Loopshaping This chapter presents a graphical technique for designing a controller to achieve robust performance for a plant that is stable and minimum-phase 7.1 The Basic Technique of Loopshaping Recall from Section 4.3 that the robust performance problem is to design a proper controller C so that the feedback system for the nominal plant is internally stable and the inequality W1S+W2Tl%<1 (7.1) is satisfied.Thus the problem input data are P,Wi,and W2;a solution of the problem is a controller C achieving robust performance. We saw in Chapter 6 that the robust performance problem is not always solvable-the tracking objective may be too stringent for the nominal plant and its associated uncertainty model.Un- fortunately,constructive (necessary and sufficient)conditions on P,Wi,and W2 for the robust performance problem to be solvable are unk nown. In this chapter we look at a graphical method that is likely to prov ide a solution when one exists. The idea is to construct the loop transfer function L to achieve (7.1)approximately,and then to get C via C=L/P.The underlying constraints are internal stability of the nominal feedback system and properness of C,so that L is not freely assignable.When P or P-is not stable,L must contain P's unstable poles and zeros (Theorem 3.2),an awkward constraint.For this reason, we assume in this chapter that P and P-l are both stable. In terms of Wi,W2,and L the robust performance inequality is I(jw):= Wi(jw) W2(jw)L(jw<1. (7.2) 1+L(0w) 1+L(w) This must hold for all w.The idea in loopshaping is to get conditions on L for (7.2)to hold,at least approximately.It is convenient to drop the argument jw. We are interested in alternative conditions under which (7.2)holds.Recall from Section 6.1 that a necessary condition is min{Wi,W2}<1, so we will assume this throughout.Thus at each frequency,either Wi<1 or W2<1.We will consider these two cases separately and derive condit ions comparable to (7.2). 93
Chapter Loopshaping This chapter presents a graphical technique for designing a controller to achieve robust performance for a plant that is stable and minimumphase The Basic Technique of Loopshaping Recall from Section that the robust performance problem is to design a proper controller C so that the feedback system for the nominal plant is internally stable and the inequality kjWSj jWT jk is satis ed Thus the problem input data are P W and W a solution of the problem is a controller C achieving robust performance We saw in Chapter that the robust performance problem is not always solvablethe tracking ob jective may be too stringent for the nominal plant and its associated uncertainty model Un fortunately constructive necessary and sucient conditions on P W and W for the robust performance problem to be solvable are unknown In this chapter we look at a graphical method that is likely to provide a solution when one exists The idea is to construct the loop transfer function L to achieve approximately and then to get C via C LP The underlying constraints are internal stability of the nominal feedback system and properness of C so that L is not freely assignable When P or P is not stable L must contain P s unstable poles and zeros Theorem an awkward constraint For this reason we assume in this chapter that P and P are both stable In terms of W W and L the robust performance inequality is j Wj Lj Wj Lj Lj This must hold for all The idea in loopshaping is to get conditions on L for to hold at least approximately It is convenient to drop the argument j We are interested in alternative conditions under which holds Recall from Section that a necessary condition is minfjWj jW jg so we will assume this throughout Thus at each frequency either jWj or jWj We will consider these two cases separately and derive conditions comparable to
94 CHAPTER 7 LOOPSHAPING Webeginby rctingthe follov irg irequalities whidh follov fom the definit icnofr: (IWl-IW)川S+IWd≤P≤(Wl+WS|+IW寸， (73) (IW寸-IwW)T+w≤T≤(IW+IwDT+Iw, (74) +r w+平业 1+ (75) ·Supposet hat|W寸<11 Thenfom(73) P<1= IWil+lWdjsI<1. (76) 1-W D<1→ IWil-IW4jsl<1. 1-|W寸 (77) Or,intemscffn(75) T<1= L W+1 1-W (78) T<1=→ 山 IW|-1 1-w牙 (7的) whenw1,thecorditiors cntherigh.hand sides of (7)and (apprcadeadhcthe, as dothosein(7)and (7),and wemay approimatetheccnditicnr<1 by W -m51<1 (710) G 工 1-W寸 (711) Nctice that (710)is like the romiral pefomance conditian Wis<1 except that the weigh Wi is ing-eased by div idingit by 1-:Rcbust perfomarceisadieved by nominal pe fo mance wit halage weigh 1 .Nov supposethat Wi<11Wemay proceed similarly tocbtain fom (74) P<1= w4+lWilrl<1 1-W P<1=→ w4-wilm<1 1-W1l a fOn (75) P<1= k 1-W W4+1 T<1=→ 1-Wil W-1
