Page - (000035) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

3.3 Proportional, Integral, andDerivativeControl 25 Perfect trackingofaconstantreferencesignalrequiresapureintegrator term,1/s. Aconstant signal has zero frequency, s=0.To tracka signal perfectly, the system transferfunction’sgainmustbeonesothattheoutputequalstheinput.Forthesimple closed loop in Eq.3.4, at zero frequency,G(0)must be one. The tracking error is 1−G=1/(1+L). The error goes to zero as the gain of the open loop goes to infinity,L(0)→∞.A transfer function requires a term1/s to approach infinity as s goes tozero. Ingeneral, highopen loopgain leads to lowtrackingerror. Third,wecanaddderivativecontrolbyincludingthetermkds.Wecanunderstand why this term differentiates the input term by following the same steps as for the analysis of integration.MultiplyingEq.2.5 by s increases the numerator’s order of its polynomial in s. That increase in the exponents of s corresponds to an increase in the order of differentiation for each termon the right side of Eq.2.4. Thus, the original input term,u(t), becomes thederivativewith respect to time, u˙(t). Differentiatingtheinputcausesthesystemtorespondtothecurrentrateofchange in the input.Thus, the systemresponds toapredictionof the future input, basedon a linear extrapolationof the recent trend. This leading, predictive response enhances sensitivity to short-term, high- frequencyfluctuations and tends to block slow, low-frequency input signals. Thus, differentiationactsasahigh-passfilterof theinputsignal.Atermsuchass+amul- tipliessignalsbya for low-frequencyinputsandmultipliessignalsbytheincreasing valueofs+a for increasinglyhigh-frequency inputs.Differentiatorsmakesystems very responsive, but also enhance sensitivity to noisyhigh-frequencyperturbations and increase the tendency for instability. Abasicproportional, integral, andderivative (PID)controllerhas the form C(s)= kp+ ki s +kds= kds 2+kps+ki s . (3.6) PIDcontrollers arewidelyusedacrossall engineeringapplications.Theywork rea- sonablywell formanycases, theyarerelativelyeasytounderstand,andtheirparam- eters are relativelyeasy to tune forvarious tradeoffs inperformance. 3.4 SensitivitiesandDesignTradeoffs Figure3.2ashowsabasicfeedbackloopwiththreeinputs: thereferencesignal,r, the loaddisturbance,d,andthesensornoise,n.Howdothesedifferentsignals influence the error between the reference signal and the systemoutput? In otherwords, how sensitive is the systemto thesevarious inputs? To derive the sensitivities, define the error in Fig.3.2a as r−η, the difference between thereference input,r, and theprocessoutput,η (ÅströmandMurray2008, Sect.11.1).Toobtainthetransferfunctionbetweeneachinputandoutput,weusethe rule fornegativefeedback:Thetransfer functionbetweenthe inputandoutput is the