112 CHAPTER 8.ADVANCED LOOPSHAPING fixed base by two springs,each with spring constant k.We assume that the beam undergoes small vertical deflections x of the center of mass and small rotations 0 about the center of mass.If we apply a vertical force u at position from the center of mass,the linearized equations of motion are M元+2kx=u, I0+2kl20=1uu. Taking M=2,I=0.5,l=2,k =0.25,and u 1,we get that the transfer functions from u to x and 0 are respectively 0.5 2 s2+0.52 s2+22 We will now measure y:=i+0,the vertical velocity of the beam at the position +1 from the center of mass.For +1,the measurement is at the same location as the force input u;this is called collocated In the -1 case the measurement is at the ot her side of the beam,noncollocated.The resulting transfer function from u to y is then 0.5s 2s(2±0.5)s(1±s2) P(8)=32+0.5±g2+2=(62+0.5)0s2+2西° (8.6) A plot of P is shown in Figure 8.6.The collocated sy stem has zeros at +j and the noncollocated 102 105 100 10 101 109 10 Figure 8.6:Bode plots of P for +(solid)and-(dash). system has zeros at +1.As expected,we shall see that the noncollocated sy stem is more difficult to control.For the collocated case it turns out that the optimal controller for the weights (8.3) is exactly C =1,which is again much simpler than the resulting loopshape.It also turns out that this result holds for all positive values of M,I,l,,and k,excluding those for which the
CHAPTER ADVANCED LOOPSHAPING xed base by two springs each with spring constant k We assume that the beam undergoes small vertical deections x of the center of mass and small rotations about the center of mass If we apply a vertical force u at position lu from the center of mass the linearized equations of motion are Mx kx u I kl luu Taking M I l k and lu we get that the transfer functions from u to x and are respectively s s We will now measure y x the vertical velocity of the beam at the position from the center of mass For the measurement is at the same location as the force input u this is called col located In the case the measurement is at the other side of the beam noncol located The resulting transfer function from u to y is then P s s s s s s s s s A plot of jP j is shown in Figure The collocated system has zeros at j and the noncollocated 10-1 100 101 102 10-1 100 101 Figure Bode plots of jP j for solid and dash system has zeros at As expected we shall see that the noncollocated system is more di cult to control For the collocated case it turns out that the optimal controller for the weights is exactly C which is again much simpler than the resulting loopshape It also turns out that this result holds for all positive values of M I l lu and k excluding those for which the
8.3.PLANTS WITH RHP POLES AND ZEROS 113 system is not mechanically possible.In each case,the optimal controller is C=1 and the optimal =v2/20.707.A proof of this,along with a discussion of the noncollocated case,is given in the last section of this chapter. It is interesting to note that both examples above have multiple crossover frequencies,that is,several distinct frequencies at which L=1.By contrast,most previous examples had only one crossover frequency.Indeed,most problems considered in classical control texts have a single crossover,and this might be considered typical.There are,however,certain application domains, such as the control of flexible struct ures,where mult iple crossovers are common.It turns out that such systems have some interesting characteristics which may appear counterintuitive to readers unfamiliar with them.These issues will be explored more fully in the last section of this chapter. It is not unusual for a reasonable controller to be much simpler than the resulting loopshape even when the problem setup is different from the one considered in the examples above.It often makes sense to begin the loopshaping design process with L equal to some constant times P and then add dynamic elements to try to get the right loopshape. Example 3 As a slightly different example,consider the same setup as in Example 1 but with 1 0.1s P(8)=3十+g2+2(18+ where =0.02 and w =0.5.Suppose also that the weight on the error e is We(s)=0.58+0.5 s+0.01 with Wu =0.5 and Wa Wn 1 as before.This gives us the performance objective as in (8.1), (8.2),and (8.4)except that now Wil=Wel(P2+1)1/2, |W2=0.5(0P|-2+1)112. The Bode magnitude plot of P is shown in Figure 8.7 and the resulting weights Wi and W2 on S and T are shown in Figure 8.8. If we compare the loopshaping constraints in Figure 8.9 with the Bode plot of P in Figure 8.7 we see immediately that C=1 will not work because there is not enough gain at low frequency. Adding the simple lag compensator co=年品 gives the loopshape shown in Figure 8.9 and (WiS2+W2T2)1/2 as shown in Figure 8.10.Fig- ure 8.10 also shows the optimal level for this problem.The simple controller found here is very close to optimal. 