Page - (000103) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

100 13 TimeDelays C y Pe- suer e- *sP* y* y ym ym Fig. 13.4 Smith predictor to compensate for time delay in the process output. Redrawn from Fig.5.1ofNormey-RicoandCamacho(2007),©Springer-Verlag a lag, and theunlaggedprocess,P,maybeunknown. Ifwecouldmodel theway in which theprocesswouldactwithout a lag,P∗, thenwecouldgenerate anestimate, y∗(t), topredict theoutput,y(t). Figure13.4showsthefeedbackpathwaythroughP∗. IfP∗ isanaccuratemodelof P, thenthefeedbackthroughP∗ shouldprovideagoodestimateofthetrackingerror. However, our goal is to control the actual output, y, rather than to consider output estimates and feedback accuracy. The Smith predictor control design in Fig.13.4 provides additional feedbacks that correct for potential errors in ourmodel of the process,P∗, and inourmodelof thedelay,δ∗. InFig.13.4, thepathway throughP∗ and then eδ∗s provides ourmodel estimate, ym, of the actual output,y. Theerrorbetween the trueoutput and themodeloutput, y−ym, is added to the estimated output, y∗, to provide the value fed back into the system to calculate the error.Byusingboth the estimatedoutput and themodeling error in the feedback, the systemcanpotentially correct discrepancies between the model and theactualprocess. The system transfer function clarifies the components of the Smith predictor system.The system transfer function isG =Y/R, from input,R, to output,Y.We canwrite the systemtransfer functionof theSmithpredictor inFig.13.4as G = ( CP 1+C (P∗+ M) ) e−δs, (13.5) inwhich themodelingerror is M =Pe−δs −P∗e−δ∗s. TheDerivationat theendof this chapter shows the steps toEq.13.5. Thestabilityofatransferfunctionsystemdependsontheformofthedenominator. In thecaseofEq.13.5, theeigenvaluesare the rootsof sobtained from1+C(P∗+ M)=0. We know the process, P∗, because that is our model to estimate the unknownsystem,P.