东南大学过程控制作业

8:AsaprocessengineerwiththeComplexPoleCorporation,youareassignedaunitwithanexothermicchemicalreactor.Inordertolearnmoreaboutthedynamicsoftheprocess,youdecidetomakeastepchangeintheinputvariable,thecoolanttemperature,from10℃to15℃.Assumethatthereactorwasinitiallyatasteadystate.Youobtainthefollowingplotfortheoutputvariable,whichisreactortemperature(noticethatthereactortemperatureisin℉).Usefigure3-9tohelpanswerthefollowingquestions.Accordingtothefollowingpicture:(i)Processgain=uy,yisthelone-termoutputchangeanduistheinputchangeSowecanconclude:uy=510-15250-275(ii)/*Thevalueofisdefinedaswhenthefirstoutputarisetothesteadyvalueof63.2%.Fromthepicturewecanseethat(275-250)*63.2%=15.8,soittakesabout4minutesormoreminutestorisefrom250to265.8*/FromthepicturewecanseethatT=26minT2-12=420.25-1*262-122T(iii)Fromethepictureof3-9wecanconcludethatwhenthemaxoutputisas1.04timesasthesteadyoutput.Inthispicturewecanconcludethatthemaxoutput/steadyoutput=288/275=1.047.Wecanconcludethat=0.25withoutanyquestion.(4)Sothedecayration=(279-275)/(288-275)=4/13(5)/*Ifwedefinethe5%isstabilitydomainthentheperiodofoscillationcanseewhentheoutputentry275*(10.05)=(261.25-288.75).Fromthepicturethereisalmostnooscillation.Ifwedefinethe2%isstabilitydomain,thestabilitydomain=(269.5-280.5).Fromthepicturewecanseewhenthesecondwavecresttheoutputentrytothestabilitydomain.*/Answer:Theperiodofoscillation=50.303–25.152=25.151minute(6)PureSecond-orderSystems:y(s)=)(*1222susskSowecanconclude:Transferfunction=121652ss11:Asaprocessengineer,youdecidetodevelopafirst-order+timedelaymodelofaprocessusingasteptest.Theprocessisinitiallyatsteadystate,withaninputflowrateof5gpmandanoutputof0.75mol/L.Youmakeastepincreaseof0.5gpmat3:00p.m,anddonotobserveanychangesuntil3:07p.m.At3:20p.m.,thevalueoftheoutputis0.8mol/L.Youbecomedistractedanddonothaveachancetolookattheoutputvariableagain,untilyouleaveforhappyhouratalocalwateringholeat6:30p.m.Younotethattheoutputhasceasedtochangeandhasachievedanewsteady-statevalueof0.85mol/L.Whatarethevaluesoftheprocessparameters,withunits?Showyourwork.Answer:Theformofafirst-order+timeprocessisshowedasfollow:)(1)(suseksypspOnaccountoftheinitialsteadystate:5.0*75.085.0pkSo2.0pkAnddonotobserveanychangesuntil7minuteslater:min7Theoutputresponsetoastepinputchange:t0)(tyt]1[)(ptpeuktySo)1(5.0*2.075.08.0pte755.18p2.0pkmol/(L*gpm)min7755.18p5:Considerthefollowingcontinuousstatespacemodel:a.Findthecontinuoustransferfunctionmodel(dothisanalytically).Answer:72.10582.60002044.02666.1)()(21sssBASICsGA=[-3.6237,0;0.8333,-2.9588];B=[5.5051;-1.2660];C=[0,1];D=0;sys=ss(A,B,C,D);syss=tf(sys)syss=-1.266s-0.0002044---------------------s^2+6.582s+10.72Continuous-timetransferfunction.b.Forasampletimeof0.25,findthediscretestatespaceandtransferfunctionmodels(use:seeModule4)Answer:1929.08814.01392.01392.0)(2zzzzGsysd=c2d(sys,0.25,'zoh')sysd=a=x1x2x10.40420x20.09160.4773b=u1x10.9052x2-0.1392c=x1x2y101d=u1y10Sampletime:0.25secondsDiscrete-timestate-spacemodel.>>sysdd=tf(sysd)sysdd=-0.1392z+0.1392-----------------------z^2-0.8814z+0.1929Sampletime:0.25secondsDiscrete-timetransferfunction.c.Comparethestepresponseofthecontinuousanddiscretemodels(usematlab).Whatdoyouobserve?Answer:[yc,tc]=step(sys)[yd,td]=step(sysdd)plot(tc,yc,td,yd,'o')>>holdon>>gridon>>xlabel('time')>>ylabel('output')>>title('stepresponse')7:consideraunitstepinputchangemadeatk=0,resultingintheoutputresponseshownintheplotandtablebelow.Estimatetheparametersforadiscretelinearmodelwiththeformcomparethestepresponseoftheestimatesmodelwiththedata.Usematlabtoconvertthemodeltoacontinuousmodel.Comparethestepresponseofthediscreteandcontinuousmodel.110.99290.9985110.98600.9929110.97900.9860110.97420.9790110.97440.9742110.98280.9744111.00230.9828111.03361.0023111.07461.0336111.11841.0746111.15311.1184111.16161.1531111.12441.1616111.02341.1244110.84941.0234110.61050.8494110.34030.6105110.10440.34031100.104401000022.19985.09929.09860.09790.09742.09744.09828.00023.10336.10746.11184.11531.11616.11244.10234.18494.06105.03403.01044.0YYTT-1）（UseMATLAB:Sowecanget:-a1=1.4138-a2=-0.6065b1=0.1044b2=0.0883Therefore:g(z)=(0.1044z+0.0883)/(z^2-1.4138z+0.6065)Usethosecommandwecangetthatsampletimeis1andassumeszero-orderholdontheinputs.Fromthosepicturewecanconcludecontinuousmodel.Thenwecomparethestepresponseofthediscretemodelandcontinuousmodels.