Page - (000026) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

16 2 ControlTheoryDynamics correspondtoparametervaluesinEq.2.11ofa=1anda=10.Allcalculationsand plots in this book are available in the accompanyingMathematica code (Wolfram Research2017)at the site listed in thePreface. In the top-left panel, at input frequencyω=1, the fast (gold) response output closely tracks the input. The slow (blue) response reduces the input by √ 2≈0.7. This output–input ratio is called the transfer function’s gain. The slow response outputalso lags the inputbyapproximately0.11ofonecompletesinewavecycleof 2π =6.28radians, thus theshift to therightof0.11×6.28≈0.7radiansalongthe x-axis. Wemay also consider the lagging shift in angular units, inwhich 2π radians is equivalent to360◦.Thelaginangularunits iscalledthephase. Inthiscase, thephase iswritten as−0.11×360◦ ≈−40◦, inwhich the negative sign refers to a lagging response. A transfer function always transformsa sinewave input into a sinewaveoutput modulated by the gain and phase. Thus, the values of gain and phase completely describe the transfer function response. Figure2.2b shows the same process but driven at a higher input frequency of ω=10. The fast response is equivalent to the slow response of the upper panel. The slow response has been reduced to a gain of approximately 0.1,with a phase of approximately−80◦. At the higher frequency ofω=100 in the bottompanel, the fast response againmatches the slowresponseof thepanel above, and the slow response’sgain is reduced toapproximately0.01. Both the slow and fast transfer functions pass low-frequency inputs into nearly unchanged outputs.At higher frequencies, theyfilter the inputs to produce greatly reduced, phase-shifted outputs. The transfer function formofEq.2.11 is therefore calleda low-passfilter,passing lowfrequenciesandblockinghighfrequencies.The twofiltersinthisexamplediffer inthefrequenciesatwhichtheyswitchfrompassing low-frequency inputs toblockinghigh-frequency inputs. 2.5 BodePlotsofGainandPhase ABodeplot showsa transfer function’sgainandphaseatvarious input frequencies. TheBode gain plot in Fig.2.2e presents the gain on a log scale, so that a value of zerocorresponds toagainofone, log(1)=0. For the systemwith the slower response, a=1 in blue, the gain is nearly one for frequencies less than a and then drops off quickly for frequencies greater than a.Similarly, thesystemwith faster response,a=10, transitions fromasystemthat passes lowfrequencies toonethatblockshighfrequenciesatapointnear itsavalue. Figure2.2f shows thephasechanges for these two low-passfilters.The slowerblue systembegins to lagat lower input frequencies.