Advances in Global Pseudospectral Methods for Optimal Control

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Advances in Global Pseudospectral Methods for Optimal Control
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Garg,Divya
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Rao, Anil
Committee Members:
Barooah, Prabir
Dixon, Warren E
Hager, William W

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Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
A new pseudospectral method that employs global collocation at the Legendre-Gauss-Radau (LGR) points is presented for direct trajectory optimization and costate estimation of finite-horizon optimal control problems. This method provides accurate state and control approximations. Furthermore, transformations are developed that relate the Karush-Kuhn-Tucker (KKT) multipliers of the discrete nonlinear programming problem (NLP) to the costate and the Lagrange multipliers of the continuous-time optimal control problem. More precisely, it is shown that the transformed KKT multipliers of the NLP correspond to the Lagrange multipliers of the continuous problem and to a pseudospectral approximation of costate that is approximated using polynomials one degree smaller than that used for the state. The relationship between the differentiation matrices for the state equation and for the costate equation is established. Next, a unified framework is presented for the numerical solution of optimal control problems based on collocation at the Legendre-Gauss (LG) and the Legendre-Gauss-Radau (LGR) points. The framework draws from the common features and mathematical properties demonstrated by the LG and the LGR methods. The framework stresses the fact that even though LG and LGR collocation appear to be only cosmetically different from collocation at Legendre-Gauss-Lobatto (LGL) points, the LG and the LGR methods are, in fact, fundamentally different from the LGL method. Specifically, it is shown that the LG and the LGR differentiation matrices are non-square and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and the LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Furthermore, it is shown that the LG and the LGR discrete costate systems have a unique solution while the LGL discrete costate system has a null space. The LGL costate approximation is found to have an error that oscillates about the exact solution, and this error is shown by example to be due to the null space in the LGL discrete costate system. Finally, it is shown empirically that the discrete state, costate and control obtained by the LG and the LGR schemes converge exponentially as a function of the number of collocation points, whereas the LGL costate is potentially non-convergent. Third, two new direct pseudospectral methods for solving infinite-horizon optimal control problems are presented that employ collocation at the LG and the LGR points. A smooth, strictly monotonic transformation is used to map the infinite time domain t belongs to 0, infinity) onto the interval tau belongs to -1, 1). The resulting problem on the interval tau belongs to -1, 1) is then transcribed to a NLP using collocation. The proposed methods provide approximations to the state and the costate on the entire horizon, including approximations at t = +infinity. These infinite-horizon methods can be written equivalently in either a differential or an implicit integral form. In numerical experiments, the discrete solution is found to converge exponentially as a function of the number of collocation points. It is shown that the mapping phi:-1, +1) right arrow 0, +infinity) can be tuned to improve the quality of the discrete approximation.
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In the series University of Florida Digital Collections.
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Includes vita.
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Statement of Responsibility:
by Divya Garg.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Rao, Anil.

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ADVANCESINGLOBALPSEUDOSPECTRALMETHODSFOROPTIMALCONTROL By DIVYAGARG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011DivyaGarg 2

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Tomyparents,AnilandAnitaandbrother,Ankur 3

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ACKNOWLEDGMENTS ThejourneyofearningthisPh.D.degreehasbeenoneofthemos timportantand fulllinglearningexperiencesofmylife.Ithasprovidedm ewithanimmensesense ofaccomplishment.Ihavecomealongwayintermsofmyprofes sionalandpersonal growth.Myadvisor,Dr.AnilV.Raohasplayedthemostinstrum entalroleinmysuccess asadoctoralcandidate.Hisattentionfordetailinevaluat ingmyworkhastaughtme tonevertakeanythingforgrantedandthatallgoodthingsin lifeareearnedwithgreat efforts.Withouthishelp,insight,andguidancethisworkwo uldnothavebeenpossible. Iwouldalsoliketothankthemembersofmycommittee:Dr.Will iamHager, Dr.WarrenDixon,andDr.PrabirBarooah.Dr.Hager'sfeedback asamathematician havebeenextremelyhelpfulinprovidingthemathematicale legancetomyresearch ofwhichIamparticularlyproud.Hisexperienceandwisdomh avegivenmenew perspectivesonhowtocontinuetogrowasaresearcher.Iamv erythankfultoDr.Dixon andDr.Barooahfortheirinvaluablesuggestionsandfortaki nganactiveinterestinmy research. TomyfellowmembersofVDOL:ChrisDarby,CamilaFrancolin, MikePatterson, PoojaHariharan,DarinToscano,BrendanMahon,andBegumSense s,Iwouldlike tosaythankyouforallthegoodtimes.Iamcondentthatwhen Iwilllookbackon thesedays,Iwillfondlyremembereachoneofyouandtherole youplayedinthis journey.ChrisDarby,youarenotonlyanidealcolleaguebut alsoaverydearfriend. YoureverwillingnesstoanswerthequestionsIhadrelatedt oresearchandunparalleled workethicsareinspiring.ThankyouChrisandCamilaforbei ngmysoundingboards wheneverIwasfeelinglow.IwouldliketothankMikeforthec ontributionshehasmade towardsmyresearch. Lastly,Icannotstressenoughupontheimportanceoftherol eplayedbymy parents,mybrotherAnkur,myfriends,andManoj.Myparentsa ndAnkurhavealways beencondentofmycapabilities,evenatthetimeswhenIwas indoubt.Somuchof 4

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whatIhaveaccomplishedisbecauseofyou.Thankyousomuchf orbeingthereto supportmeandstandingbymeatalltimes.ThankyouManojfor beingmyfamilyaway fromfamily.EverytimeIfeltlikegivingup,youwerethereto encouragemeandhelped megetthroughthis. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFFIGURES .....................................9 ABSTRACT .........................................12 CHAPTER 1INTRODUCTION ...................................14 2MATHEMATICALBACKGROUND .........................24 2.1OptimalControl .................................25 2.1.1CalculusofVariationsandNecessaryConditions ..........26 2.1.2Pontryagin'sPrinciple .........................30 2.2NumericalOptimization ............................31 2.2.1UnconstrainedOptimization ......................32 2.2.2EqualityConstrainedOptimization ..................33 2.2.3InequalityConstrainedOptimization .................35 2.3Finite-DimensionalApproximation ......................37 2.3.1PolynomialApproximation .......................37 2.3.1.1Approximationerror .....................39 2.3.1.2FamilyofLegendre-Gausspoints .............41 2.3.2NumericalSolutionofDifferentialEquations .............46 2.3.2.1Time-marchingmethods ..................46 2.3.2.2Collocation ..........................48 2.3.3NumericalIntegration ..........................50 2.3.3.1Low-orderintegrators ....................51 2.3.3.2Gaussianquadrature ....................53 3MOTIVATIONFORTHERADAUPSEUDOSPECTRALMETHOD ........59 3.1ScaledContinuous-TimeOptimalControlProblem .............62 3.2LobattoPseudospectralMethod .......................67 3.2.1NLPFormulationoftheLobattoPseudospectralMethod ......68 3.2.2NecessaryOptimalityConditions ...................71 3.3GaussPseudospectralMethod ........................75 3.3.1NLPFormulationoftheGaussPseudospectralMethod .......76 3.3.2NecessaryOptimalityConditions ...................79 3.4Summary ....................................85 4RADAUPSEUDOSPECTRALMETHOD ......................87 4.1NLPFormulationoftheRadauPseudospectralMethod ..........88 4.2NecessaryOptimalityConditions .......................92 6

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4.3FlippedRadauPseudospectralMethod ...................99 4.3.1NLPFormulationoftheFlippedRadauPseudospectralMe thod ..99 4.3.2NecessaryOptimalityConditions ...................102 4.4Summary ....................................109 5AUNIFIEDFRAMEWORKFORPSEUDOSPECTRALMETHODS .......111 5.1ImplicitIntegrationScheme ..........................112 5.1.1IntegralFormulationUsingLGCollocation ..............112 5.1.2IntegralFormulationUsingStandardLGRCollocation .......117 5.1.3IntegralFormulationUsingFlippedLGRCollocation ........121 5.1.4IntegralFormulationUsingLGLCollocation .............125 5.2CostateDynamicsforInitial-ValueProblem .................128 5.2.1GaussPseudospectralMethod ....................128 5.2.2RadauPseudospectralMethod ....................129 5.2.3FlippedRadauPseudospectralMethod ................130 5.2.4LobattoPseudospectralMethod ....................131 5.3Convergence ..................................133 5.4Summary ....................................135 6FINITE-HORIZONOPTIMALCONTROLEXAMPLES ..............136 6.1Example1:NonlinearOne-DimensionalInitial-ValueProb lem .......137 6.2Example2:NonlinearOne-DimensionalBoundary-ValueProb lem ....144 6.3Example3:Orbit-RaisingProblem ......................153 6.4Example4:BrysonMaximumRangeProblem ................157 6.5Example5:Bang-BangControlProblem ...................162 6.6Example5:SingularArcProblem .......................170 6.7Summary ....................................176 7INFINITE-HORIZONOPTIMALCONTROLPROBLEMS .............177 7.1Innite-HorizonOptimalControlProblem ...................178 7.2Innite-HorizonGaussPseudospectralMethod ...............182 7.2.1NLPFormulation ............................182 7.2.2Karush-Kuhn-TuckerConditions ....................185 7.2.3EquivalentImplicitIntegrationScheme ................189 7.3Innite-HorizonRadauPseudospectralMethod ...............191 7.3.1NLPFormulation ............................191 7.3.2Karush-Kuhn-TuckerConditions ....................193 7.3.3EquivalentImplicitIntegrationScheme ................198 7.4InapplicabilityofLobattoPseudospectralMethod ..............200 7.5Examples ....................................202 7.5.1Example1:Innite-HorizonOne-DimensionalNonlinea rProblem .202 7.5.2Example2:Innite-HorizonLQRProblem ..............213 7.6Summary ....................................226 7

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8CONCLUSION ....................................227 8.1DissertationSummary .............................227 8.2FutureWork ...................................230 8.2.1ConvergenceProofforGaussandRadauPseudospectralMe thod 230 8.2.2CostateEstimationUsingLobattoPseudospectralMetho d .....230 REFERENCES .......................................231 BIOGRAPHICALSKETCH ................................238 8

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LISTOFFIGURES Figure page 2-1Anextremalcurve y ( t ) andacomparisoncurve y ( t ) ..............27 2-2Approximationof y ( )=1 = (1+50 2 ) usinguniformdiscretizationpoints. ...40 2-3SchematicdiagramofLegendrepoints. ......................42 2-4Approximationof y ( )=1 = (1+50 2 ) using 11 and 41 LGdiscretizationpoints. 43 2-5Approximationof y ( )=1 = (1+50 2 ) using 11 and 41 LGRdiscretizationpoints. 44 2-6Errorvs.numberofdiscretizationpoints ......................45 2-7Fourintervaltrapezoidruleapproximation. .....................52 2-8Errorvs.numberofintervalsfortrapezoidruleapproxim ation ..........53 2-9Errorvs.numberofGaussianquadraturepointsforapprox imation .......58 3-1RelationshipbetweenKKTconditionsandrst-orderoptim alityconditions ...60 3-2Multiple-intervalimplementationofLGL,LG,andLGRpo ints. ..........61 3-3DiscretizationandcollocationpointsforLobattopseu dospectralmethod. ....70 3-4DiscretizationandcollocationpointsforGausspseudo spectralmethod. ....78 4-1DiscretizationandcollocationpointsforRadaupseudo spectralmethod. ....91 4-2DiscretizationandcollocationpointsforippedRadau pseudospectralmethod. 102 4-3RelationshipbetweenKKTconditionsandrst-orderoptim alityconditions ...110 6-1TheGPMsolutionforExample 1 ..........................138 6-2TheRPMsolutionforExample 1 ..........................139 6-3Thef-RPMsolutionforExample 1 .........................140 6-4TheLPMsolutionforExample 1 ..........................141 6-5LPMcostateerrorandnullspaceforExample 1 .................142 6-6Solutionerrorsvs.numberofcollocationpointsforExamp le 1 ..........143 6-7TheGPMsolutionforExample 2 ..........................146 6-8TheRPMsolutionforExample 2 ..........................147 6-9Thef-RPMsolutionforExample 2 .........................148 9

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6-10TheLPMsolutionforExample 2 ..........................149 6-11ThesolutionformodiedExample 2 ........................151 6-12Solutionerrorsvs.numberofcollocationpointsforExam ple 2 ..........152 6-13TheGPMsolutionforExample 3 ..........................154 6-14TheRPMsolutionforExample 3 ..........................155 6-15TheLPMsolutionforExample 3 ..........................156 6-16TheGPMsolutionforExample 4 ..........................158 6-17TheRPMsolutionforExample 4 ..........................159 6-18Thef-RPMsolutionforExample 4 .........................160 6-19TheLPMsolutionforExample 4 ..........................161 6-20TheGPMsolutionforExample 5 ..........................164 6-21TheRPMsolutionforExample 5 ..........................165 6-22Thef-RPMsolutionforExample 5 .........................166 6-23TheLPMsolutionforExample 5 ..........................167 6-24Solutionerrorsvs.numberofcollocationpointsforExam ple 5 ..........169 6-25TheGPMsolutionforExample 6 ..........................171 6-26TheRPMsolutionforExample 6 ..........................172 6-27Thef-RPMsolutionforExample 6 .........................173 6-28TheLPMsolutionforExample 6 ..........................174 6-29Solutionerrorsvs.numberofcollocationpointsforExam ple 6 ..........175 7-1Growthin ( ) at40LGpoints. ...........................180 7-2Growthof y ( t ) andlocationof40collocationpointsusing b ( ) ........181 7-3TheGPMsolutionusing a ( ) forinnite-horizon1-dimensionalproblem. ...204 7-4TheRPMsolutionusing a ( ) forinnite-horizon1-dimensionalproblem. ...205 7-5TheGPMsolutionusing b ( ) forinnite-horizon1-dimensionalproblem. ...206 7-6TheRPMsolutionusing b ( ) forinnite-horizon1-dimensionalproblem. ...207 7-7TheGPMsolutionusing c ( ) forinnite-horizon1-dimensionalproblem. ...208 10

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7-8TheRPMsolutionusing c ( ) forinnite-horizon1-dimensionalproblem. ...209 7-9Stateerrorsforinnite-horizon1-dimensionalproblem ..............210 7-10Controlerrorsforinnite-horizon1-dimensionalpro blem .............211 7-11Costateerrorsforinnite-horizon1-dimensionalpro blem .............212 7-12TheGPMsolutionusing a ( ) forinnite-horizonLQRproblem. .........215 7-13TheRPMsolutionusing a ( ) forinnite-horizonLQRproblem. .........216 7-14TheGPMsolutionusing b ( ) forinnite-horizonLQRproblem. .........217 7-15TheRPMsolutionusing b ( ) forinnite-horizonLQRproblem. .........218 7-16TheGPMsolutionusing c ( ) forinnite-horizonLQRproblem. .........219 7-17TheRPMsolutionusing c ( ) forinnite-horizonLQRproblem. .........220 7-18Firstcomponentofstateerrorsforinnite-horizonLQ Rproblem .........221 7-19Secondcomponentofstateerrorsforinnite-horizonLQ Rproblem .......222 7-20Controlerrorsforinnite-horizonLQRproblem ..................223 7-21Firstcomponentofcostateerrorsforinnite-horizon LQRproblem .......224 7-22Secondcomponentofcostateerrorsforinnite-horizon LQRproblem .....225 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ADVANCESINGLOBALPSEUDOSPECTRALMETHODSFOROPTIMALCONTROL By DivyaGarg August2011 Chair:AnilV.RaoMajor:MechanicalEngineering Anewpseudospectralmethodthatemploysglobalcollocatio nattheLegendre-Gauss-Radau (LGR)pointsispresentedfordirecttrajectoryoptimizati onandcostateestimation ofnite-horizonoptimalcontrolproblems.Thismethodpro videsaccuratestateand controlapproximations.Furthermore,transformationsar edevelopedthatrelatethe Karush-Kuhn-Tucker(KKT)multipliersofthediscretenonline arprogrammingproblem (NLP)tothecostateandtheLagrangemultipliersoftheconti nuous-timeoptimal controlproblem.Moreprecisely,itisshownthatthetransf ormedKKTmultipliersof theNLPcorrespondtotheLagrangemultipliersofthecontin uousproblemandtoa pseudospectralapproximationofcostatethatisapproxima tedusingpolynomialsone degreesmallerthanthatusedforthestate.Therelationshi pbetweenthedifferentiation matricesforthestateequationandforthecostateequation isestablished. Next,auniedframeworkispresentedforthenumericalsolu tionofoptimalcontrol problemsbasedoncollocationattheLegendre-Gauss(LG)an dtheLegendre-Gauss-Radau (LGR)points.Theframeworkdrawsfromthecommonfeaturesa ndmathematical propertiesdemonstratedbytheLGandtheLGRmethods.Thefr ameworkstressesthe factthateventhoughLGandLGRcollocationappeartobeonly cosmeticallydifferent fromcollocationatLegendre-Gauss-Lobatto(LGL)points, theLGandtheLGRmethods are,infact,fundamentallydifferentfromtheLGLmethod.Sp ecically,itisshownthat theLGandtheLGRdifferentiationmatricesarenon-squarea ndfullrankwhereasthe 12

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LGLdifferentiationmatrixissquareandsingular.Consequ ently,theLGandtheLGR schemescanbeexpressedequivalentlyineitherdifferenti alorintegralform,whilethe LGLdifferentialandintegralformsarenotequivalent.Fur thermore,itisshownthatthe LGandtheLGRdiscretecostatesystemshaveauniquesolutio nwhiletheLGLdiscrete costatesystemhasanullspace.TheLGLcostateapproximati onisfoundtohavean errorthatoscillatesabouttheexactsolution,andthiserr orisshownbyexampletobe duetothenullspaceintheLGLdiscretecostatesystem.Fina lly,itisshownempirically thatthediscretestate,costateandcontrolobtainedbythe LGandtheLGRschemes convergeexponentiallyasafunctionofthenumberofcolloc ationpoints,whereasthe LGLcostateispotentiallynon-convergent. Third,twonewdirectpseudospectralmethodsforsolvingin nite-horizonoptimal controlproblemsarepresentedthatemploycollocationatt heLGandtheLGRpoints. Asmooth,strictlymonotonictransformationisusedtomapt heinnitetimedomain t 2 [0, 1 ) ontotheinterval 2 [ 1,1) .Theresultingproblemontheinterval 2 [ 1,1) isthentranscribedtoaNLPusingcollocation.Theproposed methods provideapproximationstothestateandthecostateontheen tirehorizon,including approximationsat t =+ 1 .Theseinnite-horizonmethodscanbewrittenequivalentl y ineitheradifferentialoranimplicitintegralform.Innum ericalexperiments,thediscrete solutionisfoundtoconvergeexponentiallyasafunctionof thenumberofcollocation points.Itisshownthatthemapping :[ 1,+1) [0,+ 1 ) canbetunedtoimprovethe qualityofthediscreteapproximation. 13

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CHAPTER1 INTRODUCTION Optimalcontrolisasubjectthatarisesinmanybranchesofe ngineeringincluding aerospace,chemical,andelectricalengineering.Particu larlyinaerospaceengineering, optimalcontrolisusedinvariousapplicationsincludingt rajectoryoptimization,attitude control,andvehicleguidance.AsdenedbyKirk[ 1 ],“Theobjectiveofanoptimalcontrol problemistodeterminethecontrolsignalsthatwillcausea processtosatisfythe physicalconstraintsandatthesametimeminimize(ormaxim ize)someperformance index”.Possibleperformanceindicesincludetime,fuelco nsumption,oranyother parameterofinterestinagivenapplication. Exceptforspecialcases,mostoptimalcontrolproblemscann otbesolved analytically.Consequently,numericalmethodsmustbeemp loyed.Numericalmethods forsolvingoptimalcontrolproblemfallintotwocategorie s:indirectmethodsanddirect methods,assummarizedbyStryketal.,Betts,andRao[ 2 – 4 ].Inanindirectmethod,the calculusofvariations[ 1 5 ]isappliedtodeterminetherst-ordernecessaryconditio ns foranoptimalsolution.Applyingthecalculusofvariations transformstheoptimalcontrol problemtoaHamiltonianboundary-valueproblem(HBVP).Theso lutiontotheHBVP isthenapproximatedusingoneofthevariousnumericalappr oaches.Commonlyused approachesforsolvingtheHBVPareshooting,multipleshooti ng[ 6 7 ],nitedifference [ 8 ],andcollocation[ 9 10 ].Althoughusinganindirectmethodhastheadvantage thatahighlyaccurateapproximationcanbeobtainedandtha ttheproximityofthe approximationtotheoptimalsolutioncanbeestablished,i ndirectmethodshaveseveral disadvantages.First,implementinganindirectmethodreq uiresthatthecomplicated rst-ordernecessaryoptimalityconditionsbederived.Sec ond,theindirectmethods requirethataverygoodinitialguessontheunknownboundar yconditionsmustbe provided.Theseguessesincludeaguessforthecostatewhic hisamathematical quantityinherenttotheHBVP.Becausethecostateisanon-intu itiveandnon-physical 14

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quantity,providingsuchaguessisdifcult.Third,whenev eraproblemneedstobe modied(e.g.,addingorremovingaconstraint),thenecess aryconditionsneedtobe reformulated.Lastly,forproblemswhosesolutionshaveac tivepathconstraints,apriori knowledgeoftheswitchingstructureofthepathconstraint smustbeknown. Inadirectmethod,thecontinuousfunctionsoftime(thesta teand/orthecontrol) oftheoptimalcontrolproblemareapproximatedandtheprob lemistranscribedintoa nite-dimensionalnonlinearprogrammingproblem(NLP).Th eNLPisthensolvedusing welldevelopedalgorithmsandsoftware[ 11 – 14 ].Inthecasewhereonlythecontrol isapproximated,themethodiscalledacontrolparameteriz ationmethod.Whenboth thestateandthecontrolareapproximated,themethodiscal ledastateandcontrol parameterizationmethod.Directmethodsovercomethedisa dvantagesofindirect methodsbecausetheoptimalityconditionsdonotneedtobed erived,theinitialguess doesnotneedtobeasgoodasthatrequiredbyanindirectmeth od,aguessofthe costateisnotneeded,andtheproblemcanbemodiedrelativ elyeasily.Directmethods, however,arenotasaccurateasindirectmethods,requiremu chmoreworktoverify optimality,andmanydirectmethodsdonotprovideanyinfor mationaboutthecostate. Manydifferentdirectmethodshavebeendeveloped.Thetwoe arliestdeveloped directmethodsforsolvingoptimalcontrolproblemarethed irectshootingmethod andthedirectmultiple-shootingmethod[ 15 – 17 ].Bothdirectshootinganddirect multiple-shootingmethodsarecontrolparameterizationm ethodswherethecontrol isparameterizedusingaspeciedfunctionalformandthedy namicsareintegrated usingexplicitnumericalintegration(e.g.,atime-marchi ngalgorithm).Adirectshooting methodisusefulwhentheproblemcanbeapproximatedwithaf ewnumberofvariables. Asthenumberofvariablesusedinadirectshootingmethodgro ws,theabilityto successfullyuseadirectshootingmethoddeclines.Inthed irectmultiple-shooting method,thetimeintervalisdividedintoseveralsubinterv alsandthenthedirectshooting methodisusedovereachinterval.Attheinterfaceofeachsub interval,thestate 15

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continuityconditionisenforcedandthestateatthebeginn ingofeachsubintervalisa parameterintheoptimization.Thedirectmultiple-shooti ngmethodisanimprovement overthestandarddirectshootingmethodasthesensitivity totheinitialguessisreduced becauseintegrationisperformedoversignicantlysmalle rtimeintervals.Boththedirect shootingmethodandthedirectmultiple-shootingmethod,h owever,arecomputationally expensiveduetothenumericalintegrationoperationandre quireaprioriknowledgeof theswitchingstructureofinactiveandactivepathconstra ints.Well-knowncomputer implementationofdirectshootingmethodsarePOST[ 18 ]andSTOPM[ 19 ]. Anotherapproachisthatofdirectcollocationmethods[ 20 – 50 ],whereboththe stateandthecontrolareparameterizedusingasetoftrial( basis)functionsandasetof differential-algebraicconstraintsareenforcedatanit enumberofcollocationpoints.In contrasttoindirectmethodsanddirectshootingmethods,a directcollocationmethod doesnotrequireaprioriknowledgeoftheactiveandinactiv earcsforproblemswith inequalitypathconstraints.Furthermore,directcolloca tionmethodsaremuchless sensitivetotheinitialguessthaneithertheaforemention edindirectmethodsordirect shootingmethods.Someexamplesofcomputerimplementation sofdirectcollocation methodsareSOCS[ 51 ],OTIS[ 52 ],DIRCOL[ 53 ],DIDO[ 54 ]andGPOPS[ 55 56 ].The twomostcommonformsofdirectcollocationmethodsareloca lcollocation[ 20 – 30 ]and globalcollocation[ 31 – 50 ]. Inadirectlocalcollocationmethod,thetimeintervalisdi videdintosubintervals andaxedlow-degreepolynomialisusedforapproximationi neachsubinterval.The convergenceofthenumericaldiscretizationisachievedby increasingthenumberof subintervals.Twocategoriesofdiscretizationhavebeenu sedforlocalcollocation: (a)Runge-Kuttamethods[ 20 – 25 ]thatusepiecewisepolynomials;(b)orthogonal collocationmethods[ 26 – 30 ]thatuseorthogonalpolynomials.Directlocalcollocatio n leadstoasparseNLPwithmanyoftheconstraintJacobianent riesaszero.Sparsity intheNLPgreatlyincreasesthecomputationalefciency.H owever,theconvergence 16

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totheexactsolutionisatapolynomialrateandoftenanexce ssivelylargenumberof subintervalsarerequiredtoaccuratelyapproximatetheso lutiontoanoptimalcontrol problemresultinginalargeNLPwithoftentensofthousands ofvariablesormore.In adirectglobalcollocationmethod,thestateandthecontro lareparameterizedusing globalpolynomials.Incontrasttolocalmethods,theclass ofdirectglobalcollocation methodsusesasmallxednumberofapproximatingintervals (oftenonlyasingle intervalisused).Convergencetotheexactsolutionisachi evedbyincreasingthedegree ofpolynomialapproximationineachinterval. Inrecentyears,aparticularclassofmethodsthathasrecei vedagreatdeal ofattentionistheclassofpseudospectralororthogonalco llocationmethods.Ina pseudospectralmethod,thebasisfunctionsaretypicallyt heChebyshevortheLagrange polynomialsandthecollocationpointsareobtainedfromve ryaccurateGaussian quadraturerules.Thesemethodsarebasedonspectralmetho dsandtypicallyhave fasterconvergencerates(exponential)thanthetradition almethodsforasmallnumber ofdiscretizationpoints[ 57 – 59 ].Spectralmethodswereappliedtooptimalcontrol problemsinthelate1980'susingChebyshevpolynomialsbyVl assenbroecketal.in Ref.[ 31 32 ],andlateraLegendre-basedpseudospectralmethodusingL agrange polynomialsandcollocationatLegendre-Gauss-Lobatto(L GL)pointswasdeveloped byElnagaretal.inRef.[ 33 – 36 ].AnextensionoftheLegendre-basedpseudospectral methodwasperformedbyFahrooetal.inRef.[ 37 ]togeneratecostateestimates.This methodlatercametobeknownastheLobattopseudospectralm ethod(LPM)[ 37 – 47 ]. Atthesametime,anotherLegendre-basedpseudospectralmet hodcalledtheGauss pseudospectralmethod(GPM)wasdevelopedbyBensonandHunti ngtoninRef.[ 48 – 50 ].TheGPMusedLagrangepolynomialsasbasisfunctionsandLe gendre-Gauss(LG) pointsforcollocation. Despitethemanyadvantagesofdirectmethods,manyofthemd onotgiveany informationaboutthecostate.Thecostateisimportantfor verifyingtheoptimalityof 17

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thesolution,meshrenement,sensitivityanalysis,andre altimeoptimization.Recently, costateestimateshavebeendevelopedforpseudospectralm ethods.Theseestimates arederivedbyrelatingtheKarush-Kuhn-Tucker(KKT)conditio nsoftheNLPtothe continuouscostatedynamicsasdemonstratedbySeywaldandSt rykinRef.[ 60 61 ].A costatemappingprinciplehasbeenderivedbyFahrooetal.i nRef.[ 37 ]toestimatethe costatefromtheKKTmultipliersfortheLobattopseudospectr almethod.Howeverthis principledoesnotholdattheboundarypoints.Theresultin gcostateestimatesatthe boundariesdonotsatisfythecostatedynamicsorboundaryc onditions,butonlyalinear combinationofthetwo.ItwasshownbyBensoninRef.[ 48 ]thatthisisaresultofthe defectsinthediscretizationwhenusingLGLpoints. Asmentionedearlier,theGausspseudospectralmethod(GPM), whichusesthe LGpoints,wasproposedbyBensoninRef.[ 48 49 ].TheGPMdiffersfromtheLobatto pseudospectralmethod(LPM)inthefactthatthedynamicequa tionsarenotcollocated attheboundarypoints.InthisapproachtheKKTconditionsoft heNLParefoundto beexactlyequivalenttothediscretizedformoftherst-or dernecessaryconditionsof theoptimalcontrolproblem.Thispropertyallowsforacost ateestimatethatismore accuratethantheoneobtainedfromtheLPM.IntheGPM,however ,becausethe dynamicsarenotcollocatedattheinitialandthenalpoint ,thecontrolateitherthe initialorthenalpointisnotobtained. Inthisdissertation,anewmethodcalledtheRadaupseudosp ectralmethod(RPM) isproposed.TheRPMisadirecttranscriptionmethodthatuse sparameterizationofthe stateandthecontrolbyglobalpolynomials(Lagrangepolyn omials)andcollocationof differentialconstraintsattheLegendre-Gauss-Radau(LG R)points[ 62 ].Themethod developedinthisdissertationdiffersfromtheLobattopse udospectralmethodinthe factthatthedynamicequationsarenotcollocatedatthena lpoint.Itisshownthat theKKTconditionsfromtheresultingNLPareequivalenttothe discretizedformof therst-ordernecessaryconditionsoftheoptimalcontrol problem.Thismethod, 18

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therefore,providesanapproachtoobtainaccurateapproxi mationsofthecostateforthe continuousproblemusingtheKKTmultipliersoftheNLP.Also,b ecausethedynamics arenotcollocatedatthenalpoint,thecostateatthenalp ointisnotobtainedinthe NLPsolution.Itisnoted,however,thatthecostateatthen altimecanbeestimated accuratelyusingaRadauquadrature.Themethodofthisdiss ertationdiffersfromthe Gausspseudospectralmethodinthefactthatthedynamicequ ationsarecollocatedat theinitialpoint.Asaresult,themethodofthisdissertatio nprovidesanapproximationof thecontrolattheinitialpoint.ItisnotedthatLGRpointsh avepreviouslybeenusedfor localcollocationbyKameswaranetal.inRef.[ 30 ]differingfromtheglobalcollocation approachusedinthisresearch. TheRadaupseudospectralmethodderivedinthisdissertati onhasmanyadvantages overothernumericalmethodsforsolvingoptimalcontrolpr oblems.First,theimplementation ofthemethodiseasyandanychangeinconstraintscanbeinco rporatedinthe formulationwithoutmuchwork.Second,anaccuratesolution canbefoundusing well-developedsparseNLPsolverswithnoneedforaninitia lguessonthecostateor derivationofthenecessaryconditions.Third,thecostate canbeestimateddirectly fromtheKKTmultipliersoftheNLP.ThenaladvantageoftheRa daupseudospectral method,isthattheytakeadvantageofthefastexponentialc onvergencetypicalof spectralmethods.Thisrapidconvergencerateisshownempi ricallyonavarietyof exampleproblems.Therapidconvergencerateindicatestha tanaccuratesolutionto theoptimalcontrolproblemcanbefoundusingfewercolloca tionpointsandpotentially lesscomputationaltime,whencomparedwithothermethods. Therapidsolutionofthe problemalongwithanaccurateestimateforthecostateandi nitialcontrolcouldalso enablereal-timeoptimalcontrolfornonlinearsystems. Next,inthisdissertation,auniedframeworkispresented forthenumerical solutionofoptimalcontrolproblemsusingtheGausspseudo spectralmethod(GPM) andtheRadaupseudospectralmethod(RPM)[ 63 ].Theframeworkisbasedon 19

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thecommonfeaturesandmathematicalpropertiesoftheGPMan dtheRPM.The frameworkstressesthefactthat,eventhoughtheGPMandtheR PMappeartobe onlycosmeticallydifferentfromtheLobattopseudospectr almethod(LPM),theGPM andtheRPMare,infact,fundamentallydifferentfromtheLPM. Intheframework,the stateisapproximatedbyabasisofLagrangepolynomials,th esystemdynamicsand thepathconstraintsareenforcedatthecollocationpoints ,andtheboundaryconditions areappliedattheendpointsofthetimeinterval.TheGPMandt heRPMemploya polynomialapproximationthatisthesamedegreeasthenumb erofcollocationpoints whiletheLPMemploysastateapproximationthatisonedegree lessthanthenumber ofcollocationpoints.ItisshownthattheGPMandtheRPMdiffe rentiationmatrices arenon-squareandfullrank,whereastheLPMdifferentiatio nmatrixissquareand singular.Consequently,theGPMandtheRPMschemescanbeexpr essedequivalently ineitherdifferentialorintegralform,whiletheLPMdiffer entialandintegralformsare notequivalent.Furthermore,itisshownthattheGPMandtheR PMdiscretecostate systemsarefullrankwhiletheLPMdiscretecostatesystemis rank-decient.TheLPM costateapproximationisfoundtohaveanerrorthatoscilla tesabouttheexactsolution, andthiserrorisshownbyexampletobeduetothenullspacein theLPMdiscrete costatesystem.Finally,itisshownempiricallythatthedi scretesolutionsforthestate, control,andcostateobtainedfromtheGPMandtheRPMconverge exponentiallyas afunctionofthenumberofcollocationpoints,whereastheL PMcostateispotentially non-convergent.Theframeworkpresentedinthisdissertat ionprovidestherstrigorous analysisthatidentiesthekeymathematicalpropertiesof pseudospectralmethods usingcollocationatGaussianquadraturepoints,enabling aresearcherorend-userto seeclearlytheaccuracyandconvergence(ornon-convergen ce)thatcanbeexpected whenapplyingaparticularpseudospectralmethodonaprobl emofinterest. Lastly,thisdissertationpresentstwonewdirectpseudosp ectralmethodsthat employcollocationatLegendre-Gauss(LG)andLegendre-Ga uss-Radau(LGR) 20

