Page - (000043) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

34 4 PIDDesignExample 0.01 0.1 1 10 100 -80 -60 -40 -20 0 0.01 0.1 1 10 100 -25 0 25 50 75 0.01 0.1 1 10 100 -20 -10 0 0.01 0.1 1 10 100 -20 -10 0 0.01 0.1 1 10 100 -150 -100 -50 0 0.01 0.1 1 10 100 -150 -100 -50 0 0.01 0.1 1 10 100 -150 -100 -50 0 0.01 0.1 1 10 100 -150 -100 -50 0 (a) (b) (c) (d) (e) (f) (g) (h) Process PIDopen loop PID feedback PIDw/filter Fig.4.4 Bodegain(top)andphase(bottom)plotsforsystemoutput,η= y, inresponsetoreference input,r, in theabsenceof loaddisturbanceandsensornoise.Bluecurves forsystemswith thebase process, P, in Eq.4.1. Gold curves for systemswith the altered process, P˜, in Eq.4.2. a,bThe original unmodifiedprocess, P or P˜,with no controller or feedback. c,dTheopen loopwith no feedback,CPorCP˜,withthePIDcontroller,C, inEq.4.3.e,fTheclosedloopwithnofeedforward filter,F =1.g,hTheclosed loopwith the feedforwardfilter,F, inEq.4.4 PIDcontrollersaretypicallydesignedtobeusedinclosed-loopfeedbacksystems, as inFig.2.1c.Panels (e)and(f) illustrate theclosed-loopresponse.Thehighopen- loopgainof thePIDcontrollerat lowfrequencycauses thefeedbacksystemtotrack the reference input closely.That close trackingmatches the log(1)=0gain at low frequency inpanel (e).Note also the low-frequencyphasematching, or zerophase lag, showninpanel (f), furtherdemonstrating theclose trackingof reference inputs. Athigh frequency, the lowgainof theopen-loopPIDcontroller shown inpanel (c) results in theclosed-loop rejectionofhigh-frequency inputs, shownas the lowgain athigh frequency inpanel (e). Note theresonantpeakof theclosed-loopsysteminpanel (e)nearω=10for the blue curve and at a lower frequency for the alteredprocess in the gold curve.Note also that thealteredprocess, P˜, ingold, retains theexcellent low-frequencytracking andhigh-frequency input rejection, even though thecontrollerwasdesigned for the base process, P, shown in blue. ThePID feedback loop is robust to differences in theunderlyingprocess thatvaries fromtheassumedformofP. Panels(g)and(h)showthePIDclosed-loopsystemwithafeedforwardfilter,F,as inFig.3.2a.Thefeedforwardfiltersmoothsout theresonantpeakfor thebluecurve, so that systemdoesnot amplify inputs at resonant frequencies.Amplified resonant inputsmayleadtoinstabilitiesorpoorsystemperformance.Notethatthefeedforward filterdoesnothavemucheffectonthealteredprocessingold.Feedforwardmodifiers ofaprocess typicallyworkwellonly fora specificprocess.Theyoftendonotwork robustlyover avariant rangeofprocesses.