采集數據模型預測控制解析

1、Sampled-DataModelPredictiveControlforNonlinearTime-VaryingSystems:StabilityandRobustnessFernandoA.C.C.Fontes,LaloMagni,EvaGyurkovicsPhD,ImperialCollegeLondonIEEETransactionsonPowerDelivery,Vol.8,No.1,Jariuary1993.KeywordbarbalatPrincipleNonlineartime-varyingsystemsStabilityandrobustnessSummary.Wedes

2、cribehereasampled-dataModelPredictiveControlframeworkthatusescontinuous-timemodelsbutthesamplingoftheactualstateoftheplantaswellasthecomputationofthecontrollaws,arecarriedoutatdiscreteinstantsoftime.Thisframeworkcanaddressaverylargeclassofsystems,nonlinear,time-varying,andnonholonomic.Asinmanyothers

3、sampled-dataModelPredictiveControlschemes,Barbalatslemmahasanimportantroleintheproofofnominalstabilityresults.ItisarguedthatthegeneralizationofBarbalatslemma,describedhere,canhavealsoasimilarroleintheproofofrobuststabilityresults,allowingalsotoaddressaverygeneralclassofnonlinear,time-varying,nonholo

4、nomicsystems,subjecttodisturbances.Thepossibilityoftheframeworktoaccommodatediscontinuousfeedbacksisessentialtoachievebothnominalstabilityandrobuststabilityforsuchgeneralclassesofsystems.1 IntroductionManyModelPredictiveControl(MPC)schemesdescribedintheliteratureusecontinuous-timemodelsandsamplethes

5、tateoftheplantatdiscreteinstantsoftime.Seee.g.3,7,9,13andalso6.Therearemanyadvantagesinconsideringacontinuous-timemodelfortheplant.Nevertheless,anyimplementableMPCschemecanonlymeasurethestateandsolveanoptimizationproblematdiscreteinstantsoftime.Inallthereferencescitedabove,Barbalatslemma,oramodifica

6、tionofit,isusedasanimportantsteptoprovestabilityoftheMPCschemes.(Barbalat-knownmnisaowerfultooltodeduceasymptoticstabilityofnonlinearsystems,especiallytime-varyingsystems,usingLyapunov-likeapproaches;seee.g.17foradiscussionandapplications).ToshowthatanMPCstrategyisstabilizing(inthenominalcase),itiss

7、hownthatifcertaindesignparameters(objectivefunction,terminalset,etc.)areconvenientlyselected,thenthevaluefunctionismonotonedecreasing.Then,applyingBarbalatslemma,attractivenessofthetrajectoryofthenominalmodelcanbeestablished(i.e.x(t)0astf?t。,(1a)=XtQXn.(lb)&XCIRforallst。*(1c)u(s)EUaef之1.(Id)Thedatao

8、fthismodelcompriseasetcontainingallpossibleinitialstatesattheinitialtimet0,avectorxt0thatisthestateoftheplantmeasuredattimet0,agivenfunctionf : IR x IR x Rm T IRnand a set U C Rof possible controlvalues.We assume this system to be asymptotically controllable on X0 and that for all t 0 f(t, 0, 0) =0.

9、 We further assume that the function f is continuous and locally Lipschitz with respect to the second argument.The construction of the feedback law is accomplished by using a sampleddata MPC strategy. Consider a sequence of sampling instants 兀:=tii 0 with a constant inter-sampling time 8 such that t

10、i+1 = ti+ Sfor all i 0. Consider also the control horizon and predictive horizon, Tc and Tp, with Tp Tc , and an auxiliary control law kaux : IR xiRn fIRm. The feedback control is obtained by repeatedly solving online open-loop optimal control problems P(ti, xti, Tc, Tp) at each sampling instant ti

11、C 怎 every time using the current measure of the state of the plant xti .P(tT xTCr 2): Mmimize*f 1/ (s,z(a)su(s)dff + lV(i + 5H，+ 4)，subject t.o:(s) =；#=：rtl 了e xg uu(#) = A氣. T) jct + 5.ae & F Jt + rp,tor all s - |t J TJ,t.fj. 3 七0 t + TJ ae s |f + /,f + Tp”(1)Notethatintheintervalt+Tc,t+Tpthecontro

