Page - (000042) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

4.3 OutputResponse toFluctuating Input 33 4.3 OutputResponse toFluctuatingInput Figure4.3 illustrates the systemoutput in response tofluctuating input (green).The top rowshows theoutput of the systemprocess, either P (blue) or P˜ (gold), alone inanopen loop.Thesystemprocess isacascadeof twolow-passfilters,whichpass low-frequency inputs anddonot respond tohigh-frequency inputs. The upper left panel shows the response to the (green) low-frequency input, ω=0.1, inwhich the base system P (blue) passes through the inputwith a slight reduction inamplitudeand lag inphase.Thealteredsystem P˜ (gold) respondsonly weakly to the low frequency of ω=0.1, because the altered system has slower response characteristics than the base system. At a reduced input frequency of ω=0.01(notshown), thegoldcurvewouldmatchthebluecurveatω=0.1.Asfre- quencyincreasesalongthetoprow,theprocessesPand P˜blockthehigher-frequency inputs. The lower row shows the response of the full PID feedback loop system.At a low frequency ofω≤0.1, the output tracks the input nearly perfectly. That close tracking arises because of the very highgain amplification of thePIDcontroller at lowfrequency,which reduces the systemtrackingerror tozero, as inEq.3.5. Atahigherfrequencyofω=10,thesystemwiththebaseprocessP respondswith a resonant increase in amplitude anda lag inphase.The slower alteredprocess, P˜, respondsonlyweakly to inputat this frequency.Asfrequencycontinues to increase, both systems respondweaklyornot at all. Thesystemresponsetosensornoisewouldbeofequalmagnitudebutalteredsign andphase, as shown inEq.3.7. Low-frequencytrackingandhigh-frequencyrejectiontypicallyprovidethegreat- est performance benefit. The environmental references that it pays to track often change relatively slowly,whereas the noisy inputs in both the reference signal and in the sensorsoftenfluctuate relatively rapidly. 4.4 Insights fromBodeGainandPhasePlots Figure4.4 providesmoregeneral insight into theways inwhichPIDcontrol, feed- back, and inputfilteringalter systemresponse. Panels(a)and(b)showtheBodegainandphaseresponsesfortheintrinsicsystem process, P (blue), and the altered process, P˜ (gold). Low-frequency inputs pass through. High-frequency inputs cause little response. The phase plot shows that theseprocesses respondslowly, lagging the input.The lag increaseswithfrequency. Panels (c) and (d) show the responses for theopen loopwith thePIDcontroller, C, combinedwith the process, P or P˜, as in Fig.2.1b.Note the very high gain in panel (c) at lower frequenciesand the lowgainathigh frequencies.