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