Page - (000089) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter11 AdaptiveControl Theparametersofaprocessmaybeunknownormaychangeslowlyover time.How canonecontrol aprocesswithunknownparameters? Earlierchaptersdiscussedrobustmethods.Thosemethodsprovidegoodresponse toabroadsetof alternativeprocessdynamics. Thischapterpresentsadaptivecontrol,inwhichthecontrolsystemadjustsitselfby usingmeasurementsof thesystem’s response. I followtheexample fromSect.6.2.4 of IoannouandSun(2012). In adaptive control, the system repeatedly updates the controller parameters to reduce theerrorbetween thesystem’sactualoutputand theoutputofan ideal target responsemodel. Figure11.1 shows the structure of a common approach known as model referenceadaptivecontrol. Suppose theprocessdynamicsaregivenby theaffine forminEq.10.3as y˙=af(y)+bg(y)u, (11.1) whichdescribes linear systemsandalsoawidevarietyofnonlinear systems. In this example,weknowthe functions f andg, butdonotknowtheparametervalues for a andb.Thegoal is todesignacontrol input,u, that causes thesystemoutput, y, to match theoutputof a specifiedmodel. 11.1 GeneralModel Typically, onechooses a simple linearmodel for thedesign target. In this example, weuse y˙m =−amym +bmr. (11.2) Here, theparametersam andbm areknownaspectsof the targetmodelspecification, and r is the reference or external environmental input. For a constant reference input, thismodelconverges to thereferenceexponentiallyat rateam,withamplitude ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_11 85