模糊PID控制器的魯棒性研究外文文獻(xiàn)翻譯

畢業(yè)設(shè)計(jì)（論文）外文文獻(xiàn)譯文及原文基于?？刂频哪：齈ID參數(shù)的整定Xiao-GangDuan,Han-XiongLi,andHuaDengSchoolofMechanicalandElectricalEngineering,CentralSouthUniVersity,Changsha410083,China,andDepartmentofManufacturingEngineeringandEngineeringManagement,CityUniVersityofHongKong,HongKong摘要:在本文中將利用模控制的整定方法實(shí)現(xiàn)模糊PID控制。此種控制方式首次應(yīng)用于模糊PID控制器，它包括一個(gè)線性PID控制器和非線性補(bǔ)償部分。非線性補(bǔ)償部分可視為一個(gè)干擾過程，模糊PID控制器的參數(shù)可在分析的基礎(chǔ)上確定模結(jié)構(gòu)。模糊PID控制系統(tǒng)利用亞譜諾夫穩(wěn)定性理論進(jìn)行穩(wěn)定性分析。仿真結(jié)果表明利用?？刂普：齈ID控制參數(shù)是有效的。引言一般而言，傳統(tǒng)的PID控制器對(duì)于十分復(fù)雜的被控對(duì)象控制效果不太理想,如高階時(shí)滯系統(tǒng)。在這種復(fù)雜的環(huán)境下,眾所周知,模糊控制器由于其固有的魯棒性可以有更好的表現(xiàn)，因此，在過去30年中，模糊控制器，特別是,模糊PID控制器因其對(duì)于線性系統(tǒng)和非線性系統(tǒng)都能進(jìn)行簡單和有效的控制，已被廣泛用于工業(yè)生產(chǎn)過程[1-4]。模糊PID控制器有多種形式[5],如單輸入模糊PID控制器,雙輸入模糊PID控制器和三個(gè)輸入的模糊PID控制器。一般情況下，沒有統(tǒng)一的標(biāo)準(zhǔn)。單輸入可能會(huì)丟失派生信息,三輸入模糊PID控制器會(huì)產(chǎn)生按指數(shù)增長的規(guī)則。在本文中所采用的雙輸入模糊PID控制器有一個(gè)適當(dāng)?shù)慕Y(jié)構(gòu)并且實(shí)用性強(qiáng)，因此在各種研究和應(yīng)用中,是最流行的模糊PID類型。盡管業(yè)界對(duì)于應(yīng)用模糊PID有越來越大的興趣，但從控制工程的主流社會(huì)的角度來看,它仍然是一個(gè)極具爭議的話題。原因之一是模糊PID參數(shù)整定的基本理論分析方法至今仍不明確。因此，模糊PID控制器不得不進(jìn)行兩個(gè)級(jí)別的整定。在較低層次上，該整定是由調(diào)整增益獲得線性控制性能。在更高層次上的調(diào)整，是由改變知識(shí)庫參數(shù)以提高控制性能,然而調(diào)整知識(shí)庫參數(shù)很難,此外，很難通過改變參數(shù)特性改善瞬態(tài)響應(yīng)。根據(jù)知識(shí)庫傳達(dá)一般控制規(guī)則傾向于保持成員函數(shù)不變，通過離線設(shè)計(jì)和調(diào)試工作擴(kuò)大增益，然而，由于由模糊PID控制器生成非線性控制表面的復(fù)雜性，調(diào)整機(jī)制的衡量因素和穩(wěn)定性分析仍然是艱巨的任務(wù)。如果非線性能得到適當(dāng)?shù)睦茫：齈ID控制器可能得到比傳統(tǒng)PID控制器更好的系統(tǒng)性能。一些非常規(guī)的調(diào)整方法已進(jìn)行了介紹[9-12]。雖然非線性被認(rèn)為是在增益裕度和相位裕度基礎(chǔ)上獲得的，但是由于非線性因素，模糊PID控制器可能會(huì)產(chǎn)生比常規(guī)PID控制器較高的增益。而高增益可能使控制系統(tǒng)的穩(wěn)定性變差[。常規(guī)PID控制器很容易實(shí)現(xiàn)，大量的整定規(guī)則可以涵蓋廣泛的進(jìn)程規(guī)格。在常規(guī)PID控制器的整定方法中，模控制基礎(chǔ)整定是在商業(yè)PID控制軟件包中流行的方法之一，因?yàn)橹恍枵{(diào)整一個(gè)參數(shù)，便可以生產(chǎn)更好的設(shè)置點(diǎn)響應(yīng)[15]。本文提出了一種基于?？刂频腜ID控制器的整定分析方法，模糊PID控制器可分解為線性PID控制器加上非線性補(bǔ)償部分的控制器。把非線性補(bǔ)償部分近似看作一個(gè)過程干擾，模糊PID參數(shù)就可以分析設(shè)計(jì)使用?？刂?。模糊PID控制器的穩(wěn)定性分析是根據(jù)亞譜諾夫穩(wěn)定性理論。最后，通過仿真來證明此種調(diào)整方法是有效的。2問題的提出2.1常規(guī)PID控制器常規(guī)PID控制器通常被描述為下列方程[8-10]：=（1）其中E是跟蹤誤差，kp是比例增益，ki是積分增益，kd是微分增益，Ti和TD分別是積分時(shí)間常數(shù)和微分時(shí)間常數(shù)，這些控制參數(shù)的關(guān)系是KI=KP/Ti和KD=KPTd。PID控制器的傳遞函數(shù)可以表示如下：（2）在根軌跡中，PID控制器有兩個(gè)零點(diǎn)和,一個(gè)極點(diǎn)是原點(diǎn)。條件是兩個(gè)零點(diǎn)滿足大于4。CPCP+udey+_y~~r—圖1?？刂婆渲脠D(a)+yedur+yedurPP__ 圖2?？刂婆渲脠D(b)2.2?？刂圃瓌t基本的?？刂圃瓌t如圖1所示，其中P是被控對(duì)象，P?是名義上的模型對(duì)象，C是控制器，r和d是設(shè)置點(diǎn)和干擾，y和yk分別是被控對(duì)象的輸出和模型對(duì)象的輸出。