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pointsforsolvinginnite-horizonoptimalcontrolproble ms[ 64 ].Asmooth,strictly monotonictransformationisusedtomaptheinnitetimedom ain t 2 [0, 1 ) onto 2 [ 1,1) interval.Theresultingproblemontheinterval 2 [ 1,1) isthen transcribedtoanonlinearprogrammingproblemusingcollo cation.Byusingthe approachdevelopedinthisdissertation,theproposedmeth odsprovideapproximations tothestateandthecostateontheentirehorizon,including approximationsat t =+ 1 Inamannersimilartothenite-horizonGaussandRadaupseu dospectralmethods, theseinnite-horizonmethodscanbewrittenequivalently ineitheradifferentialor animplicitintegralform.Innumericalexperiments,thedi scretesolutionisfoundto convergeexponentiallyasafunctionofthenumberofcolloc ationpoints.Itisshown thatthemapping :[ 1,+1) [0,+ 1 ) canbetunedtoimprovethequalityofthe discreteapproximation.Itisalsoshownthatcollocationa tLegendre-Gauss-Lobatto (LGL)pointscannotbeusedforsolvinginnite-horizonopt imalcontrolproblems. Apseudospectralmethodtosolveinnite-horizonoptimalc ontrolproblemsusing LGRpointshasbeenpreviouslyproposedbyFahrooetal.inRe f.[ 65 ].Themethods proposedinthisdissertationaresignicantlydifferentf romthemethodofRef.[ 65 ]as themethodsofthisdissertationyieldapproximationstoth estateandcostateonthe entirehorizon,includingapproximationsat t =+ 1 whereasthemethodofRef.[ 65 ] doesnotprovidethesolutionat t = 1 .Furthermore,themethodofRef.[ 65 ]uses atransformationthatgrowsrapidlyas t !1 ,whereasinthisdissertationageneral changeofvariables t = ( ) ofaninnite-horizonproblemtoanite-horizonproblemis considered.Itisshownthatbetterapproximationstotheco ntinuous-timeproblemcan beattainedbyusingafunction ( ) thatgrowsslowlyas t !1 Thisdissertationisdividedintothefollowingchapters.C hapter 2 describesthe mathematicalbackgroundnecessarytounderstandthepseud ospectralmethods usedforsolvingoptimalcontrolproblems.Ageneralcontin uous-timeoptimalcontrol problemisdenedandtherst-ordernecessaryoptimalityc onditionsforthatproblem 21

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arederivedusingthecalculusofvariations.Thedevelopme ntandsolutionmethodsof nite-dimensionalnonlinearprogrammingproblemsaredis cussednext.Lastly,many mathematicalconceptsthatareusedintranscribingthecon tinuous-timeoptimalcontrol problemintoanite-dimensionalNLPusingtheproposedpse udospectralmethod arereviewed.Chapter 3 providesmotivationfortheRadaupseudospectralmethod. ItisdemonstratedthattheLobattopseudospectralmethodh asaninherentdefect inthecostatedynamicsattheboundariesandalthoughtheGa usspseudospectral methoddoesnotsufferfromthisdefect,itlackstheability togivetheinitialcontrol. TheRadaupseudospectralmethodposessesthesameaccuracy asthatoftheGPM, butalsoprovidestheinitialcontrolfromthesolutionofth eNLP.Chapter 4 describesa directtranscriptionmethod,calledtheRadaupseudospect ralmethodthattranscribes acontinuous-timeoptimalcontrolproblemintoadiscreten onlinearprogramming problem.ThemethodusestheLegendre-Gauss-Radau(LGR)po intsforcollocationof thedynamicconstraints,andforquadratureapproximation oftheintegratedLagrange costterm.TheLGRdiscretizationschemeresultsinasetofKKT conditionsthatare equivalenttothediscretizedformofthecontinuousrst-o rderoptimalityconditionsand, hence,providesasignicantlymoreaccuratecostateestim atethanthatobtainedusing theLobattopseudospectralmethod.Inaddition,becauseco llocationisperformedat theLGRpoints,andtheLGRpointsincludetheinitialpoint, thecontrolattheinitialtime isalsoobtainedinthesolutionoftheNLP.Lastly,theprobl emformulationfora ipped RadaupseudospectralmethodthatusestheippedLGRpoints isgiven,wherethe ippedLGRpointsarethenegativeoftheLGRpoints. Next,Chapter 5 presentsauniedframeworkfortwodifferentpseudospectr al methodsbasedoncollocationattheLegendre-Gauss(LG)and theLegendre-Gauss-Radau (LGR)points.Eachoftheseschemescanbeexpressedineither adifferentialoran integralformulation.TheLGandtheLGRdifferentiationan dintegrationmatrices areinvertible,andthedifferentialandintegralversions areequivalent.Eachofthese 22

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schemesprovideanaccuratetransformationbetweentheLag rangemultipliersofthe discretenonlinearprogrammingproblemandthecostateoft hecontinuousoptimal controlproblem.Itisshownthatbothoftheseschemesforma determinedsystem oflinearequationsforcostatedynamics.Theseschemesare differentfromthe pseudospectralmethodbasedoncollocationattheLegendre -Gauss-Lobatto(LGL) points.TheLGLdifferentiationmatrixissingularandhenc etheequivalencebetweenthe differentialandintegralformulationisnotestablished. FortheLGLscheme,thelinear systemofequationsforcostatedynamicsisunder-determin ed.Thetransformation betweentheLagrangemultipliersofthediscretenonlinear programmingproblemand thecostateofthecontinuousoptimalcontrolproblemforth eLGLschemeisfound tobeinaccurate.InChapter 6 ,avarietyofexamplesaresolvedusing,theGauss, theRadau,andtheLobattopseudospectralmethods.Threema inobservationsare madeintheexamples.First,itisseenthattheGaussandtheR adaupseudospectral methodsconsistentlygenerateaccuratestate,controland costatesolutions,while thesolutionobtainedfromtheLobattopseudospectralmeth odisinconsistentand unpredictable.Second,itisseenfortheexamplesthathavee xactanalyticalsolutions thattheerrorinthesolutionsobtainedfromtheGausspseud ospectralmethodand theRadaupseudospectralmethodgoestozeroatanexponenti alrateasthenumber ofdiscretizationpointsareincreased.Third,itisshownt hatnoneofthesemethods arewellsuitedforsolvingproblemsthathavediscontinuit iesinthesolutionorwhere thesolutionslieonasingulararc.Chapter 7 describestwodirectpseudospectral methodsforsolvinginnite-horizonoptimalcontrolprobl emsnumericallyusingthe Legendre-Gauss(LG)andtheLegendre-Gauss-Radau(LGR)co llocation.Theproposed methodsyieldapproximationstothestateandthecostateon theentirehorizon, includingapproximationsat t =+ 1 .Finally,Chapter 8 summarizesthecontributionsof thisdissertationandsuggestsfutureresearchprospects. 23

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CHAPTER2 MATHEMATICALBACKGROUND Inthischapter,rst,ageneralcontinuous-timeoptimalco ntrolproblemisdened andtherst-ordernecessaryoptimalityconditionsfortha tproblemarederivedusing thecalculusofvariations.Pontryagin'sprinciple,which isusedtosolvefortheoptimal controlinsomespecialcases,isalsodiscussed.Next,this chapterdescribesthe mathematicalbackgroundnecessarytounderstandthepseud ospectralmethodsused forsolvingtheoptimalcontrolproblems.Inapseudospectr almethod,thecontinuous functionsoftimeofanoptimalcontrolproblemareapproxim atedandtheproblemis transcribedintoanite-dimensionalnonlinearprogrammi ngproblem(NLP).TheNLP isthensolvedusingwelldevelopedalgorithmsandsoftware .Thedevelopmentand solutionmethodsofnite-dimensionalnonlinearprogramm ingproblemsarediscussed inthischapter.Unconstrained,equalityconstrained,and inequalityconstrained problemsareconsidered.Furthermore,thenecessarycondi tionsforoptimalityor theKarush-Kuhn-Tucker(KKT)conditionsoftheNLParepresent edforeachofthethree cases. Lastly,manymathematicalconceptsthatareusedintranscr ibingthecontinuous-time optimalcontrolproblemintoanite-dimensionalNLPusing theproposedpseudospectral methodarereviewedinthischapter.Therstandthemostimp ortantistheidea ofpolynomialapproximationusingabasisofLagrangepolyn omials.Polynomial approximationisusedforapproximatingthecontinuousfun ctionsoftimeoftheoptimal controlproblem.Anotherimportantconceptistheapplicati onofnumericalmethods forapproximatingthesolutiontodifferentialequations. Inapseudospectralmethod, thedifferentialequationconstraintsofanoptimalcontro lproblemaretranscribed toalgebraicequalityconstraints.Twoapproachesaredisc ussedtotranscribethe differentialequationstoalgebraicequations:time-marc hingmethodsandcollocation. Thelastconceptreviewedinthischapteristhenumericalin tegrationoffunctions. 24

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Low-ordernumericalintegratorsandintegrationusingGau ssianquadratureare discussed.Thepseudospectralmethodofthisresearchuses Legendre-Gauss-Radau quadratureforintegrationandorthogonalcollocationatL egendre-Gauss-Radaupoints forapproximatingthesolutiontodifferentialequations. 2.1OptimalControl Theobjectiveofanoptimalcontrolproblemistodeterminet hestateandthecontrol thatoptimizeaperformanceindexwhilesatisfyingthephys icalanddynamicconstraints ofthesystem.Mathematically,anoptimalcontrolproblemc anbewritteninBolzaform asfollows.Minimizethecostfunctional J =( y ( t 0 ), t 0 y ( t f ), t f )+ Z t f t 0 g ( y ( t ), u ( t ), t ) dt (2–1) subjecttothedynamicconstraints y ( t )= f ( y ( t ), u ( t ), t ), (2–2) theboundaryconditions ( y ( t 0 ), t 0 y ( t f ), t f )= 0 (2–3) andtheinequalitypathconstraints C ( y ( t ), u ( t ), t ) 0 (2–4) where y ( t ) 2 R n isthestate, u ( t ) 2 R m isthecontrol,and t 2 [ t 0 t f ] istheindependent variable.ThecostfunctionaliscomposedoftheMayercost, : R n R R n R R andtheLagrangian, g : R n R m R R .Furthermore, f : R n R m R R n denestheright-handsideofthedynamicconstraints, C : R n R m R R s denesthepathconstraints,and : R n R R n R R q denestheboundary conditions.Therst-ordernecessaryconditionsfortheop timalsolutionoftheproblem giveninEq.( 2–1 )-( 2–4 )arederivedbyusingthecalculusofvariationsasdescribe din Section. 2.1.1 below. 25

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2.1.1CalculusofVariationsandNecessaryConditions Unconstrainedoptimizationproblemsthatdependoncontin uousfunctionsoftime requirethattherstvariation, J ( y ( t )) ,ofthecostfunctional, J ( y ( t )) ,onanoptimalpath y ,vanishforalladmissiblevariations y [ 1 ].Inotherwords, J ( y y )= 0 (2–5) Foraconstrainedoptimizationproblem,anextremalsoluti onisgeneratedfromthe continuous-timerst-ordernecessaryconditionsbyapply ingthecalculusofvariations toanaugmentedcost.Theaugmentedcostisobtainedbyappen dingtheconstraintsto thecostfunctionalusingtheLagrangemultipliers.Theaug mentedcostisgivenas J a =( y ( t 0 ), t 0 y ( t f ), t f ) T ( y ( t 0 ), t 0 y ( t f ), t f ) (2–6) + Z t f t 0 [ g ( y ( t ), u ( t ), t ) T ( t )(_ y ( t ) f ( y ( t ), u ( t ), t )) r T ( t ) C ( y ( t ), u ( t ), t )] dt where ( t ) 2 R n 2 R q ,and r ( t ) 2 R s aretheLagrangemultiplierscorrespondingto Eqs.( 2–2 ),( 2–3 ),and( 2–4 ),respectively.Thequantity ( t ) iscalledthecostateorthe adjoint.Therst-ordervariationwithrespecttoallfreev ariablesisgivenas J a = @ @ y ( t 0 ) y 0 + @ @ t 0 t 0 + @ @ y ( t f ) y f + @ @ t f t f (2–7) T T @ @ y ( t 0 ) y 0 T @ @ t 0 t 0 T @ @ y ( t f ) y f T @ @ t f t f +(( g T (_ y f ) r T C ) j t = t f ) t f (( g T (_ y f ) r T C ) j t = t 0 ) t 0 + Z t f t 0 @ g @ y y + @ g @ u u T (_ y f )+ T @ f @ y y + T @ f @ u u T y r T C r T @ C @ y y r T @ C @ u u dt Fig. 2-1 showsthedifferencesbetween y f and y ( t f ) ,and y 0 and y ( t 0 ) where y ( t ) representsanextremalcurveand y ( t ) representsaneighboringcomparisoncurve.The quantities t 0 and t f aretheinitialandthenaltimes, t 0 and t f arethevariationsinthe initialandthenaltimes,and y 0 and y f arethestateattheinitialandthenaltimes, 26

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y y y y y y y y Figure2-1.Anextremalcurve y ( t ) andacomparisoncurve y ( t ) respectively.Itisnotedthat y ( t f ) isthedifferencebetween y ( t ) and y ( t ) evaluated at t f whereas y f isthedifferencebetween y ( t ) and y ( t ) evaluatedattheendofeach curve.Similarinterpretationsaretruefor y ( t 0 ) and y 0 .AscanbeseeninFig. 2-1 ,the rst-orderapproximationsto y f and y 0 aregivenas y 0 = y ( t 0 )+_ y ( t 0 ) t 0 (2–8) y f = y ( t f )+_ y ( t f ) t f (2–9) Furthermore,thetermcontaining y inEq.( 2–7 )isintegratedbypartssuchthatitcan beexpressedintermsof y ( t 0 ), y ( t f ), and y as Z t f t 0 T y dt = T ( t f ) y ( t f )+ T ( t 0 ) y ( t 0 )+ Z t f t 0 T y dt (2–10) 27

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SubstitutingEqs.( 2–8 ),( 2–9 ),and( 2–10 )intoEq.( 2–7 ),thevariationof J a isgivenas J a = @ @ y ( t 0 ) T @ @ y ( t 0 ) + T ( t 0 ) y 0 (2–11) + @ @ y ( t f ) T @ @ y ( t f ) T ( t f ) y f T + @ @ t 0 T @ @ t 0 ( g + T f r T C ) j t = t 0 t 0 + @ @ t f T @ @ t f +( g + T f r T C ) j t = t f t f + Z t f t 0 @ g @ y + T @ f @ y r T @ C @ y + y + @ g @ u + T @ f @ u r T @ C @ u u T (_ y f ) r T C i dt Therst-orderoptimalityconditionsareformedbysetting thevariationof J a equalto zerowithrespecttoeachfreevariablesuchthat y = f (2–12) @ g @ y + T @ f @ y r T @ C @ y = (2–13) @ g @ u + T @ f @ u r T @ C @ u = 0 (2–14) @ @ y ( t 0 ) + T @ @ y ( t 0 ) = T ( t 0 ), (2–15) @ @ y ( t f ) T @ @ y ( t f ) = T ( t f ), (2–16) @ @ t 0 T @ @ t 0 =( g + T f r T C ) j t = t 0 (2–17) @ @ t f + T @ @ t f =( g + T f r T C ) j t = t f (2–18) = 0 (2–19) DeningtheaugmentedHamiltonianasH ( y ( t ), u ( t ), ( t ), r ( t ), t )= g ( y ( t ), u ( t ), t )+ T ( t ) f ( y ( t ), u ( t ), t ) r T ( t ) C ( y ( t ), u ( t ), t ), (2–20) 28

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therst-orderoptimalityconditionsarethenconvenientl yexpressedas y T ( t )= @ H @ (2–21) T ( t )= @ H @ y (2–22) 0 = @ H @ u (2–23) T ( t 0 )= @ @ y ( t 0 ) + T @ @ y ( t 0 ) (2–24) T ( t f )= @ @ y ( t f ) T @ @ y ( t f ) (2–25) H j t = t 0 = T @ @ t 0 + @ @ t 0 (2–26) H j t = t f = T @ @ t f @ @ t f (2–27) = 0 (2–28) Furthermore,usingthecomplementaryslacknesscondition r takesthevalue r i ( t )=0 when C i ( y ( t ), u ( t )) < 0,1 i s (2–29) r i ( t ) < 0 when C i ( y ( t ), u ( t ))=0,1 i s (2–30) When C i < 0 thepathconstraintinEq.( 2–4 )isinactive.Therefore,bymaking r i ( t )=0 theconstraintissimplyignoredinaugmentedcost.Thenega tivityof r i when C i =0 is interpretedsuchthatimprovingthecostmayonlycomefromv iolatingtheconstraint[ 5 ]. Thecontinuous-timerst-orderoptimalityconditionsofEq s.( 2–21 )–( 2–30 )dene asetofnecessaryconditionsthatmustbesatisedforanext remalsolutionofan optimalcontrolproblem.Thisextremalsolutioncanbeamax ima,minimaorsaddle. Thesecond-ordersufciencyconditionsmustbeinspectedt odeterminewhichofthe extremalsolutionsisaglobalminima.Thederivationofthe second-ordersufciency conditions,however,isbeyondthescopeofthisdissertati on.Foralocalminima,the particularextremalwiththelowestcostischosen. 29

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2.1.2Pontryagin'sPrinciple ThePontryagin'sprincipleisusedtodeterminetheconditi onsforobtainingthe optimalcontrol.Iftheoptimalcontrolisinteriortothefe asiblecontrolset,therst-order necessaryconditionrelatedtocontrolgiveninEq.( 2–23 )isusedtoobtaintheoptimal control.Eq.( 2–23 )iscalledthestrongformofthePontryagin'sprinciple.If ,however, thesolutionliesontheboundaryofthefeasiblecontroland stateset,thestrongform ofthePontryagin'sprinciplemaynotbeusedtocomputetheo ptimalcontrolasthe differentialisone-sided.Suchacontroliscalleda“bang-b ang”control.Furthermore,if theHamiltoniandenedinEq.( 2–20 )islinearincontrol,i.e., H ( y ( t ), u ( t ), ( t ), r ( t ), t )= g ( y ( t ), t )+ T ( t ) f ( y ( t ), t )+ T ( t ) u ( t ) r T ( t ) C ( y ( t ), t ), (2–31) thederivativeinEq.( 2–23 )doesnotprovideanyinformationabouttheoptimalcontrol Forsuchproblems,theweakformofthePontryagin'sPrincipl emustbeusedto determinetheoptimalcontrol[ 66 ]. Thecontrol u thatgivesalocalminimumofthecost J isbydenition[ 1 ], J ( u ) J ( u )= J ( u u ) 0, (2–32) foralladmissiblecontrol u 2 U sufcientlycloseto u .If u isdenedas u = u + u ,then thechangeinthecostcanbeexpressedas J ( u u )= J ( u u )+ higherorderterms (2–33) If u issufcientlysmall,thenthehigherordertermsapproachz eroandthecosthasa localminimumif J ( u u ) 0. (2–34) Attheoptimalsolution, y u , r ,thedifferentialequationsalongwiththe boundaryconditionsandpathconstraintsaresatised.The refore,allthecoefcients 30

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ofthevariationtermsinEq.( 2–11 )arezero,exceptthecontrolterm.Thisleavesthe variationofthecostas J a ( u u )= Z t f t 0 @ g @ u + T @ f @ u r T @ C @ u y ,u , r u dt (2–35) = Z t f t 0 @ H @ u y ,u , r u dt Thevariationofthecostistheintegraloftherst-orderap proximationtothechangein theHamiltoniancausedbyachangeinthecontrolalone.The rst-orderapproximation ofthechangeintheHamiltonianisbydenition @ H @ u y ,u , r u = H ( y u + u , r t ) H ( y u , r t ). (2–36) Thevariationofthecostforalladmissibleandsufciently small u becomes J a ( u u )= Z t f t 0 ( H ( y u + u , r t ) H ( y u , r t ) ) dt (2–37) Inorderthat J ( u u ) isnon-negativeforanyadmissiblevariationinthecontrol ,the HamiltonianmustbegreaterthantheoptimalHamiltonianfo ralltime H ( y u + u , r t ) H ( y u , r t ). (2–38) Therefore,theoptimalcontrolistheadmissiblecontrolth atminimizestheHamiltonian. TheweakformofthePontryagin'sprincipleisstatedas[ 67 ] u ( t )= argmin u 2 U [ H ( y ( t ), u ( t ), ( t ), r ( t ), t )]. (2–39) ThePontraygin'sprincipleinEq.( 2–39 )isusedtoobtainoptimalcontrolwhenthe controlisbang-bangorwhentheHamiltonianislinearincon trol. 2.2NumericalOptimization Anonlinearprogrammingproblem(NLP)arisesinoptimalcont rolwhena continuous-timeoptimalcontrolproblemisdiscretized.I nthissection,thedevelopment ofsolutionstothenite-dimensionaloptimizationproble msornonlinearprogramming 31

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problems(NLP)arediscussed.TheobjectiveofaNLPistonda setofparameters thatminimizessomecostfunctionthatissubjecttoasetofa lgebraicequalityor inequalityconstraints.TheNLPissolvedusingwelldevelo pedalgorithmsand software.Unconstrainedminimization,equalityconstrai nedminimization,andinequality constrainedminimizationofafunctionarenowdiscussed.T henecessaryconditionsfor optimalityortheKarush-Kuhn-Tucker(KKT)conditionsoftheN LParealsopresentedfor eachofthethreecases.2.2.1UnconstrainedOptimization Theobjectiveofanunconstrainedoptimizationproblemist ondthesetof parametersthatgivesaminimumvalueofascalarfunction.C onsiderthefollowing problemofdeterminingtheminimumofafunctiondenedas[ 68 ] J ( y ), (2–40) where y =( y 1 ,..., y n ) T 2 R n .If y isalocallyminimizingsetofparametersthenthe minimumvalueoftheobjectivefunctionis J ( y ) .For y tobealocallyminimizingpoint, theobjectivefunctionevaluatedatanyneighboringpoint, y ,mustbegreaterthanthe optimalcost,i.e., J ( y ) > J ( y ). (2–41) For y tobealocallyminimizingpoint,therst-ordernecassaryc onditionisstatedas g ( y )= 0 (2–42) where g ( y ) isthegradientvectordenedas g ( y ) r y J T = 266666664 @ J @ y 1 @ J @ y 2 ... @ J @ y n 377777775 (2–43) 32

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Thegradientconditionisasetof n conditionstodeterminethe n unknownvariablesof vector y .Thenecessaryconditionbyitselfonlydenesanextremalp ointwhichcanbe alocalminimum,maximum,orsaddlepoint.Inordertodevelo pthesufcientcondition deningalocallyminimizingpoint y ,considerathreetermTaylorseriesexpansionof J ( y ) abouttheextremalpoint, y .Theobjectivefunctionat y isapproximatedas J ( y )= J ( y )+ g T ( y )( y y )+ 1 2 ( y y ) T H ( y )( y y )+ higherorderterms (2–44) where H ( y ) isthesymmetric n n Hessianmatrixdenedas H ( y ) r yy J @ 2 J @ y 2 = 266666664 @ 2 J @ y 2 1 @ 2 J @ y 1 @ y 2 ... @ 2 J @ y 1 @ y n @ 2 J @ y 2 @ y 1 @ 2 J @ y 2 2 ... @ 2 J @ y 2 @ y n ... @ 2 J @ y n @ y 1 @ 2 J @ y n @ y 2 ... @ 2 J @ y 2 n 377777775 (2–45) If y y issufcientlysmall,higherordertermscanbeignored.Also ,becauseofthe rst-ordernecessaryconditiongiveninEq.( 2–42 ), J ( y )= J ( y )+ 1 2 ( y y ) T H ( y )( y y ). (2–46) Fromtheinequalitygivenin( 2–41 ), J ( y )+ 1 2 ( y y ) T H ( y )( y y ) > J ( y ), ( y y ) T H ( y )( y y ) > 0. (2–47) Inordertoensure y isalocalminimum,theadditionalconditionin( 2–47 )mustbe satised.Eqs.( 2–42 )and( 2–47 )togetherdenenecessaryandsufcientconditionsfor alocalminimum.2.2.2EqualityConstrainedOptimization Considerndingtheminimumoftheobjectivefunction J ( y ), (2–48) 33

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subjecttoasetof m n constraints f ( y )= 0 (2–49) Similartothecalculusofvariationsapproachfordetermini ngtheextremaloffunctionals, inndingtheminimumofanobjectivefunctionsubjecttoequ alityconstraints,an augmentedcostcalledtheLagrangianisused.DenetheLagr angianas L ( y )= J ( y ) T f ( y ), (2–50) where =( 1 ,..., m ) T 2 R m aretheLagrangemultipliers.Then,thenecessary conditionsfortheminimumoftheLagrangianisthatthepoin t ( y ) satises r y L ( y )= 0 (2–51) r L ( y )= 0 (2–52) wherethegradientof L withrespectto y and is r y L = g ( y ) G T ( y ) (2–53) r L = f ( y ), (2–54) where G ( y ) istheJacobianmatrix,denedas G ( y ) @ f @ y = 266666664 @ f 1 @ y 1 @ f 1 @ y 2 ... @ f 1 @ y n @ f 2 @ y 1 @ f 2 @ y 2 ... @ f 2 @ y n ... @ f m @ y 1 @ f m @ y 2 ... @ f m @ y n 377777775 (2–55) ItisnotedthatataminimumoftheLagrangian,theequalityc onstraintofEq.( 2–49 )is satised.Next,thisnecessaryconditionalonedoesnotspe cifyaminimum,maximumor saddlepoint.Inordertospecifyaminimum,rst,denetheH essianoftheLagrangian 34

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as H L = r yy L = r yy J m X i =1 i r yy f i (2–56) Then,asufcientconditionforaminimumisthat v T H L v > 0 (2–57) foranyvector v intheconstrainttangentspace. 2.2.3InequalityConstrainedOptimization Considertheproblemofminimizingtheobjectivefunction J ( y ), (2–58) subjecttotheinequalityconstraints c ( y ) 0 (2–59) Inequalityconstrainedproblemsaresolvedbydividingthe inequalityconstraintsinto asetofactiveconstraints,andasetofinactiveconstraint s.Attheoptimalsolution y someoftheconstraintsaresatisedasequalities,thatis c i ( y )=0, i 2 A (2–60) where A iscalledtheactiveset,andsomeconstraintsarestrictlys atised,thatis c i ( y ) < 0, i 2 A 0 (2–61) where A 0 iscalledtheinactiveset. Byseparatingtheinequalityconstraintsintoanactiveseta ndaninactiveset,the activesetcanbedealtwithasequalityconstraintsasdescr ibedintheprevioussection, andtheinactivesetcanbeignored.Theaddedcomplexityofi nequalityconstrained problemsisindeterminingwhichsetofconstraintsareacti ve,andwhichareinactive.If theactiveandinactivesetsareknown,aninequalityconstr ainedproblembecomesan 35

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equalityconstrainedproblemstatedasfollows.Minimizet heobjectivefunction J ( y ), (2–62) subjecttotheconstraints c i ( y )=0, i 2 A (2–63) andthesamemethodologyisappliedasintheprevioussectio ntodeterminea minimum. Finally,considertheproblemofndingtheminimumoftheob jectivefunction, J ( y ) subjecttotheequalityconstraints f ( y )= 0 (2–64) andtheinequalityconstraints c ( y ) 0 (2–65) Theinequalityconstraintsareseparatedintoactiveandin activeconstraints.Then, denetheLagrangianas L ( y , ( A ) )= J ( y ) T f ( y ) ( A ) T c ( A ) ( y ), (2–66) where aretheLagrangemultiplierswithrespecttotheequalityco nstraintsand ( A ) aretheLagrangemultipliersassociatedwiththeactiveset ofinequalityconstraints. Furthermore,notethattheinactivesetofconstraintsarei gnoredbychoosing ( A 0 ) = 0 NecessaryconditionfortheminimumoftheLagrangianistha tthepoint ( y , ( A ) ) satises r y L ( y , ( A ) )= 0 (2–67) r L ( y , ( A ) )= 0 (2–68) r ( A ) L ( y , ( A ) )= 0 (2–69) 36

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Manynumericalalgorithms[ 69 – 71 ]existforsolvingthenonlinearprogramming problems.ExamplesincludetheNewton'smethod,conjugated irectionmethods,and gradient-basedmethodssuchasthesequentialquadraticpr ogramming(SQP)and theinterior-pointmethods.Variousrobustandversatiles oftwareprogramshavebeen developedforthenumericalsolutionoftheNLPs.Examplesofw ell-knownsoftwarethat usetheSQPmethodsincludethedenseNLPsolverNPSOL[ 11 ]andthesparseNLP solverSNOPT[ 12 ].WellknownsparseinteriorpointNLPsolversincludeKNITR O[ 13 ] andIPOPT[ 14 ]. 2.3Finite-DimensionalApproximation Inthissection,theimportantmathematicalconceptsthata reusedtotranscribea continuous-timeoptimalcontrolproblemtoanonlinearpro grammingproblem(NLP)are reviewed.Threeconceptsareimportantinconstructingadi scretizednite-dimensional optimizationproblemfromacontinuous-timeoptimalcontr olproblem.Thesethree conceptsarepolynomialapproximation,numericalsolutio nofdifferentialequations,and numericalintegration.Inordertotranscribeacontinuous -timeoptimalcontrolproblem toaNLP,theinnite-dimensionalcontinuousfunctionsoft heoptimalcontrolproblem areapproximatedbyanite-dimensionalLagrangepolynomi albasis.Furthermore,the dynamicconstraintsaretranscribedtoalgebraicconstrai ntsbysettingthederivative ofthestateapproximation(obtainedusingLagrangepolyno mialapproximation),equal totheright-handsideofdynamicconstraintsofEq.( 2–2 )ataspeciedsetofpoints calledthecollocationpoints.Lastly,theLagrangecostis approximatedbynumerical integrationusingtheGaussianquadrature.2.3.1PolynomialApproximation Inthisresearch,Lagrangepolynomialsareusedtoapproxim atecontinuous functionsoftimelikestate,controlandcostate.TheLagra ngepolynomialapproximation [ 72 ]isbasedonthefactthatgivenasetof N arbitrarysupportpoints, ( t 1 ,..., t N ) ,called thediscretizationpointsofacontinuousfunctionoftime, y ( t ) ,ontheinterval t i 2 [ t 0 t f ] 37

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thereexistsauniquepolynomial, Y ( t ) ,ofdegree N 1 suchthat Y ( t i )= y ( t i ),1 i N (2–70) Theuniquepolynomialapproximationtothefunction, y ( t ) ,isgivenbytheLagrange polynomialapproximationformula Y ( t )= N X i =1 y i L i ( t ), (2–71) where y i = y ( t i ) and L i ( t ) aretheLagrangepolynomials[ 73 ],denedas L i ( t )= N Y k =1 k 6 = i t t k t i t k (2–72) ItisnotedthattheseLagrangepolynomialssatisfytheisol ationproperty,i.e.,theyare oneatthe i th discretizationpointandzeroatallothers,sothat L i ( t k )= ik = 8><>: 1: k = i 0: k 6 = i (2–73) Thispropertyisparticularlyadvantageousinthenite-di mensionaltranscriptionof theoptimalcontrolproblem.Asnotedbefore,anoptimalcont rolproblemcomprises ofvariousfunctionals(functionsofstateandcontrol).Asw illbeseenfurtherinthis chapter,inordertotranscribethecontinuous-timeproble mtoanite-dimensional problem,thesefunctionalsareevaluatedatsomeorallofth ediscretizationpointsused inconstructingtheLagrangepolynomials.Whenthesefuncti onalsareevaluatedat adiscretizationpoint,isolatedstateandcontrolatthatp articulardiscretizationpoint appearintheexpressionandnotalinearcombinationofallt hesupportpoints.Asa resultofwhichtheJacobianmatrixoftheconstraintsofthe NLPdenedinEq.( 2–55 )is asparsematrix. 38

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2.3.1.1Approximationerror TheerrorintheLagrangeapproximationformulaforfunctio nsinwhich N derivatives existin [ t 0 t f ] isknowntobe[ 74 ] y ( t ) Y ( t )= ( t t 1 ) ( t t N ) N y N ( ), (2–74) where y N ( ) isthe N th derivativeofthefunction y ( t ) evaluatedatsome 2 [ t 0 t f ] .Itis notedthatatanysupportpoint,theerroriszero.Furthermo re,if y ( t ) isapolynomialof degreeatmost N 1 ,the N th derivativeinEq.( 2–74 )vanishes.Thus,itisseenfrom Eq.( 2–74 )thattheLagrangeinterpolationapproximationusing N discretizationpointsis exactforpolynomialsofdegreeatmost N 1 .However,thebehavioroftheinterpolation erroras N approachesinnityfornon-polynomialfunctionsoranalyt icfunctionswith singularitiesisratherinterestingandischaracterizeda stheRungephenomenon[ 75 ]. TheRungephenomenonexistsforLagrangepolynomialsforau niformlydistributed setofdiscretizationpoints.TheRungephenomenonistheoc curenceofoscillations intheapproximatingfunction, Y ( t ) ,betweendiscretizationpoints.Asthenumberof discretizationpointsgrows,themagnitudeoftheoscillat ionsbetweensupportpoints alsogrows.TheRungephenomenoncanbedemonstratedonthef ollowingfunction denedintimeinterval 2 [ 1,+1] y ( )= 1 1+50 2 2 [ 1,+1]. (2–75) Fig. 2-2 showstheLagrangepolynomialapproximationtothefunctio n y ( )= 1 = (1+50 2 ) utilizing N =11 and N =41 uniformlydistributeddiscretization points,i.e.,usinga 10 th -degreeanda 40 th -degreeLagrangepolynomialbasis.Itis seenthatasthenumberofpointsareincreased,theapproxim ationneartheends oftheintervalbecomesincreasingly worse andtheerrorattheendsismuchlarger thantheerrorinthemiddleoftheinterval.Inordertoavoid theRungephenomenon inLagrangepolynomialapproximation,thediscretization pointsmustbechosenso 39

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4 3 2 1 1 0.5 0 0 0.5 1 1 y ( ) Y ( )y ( ) AApproximationof y ( )=1 = (1+50 2 ) using 11 uniformdiscretizationpoints. 10 6 1 0.5 0.5 0 0 0.5 0.5 1 1 1.5 2 2.5 y ( ) Y ( )y ( ) BApproximationof y ( )=1 = (1+50 2 ) using 41 uniformdiscretizationpoints. Figure2-2.Approximationof y ( )=1 = (1+50 2 ) using 11 and 41 uniformlyspaced discretizationpoints. 40

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thattheerrorismoreequitablydistributedandthemaximum errorontheintervalis minimized.Non-uniformdiscretizationpointsobtainedfr omorthogonalpolynomials liketheChebyshevpolynomialsandtheLegendrepolynomial sarecommonlyusedas discretizationpointstoavoidtheRungephenomenon.2.3.1.2FamilyofLegendre-Gausspoints Inpseudospectralmethods,threesetsofpointsarecommonl yusedasdiscretization pointsinLagrangepolynomialapproximation:Legendre-Ga uss-Lobatto(LGL)points, Legendre-Gauss(LG)points,andLegendre-Gauss-Radau(LG R)points.Allthreesets ofpointsaredenedonthedomain [ 1,+1] ,butdiffersignicantlyinthattheLGpoints includeneitheroftheendpoints,theLGRpointsincludeone oftheendpoints,andthe LGLpointsincludebothoftheendpoints.Inaddition,theLG Rpointsareasymmetric relativetotheoriginandarenotuniqueinthattheycanbede nedusingeithertheinitial pointortheterminalpoint.TheLGRpointsthatincludethet erminalendpointareoften calledthe ipped LGRpoints.Aschematicrepresentationofthesepointsissh ownin Fig. 2-3 .ThesesetsofpointsareobtainedfromtherootsofaLegendr epolynomial and/orlinearcombinationsofaLegendrepolynomialandits derivatives.Denotingthe N th degreeLegendrepolynomialby P N ( ) ,then LG:Rootsobtainedfrom P N ( ) LGR:Rootsobtainedfrom P N 1 ( )+ P N ( ) LGL:Rootsobtainedfrom P N 1 ( ) togetherwiththepoints-1and1 Becausethesepointsaredenedonthedomain [ 1,+1] ,thetimedomain [ t 0 t f ] isrstmappedto [ 1,+1] ,byusingthefollowingafnetransformation t = t f t 0 2 + t f + t 0 2 (2–76) Let ( 1 ,..., N ) bethe N LGLpointssuchthat 1 = 1 and N =1 ,thenafunction y ( ) isapproximatedontheinterval [ 1,+1] usingtheLagrangepolynomialsandtheLGL 41

