8.3.PLANTS WITH RHP POLES AND ZEROS 117 This also adds the same factor to the loop transfer function L.The sign of Pp is chosen so that Pp(0)=-1.For p small,this gives the right number of encirclements for L to give closed-loop stability.Now we expect that we will have problems unlessp is smallerthan the crossover frequency: Recall that IW2Tl✉≥IW2(p)l, so that W2 must not be large at p. We may also consider the effect of the RHP pole in terms of the gain-phase relat ions.The phase at crossover will be the original phase of Lo plus the phase of Po: ∠L(jw)=∠Lo(iw)+∠Pp(jw), ∠Pp(w)=2∠(w+p). We may illustrate this with the same example.Let the initial loopshape be Lo(s)=1/s with L as above.The crossover frequency is w =1 and the closed-loop system is stable if p<1 and unstable if p >1.Similar effects hold when there is more than one RHP pole. All things being equal,we would prefer to avoid RHP poles in our plants.Even though they can be stabilized by feedback,there is a price to pay in terms of closed-loop performance.Recall that there are times when we must add RHP poles in our compensator just to stabilize the system. Are there other times when we would want to add RHP poles?Clearly no if we have only one crossover,but if there are multiple crossovers it might be advantageous.This will be discussed in a later section on optimality. Including R HP Poles and Zeros in Uncertainty Descri tion A somewhat more formal strategy for handling RHP poles and zeros is to include them in the uncertainty description as follows.Suppose that we have a RHP zero at z.Then we can factor P as P)=乃(s)2-s=乃(s1+ -2s 3+8 2+8 and cover this with a multiplicative perturbat ion to get 28 P=P(1+Wz△)， W2(s)=- 2+s A somewhat tighter cover is given by P@=BO)、-& 2+82+8 =年1+wa△， W(s)=2 Robust stability for uncertainty in this form would involve a test on W2To. Why would we want to do this?If we put all the RHP zeros into the plant uncertainty,then we can use any design technique for plants with no RHP zeros,provided that we account for the extra uncertainty.Basically,one makes sure that T is small enough where necessary.The way we have covered the RHP zero above makes the model have little uncertainty at low frequencies and considerable uncertainty at high frequencies.The transition occurs at frequencies near z.It is also
PLANTS WITH RHP POLES AND ZEROS This also adds the same factor to the loop transfer function L The sign of Pp is chosen so that Pp For p small this gives the right number of encirclements for L to give closedloop stability Now we expect that we will have problems unless p is smal ler than the crossover frequency Recall that kWT k jWp j so that jWj must not be large at p We may also consider the eect of the RHP pole in terms of the gainphase relations The phase at crossover will be the original phase of L plus the phase of Pp Lj Lj Ppj Ppj j p We may illustrate this with the same example Let the initial loopshape be Ls s with L as above The crossover frequency is and the closedloop system is stable if p and unstable if p Similar eects hold when there is more than one RHP pole All things being equal we would prefer to avoid RHP poles in our plants Even though they can be stabilized by feedback there is a price to pay in terms of closedloop performance Recall that there are times when we must add RHP poles in our compensator just to stabilize the system Are there other times when we would want to add RHP poles Clearly no if we have only one crossover but if there are multiple crossovers it might be advantageous This will be discussed in a later section on optimality Including RHP Poles and Zeros in Uncertainty Description A somewhat more formal strategy for handling RHP poles and zeros is to include them in the uncertainty description as follows Suppose that we have a RHP zero at z Then we can factor P as P s Ps z s z s Ps s z s and cover this with a multiplicative perturbation to get P P Wz Wz s s z s A somewhat tighter cover is given by P s Ps z z s s z s Ps z z s Wz s s Wz s s z Robust stability for uncertainty in this form would involve a test on kWzT k Why would we want to do this If we put all the RHP zeros into the plant uncertainty then we can use any design technique for plants with no RHP zeros provided that we account for the extra uncertainty Basically one makes sure that T is small enough where necessary The way we have covered the RHP zero above makes the model have little uncertainty at low frequencies and considerable uncertainty at high frequencies The transition occurs at frequencies near z It is also
118 CHAPTER 8.ADVANCED LOOPSHAPING easy to cover the all-pass part with an uncertainty that is large at low frequencies and small at high frequencies. We can model RHP poles similarly.Care must be taken,however,because RHP poles don't naturally go into multiplicative uncertainty.Suppose that we have a RHP pole at p.Then we can factor P as r=A=e(+)厂 and cover this with a perturbation to get 1 P=Po1+WpA W(s)=- 2p +p This introduces an additional weight on S.A somewhat tighter cover is given by -1 P(s)=Po(s) 8 s+p s+p, 8+p 1 P(s) 81+Wp(s)△(s) Wn(s)=2. 8 How conservative is this covering method?For most problems with a single crossover and for which the performance objectives are achievable,this approach will work well.Basically,we need to make sure that S is small enough where there are RHP poles and T is small enough where there are RHP zeros.There are more complicated problems,particularly those involving multiple crossovers where the impact of RHP poles and zeros might be quite different.This issue will be considered again in the final section of this chapter. Examples We will now consider several examples that illustrate loopshaping for systems with RHP zeros and poles. Example 1 Consider P(s)=乃(s)2- A同=片 As in Section 8.2, 1w=0.5P12+1)'/2, 1W2l=0.5(0P-2+1)1/2, (8.7) or W1(s)=0.5 + W2(s)=0.5(s+1) (8.8) The obvious controller for Po is C=1,which is also optimal with WiS=W2T=0.5 and (C)=v2/2.This simple controller will work fine for >1,but deteriorates as z approaches 1 and does not stabilize for z<1 because of the additional phase lag caused by the all-pass factor. Recall that for any controller Iw≥w(e=05(1+)
CHAPTER ADVANCED LOOPSHAPING easy to cover the allpass part with an uncertainty that is large at low frequencies and small at high frequencies We can model RHP poles similarly Care must be taken however because RHP poles don!t naturally go into multiplicative uncertainty Suppose that we have a RHP pole at p Then we can factor P as P s Ps s p s p Ps p s p and cover this with a perturbation to get P P Wp Wps p s p This introduces an additional weight on S A somewhat tighter cover is given by P s Ps s s p p s p Ps s p s Wps s Wps p s How conservative is this covering method For most problems with a single crossover and for which the performance ob jectives are achievable this approach will work well Basically we need to make sure that S is small enough where there are RHP poles and T is small enough where there are RHP zeros There are more complicated problems particularly those involving multiple crossovers where the impact of RHP poles and zeros might be quite dierent This issue will be considered again in the nal section of this chapter Examples We will now consider several examples that illustrate loopshaping for systems with RHP zeros and poles Example Consider P s Ps z s z s Ps s As in Section jWj jP j jWj jP j or Ws s Ws s The obvious controller for P is C which is also optimal with jWSj jWT j and C p This simple controller will work ne for z but deteriorates as z approaches and does not stabilize for z because of the additional phase lag caused by the allpass factor Recall that for any controller kWSk jWz j z