CHAPTER LOOPSHAPING We begin by noting the following inequalities which follow from the de nition of jWjjWj jSj jWj jWj jWj jSj jWj jWjjWj jT j jWj jWj jWj jT j jWj jWj jWLj jLj jWj jWLj j jLjj Suppose that jWj Then from jWj jWj jWj jSj jWjjWj jWj jSj Or in terms of L from jLj jWj jWj jLj jWj jWj When jWj the conditions on the righthand sides of and approach each other as do those in and and we may approximate the condition by jWj jWj jSj or jLj jWj jWj Notice that is like the nominal performance condition jWSj except that the weight W is increased by dividing it by jWj Robust performance is achieved by nominal performance with a larger weight Now suppose that jWj We may proceed similarly to obtain from jWj jWj jWj jT j jWjjWj jWj jT j or from jLj jWj jWj jLj jWj jWj
ZL.THE BASI CTE HNIQUE OF LOOPSHAPING 95 whw1 we ma Qina.thecaij.>by 7m> W (7.>2) 4≥3 (7.>.) ,sla.o.SPOob.d与OSi说ilag W rlaauSaQ iSSmmaLed OS W|1 >1 W4 11 W >.W4 Wl->(W >.Wil W rQgxapetr as la r tequeIS he w1>1 wticskpsChd av I Spastie w saoGeig aw-(isan eeslgu.Typaly,a freque l W 1 >1 W aa lglreque qg电r.0,s W3 PIwCurveS(S(,mTlnde ver SShreque:ir WiL >.w+ rthO-frequeragwe w1>1 w Scathg >JWil W rtbelreque rwhe w1.w. 2.ongTOn:agiGethge et t Jelfir D curve aaCbe 1 cneTrequeloy let 11e belOtaeccacurve ad @1。ddo haCOr,tArequewet ude cqudsrecdSSbed bel). .Ge.3e ml的ro BC miiae下isHcurve ju cructed,Ina necekaL(0)1 0
THE BASIC TECHNIQUE OF LOOPSHAPING When jWj we may approximate the condition by jWj jWj jT j or jLj jWj jWj Inequality says that robust performance is achieved by robust stability with a larger weight The discussion above is summarized as follows jWj jWj jLj jWj jWj jWj jWj jLj jWj jWj For example the rst row says that over frequencies where jWj jWj the loopshape should satisfy jLj jWj jWj Lets take the typical situation where jWj j is a decreasing function of and jWj j is an increasing function of Typically at low frequency jWj jWj and at high frequency jWj jWj A loopshaping design goes very roughly like this Plot two curves on loglog scale magnitude versus frequency rst the graph of jWj jWj over the lowfrequency range where jWj jWj second the graph of jWj jWj over the highfrequency range where jWj jWj On this plot t another curve which is going to be the graph of jLj At low frequency let it lie above the rst curve and also be at high frequency let it lie below the second curve and also be at very high frequency let it roll o at least as fast as does jP j so C is proper do a smooth transition from low to high frequency keeping the slope as gentle as possible near crossover the frequency where the magnitude equals the reason for this is described below Get a stable minimumphase transfer function L whose Bode magnitude plot is the curve just constructed normalizing so that L
96 CHAPTER 7.LOOPSHAPING 103 TtTTtt 102 10 100 10- 10-2 ir 10-3L 10-2 10- 100 101 102 103 104 Figure 7.1:Bode plots of L (solid),W/(1-W2)(dash),and (1-Wi)/W2|(dot) Typical curves are as in Figure 7.1.Such a curve for L will sat isfy (7.11)and (7.13),and hence (7.2)at low and high frequencies.But (7.2)will not necessarily hold at intermediate frequencies. Even worse,L may not result in nominal internal stability.If L(0)>0 and L is as just pictured (ie.,a decreasing function),then the angle of L starts out at zero and decreases (this follows from the phase formula to be derived in the next section).So the Nyquist plot of L starts out on the positive real axis and begins to move clockwise.By the Nyquist criterion,nominal internal stability will hold iff the angle of L at crossover is greater than 180(i.e.,crossover occurs in the third or fourth quadrant).But the greater the slope ofL near crossover,the smaller the angle of L (proved in the next section).So internal instability is unavoidable if L drops off too rapidly through crossover,and hence in our loopshaping we must maintain a gentle slope;a rule of thumb is that the magnit ude of the slope should not be more than 2.After doing the three steps above we must validate the design by checking that internal stability and(7.2)both hold.If not,we must go back and try again.Loopshaping therefore is a craft requiring experience for mastery. 7.2 The Phase Formula (Optional) It is a fundamental fact that if L is stable and minimum-phase and normalized so that L(0)>0, then its magnitude Bode plot uniquely determines its phase plot.The normalization is necessary, for md品 1 are stable,minimum-phase,and have the same magnit ude plot,but they have different phase plots. Our goal in this section is a formula for L in terms of L. Assume that L is proper,L and L are analytic in Res >0,and L(0)>0.Define G:=In L
