88 PERFORMANCE SPECIFICATIONS AND LIMITATIONS 1/w| u lKS(j@)l w bc Figure 6.8:Control Weight Wu and Desired KS different design objectives.It is also known that the fundamental requirements such as stability and robustness impose inherent limitations upon the feedback properties irrespective of design methods,and the design limitations become more severe in the presence of right-half plane zeros and poles in the open-loop transfer function. In the classical feedback theory,the Bode's gain-phase integral relation(see Bode [1945)has been used as an important tool to express design constraints in scalar sys- tems.This integral relation says that the phase of a stable and minimum phase transfer function is determined uniquely by the magnitude of the transfer function.More pre- cisely,let L(s)be a stable and minimum phase transfer function,then a=mw (6.11) dv 2 whe)hfanctionotIis ploted in ige.. Note that In coth decreases rapidly as w deviates from wo and hence the integral depends mostly on the behavior of din L(jw) dv near the frequency wo.This is clear from the following integration: 1.1406(rad), a=In3 65.30 a=In3 1.3146(rad, a=ln5 75.30, a =In5 1.443(rad), a=1n10 82.7°， a=In10 Note that isth ope of the Bode plot which is generally negative for almost all frequencies.It follows that L(jwo)will be large if the gain L attenuates
PERFORMANCE SPECIFICATIONS AND LIMITATIONS ε bc M 1 1 u u ω 1/|W | |KS(j ω)| Figure Control Weight Wu and Desired KS dierent design ob jectives It is also known that the fundamental requirements such as stability and robustness impose inherent limitations upon the feedback properties irrespective of design methods and the design limitations become more severe in the presence of righthalf plane zeros and poles in the openloop transfer function In the classical feedback theory the Bodes gainphase integral relation see Bode has been used as an important tool to express design constraints in scalar sys tems This integral relation says that the phase of a stable and minimum phase transfer function is determined uniquely by the magnitude of the transfer function More pre cisely let L s be a stable and minimum phase transfer function then L j Z d ln jLj d ln coth jj d where ln The function ln coth jj ln e jjejj ejjejj is plotted in Figure Note that ln coth jj decreases rapidly as deviates from and hence the integral depends mostly on the behavior of d ln jL jj d near the frequency This is clear from the following integration Z ln coth jj d rad ln rad ln rad ln ln ln ln Note that d ln jL jj d is the slope of the Bode plot which is generally negative for almost all frequencies It follows that L j will be large if the gain L attenuates
6.4.Bode's Gain and Phase Relation 89 4.5 3.5 三15 0 2 -1 Figure 6.9:The Function In coth- 2 slowly near wo and small if it attenuates rapidly near wo.For example,suppose the slope dln()i.e(-20dB per decade),in the neighborhood of wo,then it dy is reasonable to expect -l×65.3,if the slope ofL=-lfor3≤“≤3 ∠L(jwo) -l×75.30,if the slope of L=-lfor吉≤÷≤5 -l×82.70,if the sope ofL=-l for≤品≤10 The behavior of L(jw)is particularly important near the crossover frequency we where L(jwe)=1 since +L(jwe)is the phase margin of the feedback system,and further the return difference is given by +L(jwe)=+L(jwe)=2 sin+L(jw) which must not be too small for good stability robustness.If +L(jwe)is forced to be very small by rapid gain attenuation,the feedback sy stem will amplify disturbances and exhibit little uncertainty tolerance at and near w.Since it is generally required that the loop transfer function L roll off as fast as possible in the high frequency range, it is reasonable to expect that L(jwe)is at most -ex 90 if the slope of L(jw)is- near we.Thus it is important to keep the slope of L near we not much smaller than -1 for a reasonably wide range of frequencies in order to guarantee some reasonable performance.The conflict between attenuation rate and loop quality near crossover is thus clearly evident