8.3 Plants with RHP Poles and Zeros Plants with RHP Zeros Suppose that we begin a loopshaping problem with a plant Po,controller Co,and loop transfer function Lo PoCo with neither RHP poles nor zeros and a single crossover.Now consider the effect of adding a single RHP zero at s =z to the plant by multiply ing by the all-pass function P(s)=之-8 2+8
PLANTS WITH RHP POLES AND ZEROS system is not mechanically possible In each case the optimal controller is C and the optimal p A proof of this along with a discussion of the noncollocated case is given in the last section of this chapter It is interesting to note that both examples above have multiple crossover frequencies that is several distinct frequencies at which jLj By contrast most previous examples had only one crossover frequency Indeed most problems considered in classical control texts have a single crossover and this might be considered typical There are however certain application domains such as the control of exible structures where multiple crossovers are common It turns out that such systems have some interesting characteristics which may appear counterintuitive to readers unfamiliar with them These issues will be explored more fully in the last section of this chapter It is not unusual for a reasonable controller to be much simpler than the resulting loopshape even when the problem setup is dierent from the one considered in the examples above It often makes sense to begin the loopshaping design process with L equal to some constant times P and then add dynamic elements to try to get the right loopshape Example As a slightly dierent example consider the same setup as in Example but with P s s s s s where and Suppose also that the weight on the error e is Wes s s with Wu and Wd Wn as before This gives us the performance ob jective as in and except that now jWj jWejjP j jWj jP j The Bode magnitude plot of P is shown in Figure and the resulting weights W and W on S and T are shown in Figure If we compare the loopshaping constraints in Figure with the Bode plot of P in Figure we see immediately that C will not work because there is not enough gain at low frequency Adding the simple lag compensator Cs s s gives the loopshape shown in Figure and jWSj jWT j as shown in Figure Fig ure also shows the optimal level for this problem The simple controller found here is very close to optimal Plants with RHP Poles and Zeros Plants with RHP Zeros Suppose that we begin a loopshaping problem with a plant P controller C and loop transfer function L PC with neither RHP poles nor zeros and a single crossover Now consider the eect of adding a single RHP zero at s z to the plant by multiplying by the allpass function Pz s z s z s
114 CHAPTER ADVANCED LOOPSHAPING 么 100 10- 100 101 102 Figure8.上：Bode plot of P\. 102 TTTTT TTTTT TTTTT 101 100 10 102 101 100 101 102 Figure 8.8:Bode plots of W usolid)and W2 udash)
CHAPTER ADVANCED LOOPSHAPING 10-2 10-1 100 101 10-2 10-1 100 101 102 Figure Bode plot of jP j 10-1 100 101 102 10-2 10-1 100 101 102 Figure Bode plots of jWj solid and jWj dash
8.3.PLANTS WITH RHP POLES AND ZEROS 115 么 10 100 101 三 101 100 101 102 Figure 8.9:Loopshaping constraints and jLj usolid)for C us 1)-s +0.01). 0.85 TTTT TTTTTT TTTTT 0.8 0.75 0.7t 0.65 0.6 0.55 0.5 101 100 101 102 Figure 8.10:uiwi Sj2 jw2 Tj2)112 for C ws +1)-ts +0.01)usolid)and optimal C udash)
PLANTS WITH RHP POLES AND ZEROS 10-2 10-1 100 101 102 10-2 10-1 100 101 102 Figure Loopshaping constraints and jLj solid for C s s 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 10-2 10-1 100 101 102 Figure jWSj jWT j for C s s solid and optimal C dash
