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