自適應(yīng)系統(tǒng)(中文)

自適應(yīng)控制

王振雷Email:wangzl75@1什么是自動(dòng)控制？2控制理論的由來(lái)

機(jī)械化與自動(dòng)化瓦特蒸汽機(jī)的速度調(diào)節(jié)系統(tǒng)（紐克曼）

抽水馬桶的自動(dòng)送水系統(tǒng)3反饋設(shè)定目標(biāo)→測(cè)量狀態(tài)→計(jì)算偏差→確定調(diào)整量

實(shí)際執(zhí)行如果所有的事情都是在瞬時(shí)里同時(shí)發(fā)生的，那這個(gè)反饋過(guò)程就無(wú)法工作開(kāi)環(huán)與閉環(huán)——洗衣機(jī)與空調(diào)4控制系統(tǒng)設(shè)計(jì)方法

不基于模型的（偏差或偏差變化）——PID，開(kāi)關(guān)控制（空調(diào)），模糊控制，魯棒控制基于模型的控制——模型預(yù)測(cè)控制，狀態(tài)空間設(shè)計(jì)（卡爾曼），最優(yōu)控制，自適應(yīng)控制，滑模變結(jié)構(gòu)控制5第一部分什么是自適應(yīng)控制簡(jiǎn)介過(guò)程變化的影響自適應(yīng)結(jié)構(gòu)特點(diǎn)自適應(yīng)控制面臨的問(wèn)題小結(jié)6自動(dòng)控制發(fā)展歷程1961討論會(huì):基于自適應(yīng)的觀點(diǎn)進(jìn)行系統(tǒng)設(shè)計(jì)IEEE委員會(huì)1973:自組織控制（Self-organizingcontrol）參數(shù)自適應(yīng)SOC性能自適應(yīng)SOC學(xué)習(xí)控制系統(tǒng)（Learningcontrolsystem）實(shí)用性定義：一類特殊的非線性控制系統(tǒng)-自適應(yīng)控制器是一類帶有可調(diào)整參數(shù)和參數(shù)整定結(jié)構(gòu)的控制器簡(jiǎn)介7簡(jiǎn)介自適應(yīng)系統(tǒng)功能圖81950S:自動(dòng)駕駛儀；增益規(guī)劃（理論局限）1960S:Bellman提出動(dòng)態(tài)規(guī)劃;Feldbaum的雙重控制;1970S:系統(tǒng)辨識(shí);MRAS算法;STR應(yīng)用,（理論尚欠缺）;1970S后期至1980S初:穩(wěn)定性證明；自適應(yīng)控制的魯棒性；1980S后期至1990S初:非線性系統(tǒng)理論的發(fā)展加深了人們對(duì)自適應(yīng)控制的理解；計(jì)算機(jī)科學(xué)中學(xué)習(xí)的思想與自適應(yīng)控制也存在密切的聯(lián)系。主要發(fā)展階段9商業(yè)化的自適應(yīng)軟件，例如：ElectromaxV（Leeps&Northrup);ECA40（Sattcontrol）；DPR900(FisherControl)等等控制器自整定技術(shù)應(yīng)用領(lǐng)域10過(guò)程變化對(duì)系統(tǒng)的影響執(zhí)行器的非線性性能：閥門特性：11系統(tǒng)參數(shù):

K=0.15;Ti=1;G0(s)=1/(s+1)3STEPRESPONSESFORPICONTROL不同操作水平的輸出響應(yīng)曲線過(guò)程變化對(duì)系統(tǒng)的影響12濃度控制流速變化的影響13系統(tǒng)初始參數(shù):

K=0.5;Ti=1.1;q=1;τ=1Closed-loopsystemdiagram過(guò)程變化對(duì)系統(tǒng)的影響14不同流速時(shí)的階躍響應(yīng)曲線:

Referenceinputcrq=0.5q=0.9q=1.1q=2Controlsignalcinq=0.5q=0.9q=1.1q=2過(guò)程變化對(duì)系統(tǒng)的影響15常見(jiàn)的自適應(yīng)結(jié)構(gòu)GainSchedulingModel-ReferenceAdaptiveControl(MRAS)Self-tuningRegulator(STR)DualControl16I.增益規(guī)劃ExampleofschedulingvariablesProductionrateMachnumberanddynamicpressure17II.模型參考自適應(yīng)控制（MRAS）Linearfeedbackfrome=y-ymisnotadequateforparameteradjustment!AdaptiveLaw:18III.自校正調(diào)節(jié)器(STR)DirectAdaptiveControlIndirectAdaptiveControl19III.自校正調(diào)節(jié)器(STR)CertaintyEquivalencePrincipleParameterestimationGradientmethodsLeastsquaresControldesignmethodsPIDPoleplacementLQG20IV.雙重控制ConceptuallyveryinterestingUnfortunatelyverycomplicated21ProcessDescriptionsContinuoustimedomain

