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 We<z/2 (o.13) in order to guarantee the closed-loop stability and some reasonable closed-loop perfor- mance. Next suppose L has a pair of complex right half zeros at z=x+jy with x >0 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 -10<Phase 1(wo/z)due to a Real Zero z>0 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