Page - (000072) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

8.1 Varying InputFrequencies 65 20 40 60 80 100 120 -2 -1 1 2 Fig. 8.2 Reference signal, r, in gold, fromEq. 8.3, and the filtered signal, rw, in blue, from the filter inEq.8.4 applied to the reference signal.Thebluecurves inFig.8.3 showthefiltered signal moreclearly R(s)= ∑ i κiωi s2+ω2i , (8.3) inwhicheachof thethree termsinthesumexpressesasinewavewithfrequencyωi. Here, Iuseψ =10. Often, low-frequencysignals represent truechanges in theexternalenvironment. Bycontrast,high-frequencyinputsrepresentnoiseorsignals thatchangetoorapidly to trackeffectively.Thus,wemaywish tooptimize the systemwith respect to low- frequency inputs and to ignorehigh-frequency inputs. Wecanaccomplishfrequencyweightingbyusingafilterederrorsignalinthecost function, EW(s)= R(s)W(s)−Y(s), for aweighting functionW that passes low frequenciesandreduces thegainofhighfrequencies.Theweightederrorsignalasa functionof time isew(t). Inourexample, the function W(s)= ( √ ψ s+√ψ )3 (8.4) will reduce the relativeweightingof thehigh-frequency input at frequencyψ. I use thefilterederror signal,ew, for thecost function inEq.8.1,yielding J = ∫ T 0 (u2+ρ2e2w)dt. (8.5) Thegoldcurve inFig.8.2shows theenvironmental referencesignal,r, for theasso- ciatedtransferfunction,R(s).Thebluecurveshowsthefilteredreferencesignal,rw, for the filtered system, R(s)W(s). The filtered curve removes the high-frequency