Theory, Computation, and Design 2nd Edition

1 ModelPredictiveControl: Theory, Computatio n,andDesign2ndEdition9377307809759 ISBN 9780975937730 ModelPredictiveControl: Theory, Computatio n, ,Wisconsin, , ,GermanyDNobHillPublishingMadison,Wiscon sinThisbookwassetinLucidausingLATEX, certi rstedition,the eldofmodelpredictivecontrol(MPC) ,thealgorithmsandhigh-levelsoftwareavail ableforsolv-ingchallengingnonlinearoptim alcontrolproblemshaveadvancedsig-ni ,wehaveaddedanewchapter,Chapter8, NumericalOptimalControl, andcoauthor, elds:simulationofdifferentialequations, ,coveringtopicssuchasderivativecomputati ons,Hessianapproximations, ,thechap-terpresentssomeofthemanywaystha tthespeci ,andahigh-levelsetofOctave/MATLAB functions,MPCT ools, ~ ,wehaveaddedsectionscoveringthefollowing topics: economicMPC ,wehaveaddedadiscussionofstochasticMPC, ,wehaveaddedanewtreatmentofstateestimati onwithpersistent, ; rst,wewantedtomaintainamanageabletotalle ngthofthetext;second,alloftheavailablesa mplingstrate-giesinparticle lteringcomeupagainstthe curseofdimensionality, whichrendersthestateestimatesinaccuratef ordimensionhigherthanabout , , ,wesupportthestrongerKL-de nitionofasymp-toticstability,inplaceofth eclassicalde nitionusedinthe cantnotationalchangeistodenoteasequencew ith a; b ; c ; : : : insteadofwithfa; b ; c ; : : :gasinthe ,Wis.

2 ,USADQML ondon,EnglandMMDF reiburg,GermanyPrefaceOurgoalinthistexti stoprovideacomprehensiveandfoundationalt reatmentofthetheoryanddesignofmodelpredi ctivecontrol(MPC).Bynowseveralexcellentm onographsemphasizingvariousaspectsofMPCh aveappeared(alistappearsatthebeginningof Chapter1), , , , (morethan300pages) , ,butalsoviiiixcoversextendedandunscented Kalman ltering,andparticle , 'sorstudent' ,again, , , ~ ,however,thatallmaterialintheappendicesi sincludedinthebook'sprintedtableofconten ts, , ,Wisconsin,USAL ondon,EnglandAcknowledgmentsBothauthorsw ouldliketothanktheDepartmentofChemicalan dBio-logicalEngineeringoftheUniversityof WisconsinforhostingDQM' :RishiAmrit,DennisBonn e,JohnCampbell,JohnEaton,PeterFindeisen, RolfFindeisen,EricHaseltine,JohnJ rgensen,NabilLaachi,ScottMead-ows,ScottM iddlebrooks,SteveMiller,KenMuske,BrianOd elson,Mu-raliRajamani,ChrisRao,BrettStew art,KaushikSubramanian,AswinVenkat, :FrankAllg ower,TomBadgwell,BhavikBakshi,DonBartusi ak,LarryBiegler,MoritzDiehl,JimDowns,Tom Edgar,BrianFroisy,RaviGudi,StenBayJ rgensen,JayLee,FernandoLima,WolfgangMarq uardt,GabrielePannocchia,JoeQin,HarmonRa y,PierreScokaert,SigurdSkogestad,TylerSo derstrom,SteveWright, ,espe-ciallyRichardVinterandMartinClark, ;hewouldalsoliketothankmanyothercol-leag ues,especiallyKarl Astr om,RogerBrockett,LarryHo,PetarKoko-tovic ,andArtKrener, :IoannisChrysochoos,WilburLangson,Hannah Michalska,SasaRakovi c,andWarrenSchroeder.

