Page - (000099) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

96 13 TimeDelays C(s) y P(s) ur e e- s (a) (b) C(s) y P(s)e- sur e Fig.13.1 Timedelays in feedback loops.aSensordelay.Thesensor thatmeasures systemoutput and passes that value as feedback has a delay of δ time units between the system input and the measuredoutput.The transfer functione−δs passes its inputunmodifiedbutwithadelayofδ time units.bProcess delay. The systemprocess,Pe−δs, has a lag of δ time units between the time at whichacontrol input signal,u, is receivedand theassociatedsystemoutput signal,y, is produced In addition to general robust approaches, many specific design methods deal explicitlywithdelays.Thedelaysareoftencalleddeadtimeor transportlag(Åström andHägglund2006;Normey-RicoandCamacho2007;Visioli andZhong2011). The designmethods typically use a predictionmodel. A prediction allows the system to usemeasured signal values at time t−δ to estimate the signal values at time t. 13.2 SensorDelay Figure13.1a shows a standard feedback loopwith a sensor delay. The sensor that measurestheprocessoutput,y,delayspassingonthemeasuredvaluebyδ timeunits. InFig.13.1a, thetransferfunctione−δsdescribes thedelay.That transferfunction passes its input unmodified, butwith a delay of δ. Thus, themeasured output that is passed by the sensor as feedback is given by the transfer functionYe−δs, which transforms inputs,y(t), into the time-delayedoutputs,y(t−δ). We can derive how the delay influences the closed-loop system response in Fig.13.1a.Definetheopenloopof thesystemasL=CP, as inEq.3.4.Thenwecan write the systemoutput asY =LE, the error input,E,multipliedby theopen-loop systemprocess,L. The error is the difference between the reference input and the feedback output from the sensor,E =R−Ye−δs. Substituting this expression for the error intoY = LE,weobtain the transfer functionexpression for theclosed-loopsystemresponse, G =Y/R, as G(s)= L(s) 1+L(s)e−δs . (13.1)