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1 0.5 0 0.5 1 LG LGR LGL LGR-f Figure2-3.SchematicdiagramofLegendrepoints.pointsas Y ( )= N X i =1 y i L i ( ). (2–77) Next,let ( 1 ,..., N ) bethe N LGRpointssuchthat 1 = 1 and N < 1 ,thenthefunction y ( ) isapproximatedontheinterval [ 1,+1] usingtheLagrangepolynomialsandthe LGRpointsbydening N +1 =1 andusingthefollowingapproximation Y ( )= N +1 X i =1 y i L i ( ). (2–78) Lastly,let ( 1 ,..., N ) bethe N LGpointssuchthat 1 > 1 and N < 1 ,thenthefunction y ( ) isapproximatedontheinterval [ 1,+1] usingtheLagrangepolynomialsandthe LGpointsbydening 0 = 1 andusingthefollowingapproximation Y ( )= N X i =0 y i L i ( ). (2–79) 42

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1 1 0.80.6 0.5 0.4 0.2 0 0 0.5 1 y ( ) Y ( )y ( ) AApproximationof y ( )=1 = (1+50 2 ) using 11 LGdiscretizationpoints. 1 1 0.80.6 0.5 0.4 0.2 0 0 0.5 1 y ( ) Y ( )y ( ) BApproximationof y ( )=1 = (1+50 2 ) using 41 LGdiscretizationpoints. Figure2-4.Approximationof y ( )=1 = (1+50 2 ) using 11 and 41 LGdiscretization points. 43

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-0.2 1 1 0.80.8 0.5 0.4 0.2 0 0 0.5 1 y ( ) Y ( )y ( ) AApproximationof y ( )=1 = (1+50 2 ) using 11 LGRdiscretizationpoints. 1 1 0.80.6 0.5 0.4 0.2 0 0 0.5 1 y ( ) Y ( )y ( ) BApproximationof y ( )=1 = (1+50 2 ) using 41 LGRdiscretizationpoints. Figure2-5.Approximationof y ( )=1 = (1+50 2 ) using 11 and 41 LGRdiscretization points. 44

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Fig. 2-4 andFig. 2-5 showtheLagrangepolynomialapproximationtothefunction y ( )=1 = (1+50 2 ) ,utilizing N =11 and N =41 non-uniformLGandLGRdiscretization points,respectively.ItisseenthatbyusingtheLGandtheL GRdiscretizationpoints,the Rungephenomenonisavoided,andtheapproximationsbecome better asthenumber ofdiscretizationpointsisincreased. 5 10 0 0 5 10 15 20 20406080100 Uniform LGR LGE yN Figure2-6.Basetenlogarithmofinnitynormerrorvs.numbe rofdiscretizationpoints, N ,forapproximating y ( )=1 = (1+50 2 ) Letthe log 10 maximuminnitynormoferrorbedenedas E y =max k log 10 jj Y ( k ) y ( k ) jj 1 (2–80) Fig. 2-6 showstheerror, E y ,asafunctionofthenumberofdiscretizationpointsused inconstructingtheLagrangepolynomialsfromtheuniforml ydistributeddiscretization points,theLG,andtheLGRdiscretizationpointsforthefun ction y ( )=1 = (1+50 2 ) .It isseenthatasthenumberofdiscretizationpointsincrease s,theLagrangepolynomial approximationusingauniformlydistributedsetofdiscret izationpoints diverges .For 100 45

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uniformlydistributeddiscretizationpoints,erroris O (10 18 ) .ForaLagrangepolynomial denedbyeithertheLGortheLGRsupportpoints,theapproxi mation converges tothe function.For 100 LGorLGRdiscretizationpoints,theerrorinapproximation is O (10 6 ) 2.3.2NumericalSolutionofDifferentialEquations Anotherimportantconceptintranscribingthecontinuous-t imeoptimalcontrol problemtoanite-dimensionalNLPisnumericallyapproxim atingthesolutiontothe differentialequationsbytranscribingthedynamicconstr aintsinEq.( 2–2 ),toalgebraic equations.Considerthefollowingdifferentialequationw hosesolutionisdesiredinthe timeinterval [ t 0 t f ] y ( t )= f ( y ( t ), t ), y ( t 0 )= y 0 (2–81) If ( t 1 ,..., t N ) 2 [ t 0 t f ] ,thensolvingthedifferentialequationinEq.( 2–81 )numerically involvesapproximating ( y ( t 1 ),..., y ( t N )) .Twoapproachesforsolvingsuchadifferential equationarenowconsidered:time-marchingmethodsandcol location[ 4 ]. 2.3.2.1Time-marchingmethods Supposethattheinterval [ t 0 t f ] isdividedinto N intervals [ t i t i +1 ] .Numerical methodsforsolvingdifferentialequationsaresometimesi mplementedinmultiplesteps, i.e.,thesolutionattime t i +1 isobtainedfromadenedsetofpreviousvalues ( t i j ,..., t i ) where j isthenumberofsteps.Thesimplestmultiple-stepmethodis asingle-step methodwith j =1 .TheEulermethodsarethemostcommonsingle-stepmethods.T he Eulermethodshavethegeneralform[ 3 ] y i +1 = y i + h i ( f i +(1 ) f i +1 ), (2–82) where f i = f ( y ( t i ), t i ) h i = t i +1 t i ,and 2 [0,1] .Thevalues =(1,1 = 2,0) correspondrespectivelytotheparticularEulermethodscal ledtheEulerforward, theCrank-Nicolson,andtheEulerbackwardmethod.Morecomp lexmultiple-step methodsinvolvetheuseofmorethanoneprevioustimepoint, i.e., j > 1 .Thetwomost 46

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commonlyusedmultiple-stepmethodsaretheAdams-Bashforth andAdams-Moulton multiple-stepmethods. TheEulerbackwardandtheCrank-Nicolsonareexamplesofimp licitmethods becausethevalue y ( t i +1 ) appearsimplicitlyontheright-handsideofEq.( 2–82 ) whereastheEulerforwardisanexampleofanexplicitmethodb ecausethevalue y ( t i +1 ) doesnotappearontheright-handsideofEq.( 2–82 ).Whenemployinganimplicit method,thesolutionat t i +1 isobtainedusingapredictor-correctorwherethepredicto r istypicallyanexplicitmethod(e.g.,Euler-forward)while thecorrectoristheimplicit formula.Theimplicitmethodsaremorestablethantheexpli citmethods,butanimplicit methodrequiresmorecomputationateachstep. Next,supposethatwedivideeachofthe N intervals [ t i t i +1 ] ,into K subintervals [ k k +1 ] where k = t i + h i k ,1 k K (2–83) where h i = t i +1 t i and k 2 [0,1] .Then,thestateat t i +1 isapproximatedas y ( t i +1 )= y ( t i )+ h i K X k =1 k f ( y ( k ), k ),1 i N (2–84) Thevalueofstateat k 1 k K ,inturnisapproximatedas y ( k )= y ( t i )+ h i K X j =1 r kj f ( y ( j ), j ),1 k K (2–85) Therefore,thestateat t i +1 isobtainedas y ( t i +1 )= y ( t i )+ h i K X k =1 k f y ( t i )+ h i K X j =1 r kj f ( y ( j ), j ), t i + h i k = y ( t i )+ h i K X k =1 k f ik (2–86) 47

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where f ik = f y ( t i )+ h i K X j =1 r kj f ( y ( j ), j ), t i + h i k = f y ( t i )+ h i K X j =1 r kj f ij (2–87) TheapproximationobtainedinEq.( 2–86 )iscalledthe K th –orderRunge-Kuttamethod [ 72 75 ]whichisamultiple-stagemethodwith K beingthenumberofstages.A Runge-Kuttamethodiscalledexplicitif r kj =0 forall j k andiscalledimplicit otherwise.InanexplicitRunge-Kuttamethod,theapproxim ationat t i +1 iscomputed usinginformationpriorto t i +1 whereasinanimplicitRunge-Kuttamethod y ( t i +1 ) is requiredinordertodeterminethesolutionat t i +1 .IntheimplicitRunge-Kuttacase,the solutionisupdatedusingapredictor-correctorapproach. ARunge-KuttamethodiscapturedsuccinctlyusingthewellknownButcherarray [ 76 ].Themostwell-knownRunge-Kuttamethodsaretheclassica lRunge-Kuttamethod whichisa 4 th –ordermethod,theHermite-Simpsonmethodwhichisa 3 rd –ordermethod. TheEulermethodsare,infact, 1 st –orderRunge-Kuttamethods. 2.3.2.2Collocation Anotherapproachtotranscribecontinuous-timedifferenti alequationstonite-dimensional algebraicequationsisbycollocation.Supposeoveraninter val [ t 0 t f ] wechoose toapproximatethestateusingthefollowingpolynomialapp roximation[ 57 ]and ( t 1 ,..., t N ) 2 [ t 0 t f ] asdiscretizationpoints: y ( t ) Y ( t )= N X i =1 a i i ( t ), (2–88) where a i = a ( t i ) isthetestfunctionevaluatedatthe i th supportpointand i ( t ) isatrial function.Thetrialfunctionsareusedasabasisforthetrun catedseriesexpansionof thesolution,whilethetestfunctionsareusedtoensuretha tthedifferentialequationis satisedascloselyaspossible.Typicaltrialfunctionsar eorthogonalpolynomialsand 48

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trigonometricfunctions.Inspectralmethods,thetrialfu nctionschosenareinnitely differentiableglobalfunctions. Inapseudospectralmethod,thetestfunctionsusedaretheL agrangepolynomials. Furthermore,thetestfunctionisthestate, y ( t ) ,itself,sothat a i = y ( t i ),1 i N (2–89) Thecontinuous-timeapproximationtostateisthengivenas y ( t ) Y ( t )= N X i =1 y ( t i ) L i ( t ). (2–90) Thederivativeofthisstateapproximationevaluatedatase tofpoints, ( 1 ,..., K ) 2 [ t 0 t f ] isequatedtotheright-handsideofthestatedynamicsequat iondenedin Eq.( 2–81 )evaluatedatthesamesetofpoints,i.e., Y ( j )= N X i =1 y ( t i ) L i ( j )= f ( y ( j ), j ),1 j K (2–91) Equation( 2–91 )iscalledacollocationconditionbecausetheapproximati ontothe derivativeissetequaltotheright-handsideofthediffere ntialequationevaluatedateach oftheintermediatepoints, ( 1 ,..., K ) ,calledthecollocationpoints. Asubsetofcollocationmethodsthathaveseenextensiveuse inoptimalcontrol aretheorthogonalcollocationmethods[ 77 ].Thekeydifferencebetweenanorthogonal collocationmethodandastandardcollocationmethodisthe mannerinwhichthe collocationpointsarechosen.Specically,inanorthogona lcollocationmethod thecollocationpointsaretherootsofapolynomialthatisa memberofafamilyof orthogonalpolynomials.Commoncollocationpointsinorth ogonalcollocationarethose obtainedfromtherootsoftheChebyshevpolynomialsortheL egendrepolynomials. Furthermore,thestateinanorthogonalcollocationmethod istypicallyapproximated onthetimeinterval 2 [ 1,+1] asthesepointsaredenedonthedomain [ 1,+1] 49

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Inpseudospectralmethods,theLG,theLGR,andtheLGLareco mmonlyusedsetof collocationpoints.2.3.3NumericalIntegration Thelaststepinnite-dimensionaldiscretizationofanopt imalcontrolproblemisto approximatethecostfunctionofEq.( 2–1 )usingnumericalintegration.Typically,thecost isapproximatedbyusinganumericalquadrature.Numerical quadraturecangenerally bedescribedassamplinganintegrandatanitenumberofpoi nts ( t 1 ,..., t N ) 2 [ t 0 t f ] andusingaweightedsumofthesepointstoapproximatethein tegralas Z t f t 0 g ( t ) dt = N X i =1 w i g ( t i ), (2–92) where w i isthequadratureweightassociatedtothe i th samplingpoint. Thequadratureruleusedforintegrationisconsistentwith thenumericalmethod usedforsolvingthedifferentialequation.Ifoneisusinga Runge-Kuttamethodfor solvingthedifferentialequation,thecostwouldalsobeap proximatedusingRunge-Kutta integration.Inthecaseofanorthogonalcollocationmetho d,theintegrationruleisan orthogonallycollocatedquadraturerulei.e.,ifoneisusi ngLegendre-Gausspoints,then theLagrangecostisapproximatedusingaGaussquadrature. Therequirementforsuch aconsistencyintheapproximationofthedifferentialequa tionsandthecostcanbe explainedasfollows. AnyBolzacostcanbeconvertedtoaMayercostbyintroducingan ewstate y n +1 andaddingthedifferentialequation y n +1 = g ( y ( t ), u ( t )), (2–93) withtheinitialcondition y n +1 ( t 0 )=0. (2–94) 50

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ThenthecostfunctionalofEq.( 2–1 )isgiveninMayerformas J =( y ( t 0 ), t 0 y ( t f ), t f )+ y n +1 ( t f ). (2–95) Acommonschemewouldthenbeusedtosolvetheaugmentedsyst emofdifferential equations y ( t )= f ( y ( t ), u ( t )), y n +1 = g ( y ( t ), u ( t )). (2–96) ConvertingbacktoBolzaform,thequadratureapproximation totheterm Z t f t 0 g ( y ( t ), u ( t )) dt inEq.( 2–1 )mustbethesameastheschemeusedtosolvethesystemofdiff erential equationsgiveninEq.( 2–96 ). Inthissection,rst,low-ordernumericalintegratorsare brieydiscussed.Next, theGaussianquadraturerulesarepresentedanditisshownt hatthequadraturerules associatedwiththeLGpointsareexactforthepolynomialso fdegreeatmost 2 N 1 theLGRquadratureruleisexactforthepolynomialsofdegre eatmost 2 N 2 andlastly, theLGLquadratureisexactforthepolynomialsofdegreeatm ost 2 N 3 2.3.3.1Low-orderintegrators Acommontechniqueusedtoapproximatetheintegralofafunc tionistouse low-degreepolynomialapproximationsthatareeasilyinte gratedovermanysubintervals. Theapproximationtotheintegralofafunctionisthenthesu moftheintegralsineach approximatingsubinterval.Whenthesesubintervalsareuni formlydistributed,these low-orderintegrationrulesarecommonlyknownastheNewto n-Cotesintegrationrules. Awell-knownlow-ordermethodisthecompositetrapezoidal rule[ 78 ].Thecomposite trapezoidruledividesafunctionintomanyuniformlydistr ibutedsubintervalsandeach subintervalisapproximatedbyastraightlineapproximati onthatpassesthroughthe 51

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functionattheendsofthesubinterval.Fig. 2-7 shows g ( )=1 = (1+50 2 ), 2 [ 1,1], andafourintervalcompositetrapezoidruleapproximation ofthefunction. 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.5 1 g ( )g ( )Approx. Figure2-7. g ( )=1 = (1+50 2 ) vs. andafourintervaltrapezoidruleapproximation. For N approximatingintervals,thecompositetrapezoidruleisg ivenas[ 78 ] Z t f t 0 g ( t ) dt t f t 0 2 N ( g ( t 0 )+2 g ( t 1 )+2 g ( t 2 )+ +2 g ( t N 1 )+ g ( t N )), (2–97) where ( t 0 ,..., t N ) arethegridpointssuchthatthepoints ( t i 1 t i ) denethebeginning andtheendofthe i th interval.Considerusingthecompositetrapezoidruletoap proximate I = Z 1 1 1 (1+50 2 ) d (2–98) Fig. 2-8 showsthe log 10 errorfromthetrapezoidruleforapproximatingEq.( 2–98 )asa functionofthe log 10 numberofapproximatingintervals.AsseeninFig. 2-8 ,inorderto achievehighlevelsofaccuracyintheapproximation,manya pproximatingintervalsare required.For onehundredthousand approximatingintervals,theerroris O (10 12 ) .The rateofconvergenceofapproximationtotheactualintegral islinearinthiscase. 52

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TherateofconvergencefortheNewton-Cotesruleisatbestp olynomial.The compositerectanglerule,theRombergintegrationformula ,thecompositeSimpson's rule,andtheSimpson's 3 = 8 th ruleareafewotherlow-ordernumericalintegration methods.InthecompositeSimpson'srule,forexample,aseco nd-degreepolynomialis usedtoapproximatethefunctionineachsubinterval. 2 4 6 8 10 12 14 1 1.5 2 2.53 4 56 3.54.5 5.5log 10 Errorlog 10 NumberofApproximatingIntervals Figure2-8.Basetenlogarithmoferrorvs.basetenlogarithm ofnumberofintervalsfor trapezoidruleapproximationofEq.( 2–98 ). 2.3.3.2Gaussianquadrature ThetraditionalNewton-Cotesformulaebasedonevenlyspac edpointsdonot convergeformanywell-behavedintegrandsasthenumberofs upportpointsis increasedand,therefore,arenotrobustwhenbeingusedash ighdegreeapproximations. IncontrasttothecompositeNewton-Cotesrule,anotherapp roachistousehighly accurateGaussianquadratures.IncaseofGaussianquadrat ure,instabilityofthe approximationdoesnotexistasisthecasewiththeNewton-C otesrules.Furthermore, veryhighordersofaccuracyareobtainedforevenafewquadr aturepoints.Inusinga 53

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N pointGaussianquadrature,twoquantitiesmustbespecied foreachpoint,namely, thelocationofthepoint,andaweightingfactormultiplyin gthevalueofthefunction evaluatedatthispoint. ThegoalofaGaussianquadratureistoconstructan N pointapproximation Z 1 1 g ( ) d = N X i =1 w i g i (2–99) where w i isthequadratureweightassociatedwithpoint i and g i = g ( i ) ,suchthat theapproximationinEq.( 2–99 )isexactforthepolynomial g ( ) ofaslargeadegree aspossible[ 78 ].Let E N ( g ) betheerrorbetweentheintegralof g ( ) anda N point quadratureapproximationtotheintegralof g ( ) ,thatis, E N ( g )= Z 1 1 g ( ) d N X i =1 w i g i (2–100) Itisnowshownthatifthepoints, i ,arechosenastherootsofa N th –degreeLegendre polynomial,thentheerror, E N ( g ) ,iszeroforapolynomial, g ( t ) ,ofdegreeatmost 2 N 1 [ 79 ].Suppose g ( ) isapolynomialofdegreeatmost 2 N 1 .Then, g ( ) canbewritten as g ( )= P N ( ) f ( )+ h ( ), (2–101) where f ( ) and h ( ) arepolynomialsofdegreeatmost N 1 and P N ( ) isthe N th –degreeLegendrepolynomialsuchthattherootsof P N ( ) are ( 1 ,..., N ) .Then theintegralis, Z 1 1 g ( ) d = Z 1 1 ( P N ( ) f ( )+ h ( )) d = Z 1 1 P N ( ) f ( ) d + Z 1 1 h ( ) d (2–102) Since, P N ( ) isorthogonaltoallpolynomialsofdegreeatmost N 1 [ 80 ], Z 1 1 P N ( ) f ( ) d =0. (2–103) 54

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Therefore, Z 1 1 g ( ) d = Z 1 1 h ( ) d (2–104) Recallthat h ( ) isapolynomialofdegreeatmost N 1 andhencecanbeexactly approximatedusing N 1 degreeLagrangepolynomialsas h ( )= N X i =1 h ( i ) L i ( ). (2–105) SubstitutingtheLagrangepolynomialapproximationintoth eintegralgiveninEq.( 2–104 ) Z 1 1 g ( ) d = Z 1 1 N X i =1 h ( i ) L i ( ) d = N X i =1 h ( i ) Z 1 1 L i ( ) d (2–106) Evaluating g ( ) at i g ( i )= P N ( i ) f ( i )+ h ( i ) = h ( i ), (2–107) because, i isarootof P N ( ) ,thatis, P N ( i )=0,1 i N (2–108) ThustheintegralofEq.( 2–104 )isgivenas, Z 1 1 g ( ) d = N X i =1 g ( i ) Z 1 1 L i ( ) d = N X i =1 w i g ( i ), (2–109) where w i arethequadratureweightsgivenby w i = Z 1 1 L i ( ) d (2–110) 55

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Hence,itisshownthatiftherootsofthe N th degreeLegendrePolynomialare chosenasthequadraturepoints,thenthequadratureapprox imationisexactforthe polynomialsofdegreeatmost 2 N 1 .Thissetofpointswaspreviouslydenedasthe Legendre-Gauss(LG)pointsandthequadratureruleobtaine dusingLGpointsiscalled astheLegendre-Gaussquadrature.TheLGquadratureweight sareobtainedusing therelationshipgiveninEq.( 2–110 ).Alternately,LGquadratureweightscanalsobe obtainedas[ 81 ] w i = 2 1 2 i P N ( i ) 2 ,1 i N (2–111) where P N isthederivativeofthe N th –degreeLegendrepolynomial. TheLegendre-Gauss-Radau(LGR)quadratureformulaissimi lartotheLegendre-Gauss formula,exceptoneoftheboundarypointisxedat 1 .Theformulaiscreatedby choosingweights w i and N 1 remainingpoints i ,tointegratethehighestdegree polynomialpossiblewithzeroerror,sothat E N ( g )= Z 1 1 g ( ) d N X i =1 w i g i =0, 1 = 1. (2–112) Becauseonedegreeoffreedomhasbeenremoved,theLGRquadra tureisexactfor thepolynomialsofdegree 2 N 2 .TheLGRpointsaredeterminedtobethezerosof thesumoftheLegendrepolynomialsofdegree N and N 1 P N ( )+ P N 1 ( ) ,where 1 = 1 .Theweightsaredeterminedas[ 81 ] w i = 1 (1 i ) P N 1 ( i ) 2 ,2 i N (2–113) Theweightattheboundarypointis w 1 = 2 N 2 (2–114) IntheLegendre-Gauss-Lobatto(LGL)quadratureformulath eboundarypointsare xedat 1 and 1 .Theformulaiscreatedbychoosingweights w i and N 2 remaining 56

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points i ,tointegratethehighestdegreepolynomialpossiblewithz eroerror,sothat E N ( g )= Z 1 1 g ( ) d N X i =1 w i g i =0, 1 = 1, N =1. (2–115) Becausetwodegreesoffreedomhavebeenremoved,theLGLquad ratureisexactfor thepolynomialsofdegree 2 N 3 .TheLGLpointsaredeterminedtobethezerosofthe derivativeoftheLegendrepolynomialofdegree N 1 P N 1 ( ) ,plusthetwoendpoints, 1 and 1 .Theweightsaredeterminedas[ 81 ] w i = 2 N ( N 1) [ P N 1 ( i ) ] 2 ,2 i N 1. (2–116) Theweightsattheboundarypointsare w 1 = w N = 2 N ( N 1) (2–117) TheaccuracyoftheLGandtheLGRpointsarenowdemonstrated fortheintegral giveninEq.( 2–98 ).Fig. 2-9 showsthe log 10 errorfortheapproximationofEq.( 2–98 ) versusthenumberofLGandLGRpoints.Itisseenthatasthenu mberoftheLG ortheLGRpointsisincreased,theaccuracyinapproximatio nrapidlyincreases. Furthermore,therateofconvergencebytheLGandLGRquadra turestotheexact solutionisexponential.Foranaccuracyof O (10 12 ) 14 LGRpointsand 16 LGpoints arerequiredasopposedtotheonehundredthousandpointsre quiredforcomposite trapezoidalrule.Therefore,forapproximatingtheintegr alofEq.( 2–98 ),theLGorthe LGRquadraturesaresignicantlymoreaccuratethanalow-o rdermethodsuchasthe compositetrapezoidrule. 57

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0 0 24 6 810 1214 161820 Nlog 10 ErrorLGR LG 2 4 6 8 10 12 14 Figure2-9.Basetenlogarithmoferrorvs.numberofGaussian quadraturepoints, N ,for approximationofEq.( 2–98 ). 58

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CHAPTER3 MOTIVATIONFORTHERADAUPSEUDOSPECTRALMETHOD AsdescribedinChapter 2 ,inadirectmethodforsolvingoptimalcontrolproblems, thecontinuous-timeoptimalcontrolproblemistranscribe dtoadiscretenonlinear programmingproblem(NLP).TheresultingNLPcanthenbesolv edbyoneofthemany welldevelopednonlinearoptimizationalgorithms.Thedir ectmethodsaregenerally preferredovertheindirectmethodsbecauseinadirectmeth odtheoptimalityconditions donotneedtobederived,arelativelylow-qualityinitialg uessmaybeused,aguessof thecostateisnotneeded,andtheproblemcanbemodiedrela tivelyeasily. Extensiveresearchhasbeendoneonvariousdiscretizations chemes.Twoof theearlyschemes,theEulertranscriptionandtheRunge-Kut tatranscription[ 20 ]are basedonthetime-marchingmethodsforsolvingdifferentia lequations.TheEuler transcriptionandtheRunge-Kuttatranscriptionexhibitp olynomialrateofconvergence. Thecollocationbasedpseudospectralmethodsareasignic antimprovementover theEulertranscriptionandtheRunge-Kuttatranscription. Thesemethodshave manyadvantages,themostimportantbeingtheexponentialc onvergencerate.The pseudospectralmethodsemployhighaccuracyquadratureru les,makingitpossibleto usemanyfewercollocationpointsthatmaybenecessarytoac hievethesameaccuracy usinganothertypeofmethod(e.g.,EulerorRunge-Kutta).In addition,becauseoftheir mathematicalstructure,thepseudospectralmethodsoffer theopportunitytocompute highaccuracycostate. TheLobattopseudospectralmethod(LPM)[ 33 – 47 ]andtheGausspseudospectral method(GPM)[ 48 – 50 ]aretwoofthemostextensivelyresearchedpseudospectral methods.IntheLPM,collocationisperformedattheLegendre -Gauss-Lobatto(LGL) points,whereasintheGPM,collocationisperformedattheLe gendre-Gauss(LG) points.AsdescribedinChapter 2 ,theLGLpointsincludetheboundarypoints 1 and +1 .Becauseoptimalcontrolproblemsgenerallyhaveboundaryc onditionsatbothends, 59

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Continuous-Time Optimal Control Problem Discretize Continuous First-Order Optimality Conditions Discrete First-Order Optimality Conditions Transformed KKT Conditions Nonlinear Programming Problem Continuous-Time Optimal Control Problem Optimize Discretize Optimize Lobatto Gauss Figure3-1.RelationshipbetweentheKKTconditionsandthedi screterst-order optimalityconditionsfortheLobattopseudospectralmeth odandtheGauss pseudospectralmethod. usingtheLGLpointsappearstobethenaturalchoice.TheLPM, however,suffersfrom adefectintheoptimalityconditionsattheboundarypoints .Theoptimalityconditions oftheNLParenotequivalenttothediscretizedformoftheco ntinuous-timeoptimality conditions.Inparticular,usingtheLPMleadstoaninaccura tecostateestimate.On theotherhand,theLGpointsdonotincludeeitheroftheendp oints,makingtheGPM theleastnaturalchoice.Itisshown,however,thatintheGPM theoptimalityconditions oftheNLPareequivalenttothediscretizedformoftheconti nuous-timeoptimality conditionsandanaccuratecostateestimateisobtained.Ho wever,becauseintheGPM collocationisnotperformedattheinitialandthenalpoin t,thecontrolattheinitialpoint isnotobtainedfromtheNLPsolution. Themotivationfortherstpartofthisresearchistondame thodthatposesses thesameaccuracyasthatoftheGPM,butalsoprovidestheinit ialcontrolfromthe solutionoftheNLP.Theneedtondanaccuratemethodthatpr ovidestheinitialcontrol 60

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isevenmorecriticalinthecasewheremultiple-intervalor hp pseudospectralmethods areused[ 82 83 ].Inamultiple-intervalor hp -approach,thetimeintervalisdividedinto severalsubintervals,whereeachsubintervalison [ 1,+1] .Theterminalpointofone subintervalistheinitialpointofthenextintervalandisc alledameshpoint.Fig. 3-2 showsatimeintervaldividedintothreesubintervals.Furt her,eachofthesubinterval usescollocationat4LGL,LGorLGRpoints.BecausetheLGLpoi ntsincludeboth theinitialandnalpoints,aredundancyinvariablesandin formationisobserved atthemeshpoints.Ontheotherhand,incaseoftheLGpoints, noinformationon controlatthemeshpointisobtained,becausetheLGpointse xcludeboththeinitial andtheterminalpoint.Thepseudospectralmethodofthisre search,calledtheRadau pseudospectralmethod,usesLegendre-Gauss-Radau(LGR)p ointsforcollocation. TheLGRpointsincludetheinitialpoint.Therefore,byusin gtheLGRpoints,controlis obtainedateachofthemeshpointsexactlyonce.Itisnoted, however,thatthecontrolat thenalpointisnotobtained. 0 0 0 1 1 1 st Interval 2 nd Interval 3 rd Interval LGL LG LGR 1 = 1 1 = 1 Figure3-2.Multiple-intervalimplementationofLGL,LG,a ndLGRpoints. 61

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Thesecondobjectiveofthisdissertationistoprovideares earcherwithabasis andjusticationtoselectaparticularpseudospectralmet hodwhilesolvinganoptimal controlproblem.Thisobjectiveisthemotivationforprese ntingauniedframework whichprovidestherstrigorousanalysisthatidentiesth ekeymathematicalproperties ofthepseudospectralmethodsusingcollocationatGaussia nquadraturepoints, enablingaresearcherorend-usertoseeclearlytheaccurac yandconvergence(or non-convergence)thatcanbeexpectedwhenapplyingaparti cularpseudospectral methodonaproblemofinterest.Inthischapter,theLobatto pseudospectralmethodand theGausspseudospectralmethodareexplored. 3.1ScaledContinuous-TimeOptimalControlProblem Considerthecontinuous-timeoptimalcontrolproblemofSec tion 2.1 again. Minimizethecostfunctional J =( y ( t 0 ), t 0 y ( t f ), t f )+ Z t f t 0 g ( y ( t ), u ( t ), t ) dt (3–1) subjecttothedynamicconstraints y ( t )= f ( y ( t ), u ( t ), t ), (3–2) theboundaryconditions ( y ( t 0 ), t 0 y ( t f ), t f )= 0 (3–3) andtheinequalitypathconstraints C ( y ( t ), u ( t ), t ) 0 (3–4) whereitisnotedthatallvectorfunctionsoftimeare row vectors;thatis, y ( t )= [ y 1 ( t ) y n ( t ) ] 2 R n .DenetheaugmentedHamiltonianas H ( y ( t ), u ( t ), ( t ), r ( t ), t )= g ( y ( t ), u ( t ), t )+ h ( t ), f ( y ( t ), u ( t ), t ) i h r ( t ), C ( y ( t ), u ( t ), t ) i (3–5) 62

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where h a b i denotestheinnerproductbetweentwovectors, a and b ,suchthat h a b i = a T b = b T a .Next,suppose p : R n R m ,then r p isthe m by n Jacobianmatrixwhose i th rowis r p i .Inparticular,thegradientofascalar-valuedfunctionis arowvector.Then rst-orderoptimalityconditionsoftheoptimalcontrolpr obleminEqs.( 3–1 )-( 3–4 )are writtenintermsoftheHamiltonianas y ( t )= r H (3–6) ( t )= r y H (3–7) 0 = r u H (3–8) ( t 0 )= r y( t 0 ) ( h i ), (3–9) ( t f )= r y( t f ) ( h i ), (3–10) H j t = t 0 = r t 0 ( h i ), (3–11) H j t = t f = r t f ( h i ), (3–12) r i ( t )=0 when C i ( y ( t ), u ( t )) < 0,1 i s (3–13) r i ( t ) < 0 when C i ( y ( t ), u ( t ))=0,1 i s (3–14) = 0 (3–15) TheHamiltonianattheinitialtimecanbeobtainedusingthe followingexpression: H j t = t 0 = 1 t f t 0 Z t f t 0 Hdt Z t f t 0 dH dt ( t f t ) dt (3–16) sincethesecondtermontheright-handsidewhenintegrated bypartiswrittenas 1 t f t 0 Z t f t 0 Hdt +( t f t 0 ) H j t = t 0 Z t f t 0 Hdt = H j t = t 0 (3–17) Similarly,theHamiltonianatthenaltimeisobtainedfrom H j t = t f = 1 t f t 0 Z t f t 0 Hdt + Z t f t 0 dH dt ( t t 0 ) dt (3–18) 63

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becauseintegratingthesecondtermontheright-handsideb ypartgives 1 t f t 0 Z t f t 0 Hdt +( t f t 0 ) H j t = t f Z t f t 0 Hdt = H j t = t f (3–19) ThetotaltimederivativeoftheHamiltonianiswrittenas dH dt = @ H @ t + @ H @ y y + @ H @ + @ H @ u u + @ H @ r r (3–20) Substitutingtherst-orderoptimalityconditionsofEq.( 3–6 )-( 3–8 ),thetotalderivativeis reducedto dH dt = @ H @ t + @ H @ r r (3–21) Next,fromthedenitionoftheHamiltonian, @ H @ r = C ( y ( t ), u ( t ), t ). (3–22) UsingthecomplementaryslacknessconditionsinEq.( 3–13 )-( 3–14 ), @ H @ r =0 when C ( y ( t ), u ( t ), t )=0, (3–23) r ( t )=0 when C ( y ( t ), u ( t ), t ) < 0 because r ( t )=0. (3–24) Therefore,thetotaltimederivativeoftheHamiltonianise qualtothepartialderivativeof Hamiltonianwithrespecttotime. dH dt = @ H @ t (3–25) HenceEqs.( 3–16 )and( 3–18 )canberewrittenas H j t = t 0 = 1 t f t 0 Z t f t 0 Hdt Z t f t 0 @ H @ t ( t f t ) dt (3–26) H j t = t f = 1 t f t 0 Z t f t 0 Hdt + Z t f t 0 @ H @ t ( t t 0 ) dt (3–27) TheLegendre-Gauss-Lobatto(LGL)pointsusedintheLobatt opseudospectralmethod andtheLegendre-Gauss(LG)pointsusedintheGausspseudos pectralmethodare denedonthedomain [ 1,+1] .Therefore,thetimedomain [ t 0 t f ] ,ismappedto 64

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[ 1,+1] ,byusingthefollowingafnetransformation t = t f t 0 2 + t f + t 0 2 (3–28) suchthat dt d = t f t 0 2 (3–29) Themappingisusedtoconverttheoptimalcontrolproblemgi veninEqs.( 3–1 )-( 3–4 )to thetimedomain 2 [ 1,+1] ,suchthattheobjectiveistominimizethecostfunctional J =( y ( 1), t 0 y (+1), t f )+ t f t 0 2 Z 1 1 g ( y ( ), u ( ), ; t 0 t f ) d (3–30) subjecttothedynamicconstraints d y ( ) d =_ y ( )= t f t 0 2 f ( y ( ), u ( ), ; t 0 t f ), (3–31) theboundaryconditions ( y ( 1), t 0 y (+1), t f )= 0 (3–32) andtheinequalitypathconstraints t f t 0 2 C ( y ( ), u ( ), ; t 0 t f ) 0 (3–33) Itisnotedthat,inordertowritetheoptimalityconditions moresuccintly,Eq.( 3–33 )is multipliedby ( t f t 0 ) = 2 withoutactuallyaffectingtheconstraint.TheHamiltonia nforthis problemisdenedasH ( y ( ), u ( ), ( ), r ( ), ; t 0 t f )= g ( y ( ), u ( ), ; t 0 t f )+ h ( ), f ( y ( ), u ( ), ; t 0 t f ) i h r ( ), C ( y ( ), u ( ), ; t 0 t f ) i (3–34) ConsiderthefollowingderivationforobtainingtheHamilt onianattheinitialtimeandthe naltimein domain.FromEq.( 3–26 ),theHamiltonianattheinitialtimeis H j t = t 0 = 1 t f t 0 Z t f t 0 Hdt Z t f t 0 @ H @ t ( t f t ) dt (3–35) 65

