Seite - (000092) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

88 11 AdaptiveControl 0 50 100 150 -4 -2 0 2 4 0 20 40 60 80 -4 -2 0 2 4 0 20 40 60 80 -4 -2 0 2 4 0 50 100 150 -4 -2 0 2 4 0 20 40 60 80 -4 -2 0 2 4 0 20 40 60 80 -4 -2 0 2 4 0 50 100 150 -3 -2 -1 0 1 2 3 0 20 40 60 80 -3 -2 -1 0 1 2 3 0 20 40 60 80 -3 -2 -1 0 1 2 3 Fig.11.3 Rateofadaptationfortheparametersthatsetu, thecontrol input intothesystemprocess, asgiven inEq.11.3.Thecontrollerparametersadapt so that the systemoutput tracks theoutputof a referencemodel thatexpresses thedesigngoal.Theequations in the textdescribe thesystemand adaptationdynamics.The top rowshows thesystem’soutput inblueand the targetmodel’soutput ingold.Themiddlerowisthesystem’serror,whichis thedifferencebetweenthesystemandmodel responsesintheupperrow.Thebottomrowshowstheadaptationofthecontrolparametersk1,k2,and w, inblue, gold, andgreen, respectively.Thisfigureuses the referencemodelparametersam =3, bm =6, and the true underlying system process parameters, a=4, and b=2. The controller parameters thatadaptarek1,k2,andw,whichare thesystem’sestimatesfor theunderlyingprocess parametersinrelationtothemodelparameters.Therateofadaptationissetbyγ1=γ2=γ3,which are, for the columns from left to right, 0.01,0.1,1, respectively.The faster the rate of adaptation, themorequickly the systemconverges to the referencemodel 11.3 UnknownProcessDynamics TheprevioussectionassumedaparticularformfortheprocessdynamicsinEq.11.4, with unknownparametersa andb.Howcouldwehandle a processwith unknown dynamics? Onesimpleapproach is toassumeaverygeneral formfor theprocessdynamics, suchasapolynomial y˙=a0+a1y+a2y2+···+anyn +bu, andthenruntheadaptationprocessontheparameters(a0,a1,...,an,b).Onecould use other generic forms for the dynamics and estimate the parameters accordingly.