3 HannahMichalskaandSasaRakovi c,inparticular, ,nowcolleagues,aswellastoFrankAllg ower,RolfFind-eisenEricKerrigan,Konstant inosKouramus,ChrisRao,PierreScokaert, ,BobBird,EricKerrigan,KenMuske,GabrieleP annocchia, :DougAl-lan,TravisArnold,CuylerBates,Luo Ji,NishithPatel,MichaelRisbeck, ,and,inparticular,thesoftwarere-lease,th ecurrentgroupofgraduatestudentsfarexceed edexpectationstohelp ,theprojectcouldnothavebeencompletedinat imelyfashionwithouttheirgenerosity,enthu siasm,professionalism,andsel , , , , ,JochemDeSchut-ter,AndreaZanelli,Dimitri sKouzoupis,JorisGillis,JoelAndersson,and RobinVerschuerenforhelpwiththepreparatio nofexercisesandexamplesinChapter8;andals owantstoacknowledgethefollowingcurrentan dformerteammembersthatcontributedtoresea rchandteachingonoptimalandmodelpredictiv econtrolattheUniversitiesofLeuvenandFrei burg:AdrianB urger,HansJoachimFerreau,J orgFis-cher,JanickFrasch,GianlucaFrison, NielsHaverbeke,GregHorn,BorisHouska,Jona sKoenemann,AttilaKozma,VyacheslavKungurt sev,Gio-vanniLicitra,RienQuirynen,CarloS avorgnan,QuocTran-Dinh,MilanVukov, ower,Al-bertoBemporad,RolfFindeisen,Larr yBiegler,HansGeorgBock,StephenBoyd,S ebastienGros,LarsGr une,ColinJones,JohnBagterpJ rgensen,ChristianKirches,DanielLeinewebe r,KatjaMombaur,YuriiNesterov,ToshiyukiOh tsuka,GoelePipeleers,AndreasPotschka,Seb astianSager, oder,VolkerSchulz,MarcSteinbach,JanSweve rs,Phil-ippeToint,AndreaWalther,StephenW right,JoosVandewalle, , ,Disturbances, (Inherent) :DynamicProgrammingSolution Uand ,UKF, andu0.

4 X isUniqueforallx2 (LQP).. ConditionalProbabilityandBayes' , ; ,outputy,andtransferfunctionma-trixGconn ectingthem;themodelisy (a)continuousactua-torsand(b) :recedinghorizonregulator,stateestimator , owrateat10minutes;nd owrateat10minutes;nd owrateat10minutes;nd owrateat10minutes;nd , ,ellipticalcostcontoursandellipsecentera x , x (shaded)andcontrollaw 3 x (line)versusx cos ;sin , 2 ; x 8;8 withoptimalsteadystate 8;4 . ,andifon,itmustbebetweenits(possiblynonz ero) Qmin 9(right-handside),XNforN Qmin (shadedregion) , N 2 XfandterminalpenaltyVf x 1=2 x0 xandthees-timateof xsp; usp ,unreachablestagecost` x ; u ,andoptimalsteadystates xs; us ,andstagecosts`s x ; u `e x ; u 0(circles)for 0; 8; `e x ; u < ,Rb, z ; u .. 2 0;1 .. 1,jx2j 2,andjuj .. 12(left)and 8(right).. " ".Dashedlineshowstheoutcomepredictedbyfo rmula( ), ,"test ".. 0 Zk p ,underboundingpriorweighting k p ,andMHEoptimalvalue (solidline)andEKFstateestimate(dashedlin e).

5 (solidline)andUKFstateestimate(dashedlin e).. (solidline)andMHEstateestimate(dashedlin e).. , ,FSO, x t isthecenterofthetube,andthedashedlineisa sampletrajectoryofx t . , up1; up2 to up 11; up 12 .. ; ; , x1 0 ; x2 0 3; 3 .. x 0 ;u1;u2 forN ,X,andU x .. ; u 0 t andtruetrajectoryx1 t ofthe z (b).. ( ) ( ) A .. A0 .. c fory max x1; x2 .. u 6 TU u .. :ThevaluefunctionV0N nition:Input/output-to-statestable(IOSS) .. :Modi :Asymptoticstabilitywithstagecost` y ; u .. :Continuityofsystemandcost; :Propertiesofconstraintsets; :Continuoussystemsolution; :Existenceofsolutiontooptimalcontrolprob -lem; nition:Lyapunovfunction:time-varying, :Lyapunovtheoremforasymptoticstability(t ime-varying,constrained).. :Basicstabilityassumption; :Optimalcostdecrease; :Optimalvaluefunctionproperties; : :MPCstability; nition:Asymptoticstability(differenceinc lusion).. nition:Lyapunovfunction(differenceinclus ion).