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ThenusingthetimetransformationinEq.( 3–28 )thefollowingexpressionisobtained: t f t t f t 0 = 1 2 (3–36) UsingEq.( 3–36 )andtherelationshipinEq.( 3–29 ),theHamiltonianattheinitialtimeis giventhenintermsof as H j t = t 0 = 1 2 Z +1 1 Hd t f t 0 2 Z +1 1 @ H @ t 1 2 d (3–37) Again,fromthetimetransformationinEq.( 3–28 )thefollowingexpressionisobtained: @ t @ t 0 = 1 2 (3–38) Therefore,theHamiltonianattheinitialtimeisgivenas H j t = t 0 = 1 2 Z +1 1 Hd t f t 0 2 Z +1 1 @ H @ t 0 d (3–39) Similarly,theHamiltonianatthenaltimeisobtainedfromEq .( 3–27 )as H j t = t f = 1 t f t 0 Z t f t 0 Hdt + Z t f t 0 @ H @ t ( t t 0 ) dt (3–40) FromthetimetransformationinEq.( 3–28 )thefollowingexpressionisobtained: t t 0 t f t 0 = 1+ 2 (3–41) UsingEq.( 3–41 )andtherelationshipinEq.( 3–29 )theHamiltonianatthenaltimeis givenintermsof as H j t = t f = 1 2 Z +1 1 Hd + t f t 0 2 Z +1 1 @ H @ t 1+ 2 d (3–42) Again,fromthetimetransformationinEq.( 3–28 )thefollowingexpressionisobtained: @ t @ t f = 1+ 2 (3–43) 66

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Therefore,theHamiltonianatthenaltimeisgivenas H j t = t f = 1 2 Z +1 1 Hd + t f t 0 2 Z +1 1 @ H @ t f d (3–44) Therst-orderoptimalityconditionsofthescaledcontinu ous-timeoptimalcontrol probleminEqs.( 3–30 )-( 3–33 )arethengivenintermsoftheHamiltonianas y ( )= t f t 0 2 r H (3–45) ( )= t f t 0 2 r y H (3–46) 0 = r u H (3–47) ( 1)= r y( 1) ( h i ), (3–48) (+1)= r y(+1) ( h i ), (3–49) r t 0 ( h i )= 1 2 Z +1 1 Hd t f t 0 2 Z +1 1 @ H @ t 0 d (3–50) r t f ( h i )= 1 2 Z +1 1 Hd + t f t 0 2 Z +1 1 @ H @ t f d (3–51) r i ( )=0 when C i ( y ( ), u ( )) < 0,1 i s (3–52) r i ( ) < 0 when C i ( y ( ), u ( ))=0,1 i s (3–53) = 0 (3–54) 3.2LobattoPseudospectralMethod TheLobattopseudospectralmethodisadirecttranscriptio nmethodthatconverts acontinuous-timeoptimalcontrolproblemintoadiscreten onlinearprogramming problem.ThemethodusestheLegendre-Gauss-Lobatto(LGL) pointsfortheLagrange polynomialapproximationofcontinuousfunctionsoftime, forcollocationofthe differentialdynamicconstraints,andforquadratureappr oximationoftheintegrated Lagrangecostterm. 67

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3.2.1NLPFormulationoftheLobattoPseudospectralMethod Considerthe N LGLpoints, ( 1 2 ,..., N ) ,where 1 = 1 and N =+1 .Next,let L i ( ) ( i =1,..., N ) ,betheLagrangepolynomialsofdegree N 1 givenas L i ( )= N Y j =1 j 6 = i j i j ,( i =1,..., N ). (3–55) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N 1 usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =1 Y i L i ( ), (3–56) where Y i = Y ( i ) .Thepoints, ( 1 ,..., N ) ,thatareusedinstateapproximationare calledthediscretizationpoints.Next,anapproximationt othederivativeofthestatein domainisgivenbydifferentiatingtheapproximationofEq.( 3–56 )withrespectto y ( ) Y ( )= N X i =1 Y i L i ( ). (3–57) Thefollowingcollocationconditionsarethenformedbyequ atingthederivativeofthe stateapproximationinEq.( 3–57 )totheright-handsideofthestatedynamicconstraints inEq.( 3–31 )atthe N LGLpoints: N X i =1 Y i L i ( k )= t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (3–58) N X i =1 D ki Y i = t f t 0 2 f ( Y k U k k ; t 0 t f ), D ki = L i ( k ), (3–59) where U k = U ( k ) and D =[ D ki ],(1 k N ),(1 i N ) isa N N square matrixandiscalledthe Lobattopseudospectraldifferentiationmatrix .Thematrix D is squarebecausethecollocationpointsarethesameasthedis cretizationpointsusedin 68

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theLagrangepolynomialapproximationofstate.Let Y LGL bedenedas Y LGL = 266664 Y 1 ... Y N 377775 Usingthematrix Y LGL ,thecollocateddynamicsatthe N LGLcollocationpointsin Eq.( 3–59 )areexpressedas D k Y LGL = t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (3–60) where D k isthe k th rowofthedifferentiationmatrix D .Next,thepathconstraintsin Eq.( 3–33 )areenforcedatthe N LGLcollocationpointsas t f t 0 2 C ( Y k U k k ; t 0 t f ) 0 ,( k =1,..., N ). (3–61) Lastly,thecostfunctionalisapproximatedusingtheLGLqu adratureas J =( Y ( 1 ), 1 Y ( N ), N )+ t f t 0 2 N X k =1 w k g ( Y k U k k ; t 0 t f ), (3–62) where w k isthequadratureweightassociatedwiththe k th LGLcollocationpoint. Thenite-dimensionalnonlinearprogrammingproblem(NLP) correspondingtothe Lobattopseudospectralmethodisthengivenasfollows.Min imizetheobjectivefunction J =( Y ( 1 ), 1 Y ( N ), N )+ t f t 0 2 N X k =1 w k g ( Y k U k k ; t 0 t f ), (3–63) subjecttothefollowingequalityandinequalityconstrain ts: D k Y LGL t f t 0 2 f ( Y k U k k ; t 0 t f )= 0 ,( k =1,..., N ), (3–64) ( Y ( 1 ), 1 Y ( N ), N )= 0 (3–65) t f t 0 2 C ( Y k U k k ; t 0 t f ) 0 ,( k =1,..., N ), (3–66) 69

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0 00.5 1 5 10 15 20 1 0.5 NumberofDiscretizationPointsDiscretizationPoint CollocationPoint Figure3-3.DiscretizationandcollocationpointsforLoba ttopseudospectralmethod. wheretheNLPvariablesare ( Y 1 ,..., Y N ) ( U 1 ,..., U N ) t 0 ,and t f .Theproblemdened byEqs.( 3–63 )-( 3–66 )isthediscreteLobattopseudospectralapproximationtot he continuous-timeoptimalcontrolproblemdenedbyEqs.( 3–30 )-( 3–33 ). AfewkeypropertiesoftheLobattopseudospectralmethodar enowstated. Thediscretizationpoints,atwhichthestateisapproximat edarethe N LGLpoints, ( 1 ,..., N ) .ThestateapproximationusestheLagrangepolynomialsofd egree N 1 whichisonelessthanthenumberofdiscretizationpoints.F urthermore,thestate dynamicsarecollocatedatthesame N LGLpoints, ( 1 ,..., N ) .Fig. 3-3 shows thediscretizationandcollocationpointsfortheLobattop seudospectralmethod.A consequenceofapproximatingthestateandcollocatingthe dynamicsatthesame N LGLpointsisthattheLobattopseudospectraldifferentiat ionmatrixisasquare, N N matrix. 70

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3.2.2NecessaryOptimalityConditions Thenecessaryoptimalityconditions,alsocalledtheKarush -Kuhn-Tucker(KKT) conditions,oftheNLPgiveninEqs.( 3–63 )-( 3–66 )arenowderived.TheLagrangian associatedwiththeNLPis L =( Y ( 1 ), 1 Y ( N ), N ) h ( Y ( 1 ), 1 Y ( N ), N ) i + t f t 0 2 N X k =1 ( w k g ( Y k U k k ; t 0 t f ) h k C ( Y k U k k ; t 0 t f ) i ) N X k =1 h k D k Y LGL t f t 0 2 f ( Y k U k k ; t 0 t f ) i (3–67) where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 3–64 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 3–65 ),and k isthe k th rowoftheLagrangemultipliersmatrix 2 R N s associatedwiththeconstraintsinEq.( 3–66 ).TheKKToptimalityconditions arethenobtainedbydifferentiatingtheLagrangianwithre specttoeachofthevariables andequatingthesederivativestozerosuchthat t f t 0 2 r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i ) D T1 = r Y 1 ( h i ), (3–68) t f t 0 2 r Y N ( w N g N + h N f N ih N C N i ) D TN = r Y N ( h i ), (3–69) t f t 0 2 r Y k ( w k g k + h k f k ih k C k i )= D Tk ,2 k N 1, (3–70) t f t 0 2 N X k =1 r t 0 ( w k g k + h k f k ih k C k i ) 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t 0 ( h i ), (3–71) t f t 0 2 N X k =1 r t f ( w k g k + h k f k ih k C k i ) + 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t f ( h i ), (3–72) r U k ( w k g k + h k f k ih k C k i )= 0 ,1 k N (3–73) 71

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D k Y LGL t f t 0 2 f ( Y k U k k ; t 0 t f )= 0 ,1 k N (3–74) ki =0 when C ki < 0,1 i s ,1 k N (3–75) ki < 0 when C ki =0,1 i s ,1 k N (3–76) ( Y ( 1 ), 1 Y ( N ), N )= 0 (3–77) where D Ti isthe i th rowof D T g k = g ( Y k U k k ; t 0 t f ) f k = f ( Y k U k k ; t 0 t f ) and C k = C ( Y k U k k ; t 0 t f ) TheKKTconditionsinEqs.( 3–68 )-( 3–77 )arereformulatedsothattheybecomea discretizationoftherst-orderoptimalityconditionsgi veninEqs.( 3–45 )-( 3–54 )forthe continuouscontrolproblemofEqs.( 3–30 )-( 3–33 ).Let D y =[ D y ij ],(1 i N ),(1 j N ) bethe N N matrixdenedasfollows: D y ii = D ii =0,2 i N 1, (3–78) D y 11 = D 11 1 w 1 (3–79) D y NN = D NN + 1 w N (3–80) D y ij = w j w i D ji ,1 i j N i 6 = j (3–81) Accordingtothedenitionof D y D T1 = w 1 D y1 W 1 1 w 1 e 1 (3–82) D TN = w N D yN W 1 + 1 w N e N (3–83) D Tk = w k D yk W 1 ,2 k N 1, (3–84) where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal, e i isthe i th rowof N N identitymatrix.SubstitutingEqs.( 3–82 )-( 3–84 )inEqs.( 3–68 )-( 3–70 ),we 72

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getthefollowingrelationships r Y 1 ( h i )= t f t 0 2 r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i ) + w 1 D y1 W 1 + 1 w 1 e 1 (3–85) r Y N ( h i )= t f t 0 2 r Y N ( w N g N + h N f N ih N C N i ) + w N D yN W 1 1 w N e N (3–86) w k D yk W 1 = t f t 0 2 r Y k ( w k g k + h k f k ih k C k i ), (3–87) 2 k N 1. Next,deningthefollowingchangeofvariables: ~ k = k w k ,1 k N (3–88) ~ r k = k w k ,1 k N (3–89) ~ = (3–90) SubstitutingEqs.( 3–88 )-( 3–90 )inEqs.( 3–73 )-( 3–77 )andinEqs.( 3–85 )-( 3–87 ),the transformedKKTconditionsfortheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (3–91) 0 = D k Y LGL t f t 0 2 f ( Y k U k k ; t 0 t f ),1 k N (3–92) ~ r ki =0 when C ki < 0,1 i s ,1 k N (3–93) ~ r ki < 0 when C ki =0,1 i s ,1 k N (3–94) 0 = ( Y ( 1 ), 1 Y ( N ), N ), (3–95) r t 0 ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) t f t 0 2 N X k =1 w k r t 0 ( g k + h ~ k f k ih ~ r k C k i ), (3–96) 73

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r t f ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) + t f t 0 2 N X k =1 w k r t f ( g k + h ~ k f k ih ~ r k C k i ), (3–97) D y1 ~ = t f t 0 2 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) 1 w 1 ~ 1 + r Y 1 ( h ~ i ) (3–98) D yN ~ = t f t 0 2 r Y N ( g N + h ~ N f N ih ~ r N C N i ) + 1 w N ~ N r Y N ( h ~ i ) (3–99) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (3–100) 2 k N 1. Now,consideracomparisonofthetransformedKKTconditionsi nEqs.( 3–91 )-( 3–100 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 3–45 )-( 3–54 )ofthe continuous-timeoptimalcontrolproblem.Itisnotedthatt hetransformedKKTconditions inEqs.( 3–91 )-( 3–95 )arethediscretizedformsofthecontinuous-timerst-ord er optimalityconditionsinEq.( 3–47 ),Eq.( 3–45 ),Eq.( 3–52 ),Eq.( 3–53 ),andEq.( 3–54 ), respectively.Next,theright-handsideofEq.( 3–96 )andEq.( 3–97 )isthequadrature approximationoftheright-handsideofEq.( 3–50 )andEq.( 3–51 ),respectively. Therefore,thetransformedKKTconditionsinEq.( 3–96 )andEq.( 3–97 )arethe discretizedversionofcontinuous-timerst-orderoptima lityconditionsinEq.( 3–50 ) andEq.( 3–51 ).Furthermore,ithasbeenshowninRef.[ 42 ]that D y = D forLGL collocation,thusmaking D y adifferentiationmatrixconnectedwiththeLGLpoints. Therefore,thelefthandsideofEqs.( 3–98 )-( 3–100 )isanapproximationofthecostate dynamicsatthe k th collocationpoint,i.e., D yk ~ = ~ k ,1 k N (3–101) 74

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Asaresult,Eq.( 3–100 )representsthediscretizedversionofthecostatedynamic sin Eq.( 3–46 )at k =(2,..., N 1) .Itisnoted,however,thatattheboundarypoints,the discreteequivalentsofcontinuousboundaryconditions( 3–48 )and( 3–49 )arecoupledin thediscretecostatedynamicsin( 3–98 )and( 3–99 ),respectively.Hence,thesystemof thetransformedKKTconditionsoftheNLPis not equivalenttotherst-orderoptimality conditionsofthecontinuous-timeoptimalcontrolproblem ThetransformedKKTconditionsfortheLobattopseudospectra lmethoddene thesetofconditionsthatapproximatethecontinuousrstorderoptimalityconditions. ThecostatecanbeestimatedfromtheKKTmultipliersusingthe relationshipgivenin Eq.( 3–88 ).Thecostateestimate,however,doesnotsatisfythediscr eteformofthe costatedynamicsattheboundaries.Thisdefectleadstosig nicanterrorsinthecostate estimate.AnalysisandimprovementofthisdefectoftheLoba ttopseudospectralmethod istheoneofthemotivationsforthisdissertation. 3.3GaussPseudospectralMethod TheGausspseudospectralmethodisalsoadirecttranscript ionmethodthat convertsacontinuous-timeoptimalcontrolproblemintoad iscretenonlinearprogramming problem(NLP).ThemethodusestheLegendre-Gauss(LG)point sforcollocation ofthedifferentialdynamicconstraintsandforquadrature approximationofthe integratedLagrangecostterm.TheLagrangepolynomialapp roximationsofthe continuousfunctionsoftime,however,utilizetheLGpoint splustheinitialpoint.This discretizationschemeresultsinasetofKKTconditionsthata reexactlyequivalent tothediscretizedformofthecontinuousrst-orderoptima lityconditions.Hence,a signicantlybettercostateestimatethantheLobattopseu dospectralmethodisobtained. Themethod,however,doesnotprovideanyinformationabout theinitialcontrolandthe implementationofthemethodismorecomplexthanthatofthe methodofthisresearch, theRadaupseudospectralmethod. 75

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3.3.1NLPFormulationoftheGaussPseudospectralMethod Considerthe N LGpoints, ( 1 2 ,..., N ) ,where 1 > 1 and N < +1 .Dene twonewpointssuchthat 0 = 1 and N +1 =1 .Next,let L i ( ) ( i =0,..., N ) ,bethe Lagrangepolynomialsofdegree N givenby L i ( )= N Y j =0 j 6 = i j i j ,( i =0,..., N ). (3–102) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =0 Y i L i ( ), (3–103) where Y i = Y ( i ) .Itisimportanttonotethat 0 = 1 isnotaLGpointbutisusedin stateapproximation.Next,anapproximationtothederivat iveofthestatein domainis givenbydifferentiatingtheapproximationofEq.( 3–103 )withrespectto y ( ) Y ( )= N X i =0 Y i L i ( ). (3–104) Thefollowingcollocationconditionsarethenformedbyequ atingthederivativeofthe stateapproximationinEq.( 3–104 )totheright-handsideofstatedynamicconstraintsin Eq.( 3–31 )atthe N LGpoints, ( 1 ,..., N ) N X i =0 Y i L i ( k )= t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (3–105) N X i =0 D ki Y i = t f t 0 2 f ( Y k U k k ; t 0 t f ), D ki = L i ( k ), (3–106) where U k = U ( k ) .Itisnotedthat 0 isnotacollocationpoint.Thematrix D = [ D ki ],(1 k N ),(0 i N ) isa N ( N +1) non-square matrixandiscalledthe Gausspseudospectraldifferentiationmatrix .Thematrix D isnon-squarebecausethe stateapproximationuses N +1 points, ( 0 ,..., N ) ,butthecollocationisdoneatonlythe 76

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N LGpoints, ( 1 ,..., N ) .Let Y LG bedenedas Y LG = 266664 Y 0 ... Y N 377775 Usingthematrix Y LG ,thecollocateddynamicsatthe N LGcollocationpointsin Eq.( 3–106 )areexpressedas D k Y LG = t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (3–107) where D k isthe k th rowofthedifferentiationmatrix D .Itisnotedherethatthestateat thenalpointisnotpresentintheformulation.Itisapprox imatedasfollows:First, y (+1)= y ( 1)+ Z +1 1 y ( ) d (3–108) TheLGquadratureapproximationofEq.( 3–108 )isthengivenas Y ( N +1 )= Y ( 0 )+ N X k =1 w k Y ( k ) = Y ( 0 )+ t f t 0 2 N X k =1 w k f ( Y k U k k ; t 0 t f ), (3–109) where w k 1 k N ,aretheLGquadratureweights.Next,thepathconstraintsi n Eq.( 3–33 )areenforcedatthe N LGcollocationpointsas t f t 0 2 C ( Y k U k k ; t 0 t f ) 0 ,( k =1,..., N ). (3–110) Lastly,thecostfunctionalisapproximatedusingLGquadra tureas J =( Y ( 0 ), 0 Y ( N +1 ), N +1 )+ t f t 0 2 N X k =1 w k g ( Y k U k k ; t 0 t f ). (3–111) 77

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0 00.5 1 5 10 15 20 1 0.5 NumberofDiscretizationPointsDiscretizationPoint CollocationPoint Figure3-4.DiscretizationandcollocationpointsforGaus spseudospectralmethod. Thenite-dimensionalnonlinearprogrammingproblemcorr espondingtotheGauss pseudospectralmethodisthengivenasfollows.Minimizeth ecostfunction J =( Y ( 0 ), 0 Y ( N +1 ), N +1 )+ t f t 0 2 N X k =1 w k g ( Y k U k k ; t 0 t f ), (3–112) subjecttothefollowingequalityandinequalityconstrain ts: D k Y LG t f t 0 2 f ( Y k U k k ; t 0 t f )= 0 ,( k =1,..., N ), (3–113) Y ( N +1 ) Y ( 0 ) t f t 0 2 N X k =1 w k f ( Y k U k k ; t 0 t f )= 0 (3–114) ( Y ( 0 ), 0 Y ( N +1 ), N +1 )= 0 (3–115) t f t 0 2 C ( Y k U k k ; t 0 t f ) 0 ,( k =1,..., N ), (3–116) wheretheNLPvariablesare ( Y 0 ,..., Y N +1 ) ( U 1 ,..., U N ) t 0 and t f .Itisnotedthat initialandterminalcontrol, U 0 and U N +1 arenotobtainedinthesolutionoftheNLP. TheproblemdenedbyEqs.( 3–112 )-( 3–116 )isthediscreteGausspseudospectral 78

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approximationtothecontinuous-timeoptimalcontrolprob lemdenedbyEqs.( 3–30 )-( 3–33 ). AfewkeypropertiesoftheGausspseudospectralmethodaren owstated.The discretizationpoints,atwhichthestateisapproximatedu singLagrangepolynomials, arethe N LGpointsplustheinitialpoint, ( 0 ,..., N ) .Thestateapproximationuses Lagrangepolynomialsofdegree N .Thestatedynamicsarecollocatedatonlythe N LG points, ( 1 ,..., N ) .Asaconsequence,theGausspseudospectraldifferentiatio nmatrixis anon-square, N ( N +1) matrix.Fig. 3-4 showsthediscretizationandthecollocation pointsfortheGausspseudospectralmethod.3.3.2NecessaryOptimalityConditions ThenecessaryoptimalityconditionsortheKarush-Kuhn-Tuc ker(KKT)conditions, oftheNLPgiveninEqs.( 3–112 )-( 3–116 )arenowderived.TheLagrangianassociated withtheNLPis L =( Y ( 0 ), 0 Y ( N +1 ), N +1 ) h ( Y ( 0 ), 0 Y ( N +1 ), N +1 ) i + t f t 0 2 N X k =1 ( w k g ( Y k U k k ; t 0 t f ) h k C ( Y k U k k ; t 0 t f ) i ) N X k =1 h k D k Y LG t f t 0 2 f ( Y k U k k ; t 0 t f ) i h N +1 Y ( N +1 ) Y ( 0 ) t f t 0 2 N X k =1 w k f ( Y k U k k ; t 0 t f ), i (3–117) where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 3–113 ), N +1 2 R n aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 3–114 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 3–115 ),and k isthe k th rowoftheLagrangemultipliersmatrix 2 R N s associatedwiththeconstraintsinEq.( 3–116 )TheKKToptimalityconditions arethenobtainedbydifferentiatingtheLagrangianwithre specttoeachofthevariables 79

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andequatingthesederivativestozerosuchthat t f t 0 2 r Y k ( w k g k + h k + w k N +1 f k ih k C k i )= D Tk ,1 k N (3–118) r Y 0 ( h i )= N +1 D T0 (3–119) r Y N +1 ( h i )= N +1 (3–120) t f t 0 2 N X k =1 r t 0 ( w k g k + h k + w k N +1 f k ih k C k i ) 1 2 N X k =1 ( w k g k + h k + w k N +1 f k ih k C k i )= r t 0 ( h i ), (3–121) t f t 0 2 N X k =1 r t f ( w k g k + h k + w k N +1 f k ih k C k i ) + 1 2 N X k =1 ( w k g k + h k + w k N +1 f k ih k C k i )= r t f ( h i ), (3–122) r U k ( w k g k + h k + w k N +1 f k ih k C k i )= 0 ,1 k N (3–123) D k Y LG t f t 0 2 f ( Y k U k k ; t 0 t f )= 0 ,1 k N (3–124) ki =0 when C ki < 0,1 i s ,1 k N (3–125) ki < 0 when C ki =0,1 i s ,1 k N (3–126) ( Y ( 0 ), 0 Y ( N +1 ), N +1 )= 0 (3–127) where D Ti isthe i th rowof D T g k = g ( Y k U k k ; t 0 t f ) f k = f ( Y k U k k ; t 0 t f ) and C k = C ( Y k U k k ; t 0 t f ) Next,theKKTconditionsgiveninEqs.( 3–118 )-( 3–127 )arereformulatedso thattheybecomeadiscretizationoftherst-orderoptimal ityconditionsgivenin Eqs.( 3–45 )-( 3–54 )forthecontinuouscontrolproblemgiveninEqs.( 3–30 )-( 3–33 ). Let D y =[ D y ij ],(1 i N ),(1 i N +1) bethe N ( N +1) matrixdenedasfollows: D y ij = w j w i D ji ,( i j )=1,..., N (3–128) D y i N +1 = N X j =1 D y ij i =1,..., N (3–129) 80

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Theorem1. Thematrix D y denedin ( 3–128 ) and ( 3–129 ) isadifferentiationmatrixfor thespaceofpolynomialsofdegree N .Moreprecisely,if p isapolynomialofdegreeat most N and p 2 R N +1 isthevectorwith i th component p i = p ( i ) 1 i N +1 ,then ( D y p ) i =_ p ( i ),1 i N ( pofdegree N ). ProofofTheorem1. Let E denotethedifferentiationmatrixdenedinthestatemento f thetheorem.Thatis, E isan N ( N +1) matrixwiththepropertythatforall p 2 R N +1 wehave ( Ep ) i =_ p ( i ),1 i N where p isthepolynomialofdegreeatmost N whichsatises p j = p ( j ) 1 j N +1 If p and q aresmooth,real-valuedfunctionswith q ( 1)= p (1)=0 ,thenintegrationby partsgives Z 1 1 p ( ) q ( ) d = Z 1 1 p ( )_ q ( ) d (3–130) Suppose p and q arepolynomialsofdegreeatmost N ,with N 1 ;inthiscase, pq and p q arepolynomialsofdegreeatmost 2 N 1 .SinceGaussquadratureisexactfor polynomialsofdegreeatmost 2 N 1 ,theintegralsin( 3–130 )canbereplacedbytheir quadratureequivalentstoobtain N X j =1 w j p j q j = N X j =1 w j p j q j (3–131) where p j = p ( j ) and p j =_ p ( j ) 1 i N p isanypolynomialofdegreeatmost N whichvanishesat 1 ,and q isanypolynomialofdegreeatmost N whichvanishesat +1 .Apolynomialofdegree N isuniquelydenedbyitsvalueat N +1 points.Let p be thepolynomialofdegreeatmost N whichsatises p ( 1)=0 and p j = p ( j ) 1 j N Let q bethepolynomialofdegreeatmost N whichsatises q (+1)=0 and q j = q ( j ) 1 j N .Substituting _p = Dp and _q = Eq in( 3–131 )gives ( WDp ) T q 1: N = ( Wp 1: N ) T Eq 81

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where W isthediagonalmatrixofquadratureweights.Sincetherstc omponentof p andthelastcomponentof q vanish,thisreducesto p T1: N ( D T1: N W + WE 1: N ) q 1: N =0. Since p 1: N and q 1: N arearbitrary,wededucethat D T1: N W + WE 1: N = 0 whichimpliesthat E ij = w j w i D ji ,( i j )=1,..., N (3–132) Since E isadifferentiationmatrix, E1 = 0 ,whichyields E i N +1 = N X j =1 E ij ,1 i N (3–133) Comparing( 3–128 )with( 3–132 )and( 3–129 )with( 3–133 ),weseethat D y = E .The matrix D y denedin ( 3–128 ) and ( 3–129 ) isadifferentiationmatrixforthespaceof polynomialsofdegree N Accordingtothedenitionof D y D Tk = w k D yk ,1: N W 1 ,1 k N (3–134) where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal.Substituting Eq.( 3–134 )inEq.( 3–118 ), w k D yk ,1: N W 1 = t f t 0 2 r Y k ( w k g k + h k f k ih k C k i ),1 k N (3–135) 82

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Next,deningthefollowingchangeofvariables: ~ N +1 = N +1 (3–136) ~ 0 = N +1 D T0 (3–137) ~ k = k w k + N +1 ,1 k N (3–138) ~ r k = k w k ,1 k N (3–139) ~ = (3–140) SubstitutingEqs.( 3–136 )-( 3–140 )inEqs.( 3–119 )-( 3–127 )andinEq.( 3–135 ),the transformedKKTconditionsfortheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (3–141) 0 = D k Y LG t f t 0 2 f ( Y k U k k ; t 0 t f ),1 k N (3–142) ~ r ki =0 when C ki < 0,1 i s ,1 k N (3–143) ~ r ki < 0 when C ki =0,1 i s ,1 k N (3–144) 0 = ( Y ( 0 ), 0 Y ( N +1 ), N +1 ), (3–145) r t 0 ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) t f t 0 2 N X k =1 w k r t 0 ( g k + h ~ k f k ih ~ r k C k i ), (3–146) r t f ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) + t f t 0 2 N X k =1 w k r t f ( g k + h ~ k f k ih ~ r k C k i ), (3–147) ~ 0 = r Y 0 ( h ~ i ), (3–148) ~ N +1 = r Y N +1 ( h ~ i ), (3–149) D yk ,1: N ~ + D y k N +1 ~ N +1 = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (3–150) 1 k N 83

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Now,consideracomparisonofthetransformedKKTconditionsi nEqs.( 3–141 )-( 3–150 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 3–45 )-( 3–54 ) ofthecontinuous-timeoptimalcontrolproblem.Itisnoted thatthetransformedKKT conditionsinEqs.( 3–141 )-( 3–145 )arethediscretizedformsofthecontinuous-time rst-orderoptimalityconditionsinEq.( 3–47 ),Eq.( 3–45 ),Eq.( 3–52 ),Eq.( 3–53 ),and Eq.( 3–54 ),respectively.Next,theright-handsideofEq.( 3–146 )andEq.( 3–147 ) isthequadratureapproximationoftheright-handsideofEq. ( 3–50 )andEq.( 3–51 ), respectively.Therefore,thesetoftransformedKKTconditio nsinEq.( 3–146 )and Eq.( 3–147 )isthediscretizedversionofthesetofcontinuous-timer st-orderoptimality conditionsinEq.( 3–50 )andEq.( 3–51 ).Furthermore,itisshowninTheorem 1 thatthe system( 3–150 )isapseudospectralschemeforthecostatedynamics,i.e. D yk ,1: N ~ + D y k N +1 ~ N +1 = ~ k ,1 k N (3–151) Therefore,thelefthandsideofEq.( 3–150 )isanapproximationofthecostatedynamics atthe k th collocationpoint.Asaresult,Eq.( 3–150 )representsthediscretizedversion ofthecostatedynamicsinEq.( 3–46 )at k =(1,..., N ) .Lastly,itisnotedthat,atthe boundarypoints,thediscreteequivalentsofcontinuousbo undaryconditions( 3–48 ) and( 3–49 )arethesameasthediscretecostateattheboundarypointsi n( 3–148 )and ( 3–149 ),respectively.Hence,thesystemoftransformedKKTconditi onsoftheNLPis exactlyequivalenttotherst-orderoptimalitycondition softhecontinuous-timeoptimal controlproblem.Therefore,anaccuratecostateestimatec anbeobtainedfromtheKKT multipliersusingtherelationshipsgiveninEqs.( 3–136 )-( 3–138 ). Itisnowshownthattheinitialcostatecanalsobeestimated fromtheGauss quadratureapproximationofcostatedynamics.Let D = [ D 0 D 1: N ] where D 0 istherst columnof D and D 1: N aretheremainingcolumns.Then D issuchthat: D 0 = D 1: N 1 where 1 isacolumnvectorofallones. Proposition1. D 0 = D 1: N 1 ;equivalently, D 1 1: N D 0 = 1 84

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ProofofProposition1. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 0 + D 1: N 1 = 0 .Rearranging,weobtain D 0 = D 1: N 1 (3–152) Returningtothedenitionof ~ 0 in( 3–137 ),weobtain ~ 0 = N +1 D T0 = N +1 N X i =1 i D i ,0 = N +1 + N X i =1 N X j =1 i D ij (3–153) = N +1 N X i =1 N X j =1 i D y ji w j w i = N +1 N X i =1 N X j =1 j D y ij w i w j (3–154) = ~ N +1 N X i =1 N X j =1 w i ( ~ j ~ N +1 ) D y ij (3–155) = ~ N +1 N X i =1 w i D yi ,1: N ~ N X i =1 w i D yi N +1 ~ N +1 (3–156) = ~ N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ), (3–157) where( 3–153 )followsfromtheidentity( 3–152 )giveninProposition 1 ,( 3–154 )isthe denition( 3–128 )of D y ,( 3–155 )isthedenition( 3–138 )of ~ j ,( 3–156 )isthedenition ( 3–129 )of D y and( 3–157 )istherst-orderoptimalitycondition( 3–150 ). 3.4Summary TheLPMandtheGPMaretwocollocationbaseddirecttranscript ionmethods thattranscribethecontinuous-timeoptimalcontrolprobl emintoadiscreteNLP.The mostimportantadvantageofthesemethodsistheirexponent ialconvergencerate.The LPM,however,suffersfromadefectintheoptimalityconditi onsattheboundarypoints. TheoptimalityconditionsoftheNLParenotequivalenttoth ediscretizedformofthe continuous-timeoptimalityconditionswhichresultsinan inaccuratecostateestimation. TheGPMusesacollocationschemesuchthattheoptimalitycon ditionsoftheNLPare 85

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exactlyequivalenttothediscretizedformofthecontinuou s-timeoptimalityconditions. Inthiscollocationscheme,however,controlattheinitial boundaryisnotobtainedfrom theNLPsolution.TheRadaupseudospectralmethodofthisre search,discussedin Chapter 4 ,doesnotsufferfromthedefectinoptimalityconditionsan dhastheabilityto computetheinitialcontrol.TheimplementationoftheRada upseudospectralmethodis signicantlylesscomplexthanthatoftheGPM. 86

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CHAPTER4 RADAUPSEUDOSPECTRALMETHOD TheRadaupseudospectralmethodisadirecttranscriptionm ethodthattranscribes acontinuous-timeoptimalcontrolproblemintoadiscreten onlinearprogramming problem.TheresultingNLPcanthenbesolvedbyoneoftheman ywelldeveloped nonlinearoptimizationalgorithms.ThemethodusestheLeg endre-Gauss-Radau(LGR) pointsforcollocationofthedynamicconstraints,andforq uadratureapproximation oftheintegratedLagrangecostterm.TheLagrangepolynomi alapproximationof thestate,however,usestheLGRpointsplusthenalpoint.T heLGRdiscretization schemedevelopedhereresultsinasetofKKTconditionsthatar eequivalenttothe discretizedformofthecontinuousrst-orderoptimalityc onditionsand,hence,provides asignicantlymoreaccuratecostateestimatethanthatobt ainedusingtheLobatto pseudospectralmethod.Inaddition,becausecollocationi sperformedattheLGRpoints, andtheLGRpointsincludetheinitialpoint,thecontrolatt heinitialtimeisalsoobtained inthesolutionoftheNLP. Inthischapter,theNLPisgiventhatarisesfromthediscret izationofacontinuous-time optimalcontrolproblemusingtheRadaupseudospectralmet hod.Next,thekey propertiesoftheRadaupseudospectraldiscretizationare statedandtherst-order necessaryoptimalityconditions,calledtheKarush-Kuhn-T ucker(KKT)conditions,ofthe NLParederived.Itisthenshown,thatafterachangeofvaria bles,theseKKTconditions areequivalenttothediscretizedformofthecontinuousrs t-ordernecessaryoptimality conditions.Thechangeofvariablesprovidesanaccuratedi screteapproximationtothe Lagrangemultipliersofthecontinuousproblem.Lastly,th eproblemformulationfora ipped RadaupseudospectralmethodthatusestheippedLGRpoints isgiven,where theippedLGRpointsarethenegativeoftheLGRpoints. 87

