Page - (000090) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

86 11 AdaptiveControl Controller y Process u r Adjustment mechanism Model ym Controller parameters Fig.11.1 Modelreferenceadaptivecontrol.Thegoal is toconstructacontrollersothat thesystem output,y,matchestheoutputofaspecifiedmodel,ym.Toachievethatgoal, thelowerfeedbackloop withcontroller andprocessmust together formasystemthathas thesamedynamicsas themodel. If parameters of the process are unknown, one can use measurement of the error, e= y− ym, to adaptively adjust the parameters of the controller in response to the error. Ideally, the system learnscontrollerparameterssuchthat theoutput,y,convergestomatchthetargetmodeloutput,ym. RedrawnfromFig.5.1ofÅströmandWittenmark(2008),©KarlJ.ÅströmandBjörnWittenmark oftheresponserelativetotheinputofbm/am.Figure11.2illustratesthedesigntarget response for a sinusoidal input,r. Forgivenvaluesofa andb, thecontrol input u= 1 g(y) [ k∗1 f(y)+k∗2y+w∗r ] k∗1 =− a b k∗2 =− am b w∗ = bm b (11.3) transforms theprocessmodel inEq.11.1 into the targetmodel inEq.11.2. If the parametersa andb are unknown, then the input,u,must be based on the estimatesfork1(t),k2(t),andw(t).Theestimatesareupdatedbyanadaptiveprocess in response to the error difference between systemandmodel output, e= y− ym. Thedynamicsof theerror are e˙= y˙− y˙m. Toobtainanexpressionfor e˙,weneedamodifiedformof y˙ thatcontainsonlythe knownparametersam andbm andtheestimatesk1,k2,andw.Thefirststepexpresses theprocessdynamics inEq.11.1byaddingandsubtractingb [ k∗1 f(y)+k∗2y+w∗r ] andusing the identitiesbk∗1 =−a andbk∗2 =−am andbw∗ =bm, yielding y˙=−amy+bmr +b[−k∗1 f(y)−k∗2y−w∗r+ug(y) ] . Write the tracking errors as k˜1= k1−k∗1 and k˜2 = k2−k∗2 and w˜=w−w∗. The errordynamicscan thenbewrittenas