6 :Asymptoticstability(differenceinclusion ).. : :Basicstabilityassumption; nition: :Bounded,convergentsequencesare -convergent : :Positivede :Boundednessandconvergenceofestimateerro r :FIEwith :ObservableandglobalK-continuousimplyFSO nition:RGAS estimation(observablecase).. :MHEisRGAS(observablecase) nition:i-IOSS(maxform).. :Positivede :EKF,UKF, : : nition:Positiveinvariance; :Robustcontrolalgorithm(linearconstraine dsystems).. :Robustcontrolalgorithm(offset-freeMPC). . :SuboptimalMPC(simpli ed).. :Basicstabilityassumption(distributed).. nition:Polytopic(polyhedral) nition:Piecewiseaf :Farkas' :SolutionofP w , :Piecewisequadratic(af ne)cost(solution).. :Continuous, nition:Activepolytope(polyhedron).. :Optimalityofu0x w :Piecewisequadratic(af ne) :Piecewiseaf :Findinga :ConvergenceofexactNewton' :Strongsecond-ordersuf :Doesuncorrelatedimplyindependent?

7 Nition:K,K1,K L,andP nition:Globalasymptoticstability(KLversi on).. nition:Variousformsofstability(constrain ed).. nition:Asymptoticstability(constrained,K Lversion). nition:Lyapunovfunction(unconstrainedand con-strained).. :LyapunovfunctionandGAS(classicalde nition) :FromP DtoK1function(JiangandWang(2002)) :Lyapunovfunctionandglobalasymptoticsta- bility(KLde nition).. :Improvingconvergence(Sontag(1998b)).. :Lyapunovfunctionforasymptoticstability( con-strained).. nition:Asymptoticstability(time-varying, constrained) nition:Lyapunovfunction:time-varying, :Lyapunovtheoremforasymptoticstability(t ime-varying,constrained).. nition:Localstability(disturbances).. nition:Globalattraction(disturbances).. nition:GAS(disturbances).. nition:Lyapunovfunction(disturbances).. :Lyapunovfunctionforglobalasymptoticsta- bility(disturbances).. nition:GlobalcontrolLyapunovfunction(CLF ).. nition:Positiveinvariance(disturbanceand control).. nition:CLF(disturbanceandcontrol).

8 Nition:Positiveinvariance(constrained).. nition:CLF(constrained).. nition:Controlinvariance(disturbances,co nstrained) nition:CLF(disturbances,constrained).. nition:Input-to-statestable(ISS).. nition:ISS(constrained).. nition:ISS-Lyapunovfunction(constrained) .. :ISS-LyapunovfunctionimpliesISS(constrai ned). nition:Output-to-statestable(OSS).. nition:Input/output-to-statestable(IOSS) .. :Modi nition: :Optimalityconditions :Optimalityconditions :Optimalityconditions :Optimalityconditions :Lipschitzcontinuityofthevaluefunction, :Clarkeetal.(1998).. :Aboundond u; U x0 foru2U x .. :ContinuityofU .. :Lipschitzcontinuityofthevaluefunction U x 766 NotationMathematicalnotation9thereexists 2isanelementof8forall=)(=implies;isimpli edby6=)6(=doesnotimply;isnotimpliedbya: baisde :bbisde nedtobeequaltoa. approximatelyequalV functionVV:A!BVisafunctionmappingsetAint osetBx,V x functionVmapsvariablextovalueV x x valueofxatnextsampletime(discretetimesys tem) xtimederivativeofx(continuoustimesystem) fxpartialderivativeoff x withrespecttoxrnablaordeloperator unitimpulseordeltafunctionjxjabsoluteval ueofscalar;normofvector(two-normunlessst atedotherwise);inducednormofmatrixxseque nceofvector-valuedvariablex, x 0 ; x 1 ; : : : kxksupnormoverasequence,supi 0jx i jkxka:bmaxa i bjx i jtr A traceofmatrixAdet A determinantofmatrixAeig A setofeigenvaluesofmatrixA A spectralradiusofmatrixA,maxij ijfor i2eig A A 1inverseofmatrixAAypseudo-inverseofmatri xAA0transposeofmatrixAinfin 0nonnegativeintegersIn:mintegersintheint erval n; m RrealnumbersR 0nonnegativerealnumbersRnreal-valuedn-ve ctorsRm nreal-valuedm nmatricesCcomplexnumbersBballinRnofunitr adiusx pxrandomvariablexhasprobabilitydensitypx E x expectationofrandomvariablexvar x varianceofrandomvariablexcov x.

9 Y covarianceofrandomvariablesxandyN m; P normaldistribution(meanm,covarianceP),x N m; P n x ; m; P normalprobabilitydensity,px x n x ; m; P ;theemptysetaff A af nehullofsetAint A interiorofsetAco A convexhullofthesetAAclosureofsetAepi f epigraphoffunctionflevaVsublevelsetoffun ctionV,fxjV x agf gcompositionoffunctionsfandg,f g s : f g s a bmaximumofscalarsaandb,Chapter4A BsetadditionofsetsAandB,Chapters3and5A BsetsubtractionofsetBfromsetAAnBelements ofsetAnotinsetBA[BunionofsetsAandBA\Bint ersectionofsetsAandBA BsetAisasubsetofsetBA BsetAisasupersetofsetBA BsetAisaproper(orstrict)subsetofsetBA BsetAisaproper(orstrict)supersetofsetBd a;B DistancebetweenelementaandsetBdH A;B HausdorffdistancebetweensetsAandBx&y x%y xconvergestoyfromabove(below)sat x saturation,sat x xifjxj 1; 1ifx< 1;1ifx>1 NotationxliSymbolsA; B ; Csystemmatrices,discretetime,x Ax B u,y C xAc; Bcsystemmatrices,continuoustime, x Acx BcuAijstatetransitionmatrixforplayeritop layerjAistatetransitionmatrixforplayeriA LiestimateerrortransitionmatrixAi LiCiBdinputdisturbancematrixBijinputmatr ixofplayeriforplayerj'sinputsBiinputmatr ixofplayeriCijoutputmatrixofplayeriforpl ayerj'sinteractionstatesCioutputmatrixof playeriCdoutputdisturbancematrixCcontrol labilitymatrixC polarconeofconeCdintegratingdisturbanceE ; Fconstraintmatrices,F x E u ef ; hsystemfunctions,discretetime,x f x ; u ,y h x fc x ; u systemfunction,continuoustime, x fc x ; u F x ; u differenceinclusion,x 2F x ; u ,FissetvaluedGinputnoise-shapingmatrixGi jsteady-stategainofplayeritoplayerjHcont rolledvariablematrixI x ; u indexsetofconstraintsactiveat x ; u I0 x indexsetofconstraintsactiveat x ; u0 x ksampletimeKoptimalcontrollergain` x ; u stagecost`N x.]

10 U nalstagecostLoptimalestimatorgainminputd imensionMcross-termpenaltymatrixx0M uMnumberofplayers,Chapter6 Mclassofadmissibleinputpolicies, 2 MnstatedimensionNhorizonlengthOobservabi litymatrix,Chapters1and4 Ocompactrobustcontrolinvariantsetcontain ingtheorigin,Chapter3poutputdimensionxli iNotationpoptimizationiterate,Chapter6p probabilitydensityofrandomvariable ps x sampledprobabilitydensity,ps x Piwi x xi PcovariancematrixintheestimatorPftermina lpenaltymatrixPpolytopicpartition,Chapte r3 Ppolytopicpartition,Chapter7PN x MPCoptimizationproblem;horizonNandinitia lstatexqimportancefunctioninimportancesa mplingQstatepenaltymatrixrcontrolledvari able,r H yRinputpenaltymatrixsnumberofsamplesinas ampledprobabilitydensitySinputrateofchan gepenaltymatrixS x ; u indexsetofactivepolytopesat x ; u S0 x indexsetofactivepolytopesat x ; u0 x ttimeTcurrenttimeinestimationproblemuinp ut(manipulatedvariable)vectorue warmstartforinputsequenceu improvedinputsequenceUN x controlconstraintsetUinputconstraintsetv outputdisturbance,Chapters1and4vnominalc ontrolinput,Chapters3and5VN x ;u MPCobjectivefunctionV0N x MPCoptimalvaluefunctionVT ;!