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4.1NLPFormulationoftheRadauPseudospectralMethod Consideragainthecontinuous-timeoptimalcontrolproble mscaledtothetime domain 2 [ 1,+1] ofSection 3.1 ,suchthattheobjectiveistominimizethecost functional J =( y ( 1), t 0 y (+1), t f )+ t f t 0 2 Z 1 1 g ( y ( ), u ( ), ; t 0 t f ) d (4–1) subjecttothedynamicconstraints d y ( ) d =_ y ( )= t f t 0 2 f ( y ( ), u ( ), ; t 0 t f ), (4–2) theboundaryconditions ( y ( 1), t 0 y (+1), t f )= 0 (4–3) andtheinequalitypathconstraints t f t 0 2 C ( y ( ), u ( ), ; t 0 t f ) 0 (4–4) whereitisagainnotedthatallvectorfunctionsoftimeare row vectors;thatis, y ( )= [ y 1 ( ) y n ( ) ] 2 R n .Therst-orderoptimalityconditionsofthescaledcontin uous-time optimalcontrolprobleminEqs.( 4–1 )-( 4–4 )arestatedusingthedenitionofHamiltonian H ( y ( ), u ( ), ( ), r ( ), ; t 0 t f )= g ( y ( ), u ( ), ; t 0 t f )+ h ( ), f ( y ( ), u ( ), ; t 0 t f ) i h r ( ), C ( y ( ), u ( ), ; t 0 t f ) i (4–5) Therst-orderoptimalityconditionsare y ( )= t f t 0 2 r H (4–6) ( )= t f t 0 2 r y H (4–7) 0 = r u H (4–8) ( 1)= r y( 1) ( h i ), (4–9) (+1)= r y(+1) ( h i ), (4–10) 88

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r t 0 ( h i )= 1 2 Z +1 1 Hd t f t 0 2 Z +1 1 @ H @ t 0 d (4–11) r t f ( h i )= 1 2 Z +1 1 Hd + t f t 0 2 Z +1 1 @ H @ t f d (4–12) r i ( )=0 when C i ( y ( ), u ( )) < 0,1 i s (4–13) r i ( ) < 0 when C i ( y ( ), u ( ))=0,1 i s (4–14) = 0 (4–15) Considerthe N LGRpoints, ( 1 2 ,..., N ) ,where 1 = 1 and N < +1 .Dene anewpointsuchthat N +1 =1 .Next,let L i ( ) ( i =1,..., N +1) ,betheLagrange polynomialsofdegree N givenby L i ( )= N +1 Y j =1 j 6 = i j i j ,( i =1,..., N +1). (4–16) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N +1 X i =1 Y i L i ( ), (4–17) where Y i = Y ( i ) .Itisimportanttonotethat N +1 =1 isnotaLGRpointbutisusedin stateapproximation.Next,anapproximationtothederivat iveofthestatein domainis givenbydifferentiatingtheapproximationofEq.( 4–17 )withrespectto y ( ) Y ( )= N +1 X i =1 Y i L i ( ). (4–18) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 4–18 )totheright-handsideofthestatedynamicconstraintsin Eq.( 4–2 )atthe N LGRpoints, ( 1 ,..., N ) andaregivenas N +1 X i =1 Y i L i ( k )= t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (4–19) N +1 X i =1 D ki Y i = t f t 0 2 f ( Y k U k ; t 0 t f ), D ki = L i ( k ), (4–20) 89

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where U k = U ( k ) .Itisnotedthat N +1 isnotacollocationpoint.Thematrix D = [ D ki ],(1 k N ),(1 i N +1) isa N ( N +1) non-square matrixandiscalledthe Radaupseudospectraldifferentiationmatrix .Thematrix D isnon-squarebecausethe stateapproximationuses N +1 points, ( 1 ,..., N +1 ) ,butthecollocationisdoneatonly the N LGRpoints, ( 1 ,..., N ) .Let Y LGR bedenedas Y LGR = 266664 Y 1 ... Y N +1 377775 Usingthematrix Y LGR ,thecollocateddynamicsatthe N LGRcollocationpointsin Eq.( 4–20 )areexpressedas D k Y LGR = t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (4–21) where D k isthe k th rowofdifferentiationmatrix D .Next,thepathconstraintsin Eq.( 4–4 )areenforcedatthe N LGRcollocationpointsas t f t 0 2 C ( Y k U k ; t 0 t f ) 0 ,( k =1,..., N ). (4–22) Lastly,thecostfunctionalisapproximatedusingLGRquadr atureas J =( Y ( 1 ), 1 Y ( N +1 ), N +1 )+ t f t 0 2 N X k =1 w k g ( Y k U k ; t 0 t f ), (4–23) where w k isthequadratureweightassociatedwiththe k th LGRcollocationpoint. Thenite-dimensionalnonlinearprogrammingproblemcorr espondingtotheRadau pseudospectralmethodisthengivenasfollows.Minimizeth ecostfunction J =( Y ( 1 ), 1 Y ( N +1 ), N +1 )+ t f t 0 2 N X k =1 w k g ( Y k U k ; t 0 t f ), (4–24) 90

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0 00.5 1 5 10 15 20 1 0.5 NumberofDiscretizationPointsDiscretizationPoint CollocationPoint Figure4-1.DiscretizationandcollocationpointsforRada upseudospectralmethod. subjecttothefollowingequalityandinequalityconstrain ts: D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f )= 0 ,( k =1,..., N ), (4–25) ( Y ( 1 ), 1 Y ( N +1 ), N +1 )= 0 (4–26) t f t 0 2 C ( Y k U k ; t 0 t f ) 0 ,( k =1,..., N ), (4–27) wheretheNLPvariablesare ( Y 1 ,..., Y N +1 ) ( U 1 ,..., U N ) t 0 ,and t f .Itisnotedthat theinitialcontrol, U 1 ,isobtainedinthesolutionoftheNLP.Theproblemdened byEqs.( 4–24 )-( 4–27 )isthediscreteRadaupseudospectralapproximationtothe continuous-timeoptimalcontrolproblemdenedbyEqs.( 4–1 )-( 4–4 ). AfewkeypropertiesoftheRadaupseudospectralmethodaren owstated.The discretizationpoints,atwhichthestateisapproximatedu singtheLagrangepolynomials, arethe N LGRpointsplusthenalpoint, ( 1 ,..., N +1 ) .Thestateapproximationuses theLagrangepolynomialsofdegree N .Thestatedynamicsarecollocatedatonlythe N LGRpoints, ( 1 ,..., N ) .Asaconsequence,theRadaupseudospectraldifferentiatio n 91

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matrixisanon-square, N ( N +1) matrix.Fig. 4-1 showsthediscretizationandthe collocationpointsfortheRadaupseudospectralmethod. 4.2NecessaryOptimalityConditions Thenecessaryoptimalityconditions,ortheKarush-Kuhn-Tu cker(KKT)conditions, oftheNLPgiveninEqs.( 4–24 )-( 4–27 )arenowderived.TheLagrangianassociated withtheNLPis L =( Y ( 1 ), 1 Y ( N +1 ), N +1 ) h ( Y ( 1 ), 1 Y ( N +1 ), N +1 ) i + t f t 0 2 N X k =1 ( w k g ( Y k U k ; t 0 t f ) h k C ( Y k U k ; t 0 t f ) i ) N X k =1 h k D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f ) i (4–28) where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 4–25 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 4–26 ),and k isthe k th rowoftheLagrangemultipliersmatrix 2 R N s associatedwiththeconstraintsinEq.( 4–27 ).TheKKToptimalityconditions arethenobtainedbydifferentiatingtheLagrangianwithre specttoeachofthevariable andequatingittozero,suchthat t f t 0 2 r Y k ( w k g k + h k f k ih k C k i )= D Tk ,2 k N (4–29) t f t 0 2 r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i )= D T1 r Y 1 ( h i ), (4–30) r Y N +1 ( h i )= D TN +1 (4–31) t f t 0 2 N X k =1 r t 0 ( w k g k + h k f k ih k C k i ) 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t 0 ( h i ), (4–32) t f t 0 2 N X k =1 r t f ( w k g k + h k f k ih k C k i ) + 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t f ( h i ), (4–33) 92

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r U k ( w k g k + h k f k ih k C k i )= 0 ,1 k N (4–34) D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f )= 0 ,1 k N (4–35) ki =0 when C ki < 0,1 i s ,1 k N (4–36) ki < 0 when C ki =0,1 i s ,1 k N (4–37) ( Y ( 1 ), 1 Y ( N +1 ), N +1 )= 0 (4–38) where D Ti isthe i th rowof D T g k = g ( Y k U k ; t 0 t f ) f k = f ( Y k U k ; t 0 t f ) and C k = C ( Y k U k ; t 0 t f ) Next,theKKTconditionsgiveninEqs.( 4–29 )-( 4–38 )arereformulatedsothatthey becomeadiscretizationoftherst-orderoptimalitycondi tionsgiveninEqs.( 4–6 )-( 4–15 ) forthecontinuouscontrolproblemgiveninEqs.( 4–1 )-( 4–4 ).Let D y =[ D y ij ],(1 i N ),(1 i N ) bethe N N matrixdenedasfollows: D y 11 = D 11 1 w 1 (4–39) D y ij = w j w i D ji otherwise (4–40) Theorem2. Thematrix D y denedin ( 4–39 ) and ( 4–40 ) isadifferentiationmatrixfor thespaceofpolynomialsofdegree N 1 .Moreprecisely,if q isapolynomialofdegree atmost N 1 and q 2 R N isthevectorwith i th component q i = q ( i ) 1 i N ,then ( D y q ) i =_ q ( i ),1 i N ( qofdegree N 1). ProofofTheorem2. Let E denotethedifferentiationmatrixdenedinthestatemento f thetheorem.Thatis, E isan N N matrixwiththepropertythatforall q 2 R N ,wehave ( Eq ) i =_ q ( i ),1 i N where q isthepolynomialofdegreeatmost N 1 whichsatises q j = q ( j ) 1 j N If p and q aresmooth,real-valuedfunctionswith p (1)=0 ,thenintegrationbyparts 93

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gives Z 1 1 p ( ) q ( ) d = p ( 1) q ( 1) Z 1 1 p ( )_ q ( ) d (4–41) Suppose p isapolynomialofdegreeatmost N and q isapolynomialofdegreeatmost N 1 with N 1 ;inthiscase, pq and p q arepolynomialsofdegreeatmost 2 N 2 BecauseGauss-Radauquadratureisexactforpolynomialsofd egreeatmost 2 N 2 theintegralsin( 4–41 )canbereplacedbytheirquadratureequivalentstoobtain N X j =1 w j p j q j = p 1 q 1 N X j =1 w j p j q j (4–42) where p j = p ( j ) and p j =_ p ( j ) 1 i N p isanypolynomialofdegreeatmost N whichvanishesat +1 ,and q isanypolynomialofdegreeatmost N 1 .Apolynomial ofdegree N isuniquelydenedbyitsvalueat N +1 points.Let p bethepolynomial ofdegreeatmost N whichsatises p (1)=0 and p j = p ( j ) 1 j N .Let q be thepolynomialofdegreeatmost N 1 suchtha q j = q ( j ) 1 j N .Substituting _p = D 1: N p and _q = Eq in( 4–42 )gives ( WD 1: N p ) T q = p 1 q 1 ( Wp ) T Eq p T D T1: N Wq = p 1 q 1 p T WEq where W isthediagonalmatrixofRadauquadratureweights.Thiscan berearranged intothefollowingform: p T ( D T1: N W + WE + e 1 e T1 ) q =0, where e 1 istherstcolumnoftheidentitymatrix.Since p and q arearbitrary,wededuce that D T1: N W + WE + e 1 e T1 = 0 94

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whichimpliesthat E 11 = D 11 1 w 1 (4–43) E ij = w j w i D ji (4–44) Comparing( 4–40 )with( 4–44 )and( 4–39 )with( 4–43 ),weseethat D y = E .Thematrix D y denedin ( 4–40 ) and ( 4–39 ) isadifferentiationmatrixforthespaceofpolynomials ofdegree N 1 Accordingtothedenitionof D y D T1 = w 1 D y1 W 1 1 w 1 e 1 (4–45) D Tk = w k D yk W 1 ,2 k N (4–46) where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal.Thevector e 1 isthe 1 st rowof N N identitymatrix.SubstitutingEqs.( 4–45 )and( 4–46 )inEqs.( 5–26 ) and( 4–29 ), r Y 1 ( h i )= t f t 0 2 r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i ) + w 1 D y1 W 1 + 1 w 1 e 1 (4–47) w k D yk W 1 = t f t 0 2 r Y k ( w k g k + h k f k ih k C k i ),2 k N (4–48) Next,deningthefollowingchangeofvariables: ~ N +1 = D TN +1 (4–49) ~ k = k w k ,1 k N (4–50) ~ r k = k w k ,1 k N (4–51) ~ = (4–52) 95

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SubstitutingEqs.( 4–49 )-( 4–52 )inEqs.( 4–31 )-( 4–38 )andinEqs.( 4–47 )-( 4–48 ),the transformedKKTconditionsfortheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (4–53) 0 = D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f ),1 k N (4–54) ~ r ki =0 when C ki < 0,1 i s ,1 k N (4–55) ~ r ki < 0 when C ki =0,1 i s ,1 k N (4–56) 0 = ( Y ( 1 ), 1 Y ( N +1 ), N +1 ), (4–57) r t 0 ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) t f t 0 2 N X k =1 w k r t 0 ( g k + h ~ k f k ih ~ r k C k i ), (4–58) r t f ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) + t f t 0 2 N X k =1 w k r t f ( g k + h ~ k f k ih ~ r k C k i ), (4–59) D y1 ~ = t f t 0 2 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) 1 w 1 ~ 1 + r Y 1 ( h ~ i ) (4–60) ~ N +1 = r Y N +1 ( h ~ i ), (4–61) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (4–62) 2 k N Now,consideracomparisonofthetransformedKKTconditionsi nEqs.( 4–53 )-( 4–62 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 4–6 )-( 4–15 )of thecontinuous-timeoptimalcontrolproblem.Itisnotedth atthetransformedKKT conditionsinEqs.( 4–53 )-( 4–57 )arethediscretizedformsofthecontinuous-time rst-orderoptimalityconditionsinEq.( 4–8 ),Eq.( 4–6 ),Eq.( 4–13 ),Eq.( 4–14 ),and Eq.( 4–15 ),respectively.Next,theright-handsideofEq.( 4–58 )andEq.( 4–59 )is 96

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thequadratureapproximationoftheright-handsideofEq.( 4–11 )andEq.( 4–12 ), respectively.Therefore,thesetoftransformedKKTconditio nsinEq.( 4–58 )and Eq.( 4–59 )isthediscretizedversionofthesetofcontinuous-timer st-orderoptimality conditionsinEq.( 4–11 )andEq.( 4–12 ).Furthermore,itisshowninTheorem 2 thatthe system( 4–62 )isapseudospectralschemeforthecostatedynamics,i.e. D yk ~ = ~ k 1 k N (4–63) Therefore,thelefthandsideofEq.( 4–62 )isanapproximationofcostatedynamics atthe k th collocationpoint, k =(2,..., N ) .Asaresult,Eq.( 4–62 )representsthe discretizedversionofthecostatedynamicsinEq.( 4–7 )at k =(2,..., N ) .Next,itis notedthatatthenalboundarypoint,thediscreteequivale ntofcontinuousboundary conditions( 4–10 )isthesameasthediscretecostateatthenalboundarypoin tin ( 4–61 ).However,attheinitialboundarypoint,thediscreteequi valentofthecontinuous boundarycondition( 4–9 )iscoupledinthediscretecostatedynamicsattheinitial boundarypointin( 4–60 ).TheequivalenceofthetransformedKKTconditionatthe initialboundary,( 4–60 ),oftheNLPtothediscretizedformofthecontinuousrst-o rder optimalityconditionin( 4–9 )isnowestablishedbymanipulating( 4–49 ). Let D = [ D 1: N D N +1 ] where D 1: N iscomposedofrst N columnsand D N +1 isthe lastcolumnof D .Then D issuchthat: D N +1 = D 1: N 1 ,where 1 isacolumnvectorofall ones.Proposition2. D N +1 = D 1: N 1 ProofofProposition2. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 1: N 1 + D N +1 = 0 .Rearranging,weobtain D N +1 = D 1: N 1 (4–64) 97

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Returningtothedenitionof ~ N +1 in( 4–49 ),weobtain ~ N +1 = D TN +1 = N X i =1 i D i N +1 = N X i =1 N X j =1 i D ij (4–65) = 1 w 1 + N X i =1 N X j =1 i D y ji w j w i = 1 w 1 + N X i =1 N X j =1 j D y ij w i w j (4–66) = ~ 1 + N X i =1 N X j =1 w i ~ j D y ij = ~ 1 + N X i =1 w i D yi ~ (4–67) = r Y 1 ( h ~ i ) t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ), (4–68) where( 4–65 )followsfromtheidentity( 4–64 )giveninProposition 2 ,( 4–66 )isthe denition( 4–39 )and( 4–40 )of D y ,( 4–67 )isthedenition( 4–50 )of ~ i ,and( 4–68 )isthe rst-orderoptimalitycondition( 4–62 )and( 4–60 ).Rearranging( 4–68 )suchthat r Y 1 ( h ~ i )= ~ N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (4–69) Next,thecontinuouscostatedynamicsinEq.( 4–7 )are ( )= t f t 0 2 r y H = t f t 0 2 r y ( g + h f ih r C i ). (4–70) Integratingthecontinuouscostatedynamicsin( 4–70 )usingRadauquadrature, ~ 1 = ~ N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (4–71) Comparing( 4–69 )with( 4–71 )gives ~ 1 = r Y 1 ( h ~ i ). (4–72) Equation( 4–72 )isthemissingboundaryconditionattheinitialpointthat wascoupled withthediscretecostatedynamicsattheinitialpointin( 4–60 ).Itisalsoimpliedby Eq.( 4–72 )thattheextratermin( 4–60 )isinfact zero ,thereby,making( 4–60 )consistent 98

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withdiscretecostatedynamicsattheinitialpoint.Hence, thesystemoftransformed KKTconditionsoftheNLPisequivalenttotherst-orderoptim alityconditionsofthe continuous-timeoptimalcontrolproblemandaccuratecost ateestimatesareobtained fromtheKKTmultipliersusingtherelationshipgiveninEqs.( 4–49 )-( 4–50 ). 4.3FlippedRadauPseudospectralMethod TheLegendre-Gauss-Radau(LGR)points,liketheLegendreGauss-Lobatto(LGL) pointsandLegaendre-Gauss(LG)pointsaredenedontheint erval [ 1,+1] .However, unlikeLGandLGLpoints,LGRpointsareasymmetricaboutthe origin.Byippingthe pointsabouttheorigin,i.e.,bytakingthenegativeofLGRp oints,anewsetofpoints, calledthe ipped LGRpoints,isobtained.Radaupseudospectralmethodcanbe implementedusingthissetofippedLGRpoints.Inthissect ion,problemformulation andKKTconditionsforaippedRadaupseudospectralmethodar epresented.The ippedRadaupseudospectralmethoddiffersfromthestanda rdRadaupseudospectral methodinthatthenaltimeisacollocationpointinippedR adaupseudospectral methodwhereasinstandardRadaupseudospectralmethod,in itialtimeisacollocation point.Asaresult,inippedRadaupseudospectralmethod,co ntrolatthenaltimeis obtained.Furthermore,itisshownthattheippedRadaupse udospectralmethodalso resultsinasystemoftransformedKKTconditionsthatisequiv alenttothediscretized formofcontinuousrst-orderoptimalityconditions.4.3.1NLPFormulationoftheFlippedRadauPseudospectralMe thod Considerthe N ippedLGRpoints, ( 1 2 ,..., N ) ,where 1 > 1 and N =+1 Deneanewpointsuchthat 0 = 1 .Next,let L i ( ) ( i =0,..., N ) ,betheLagrange polynomialsofdegree N givenby L i ( )= N Y j =0 j 6 = i j i j ,( i =0,..., N ). (4–73) 99

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Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =0 Y i L i ( ), (4–74) where Y i = Y ( i ) .Itisimportanttonotethat 0 = 1 isnotaippedLGRpointbutis usedinstateapproximation.Next,anapproximationtothed erivativeofthestatein domainisgivenbydifferentiatingtheapproximationofEq.( 4–74 )withrespectto y ( ) Y ( )= N X i =0 Y i L i ( ). (4–75) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 4–75 )totheright-handsideofstatedynamicconstraintsin Eq.( 4–2 )atthe N ippedLGRpoints, ( 1 ,..., N ) andaregivenas N X i =0 Y i L i ( k )= t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (4–76) N X i =0 D ki Y i = t f t 0 2 f ( Y k U k ; t 0 t f ), D ki = L i ( k ), (4–77) where U k = U ( k ) .Itisnotedthat 0 isnotacollocationpoint.Thematrix D = [ D ki ],(1 k N ),(0 i N ) isa N ( N +1) non-square matrixandiscalled the ippedRadaupseudospectraldifferentiationmatrix .Thematrix D isnon-square becausethestateapproximationuses N +1 points, ( 0 ,..., N ) ,butthecollocationis doneatonly N ippedLGRpoints, ( 1 ,..., N ) .Let Y LGR bedenedas Y LGR = 266664 Y 0 ... Y N 377775 Usingthematrix Y LGR ,thecollocateddynamicsatthe N LGRcollocationpointsin Eq.( 4–77 )areexpressedas D k Y LGR = t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (4–78) 100

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where D k isthe k th rowofdifferentiationmatrix D .Next,thepathconstraintsin Eq.( 4–4 )areenforcedatthe N ippedLGRcollocationpointsas t f t 0 2 C ( Y k U k ; t 0 t f ) 0 ,( k =1,..., N ). (4–79) Lastly,thecostfunctionalisapproximatedusingLGRquadr atureas J =( Y ( 0 ), 0 Y ( N ), N )+ t f t 0 2 N X k =1 w k g ( Y k U k ; t 0 t f ), (4–80) where w k isthequadratureweightassociatedwiththe k th ippedLGRcollocationpoint. Thenite-dimensionalnonlinearprogrammingproblemcorr espondingtotheipped Radaupseudospectralmethodisthengivenasfollows.Minim izetheobjectivecost function J =( Y ( 0 ), 0 Y ( N ), N )+ t f t 0 2 N X k =1 w k g ( Y k U k ; t 0 t f ), (4–81) subjecttothefollowingequalityandinequalityconstrain ts: D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f )= 0 ,( k =1,..., N ), (4–82) ( Y ( 0 ), 0 Y ( N ), N )= 0 (4–83) t f t 0 2 C ( Y k U k ; t 0 t f ) 0 ,( k =1,..., N ). (4–84) wheretheNLPvariablesare ( Y 0 ,..., Y N ) ( U 1 ,..., U N ) t 0 ,and t f .Itisnotedthatthe initialcontrol, U 0 ,isnotobtainedinthesolutionoftheNLP.However,thecont rolatthe nalpoint, U N ,isobtained.TheproblemdenedbyEqs.( 4–81 )-( 4–84 )isthediscrete ippedRadaupseudospectralapproximationtothecontinuo us-timeoptimalcontrol problemdenedbyEqs.( 4–1 )-( 4–4 ). AfewkeypropertiesoftheippedRadaupseudospectralmeth odarenow stated.Thediscretizationpoints,atwhichthestateisapp roximatedusingLagrange polynomials,arethe N ippedLGRpointsplustheinitialpoint, ( 0 ,..., N ) .Thestate approximationusesLagrangepolynomialsofdegree N .Thestatedynamicsare 101

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0 0 0.5 1 5 10 15 20 1 0.5 NumberofDiscretizationPointsDiscretizationPoint CollocationPoint Figure4-2.Discretizationandcollocationpointsforipp edRadaupseudospectral method. collocatedatonlythe N ippedLGRpoints, ( 1 ,..., N ) .Asaconsequence,theipped Radaupseudospectraldifferentiationmatrixisanon-squa re, N ( N +1) matrix.Fig. 4-2 showsthediscretizationandcollocationpointsfortheip pedRadaupseudospectral method.4.3.2NecessaryOptimalityConditions Thenecessaryoptimalityconditions,ortheKarush-Kuhn-Tu cker(KKT)conditions, oftheNLPgiveninEqs.( 4–81 )-( 4–84 )arenowderived.TheLagrangianassociated withtheNLPis L =( Y ( 0 ), 0 Y ( N ), N ) h ( Y ( 0 ), 0 Y ( N ), N ) i + t f t 0 2 N X k =1 ( w k g ( Y k U k ; t 0 t f ) h k C ( Y k U k ; t 0 t f ) i ) N X k =1 h k D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f ) i (4–85) 102

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where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 4–82 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 4–83 ),and k isthe k th rowoftheLagrangemultipliersmatrix 2 R N s associatedwiththeconstraintsinEq.( 4–84 )TheKKToptimalityconditionsare thenobtainedbydifferentiatingtheLagrangianwithrespe cttoeachofthevariableand equatingittozerosuchthat t f t 0 2 r Y k ( w k g k + h k f k ih k C k i )= D Tk ,1 k N 1, (4–86) t f t 0 2 r Y N ( w N g N + h N f N ih N C 1 i )= D TN r Y N ( h i ), (4–87) r Y 0 ( h i )= D T0 (4–88) t f t 0 2 N X k =1 r t 0 ( w k g k + h k f k ih k C k i ) 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t 0 ( h i ), (4–89) t f t 0 2 N X k =1 r t f ( w k g k + h k f k ih k C k i ) + 1 2 N X k =1 ( w k g k + h k f k ih k C k i )= r t f ( h i ), (4–90) r U k ( w k g k + h k f k ih k C k i )= 0 ,1 k N (4–91) D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f )= 0 ,1 k N (4–92) ki =0 when C ki < 0,1 i s ,1 k N (4–93) ki < 0 when C ki =0,1 i s ,1 k N (4–94) ( Y ( 0 ), 0 Y ( N ), N )= 0 (4–95) where D Ti isthe i th rowof D T g k = g ( Y k U k ; t 0 t f ) f k = f ( Y k U k ; t 0 t f ) and C k = C ( Y k U k ; t 0 t f ) Next,theKKTconditionsgiveninEqs.( 4–86 )-( 4–95 )arereformulatedsothatthey becomeadiscretizationoftherst-orderoptimalitycondi tionsgiveninEqs.( 4–6 )-( 4–15 ) forthecontinuouscontrolproblemgiveninEqs.( 4–1 )-( 4–4 ).Let D y =[ D y ij ],(1 i 103

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N ),(1 i N ) bethe N N matrixdenedasfollows: D y NN = D NN + 1 w N (4–96) D y ij = w j w i D ji otherwise (4–97) Theorem3. Thematrix D y denedin ( 4–96 ) and ( 4–97 ) isadifferentiationmatrixfor thespaceofpolynomialsofdegree N 1 .Moreprecisely,if q isapolynomialofdegree atmost N 1 and q 2 R N isthevectorwith i th component q i = q ( i ) 1 i N ,then ( D y q ) i =_ q ( i ),1 i N ( qofdegree N 1). ProofofTheorem3. Let E denotethedifferentiationmatrixdenedinthestatemento f thetheorem.Thatis, E isan N N matrixwiththepropertythatforall q 2 R N ,wehave ( Eq ) i =_ q ( i ),1 i N where q isthepolynomialofdegreeatmost N 1 whichsatises q j = q ( j ) 1 j N If p and q aresmooth,real-valuedfunctionswith p ( 1)=0 ,thenintegrationbyparts gives Z 1 1 p ( ) q ( ) d = p (1) q (1) Z 1 1 p ( )_ q ( ) d (4–98) Suppose p isapolynomialofdegreeatmost N and q isapolynomialofdegreeatmost N 1 with N 1 ;inthiscase, pq and p q arepolynomialsofdegreeatmost 2 N 2 SinceGauss-Radauquadratureisexactforpolynomialsofdeg reeatmost 2 N 2 ,the integralsin( 4–98 )canbereplacedbytheirquadratureequivalentstoobtain N X j =1 w j p j q j = p N q N N X j =1 w j p j q j (4–99) where p j = p ( j ) and p j =_ p ( j ) 1 i N p isanypolynomialofdegreeatmost N whichvanishesat +1 ,and q isanypolynomialofdegreeatmost N 1 .Apolynomial ofdegree N isuniquelydenedbyitsvalueat N +1 points.Let p bethepolynomial ofdegreeatmost N whichsatises p ( 1)=0 and p j = p ( j ) 1 j N .Let q be 104

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thepolynomialofdegreeatmost N 1 suchtha q j = q ( j ) 1 j N .Substituting _p = D 1: N p and _q = Eq in( 4–99 )gives ( WD 1: N p ) T q = p N q N ( Wp ) T Eq p T D T1: N Wq = p N q N p T WEq where W isthediagonalmatrixofRadauquadratureweights.Thiscan berearranged intothefollowingform: p T ( D T1: N W + WE e N e TN ) q =0, where e N isthe N th columnoftheidentitymatrix.Since p and q arearbitrary,wededuce that D T1: N W + WE e N e TN = 0 whichimpliesthat E NN = D NN + 1 w N (4–100) E ij = w j w i D ji (4–101) Comparing( 4–97 )with( 4–101 )and( 4–96 )with( 4–100 ),weseethat D y = E .The matrix D y denedin ( 4–97 ) and ( 4–96 ) isadifferentiationmatrixforthespaceof polynomialsofdegree N 1 Accordingtothedenitionof D y D TN = w N D yN W 1 + 1 w N e N (4–102) D Tk = w k D yk W 1 ,1 k N 1, (4–103) where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal.Vector e N isthe N th rowof N N identitymatrix.SubstitutingEqs.( 4–102 )and( 4–103 )in 105

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Eqs.( 4–87 )and( 4–86 ), r Y N ( h i )= t f t 0 2 r Y N ( w N g N + h N f N ih N C N i ) + w N D yN W 1 1 w N e N (4–104) w k D yk W 1 = t f t 0 2 r Y k ( w k g k + h k f k ih k C k i ),1 k N 1. (4–105) Next,deningthefollowingchangeofvariables: ~ 0 = D T0 (4–106) ~ k = k w k ,1 k N (4–107) ~ r k = k w k ,1 k N (4–108) ~ = (4–109) SubstitutingEqs.( 4–106 )-( 4–109 )inEqs.( 4–88 )-( 4–95 )andinEqs.( 4–104 )-( 4–105 ), thetransformedKKTconditionsfortheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (4–110) 0 = D k Y LGR t f t 0 2 f ( Y k U k ; t 0 t f ),1 k N (4–111) ~ r ki =0 when C ki < 0,1 i s ,1 k N (4–112) ~ r ki < 0 when C ki =0,1 i s ,1 k N (4–113) 0 = ( Y ( 0 ), 0 Y ( N ), N ), (4–114) r t 0 ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) t f t 0 2 N X k =1 w k r t 0 ( g k + h ~ k f k ih ~ r k C k i ), (4–115) r t f ( h ~ i )= 1 2 N X k =1 w k ( g k + h ~ k f k ih ~ r k C k i ) + t f t 0 2 N X k =1 w k r t f ( g k + h ~ k f k ih ~ r k C k i ), (4–116) 106

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D yN ~ = t f t 0 2 r Y N ( g N + h ~ N f N ih ~ r N C N i ) + 1 w N ~ N r Y N ( h ~ i ) (4–117) ~ 0 = r Y 0 ( h ~ i ), (4–118) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (4–119) 1 k N 1. NowconsideracomparisonofthetransformedKKTconditionsin Eqs.( 4–110 )-( 4–119 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 4–6 )-( 4–15 )ofthe continuous-timeoptimalcontrolproblem.Itisnotedthatt hetransformedKKTconditions inEqs.( 4–110 )-( 4–114 )arethediscretizedformsofthecontinuous-timerst-ord er optimalityconditionsinEq.( 4–8 ),Eq.( 4–6 ),Eq.( 4–13 ),Eq.( 4–14 ),andEq.( 4–15 ), respectively.Next,theright-handsideofEq.( 4–115 )andEq.( 4–116 )isthequadrature approximationoftheright-handsideofEq.( 4–11 )andEq.( 4–12 ),respectively. Therefore,thetransformedKKTconditionsinEq.( 4–115 )andEq.( 4–116 )arethe discretizedversionofcontinuous-timerst-orderoptima lityconditionsinEq.( 4–11 ) andEq.( 4–12 ).Furthermore,itisshowninTheorem 3 thatthesystem( 4–119 )isa pseudospectralschemeforthecostatedynamics,i.e. D yk ~ = ~ k 1 k N (4–120) Therefore,thelefthandsideofEq.( 4–119 )isanapproximationofcostatedynamics atthe k th collocationpoint, k =(1,..., N 1) .Asaresult,Eq.( 4–119 )representsthe discretizedversionofthecostatedynamicsinEq.( 4–7 )at k =(1,..., N 1) .Next, itisnotedthatattheinitialboundarypoint,thediscretee quivalentofthecontinuous boundarycondition( 4–9 )isthesameasthediscretecostateattheinitialboundary pointin( 4–118 ).However,atthenalboundarypoint,thediscreteequival entofthe continuousboundarycondition( 4–10 )iscoupledinthediscretecostatedynamicsatthe nalboundarypointin( 4–117 ). 107

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TheequivalenceofthetransformedKKTcondition,atthenalb oundary( 4–117 ),of theNLPtothediscretizedformofcontinuousrst-orderopt imalityconditionin( 4–10 )is nowestablishedbymanipulating( 4–106 ). Let D = [ D 0 D 1: N ] where D 0 istherstcolumnof D and D 1: N istheremaining columns.Then D issuchthat: D 0 = D 1: N 1 ,where 1 isacolumnvectorofallones. Proposition3. D 0 = D 1: N 1 ;equivalently, D 1 1: N D 0 = 1 ProofofProposition3. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 0 + D 1: N 1 = 0 .Rearranging,weobtain D 0 = D 1: N 1 (4–121) Returningtothedenitionof ~ 0 in( 4–106 ),weobtain ~ 0 = D T0 = N X i =1 i D i ,0 = N X i =1 N X j =1 i D ij (4–122) = N w N N X i =1 N X j =1 i D y ji w j w i = N w N N X i =1 N X j =1 j D y ij w i w j (4–123) = ~ N N X i =1 N X j =1 w i ~ j D y ij = ~ N N X i =1 w i D yi ~ (4–124) = r Y N ( h ~ i )+ t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ), (4–125) where( 4–122 )followsfromtheidentity( 4–121 )giveninProposition 3 ,( 4–123 )isthe denition( 4–96 )and( 4–97 )of D y ,( 4–124 )isthedenition( 4–107 )of ~ i ,and( 4–125 )is therst-orderoptimalitycondition( 4–119 )and( 4–117 ).Rearranging( 4–125 )suchthat r Y N ( h ~ i )= ~ 0 t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (4–126) 108

