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