DiscretetimedomainControllerstructure

DirectAdaptiveControl

IndirectAdaptivecontrolConstructionofanAdaptiveControlSteps1)Characterizethedesiredbehavioroftheclosed-loopsystem2)Determineasuitablecontrollawwithadjustableparameters3)Findamechanismforadjustingtheparameters4)Implementthecontrollaw

自適應(yīng)控制系統(tǒng)設(shè)計(jì)22應(yīng)用領(lǐng)域

自整定

增益規(guī)劃

連續(xù)自適應(yīng)Proceduretodecidewhattypeofcontrollertouse23

自適應(yīng)控制主要用來(lái)處理：--過(guò)程動(dòng)態(tài)變化--擾動(dòng)信號(hào)變化

自適應(yīng)系統(tǒng)是非線性系統(tǒng)

原理很好理解

來(lái)源于工業(yè)過(guò)程，應(yīng)用于工業(yè)過(guò)程

常見(jiàn)應(yīng)用----自整定,增益規(guī)劃尚未解決的問(wèn)題--系統(tǒng)特性的理論方法--自適應(yīng)調(diào)整因子的確定小結(jié)241.簡(jiǎn)介2.最小二乘方法和回歸3.系統(tǒng)動(dòng)態(tài)4.輸入信號(hào)條件5.范例6.結(jié)論第二講

實(shí)時(shí)參數(shù)估計(jì)提綱251、如何獲得過(guò)程模型?

機(jī)理模型(白箱模型)

實(shí)驗(yàn)?zāi)Ｐ?黑箱模型)

混和模型(灰箱模型)2、實(shí)驗(yàn)設(shè)計(jì)——激勵(lì)信號(hào)選擇3、選擇合適的模型結(jié)構(gòu)

傳遞函數(shù)

脈沖響應(yīng)模型

狀態(tài)空間模型4、參數(shù)估計(jì)——最小二乘方法5、校驗(yàn)系統(tǒng)辨識(shí)26最小二乘方法和回歸模型問(wèn)題來(lái)源:行星和小行星的軌道計(jì)算解決問(wèn)題的人:KarlFriedrichGauss

最小二乘方法的原理:“構(gòu)建一個(gè)數(shù)學(xué)模型，使得觀測(cè)數(shù)據(jù)和模型計(jì)算值之間的偏差的平方和最小"

最小二乘方法被廣泛應(yīng)用于多個(gè)領(lǐng)域。27

回歸模型:問(wèn)題的數(shù)學(xué)表達(dá)：28符號(hào)說(shuō)明:問(wèn)題的數(shù)學(xué)表達(dá)：29求取θ使下式達(dá)到最小值:LS問(wèn)題求解損失函數(shù)!30下列正則方程可以使損失函數(shù)取最小值：如果矩陣A是非奇異矩陣，則參數(shù)θ的最小值存在并唯一：LS問(wèn)題求解關(guān)于有偏估計(jì)和無(wú)偏估計(jì)的討論－－增廣最小二乘估計(jì)31范例32范例33范例34如果使用過(guò)于復(fù)雜的模型，則會(huì)出現(xiàn)過(guò)擬合現(xiàn)象（Overfitting）!范例35最小二乘法的幾何解釋36在什么時(shí)候E會(huì)最小?正交!最小二乘法的幾何解釋找出向量的線性組合系數(shù)使向量的線性組合盡量逼近Y。

當(dāng)向量線性獨(dú)立時(shí)，系數(shù)值唯一。37遞歸最小二乘算法新的數(shù)據(jù)源到來(lái)時(shí)，常規(guī)的LS方法需要重新對(duì)所有數(shù)據(jù)進(jìn)行回歸運(yùn)算，運(yùn)算量增長(zhǎng)很快遞歸循環(huán)計(jì)算.