12、lvalueisselectedfromasingletonandthereforetheoptimizationdecisionsareallcarriedoutintheintervalt,t+Tcwiththeexpectedbenefitsinthecomputationaltime.Thenotationadoptedhereisasfollows.Thevariabletrepresentsrealtimewhilewereservestodenotethetimevariableusedinthepredictionmodel.Thevectorxtdenotestheactua

13、lstateoftheplantmeasuredattimet.Theprocess(x,u)isapairtrajectory/controlobtainedfromthemodelofthesystem.Thetrajectoryissometimesdenotedass_fxs;t,xt,u)whenwewanttomakeexplicitthedependenceontheinitialtime,initialstate,andcontrolfunction.Thepair(x,u)denotesouroptimalsolutiontoanopen-loopoptimalcontrol

14、problem.Theprocess(x?,u?)istheclosed-looptrajectoryandcontrolresultingfromtheMPCstrategy.Wecalldesignparametersthevariablespresentintheopen-loopoptimalcontrolproblemthatarenotfromthesystemmodel(i.e.variablesweareabletochoose);thesecomprisethecontrolhorizonTgthepredictionhorizonTp,therunningcostandte

15、rminalcostsfunctionsLandW,theauxiliarycontrollawkaux,andtheterminalconstraintsetS?IRn.Theresultantcontrollawu?isasamp-lfnedbackcontrolsinceduringeachsamplinginterval,thecontrolu?isdependentonthestatex?(ti).Morepreciselytheresultingtrajectoryisgivenbyar*(0)=史率團=/(t,ar*(t)twherew*(t)=寓)：=u(t;團律*(曲)ii(

16、nandthefunctiont_f_t_givesthelastsamplinginstantbeforet,thatis比17r:=inaxtiG7F: OsmallAstabilityanalysiscanbecarriedouttoshowthatifthedesignparametersareconvenientlyselected(i.e.selectedtosatisfyacertainsufficientstabilitycondition,seee.g.7),thenacertainMPCvaluefunctionVisshowntobemonotonedecreasing.

17、Moreprecisely,forsomeenoughandforany。3/(,r(s I )(ls.where M is a continuous, radially unbounded, positive definite function. TheMPC value function V is defined asV(t.r) :=Vx)I； f / r*Tf, Tc whereis the value function for the optimal control problemthe optimal control problem defined where the horizo

18、n isshrankin its initial part by).From we can then write that for any t 書 tAf(D)ds.Sinceisfinite,weconcludethatthefunctionist-r+(nboundedandthenthatdsisalsobounded.Thereforeisboundedand,sincefiscontinuousandtakesvaluesonboundedsetsofisalsobounded.AlltheconditionstoapplyBarbalatlemma2aremet,yieldingt

19、hatthetrajectoryasymptoticallyconvergestotheorigin.NotethatthisnotionofstabilitydoesnotnecessarilyincludetheLyapunovstabilitypropertyasisusualinothernotionsofstability;see8foradiscussion.6 RobustStabilityInthelastyearsthesynthesisofrobustMPClawsisconsideredindifferentworks14.Theframeworkdescribedbel

20、owisbasedontheonein9,extendedtotimevaryingsystems.the state of the nonlinear system9(cIRTl)Ourobjectiveistodrivetoagiventargetsetsubjecttoboundeddisturbancesi(t)=ft,w(td(t)attoi(Sa)了(加)=3口WX。，(8b)rr(2)fXforallt(8e)u(t)GUttOi(8dIrf(t)Da.e.ttq?(8e)u(4Tt2|)：=(u:ha一口穆：ugc&石h聞)t0(，zD：=：h匕-IRP:d(s)ense卜*(

21、，工/,1f(f+t+TJ)f.Tr)j(rsh忒.)入+，+Tpl;K)is-Q*)Sinceisfinite,weconcludethatthefunctionbounded and then thatisf;tIiff)isalsobounded.Thereforer詼tZ,d)Jj-boundedand,sincefiscontinuousandtakesvaluesonboundedsetsofisalsobounded.Usingthefactthatx?isabsolutelycontinuousandcoincideswith?xatallt.Ii*(t)andtIT*(tsa

22、mplinginstants,wemaydeducethatarealsobounded.WeareintheconditionstoapplythepreviouslyestablishedGeneralizationofBarbalatsLemmayieldingtheassertionofthetheorem.7FiniteParameterizationsoftheControlFunctionsTheresultsonstabilityandrobuststabilitywereprovedusinganoptimalcontrolproblemwherethecontrolsare

