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 o20, 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
98 CHAPTER-LOOPSHAPING Proof Deone the function F(s→：= G(s+ReG(Gjwo→G(s+ReG(jwo-→ s-jwo s+jwo =2jw0 G(s-ReG(jwo (214→ s"+w6 Then F is analytic in the right halfplane and on the imaginary axis except for poles at +jwo. Bring in the usual Nyquist contour:Go up the imaginary axis indenting to the right at the points -jwo and jwo along semicircles of radius rathen close the contour by a large semicircle of radius R in the right halfplane.The integral of F around this contour equals zero (Cauchy's theorem- This integral equals the sum of six separate integrals corresponding to the three intervals on the imaginary axisthe two smaller semicircles cand the larger semicircle.Let I7 denote the sum of the three integrals along the intervals on the imaginary axis.the integral around the lower small semicircle 2around the upper small semicirclesand I around the large semicircle.We show that lim I7=2wo ZReG(jw->ReG(jw0-+ (215→ R→∠→0 J2∠ w.-wi limI.=-(Im G(jwo+ (≥16→ T→0 limI2 =-(Im G(jwo (212→ 0 21(=03 (≥18→ The lemma follows immediately from these four equations and the fact that ReG(jw-is even. First≈ I7= jF(jw-+ where the integral is over the set [-R+Wo-r]U[-Wo+rto-r]Uwo +r+R]3 (19→ AsR+o and r→0≈this set becomes the interval(-o+0-Also≈from(≥l4+ jF(jw-=2wo G(jw-ReG(jwo w·-6 Since ImG(Gw→ w·-wi is an odd functionaits integral over set (219-equals zerosand we therefore get (215-> Second≈ I.= G-jwo+re9→ReG(j°an 2-0. -jwo +rej0-jwo -厂e-io+ro+re6iojre"an3 J2-0. -jwo+rej+jwo As r-Oathe orst integral tends to 0 while the second tends to [G(-jwo-ReG(jwo-d7 =(Im G(jwo-3
CHAPTER LOOPSHAPING Proof De ne the function F s Gs ReGj s j Gs ReGj s j j Gs ReGj s Then F is analytic in the right halfplane and on the imaginary axis except for poles at j Bring in the usual Nyquist contour Go up the imaginary axis indenting to the right at the points j and j along semicircles of radius r then close the contour by a large semicircle of radius R in the right halfplane The integral of F around this contour equals zero Cauchys theorem This integral equals the sum of six separate integrals corresponding to the three intervals on the imaginary axis the two smaller semicircles and the larger semicircle Let I denote the sum of the three integrals along the intervals on the imaginary axis I the integral around the lower small semicircle I around the upper small semicircle and I around the large semicircle We show that lim Rr I Z ReGj ReGj d lim r I Im Gj lim r I Im Gj lim R I The lemma follows immediately from these four equations and the fact that ReGj is even First I Z jF j d where the integral is over the set R r r r r R As R and r this set becomes the interval Also from jF j Gj ReGj Since Im Gj is an odd function its integral over set equals zero and we therefore get Second I Z Gj re j ReGj j rej j jre jd Z Gj re j ReGj j rej j jre jd As r the rst integral tends to while the second tends to Gj ReGj j Z d Im Gj
-.3.THE PHASE FORMULA HOPTIONAL1 99 This proves (76)/Verification of (7>7)is similar/ Finally, I4=· F(Rej)jReid0, 2-7 SO I40 sup 色G(ci9.ReG0】 T (Rej0)7+w, 220002 Thus 4→(const)sup G(Rei)I→0. R This proves(78)/■ Rewriting the formula in the lemma in terms of L we get ∠L(0w.)= 2 r21 L(j).In.)d. (720) wr.w This is now manipulated to get the phase formula Theorem 1 For every frequency w. LL(jw.)= In coth 2 where the integration variable v=In(w/w.) Proof Change variables of integration in(7/20)to get LL(jw.)= m m.dv. sinhv Note that in this integral In L is really In L(jw.e")considered as a function of v/Now integrate by parts,from.oo to 0 and from 0 to oo: L60o,）=·l.az6ia.0homh引 di mncothd V +dv 2 +m4.hlio.in岁f X d cothdv. dv 2 The first and third terms sum to zero/ Example Suppose that InL has constant slope, din L 三C dv