和之間是否可以找到遞歸公式?38遞歸最小二乘算法39時(shí)變參數(shù)（突變和緩變）帶“折扣”的損失函數(shù)遞歸LS形式演化為：遺忘因子40連續(xù)模型帶“折扣”的損失函數(shù)正則方程41連續(xù)模型回歸模型遞歸方程42動(dòng)態(tài)系統(tǒng)參數(shù)估計(jì)基本思路:把相關(guān)方程表示成回歸模型形式!動(dòng)態(tài)系統(tǒng)

有限脈沖響應(yīng)（FIR）模型

連續(xù)模型

非線性模型實(shí)驗(yàn)條件

激勵(lì)

閉環(huán)辨識(shí)43有限脈沖響應(yīng)模型采用回歸模型：44有限脈沖響應(yīng)模型45ARMA模型模型形式：寫(xiě)作：其中46傳遞函數(shù)模型模型寫(xiě)作：47F(p)是極點(diǎn)個(gè)數(shù)超過(guò)

n的穩(wěn)定傳遞函數(shù)。傳遞函數(shù)模型也是一個(gè)遞歸模型形式！48非線性模型考慮下列模型線性參數(shù)!49實(shí)驗(yàn)條件激勵(lì)---

激勵(lì)信號(hào)性質(zhì)非常關(guān)鍵!閉環(huán)辨識(shí)

由于反饋的存在使辨識(shí)變得更加復(fù)雜!50激勵(lì)條件（FIR）滿秩!51持續(xù)激勵(lì)定義：如果上式的極限存在并且矩陣Cn是正定矩陣，則信號(hào)u稱為n階持續(xù)激勵(lì)信號(hào)。52反饋帶來(lái)的辨識(shí)損失在自適應(yīng)控制中，系統(tǒng)辨識(shí)經(jīng)常在閉環(huán)條件下進(jìn)行，這增加了了辨識(shí)難度。下面估計(jì)ARMA模型參數(shù)的例子說(shuō)明了這一問(wèn)題：53考慮下列閉環(huán)系統(tǒng):反饋帶來(lái)的辨識(shí)損失（1）（2）把（2）式兩端乘以α，并加到（1）式兩端，變換后，式（1）變?yōu)椋簼M足下列關(guān)系的參數(shù)可以得到相同的輸入輸出關(guān)系：54反饋帶來(lái)的辨識(shí)損失55范例考慮下列模型：過(guò)程參數(shù):56輸入：范例57范例----有色噪聲考慮下列模型:模型形式:58范例輸入：59小結(jié)需要掌握的內(nèi)容:

遞歸形式---RLS

激勵(lì)的作用在自適應(yīng)控制中的作用

遞歸估計(jì)在自適應(yīng)控制中的關(guān)鍵部分

遞歸最小二乘是一種非常有用的方法60第三講確定系統(tǒng)的自校正調(diào)節(jié)器1.基本思想2.極點(diǎn)配置設(shè)計(jì)方法3.間接型STR4.直接型STR5.結(jié)論61基本思想自動(dòng)參數(shù)估計(jì)和控制器設(shè)計(jì)62基本思想分離原理估計(jì)參數(shù)將未知參數(shù)估計(jì)與控制器設(shè)計(jì)分開(kāi)獨(dú)立進(jìn)行，采用遞推方法在線估計(jì)未知參數(shù)，估計(jì)出的參數(shù)就看成真實(shí)參數(shù)而不考慮誤差，然后設(shè)計(jì)控制器

（確定性等價(jià)原理）——間接自適應(yīng)控制方法將估計(jì)模型按照控制器參數(shù)重新參數(shù)化，則不需要估計(jì)過(guò)程參數(shù)而直接估計(jì)控制器參數(shù)——直接自校正控制63極點(diǎn)配置設(shè)計(jì)算法過(guò)程模型因果律條件范例極點(diǎn)多項(xiàng)式Ao的理解總結(jié)64過(guò)程模型或者相對(duì)階連續(xù)系統(tǒng)模型65閉環(huán)系統(tǒng)66閉環(huán)系統(tǒng)閉環(huán)系統(tǒng)表述：閉環(huán)特征多項(xiàng)式丟番圖方程!如果多項(xiàng)式A和B沒(méi)有公因數(shù)，則該方程有解！67模型變換閉環(huán)響應(yīng)方程滿足下列方程可以實(shí)現(xiàn)最有效的跟蹤目標(biāo)響應(yīng)方程68模型變換因此?69因果律因此控制器丟番圖方程有解：已知degA>degB70因果律滿足條件等式2端同時(shí)加上deg(B-),則上式變?yōu)椋阂蚬P(guān)系式可以寫(xiě)作:最小階極點(diǎn)配置(blackboad)71Example3.1帶有零點(diǎn)對(duì)消采樣周期