23、functionsselectedfromaverygeneralset(thesetofmeasurablefunctionstakingvaluesonasetU,subsetofRm).Thisisadequatetoprovetheoreticalstabilityresultsanditevenpermitstousetheresultsonexistenceofaminimizingsolutiontooptimalcontrolproblems(e.g.Proposition2).However,forimplementation,usinganyoptimizationalgo

24、rithm,thecontrolfunctionsneedtobedescribedbyafinitenumberofparameters(thesocalledfiniteparameterizationsofthecontrolfunctions).Thecontrolcanbeparameterizedaspiecewiseconstantcontrols(e.g.13),polynomialsorsplinesdescribedbyafinitenumberofcoeficients,bang-bangcontrols(e.g.9,10),etc.Notethatwearenotcon

25、sideringdiscretizationofthemodelorthedynamicequation.Theproblemsofdiscreteapproximationsarediscussedindetaile.g.in16and12.But,intheproofofstability,wejusthavetoshowatsomepointthattheoptimalcost(thevaluefunction)islowerthanthecostofusinganotheradmissiblecontrol.So,aslongasthesetofadmissiblecontrolval

26、uesUisconstantforalltime,aneasy,butneverthelessimportant,corollaryofthepreviousstabilityresultsfollowsIfweconsiderthesetofadmissiblecontrolfunctions(includingtheauxiliarycontrollaw)tobeafinitelyparameterizablesetsuchthatthesetofadmissiblecontrolvaluesisconstantforalltime,thenboththenominalstabilitya

27、ndrobuststabilityresultsheredescribedremainvalid.Anexample,istheuseofdiscontinuousfeedbackcontrolstrategiesofbang-bangtype,whichcanbedescribedbyasmallnumberofparametersandsomaketheproblemcomputationallytractable.Inbang-bangfeedbackstrategies,thecontrolsvaluesofthestrategyareonlyallowedtobeatoneofthe

28、extremesofitsrange.Manycontrolproblemsofinterestadmitabang-bangstabilizingcontrol.FontesandMagni9describetheapplicationofthisparameterizationtoaunicyclemobilerobotsubjecttoboundeddisturbances.Sampled-DataModelPredictiveControlforNonlineaTime-VaryingSystems:StabilityandRobustness.IEEETransactionsonPo

29、werDeliveryVol.8,No.1,Jariuary1993采集數據模型預測控制非線性時變系統(tǒng)的穩(wěn)定性和魯棒性FernandoA.C.C.Fontes英國倫敦皇家學院摘要我們這里所敘述的是一采樣數據模型預測控制的框架，使用連續(xù)時間模型，采樣的實際狀況以及計算控制的狀態(tài)，在離散時刻的時間進行。這個框架可以解決一個非常大的非線性、時變、非完整系統(tǒng)。像許多其他采樣數據模型預測控制計劃，barbalat引理一個重要的角色證明名義穩(wěn)定的結果。這是認為泛為barbalat的引理，這里所描述的在證明的魯棒穩(wěn)定性的結果也有類似的的作用，也可以解決一個非常一般的一類非線性，時變，非完整受到擾動的系統(tǒng)。在

30、那個可能性的框架內，以容納間斷的意見是必要的實現名義的穩(wěn)定性和魯棒穩(wěn)定性，例如一般類別的系統(tǒng)。關鍵詞barbalat引理、非線性時變系統(tǒng)、穩(wěn)定性和魯棒性1、引言多模型預測控制（MPC計劃使用連續(xù)時間描述的文獻模型和樣本工廠在離散時刻的狀態(tài)的時間。例如3,7,9,13和6。在考慮連續(xù)時間模型時有許多好處。盡管如此，任何可執(zhí)行的模型預測控制計劃只能衡量和解決再離散時刻時間的優(yōu)化問題。引用上述所提到的情況，barbalat的引理，或者修改它，使其用來作為一個重要步驟，以證明穩(wěn)定的MPCJ計劃。（barbalat引理是眾所周知的并且強有力以推斷的漸近穩(wěn)定性的非線性系統(tǒng)的工具，尤其是在推斷時間變量系統(tǒng)上

