-,-PERFORMANCE 35 Ex ane To see hav this workstakethe simplet possibeexample C+13 Then thetransfer function from Ito eequals 1中=3 1 Sothe cpenloop pole at s =0 becomes a dosedloopzerocf theerror transfer fun(tion+then this zeo cancels the pale cf Is=>resulting in no unstable poles in eSimilar remarks apply for a ramp input. Theorem 3 is a special case cf an elemetary principle For perfect asymptctic tracking=the looptransfer function L must COntain an internal model cf theunstable pcles cf I. A similar,analysis can be done for the situation where I=n =0 and d is a sinusoid-say d(sin(,t(1 is the unit step You can shov this If the feedbad system is internally stablezthe y(ti0st、1,ir therP has azero at s=j,or Chas apale at s=方， (Exerase3→ 3.4 p erform anCe In this section we again lock at tradking a reference signal>but whereas in the preceding section weconsidered perfect aymptctictradking cf asingle signal-wewill now consider aset cf reference Signals and a bcund on the steadyktate error.This performance ckjective will be quanti/ed in terms cf aweighted nom bound. As beforeleL dencte the loop transrer functionL=P Thetrasrer function fron rEference input I'totradking error eis 1 S:=1+L calledthe sensitivity function.In the analysistofcllov>it will always be assumedthat thefeedbad System isinternally stable sos is astable-proper transfer function.Observethat sinceL isstrictly proper (since P is->S (j1. Thenamesensitivity function comnesfron thefcllowing idea.Let T denctethetransfer function from I'toy: Oneway toquantify hov sensitive T istovariations in P is totake the limiting ratiocf arelative peturbation in T(i.e≥△T;T-to a relative perturbation in P(i.e>△P:P→Thinking Cf P as a variable and T as afunction cf it>weget 票3 lim Theright hhand side is easily evaluated to bes.In this ways isthe sensitivity cf the clcsedlocp transfer fun(tiOn T to an in/nitesimal peturkation in P. Now wehaetodecide on aperformancespeci/cation-ameasurecf goodhess cf tracking.This decision depends on twothings what we know about I'and what measurewe choosetoassign to thetracking error.USually>Iis nct knovn in advance-few cOntrcl Systems are designed for Ce PERFORMANCE  Example To see how this works take the simplest possible example P s  s C s Then the transfer function from r to e equals  s￾￾  s s  So the openloop pole at s   becomes a closedloop zero of the error transfer function then this zero cancels the pole of r s resulting in no unstable poles in e s Similar remarks apply for a ramp input Theorem  is a special case of an elementary principle For perfect asymptotic tracking the loop transfer function L must contain an internal model of the unstable poles of r A similar analysis can be done for the situation where r  n   and d is a sinusoid say dt  sint t  is the unit step You can show this If the feedback system is internally stable then yt  as t i either P has a zero at s  j or C has a pole at s  j Exercise  ￾ Performance In this section we again look at tracking a reference signal but whereas in the preceding section we considered perfect asymptotic tracking of a single signal we will now consider a set of reference signals and a bound on the steadystate error This performance ob jective will be quantied in terms of a weighted norm bound As before let L denote the loop transfer function L  P C The transfer function from reference input r to tracking error e is S   L called the sensitivity function In the analysis to follow it will always be assumed that the feedback system is internally stable so S is a stable proper transfer function Observe that since L is strictly proper since P is Sj  The name sensitivity function comes from the following idea Let T denote the transfer function from r to y T  P C  P C One way to quantify how sensitive T is to variations in P is to take the limiting ratio of a relative perturbation in T ie T T  to a relative perturbation in P ie P P  Thinking of P as a variable and T as a function of it we get lim P T T P P  dT dP P T The righthand side is easily evaluated to be S In this way S is the sensitivity of the closedloop transfer function T to an innitesimal perturbation in P Now we have to decide on a performance specication a measure of goodness of tracking This decision depends on two things what we know about r and what measure we choose to assign to the tracking error Usually r is not known in advancefew control systems are designed for one