Bodes Gain and Phase Relation −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ν ln coth |ν |/ 2 Figure The Function ln coth jj vs slowly near and small if it attenuates rapidly near For example suppose the slope d ln jL jj d ie dB per decade in the neighborhood of then it is reasonable to expect L j if the slope of L for if the slope of L for if the slope of L for The behavior of L j is particularly important near the crossover frequency c where jL jcj since L jc is the phase margin of the feedback system and further the return dierence is given by j L jcj j L jcj sin L jc which must not be too small for good stability robustness If L jc is forced to be very small by rapid gain attenuation the feedback system will amplify disturbances and exhibit little uncertainty tolerance at and near c Since it is generally required that the loop transfer function L roll o as fast as possible in the high frequency range it is reasonable to expect that L jc is at most if the slope of L j is near c Thus it is important to keep the slope of L near c not much smaller than for a reasonably wide range of frequencies in order to guarantee some reasonable performance The conict between attenuation rate and loop quality near crossover is thus clearly evident
90 PERFORMANCE SPECIFICATIONS AND LIMITATIONS The Bode's gain and phase relation can be extended to stable and nonminimum phase transfer functions easily.Let 21,22,...,zk be the right-half plane zeros of L(s), then L can be factorized as o)=二s+as+丝二s+丝Lm( s+刘18+2 8十2k where Lmp is stable and minimum phase and (jw)=Lmp(jw).Hence ∠L(io）=∠Lmp(jwo)+Ⅱjo+a 0+2i =1 1 ncoth+∑∠n+ jwo+zi which gives ZL(jwo)= 1 lnoth以d+∑∠io+产 dv (o.12) jwo+zi Since∠二jwo+名≤0 for each,anon-minimumphaseeonibutesan additional w0+2i phase lag and imposes limitations upon the rolloff rate of the open-loop gain.For example,suppose L has a zero at z>0 then (w/2):=∠jiwo+2 =-90°，-53.130，-280 1w0+2 lw-z,z/2,3/4 as shown in Figure 0.10.Since the slope of L near the crossover frequency is in general no greater than-1 which means that the phase due to the minimum phase part,Lmp, of L will in general be no greater than-900,the crossover frequency (or the cosed-loop bandwidth)must satisfy We0 then (wo/小el0:=∠二iwo+之二jwo+z jw0+2jw0+2 w=|z,lz/2,|zl/3,l/4 -180°，-10a.2.°，-73.70， -5,Re(z)≥S(z) -180°， -8.70,-55.9° -41.3,Re(z)≈S(z -30°， 00, 0. 0°，Re(z)≤S(z)
PERFORMANCE SPECIFICATIONS AND LIMITATIONS The Bodes gain and phase relation can be extended to stable and nonminimum phase transfer functions easily Let z zzk be the righthalf plane zeros of L s then L can be factorized as L s s z s z s z s z s zk s zk Lmp s where Lmp is stable and minimum phase and jL jj jLmp jj Hence L j Lmp j Y k i j zi j zi Z d ln jLmpj d ln coth jj d X k i j zi j zi which gives L j Z d ln jLj d ln coth jj d X k i j zi j zi Since j zi j zi for each i a nonminimum phase zero contributes an additional phase lag and imposes limitations upon the rollo rate of the openloop gain For example suppose L has a zero at z then z j z j z zzz as shown in Figure Since the slope of jLj near the crossover frequency is in general no greater than which means that the phase due to the minimum phase part Lmp of L will in general be no greater than the crossover frequency or the closedloop bandwidth must satisfy c z in order to guarantee the closedloop stability and some reasonable closedloop perfor mance Next suppose L has a pair of complex right half zeros at z x jy with x then jzj j z j z j z j z jzjjzjjzjjzj Re z z Re z z Re z z
