Page - (000039) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

30 4 PIDDesignExample 0 5 10 15 20 25 30 0.5 1.0 1.5 0 1 2 3 4 5 0.5 1.0 1.5 0 1 2 3 4 5 0.5 1.0 1.5 (a) (b) (c) Process PID feedback PIDw/filter Fig.4.1 Responseofthesystemoutput,η= y, toasuddenunitstepincreaseinthereferenceinput, r, in theabsenceofdisturbanceandnoiseinputs,d andn.Thex-axisshowsthetime,andthey-axis shows the systemoutput.aResponseof the original process, P(s), inEq.4.1 (blue curve) andof theprocesswithalteredparameters, P˜(s) inEq.4.2 (goldcurve).bSystemwith thePIDcontroller embedded inanegative feedback loop,withno feedforwardfilter,F(s)=1, as inFig.3.2a.cPID feedback loopwith feedforwardfilter,F, inEq.4.4 when itwas designed, or evolved, in response to a process, P, and the underlying systembecomes P˜. In this example, theproblemconcerns thedesignofanegative feedback loop, as in Fig.3.2a, that uses a controllerwith proportional, integral, and derivative (PID) action.Manymethods derive PIDcontrollers by tuning the various sensitivity and performance tradeoffs (ÅströmandHägglund2006;Garpinger et al. 2014). I obtained theparameters for thePIDcontroller inEq.3.6byusing theZiegler– Nicholsmethod inMathematica, yielding C(s)= 6s 2+121s+606 s . (4.3) I alsousedMathematica tocalculate the feedforwardfilter inFig.3.2a,yielding F(s)= s 2+10.4s+101 s2+20.2s+101. (4.4) 4.1 OutputResponse toStepInput Figure4.1illustratesvarioussystemresponsestoaunitstepincreasefromzerotoone inthereferenceinputsignal,r.Panel(a)showstheresponseofthebaseprocess,P,by itself.ThebluecurveisthedoubleexponentialdecayprocessofEq.4.1.Thatprocess responds slowly because of the first exponential processwith time decay a=0.1, whichaverages inputsover a timehorizonwithdecay time1/a=10, as inEq.2.8.