THE PHASE FORMULA OPTIONAL This proves Veri cation of is similar Finally I Z F Re j jRe jd so jIj sup GRe j ReGj Rej R Thus jIj const sup jGRe j j R This proves Rewriting the formula in the lemma in terms of L we get Lj Z ln jLj j ln jLj j d This is now manipulated to get the phase formula Theorem For every frequency Lj Z d ln jLj d ln coth j j d where the integration variable ln Proof Change variables of integration in to get Lj Z ln jLj ln jLj j sinh d Note that in this integral ln jLj is really ln jLje j considered as a function of Now integrate by parts from to and from to Lj h ln jLj ln jLj j ln coth i Z d ln jLj d ln coth d ln jLj ln jLj j ln coth Z d ln jLj d ln coth d The rst and third terms sum to zero Example Suppose that ln jLj has constant slope d ln jLj d c
100 CHAPTER-/LOOPSHAPING Then that is,the phase shift is constant at -90c degrees. In the phase formula,the slope function dln L.d-is weighted by the function 4+405 lncoth之=n-。之 2 This function is symmetric about 4 =40 (In scale on the horizontal axis),positive,infinite at 4 =40,increasing from 4 =0 to 4 =40,and decreasing from 4 =40 to 4 =oo.In this way,the values of dInL.d-are more heavily weighted near 4 =40.We conclude,roughly speaking,that the steeper the graph of L near the frequency 40,the smaller the value of 1 L. 7.3 Examples This section presents three simple examples of loopshaping. Example 1 In principle the only informat ion we need to know about P right now is its relative degree,degree of denominator minus degree of numerator.This determines the high-frequency slope on its Bode magnitude plot.We have to let L have at least equal relat ive degree or else C will not be proper.Assume that the relative degree of P equals 1.The actual plant transfer function enters into the picture only at the very end when we get C from L via C=L.P. Take the weight ing function W.to be W.(s)= s+1 200401s+D4 See Figure 7.2 for the Bode magnitude plot.Remember (Section w.2)that W.(4)is an upper bound on the magnitude of the relative plant perturbation at frequency 4.For this example,W. starts at 0405 and increases monotonically up to 5,crossing 1 at 20 rad/s. Let the performance objective be to track sinusoidal reference signals over the frequency range from 0 to 1 rad/s.Let's not say at the start what maximum tracking error we will tolerate;rather, let's see what tracking error is incurred for a couple of loopshapes Ideally,we would take W7 to have constant magnitude over the frequency range 0+and zero magnitude beyond Such a magnitude characteristic cannot come from a rational function.Nevert heless,you can check that Theoremw.2 cont inues to be valid for such W7 that is,if the nominal feedback system is internally stable,then W7S W.Tl 7 1 and W.T 71+△ 证 W+W.TIl 7 14 With this just ification,we can take a+if0≤4≤1 [W714)1=0+else, where a is as yet unspecified
CHAPTER LOOPSHAPING Then Lj c Z ln coth j j d c that is the phase shift is constant at c degrees In the phase formula the slope function d ln jLjd is weighted by the function ln coth j j ln This function is symmetric about ln scale on the horizontal axis positive in nite at increasing from to and decreasing from to In this way the values of d ln jLjd are more heavily weighted near We conclude roughly speaking that the steeper the graph of jLj near the frequency the smaller the value of L Examples This section presents three simple examples of loopshaping Example In principle the only information we need to know about P right now is its relative degree degree of denominator minus degree of numerator This determines the highfrequency slope on its Bode magnitude plot We have to let L have at least equal relative degree or else C will not be proper Assume that the relative degree of P equals The actual plant transfer function enters into the picture only at the very end when we get C from L via C LP Take the weighting function W to be Ws s s See Figure for the Bode magnitude plot Remember Section that jWj j is an upper bound on the magnitude of the relative plant perturbation at frequency For this example jWj starts at and increases monotonically up to crossing at rad s Let the performance ob jective be to track sinusoidal reference signals over the frequency range from to rad s Lets not say at the start what maximum tracking error we will tolerate rather lets see what tracking error is incurred for a couple of loopshapes Ideally we would take W to have constant magnitude over the frequency range and zero magnitude beyond Such a magnitude characteristic cannot come from a rational function Nevertheless you can check that Theorem continues to be valid for such W that is if the nominal feedback system is internally stable then kWT k and WS WT i kjWSj jWT jk With this justi cation we can take jWj j a if else where a is as yet unspeci ed
