X Bodefs Sensitivity Integral 93 9 0 Figure 6.12:Water Bed Effect of Sensitivity Function and 8(5 容（告（台 1专n1-). Then dri)e -62+ns( 6 ac+(小hs6p-aa1- +(-学hs6-专1-司 which gives 25a4
Bodes Sensitivity Integral |S| 1 ω + − Figure Water Bed Eect of Sensitivity Function and Z h ln Mh d X i Z h i Mh i d X i i h i Mh h i h X i i Mh h i h ln Mh h h ln Then Xm i Re pi Z ln jS jjd Z l ln jS jjd Z h l ln jS jjd Z h ln jS jjd l ln h l max l h ln jS jj Z h ln Mh d l ln h l max l h ln jS jj h ln which gives max l h jS jj e l hl h hl
94 PERFORMANCE SPECIFICATIONS AND LIMITATIONS where e∑vRe(pi wh·wl The above lower bound shows that the sensitivity can be very significant in the transition bandx Next,we investigate the design constraints imposed by open-loop non-minimum phase zeros upon sensitivity properties using the Poisson integral relationx Suppose L has at least one more poles than zeros and suppose z=zo+yo with zo 0is a right half plane zero of LxThen m To (-17) This result implies that the sensitivity reduction ability of the system may be severely limited by the open-loop unstable poles and non-minimum phase zeros,especially when these poles and zeros are close to each otherx Define (z)车 To Then (a·8(z)ln‖S(|w)川∞+(z)ln(e) which gives ’1’ *-0 IS(s)‖∞≥ Di This lower bound on the maximum sensitivity shows that for a non-minimum phase system,its sensitivity must increase significantly beyond one at certain frequencies if the sensitivity reduction is to be achieved at other frequenciesx 22 Analyticity Constraints Let prapu..spm and..be the open right half plane poles and zeros of L, respectivelyxSuppose that the closed loop system is stablexThen S(pi)=0NT(pi)=1≈i=12≈.m and S()=1≈T(z)=0N|=12≈.k
PERFORMANCE SPECIFICATIONS AND LIMITATIONS where Pm i Re pi h l The above lower bound shows that the sensitivity can be very signicant in the transition band Next we investigate the design constraints imposed by openloop nonminimum phase zeros upon sensitivity properties using the Poisson integral relation Suppose L has at least one more poles than zeros and suppose z x jy with x is a right half plane zero of L Then Z ln jS jj x x y d lnYm i z pi z pi This result implies that the sensitivity reduction ability of the system may be severely limited by the openloop unstable poles and nonminimum phase zeros especially when these poles and zeros are close to each other Dene z Z l l x x y d Then lnYm i z pi z pi Z ln jS jj x x y d z ln kS jk z ln which gives kS sk z z Ym i z pi z pi z This lower bound on the maximum sensitivity shows that for a nonminimum phase system its sensitivity must increase signicantly beyond one at certain frequencies if the sensitivity reduction is to be achieved at other frequencies Analyticity Constraints Let p ppm and z zzk be the open right half plane poles and zeros of L respectively Suppose that the closed loop system is stable Then S pi T pi i m and S zj T zj j k
646+Analyticity Constraints 95 The internal stability of the feedback system is guaranteed by satisfying these analyticity (or interpolation)conditions.On the other hand,these conditions also impose severe limitations on the achievable performance of the feedback sy stem. Suppose S=(I+L)-1 and T=L(I+L)-1 are stable.Then p1,p2,...,pm are the RHP zeros of S and z1,22,...,z are the RHP zeros of T.Let B,(=ⅡB，B.()=Ⅱ之 s+pi s+2 i=1 j=1 Then Bp(jw)=1 and |B:(jw)|=1 for all frequencies and moreover B'(s)S(s)∈Ho,B1(s)T(s)eH· Hence by Maximum Modulus Theorem,we have Is(s)‖。=‖B,'(s)S(s)‖∞≥B(2)S(2川 for any z with Re(2)>0.Let z be a RHP zero of L,then ‖s(训≥IB1(2=] Similarly,one can obtain IT(s)训≥B'(pl= where p is a RHP pole of L. The weighted problem can be considered in the same fashion.Let We be a weight such that WeS is stable.Then IW.(s)S(s)‖。≥1w.(2 z-Di Now suppose We(s)= 马M,+,Iw.Slo≤1 and is a ra RHP.Then s+Wbe z/Ms +wb + =:a z+Wbe which gives aa-)≈a- 1 wb≤1-a where a =1 if L has no RHP poles.This shows that the bandwidth of the closed-loop must be much smaller than the right half plane zero.Similar conclusions can be arrived for complex RHP zeros
Analyticity Constraints The internal stability of the feedback system is guaranteed by satisfying these analyticity or interpolation conditions On the other hand these conditions also impose severe limitations on the achievable performance of the feedback system Suppose S I L and T L I L are stable Then p ppm are the RHP zeros of S and z zzk are the RHP zeros of T Let Bp s Ym i s pi s pi Bz s Y k j s zj s zj Then jBp jj and jBz jj for all frequencies and moreover B p sS s H B z sT s H Hence by Maximum Modulus Theorem we have kS sk B p sS s jB p zS zj for any z with Re z Let z be a RHP zero of L then kS sk jB p zj Ym i z pi z pi Similarly one can obtain kT sk jB z pj Y k j p zj p zj where p is a RHP pole of L The weighted problem can be considered in the same fashion Let We be a weight such that WeS is stable Then kWe sS sk jWe zj Ym i z pi z pi Now suppose We s sMs b s b kWeSk and z is a real RHP zero Then zMs b z b Ym i z pi z pi which gives b z Ms z Ms where if L has no RHP poles This shows that the bandwidth of the closedloop must be much smaller than the right half plane zero Similar conclusions can be arrived for complex RHP zeros