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Next,thecontinuouscostatedynamicsinEq.( 4–7 )are ( )= t f t 0 2 r y H = t f t 0 2 r y ( g + h f ih r C i ). (4–127) Integratingthecontinuouscostatedynamicsin( 4–127 )usingRadauquadrature, ~ N = ~ 0 t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (4–128) Comparing( 4–126 )with( 4–128 )gives ~ N = r Y N ( h ~ i ). (4–129) Equation( 4–129 )isthemissingboundaryconditionatthenalpointthatwas coupled withthediscretecostatedynamicsatthenalpointin( 4–117 ).Itisalsoimpliedby Eq.( 4–129 )thattheextratermin( 4–117 )isinfact zero ,thereby,making( 4–117 ) consistentwithdiscretecostatedynamicsatthenalpoint .Hence,thesystemof transformedKKTconditionsoftheNLPisequivalenttotherst -orderoptimality conditionsofthecontinuous-timeoptimalcontrolproblem andanaccuratecostate estimateisobtainedfromtheKKTmultipliersusingtherelati onshipgiveninEq.( 4–106 ) andEq.( 4–107 ). 4.4Summary TheRadauandtheippedRadaupseudospectralmethodsbased onthecollocation attheLGRandtheippedLGRpoints,respectively,havebeen presented.The continuous-timeoptimalcontrolproblemistranscribedin toadiscretenonlinear programmingproblem(NLP)usingeitherofthetwomethods.It hasbeenshownthatthe transformedKKTconditionsobtainedincaseofboththeRadaup seudospectralmethod andtheippedRadaupseudospectralmethodareequivalentt othediscretizedversion ofthecontinuous-timerst-orderoptimalityconditions. WhileRadaupseudospectral methodhastheabilitytocomputetheinitialcontrol,contr olatthenalpointisnot 109

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Continuous-Time Optimal Control Problem Discretize Continuous First-Order Optimality Conditions Discrete First-Order Optimality Conditions Transformed KKT Conditions Nonlinear Programming Problem Continuous-Time Optimal Control Problem Optimize Discretize Optimize Radau flippedRadau Figure4-3.RelationshipbetweentheKKTconditionsandthedi screterst-order optimalityconditionsforRadaupseudospectralmethodand ipped Radau pseudospectralmethod. obtained.InippedRadaupseudospectralmethod,controla tthenalpointisobtained butnotthecontrolattheinitialpoint.Theimplementation oftheRadaupseudospectral methodissignicantlylesscomplexthantheGausspseudosp ectralmethodbecause intheRadaupseudospectralmethodimplementation,therei snoneedtocomputethe stateatthenalpointusingquadratureruleasisthecasein theGausspseudospectral method. 110

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CHAPTER5 AUNIFIEDFRAMEWORKFORPSEUDOSPECTRALMETHODS Auniedframeworkispresentedfortwodifferentpseudospe ctralmethodsbased oncollocationattheLegendre-Gauss(LG)andtheLegendreGauss-Radau(LGR) points.Foreachoftheschemesinthisframework,(1)thesta teisapproximatedusing collocationpointsandnoncollocatedpoints.(2)thestate dynamicsareevaluatedatonly thecollocationpoints.Asaresult,schemeswithinthisfram eworkemploypolynomials toapproximatethestatethatarethesamedegreeasthenumbe rofcollocation points.Eachoftheseschemescanbeexpressedineitheradiff erentialoranintegral formulation.TheLGandtheLGRdifferentiationandintegra tionmatricesareinvertible, andthedifferentialandintegralversionsareequivalent. Eachoftheseschemesprovide anaccuratetransformationbetweentheLagrangemultiplie rsofthediscretenonlinear programmingproblemandthecostateofthecontinuousoptim alcontrolproblem.It isshownthatbothoftheseschemesformadeterminedsystemo flinearequations forcostatedynamics.Lastly,itisshownempiricallythatt hestate,thecontrolandthe costatesolutionsobtainedbytheseschemesconverge.Thes eschemesaredifferent fromthepseudospectralmethodbasedoncollocationattheL egendre-Gauss-Lobatto (LGL)points.First,bothstateapproximationanddynamics collocationusethesame setofcollocationpoints.Asaresult,intheLGLschemethede greeofpolynomialsused toapproximatethestateisonelessthanthenumberofcolloc ationpoints.TheLGL differentiationmatrixissingularandhencetheequivalen cebetweenthedifferential andintegralformulationisnotestablished.FortheLGLsch eme,thelinearsystem ofequationsforcostatedynamicsisunder-determined,the implicationsofwhichare shownbymeansofanexample.Thetransformationbetweenthe Lagrangemultipliersof thediscretenonlinearprogrammingproblemandthecostate ofthecontinuousoptimal controlproblemfortheLGLschemeisfoundtobeinaccurate. Lastly,thesolutions 111

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obtainedfromtheLGLschemearetypicallynon-convergento rhavehigherordersof magnitudeerrorthanthesolutionobtainedfromtheLGorthe LGRscheme. 5.1ImplicitIntegrationScheme Inallthepseudospectralmethods,thesolutionstothedyna micconstraintsare approximatednumericallyusingcollocation.Considerthe dynamicconstraintsofthe form d y ( ) d =_ y ( )= t f t 0 2 f ( y ( ), u ( ), ; t 0 t f ). (5–1) Incollocation,rst,thestate, y ( ) ,isapproximatedusingLagrangepolynomials.Let Y ( ) betheLagrangepolynomialstateapproximation.Thenthede rivativeofthestate approximation, Y ( ) ,issetequaltotheright-handsideofthedifferentialequa tionin ( 5–1 )evaluatedatasetofintermediatepoints, ( 1 ,..., K ) ,calledthecollocationpoints, resultinginasystemofalgebraicconstraints.Thesealgeb raicconstraintsarethe equalityandconstraintsoftheNLP. Analternativeapproachtosolvethedifferentialequations in( 5–1 )istouse numericalintegration.Inthisapproach,therighthandsid eofthedynamicconstraintsin Eq.( 5–1 )isrstapproximatedusingLagrangepolynomials.Thenthe approximationis numericallyintegratedtoobtainthesolutionofthestatea tthesetofintermediatepoints. Inthissection,itisshownthatthedifferentialandtheint egralschemesareequivalent forthemethodsthatareapartoftheframework,i.e.,theGau sspseudospectralmethod, theRadaupseudospectralmethod,andtheippedRadaupseud ospectralmethod.It isalsoshownthatfortheLobattopseudospectralmethod,th edifferentialandintegral schemesarenotequivalent.5.1.1IntegralFormulationUsingLGCollocation Considerthe N LGpoints, ( 1 2 ,..., N ) ,where 1 > 1 and N < +1 .Dene twonewpointssuchthat 0 = 1 and N +1 =1 .Next,let L i ( ) ( i =0,..., N ) ,bethe 112

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Lagrangepolynomialsofdegree N givenby L i ( )= N Y j =0 j 6 = i j i j ,( i =0,..., N ). (5–2) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =0 Y i L i ( ), (5–3) where Y i = Y ( i ) .Itisimportanttonotethat 0 = 1 isnotaLGpointbutisusedinthe stateapproximation.Next,anapproximationtothederivat iveofthestatein domainis givenbydifferentiatingtheapproximationofEq.( 5–3 )withrespectto y ( ) Y ( )= N X i =0 Y i L i ( ). (5–4) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 5–4 )totheright-handsideofstatedynamicconstraintsin Eq.( 5–1 )atthe N LGpoints, ( 1 ,..., N ) N X i =0 Y i L i ( k )= t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (5–5) N X i =0 D ki Y i = t f t 0 2 f k D ki = L i ( k ), (5–6) where U k = U ( k ) f k = f ( Y k U k k ; t 0 t f ) and D =[ D ki ],(1 k N ),(0 i N ) is a N ( N +1) non-square matrixandiscalledthe Gausspseudospectraldifferentiation matrix .Thematrix D isnon-squarebecausethestateapproximationuses N +1 points, ( 0 ,..., N ) ,butthecollocationisdoneatonly N LGpoints, ( 1 ,..., N ) Let D = [ D 0 D 1: N ] where D 0 istherstcolumnof D and D 1: N aretheremaining columns.Then D issuchthat:(a) D 1: N isnonsingularand(b) D 0 = D 1: N 1 ;equivalently, D 1 1: N D 0 = 1 ,where 1 isacolumnvectorofallones. 113

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Proposition4. Thematrix D 1: N obtainedbydeletingtherstcolumnoftheGauss pseudospectraldifferentiationmatrix D isinvertible. ProofofProposition4. Supposethatforsomenonzero p 2 R N +1 with p 0 =0 ,wehave Dp = 0 .Let p betheuniquepolynomialofdegreeatmost N whichsatises p ( i )= p i 0 i N .Sincethecomponentsof Dp arethederivativesof p evaluatedatthe collocationpoints,wehave 0=( Dp ) i =_ p ( i ),1 i N Since p isapolynomialofdegreeatmost N 1 ,itmustbeidenticallyzerosinceit vanishesat N points.Hence, p isconstant.Since p ( 1)=0 and p isconstant,itfollows that p isidentically0.Thisshowsthat p i = p ( i )=0 foreach i .Sincetheequation Dp = 0 with p 0 =0 hasnononzerosolution, D 1: N isnonsingular. Proposition5. D 0 = D 1: N 1 ;equivalently, D 1 1: N D 0 = 1 ProofofProposition5. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 0 + D 1: N 1 = 0 .Rearranging,weobtain D 0 = D 1: N 1 (5–7) Multiplyingby D 1 1: N gives D 1 1: N D 0 = 1 Usingthematrix D = [ D 0 D 1: N ] ,thecollocateddynamicsatthe N LGcollocation pointsinEq.( 5–6 )areexpressedas D 0 Y 0 + D 1: N 266664 Y 1 ... Y N 377775 = t f t 0 2 266664 f 1 ... f N 377775 (5–8) 114

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MultiplyingEq.( 5–8 )by B = D 1 1: N andutilizingProposition 5 toobtain Y k = Y 0 + t f t 0 2 N X i =1 B ki f i ,1 k N (5–9) Next,theapproachofnumericalintegrationisappliedtoso lvethedifferentialequations in( 5–1 )byapproximatingthederivativeontheright-handsidebyL agrangepolynomials. Let L yi ( ) betheLagrangepolynomialsassociatedwiththecollocatio npoints: L yi = N Y j =1 j 6 = i j i j ,1 i N (5–10) NoticethattheLagrangepolynomials L i denedin( 5–2 )aredegree N whilethe Lagrangepolynomials L yi aredegree N 1 .Thenthestatedynamicsontheright-hand sideofEq.( 5–1 )areapproximatedbytheLagrangepolynomialsas F ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–11) EquatingtheapproximationinEq.( 5–11 )tothetimederivativeofstatesuchthat Y ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–12) IntegratingEq.( 5–12 )from 0 to k ,thefollowingrelationshipisobtained Y k = Y 0 + t f t 0 2 N X i =1 f i A ki A ki = Z k 1 L yi ( ) d ,1 k N (5–13) where A =[ A ki ],(1 k N ),(1 i N ) isa N N matrixcalledthe Gauss pseudospectralintegrationmatrix Itisnowdemonstratedthatthedifferentialschemeandthei ntegralschemeare equivalentbyshowingthatthematrix B = D 1 1: N inEq.( 5–9 )isequaltothematrix A in Eq.( 5–13 ). 115

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Let p beanypolynomialofdegreeatmost N .Bytheconstructionofthe N ( N +1) matrix D ,wehave Dp =_ p where p k = p ( k ),0 k N (5–14) p k =_ p ( k ),1 k N (5–15) Theidentity p = Dp canbewrittenas p = D 0 p 0 + D 1: N p 1: N (5–16) Multiplyingby D 1 1: N andutilizingProposition 5 gives p k = p 0 + D 1 1: N p k ,1 k N (5–17) Next,adifferentexpressionfor p k p 0 basedontheLagrangepolynomialapproximation ofthederivativeisobtained.Because p isapolynomialofdegreeatmost N 1 ,itcan beapproximatedexactlybytheLagrangepolynomials L yi as p = N X i =1 p i L yi ( ). (5–18) Integrating p from 1 to k ,thefollowingrelationshipisobtained p ( k )= p ( 1)+ N X i =1 p i A ki A ki = Z k 1 L yi ( ) d ,1 k N (5–19) Utilizingthenotation( 5–14 )and( 5–15 ),wehave p k = p 0 + ( A p ) k ,1 k N (5–20) Therelations( 5–17 )and( 5–20 )aresatisedforanypolynomialofdegreeatmost N Equating( 5–17 )and( 5–20 )toobtain A p = D 1 1: N p 116

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Choose p fromthecolumnsoftheidentitymatrixtodeducethat A = D 1 1: N .Thus,it isshownthatthedifferentialandintegralformsofthestat edynamicsintheGauss pseudospectralmethodare equivalent 5.1.2IntegralFormulationUsingStandardLGRCollocation Considerthe N LGRpoints, ( 1 2 ,..., N ) ,where 1 = 1 and N < +1 .Dene anewpointsuchthat N +1 =1 .Next,let L i ( ) ( i =1,..., N +1) ,betheLagrange polynomialsofdegree N givenby L i ( )= N +1 Y j =1 j 6 = i j i j ,( i =1,..., N +1). (5–21) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N +1 X i =1 Y i L i ( ), (5–22) where Y i = Y ( i ) .Itisimportanttonotethat N +1 =1 isnotaLGRpointbutisusedin stateapproximation.Next,anapproximationtothederivat iveofthestatein domainis givenbydifferentiatingtheapproximationofEq.( 5–22 )withrespectto y ( ) Y ( )= N +1 X i =1 Y i L i ( ). (5–23) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 5–23 )totheright-handsideofthestatedynamicconstraintsin Eq.( 5–1 )atthe N LGRpoints, ( 1 ,..., N ) N +1 X i =1 Y i L i ( k )= t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (5–24) N +1 X i =1 D ki Y i = t f t 0 2 f k D ki = L i ( k ), (5–25) where U k = U ( k ) f k = f ( Y k U k k ; t 0 t f ) and D =[ D ki ],(1 k N ),(1 i N +1) isa N ( N +1) non-square matrixandiscalledthe Radaupseudospectraldifferentiation 117

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matrix .Thematrix D isnon-squarebecausethestateapproximationuses N +1 points, ( 1 ,..., N +1 ) ,butthecollocationisdoneatonly N LGRpoints, ( 1 ,..., N ) Let D = [ D 1 D 2: N +1 ] where D 1 istherstcolumnof D and D 2: N +1 aretheremaining columns.Then D issuchthat:(a) D 2: N +1 isnonsingularand(b) D 1 = D 2: N +1 1 ; equivalently, D 1 2: N +1 D 1 = 1 ,where 1 isacolumnvectorofallones. Proposition6. Thematrix D 2: N +1 obtainedbydeletingtherstcolumnoftheRadau pseudospectraldifferentiationmatrix D isinvertible. ProofofProposition6. Supposethatforsomenonzero p 2 R N +1 with p 1 =0 ,wehave Dp = 0 .Let p betheuniquepolynomialofdegreeatmost N whichsatises p ( i )= p i 1 i N +1 .Sincethecomponentsof Dp arethederivativesof p evaluatedatthe collocationpoints,wehave 0=( Dp ) i =_ p ( i ),1 i N Since p isapolynomialofdegreeatmost N 1 ,itmustbeidenticallyzerosinceit vanishesat N points.Hence, p isconstant.Since p ( 1)=0 and p isconstant,itfollows that p isidentically0.Thisshowsthat p i = p ( i )=0 foreach i .Sincetheequation Dp = 0 with p 1 =0 hasnononzerosolution, D 2: N +1 isnonsingular. Proposition7. D 1 = D 2: N +1 1 ;equivalently, D 1 2: N +1 D 1 = 1 ProofofProposition7. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 1 + D 2: N +1 1 = 0 .Rearranging,weobtain D 1 = D 2: N +1 1 (5–26) Multiplyingby D 1 2: N +1 gives D 1 2: N +1 D 1 = 1 118

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Usingthematrix D = [ D 1 D 2: N +1 ] ,thecollocateddynamicsatthe N LGRcollocation pointsinEq.( 5–25 )areexpressedas D 1 Y 1 + D 2: N +1 266664 Y 2 ... Y N +1 377775 = t f t 0 2 266664 f 1 ... f N 377775 (5–27) MultiplyingEq.( 5–27 )by B = D 1 2: N +1 andutilizingProposition 7 gives Y k = Y 1 + t f t 0 2 N X i =1 B ki f i ,2 k N +1. (5–28) Next,theapproachofnumericalintegrationisappliedtoso lvethedifferentialequations in( 5–1 )byapproximatingthederivativeontheright-handsidebyL agrangepolynomials. Let L yi ( ) betheLagrangepolynomialsassociatedwiththecollocatio npoints: L yi = N Y j =1 j 6 = i j i j ,1 i N (5–29) NoticethattheLagrangepolynomials L i denedin( 5–21 )aredegree N whilethe Lagrangepolynomials L yi aredegree N 1 .Thenthestatedynamicsontheright-hand sideofEq.( 5–1 )areapproximatedbytheLagrangepolynomialsas F ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–30) EquatingtheapproximationinEq.( 5–30 )tothetimederivativeofstatesuchthat Y ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–31) IntegratingEq.( 5–31 )from 1 to k ,thefollowingrelationshipisobtained: Y k = Y 1 + t f t 0 2 N X i =1 f i A ki A ki = Z k 1 L yi ( ) d ,2 k N +1, (5–32) 119

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where A =[ A ki ],(2 k N +1),(1 i N ) isa N N matrixcalledthe Radau pseudospectralintegrationmatrix Itisnowdemonstratedthatthedifferentialschemeandthei ntegralschemeare equivalentbyshowingthatthematrix B = D 1 2: N +1 inEq.( 5–28 )isequaltothematrix A inEq.( 5–32 ). Let p beanypolynomialofdegreeatmost N .Bytheconstructionofthe N ( N +1) matrix D ,wehave Dp =_ p where p k = p ( k ),1 k N +1, (5–33) p k =_ p ( k ),1 k N (5–34) Theidentity p = Dp canbewrittenas p = D 1 p 1 + D 2: N +1 p 2: N +1 (5–35) Multiplyingby D 1 2: N +1 andutilizingProposition 7 gives p k = p 1 + D 1 2: N +1 p k ,2 k N +1. (5–36) Next,adifferentexpressionfor p k p 1 basedontheLagrangepolynomialapproximation ofthederivativeisobtained.Because p isapolynomialofdegreeatmost N 1 ,itcan beapproximatedexactlybytheLagrangepolynomials L yi : p = N X i =1 p i L yi ( ). (5–37) Integrating p from 1 to k ,thefollowingrelationshipisobtained p ( k )= p ( 1)+ N X i =1 p i A ki A ki = Z k 1 L yi ( ) d ,2 k N +1. (5–38) 120

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Utilizingthenotation( 5–33 )and( 5–34 ),wehave p k = p 1 + ( A p ) k ,2 k N +1. (5–39) Therelations( 5–36 )and( 5–39 )aresatisedforanypolynomialofdegreeatmost N Equating( 5–36 )and( 5–39 )toobtain A p = D 1 2: N +1 p Choose p fromthecolumnsoftheidentitymatrixtodeducethat A = D 1 2: N +1 .Thus, itisshownthatthedifferentialandintegralformsofthest atedynamicsintheRadau pseudospectralmethodare equivalent 5.1.3IntegralFormulationUsingFlippedLGRCollocation Considerthe N ippedLGRpoints, ( 1 2 ,..., N ) ,where 1 > 1 and N =+1 Deneanewpointsuchthat 0 = 1 .Next,let L i ( ) ( i =0,..., N ) ,betheLagrange polynomialsofdegree N givenby L i ( )= N Y j =0 j 6 = i j i j ,( i =0,..., N ). (5–40) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =0 Y i L i ( ), (5–41) where Y i = Y ( i ) .Itisimportanttonotethat 0 = 1 isnotaippedLGRpointbutis usedinstateapproximation.Next,anapproximationtothed erivativeofthestatein domainisgivenbydifferentiatingtheapproximationofEq.( 5–41 )withrespectto y ( ) Y ( )= N X i =0 Y i L i ( ). (5–42) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 5–42 )totheright-handsideofthestatedynamicconstraintsin 121

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Eq.( 5–1 )atthe N ippedLGRpoints, ( 1 ,..., N ) N X i =0 Y i L i ( k )= t f t 0 2 f ( Y k U k ; t 0 t f ),( k =1,..., N ), (5–43) N X i =0 D ki Y i = t f t 0 2 f k D ki = L i ( k ), (5–44) where U k = U ( k ) f k = f ( Y k U k k ; t 0 t f ) and D =[ D ki ],(1 k N ),(0 i N ) isa N ( N +1) non-square matrixandiscalledthe ippedRadaupseudospectral differentiationmatrix .Thematrix D isnon-squarebecausethestateapproximation uses N +1 points, ( 0 ,..., N ) ,butthecollocationisdoneatonly N ippedLGRpoints, ( 1 ,..., N ) Let D = [ D 0 D 1: N ] where D 0 istherstcolumnof D and D 1: N aretheremaining columns.Then D issuchthat:(a) D 1: N isnonsingularand(b) D 1 1: N D 0 = 1 ,where 1 isa columnvectorofallones.Proposition8. Thematrix D 1: N obtainedbydeletingtherstcolumnoftheipped Radaupseudospectraldifferentiationmatrix D isinvertible. ProofofProposition8. Supposethatforsomenonzero p 2 R N +1 with p 0 =0 ,wehave Dp = 0 .Let p betheuniquepolynomialofdegreeatmost N whichsatises p ( i )= p i 0 i N .Sincethecomponentsof Dp arethederivativesof p evaluatedatthe collocationpoints,wehave 0=( Dp ) i =_ p ( i ),1 i N Since p isapolynomialofdegreeatmost N 1 ,itmustbeidenticallyzerosinceit vanishesat N points.Hence, p isconstant.Since p ( 1)=0 and p isconstant,itfollows that p isidentically0.Thisshowsthat p i = p ( i )=0 foreach i .Sincetheequation Dp = 0 with p 0 =0 hasnononzerosolution, D 1: N isnonsingular. 122

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Proposition9. D 1 1: N D 0 = 1 ProofofProposition9. Thecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,whichimplies that D1 = D 0 + D 1: N 1 = 0 .Rearranging,weobtain D 0 = D 1: N 1 (5–45) Multiplyingby D 1 1: N gives D 1 1: N D 0 = 1 Usingthematrix D = [ D 0 D 1: N ] ,thecollocateddynamicsatthe N LGcollocation pointsinEq.( 5–44 )areexpressedas D 0 Y 0 + D 1: N 266664 Y 1 ... Y N 377775 = t f t 0 2 266664 f 1 ... f N 377775 (5–46) MultiplyingEq.( 5–46 )by B = D 1 1: N andutilizingProposition 5–45 toobtain Y k = Y 0 + t f t 0 2 N X i =1 B ki f i ,1 k N (5–47) Next,theapproachofnumericalintegrationisappliedtoso lvethedifferentialequations in( 5–1 )byapproximatingthederivativeontheright-handsidebyL agrangepolynomials. Let L yi ( ) betheLagrangepolynomialsassociatedwiththecollocatio npoints: L yi = N Y j =1 j 6 = i j i j ,1 i N (5–48) NoticethattheLagrangepolynomials L i denedin( 5–40 )aredegree N whilethe Lagrangepolynomials L yi aredegree N 1 .Thenthestatedynamicsontheright-hand sideofEq.( 5–1 )areapproximatedbytheLagrangepolynomialsas F ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–49) 123

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EquatingtheapproximationinEq.( 5–49 )tothetimederivativeofstatesuchthat Y ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–50) IntegratingEq.( 5–50 )from 0 to k ,followingrelationshipisobtained Y k = Y 0 + t f t 0 2 N X i =1 f i A ki A ki = Z k 1 L yi ( ) d ,1 k N (5–51) where A =[ A ki ],(1 k N ),(1 i N ) isa N N matrixcalledthe ippedRadau pseudospectralintegrationmatrix Itisnowdemonstratedthatthedifferentialschemeandthei ntegralschemeare equivalentbyshowingthatthematrix B = D 1 1: N inEq.( 5–47 )isequaltothematrix A in Eq.( 5–51 ). Let p beanypolynomialofdegreeatmost N .Bytheconstructionofthe N ( N +1) matrix D ,wehave Dp =_ p where p k = p ( k ),0 k N (5–52) p k =_ p ( k ),1 k N (5–53) Theidentity p = Dp canbewrittenas p = D 0 p 0 + D 1: N p 1: N (5–54) Multiplyingby D 1 1: N andutilizingProposition 5–45 gives p k = p 0 + D 1 1: N p k ,1 k N (5–55) Next,adifferentexpressionfor p k p 0 basedontheLagrangepolynomialapproximation ofthederivativeisobtained.Because p isapolynomialofdegreeatmost N 1 ,itcan 124

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beapproximatedexactlybytheLagrangepolynomials L yi as p = N X i =1 p i L yi ( ). (5–56) Integrating p from 1 to k ,thefollowingrelationshipisobtained p ( k )= p ( 1)+ N X i =1 p i A ki A ki = Z k 1 L yi ( ) d ,1 k N (5–57) Utilizingthenotation( 5–52 )and( 5–53 ),wehave p k = p 0 + ( A p ) k ,1 k N (5–58) Therelations( 5–55 )and( 5–58 )aresatisedforanypolynomialofdegreeatmost N Equating( 5–55 )and( 5–58 )toobtain A p = D 1 1: N p Choose p fromthecolumnsoftheidentitymatrixtodeducethat A = D 1 1: N .Thus,itis shownthatthedifferentialandintegralformsofthestated ynamicsintheippedRadau pseudospectralmethodare equivalent 5.1.4IntegralFormulationUsingLGLCollocation Considerthe N LGLpoints, ( 1 2 ,..., N ) ,where 1 = 1 and N =+1 .Next,let L i ( ) ( i =1,..., N ) ,betheLagrangepolynomialsofdegree N 1 givenby L i ( )= N Y j =1 j 6 = i j i j ,( i =1,..., N ). (5–59) Thestate, y ( ) ,isapproximatedbyapolynomialofdegreeatmost N 1 usingthe Lagrangepolynomialsas y ( ) Y ( )= N X i =1 Y i L i ( ), (5–60) 125

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where Y i = Y ( i ) .Thepoints, ( 1 ,..., N ) ,thatareusedinstateapproximationare calledthediscretizationpoints.Next,anapproximationt othederivativeofthestatein domainisgivenbydifferentiatingtheapproximationofEq.( 5–60 )withrespectto y ( ) Y ( )= N X i =1 Y i L i ( ). (5–61) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 5–61 )totheright-handsideofstatedynamicconstraintsin Eq.( 5–1 )atthe N LGLpoints.The N LGLpointsatwhichthecollocationisdoneare calledthecollocationpoints. N X i =1 Y i L i ( k )= t f t 0 2 f ( Y k U k k ; t 0 t f ),( k =1,..., N ), (5–62) N X i =1 D ki Y i = t f t 0 2 f k D ki = L i ( k ), (5–63) where U k = U ( k ) f k = f ( Y k U k k ; t 0 t f ) and D =[ D ki ],(1 k N ),(1 i N ) isa N N square matrixandiscalledthe Lobattopseudospectraldifferentiationmatrix Thematrix D issquarebecausethecollocationpointsarethesameasthed iscretization pointsusedinLagrangepolynomialapproximationofstate. Usingthematrix D ,the collocateddynamicsatthe N LGLcollocationpointsinEq.( 5–63 )areexpressedas D 266664 Y 1 ... Y N 377775 = t f t 0 2 266664 f 1 ... f N 377775 (5–64) ItisnowshownthattheLobattopseudospectraldifferentia tionmatrixisasingular matrix.Proposition10. TheLobattopseudospectraldifferentiationmatrix D issingular. ProofofProposition10. Let p ( )=1 beaconstantpolynomial.Next,let p 2 R N be suchthat p i = p ( i )=1 1 i N .Thecomponentsofthevector Dp arethederivatives 126

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atthecollocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,which impliestheequation Dp = 0 hasanonzerosolution.Hence, D issingular. Next,theapproachofnumericalintegrationisappliedtoso lvethedifferential equationsin( 5–1 )byapproximatingthederivativeontheright-handsidebyL agrange polynomials.Let L yi ( ) betheLagrangepolynomialsassociatedwiththecollocatio n points: L yi = N Y j =1 j 6 = i j i j ,1 i N (5–65) NoticethattheLagrangepolynomials L i denedin( 5–59 )andtheLagrangepolynomials L yi aredegree N 1 .Thenthestatedynamicsontheright-handsideofEq.( 5–1 )are approximatedbytheLagrangepolynomialsas F ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–66) EquatingtheapproximationinEq.( 5–66 )tothetimederivativeofstatesuchthat Y ( )= t f t 0 2 N X i =1 f i L yi ( ). (5–67) IntegratingEq.( 5–67 )from 1 to k ,followingrelationshipisobtained Y k = Y 1 + t f t 0 2 N X i =1 f i A ki A ki = Z k 1 L yi ( ) d ,2 k N (5–68) where A =[ A ki ],(2 k N ),(1 i N ) isa N N matrixcalledthe Lobatto pseudospectralintegrationmatrix .TheLobattointegrationmatrixisthesameasthat foundinRef.[ 84 ] ItisnotedthattheintegratedschemeinEq.( 5–68 )isnotequivalenttothe differentialschemeinEq.( 5–64 ).Boththeschemesarealtogetherdifferentschemes. 127

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Thedifferentialschemeconsistsof N equationswhiletheintegralschemerepresents N 1 equations. 5.2CostateDynamicsforInitial-ValueProblem Averyinterestingfeatureofthecostatedynamicsobtained inthepseudospectral methodsisobservedinthecaseofinitial-valueproblems.I naninitial-valueproblem, theboundaryconditionsatonlytheinitialendpointarespe cied.Thisfeatureunies theGausspseudospectralmethod,theRadaupseudospectral methodandtheipped Radaupseudospectralmethodinthesensethatthecostatedy namicsinallthese threemethodsformadeterminedsystemofequationswhereas ,incaseoftheLobatto pseudospectralmethod,thesystemofequationsformedbyth ecostatedynamicsis under-determined.Thediscretecostatedynamicsformalin earsystemofequationsin thecostate, .ItisshownthatincaseofLobattopseudospectralmethod,t hematrix ofthelinearsystemhasanullspaceandthereexistsaninni tenumberofsolutionsto thecostatedynamics.Itisshownempiricallythatincaseof aone-dimensionaloptimal controlproblemthecostatesolutionobtainedfromtheLoba ttopseudospectralmethod oscillatesabouttheactualcostateandthattheoscillatio nshavethesamepatternasthe nullspaceofthesystem.5.2.1GaussPseudospectralMethod ConsiderthecostatedynamicsobtainedinthetransformedKKT conditionsforthe Gausspseudospectralmethod D yk ,1: N ~ + D y k N +1 ~ N +1 = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (5–69) 1 k N Itisobservedthat,given Y k U k ~ r k ( k =1,..., N ) t 0 ,and t f ,( 5–69 )represents Nn equationsin ( N +1) n variables, ~ i 2 R n ( i =1,..., N +1) .Foraninitial-valueproblem, 128

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costateattheterminalendpoint, ~ N +1 isobtainedas ~ N +1 = r Y N +1 (5–70) Substituting( 5–70 )inEq.( 5–69 ),thecostatedynamicsareobtainedas D y1: N ~ + D yN +1 r Y N +1 = t f t 0 2 266664 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) ... r Y N ( g N + h ~ N f N ih ~ r N C N i ) 377775 (5–71) where D y1: N aretherst N columnsof D y and D yN +1 isthelastcolumnof D y .Itis observedthat( 5–71 )represents Nn linearequationsin Nn variables, ~ i 2 R n ( i =1,..., N ) .Hence,itisadeterminedsystemofequations. 5.2.2RadauPseudospectralMethod ConsiderthecostatedynamicsobtainedinthetransformedKKT conditionsforthe Radaupseudospectralmethod D y1 ~ = t f t 0 2 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) 1 w 1 ~ 1 + r Y 1 ( h ~ i ) (5–72) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (5–73) 2 k N UsingtheresultofEq.( 4–69 ),weknowthat r Y 1 ( h ~ i )= ~ N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (5–74) Foraninitial-valueproblem,costateattheterminalendpo int, ~ N +1 isobtainedas ~ N +1 = r Y N +1 (5–75) 129

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Substituting( 5–75 )inEq.( 5–74 ),followingresultisobtained r Y 1 ( h ~ i )= r Y N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (5–76) Substituting( 5–76 )inEq.( 5–72 ),thecostatedynamicsareobtainedas D y ~ + 1 w 1 e 1 e T1 ~ = t f t 0 2 266664 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) ... r Y N ( g N + h ~ N f N ih ~ r N C N i ) 377775 (5–77) + 1 w 1 e 1 r Y N +1 + t f t 0 2 N X i =1 w i r Y i ( g i + h ~ i f i ih ~ r i C i i ) where e 1 istherstcolumnof N N Identitymatrix.Itisobservedthat,given Y k U k ~ r k ( k =1,..., N ) t 0 ,and t f ,( 5–77 )represents Nn linearequationsin Nn variables, ~ i 2 R n ( i =1,..., N ) .Hence,itisadeterminedsystemofequations. 5.2.3FlippedRadauPseudospectralMethod ConsiderthecostatedynamicsobtainedinthetransformedKKT conditionsforthe ippedRadaupseudospectralmethod D yN ~ = t f t 0 2 r Y N ( g N + h ~ N f N ih ~ r N C N i ) + 1 w N ~ N r Y N (5–78) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (5–79) 1 k N 1. Rewritingthecostatedynamicsas D y ~ 1 w N e N e TN ~ = t f t 0 2 266664 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) ... r Y N ( g N + h ~ N f N ih ~ r N C N i ) 377775 1 w N e N r Y N (5–80) where e N isthe N th columnof N N Identitymatrix.Itisobservedthat,given Y k U k ~ r k ( k =1,..., N ) t 0 ,and t f ,Eq.( 5–80 )represents Nn linearequationsin Nn variables, 130

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~ i 2 R n ( i =1,..., N ) .Hence,thesystemofequationsin( 5–80 )isadeterminedsystem ofequations.5.2.4LobattoPseudospectralMethod ConsiderthecostatedynamicsobtainedinthetransformedKKT conditionsforthe Lobattopseudospectralmethod D y1 ~ = t f t 0 2 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) 1 w 1 ~ 1 + r Y 1 ( h ~ i ) (5–81) D yN ~ = t f t 0 2 r Y N ( g N + h ~ N f N ih ~ r N C N i ) + 1 w N ~ N r Y N (5–82) D yk ~ = t f t 0 2 r Y k ( g k + h ~ k f k ih ~ r k C k i ), (5–83) 2 k N 1. Becauseforthisdiscussion,theoptimalcontrolproblemisb eingdenedasan initial-valueproblem,considerthefollowingformofinit ialboundarycondition: ( x ( 1)) x ( 1) x 0 = 0 x ( 1 ) x 0 = 0 (5–84) Therefore, r Y 1 h ~ i = r Y 1 h ~ x ( 1 ) x 0 i = ~ (5–85) where ~ 2 R n 131