CHAPTER LOOPSHAPING 10-3 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 104 Figure Bode plots of jLj solid jWj jWj dash and jWj jW j dot Typical curves are as in Figure Such a curve for jLj will satisfy and and hence at low and high frequencies But will not necessarily hold at intermediate frequencies Even worse L may not result in nominal internal stability If L and jLj is as just pictured ie a decreasing function then the angle of L starts out at zero and decreases this follows from the phase formula to be derived in the next section So the Nyquist plot of L starts out on the positive real axis and begins to move clockwise By the Nyquist criterion nominal internal stability will hold i the angle of L at crossover is greater than ie crossover occurs in the third or fourth quadrant But the greater the slope of jLj near crossover the smaller the angle of L proved in the next section So internal instability is unavoidable if jLj drops o too rapidly through crossover and hence in our loopshaping we must maintain a gentle slope a rule of thumb is that the magnitude of the slope should not be more than After doing the three steps above we must validate the design by checking that internal stability and both hold If not we must go back and try again Loopshaping therefore is a craft requiring experience for mastery The Phase Formula Optional It is a fundamental fact that if L is stable and minimumphase and normalized so that L then its magnitude Bode plot uniquely determines its phase plot The normalization is necessary for s and s are stable minimumphase and have the same magnitude plot but they have dierent phase plots Our goal in this section is a formula for L in terms of jLj Assume that L is proper L and L are analytic in Res and L De ne G ln L
-.3.THE PHASE FORMULA OPTIONAL1 97 Then ReG=InL,ImG 7L, and G has the following three properties: >/G is analytic in some right half-plane containing the imaginary axis/Instead of a formal proof,one way to see why this is true is to look at the derivative of G: 0=名 Since L is analytic in the right half-plane,so is L/Then since L has no zeros in the right half-plane,G exists at all points in the right half-plane,and hence at points a bit to the left of the imaginary axis/ 2/ReG(jw)is an even function of w and ImG(jw)is an odd function of w/ 3/sG(s)tends to zero uniformly on semicircles in the right half-plane as the radius tends to infinity,that is, G(Rej lim sup o2<0<-o2 Rejd =0. Proof Since G(Rej)=In L(Rei)+j7L(Rej) and 7L(Re)is bounded as R-oo,we have G(Rej) IIn (Rejl Rejo R Now L is proper,so for some c andk>0, o)≈as→6o. Thus G(Rej) Inlc/R*ll Rejo R In c 7 kInR R n →k R →0.· Next,we obtain an expression for the imaginary part of G in terms of its real part/ Lemma 1 For each frequency wo Im G(jwo)= 2wo ReG(jw)7 ReG(jw). w27w哈
THE PHASE FORMULA OPTIONAL Then ReG ln jLj ImG L and G has the following three properties G is analytic in some right halfplane containing the imaginary axis Instead of a formal proof one way to see why this is true is to look at the derivative of G G L L Since L is analytic in the right halfplane so is L Then since L has no zeros in the right halfplane G exists at all points in the right halfplane and hence at points a bit to the left of the imaginary axis ReGj is an even function of and ImGj is an odd function of sGs tends to zero uniformly on semicircles in the right halfplane as the radius tends to in nity that is lim R sup GRe j Rej Proof Since GRe j ln jLRe j j jLRe j and LRe j is bounded as R we have GRe j Rej j ln jLRe j jj R Now L is proper so for some c and k Ls c sk as jsj Thus GRe j Rej j ln jcRk jj R j ln jcj k ln jRjj R k ln R R Next we obtain an expression for the imaginary part of G in terms of its real part Lemma For each frequency Im Gj Z ReGj ReGj d