h=0.5考慮下列過(guò)程:72Example3.1帶零點(diǎn)對(duì)消選擇

Ao

=1:73Example3.2–無(wú)零點(diǎn)對(duì)消選擇

Ao

=1:丟番圖方程:74Example3.2–無(wú)零點(diǎn)對(duì)消令q=-b1/b0

可以解出

r1其中

進(jìn)而可以求得s0和s1。75總結(jié)極點(diǎn)配置方法容易使用

采樣周期的選擇至關(guān)重要對(duì)于高階系統(tǒng)應(yīng)用存在困難（太多極點(diǎn)需要配置）76間接STR

參數(shù)估計(jì)

算法介紹

范例77參數(shù)估計(jì)過(guò)程模型(不含擾動(dòng)信號(hào)!)引入：則變?yōu)榛貧w模型:78參數(shù)估計(jì)通過(guò)下列方程進(jìn)行參數(shù)估計(jì)計(jì)算79間接STR算法流程步驟:80范例–帶有零點(diǎn)對(duì)消過(guò)程模型：控制律：81仿真結(jié)果82仿真結(jié)果83范例–不帶零點(diǎn)抵消過(guò)程模型控制律84仿真結(jié)果85仿真結(jié)果86帶負(fù)載干擾

有擾動(dòng)時(shí)系統(tǒng)響應(yīng)變化?

設(shè)想一下會(huì)發(fā)生什么?

如何改進(jìn)系統(tǒng)？87SimulationCompareExample3.588SimulationCompareExample3.589Whathappens?Whathappenstoestimator?ProcessmodelActualmodelStructureofthecontrollawNointegralaction!Noticewhattheadaptivecontrollerdoes!ParametersdonotsettleNosteadystateerrorifsetpointconstant90HowtoDealwithDisturbancesCharacterizedisturbancesasfiltersdriven

by

e

Theonlythingthatmattersisthepoles

ofthefilter!Thinkaboutthisasthe

disturbanceannihilatorwhichwipesout

asmuchaspossibleofthedisturbance!Animpulse(pulse)Sequencesofimpulses(pulses)WhitenoiseHowwillthedisturbanceinfluencethesystem?Howshouldthecontrolsystembemodified?91ModifiedControlDesignProcessmodelDisturbancemodelHenceTheeffectofthedisturbancescanbereducedbyrequiringthatAdisafactorofR!!92DirectSTRWhy?

Avoiddesigncalculationtime-consumingTheIdeaMinimum-phaseSystemsNon-minimum-phasesystemsExamples93TheIdeaProcessmodelDesiredresponseDiophantineequationLetthisoperateony(t)!ButHenceEstimateparametersinthisequationinstead.94MinimumPhaseSystemsConsidertheequationMinimumphasesystem

!!NaturaltochooseIntroduceparametervectorandregressionvectorHence95AnAlternativeAsbeforeFiltersignalssothatwecantakederivativesIntroduceparametervector96AnAlternativeandregressionvectorThemodelthenbecomesAstandardregressionmodel!!97AndirectSTRDesignStepWhere98ExampleWithdegAm=2andd0=1wehavedegA0=d0-1=0.FilteringRegressionmodelControlLawwhere99Simulation(d0=1)100Whathappenswithd0>1?101Whathappenswithd0>1?102Non-minimum-Phase(NMP)SystemsThecaseinwhichprocesszeroscan’tbecanceledwillnowbediscussed.Considerthetransformedprocessmodel:whereIntroduce:Theequationcanbewrittenas:103Directself-tuningregulatorforNMPsystemsData:GivenspecificationsintermsofAm,BmandA0andtherelativedegreed0ofthesystem.Step1:EstimatethecoefficientsofthepolynomialsRandS.Step2:CancelpossiblecommonfactorsinRandS

toobtainRandS.Step3:CalculatethecontrolsignalfromcontrollawequationwhereRandSarethoseobtainedinStep2andTisgivenby.RepeatStep1,2and3ateachsamplingperiod.104AmethodtoestimatethepolynomialTNoticethat:Theerrore=y-ym