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Rewritingthecostatedynamicsas D y ~ + 1 w 1 e 1 e T1 ~ 1 w N e N e TN ~ 1 w 1 e 1 ~ = t f t 0 2 266664 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) ... r Y N ( g N + h ~ N f N ih ~ r N C N i ) 377775 1 w 1 e 1 r Y 1 1 w N e N r Y N (5–86) D y + 1 w 1 e 1 e T1 1 w N e N e TN 1 w 1 e 1 264 ~ ~ 375 = t f t 0 2 266664 r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) ... r Y N ( g N + h ~ N f N ih ~ r N C N i ) 377775 1 w 1 e 1 r Y 1 1 w N e N r Y N (5–87) where e 1 and e N aretherstandthe N th columns,respectively,of N N Identitymatrix. Itisobservedthat,given Y k U k ~ r k ( k =1,..., N ) t 0 ,and t f ,Eq.( 5–87 )represents Nn linearequationsin ( N +1) n variables, ~ i 2 R n ( i =1,..., N ) and ~ 2 R n .Hence, thematrixofthelinearsystemhasanullspaceandthereexis tsaninnitenumberof solutionstotheLGLcostatedynamics.Thedimensionofthen ullspaceisatleast n since ~ i 2 R n DespitethenullspaceintheLGLcostatedynamics,awealtho fnumerical examplespublishedintheliterature[ 37 40 42 – 45 65 ],alongwiththeconvergence theoryofKang[ 85 ],alldemonstratethattheLobattopseudospectralmethodl eadsto convergentapproximationstothestateandcontrolvariabl e.However,duetothenull spaceinthediscretecostatedynamics,theconceptofconve rgenceforthecostateis opentointerpretation[ 63 ].InRef.[ 44 ],Gongetal.haveshownin“CovectorMapping Theorem”thatanysolutiontotherst-orderoptimalitycon ditionsforthecontinuous controlproblem,approximatelysatisesthetransformedKKT conditionsofthediscrete LGLproblem,andtheerrortendstozeroas N !1 .Thisprovidesevidencethatamong theinnitesetofsolutionsassociatedwiththediscreteco statedynamics,thereexists agoodapproximationtothecontinuouscostate[ 63 ].Moreover,a closurecondition is 132

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proposedforselectingagoodapproximationtothecontinuo uscostatefromamongthe innitesetofsolutionstothecostatedynamics.Inthecont extofainitial-valuecontrol problem,theclosureconditionamountstochoosingasoluti ontothediscretecostate equation( 5–87 )whichsatisestheconditions k ~ 1 ~ k k ~ N r X ( X N ) k (5–88) where issomegivenerrortolerance.In[ 44 ,Thm.4]itisshownthatanasymptotically validchoicefor isoftheform = N (1.5 m ) ,where m 4 ,independentof N ,depends onthenumberofderivativesofsmoothnessofanoptimalsolu tion.Inpractice,thenull spacefortheLGLcostatedynamicsisoftenobservedtobehig hlyoscillatory[ 63 ].Asa result,inRef.[ 37 ](page276),Fahrooetal.suggestedthatthecomputedcosta tecanbe post-processedusingaltertoobtainagoodapproximation tothecontinuouscostate. Veryrecently,inRef.[ 86 ],Gongetal.proposedareplacementtotheclosureconditio n ( 5–88 )whichentailsrelaxingthecollocateddynamicsatthebegi nningandendofthe timeinterval. 5.3Convergence TheequivalencebetweenthetransformedKKTconditionsofthe NLPandthe discretizedformofthecontinuous-timerst-orderoptima lityconditionsprovidesaway toapproximatethecostateandtheLagrangemultipliersofa noptimalcontrolproblem. Furthermore,theequivalencebetweenthedifferentialand implicitintegrationscheme establishesthefactthatinusingthedifferentialformofa pseudospectralmethodto solveanoptimalcontrolproblem,thestatedynamicsareact uallybeingintegrated numerically.However,aformalmathematicalproofthatsho wsthatthesolutiontothe discreteNLPconvergestotheoptimalsolutionoftheorigin alcontinuous-timeoptimal controlproblemisstillmissing. 133

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Ithasbeenshowninliteraturethatmanynumericalintegrat ionschemes(for e.g.,Eulermethods,trapezoidalmethodandRunge-Kuttamet hods)havecertain convergenceproperties.However,convergenceofanintegr ationmethoddoesnot necessarilymeanthatthesamediscretizationschemewillc onvergewhenusedtosolve anoptimalcontrolproblem.InRef.[ 21 ],itisshownthataconvergentRunge-Kutta methodmaynotactuallyconvergetothecontinuousoptimals olutionwhenusedtosolve anoptimalcontrolproblem.Withtheinclusionofacostfunct ion,theproblemceases tobemerelyasystemofdifferentialequationsthatcanbeso lvedusingconvergent numericalintegrationschemes.Thus,theconvergenceproo fforpseudospectral methodsisanon-trivialproof,andiscurrentlybeingstudi edbyseveralresearchers [ 30 44 45 ].Ref.[ 30 ]implementstheRadaupseudospectralmethodlocally,i.e. ,the timeintervalisbrokenintosubintervalsandsolutioninea chsubintervalisapproximated usingtheRadaupseudospectralmethod.Convergencetothel ocalapproachisshown usingthemoretraditionalapproachtoconvergence,namely theerrortendstozeroas thenumberofsubintervalsapproachesinnity. Forglobalpseudospectralapproachesdiscussedinthisdis sertation,anapproximation tothesolutionisobtainedoverthetimeintervaltreatedas asingleinterval,so someotherapproachtoconvergencemustbeused.Oneapproac hwouldbeto showconvergenceasthenumberofdiscretizationpointsinc reasestoinnity.In Ref.[ 44 45 ],itisdemonstratedthattheLobattopseudospectralmetho dleadsto convergentapproximationstothestateandcontrolvariabl e.InRef.[ 44 ],itisshownin “CovectorMappingTheorem”thatanysolutiontotherst-or deroptimalityconditionsfor thecontinuouscontrolproblem,approximatelysatisesth etransformedKKTconditions ofthediscreteLGLproblem,andtheerrortendstozeroas N !1 .However,duetothe nullspaceinthediscretecostatedynamics,theconceptofc onvergenceforthecostate isopentointerpretation. 134

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InthenumericalexperimentsdiscussedinChapter 6 ,itisseenfortheexamples thathaveexactanalyticalsolutionsthattheerrorintheso lutionsobtainedfromthe GausspseudospectralmethodandtheRadaupseudospectralm ethodgoestozeroat anexponentialrateasthenumberofdiscretizationpointsa reincreased.Furthermore, thenumberofdiscretizationpointsatwhichtheerrorgoest ozeroisshowntobesmall, supportingthefactthatacoarsediscretizationmaybeused toobtainanaccurate solutionusingeithertheGausspseudospectralmethodorth eRadaupseudospectral method.Ontheotherhand,itisshownthatthecostatesoluti onobtainedfromthe Lobattopseudospectralmethodistypicallynonconvergent .InthecasewheretheLPM solutiondoesconverge,theerrorsarefoundtobetwotothre eordersofmagnitude higherfortheLPMthanfortheGPMortheRPM.Thenumberofdiscre tizationpointsat whichmachineprecisionisachievedfortheLPMisusuallyhig herthanthatfortheGPM ortheRPM. 5.4Summary AuniedframeworkhasbeenpresentedbasedontheGaussandt heRadau pseudospectralmethods.Itwasdemonstratedthateachofth eseschemescanbe expressedineitheradifferentialoranintegralformulati onthatareequivalenttoeach other.ItwasalsodemonstratedthattheLobattopseudospec tralmethoddoesnotexhibit suchanequivalence.Furthermore,itwasshownthatthedisc retecostatesystems inboththeGaussandtheRadaupseudospectralmethodsarefu llrankwhilethe discretecostatesysteminLobattopseudospectralmethodh asanullspace.Arigorous mathematicalproofofconvergencefortheGaussandtheRada upseudospectral methodsisstillmissing.IntheChapter 6 ,however,itisshownempiricallythatthe errorinthesolutionsobtainedfromtheGausspseudospectr almethodandthe Radaupseudospectralmethodgoestozeroatanexponentialr ateasthenumberof discretizationpointsareincreased. 135

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CHAPTER6 FINITE-HORIZONOPTIMALCONTROLEXAMPLES Inthischapter,avarietyofexamplesaresolvedusing,theG auss,theRadau, andtheLobattopseudospectralmethods.Threemainobserva tionsaremadeinthe examples.First,itisseenthattheGaussandtheRadaupseud ospectralmethods consistentlygenerateaccuratestate,controlandcostate solutions,whilethesolution obtainedfromtheLobattopseudospectralmethodisinconsi stentandunpredictable. Specically,ItisshownthatthecostateobtainedfromtheLo battopseudospectral methodtypicallyoscillatesabouttheexactcostate.Inone oftheexamples,itis shownthattheoscillationhasapatternsimilartothepatte rnofthenullspaceofthe transformedLPMcostatedynamics.Furthermore,itisdemons tratedthat,whenan optimalcontrolproblemhasanincompletesetofboundaryco nditions,thecostate fromtheLPMoscillateswhereastheseoscillationsareareno tpresentwhenaproblem consistsofafullydenedsetofboundaryconditions.Thedi sadvantageoftheLPMnot beinganimplicitintegrationschemeisalsodemonstratedi noneexample.Second,itis seenfortheexamplesthathaveexactanalyticalsolutionst hattheerrorinthesolutions obtainedfromtheGausspseudospectralmethodandtheRadau pseudospectral methodgoestozeroatanexponentialrateasthenumberofdis cretizationpointsare increased.Furthermore,thenumberofdiscretizationpoin tsatwhichtheerrorgoesto zeroisshowntobesmall,supportingthefactthatacoarsedi scretizationmaybeused toobtainanaccuratesolutionusingeithertheGausspseudo spectralmethodorthe Radaupseudospectralmethod.Ontheotherhand,itisshownt hatthecostatesolution obtainedfromtheLobattopseudospectralmethodistypical lynonconvergent.Inthe casewheretheLPMsolutiondoesconverge,theerrorsarefoun dtobetwotothree ordersofmagnitudehigherfortheLPMthanfortheGPMortheRPM. Thenumber ofdiscretizationpointsatwhichmachineprecisionisachi evedfortheLPMisusually higherthanthatfortheGPMortheRPM.Third,itisshownthatno neofthesemethods 136

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arewellsuitedforsolvingproblemsthathavediscontinuit iesinthesolutionorwherethe solutionslieonasingulararc. 6.1Example1:NonlinearOne-DimensionalInitial-ValueProb lem Therstexampleconsideredisanonlinearone-dimensional Mayercostinitial-value problem[ 63 ].Determinethestate, y ( t ) ,andthecontrol, u ( t ) ,on t 2 [0, t f ] thatminimize thecostfunctional J = y ( t f ), (6–1) subjecttothedynamicconstraint y = 5 2 ( y + yu u 2 ), (6–2) andtheboundarycondition y (0)=1, (6–3) where t f =2 .Theoptimalsolutiontothisproblemis y ( t )=4 = (1+3exp(5 t = 2)), u ( t )= y ( t ) = 2, y ( t )= exp(2ln(1+3exp(5 t = 2)) 5 t = 2) = (exp( 5)+6+9exp(5)). (6–4) Itisnotedthatthegivenoptimalcontrolproblemisone-dim ensionalinitial-value problem,thestatedynamicsarenonlinear,naltimeisxed andthecostfunctional consistsofonlytheMayercost. TheexamplewassolvedusingtheGauss,Radau,ippedRadau, andtheLobatto pseudospectralmethodsfor N =30 collocationpointsusingtheNLPsolverSNOPT withoptimalityandfeasibilitytolerancesof 10 15 and 2 10 15 ,respectively.Foreach method,theinitialguesswaszero.Figs. 6-1 6-2 6-3 6-4 showthesolutionobtained fromeachofthemethodsalongwiththeexactsolution.Itiss eenthattheLobatto costateoscillatesabouttheexactsolutionwhiletheGauss andRadaucostateare indistinguishablefromtheoptimalsolution.Theoscillat ionoftheLobattocostateisdue 137

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0 0 0.2 0.4 0.5 0.6 0.8 1 1 1.52 tState y ( t ) y ( t ) AStatesolution. 0 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.52 t Control u ( t ) u ( t ) BControlsolution. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.52 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-1.SolutionobtainedfromtheGausspseudospectral methodforExample 1 tothenullspaceintheLobattocostatedynamicsdiscussedi nChapter 5 .Since n =1 in thisproblem,thedimensionofthenullspaceis1.InFig. 6-5A ,thedifferencebetween theexactcostateandthecostateobtainedfromtheLPMisplot ted.InFig. 6-5B a vectorinthenullspaceisplotted.ComparingFig. 6-5A toFig. 6-5B ,weseethatthe oscillationsintheLobattocostatearoundthecorrectcost ateareessentiallyduetothe additionofavectorinthenullspace.Forthisparticularex ampleanimprovedLobatto 138

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0 0 0.2 0.4 0.5 0.6 0.8 1 1 1.52 tState y ( t ) y ( t ) AStatesolution. 0 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.52 t Control u ( t ) u ( t ) BControlsolution. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-2.SolutionobtainedfromtheRadaupseudospectral methodforExample 1 costatecanbeobtainedbyadding0.4timesthenullspacevec torgiveninFig. 6-5B to theLPMcostateestimateobtainedfromtheNLPsolver. Next,theoptimalcontrolproblemwassolvedfor N =(5,10,15,...,30) collocation pointsfortheGauss,theRadau,andtheLobattopseudospect ralmethodswith exactsolutionastheinitialguess.Figs. 6-6A – 6-6C showthebase 10 logarithmof 139

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0 0 0.2 0.4 0.5 0.6 0.8 1 1 1.52 tState y ( t ) y ( t ) AStatesolution. 0 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.52 t Control u ( t ) u ( t ) BControlsolution. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-3.SolutionobtainedfromtheippedRadaupseudosp ectralmethodfor Example 1 the L 1 -normerrorsforthestate,control,andcostate,respectiv ely,denedasfollows: E y =max k log 10 jj y ( k ) y ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E y =max k log 10 y ( k ) y ( k ) 1 (6–5) 140

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0 0 0.2 0.4 0.5 0.6 0.8 1 1 1.52 tState y ( t ) y ( t ) AStatesolution. 0 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 1 1.5 2 t Control u ( t ) u ( t ) BControlsolution. 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.5 1 1.52 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-4.SolutionobtainedfromtheLobattopseudospectr almethodforExample 1 Fig. 6-6A showsthatthestateerrorusingeithertheGaussorRadaupse udospectral methodsisapproximatelytwotofourordersofmagnitudesma llerthanthestateerrorfor theLobattopseudospectralmethodfor N 15 .InFigure 6-6B ,itisseenthattheGauss andRadaucontrolisbetweentwoandsevenordersofmagnitud emoreaccuratethan thecorrespondingLobattocontrolsfor N 15 .For N > 15 ,theGaussandRadaustate andcontrolerrorsdroptomachineprecision(approximatel y 10 16 ),whiletheLobatto errorsachievemachineprecisionat N =30 .InFig. 6-6C itisseenthattheGaussand 141

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0.2 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.5 1 1.5 2 t y ( t ) y ( t ) ALPMcostateerrorforExample 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.5 0.6 1 1.5 2 t NullSpaceBNullspaceofLobattotransformedcostatedynamicssystem Figure6-5.LPMcostateerrorandnullspaceforExample 1 142

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30 25 20 15 10 5 0 2 4 6 8 10 12 14 16 GPM RPM LPME y N AStateerrorsforExample 1 30 25 20 15 10 5 0 2 4 6 8 10 12 14 16 GPM RPM LPM E uN BControlerrorsforExample 1 30 25 20 15 10 10 5 5 0 5 10 15 GPM RPM LPM E y N CCostateerrorsforExample 1 Figure6-6.Solutionerrorvs.numberofcollocationpoints, N ,forExample 1 theRadaucostateerrorsdecreasetoneartheoptimizertole rances(approximately 10 15 )whiletheLobattocostateerrorremainsabove 10 2 143

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6.2Example2:NonlinearOne-DimensionalBoundary-ValuePro blem Thesecondexampleconsideredisanonlinearone-dimension alBolzacost boundary-valueproblem[ 62 ].Minimizethecostfunctional J = 1 2 Z t f 0 ( y + u 2 ) dt (6–6) subjecttothedynamicconstraint y =2 y +2 u p y (6–7) andtheboundaryconditions y (0)=2, y ( t f )=1, t f =5. (6–8) Itisnotedthatthegivenoptimalcontrolproblemisone-dim ensionalinthestateand thecontrol,thestatedynamicsarenonlinearandtherearen opathconstraintsinthe problem.Furthermore,thereisanalboundarycondition,t hecostfunctionalconsistsof aLagrangecostandthenaltimeisxed. Theexactsolutiontotheoptimalcontrolproblemof( 6–6 )–( 6–8 )isgivenas y ( t )= x 2 ( t ), y ( t )= x ( t ) 2 p y u ( t )= x ( t ), (6–9) where x ( t ) and x ( t ) aregivenas 264 x ( t ) x ( t ) 375 =exp( A t ) 264 x 0 x 0 375 (6–10) 144

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where A = 264 1 1 1 1 375 x 0 = p 2 x f =1 x 0 = x f B 11 x 0 B 12 (6–11) and B = 264 B 11 B 12 B 21 B 22 375 =exp( A t f ) (6–12) TheexamplewassolvedusingtheGauss,Radau,ippedRadau, andtheLobatto pseudospectralmethodsfor N =40 collocationpointsusingtheNLPsolverSNOPT withoptimalityandfeasibilitytolerancesof 10 10 .Foreachmethod,theinitialguess waszero.Figs. 6-7 6-8 6-9 6-10 showthesolutionobtainedfromeachofthemethod alongwiththeexactsolution.Itisseenthatthestate,cont rol,andthecostatesolution obtainedfromtheNLPforeachofthemethodsisindistinguis hablefromtheexact solution.Itisinterestingtonotethatinthisproblemwher eboththeboundaryconditions attheinitialandthenaltimesareknown,theLPMcostatedoe snotdemonstrate oscillationsabouttheexactsolutionaswasseenintheprev iousexample. Next,considerthefollowingformulationoftheoriginalLa grangecostproblemtoa Mayercostproblem.Let v ( t ) beacomponentofthestatevectorsuchthatthedynamics aredenedas v = 1 2 ( y + u 2 ), (6–13) withtheinitialcondition v (0)= 2. (6–14) 145

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0 0 0.5 1 1 1.5 2 2 2.5 3 4 5 tState y ( t ) y ( t ) AStatesolution. 4 3 2 1 0 0 1 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.5 0 0 0.5 1 1 1.5 2 3 4 5 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-7.SolutionobtainedfromtheGausspseudospectral methodforExample 2 Asaresult,theoriginalone-dimensionalLagrangecostprob lemcanbewrittenasthe followingtwo-dimensionalMayercostproblem.Minimizeth ecostfunctional J = v ( t f ) v (0), (6–15) 146

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0 0 0.5 1 1 1.5 2 2 2.5 3 4 5 tState y ( t ) y ( t ) AStatesolution. 4 3 2 1 0 0 1 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.5 0 0 0.5 1 1 1.5 2 3 4 5 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-8.SolutionobtainedfromtheRadaupseudospectral methodforExample 2 subjecttothedynamicconstraints y =2 y +2 u p y v = 1 2 ( y + u 2 ), (6–16) 147

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0 0 0.5 1 1 1.5 2 2 2.5 3 4 5 tState y ( t ) y ( t ) AStatesolution. 4 3 2 1 0 0 1 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.5 0 0 0.5 1 1 1.5 2 3 4 5 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-9.SolutionobtainedfromtheippedRadaupseudosp ectralmethodfor Example 2 andtheboundaryconditions y (0)=2, v (0)= 2, y ( t f )=1, t f =5. (6–17) 148

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0 0 0.5 1 1 1.5 2 2 2.5 3 4 5 tState y ( t ) y ( t ) AStatesolution. 4 3 2 1 0 0 1 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.5 0 0 0.5 1 1 1.5 2 3 4 5 t Costate y ( t ) y ( t ) CCostatesolution. Figure6-10.SolutionobtainedfromtheLobattopseudospect ralmethodforExample 2 ThismodiedproblemwassolvedusingtheGPM,theRPM,andtheL PMfor N =40 collocationpointsusingtheNLPsolverSNOPTwithoptimality andfeasibilitytolerances of 10 15 .Figs. 6-11A 6-11E showthesolutionobtainedfortheproblem.Itisseenthat thestate,control,andtherstcomponentofthecostatesol utionobtainedfromthe NLPforeachofthemethodsisindistinguishablefromtheexa ctsolution.Thesecond componentofthecostateobtainedfromtheLPM,however,show soscillationsaboutthe optimalcostate.Themagnitudeoferrorissmall( 10 7 )ascomparedtothemagnitude 149

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oferrorobtainedinthepreviousexample.Thiserrorinthec ostateobtainedfromthe LPMcanbeattributedtothefactthatthemodiedproblemhasa nincompletesetof boundaryconditionsandthattheLPMschemeisnotanimplicit integrator.IftheLPM hadbeenanimplicitintegrator,thecostwrittenasaMayerc ostoraLagrangecost wouldnothavemadeanydifferenceinthesolution.Unliketh eresultobtainedusingthe LPM,theGPMortheRPMcostateisindistinguishablefromtheexa ctcostate. Next,theoptimalcontrolproblemwassolvedfor N =(5,10,15,...,50) collocation pointsfortheGauss,theRadau,andtheLobattopseudospect ralmethodswiththe exactsolutionastheinitialguess.Figs. 6-12A – 6-12E showthebase 10 logarithmofthe L 1 -normerrorsforthestate,control,andcostate,respectiv ely,denedasfollows: E y =max k log 10 jj y ( k ) y ( k ) jj 1 E v =max k log 10 jj v ( k ) v ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E y =max k log 10 y ( k ) y ( k ) 1 E v =max k log 10 jj v ( k ) v ( k ) jj 1 (6–18) Itisseenthat E y E v E u ,and E y decreaseinalinearmannerfromN=5to50for allthreemethods.Moreover,for N > 40 E v and E u reachedmachineprecision (approximately 10 16 ).Again,thelinearrateofdecreaseofthebase 10 logarithm error, E ,forthelowernumberofnodesdemonstratesanexponentialc onvergencerate. Fig. 6-12D showsthesecondcomponentofcostateerror, E v .For N 30 ,thecostate fromtheGPMandtheRPMareupto 10 ordersofmagnitudemoreaccuratethanthat fromtheLPM.For N > 30 ,thecostatefromtheGPMandtheRPMareupto 5 ordersof magnitudemoreaccuratethanthecostatefromtheLPM. 150

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0 0 0.5 1 1 1.5 2 2 2.5 3 4 5 ty ( t ) Exact GPM RPM LPM AFirstcomponentofstate. 2 1.5 1 0.5 0 0 0.5 1 1 2 3 4 5 t v ( t ) Exact GPM RPM LPM BSecondcomponentofstate. 0.5 0 0 0.5 1 1 1.5 2 3 4 5 t y ( t ) Exact GPM RPM LPM CFirstcomponentofcostate. 0.9999996 0.9999998 1.0000000 1.0000002 1.0000004 1.0000006 0 1 2 3 4 5 t v ( t )Exact GPM RPM LPM DSecondcomponentofcostate. 4 3 2 1 0 0 1 1 2 3 4 5 t u ( t ) Exact GPM RPM LPM EControlsolution. Figure6-11.SolutionobtainedfromtheGauss,theRadau,and theLobatto pseudospectralmethodfor N =40 formodiedExample 2 151

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50 40 30 20 10 0 0 2 4 6 8 10 12 GPM RPM LPME y N AFirstcomponentofstateerrorsforExample. 50 40 30 20 10 0 0 5 10 15 20 GPM RPM LPM E v N BSecondcomponentofstateerrorsforExample. 50 40 30 25 20 10 0 0 2 4 6 8 10 GPM RPM LPM E y N CFirstcomponentofcostateerrorsforExample. 50 40 30 20 10 0 0 5 10 15 20 GPM RPM LPM E v N DSecondcomponentofcostateerrorsforExample. 50 40 30 20 10 5 0 0 5 10 15 20 GPM RPM LPM E uN EControlerrorsforExample. Figure6-12.Solutionerrorvs.numberofcollocationpoints N ,forExample 2 152

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6.3Example3:Orbit-RaisingProblem Thenextexampleconsideredismulti-dimensionalinthesta tewithanincomplete setofboundaryconditions.Considerthefollowingorbit-r aisingoptimalcontrolproblem foundinRef.[ 5 ].Minimizethecostfunctional J = r ( t f ) (6–19) subjecttothedynamicconstraints r = v r = v = r v r = v 2 = r = r 2 + a sin v = v r v = r + a cos (6–20) andtheboundaryconditions ( r (0), (0), v r (0), v (0))=(1,0,0,1), ( v r ( t f ), v ( t f ))=(0, p = r ( t f )), (6–21) where a a ( t )= T m 0 j m j t (6–22) Itisnotedforthisexamplethat =1 T =0.1405 m 0 =1 m =0.0749 ,and t f =3.32 Theorbit-raisingproblemwassolvedusingtheGPM,RPM,andLPM for N = 64 .Forthisproblem,theoptimalsolutionisnotknown.Theref ore,onlyaqualitative comparisonbetweenthesolutionsisperformed.Thestate,c ontrol(afteranupwrapping oftheangle),andcostatesolutionsforallthemethodsares howninFigs. 6-13 6-15 First,itisobservedthateachofthemethodsproducequalit ativelysimilarvalues forthestateandthecontrol.Oncomparingthecostateappro ximationsfordifferent methods,however,weobserveanoscillationintheLPMcostat e,whichislikelydue tothecontributionsinthenullspaceassociatedwiththedi scretecostatedynamics. Inparticular,itisseenthattheGPMandtheRPMproduceaverya ccurateresultfor 153

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0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3.5 tState r ( t ) ( t ) v r ( t ) v ( t ) AStatesolution. 0 0 0.5 1 1.5 2 2.5 3 3.5 50 100 150 200 250 300 350 t Control BControlsolution. 2.5 2 1.5 1 0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2.5 3 3.5 t Costate r ( t ) ( t ) v r ( t ) v ( t ) CCostatesolution. Figure6-13.SolutionobtainedfromtheGausspseudospectra lmethodforExample 3 ( t ) whileLPMproducesavaluefor ( t ) thatoscillatesaroundzero.Inaddition,itis seenthat r ( t ) forLPMalsooscillates(unlikethesmoothbehaviorshownbyt hecostate obtainedfromtheGPMandtheRPM).Thus,theGPMandtheRPMdiffer signicantly fromtheLPMincostateaccuracy,demonstratingafundamenta ldifferenceinthenature ofthecostateestimatesobtainedusingeithertheGPMortheR PMascomparedwith theLPM. 154

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0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3.5 tState r ( t ) ( t ) v r ( t ) v ( t ) AStatesolution. 0 0 0.5 1 1.5 2 2.5 3 3.5 50 100 150 200 250 300 350 t Control BControlsolution. 2.5 2 1.5 1 0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2.5 3 3.5 t Costate r ( t ) ( t ) v r ( t ) v ( t ) CCostatesolution. Figure6-14.SolutionobtainedfromtheRadaupseudospectra lmethodforExample 3 155

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0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3.5 tState r ( t ) ( t ) v r ( t ) v ( t ) AStatesolution. 0 0 0.5 1 1.5 2 2.5 3 3.5 50 100 150 200 250 300 350 t Control BControlsolution. 4 2 3 1 0 0 0.5 1 1 1.5 2 2 2.5 3 3 3.5 4 t Costate r ( t ) ( t ) v r ( t ) v ( t ) CCostatesolution. Figure6-15.SolutionobtainedfromtheLobattopseudospect ralmethodforExample 3 156

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6.4Example4:BrysonMaximumRangeProblem ThenextexampleconsideredisavariationoftheBrysonmaxim umrangeproblem takenfromRef.[ 5 ].Inthisexample,boththestateandthecontrolaremulti-d imensional andtheproblemcontainsanequalitycontrolpathconstrain t.Theproblemisstatedas follows:Maximize J = x ( t f ), (6–23) subjecttothedynamicconstraints x = vu 1 y = vu 2 (6–24) v = g = 2 gu 2 theboundaryconditions ( x (0), y (0), v (0))=(0,0,0), y ( t f )=0.1, (6–25) andthepathconstraint u 2 1 + u 2 2 =1. (6–26) Thisexamplewassolvedfor g =1 ,usingtheGPM,RPM,andLPMfor N =40 .The state,control,andcostateforeachmethodareshowninFigs 6-16 – 6-19 .Again,for thisproblem,theoptimalsolutionisnotknown.Therefore, onlyaqualitativecomparison betweenthesolutionsisperformed.First,itisobservedth ateachofthemethods producequalitativelysimilarvaluesforthestate.Oncomp aringthecontrolandthe costateapproximationsfordifferentmethods,however,we observethattheLPMcontrol isquitedifferentfromtheGPMortheRPMcontrol.Furthermore ,oscillationsintheLPM costatearealsoobserved.Aswiththepreviousexamples,thi sexampledemonstrates thefundamentaldifferenceinaccuracybetweentheGPMorRPMa ndtheLPM.Inthis exampleitwasfoundthatthestateforallthreemethodsmatc hed,butboththeLPM 157

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0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 tState x ( t ) y ( t ) v ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.5 2 t Control u 1 ( t ) u 2 ( t ) BControlsolution. 1.5 1 0.5 0 0 0.5 1 1.5 2 t Costate x ( t ) y ( t ) v ( t ) CCostatesolution. Figure6-16.SolutionobtainedfromtheGausspseudospectra lmethodforExample 4 costateandLPMcontrolweresignicantlydifferentfromthe controlandthecostate solutionsfromtheGPMandtheRPM.TheerrorsintheLPMcontrola ndcostateare attributedtothefactthattheLPMtransformedadjointsyste mhasacouplingbetween thecollocatedcostatedynamicsandthetransversalitycon dtions,theLPMcostate dynamicssystemhasanullspaceassociatedwithit,andthat theLPMisnotanimplicit integrator. 158

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replacements 0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 tState x ( t ) y ( t ) v ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.5 2 t Control u 1 ( t ) u 2 ( t ) BControlsolution. 1.5 1 0.5 0 0 0.5 0.5 1 1.52 t Costate x ( t ) y ( t ) v ( t ) CCostatesolution. Figure6-17.SolutionobtainedfromtheRadaupseudospectra lmethodforExample 4 159

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0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 tState x ( t ) y ( t ) v ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.52 t Control u 1 ( t ) u 2 ( t ) BControlsolution. 1.5 1 0.5 0 0 0.5 0.5 1 1.5 2 t Costate x ( t ) y ( t ) v ( t ) CCostatesolution. Figure6-18.SolutionobtainedfromtheippedRadaupseudos pectralmethodfor Example 4 160

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2 0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 tState x ( t ) y ( t ) v ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.5 2 t Control u 1 ( t ) u 2 ( t ) BControlsolution. 6 4 2 0 0 0.5 1 1.5 2 2 4 t Costate x ( t ) y ( t ) v ( t ) CCostatesolution. Figure6-19.SolutionobtainedfromtheLobattopseudospect ralmethodforExample 4 161

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6.5Example5:Bang-BangControlProblem ThenextexampleconsideredistakenfromRef.[ 48 ]andhasa“Bang-Bang” optimalcontrol.Minimizethenaltime, J = t f (6–27) subjecttothedynamicconstraints x = y ( t ), y = u ( t ), (6–28) theboundaryconditions x (0)= x 0 y (0)= y 0 x ( t f )=0, y ( t f )=0, (6–29) andthecontrolinequalityconstraint j u ( t ) j u max =1. (6–30) Itisnotedthattheexactsolutiontotheoptimalcontrolpro blemof( 6–27 )–( 6–30 )is obtainedusingtheweakformofPontryagin'sprinciple.The exactsolutionisgivenas x = 8><>: t 2 2 + y 0 t + x 0 : t t 1 t 2 2 t f t + t 2 f 2 : t > t 1 (6–31) y = 8><>: t + y 0 : t t 1 t t f : t > t 1 (6–32) u = 8><>: 1: t t 1 1: t t 1 (6–33) x = c 1 (6–34) y = c 1 t + c 2 (6–35) 162

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wheretheswitchingtime t 1 ,naltime t f ,constants c 1 and c 2 aregivenas t 1 = y 0 + p 0.5 y 2 0 + x 0 t f =2 t 1 y 0 c 1 = 1 t 1 t f c 2 = c 1 t 1 (6–36) TheexamplewassolvedusingtheGauss,Radau,ippedRadau, andLobatto pseudospectralmethodsfor x 0 =1 y 0 =3 ,and N =40 collocationpointsusing theNLPsolverSNOPTwithdefaultoptimalityandfeasibilityt olerancesof 10 6 and 2 10 6 ,respectively.Foreachmethod,theinitialguessforther stcomponentof statewas 1 ,thesecondcomponentofstatewas 3 ,thecontrolwas 1 andthatfornal timewas 7 .Figs. 6-20 6-21 6-22 6-23 showthesolutionobtainedfromeachofthe methodalongwiththeexactsolution.Itisseenthatthestat esolutionobtainedfrom theNLPforeachofthemethodisindistinguishablefromthee xactsolution.Thecontrol andthecostate,however,arenotexactlythesameasthoseof theoptimalsolution. TheerrorinthecostateobtainedfromtheGauss,theRadau,a ndtheippedRadau pseudospectralmethodsissmallerthantheerrorinthecost ateobtainedfromthe Lobattopseudospectralmethod.ThecostatefromtheLPMosci llatesabouttheoptimal costate. 163

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4 2 0 0 0.5 2 2 4 4 6 6 8 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 1 2 4 6 8 t Control u ( t ) u ( t ) BControlsolution. 2 1 0 0 1 2 2 3 4 6 8 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-20.SolutionobtainedfromtheGausspseudospectra lmethodforExample 5 164

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4 2 0 0 0.5 2 2 4 4 6 6 8 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 1 2 4 6 8 t Control u ( t ) u ( t ) BControlsolution. 2 1 0 0 1 2 2 3 4 6 8 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-21.SolutionobtainedfromtheRadaupseudospectra lmethodforExample 5 165

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4 2 0 0 2 2 4 4 6 6 8 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 1 2 4 6 8 t Control u ( t ) u ( t ) BControlsolution. 2 1 0 0 1 2 2 3 4 6 8 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-22.SolutionobtainedfromtheippedRadaupseudos pectralmethodfor Example 5 166

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4 2 0 0 2 2 3 4 4 6 6 8 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 1 2 4 6 8 t Control u ( t ) u ( t ) BControlsolution. 15 10 5 0 0 2 4 5 6 8 10 15 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-23.SolutionobtainedfromtheLobattopseudospect ralmethodforExample 5 167

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Next,theoptimalcontrolproblemwassolvedfor N =(5,10,15,...,50) collocation pointsfortheGauss,Radau,andLobattopseudospectralmet hods.Figs. 6-24A – 6-24E showthebase 10 logarithmofthe L 1 -normerrorsforthestate,control,andcostate, respectively,denedasfollows: E x =max k log 10 jj x ( k ) x ( k ) jj 1 E y =max k log 10 jj y ( k ) y ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E x =max k log 10 jj x ( k ) x ( k ) jj 1 E y =max k log 10 y ( k ) y ( k ) 1 (6–37) Itisseenthat E x E y E x E y ,and E u showsignicanterrorsinthesolutionfor allthreemethods.Although,theGaussandtheRadaupseudosp ectralmethods outperformtheLobattopseudospectralmethod,neitherone ofthemethodsshows muchimprovementasthenumberofcollocationpointsareinc reased.Thedivergence fromthetypicalexponentialbehaviorcanbeexplainedbyex aminingthediscontinuities inthesolution.Attheswitchingpoint,thecontrolisdiscon tinuous,therststatehas adiscontinuoussecondderivative,andthesecondstatehas adiscontinuousrst derivative.Pseudospectralmethodsusepolynomialstoappr oximatethestate.The polynomialapproximationsareunabletoapproximatenon-s moothfunctionsaccurately. Asaconsequence,globalpseudospectralmethodsarenotwell suitedforproblemswith discontinuitiesinthesolutionorinthederivativesofthe solution. 168

