采集数据模型预测控制解析

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气. 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)=史率团=/(t,ar*(t)twherew*(t)=寓)：=u(t;团律*(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 书 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闻)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诙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采集数据模型预测控制非线性时变系统的稳定性和鲁棒性FernandoA.C.C.Fontes英国伦敦皇家学院摘要我们这里所叙述的是一采样数据模型预测控制的框架，使用连续时间模型，采样的实际状况以及计算控制的状态，在离散时刻的时间进行。这个框架可以解决一个非常大的非线性、时变、非完整系统。像许多其他采样数据模型预测控制计划，barbalat引理一个重要的角色证明名义稳定的结果。这是认为泛为barbalat的引理，这里所描述的在证明的鲁棒稳定性的结果也有类似的的作用，也可以解决一个非常一般的一类非线性，时变，非完整受到扰动的系统。在

30、那个可能性的框架内，以容纳间断的意见是必要的实现名义的稳定性和鲁棒稳定性，例如一般类别的系统。关键词barbalat引理、非线性时变系统、稳定性和鲁棒性1、引言多模型预测控制（MPC计划使用连续时间描述的文献模型和样本工厂在离散时刻的状态的时间。例如3,7,9,13和6。在考虑连续时间模型时有许多好处。尽管如此，任何可执行的模型预测控制计划只能衡量和解决再离散时刻时间的优化问题。引用上述所提到的情况，barbalat的引理，或者修改它，使其用来作为一个重要步骤，以证明稳定的MPCJ计划。（barbalat引理是众所周知的并且强有力以推断的渐近稳定性的非线性系统的工具，尤其是在推断时间变量系统上

31、面，利用Lyapunov办法；见例如17的讨论和应用）。显示模型预测控制的一项战略是稳定（在名义上是如此），这表明，如果某些设计参数（例如目标函数，码头设置等）方便选定，那么价值函数就是单调递减的。然后运用barbalat引理，该吸引力轨迹的模型名义上可以建立为（i.e.x（t）一0ast一）.从这种稳定的状态可以推断，一个很笼统的非线性系统：包括时变系统，非完整系统，系统允许间断意见等除外，如果信函数具有一定的连续性属性，然后Lyapunov稳定性（即轨迹停留任意接近的起源提供了足够向原产地的密切开始）也可以得到保障（见例如11）。不过，这为某些类别系统的最后状态可能不会实现，例如轮式车的例

32、子（见例8所讨论的问题）。类似的做法，建立单调减少的价值功能之后，我们想要设置一些包含原点以保证状态轨迹的渐近方法。但是，困难所在是预测的轨迹只有刚好与由此产生的轨迹在特定的抽样instants时鲁棒稳定性能才可以得到，因为我们用一种广义版本的barbalat的引理。这些鲁棒稳定性结果也有效期为一个很一般类非线性时变系统的允许间断的意见。最优控制有待解决的问题与模型预测控制的战略是在这里制定了非常笼统的受理套管制（例如，可衡量的控制职能），使结果更容易得到保证。在理论上讲，存在解决办法。不过，某种形式的有限参数的控制功能需要/可取的解决上线的优化问题。它可以证明即稳定或鲁棒性的结果在这里所描述

33、的仍然有效，当优化进行了有限的参数化的管制，如分段常数控制（如在13）,或帮邦间断反馈（如在9）。2、数据采样MPC1架我们会考虑一种非线性的静态具有输入与状态的限制，凡变化状态后，时间t0,预计由以下模型。i(s)=凡工户)82,口,(la)工口)=/w(lb)rs)e-Vcnr1foralls(1c)tua.e,s&).(Id)这个模型的数据包括了一套X。二般”所有可能的初始状态，在最初的时间电矢量。口这是状态的测量时间某一函数f：/：出父U广XET口一套俄巾的尽可能控制值。我们假设这个制度，以渐近的可控性对X。,并为所有）“七0，。）=口，我们进一步假设函数f是连续的和局部Lipschi

34、tz方面的第二个论点。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，在区间仗+二二+七控制值的选定是由一个单值决定，因此优化的设定都是进行在区间1M+八1内的。这里采用的符号如下所示。实时变量t代表我们的储备，我们保留S来表示时间变量，用于在预测模型。向量xt是指的实际状态。核电厂的测量时间t过程（工,刈的是一对从系统模型取得的弹道/控制。那个轨迹，有时是标注为&武,:心口,口）的，当我们想作明确地依赖于初始时间、初始状态和控制功能。两变量（心可是指我们的最优解，这是一个开放的闭环优化控制问题。过程中是闭环系统的轨迹和控制造成的从MPCJ策略。我们要求设计参数的变数，目前，在开环最优控制问题是没有从系统模型（即变量，我们可以选择）；这些包括控制TC,该预测地平线总磷，运行成本和终端成本的职能开和W辅助控制律kaux,和终端约束集5U0,是由此产生的轨迹是由/一=