模控制結(jié)構(gòu)相當(dāng)于古典單閉環(huán)反饋控制器如圖1（b）所示，如果單閉環(huán)控制器如下：（3）及（4）其中(s)是被控模型的最小相位部分，包含任何時(shí)間延遲和右零點(diǎn)，f(s)是一個(gè)低通濾波器，一般形式是：（5）調(diào)整參數(shù)tc是理想閉環(huán)時(shí)間常數(shù)n是一個(gè)待定的正整數(shù)。KiKdKiKdRuleBasesERu圖3模糊PID控制器結(jié)構(gòu)2.3模糊PID控制器模型模糊PID控制器如圖2所示，形式為：及（6）是一種非線性的時(shí)間變量參數(shù)(),A和B分別是每個(gè)輸入和輸出的成員函數(shù)一半的外延。模糊PID控制實(shí)際上有兩個(gè)層次的增益。擴(kuò)大增益(Ke,Kd,K0,和K1)處于較低的水平。擴(kuò)大增益的調(diào)整將會(huì)影響模糊PID控制器效果，造成控制參數(shù)的不斷變化。作為控制行為的模糊耦合控制，Ke,Kd,K0,和K1以何種不同的控制行動(dòng)仍然沒有非常清楚，這使得實(shí)際設(shè)計(jì)和調(diào)試過程相當(dāng)困難。3基于?？刂频哪：齈ID整定在模糊PID控制器整定的基礎(chǔ)上的?？刂品椒?，通過分析模糊PID控制模型得到第一個(gè)簡單推導(dǎo)。然后，參數(shù)模糊PID控制器可在?？刂频幕A(chǔ)上確定參數(shù)。假設(shè)一個(gè)工業(yè)過程可以模仿成一階加上延遲（FOPDT）環(huán)節(jié)，傳遞函數(shù)如下：(7)其中K、T和L分別是穩(wěn)態(tài)增益，時(shí)間常數(shù)，和延遲時(shí)間，這些參數(shù)通過階躍響應(yīng)法，頻率響應(yīng)，和閉環(huán)繼電反饋等方法來描述的，F(xiàn)OPDT是一種最常見最實(shí)用的模型，尤其是在過程控制中[18]。通過式（6）可以得到：（8）（9）（10）是一個(gè)非線性項(xiàng)，沒有明確的分析表達(dá)。顯然，模糊PID控制可視為常規(guī)PID的非線性補(bǔ)償。常規(guī)PID控制部分是UPID(s)，非線性補(bǔ)償部分是UN(s)?；谀？刂频哪：齈ID整定。如果我們考慮非線性補(bǔ)償U(kuò)N(s)作為一個(gè)過程的干擾，并設(shè)置為Gf(s)如圖3，基于?？刂频哪：齈ID控制器可簡化如下：(11)因此，為可以分解為=，其中(12)從而得到（13）模糊PID在第k水平上的帶寬可以通過適合的來控制。帶寬和快速的反應(yīng)，的值越小可得到較大的帶寬和較快的響應(yīng)速度，否則帶寬變小，響應(yīng)緩慢，因此，為了提高上升時(shí)間，的值應(yīng)該小，所以，兩個(gè)參數(shù)和可得到確定。備注：模糊PID控制實(shí)際上是一個(gè)傳統(tǒng)PID控制器uPID加上滑動(dòng)控制δ。由于滑?？刂剖且环N魯棒控制所以模糊PID控制是力的比傳統(tǒng)的PID控制有更好的魯棒性。4控制仿真在這一節(jié)中，通過上述方法進(jìn)行模糊PID整定的控制性能與常規(guī)PID的比較，選擇IEA和ITAE作為標(biāo)準(zhǔn)，數(shù)值越小意味著控制性能越好。（14）在所有控制仿真中常規(guī)PID控制參數(shù)是由模控制方法決定的，模糊PID控制參數(shù)是由上述整定方法確定的。例1考慮一個(gè)工業(yè)過程，所描述的一階延遲環(huán)節(jié)，模型函數(shù)如下：（15）線性部分在過程中占主導(dǎo)地位。小延遲時(shí)間意味著弱非線性特性。由圖5可以看出，由于延遲時(shí)間小，常規(guī)PID控制和模糊PID控制差異不大。然而，當(dāng)延遲時(shí)間增加至L=0.6，如圖6，模糊PID控制實(shí)現(xiàn)了優(yōu)于常規(guī)PID控制控制性能。此外模糊PID控制器增益低于常規(guī)PID控制器。圖4例1中模糊PID控制（實(shí)線）和常規(guī)圖5延遲時(shí)間增加至L=0.6，模糊PID控PID控制（虛線）性能比較制（實(shí)線）和常規(guī)PID控制（虛線）性能比較例2假設(shè)一工業(yè)過成描述如下：（16）其中a=1，假設(shè)不存在建模誤差，在階躍響應(yīng)和奈奎斯特工業(yè)過程曲線基礎(chǔ)上可獲得逼近模型如下：（17）如圖7所示，常規(guī)PID控制和模糊PID控制差異不大。因?yàn)樵撃Ｐ褪钦_的。但是，假設(shè)有建模誤差和參數(shù)a的實(shí)際值是0.95。如圖8，模糊PID控制比常規(guī)PID控制實(shí)現(xiàn)更好的控制性能。此外，由圖8可以看出模糊PID控制器增益低于常規(guī)PID控制器。

圖6a=1時(shí)，模糊PID控制（實(shí)線）和常規(guī)圖7a=0.95時(shí)，模糊PID控制（實(shí)線）和常規(guī)PID控制（虛線）性能比較PID控制（虛線）性能比較5結(jié)論本文介紹了一種基于?？刂频哪：齈ID控制器的整定分析方法。解析模型是第一次應(yīng)用于模糊PID控制器的整定。分析模型包括一個(gè)線性PID控制及非線性補(bǔ)償部分。在?？刂品椒ɑA(chǔ)上，模糊PID控制器的參數(shù)可由過程干擾的補(bǔ)償部分來分析確定。雖然擴(kuò)大收益和是耦合的，這一程序是在解耦基礎(chǔ)上的滑動(dòng)模型控制。穩(wěn)定性分析表明，該控制系統(tǒng)是全局漸近穩(wěn)定的。

