Page - (000044) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

4.5 Sensitivities inBodeGainPlots 35 4.5 Sensitivities inBodeGainPlots Figure4.5illustratesthesensitivitiesofthesystemerroroutput,r−η, toinputsfrom the reference, r, sensor noise, n, and load disturbance, d, signals, calculated from Eq.3.9.Figure3.2a shows the inputs and loopstructure. Thebluecurveofpanel(a)showstheerrorsensitivity tothereferenceinput.That sensitivity is approximately themirror image of the systemoutput response to the reference input, as shown in Fig.4.4e (note the different scale). The duality of the error response and the system response arises from the fact that the error is r−η, and the systemresponse isη. Perfect trackingmeans that the outputmatches the input, r =η. Thus, a small error corresponds to a lowgain of the error in response to input, as occurs at low frequencyfor thebluecurveofFig.4.5a. In thesameway,asmallerrorcorresponds toagainofonefortherelationbetweenthereferenceinput,r,andthesystemoutput, η, asoccursat lowfrequency for thebluecurveofFig.4.4e. Thenoisesensitivity in thegreencurveofFig.4.5ashowsthat thesystemerror is sensitivetolow-frequencybiasinthesensormeasurements,y,ofthesystemoutput,η. Whenthesensorproducesalow-frequencybias, thatbiasfeedsbackintothesystem and creates a bias in the error estimate, thus causing an error mismatch between the reference input and the systemoutput. Inotherwords, the systemis sensitive to errorswhen thesensor suffers low-frequencyperturbations.ThePIDsystemrejects high-frequencysensornoise,leadingtothereducedgainathighfrequencyillustrated by thegreencurve. Thedisturbance load sensitivity in the red curveofFig.4.5a shows the lowsen- sitivityof thisPIDfeedbacksystemtoprocessvariations. This PID feedback system is very robust to an altered underlying process, as shown in earlier figures. Here, Fig.4.5b illustrates that robustness by showing the relativelyminorchangesinsystemsensitivitieswhentheunderlyingprocesschanges 0.01 0.1 1 10 100 1000 -100 -75 -50 -25 0 0.01 0.1 1 10 100 1000 -100 -75 -50 -25 0 (a) (b) Fig.4.5 Bodegainplots for theerroroutput,r−η, in response toreferenceinput,r (blue), sensor noise,n (green),andloaddisturbance,d (red),fromEq.3.9.ThesystemsarethefullPID-controlled feedback loops as inFig.3.2a,withno feedforwardfilter.ThePIDcontroller is given inEq.4.3.a Systemwith thebaseprocess,P, fromEq.4.1.bSystemwith thealteredprocess, P˜, fromEq.4.2