canbewrittenas:105MixedDirectandIndirectAlgorithmsByBlackboad106Conclusion107Lecture4Model-ReferenceAdaptiveSystems1.Theidea2.TheMITRule3.Determinationoftheadaptivegain4.Conclusions108Introduction109IntroductionFlightcontrolinthe1950sTwoideasPhilWhitakerMIT–TheReferenceModel–ParameteradjustmentruleModifiedadjustmentrules110TheMITRuleTrackingerrorIntroduceChangeparameterssuchthatManyalternativesgives111AdaptionofFeedforwardGainProcessDesiredresponseControllerSensitivityderivativeMITrule112AdaptionofFeedforwardGainBlockDiagram113AdaptionofFeedforwardGainExampleG(s)=1/(s+1)114AFirstOrderSystemProcessModelControlleridealparameters115AFirstOrderSystemTheerrorApproximateHence116AFirstOrderSystem117AFirstOrderSystemSimulationInputandoutput118AFirstOrderSystemSimulationParameters119DeterminationofAdaptationGainFeedforwardgainsystemParameterequationEq.isalineartime-varyingordinarydifferentialequation.Athoughtexperiment120DeterminationofAdaptationGainCharacteristicequationKeyparameterNormalizedAlgorithms(Eq.1)Eq.1thenbecomes121DeterminationofAdaptationGainStabilitycondition(Eq.2)Commandsignalamplitude:0.113.5122DeterminationofAdaptationGainMITrule(Eq.3)Normalizedadaptationlaw(Eq.4)123DeterminationofAdaptationGainMITrule(Eq.3)Commandsignalamplitude:0.113.5Normalizedadaptationlaw(Eq.4)124Conclusion125Lecture5AUTO-TUNING1.Introduction2.PIDControl3.Auto-tuning4.Conclusions126Introduction1.ThedrawbackofMRASorSTRPrioriinformationabouttheprocessdynamics2.PIDregulatorsareusedwidelyforindustrialautomationMaindifficultyindesigncontrolsystembasedonPID127PIDregulatorTextbookversion:Parameters:Kc

Ti

Td128PIDregulatorIndustrialversion:129Auto-tuningtechniquesTransientresponsemethodsTheZiegler-Nicholsmethod130Auto-tuningtechniquesTransientResponseMethodsThethreeparametermodela=k*L/T131Auto-tuningtechniquesTheZiegler-NicholsStepResponseMethodControllerKcTiTdP1/aPI0.9/a3LPID1.2/a2LL/2

Condition:

0.1<L/T<0.6132Auto-tuningtechniquesTheZiegler-NicholsStepResponseMethodAreamethodParametersaregivenby133Auto-tuningtechniquesDrawbacksoftheareamethod

WhetherasteadystatehasbeenreachedTheamplitudeofstepsignalisdifficulttoselectDisturbances134Auto-tuningtechniquesManyprocesseshavelimitcycleoscillationsunderrelayfeedback.RelayFeedback135Auto-tuningtechniquesRelayFeedbackyudaTu136Auto-tuningtechniquesZ-NClosed-LoopMethodControllerKcTiTdP0.5KuPI0.4Ku0.8TuPID0.6Ku0.5Tu0.12Tu137Auto-tuningtechniquesZ-NClosed-LoopMethodTherewillnotbeuniquelimitcycleoscillationsforanarbitrarytransferfunctionPIDcontrolisnotappropriateforallprocesses138Auto-tuningtechniquesExampleAuto-tuningofcascadedtanks139Lecture6GAINSCHEDULING1.Introduction2.Theprinciple3.Designofgainschedulingcontroller4.Conclusions140IntroductionHowthedynamicsofaprocesschangewiththeoperatingconditionsoftheprocessisknownChangingthecontrollerparametersbymonitoringtheoperatingconditionsoftheprocessAccommodatechangesinprocessgainonlyTheideawasoldbuttheapplicationwaslaterGainschedulingbasedonmeasurementsofoperatingconditionsoftheprocess141ThePrincipleFindtheauxiliaryvariablesthatcorrelatewellwiththechangesinprocessdynamicsChangethecontrollerparametersasfunctionsoftheauxiliaryvariablesOriginatedinconnectionwithflightcontrolsystems-----MachnumberanddynamicpressureInprocesscontroltheproductionratecanoftenbechosenasaschedulingvariable,sincetimeconstantsandtimedelaysareofteninverselyproportionaltoproductionrate.142ThePrincipleThereisnofeedbackfromtheperformanceoftheclosed-loopsystemtothecontrollerparameters143ThePrincipleDrawbacks1.Open-loopcompensation:nofeedbacktocompensateforanincorrectschedule.2.Thedesignmaybetime-consuming:manyoperatingconditions,extensivesimulationsforperformancecheckingAdvantageThecontrollerparameterscanbechangedveryqui