模糊PID控制器采用此種整定方法比傳統(tǒng)的PID控制器有更的魯棒性強(qiáng)大。仿真結(jié)果表明，模糊PID控制器通過此種整定方法，與傳統(tǒng)的PID控制器相比在動(dòng)態(tài)和靜態(tài)上都實(shí)現(xiàn)更好的控制性能和更好的魯棒性。參考文獻(xiàn)(1)Sugeno.M.IndustrialApplicationsofFuzzyControl;Elsevier:Amsterdam,TheNetherlands,1985.(2)Manel,A.;Albert,A.;Jordi,A.;Manel,P.WastewaterNeutralization.ControlBasedonFuzzyLogic:ExperimentalResults.Ind.Eng.Chem.Res.1999,38,2709–2719.(3)Zhang,J.ANonlinearGainSchedulingControlStrategyBasedonNeuro-fuzzyNetworks.Ind.Eng.Chem.Res.2001,40,3164–3170.(4)Hojjati,H.;Sheikhzadeh,M.;Rohani,S.ControlofSupersaturationinaSemibatchAntisolventCrystallizationProcessUsingaFuzzyLogicController.Ind.Eng.Chem.Res.2007,46,1232–1240.(5)George,K.I.M.;Hu,B.G.;Raymond,G.G.AnalysisofDirectActionFuzzyPIDControllerStructures.IEEETrans.Syst.,Man,Cybernetics,PartB1999,29(3),371–388.(6)Li,H.X.;Gatland,H.ConventionalFuzzyLogicControlandItsEnhancement.IEEETrans.Syst.,Man,Cybernetics1996,26(10),791–797.(7)George,K.I.M.;Hu,B.G.;Raymond,G.G.Two-LevelTuningofFuzzyPIDCotrollers.IEEETrans.Syst.,Man,Cybernetics,PartB2001,31(2),263–269.(8)Woo,Z.W.;Chung,H.Y.;Lin,J.J.APIDTypeFuzzyControllerwithSelf-TuningScalingFactors.FuzzySetsSyst.2000,115,321–326.(9)Vega,P.;Prada,C.;Aleixander,V.Self-TuningPredictivePIDController.IEEPro.D1991,(3),303–311.(10)Rajani,K.M.;Nikhil,R.P.ARobustSelf-TuningSchemeforPIandPD-typeFuzzyControllers.IEEETtrans.FuzzySyst.1999,7(1),2–16.(11)Rajani,K.M.;Nikhil,R.P.ASelf-TuningFuzzyPIController.FuzzySetsSyst.2000,115,327–338.(12)Yesil,E.;Guzelkaya,M.;Eksin,I.SelfTuningFuzzyPIDTypeLoadandFrequencyController.EnergyConVers.Manage.2004,45,377–390.(13)Xu,J.X.;Pok,Y.M.;Liu,C.;Hang,C.C.TuningandAnalysisofaFuzzyPIControllerBasedonGainandPhaseMargins.IEEETrans.Syst.,Man,Cybernetics,PartA1998,28(5),685–691.(14)Xu,J.X.;Hang,C.C.;Liu,C.ParallelStructureandTuningofaFuzzyPIDController.Automatica2000,36,673–684.(15)Kaya,I.ObtainingControllerParametersforaNewPI-PDSmithPredictorUsingAutotuning.J.ProcessControl2000,13,465–472.(16)Li,Y.;Kiam,H.A.;Gregory,C.Y.Patents,Software,andHardwareforPIDControl.IEEEControlSyst.Mag.2006,42–54.(17)Cha,S.Y.;Chun,D.W.;Lee,J.t.Two-StepIMC-PIDMethodforMultiloopControlSystemDesign.Ind.Eng.Chem.Res.2002,41,3037–3041.(18)Li,H.X.;Gatland,H.B.;Green,A.W.FuzzyVariableStructureControl.IEEETrans.Syst.,Man,Cybernetics,PartB1997,27(2),306–312.EffectiveTuningMethodforFuzzyPIDwithInternalModelControlXiao-GangDuan,Han-XiongLi,andHuaDengSchoolofMechanicalandElectricalEngineering,CentralSouthUniVersity,Changsha410083,China,andDepartmentofManufacturingEngineeringandEngineeringManagement,CityUniVersityofHongKong,HongKongAninternalmodelcontrol(IMC)basedtuningmethodisproposedtoautotunethefuzzyproportionalintegralderivative(PID)controllerinthispaper.AnanalyticalmodelofthefuzzyPIDcontrollerisfirstderived