96 PERFORMANCE SPECIFICATIONS AND LIMITATIONS 6.7 Notes and References The loop shaping design is well known for SISO systems in the classical control theory. The idea was extended to MIMO systems by Doyle and Stein 1981]using LQG design technique.The limitations of the loop shaping design are discussed in detail in Stein and Doyle [1991].Chapter 16 presents another loop shaping method using Hoo control theory which has the potential to overcome the limitations of the LQG/LTR method. The design tradeoffs and limitations for SISO systems are discussed in detail in Bode 1945,Horowitz 1963,and Doyle,Francis,and Tannenbaum [1992.The monograph by Freudenberg and Looze [1988 contains many multivariable generalizations.The multivariable generalization of Bode's integral relation can be found in Chen 1995]. Some related results can be found in Boyd and Desoer 1985.Additional related results can be found in a recent book by Seron,Braslavsky and Goodwin [1997. 6.8 Problems Problem 6.1 Let P be an open loop plant.It is desired to design a controller so that the overshoot 10%and settling time <10sec.Estimate the allowable peak sensitivity Ms and the closed-loop bandwidth. Problem 6.2 LetL be an open loop transfer function of a unity feedback system.Find the phase margin,overshoot,settling time,and the corresponding Ms. Problem 6.3 Repeated the last problem with 100(s+10) L2=8+1)(8+2)8+20 Problem6.4LletP=.tseclasicalopshapingmdhodtodesigmaomntrolilg so that the system has at least 300 phase margin and as large crossover frequency as possible. Problem 6.5 Use root locus method to show that a nonminimum phase system cannot be stabilized by a high gain controller. Problem 6.6 Let P-Design a controller so that the system has at least 5 300 phase margin and the smallest possible bandwith (or crossover frequency). Problem 6.7 Use root locus method to show that a unstable system cannot be stabilized by a low gain controller. Problem 6.8 Consider the unity-feedback loop with proper controller K(s)and strictly proper plant P(s),both assumed square.Assume internal stability
PERFORMANCE SPECIFICATIONS AND LIMITATIONS Notes and References The loop shaping design is well known for SISO systems in the classical control theory The idea was extended to MIMO systems by Doyle and Stein using LQG design technique The limitations of the loop shaping design are discussed in detail in Stein and Doyle Chapter presents another loop shaping method using H control theory which has the potential to overcome the limitations of the LQGLTR method The design tradeos and limitations for SISO systems are discussed in detail in Bode Horowitz and Doyle Francis and Tannenbaum The monograph by Freudenberg and Looze contains many multivariable generalizations The multivariable generalization of Bodes integral relation can be found in Chen Some related results can be found in Boyd and Desoer Additional related results can be found in a recent book by Seron Braslavsky and Goodwin Problems Problem Let P be an open loop plant It is desired to design a control ler so that the overshoot and settling time sec Estimate the al lowable peak sensitivity Ms and the closedloop bandwidth Problem Let L s s be an open loop transfer function of a unity feedback system Find the phase margin overshoot settling time and the corresponding Ms Problem Repeated the last problem with L s s s s Problem Let P s s s Use classical loop shaping method to design a control ler so that the system has at least phase margin and as large crossover frequency as possible Problem Use root locus method to show that a nonminimum phase system cannot be stabilized by a high gain control ler Problem Let P s s Design a control ler so that the system has at least phase margin and the smal lest possible bandwith or crossover frequency Problem Use root locus method to show that a unstable system cannot be stabilized by a low gain control ler Problem Consider the unityfeedback loop with proper control ler K s and strictly proper plant P s both assumed square Assume internal stability
6.8.Problems 97 1.Letw(s)heascalar weightirg fircticn asumedinRH.Defire e=kw(I+PK)4 8=kK(I+PK)4e. so e neses say,cstubarce attenaticnards neeres say,crtd effcrt Derive the folloirg ireuality,that shois that e ards carrct bthe snall si- mltarecusly ingereral:Fareery reso5 0 lw(so)I<e+w(so)lomin[P(so)]8. 2.Ifwearty odcsturlarceateraticnat aparticlar frererc,yo nigt quessthat wereedhigh crticller gainat that frepuercy.Fir-withj-rct apdle cf P(s),ardsuppcse e:=omazl(I+PK)4(j-)]<1. Derivealcuer brdfrminK(j-).This ler burdshculdlc up ase0
Problems Let w s be a scalar weighting function assumed in RH Dene kw I P Kk kK I P Kk So measures say disturbance attenuation and measures say control e ort Derive the fol lowing inequality that shows that and cannot both be smal l si multaneously in general For every Re s jw sj jw sjmin P s If we want very good disturbance attenuation at a particular frequency you might guess that we need high control ler gain at that frequency Fix with j not a pole of P s and suppose max I P K j Derive a lower bound for min K j This lower bound should blow up as