31、面，利用Lyapunov辦法；見例如17的討論和應用）。顯示模型預測控制的一項戰(zhàn)略是穩(wěn)定（在名義上是如此），這表明，如果某些設計參數（例如目標函數，碼頭設置等）方便選定，那么價值函數就是單調遞減的。然后運用barbalat引理，該吸引力軌跡的模型名義上可以建立為（i.e.x（t）一0ast一）.從這種穩(wěn)定的狀態(tài)可以推斷，一個很籠統(tǒng)的非線性系統(tǒng)：包括時變系統(tǒng)，非完整系統(tǒng)，系統(tǒng)允許間斷意見等除外，如果信函數具有一定的連續(xù)性屬性，然后Lyapunov穩(wěn)定性（即軌跡停留任意接近的起源提供了足夠向原產地的密切開始）也可以得到保障（見例如11）。不過，這為某些類別系統(tǒng)的最后狀態(tài)可能不會實現，例如輪式車的例

32、子（見例8所討論的問題）。類似的做法，建立單調減少的價值功能之后，我們想要設置一些包含原點以保證狀態(tài)軌跡的漸近方法。但是，困難所在是預測的軌跡只有剛好與由此產生的軌跡在特定的抽樣instants時魯棒穩(wěn)定性能才可以得到，因為我們用一種廣義版本的barbalat的引理。這些魯棒穩(wěn)定性結果也有效期為一個很一般類非線性時變系統(tǒng)的允許間斷的意見。最優(yōu)控制有待解決的問題與模型預測控制的戰(zhàn)略是在這里制定了非?；\統(tǒng)的受理套管制（例如，可衡量的控制職能），使結果更容易得到保證。在理論上講，存在解決辦法。不過，某種形式的有限參數的控制功能需要/可取的解決上線的優(yōu)化問題。它可以證明即穩(wěn)定或魯棒性的結果在這里所描述

33、的仍然有效，當優(yōu)化進行了有限的參數化的管制，如分段常數控制（如在13）,或幫邦間斷反饋（如在9）。2、數據采樣MPC1架我們會考慮一種非線性的靜態(tài)具有輸入與狀態(tài)的限制，凡變化狀態(tài)后，時間t0,預計由以下模型。i(s)=凡工戶)82,口,(la)工口)=/w(lb)rs)e-Vcnr1foralls(1c)tua.e,s&).(Id)這個模型的數據包括了一套X。二般”所有可能的初始狀態(tài)，在最初的時間電矢量。口這是狀態(tài)的測量時間某一函數f：/：出父U廣XET口一套俄巾的盡可能控制值。我們假設這個制度，以漸近的可控性對X。,并為所有）“七0，。）=口，我們進一步假設函數f是連續(xù)的和局部Lipschi

34、tz方面的第二個論點。Pit.Tf.Tc,Tp):Minimizej工工(S，u(s)ds+Wt+Tp?ir(i+弓)入tsubjectto:i(s)=fis,1(s),“.(口x(t)=力，工(s)Xu(s)eUu(s)=妒3(即工(占)T(t+Tp)es.a.e.st,t4TJforallsG1J+Tpfa.e.sEf,t+TJsC，十7t+J注意到，在區(qū)間仗+二二+七控制值的選定是由一個單值決定，因此優(yōu)化的設定都是進行在區(qū)間1M+八1內的。這里采用的符號如下所示。實時變量t代表我們的儲備，我們保留S來表示時間變量，用于在預測模型。向量xt是指的實際狀態(tài)。核電廠的測量時間t過程（工,刈的是一對從系統(tǒng)模型取得的彈道/控制。那個軌跡，有時是標注為&武,:心口,口）的，當我們想作明確地依賴于初始時間、初始狀態(tài)和控制功能。兩變量（心可是指我們的最優(yōu)解，這是一個開放的閉環(huán)優(yōu)化控制問題。過程中是閉環(huán)系統(tǒng)的軌跡和控制造成的從MPCJ策略。我們要求設計參數的變數，目前，在開環(huán)最優(yōu)控制問題是沒有從系統(tǒng)模型（即變量，我們可以選擇）；這些包括控制TC,該預測地平線總磷，運行成本和終端成本的職能開和W輔助控制律kaux,和終端約束集5U0,是由此產生的軌跡是由/一=