Page - (000053) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

Chapter6 Regulation Theregulationproblemanalyzeshowquicklyaperturbedsystemreturns to itsequi- libriumsetpoint.For thisproblem,weassumethat thesetpointdoesnotchange.We can,without lossofgenerality, assume that theexternal reference signal isr =0. Withnoexternalreferencesignal,wecanexpressthegeneralformoftheregulation problemas inFig.6.1.We take theprocess,P, as given, subject to uncertainties or disturbances represented by the input, d. We seek an optimal controller,C, with respect toparticulardesign tradeoffs. 6.1 CostFunction Thecost functionsummarizes thedesign tradeoffs.Weuseacost functionbasedon theH2 norm, similar toEq.5.5.TheH2 normdescribes the responseof the system toperturbationswhenaveragedoverall input frequencies.Minimizing theH2 norm minimizes theextent towhich the systemresponds toperturbations.Recall that the H2 normisoftenequivalent tothesignalenergy,whichis thetotalsquareddeviation of a signal fromzerowhenmeasured from the timeof an initial perturbation until the timewhen thesignal returns tozero. FromFig.6.1, the twoinputsare the loaddisturbance,d, and thesensornoise,n. The twooutputs are the process output,η, and the control signal, u.We canwrite theoutputs as transfer functions,η(s)andU(s), and thecost function inEq.5.5as J =||U(s)||22+ρ2||η(s)||22. In this case, we need to relate each of the two outputs to each of the two inputs. We require four transfer functions to describe all of the input–output connections. For the transfer functionbetween the inputd and theoutputη,wewriteGηd(s), for ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_6 45