,whichconsistsofalinearPIDcontrollerandanonlinearcompensationitem.Thenonlinearcompensationitemcanbeconsideredasaprocessdisturbance,andthenparametersofthefuzzyPIDcontrollercanbeanalyticallydeterminedonthebasisoftheIMCstructure.ThestabilityofthefuzzyPIDcontrolsystemisanalyzedusingtheLyapunovstabilitytheory.Thesimulationresultsdemonstratetheeffectivenessoftheproposedtuningmethod.1.IntroductionGenerallyspeaking,conventionalproportionalintegralderivative(PID)controllersmaynotperformwellforthecomplexprocess,suchasthehigh-orderandtimedelaysystems.Underthiscomplexenvironment,itiswell-knownthatthefuzzycontrollercanhaveabetterperformanceduetoitsinherentrobustness.Thus,overthepastthreedecades,fuzzycontrollers,especially,fuzzyPIDcontrollershavebeenwidelyusedforindustrialprocessesduetotheirheuristicnaturesassociatedwithsimplicityandeffectivenessforbothlinearandnonlinearsystems.1-4TherearetoomanyvariationsoffuzzyPIDcontrollers,suchas,one-input,two-input,andthree-inputPIDtypefuzzycontrollers.Ingeneral,thereisnostandardbenchmark.Theone-inputmaymissmoreinformationonthederivativeaction,andthethree-inputfuzzyPIDcontrollersmaycauseexponentialgrowthofrules.Thetwo-inputfuzzyPID,asweusedinthepaper,hasaproperstructureandthemostpracticaluse,andthusisthemostpopulartypeoffuzzyPIDusedinvariousresearchandapplication.DespitethefactthatindustryshowsgreaterandgreaterinterestintheapplicationsoffuzzyPID,itisstillahighlycontroversialtopicfromthepointofviewofthemainstreamcontrolengineeringcommunity.OnereasonisthatthefundamentaltheoryfortheanalyticaltuningmethodsoffuzzyPIDisstillmissing.Therefore,fuzzyPIDcontrollershadtobetunedqualitativelybytwo-leveltuning.Atalowerlevel,thetuningisperformedbyadjustingthescalinggainstoobtainoveralllinearcontrolperformance.Atahigherlevel,thetuningisperformedbychangingtheknowledgebaseparameterstoenhancethecontrolperformance.However,itisdifficulttotunetheknowledgebaseparameters.Moreover,itishardtoimprovethetransientresponsebychangingthememberfunction.Astheknowledgebaseconveysageneralcontrolpolicy,itispreferredtokeepthememberfunctionunchangedandtoleavethedesignandtuningexercisestoscalinggains.However,thetuningmechanismofscalingfactorsandthestabilityanalysisarestilldifficulttasksduetothecomplexityofthenonlinearcontrolsurfacethatisgeneratedbyfuzzyPIDcontrollers.Ifthenonlinearitycanbesuitablyutilized,fuzzyPIDcontrollersmayposethepotentialtoachievebettersystemperformancethanconventionalPIDcontrollers.Somenonanalyticaltuningmethodswereintroduced.9-12Althoughthenonlinearitywasconsideredonthebasisofgainmarginandphasemarginspecifications,thefuzzyPIDcontrollermayproducehighergainsthanconventionalPIDcontrollersduetothenonlinearfactor.Ahighgaincoulddeterioratethestabilityofthecontrolsystem.15TheconventionalPIDcontrolleriseasytoimplement,andlotsoftuningrulesareavailabletocoverawiderangeofprocessspecifications.AmongtuningmethodsoftheconventionalPIDcontroller,theinternalmodelcontrol(IMC)basedtuningisoneofthepopularmethodsincommercialPIDsoftwarepackagesbecauseonlyonetuningparameterisrequiredandbettersetpointresponsecanbeproduced.17AnanalyticaltuningmethodbasedonIMCtotunefuzzyPIDcontrollersisproposedinthispaper.ThefuzzyPIDcontrollerisfirstdecomposedasalinearPIDcontrollerplusanonlinearcompensationitem.Whenthenonlinearcompensationitemisapproximatedasaprocessdisturbance,thefuzzyPIDscalingparameterscanthenbeanalyticallydesignedusingtheIMCscheme.ThestabilityanalysisofthefuzzyPIDcontrollersisgivenonthebasisoftheLyapunovstabilitytheory.Finally,theeffectivenessofthetuningmethodologyisdemonstratedbysimulations.2ProblemFormulation2.1ConventionalPIDControllerTheconventionalPIDcontrollerisoftendescribedbythefollowingequation:20,21=（1）whereeisthetrackingerror,KPistheproportionalgain,KIistheintegralgain,KDisthederivativegain,andTiandTdaretheintegraltimeconstantandthederivativetimeconstant,respectively.TherelationshipsbetweenthesecontrolparametersareKI=KP/TiandKD=KPTd.ThetransferfunctionofthePIDcontroller(1)canbeexpressedasfollows:（2）Ontheroot-locusplane,thePIDcontrollerhastwozerostiandtd,andonepoleattheorigin.TheconditiontohaverealzerosisthatTi＞4Td.CCP+udey+_y~~r_Figure1IMCconfiguration（a）PPre_+udyFigure2IMCconfiguration(b)2.2PrincipleofIMCThebasicIMCprincipleisshowninFigure1a,wherePistheplant,P?isanominalmodeloftheplant,Cisacontroller;randdarethesetpointandthedisturbance,andyandykaretheoutputsoftheplantanditsnominalmodel,respectively.TheIMCstructureisequivalenttotheclassicalsingle-loopfeedbackcontrollershowninFigure1b.Ifthesingle-loopcontrollerCIMCisgivenby （3）with（4）whereP?(s)=P?-(s)P?+(s),P?-(s)istheminimumphasepartoftheplantmodel,P?+(s)containsanytimedelaysandright-halfplanezeros,andf(s)isalow-passfilterwithasteady-stategainofone,whichtypicallyhastheform:（5）Thetuningparametertcisthedesiredclosed-looptimeconstant,andnisapositiveintegertobedetermined.KiKdRuleKiKdRuleBasesERu2.3ModelofFuzzyPIDControllerThefuzzyPIDcontroller,asshowninFigure2,isdescribedasfollows:(6)withγisanonlineartimevaryingparameter(),AandBarehalfofthespreadofeachinputandoutmemberfunction,respectively.ThefuzzyPIDcontrolactuallyhastwolevelsofgains.6Thescalinggains(Ke,Kd,K0,andK1)areatthelowerlevel.ThetuningofthesescalinggainswillaffectthegainsoffuzzyPIDThefuzzyPIDcontrolactuallyhastwolevelsofgains.6Thescalinggains(Ke,Kd,K0,andK1)areatthelowerlevel.ThetuningofthesescalinggainswillaffectthegainsoffuzzyPIDcontrollers,resultinginthechangingofthecontrolperformance.Asthecontrolactionsarefuzzilycoupled,thecontributionofeachKe,Kd,K0,andK1todifferentcontrolactionsisstillnotveryclear,whichmakesthepracticaldesignandtuningprocessratherdifficult.3TuningFuzzyPIDBasedontheIMCTotunethefuzzyPIDcontrollerbasedontheIMCmethod,ananalyticalmodelofthefuzzyPIDcontrollerisobtainedfirstbysimplederivation.Then,theparametersofthefuzzyPIDcontrollercanbedeterminedonthebasisoftheIMCprinciple.Supposethatanindustrialprocesscanbemodeledbyafirstorderplusdelaytime(FOPDT)structurethathasthetransferfunctionasfollows:(7)whereK,T,andLarethesteady-stategain,thetimeconstant,andthetimedelay,respectively.Theestimationoftheseparametersusingthestepresponsemethod,frequencyresponse,andclosed-looprelayfeedback,etc.,iswell-described.TheFOPDTmodelisoneofthemostcommonandadequateonesused,especiallyintheprocesscontrolindustries.18Oneobtainsfrom(6):（8）（9）（10）withδ(s)beinganonlineartermwithoutanexplicitanalyticalexpression.Obviously,thefuzzyPIDcontrolcanbeconsideredasaconventionalPIDwithanonlinearcompensation.TheconventionalPIDcontroltermisuPID(s)andthenonlinearcompensationisuN(s).TuningofFuzzyPIDControllerBasedonIMC.IfweconsiderthenonlinearcompensationuNasaprocessdisturbanceandsetGf(s))=CIMC(s),whichisshowninFigure3,theIMCbasedtuningforfuzzyPIDcontrollerscanbesimplifiedasfollows.Bythefirst-orderPade′approximation,thedelaytimeisapproximatedasfollows:(11)Therefore,theP?(s)canbefactorizedasP?(s))P?+(s)P?-(s),其中(12)Wecanachieve（13）ThebandwidthofthefuzzyPIDatthekthlevelcanbecontrolledbyadjustingR.AsmallvalueofRgiveswidebandwidthandfastresponse;otherwise,itgivesalowbandwidthandsluggishresponse.Toimprovetherisetime,thevalueofRshouldbesmall.Therefore,thetwoparametersandcanbedetermined.Remark:Thefuzzy-PIDcontrol(11)isactuallyaconventionalPIDcontroluPIDplusapseudo-slidingmodecontrolδ.Becausetheslidingmodecontrolisarobustcontrol,thefuzzyPIDcontrolismorerobustthanaconventionalPIDcontrol.4ControlSimulationsInthissection,thecontrolperformanceoffuzzyPIDtunedbytheproposedmethodiscomparedwiththatofconventionalPIDcontrol.QuantitativecriteriaformeasuringtheperformancearechosenasIAEandITAE.Smallernumbersimplybetterperformance.