116 CHAPTER 2.ADVANCED LOOPSHAPING where the sign of Pz is chosen so that Pa(0)=1.This also adds the same factor to the loop transfer function,which becomes L(s)=Lo(s)2-3 2+8 From the results of Chapter 6 we would expect problems unless z is larger than the crossover frequency:Recall that ‖WSl1≥IW(z)l, so that |Wi must not be large at z. We may also consider the effect of the RHP zero in terms of the gain-phase relations from Section 7.2.The phase at crossover will be the original phase of Lo plus the phase of P: L(w)=Lo(lw)+/P2(w), /P(|w)=21(-lw+z). The phase from Pz is negative and can be substantial as w approaches z,with/P2(z)=-T82. Again,we see that if z approaches the crossover frequency,the additional phase will degrade the closed-loop system performance or even lead to instability.Thus RHP zeros make the problem worse for systems with one crossover.As an example,let the initial loopshape be Lo(s)=18s with L as above.The crossover frequency is w=1 and the closed-loop system is stable if z>1 and unstable if z≤1. If Pz has a complex pair of RHP zeros,then Pa(s)= ωz-2(2w28+8 wz+2(2w28+s and /P(|w）=21（-2ww+wz-w. Observe that for lightly damped zeros where z is small,the phase changes abruptly near w=wz. Otherwise,the same remarks apply here as in the case of one RHP zero.The simplest strategy to adopt in doing loopshaping designs for systems with RHP zeros is to proceed as usual while keeping track of the extra phase. It would seem from this discussion that RHP zeros are always undesirable,that we would always avoid them if possible,and that we would never deliberately introduce them in our controllers.It is clearly true that all other things being equal,there is no advantage in having RHP zeros in the plant,because we could always add them in the controller if they were desirable.The issue of using RHP zeros in the controller is more subtle.Basically,they are clearly undesirable when there is only one crossover,but may be useful when there are multiple crossovers.This issue will be considered more fully in later sections on optimality. Plants with RHP Poles The problems created by RHP poles are similar to those created by RHP zeros,but there are important differences.Suppose that we begin again with a loopshaping problem with a plant Po, controller Co,and loop transfer function Lo PoCo with neither RHP poles nor zeros and a single crossover.Now consider the effect of adding a single RHP pole at s=p to the plant by mult iplying it by the all-pass function B(s)=8+卫 8-D
CHAPTER ADVANCED LOOPSHAPING where the sign of Pz is chosen so that Pz This also adds the same factor to the loop transfer function which becomes Ls Ls z s z s From the results of Chapter we would expect problems unless z is larger than the crossover frequency Recall that kWSk jWz j so that jWj must not be large at z We may also consider the eect of the RHP zero in terms of the gainphase relations from Section The phase at crossover will be the original phase of L plus the phase of Pz Lj Lj Pz j Pz j j z The phase from Pz is negative and can be substantial as approaches z with Pz jz Again we see that if z approaches the crossover frequency the additional phase will degrade the closedloop system performance or even lead to instability Thus RHP zeros make the problem worse for systems with one crossover As an example let the initial loopshape be Ls s with L as above The crossover frequency is and the closedloop system is stable if z and unstable if z If Pz has a complex pair of RHP zeros then Pz s z zzs s z zzs s and Pz j zzj z Observe that for lightly damped zeros where z is small the phase changes abruptly near z Otherwise the same remarks apply here as in the case of one RHP zero The simplest strategy to adopt in doing loopshaping designs for systems with RHP zeros is to proceed as usual while keeping track of the extra phase It would seem from this discussion that RHP zeros are always undesirable that we would always avoid them if possible and that we would never deliberately introduce them in our controllers It is clearly true that all other things being equal there is no advantage in having RHP zeros in the plant because we could always add them in the controller if they were desirable The issue of using RHP zeros in the controller is more subtle Basically they are clearly undesirable when there is only one crossover but may be useful when there are multiple crossovers This issue will be considered more fully in later sections on optimality Plants with RHP Poles The problems created by RHP poles are similar to those created by RHP zeros but there are important dierences Suppose that we begin again with a loopshaping problem with a plant P controller C and loop transfer function L PC with neither RHP poles nor zeros and a single crossover Now consider the eect of adding a single RHP pole at s p to the plant by multiplying it by the allpass function Pps s p s p