Page - (000073) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

66 8 Tracking 20 40 60 80 100 120 -2 -1 1 2 20 40 60 80 100 120 -1.5 -1.0 -0.5 0.5 1.0 1.5 20 40 60 80 100 120 -2 -1 1 2 20 40 60 80 100 120 -5 5 20 40 60 80 100 120 -2 -1 1 2 20 40 60 80 100 120 -15 -10 -5 5 10 15 Fig.8.3 Optimizationof thecost function inEq.8.5 for thecontroller inEq.8.2.The left column shows the trackingperformance.Thebluecurve traces thefiltered reference signal,rw, associated withR(s)W(s). Thegold curve traces the systemoutput, y, associatedwithY(s). Thedifference between thecurves is theerror,ew = rw− y.The rightcolumnshows theerror,ew, in red, and the control signal,u forU(s), ingreen.The rowsshow, fromtop tobottom,an increasedweightingof the error versus the control signal in the cost, J, inEq. 8.5,withρ= (1,10,100). Theoptimized controllersmayrepresent local rather thanglobaloptima.See thesupplementalMathematicacode noiseof thereferencesignalandcloselymatchesthefluctuationsfromthetwolower frequencysinewave inputs. Figure8.3 illustrates the tradeoff between the trackingperformanceand the cost ofthecontrolsignalenergytodrivethesystem.ThecostfunctioninEq.8.5describes the tradeoff between tracking,measured by the squared error between the filtered reference signal and the systemoutput,e2w, and thecontrol signal energy,u 2. Theparameterρ sets therelativebalancebetweentheseopposingcosts.Ahigher ρ value favors closer trackingand smaller error because ahighvalueofρ puts less weighton thecostof thecontrol signal.Witha lowercost forcontrol, thecontroller canoutputastrongersignal todrivethesystemtowardaclosermatchwiththetarget reference signal.