Page - (000091) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

11.1 GeneralModel 87 0 20 40 60 80 100 -2 0 2 4 0 20 40 60 80 100 -2 0 2 4 0 20 40 60 80 100 -2 0 2 4 (a) (b) (c) Fig. 11.2 Illustrations of themodel response (gold curve) in Eq.11.2 for reference input (blue curve) given by r(t)= sin(t/10)+sin(t). The speed of the tracking response increaseswith the parameteram, andtheamplitudeof theresponse increaseswithbm/am.aAslow, laggingresponse witham =0.3anda twofoldamplitudeenhancementwithbm/am =2.bAfast tracking response with am =3 and a twofold amplitude enhancementwith bm/am =2. cAfast tracking response witham =3anda twofoldamplitude reductionwithbm/am =1/2 e˙=−ame+b [ k˜1 f(y)+ k˜2y+ w˜r ] . Toanalyzetheerrordynamics,weneedexpressionsfor theprocessesusedtoupdate theparameter estimates.Acommonchoice is k˙1=−γ1ef(y) k˙2 =−γ2ey w˙=−γ3er, inwhich Ihaveassumed thatb>0. 11.2 ExampleofNonlinearProcessDynamics Thegeneral resultsof thepriorsectioncanbeapplied toanylinearprocessor toany nonlinear process that canbe approximatedby the affine formofEq.11.1. For this nonlinear example, let y˙=ay2+bu, (11.4) with f(y)= y2 andg(y)=1. Figure11.3 illustrates the rateofadaptation forvariousparameters.As theadap- tation parameters,γ , increase, the systemoutput converges increasingly rapidly to the targetmodeloutput.