6.5.Bode's Sensitivity Integral 91 -10 -20 -40 50 -70 00.1 02o3&040506o7o8o9 Figure -100 as shown in Figure -11x In this case we conclude that the crossover frequency must satisfy |zl/4,Re(z)≥3(z) z/3,Re(z≈S(z) (-14) Re(z)≤S(z in order to guarantee the closed-loop stability and some reasonable closed-loop perfor- mancex ∠.5 Bode's Sensitivity Integral In this section,we consider the design limitations imposed by the bandwidth constraints and the right half plane poles and zeros using the Bode's sensitivity integral and Poisson integralx Let L be the open loop transfer function with at least two more poles than zeros and let pi,p...,Pm be the open right half plane poles of LxThen the following Bode's sensitivity integral holds m lnlS(jw)ldw=x∑Re(i) (-15) i=1 In the case where L is stable,the integral simplifies to 8 In S(jw)ldw =0. (-1-）
Bodes Sensitivity Integral 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 ω0 / |z| phase φ1(ω0 /z) (in degree) Figure Phase z due to a Real Zero z as shown in Figure In this case we conclude that the crossover frequency must satisfy c jzj Re z z jzj Re z z jzj Re z z in order to guarantee the closedloop stability and some reasonable closedloop perfor mance Bodes Sensitivity Integral In this section we consider the design limitations imposed by the bandwidth constraints and the right half plane poles and zeros using the Bodes sensitivity integral and Poisson integral Let L be the open loop transfer function with at least two more poles than zeros and let p ppm be the open right half plane poles of L Then the following Bodes sensitivity integral holds Z ln jS jjd Xm i Re pi In the case where L is stable the integral simplies to Z ln jS jjd
92 PERFORMANCE SPECIFICATIONS AND LIMITATIONS yk=10 40 y/x=100 -60 (xap u)(seyd -80 y←3 140 yk-0.01 180 20 0.1 0203 品a05 0.60.7080.9 1 Figure -11<Phase 9_(6off)due to a Pair of Complex RHP Zeros<z=x+jy and x10 These integrals show that there will exist a frequency range over which the magnitude of the sensitivity function exceeds one if it is to be kept below one at other frequencies as illustrated in Figure -12xThis is the so-called water bed effect x Suppose that the feedback sy stem is designed such that the level of sensitivity re- duction is given by |s061’e<1,6∈0,6， where e1 0is a given constantx Bandwidth constraints in feedback design typically require that the open-loop trans- fer function be small above a specified frequency,and that it roll off at a rate of more than one pole-zero exoess above that frequencyxThese constraints are commonly needed to ensure stability robustness despite the presence of modeling uncertainty in the plant model,particularly at high frequenciesx One way of quantifying such bandwidth con- straints is by requiring the open-loop transfer function to satisfy 1U61'’<1,v6∈6 61+, where 61 61,and Mn 1 0,5 1 0 are some given constantsx Note that for6≥6h, 1s(661'.0o·元 1
PERFORMANCE SPECIFICATIONS AND LIMITATIONS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −200 −180 −160 −140 −120 −100 −80 −60 −40 −20 0 ω0 / |z| phase φ2 ( ω0 / |z| ) (in degree) y/x=0.01 y/x=1 y/x=3 y/x=10 y/x=100 Figure Phase jzj due to a Pair of Complex RHP Zeros z x jy and x These integrals show that there will exist a frequency range over which the magnitude of the sensitivity function exceeds one if it is to be kept below one at other frequencies as illustrated in Figure This is the socalled water bed eect Suppose that the feedback system is designed such that the level of sensitivity re duction is given by jS jj l where is a given constant Bandwidth constraints in feedback design typically require that the openloop trans fer function be small above a specied frequency and that it roll o at a rate of more than one polezero excess above that frequency These constraints are commonly needed to ensure stability robustness despite the presence of modeling uncertainty in the plant model particularly at high frequencies One way of quantifying such bandwidth con straints is by requiring the openloop transfer function to satisfy jL jj Mh h where h l and Mh are some given constants Note that for h jS jj jL jj Mh
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