(14)Inallcontrolsimulations,parametersofconventionalPIDcontrolaredeterminedbyIMC-basedmethodandtheparametersoffuzzyPIDcontrolaredeterminedbytheproposedtuningmethod.Example1.Consideranindustrialprocessthatisapproximatelydescribedbyafirst-orderrationaltransferfunctionmodelwithadelaytimeasfollows:(15)Thelinearpartisthedominantprocess.Thesmalldelaytimeimpliesweaknonlinearfeatures.AsshowninFigure5,littledifferenceisobservedbetweentheconventionalPIDcontrolandfuzzyPIDcontrolduetothesmalldelaytime.However,whenthedelaytimeisincreasedtoL=0.6,therewillbelargemodelerrorcausedbyapproximatingthedelaytimewithafirst-orderPade′approximationin(15).AsshowninFigure6,fuzzyPIDcontrolachievesbettercontrolperformancethanconventionalPIDcontrol.Morever,thegainofthefuzzyPIDcontrollerislowerthanthatofconventionalPIDcontroller.Figure4ControlperformanceoffuzzyPIDFigure5PerformanceoffuzzyPIDandPIDandPIDforexample1,fuzzyPID(solidline),fordelayL=0.6,fuzzyPID(solidline),andconventionalPID(dottedline).andconventionalPID(dottedline).Example2.Assumethatanindustrialprocessisdescribedby(16)wherea=1,Supposethatthereisnomodelingerrorintheprocess.OnthebasisofstepresponseandNyquistcurvesoftheindustrialprocess,theapproximationmodelcanbeobtainedasfollows:(17)AsshowninFigure7,littledifferenceisobservedbetweentheconventionalPIDcontrolandfuzzyPIDcontrolbecausethemodelisaccurate.However,supposethatthereismodelingerrorandthepracticalvalueoftheparameterais0.95..AsshowninFigure8,fuzzyPIDcontrolachievesbettercontrolperformancethanconventionalPIDcontrol.Morever,thegainofthefuzzyPIDcontrollerislowerthanthatoftheconventionalPIDcontroller,whichisshowninFigure8.Figure6.ControlperformanceoffuzzyPIDFigure7.ControlperformanceoffuzzyPIDandPIDandPIDfora)1.FuzzyPID(solidline)andforprocessa=0.95.FuzzyPIDconventionalPID(dottedline).(solidline)andconventionalPID(dottedline)5ConclusionAneffectivetuningmethodforfuzzyPIDcontrollersbasedonIMCispresentedinthispaper.AnanalyticalmodelisfirstdevelopedforthetuningoffuzzyPIDcontrollers.TheanalyticalmodelincludesalinearPIDcontrolandanonlinearcompensationitem.OnthebasisoftheIMCmethod,theparametersoffuzzyPIDcontrollercanbeanalyticallydeterminedbyregardingthecompensationitemasaprocessdisturbance.Althoughthescalinggainsandarecoupled,aprocedureisusedtodecouplethemonthebasisoftheslidingmodecontrol.Thestabilityanalysisshowsthatthecontrolsystemisgloballyasymptoticallystable.FuzzyPIDcontrollerstunedbytheproposedmethodaremorerobustthantheconventionalPIDcontroller.ThesimulationresultsshowthatfuzzyPIDcontrollerstunedbytheproposedmethodachievebettercontrolperformanceinboththetransientandsteadystatesandaremorerobustthanconventionalPIDcontrollers.LiteratureCited(1)Sugeno.M.IndustrialApplicationsofFuzzyControl;Elsevier:Amsterdam,TheNetherlands,1985.(2)Manel,A.;Albert,A.;Jordi,A.;Manel,P.WastewaterNeutralizationControlBasedonFuzzyLogic:ExperimentalResults.Ind.Eng.Chem.