-.(EXAMPLES 101 Let's orst try a onst order low pass loop transfer functionathat is of the form L(s-b 3 8+1 It is reasonable to take c=1 so that starts rolling off near the upper end of the operating band [0-].We want b as large as possible for good tracking.The largest value of b so that L o 1.Wb1 1W+=w+ω≥20 is 20.So we have s-3 20 See Figure >2.For this L the nominal feedback system is internally stable. It remains to check what robust performance level we have achieved.For this we choose the largest value of a so that a 1≥1.ww018 The function a 1.W(jw is increasing over the range [04]while L(jw-is decreasing.So a can be got by solving a IL(01+=1.w61+3 This gives a 14315. Now to verify robust performanceagraph the function |Wws(jw卡+IW(0w(jw+ (Figure >2 Its maximum value is about 032.Since this is less than 1robust performance is verioed.(We could also have determined as in Section 4.4 the largest a for which the robust performance condit ion holds. Let's recap.For the performance weight 14315+if00w01 Wsjw= 0+else≈ we can take L(s-=20/(s+1-to achieve robust performance.The tracking error is then o 1/14315 3%. Suppose that a tracking error is too large.To reduce the error make larger over the frequency range [04].For exampleawe could try L(s÷8+1020 3 8+18+1 The new factors+10-(s+1-has magnitude nearly 10 over [0 and rolls off to about 1 above 10 rad/s.See Figure 24.Againthe nominal feedback system is internally stable.If we take Woo as before and compute a again we get a =94316.The robust performance inequality is checked graphically (Figure 24->Now the tracking error is o 1/94316 =13%. The problem above is quite easy because W-is small on the operat ing band [041;the require7 ments of performance and robust stability are only weakly compet itive. Example o This example examines the pit ch rate control of an aircraft.The signals are
EXAMPLES Lets rst try a rstorder lowpass loop transfer function that is of the form Ls b cs It is reasonable to take c so that jLj starts rolling o near the upper end of the operating band We want b as large as possible for good tracking The largest value of b so that jLj jWj jWj jWj is So we have Ls s See Figure For this L the nominal feedback system is internally stable It remains to check what robust performance level we have achieved For this we choose the largest value of a so that jLj a jWj The function a jWj j is increasing over the range while jLj j is decreasing So a can be got by solving jLj j a jWj j This gives a Now to verify robust performance graph the function jWj Sj j jWj T j j Figure Its maximum value is about Since this is less than robust performance is veri ed We could also have determined as in Section the largest a for which the robust performance condition holds Lets recap For the performance weight jWj j if else we can take Ls s to achieve robust performance The tracking error is then Suppose that a tracking error is too large To reduce the error make jLj larger over the frequency range For example we could try Ls s s s The new factor s s has magnitude nearly over and rolls o to about above rad s See Figure Again the nominal feedback system is internally stable If we take W as before and compute a again we get a The robust performance inequality is checked graphically Figure Now the tracking error is The problem above is quite easy because jWj is small on the operating band the require ments of performance and robust stability are only weakly competitive Example This example examines the pitch rate control of an aircraft The signals are
102 CHAPTER LOOPSHAPING 102 E 101 100 10 10 02 10 100 10 102 103 Figure≥2：Bode plots of jLj(solid-jW2j(dash-≈and jW Sj+jW2Tj(dot 103 么 101 四 10- 10 100 10 102 Figure≥.4：Bode plots of jLj(solid-jW2j(dash-and jW Sj+jW2Tj(dot→
CHAPTER LOOPSHAPING 10-2 10-1 100 101 102 10-2 10-1 100 101 102 103 Figure Bode plots of jLj solid jWj dash and jWSj jWT j dot 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103 Figure Bode plots of jLj solid jWj dash and jWSj jWT j dot