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50 40 30 20 10 0.5 0 0 0.5 1 1.5 2 2.5 3 GPM RPM LPME x N AFirstcomponentofstateerror. 50 40 30 20 10 0.5 0 0 0.5 1 1.5 2 GPM RPM LPM E y N BSecondcomponentofstateerror. 50 40 30 20 10 21 0 0 1 2 3 4 GPM RPM LPM E x N CFirstcomponentofcostateerror. 50 40 30 20 10 1 0.5 0 0 0.5 1 1.5 2 2.5 3 GPM RPM LPM E y N DSecondcomponentofcostateerror. 50 40 30 20 10 0.5 0 0 0.5 1 1.5 GPM RPM LPM E uN EControlerrors. Figure6-24.Solutionerrorvs.numberofcollocationpoints N ,forExample 5 169

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6.6Example5:SingularArcProblem Thenextexampleconsideredisaproblemwhoseoptimalsolut ionliesonasingular arcandistakenfromRef.[ 48 ].Theproblemisstatedasfollows.Minimize J = 1 2 Z t f 0 ( x 2 + y 2 ) dt (6–38) subjecttothedynamicconstraints x = y ( t ), y = u ( t ), (6–39) theboundaryconditions x (0)=0.8, y (0)=0.8, (6–40) x ( t f )=0.01, y ( t f )= 0.01, (6–41) andthecontrolinequalityconstraint j u ( t ) j u max =1. (6–42) Theexactsolutionisgivenas u ( t )=0.8exp( t ), x ( t )=0.8exp( t ), y ( t )= 0.8exp( t ), (6–43) x ( t )=0.8exp( t ), v ( t )=0. Theoptimaltimeisobtainedas t f = log 1 80 (6–44) 170

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1 0.5 0 0 0.5 1 1 2 3 4 5 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 1 1.2 2 3 4 5 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-25.SolutionobtainedfromtheGausspseudospectra lmethodforExample 6 TheexamplewassolvedusingtheGauss,Radau,ippedRadau, andLobatto pseudospectralmethodsfor N =30 collocationpointsusingtheNLPsolverSNOPTwith defaultoptimalityandfeasibilitytolerancesof 10 6 and 2 10 6 ,respectively.Foreach method,theinitialguessfortherstcomponentofstatewas 0.8 ,thesecondcomponent ofstatewas 0.8 ,thecontrolwas 1 andthatfornaltimewas 4 .Figs. 6-25 – 6-28 show thesolutionobtainedfromeachofthemethodalongwiththee xactsolution.Itisseen 171

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1 0.5 0 0 0.5 1 1 2 3 4 5 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 1 1.2 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 1 1.2 2 3 4 5 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-26.SolutionobtainedfromtheRadaupseudospectra lmethodforExample 6 thattheapproximatesolutionforthestateandcostateisar elativelygoodapproximation totheexactsolutionwhilethecontrolobtainedfromeachof themethodsisinaccurate. Next,theoptimalcontrolproblemwassolvedfor N =(5,10,15,...,50) collocation pointsfortheGauss,Radau,andLobattopseudospectralmet hods.Figs. 6-29A – 6-29E showthebase 10 logarithmofthe L 1 -normerrorsforthestate,control,andcostate, 172

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1 0.5 0 0 0.5 1 1 2 3 4 5 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 1 1.2 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 1 1.2 2 3 4 5 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-27.SolutionobtainedfromtheippedRadaupseudos pectralmethodfor Example 6 respectively,denedasfollows: E x =max k log 10 jj x ( k ) x ( k ) jj 1 E y =max k log 10 jj y ( k ) y ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E x =max k log 10 jj x ( k ) x ( k ) jj 1 E y =max k log 10 y ( k ) y ( k ) 1 (6–45) 173

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1 0.5 0 0 0.5 1 1 2 3 4 5 tState x ( t ) y ( t ) x ( t ) y ( t ) AStatesolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 2 3 4 5 t Control u ( t ) u ( t ) BControlsolution. 0.2 0 0 0.2 0.4 0.6 0.8 1 2 3 4 5 t Costate x ( t ) y ( t ) x ( t ) y ( t ) CCostatesolution. Figure6-28.SolutionobtainedfromtheLobattopseudospect ralmethodforExample 6 Itisseenthat E x E y E x E y ,and E u arenotconvergingas N isincreaseddemonstrating thatthesemethodsarenotsuitabletosolveprobemswhereop timalsolutionliesona singulararc. 174

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50 40 30 20 10 0 2 3 4 5 6 7 GPM RPM LPME x N AFirstcomponentofstateerror. 50 40 30 20 10 0 1 2 3 4 5 6 GPM RPM LPM E y N BSecondcomponentofstateerror. 50 40 30 20 10 0 1 2 3 4 5 6 GPM RPM LPM E x N CFirstcomponentofcostateerror. 50 40 30 20 10 0 3 4 5 6 7 8 9 10 GPM RPM LPM E y N DSecondcomponentofcostateerror. 50 40 30 20 10 0 0 1 2 3 4 5 GPM RPM LPM E uN EControlerrors. Figure6-29.Solutionerrorvs.numberofcollocationpoints N ,forExample 6 175

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6.7Summary IthasbeenshownthattypicallytheLPMcostateapproximatio nisfoundtohave anerrorthatoscillatesabouttheexactsolution,andthise rrorwasshownbyexample tobeduetothenullspaceintheLPMdiscretecostatedynamics .Empiricalevidence hassuggestedthattheGaussandtheRadaupseudospectralme thodsconvergerapidly (exponentially)foralargeclassofproblemsandgiveabett ercostateestimatethan theLobattopseudospectralmethod.However,ithasbeensho wnthatthesemethods arenotwellsuitedforsolvingproblemsthathavediscontin uitiesinthesolutionor discontinuitiesinthederivativesofthesolution,andpro blemsthatcontainsingulararcs. 176

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CHAPTER7 INFINITE-HORIZONOPTIMALCONTROLPROBLEMS Certainprocessesmustbecontrolledindenitely.Suchproc essesaredescribed onaninnitetimedomain, [0, 1 ) .Inthischapter,twodirectpseudospectralmethods aredescribedforsolvinginnite-horizonoptimalcontrol problemsnumericallyusingthe Legendre-Gauss(LG)andtheLegendre-Gauss-Radau(LGR)co llocation.Animportant aspectofnumericallyapproximatingthesolutionofaninn ite-horizonoptimalcontrol problemisthemannerinwhichthehorizonistreated.Findin gthestateofthesystemat t = 1 isachallengewhilesolvinginnite-horizonoptimalcontr olproblemsnumerically. Tosolveaninnite-horizonoptimalcontrolproblemnumeri cally,theinnite-horizon optimalcontrolproblemmustbetransformedtoanite-hori zonproblem.Inthis research,asmooth,strictlymonotonictransformation t = ( ) isusedtomapthe innitetimedomain t 2 [0, 1 ) ontothedomain 2 [ 1,1) .Theresultingproblemonthe niteintervalistranscribedtoanonlinearprogrammingpr oblemusingcollocation.Two methodsthatemploycollocationattheLGandtheLGRpoints, calledtheinnite-horizon Gausspseudospectralmethodandtheinnite-horizonRadau pseudospectralmethod, aredescribedinthischapter.Furthermore,itisshownthat collocationbasedontheLGL pointsisnotsuitableforsolvinginnite-horizonoptimal controlproblems. ItisnotedthatanLGRpseudospectralmethodforapproximat ingthesolutionof nonlinearinnite-horizonoptimalcontrolproblemshasbe enpreviouslydeveloped inRef.[ 65 ].Themethodspresentedinthisdissertationarefundament allydifferent fromthemethodofRef.[ 65 ].First,theproposedmethodsyieldapproximationsto thestateandthecostateontheentirehorizon,includingap proximationsat t =+ 1 Second,similartotheGaussandtheRadaupseudospectralmet hodsfornite-horizon problems,thesemethodscanalsobewrittenequivalentlyin eitheradifferentialoran implicitintegralform.Third,intheproposedmethodsagen eralchangeofvariables t = ( ) ofaninnite-horizonproblemtoanite-horizonproblemis considered.Itis 177

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shownthatthemap :[ 1,+1) [0,+ 1 ) canbetunedtoimprovethequalityofthe discreteapproximation.Innumericalexamples,thediscre tesolutionexhibitsexponential convergenceasafunctionofthenumberofcollocationpoint s. 7.1Innite-HorizonOptimalControlProblem Considertheinnite-horizonoptimalcontrolproblem:Min imizethecostfunctional J = Z 1 0 g ( y ( t ), u ( t )) dt (7–1) subjecttothedynamicconstraints y ( t )= f ( y ( t ), u ( t )), (7–2) theboundaryconditions y (0) y 0 = 0 (7–3) andtheinequalitypathconstraints C ( y ( t ), u ( t )) 0 (7–4) whereallvectorfunctionsoftimeare row vectors;thatis, y ( t )= [ y 1 ( t ) y n ( t ) ] 2 R n g : R n R m R f : R n R m R n C : R n R m R s ,and y denotesthetime derivativeof y .Itisnotedthatgenerallyaninnite-horizonoptimalcont rolproblemdoes notconsistofaMayercosttermandtheindependentvariable ,i.e.,timeisnotpresent asanexplicitvariableintheproblem.Inthisproblem,thei nitialandthenaltimesare treatedasxedat 0 and 1 ,respectively. Theinnitetimedomain, [0, 1 ) isnowmappedto [ 1,1) usingthechangeof variables t = ( ) where isadifferentiable,strictlymonotonicfunctionof thatmaps 178

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theinterval [ 1,1) onto [0, 1 ) .Threeexamplesofsuchafunctionare a ( )= 1+ 1 (7–5) b ( )=log e 2 1 (7–6) c ( )=log e 4 (1 ) 2 (7–7) Thechangeofvariables a ( ) wasoriginallyproposedinRef.[ 65 ],whilethetransformations b ( ) and c ( ) areintroducedinthisresearch.Theselatterchangesofvar iables produceslowergrowthin t = ( ) as approaches +1 ,thanthatof a ( ) asshownin Figure 7-1 .Betterdiscretizationcanbeachievedbytuningthechangeo fvariablesto theproblem.Forexample,supposetheexactstateofaproble misgivenas y ( t )=exp 1 2 e t (7–8) Figure 7-2 showsthegrowthof y ( t ) .AsisseenintheFigure, y ( t ) changesslowly when t islarge.Thelogarithmicfunctionsof b ( ) and c ( ) essentiallymove collocationpointsassociatedwithlargevaluesof t totheleft.IntheFigure,collocation pointscorrespondingto b ( ) areshown.Thisleftwardmovementofthecollocation pointsisbenecialsincemorecollocationpointsaresitua tedwherethesolutionis changingmostrapidly. Next,dene T ( )= d d 0 ( ). (7–9) Afterchangingvariablesfrom t to ,theinnite-horizonoptimalcontrolproblem becomes J = Z +1 1 T ( ) g ( y ( ), u ( )) d (7–10) subjecttothedynamicconstraints d y ( ) d =_ y ( )= T ( ) f ( y ( ), u ( )), (7–11) 179

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0 00.5 1 1 0.5 400 800 1200 a ( ) AGrowthin a ( ) at40LGpoints. 0 00.5 1 1 0.5 t5 10 15 b ( ) c ( ) BGrowthin b ( ) and c ( ) at40LGpoints. Figure7-1.Growthin ( ) at40LGpoints. 180

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y ( t )y ( t )CollocationPoint t 0 0 1 2 2 3 4 68 Figure7-2.Growthof y ( t ) andlocationof40collocationpointsusing b ( ) theboundaryconditions y ( 1) y 0 = 0 (7–12) andtheinequalitypathconstraints T ( ) C ( y ( ), u ( )) 0 (7–13) Itisnotedthat, T ( ) ,ismultipliedtoEq.( 7–13 )withoutactuallyaffectingtheconstraint sothattheoptimalityconditionscanbeposedinasuccintma nner.Here y ( ) and u ( ) denotethestateandthecontrolasafunctionofthenewvaria ble 181

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Therst-orderoptimalityconditionsforthenitehorizon controlproblemin Eqs.( 7–10 )-( 7–13 ),are y ( )= T ( ) r H (7–14) ( )= T ( ) r y H (7–15) 0 = r u H (7–16) ( 1)= (7–17) (+1)= 0 (7–18) r i ( )=0 when C i ( y ( ), u ( )) < 0,1 i s (7–19) r i ( ) < 0 when C i ( y ( ), u ( ))=0,1 i s (7–20) = 0 (7–21) wheretheHamiltonian H isdenedas H ( y ( ), u ( ), ( ), r ( ))= g ( y ( ), u ( ))+ h ( ), f ( y ( ), u ( )) i h r ( ), C ( y ( ), u ( )) i (7–22) 7.2Innite-HorizonGaussPseudospectralMethod Inthissection,thediscreteapproximationtothenonlinea rinnite-horizonoptimal controlproblemisformulatedusingglobalcollocationatL egendre-Gausspoints.The stateatthehorizonisincludedbyquadrature.Equivalenceb etweenthetransformed KKTconditionsanddiscreteversionoftherst-orderoptimal ityconditionsisestablished. Theequivalentimplicitintegrationschemeisalsoderived 7.2.1NLPFormulation ConsidertheLGcollocationpoints ( 1 ,..., N ) ontheinterval ( 1,1) andtwo additional noncollocated points 0 = 1 (theinitialtime)and N +1 =1 (theterminaltime, correspondingto t =+ 1 ).Thestateisapproximatedbyapolynomialofdegreeatmost 182

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N as y ( ) Y ( )= N X i =0 Y i L i ( ), (7–23) where Y i 2 R n and L i ( ) isaLagrangepolynomialofdegree N denedas L i ( )= N Y j =0 j 6 = i j i j ,( i =0,..., N ). (7–24) Thebasisincludesthefunction L 0 correspondingtotheinitialtime 0 = 1 ,butnota functioncorrespondingto N +1 =+1 .DifferentiatingtheseriesofEq.( 7–23 )withrespect to ,anapproximationtothederivativeofthestatein domainisobtainedas y ( ) Y ( )= N X i =0 Y i L i ( ). (7–25) Thefollowingcollocationconditionsarethenformedbyequ atingthederivativeofthe stateapproximationinEq.( 7–25 )totheright-handsideofthestatedynamicconstraints inEq.( 7–11 )atthe N LGpoints, ( 1 ,..., N ) : N X i =0 Y i L i ( k )= T ( k ) f ( Y k U k ),( k =1,..., N ), (7–26) N X i =0 D ki Y i = T ( k ) f ( Y k U k ), D ki = L i ( k ), (7–27) whereitisnotedthat 0 isnotacollocationpoint.Hence,thediscreteapproximati onto thesystemdynamicsis D k Y LG = T ( k ) f ( Y k U k ),1 k N (7–28) where D k isthe k th rowofdifferentiationmatrix D =[ D ki ],(1 k N ),(0 i N ) calledthe Gausspseudospectraldifferentiationmatrix and Y LG isdenedas Y LG = 266664 Y 0 ... Y N 377775 183

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Thus,thediscreteapproximationtothesystemdynamics y ( )= T ( ) f ( y ( ), u ( )) is obtainedbyevaluatingthesystemdynamicsateachcollocat ionpointandreplacing y ( k ) byitsdiscreteapproximation D k Y LG .Itisimportanttoobservethattheleft-hand sideofEq.( 7–28 )containsapproximationsforthestateattheinitialpoint plustheLG pointswhiletheright-handsidecontainsapproximationsf orthestate(andcontrol)at onlytheLGpoints.Next,itisagainnotedthattheGaussdiff erentiationmatrix D hasthe followingproperties:if D = [ D 0 D 1: N ] where D 0 istherstcolumnof D and D 1: N arethe remainingcolumns,then D issuchthat:(a) D 1: N isnonsingularand(b) D 0 = D 1: N 1 ; equivalently, D 1 1: N D 0 = 1 ,where 1 isacolumnvectorofallones. TheobjectivefunctioninEq.( 7–10 )isapproximatedbytheLegendre-Gauss quadratureas J = Z +1 1 T ( ) g ( y ( ), u ( )) d N X k =1 T ( k ) w k g ( Y k U k ), (7–29) where w k isthequadratureweightassociatedwith k .Next,wehave y (+1)= y ( 1)+ Z +1 1 T ( ) f ( y ( ), u ( )) d (7–30) Eq.( 7–30 )canbeapproximatedusingtheLGquadratureas Y N +1 = Y 0 + N X k =1 w k T ( k ) f ( Y k U k ), (7–31) where Y N +1 istreatedasanadditionalvariable.RearrangingEq.( 7–31 ),thefollowing equalityconstraintisthenaddedinthediscreteapproxima tion: Y N +1 Y 0 N X k =1 w k T ( k ) f ( Y k U k )= 0 (7–32) Lastly,thepathconstraintsinEq.( 7–13 )areenforcedatthe N LGcollocationpointsas T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ). (7–33) 184

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Thecontinuous-timenonlinearinnite-horizonoptimalco ntrolproblemofEqs.( 7–10 )-( 7–13 ) isthenapproximatedbythefollowingNLP:Minimizethecost J = N X k =1 T ( k ) w k g ( Y k U k ), (7–34) subjecttothefollowingequalityandinequalityconstrain ts D k Y LG T ( k ) f ( Y k U k )= 0 ,( k =1,..., N ), (7–35) Y N +1 Y 0 N X k =1 w k T ( k ) f ( Y k U k )= 0 (7–36) Y 0 y 0 = 0 (7–37) T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ), (7–38) wheretheNLPvariablesare ( Y 0 ,..., Y N +1 ) and ( U 1 ,..., U N ) .Itisnotedthat,the singularityinthechangeofvariables t = ( ) andin T ( )= 0 ( ) at =+1 is avoidedintheNLPformulation. T ( )= 0 ( ) isneverevaluatedatthesingularityin Eqs.( 7–34 )-( 7–38 ),rather T ( ) isevaluatedatthequadraturepointsonlywhichareall strictlylessthan 1 .Furthermore,because Y N +1 isaNLPvariable,thesolutionofstateat thenalpoint,i.e.,at t = 1 isalsoobtained. 7.2.2Karush-Kuhn-TuckerConditions ThenecessaryoptimalityconditionsortheKarush-Kuhn-Tuc ker(KKT)conditions,of theNLPgiveninEqs.( 7–34 )-( 7–38 )arenowderived.TheLagrangianassociatedwith theNLPis L = h Y 0 y 0 i + N X k =1 T ( k )( w k g ( Y k U k ) h k C ( Y k U k ) i ) N X k =1 h k D k Y LG T ( k ) f ( Y k U k ) i h N +1 Y N +1 Y 0 N X k =1 T ( k ) w k f ( Y k U k ) i (7–39) 185

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where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 7–35 ), N +1 2 R n aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 7–36 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 7–37 ),and k isthe k th rowoftheLagrangemultipliersmatrix 2 R N s associatedwiththeconstraintsinEq.( 7–38 )TheKKToptimalityconditionsare thenobtainedbydifferentiatingtheLagrangianwithrespe cttoeachofthevariableand equatingthederivativetozero,suchthat T ( k ) r Y k ( w k g k + h k + w k N +1 f k ih k C k i )= D Tk ,1 k N (7–40) = N +1 D T0 (7–41) 0 = N +1 (7–42) r U k ( w k g k + h k + w k N +1 f k ih k C k i )= 0 ,1 k N (7–43) D k Y LG T ( k ) f ( Y k U k )= 0 ,1 k N (7–44) ki =0 when C ki < 0,1 i s ,1 k N (7–45) ki < 0 when C ki =0,1 i s ,1 k N (7–46) Y 0 y 0 = 0 (7–47) where D Ti isthe i th rowof D T g k = g ( Y k U k ) f k = f ( Y k U k ) and C k = C ( Y k U k ) Next,theKKTconditionsgiveninEqs.( 7–40 )-( 7–47 )arereformulatedso thattheybecomeadiscretizationoftherst-orderoptimal ityconditionsgivenin Eqs.( 7–14 )-( 7–21 )forthecontinuouscontrolproblemgiveninEqs.( 7–10 )-( 7–13 ). Let D y =[ D y ij ],(1 i N ),(1 i N +1) bethe N ( N +1) matrixdenedasfollows: D y ij = w j w i D ji ,( i j )=1,..., N (7–48) D y i N +1 = N X j =1 D y ij i =1,..., N (7–49) 186

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Itisnotedagainthatthematrix D y denedin ( 7–48 ) and ( 7–49 ) isadifferentiation matrixforthespaceofpolynomialsofdegree N .Accordingtothedenitionof D y D Tk = w k D yk ,1: N W 1 ,1 k N (7–50) where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal.Substituting Eq.( 7–50 )inEq.( 7–40 ), w k D yk ,1: N W 1 = T ( k ) r Y k ( w k g k + h k f k ih k C k i ),1 k N (7–51) Next,deningthefollowingchangeofvariables: ~ N +1 = N +1 (7–52) ~ 0 = N +1 D T0 (7–53) ~ k = k w k + N +1 ,1 k N (7–54) ~ r k = k w k ,1 k N (7–55) ~ = (7–56) SubstitutingEqs.( 7–52 )-( 7–56 )inEqs.( 7–41 )-( 7–47 )andinEq.( 7–51 ),thetransformed KKTconditionsoftheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (7–57) 0 = D k Y LG T ( k ) f ( Y k U k ),1 k N (7–58) ~ r ki =0 when C ki < 0,1 i s ,1 k N (7–59) ~ r ki < 0 when C ki =0,1 i s ,1 k N (7–60) 0 = Y 0 y 0 (7–61) ~ 0 = ~ (7–62) ~ N +1 = 0 (7–63) D yk ,1: N ~ + D y k N +1 ~ N +1 = T ( k ) r Y k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (7–64) 187

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Now,consideracomparisonofthetransformedKKTconditionsi nEqs.( 7–57 )-( 7–64 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 7–14 )-( 7–21 ) ofthecontinuous-timeoptimalcontrolproblem.Itisnoted thatthetransformedKKT conditionsinEqs.( 7–57 )-( 7–61 )arethediscretizedformsofthecontinuous-time rst-orderoptimalityconditionsinEq.( 7–16 ),Eq.( 7–14 ),Eq.( 7–19 ),Eq.( 7–20 ),and Eq.( 7–21 ),respectively.Furthermore,itisshowninTheorem 1 thatthesystem( 7–64 ) isapseudospectralschemeforthecostatedynamics,i.e. D yk ,1: N ~ + D y k N +1 ~ N +1 = ~ k ,1 k N (7–65) Therefore,thelefthandsideofEq.( 7–64 )isanapproximationofcostatedynamics atthe k th collocationpoint.Asaresult,Eq.( 7–64 )representsthediscretizedversion ofthecostatedynamicsinEq.( 7–15 )at k =(1,..., N ) .Lastly,itisnotedthatatthe boundarypoints,thediscreteequivalentsofcontinuousbo undaryconditions( 7–17 ) and( 7–18 )arethesameasthediscretecostateattheboundarypointsi n( 7–62 )and ( 7–63 ),respectively.Hence,thesystemoftransformedKKTconditi onsoftheNLPis exactlyequivalenttotherst-orderoptimalitycondition softhecontinuous-timeoptimal controlproblem.Therefore,accuratecostateestimatesca nbeobtainedfromtheKKT multipliersusingtherelationshipgiveninEqs.( 7–52 )-( 7–54 )as N +1 ~ N +1 = N +1 (7–66) 0 ~ 0 = N +1 D T0 (7–67) k ~ k = k w k + N +1 ,1 k N (7–68) r k ~ r k = k w k ,1 k N (7–69) ~ = (7–70) 188

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7.2.3EquivalentImplicitIntegrationScheme ItisnowshownthattheLGpseudospectraldiscretizationof thestateequationhas anequivalentintegratedformulation.Let p beanypolynomialofdegreeatmost N .By theconstructionofthe N ( N +1) matrix D ,wehave Dp =_ p where p k = p ( k ),0 k N (7–71) p k =_ p ( k ),1 k N (7–72) Let D = [ D 0 D 1: N ] where D 0 istherstcolumnof D and D 1: N aretheremainingcolumns. Thentheidentity p = Dp canbewrittenas p = D 0 p 0 + D 1: N p 1: N (7–73) Multiplyingby D 1 1: N andutilizingProposition 5 gives p k = p 0 + D 1 1: N p k ,1 k N (7–74) Next,adifferentexpressionfor p k p 0 basedontheLagrangepolynomialapproximation ofthederivativeisobtained.Let L yi ( ) betheLagrangeinterpolationpolynomials associatedwiththecollocationpoints: L yi = N Y j =1 j 6 = i j i j ,1 i N (7–75) NoticethattheLagrangepolynomials L i denedin( 7–24 )aredegree N whilethe Lagrangepolynomials L yi aredegree N 1 .Thenbecause p isapolynomialofdegreeat most N 1 ,itcanbeapproximatedexactlybytheLagrangepolynomials L yi : p = N X i =1 p i L yi ( ). (7–76) 189

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Integrating p from 1 to k ,followingrelationshipisobtained p ( k )= p ( 1)+ N X i =1 p i A ki A ki = Z k 1 L yi ( ) d ,1 k N (7–77) Utilizingthenotation( 7–71 )and( 7–72 ),wehave p k = p 0 + ( A p ) k ,1 k N (7–78) Therelations( 7–74 )and( 7–78 )aresatisedforanypolynomialofdegreeatmost N Equating( 7–74 )and( 7–78 )toobtain A p = D 1 1: N p Choose p fromthecolumnsoftheidentitymatrixtodeducethat A = D 1 1: N .Rewriting Eq.( 7–28 )suchthat D 0 Y 0 + D 1: N 266664 Y 1 ... Y N 377775 = 266664 T ( 1 ) f ( Y 1 U 1 ) ... T ( N ) f ( Y N U N ) 377775 (7–79) Multiplyingby A = D 1 1: N andutilizingProposition 5 toobtain Y k = Y 0 + N X i =1 A ki T ( i ) f ( Y i U i ),1 k N (7–80) Hence,theEq.( 7–80 )istheequivalentimplicitintegralformofthestateequat ionin ( 7–28 ).Theelementsof A aretheintegralsoftheLagrangebasis L yi ,whiletheelements of D arethederivativesoftheLagrangebasis L i denedin( 7–24 ).Intheimplicit integrationscheme,thestateatthehorizonisapproximate dusingGaussquadraturein Eq.( 7–36 )asinthedifferentialscheme.Computationally,thediffe rentialformulationof Eq.( 7–28 )ofthesystemdynamicsismoreconvenientsinceanynonline artermsin f 190

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retaintheirsparsityinthediscretization,whileforthei ntegratedversionofEq.( 7–80 ), thenonlineartermsarenonsparseduetomultiplicationbyt hedensematrix A 7.3Innite-HorizonRadauPseudospectralMethod Inthissectiondiscreteapproximationtothenonlinearin nite-horizonoptimalcontrol problemisformulated.Thisdiscreteschemeisbasedonglob alcollocationusingthe Legendre-Gauss-Radaucollocationpoints.Thestateatthe horizonisincludedinthe stateapproximation.EquivalencebetweenthetransformedKKT conditionsanddiscrete versionoftherst-orderoptimalityconditionsisestabli shed.Theequivalentimplicit integrationschemeisalsoderived.7.3.1NLPFormulation ConsidertheLGRcollocationpoints ( 1 ,..., N ) ontheinterval [ 1,1) where 1 = 1 andoneadditional noncollocated point N +1 =1 (theterminaltime,correspondingto t =+ 1 ).Thestateisapproximatedbyapolynomialofdegreeatmost N as y ( ) Y ( )= N +1 X i =1 Y i L i ( ), (7–81) where Y i 2 R n and L i ( ) isaLagrangepolynomialofdegree N denedas L i ( )= N +1 Y j =1 j 6 = i j i j ,( i =1,..., N +1). (7–82) Thebasisincludesthefunction L N +1 correspondingtothenaltime N +1 = 1 DifferentiatingtheseriesofEq.( 7–81 )withrespectto ,anapproximationtothe derivativeofthestatein domainisobtainedas y ( ) Y ( )= N +1 X i =1 Y i L i ( ). (7–83) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 7–83 )totheright-handsideofthestatedynamicconstraintsin 191

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Eq.( 7–11 )atthe N LGRpoints, ( 1 ,..., N ) N +1 X i =1 Y i L i ( k )= T ( k ) f ( Y k U k ),( k =1,..., N ), (7–84) N +1 X i =1 D ki Y i = T ( k ) f ( Y k U k ), D ki = L i ( k ). (7–85) Itisnotedthat N +1 isnotacollocationpoint.Hence,thediscreteapproximati ontothe systemdynamicsis D k Y LGR = T ( k ) f ( Y k U k ),1 k N (7–86) where D k isthe k th rowofdifferentiationmatrix D =[ D ki ],(1 k N ),(1 i N +1) calledthe Radaupseudospectraldifferentiationmatrix and Y LGR isdenedas Y LGR = 266664 Y 1 ... Y N +1 377775 Thus,thediscreteapproximationtothesystemdynamics y ( )= T ( ) f ( y ( ), u ( )) is obtainedbyevaluatingthesystemdynamicsateachcollocat ionpointandreplacing y ( k ) byitsdiscreteapproximation D k Y LGR .Itisimportanttoobservethattheleft-hand sideofEq.( 7–86 )containsapproximationsforthestateattheLGRpointsplu sthenal pointwhiletheright-handsidecontainsapproximationsfo rthestate(andcontrol)atonly theLGRpoints. TheobjectivefunctioninEq.( 7–10 )isapproximatedbytheLegendre-Gauss-Radau quadratureas J = Z +1 1 T ( ) g ( y ( ), u ( )) d N X k =1 T ( k ) w k g ( Y k U k ), (7–87) where w k isthequadratureweightassociatedwith k .Lastly,thepathconstraintsin Eq.( 7–13 )areenforcedat N LGRcollocationpointsas T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ). (7–88) 192

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Thecontinuous-timenonlinearinnite-horizonoptimalco ntrolproblemofEqs.( 7–10 )-( 7–13 ) isthenapproximatedbythefollowingNLP:Minimizethecost J = N X k =1 T ( k ) w k g ( Y k U k ), (7–89) subjecttothefollowingequalityandinequalityconstrain ts D k Y LGR T ( k ) f ( Y k U k )= 0 ,( k =1,..., N ), (7–90) Y 1 y 0 = 0 (7–91) T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ), (7–92) wheretheNLPvariablesare ( Y 1 ,..., Y N +1 ) and ( U 1 ,..., U N ) .Itisnotedthat,the singularityinthechangeofvariables t = ( ) andin T ( )= 0 ( ) at =+1 is avoidedintheNLPformulation. T ( )= 0 ( ) isneverevaluatedatthesingularityin Eqs.( 7–89 )-( 7–92 ),rather T ( ) isevaluatedatthequadraturepointsonly,whichareall strictlylessthan 1 .Furthermore,because Y N +1 isaNLPvariable,thesolutionofstateat thenalpoint,i.e.,at t = 1 isalsoobtained. 7.3.2Karush-Kuhn-TuckerConditions ThenecessaryoptimalityconditionsortheKarush-Kuhn-Tuc ker(KKT)conditions,of theNLPgiveninEqs.( 7–89 )-( 7–92 )arenowderived.TheLagrangianassociatedwith theNLPis L = h Y 1 y 0 i + N X k =1 T ( k )( w k g ( Y k U k ) h k C ( Y k U k ) i ) N X k =1 h k D k Y LGR T ( k ) f ( Y k U k ) i (7–93) where k isthe k th rowoftheLagrangemultipliersmatrix 2 R N n associatedwith theconstraintsinEq.( 7–90 ), 2 R q aretheLagrangemultipliersassociatedwith theconstraintsinEq.( 7–91 ),and k isthe k th rowoftheLagrangemultipliersmatrix 193

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2 R N s associatedwiththeconstraintsinEq.( 7–92 )TheKKToptimalityconditionsare thenobtainedbydifferentiatingtheLagrangianwithrespe cttoeachofthevariableand equatingthederivativetozero,suchthat T ( k ) r Y k ( w k g k + h k f k ih k C k i )= D Tk ,2 k N (7–94) T ( k ) r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i )= D T1 + (7–95) 0 = D TN +1 (7–96) r U k ( w k g k + h k f k ih k C k i )= 0 ,1 k N (7–97) D k Y LGR T ( k ) f ( Y k U k )= 0 ,1 k N (7–98) ki =0 when C ki < 0,1 i s ,1 k N (7–99) ki < 0 when C ki =0,1 i s ,1 k N (7–100) Y 1 y 0 = 0 (7–101) where D Ti isthe i th rowof D T g k = g ( Y k U k ) f k = f ( Y k U k ) and C k = C ( Y k U k ) Next,theKKTconditionsgiveninEqs.( 7–94 )-( 7–101 )arereformulatedso thattheybecomeadiscretizationoftherst-orderoptimal ityconditionsgivenin Eqs.( 7–14 )-( 7–21 )forthecontinuouscontrolproblemgiveninEqs.( 7–10 )-( 7–13 ). Let D y =[ D y ij ],(1 i N ),(1 i N ) bethe N N matrixdenedasfollows: D y 11 = D 11 1 w 1 (7–102) D y ij = w j w i D ji otherwise (7–103) Thematrix D y denedin ( 7–103 ) and ( 7–102 ) isadifferentiationmatrixforthespaceof polynomialsofdegree N 1 .Accordingtothedenitionof D y D T1 = w 1 D y1 W 1 1 w 1 e 1 (7–104) D Tk = w k D yk W 1 ,2 k N (7–105) 194

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where W isadiagonalmatrixwithweights w k 1 k N ,onthediagonal. e 1 isthe 1 st rowof N N identitymatrix.SubstitutingEqs.( 7–104 )and( 7–105 )inEqs.( 7–95 )and ( 7–94 ), = T ( k ) r Y 1 ( w 1 g 1 + h 1 f 1 ih 1 C 1 i ) + w 1 D y1 W 1 + 1 w 1 e 1 (7–106) w k D yk W 1 = T ( k ) r Y k ( w k g k + h k f k ih k C k i ),2 k N (7–107) Next,deningthefollowingchangeofvariables: ~ N +1 = D TN +1 (7–108) ~ k = k w k ,1 k N (7–109) ~ r k = k w k ,1 k N (7–110) ~ = (7–111) SubstitutingEqs.( 7–108 )-( 7–111 )inEqs.( 7–96 )-( 7–101 )andinEqs.( 7–106 )-( 7–107 ), thetransformedKKTconditionsfortheNLParegivenas 0 = r U k ( g k + h ~ k f k ih ~ r k C k i ),1 k N (7–112) 0 = D k Y LGR T ( k ) f ( Y k U k ),1 k N (7–113) ~ r ki =0 when C ki < 0,1 i s ,1 k N (7–114) ~ r ki < 0 when C ki =0,1 i s ,1 k N (7–115) 0 = Y 1 y 0 (7–116) D y1 ~ = T ( k ) r Y 1 ( g 1 + h ~ 1 f 1 ih ~ r 1 C 1 i ) 1 w 1 ~ 1 ~ (7–117) ~ N +1 = 0 (7–118) D yk ~ = T ( k ) r Y k ( g k + h ~ k f k ih ~ r k C k i ), (7–119) 2 k N 195

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Now,consideracomparisonofthetransformedKKTconditionsi nEqs.( 7–112 )-( 7–119 ) oftheNLPtotherst-ordernecessaryoptimalitycondition sinEqs.( 7–14 )-( 7–21 )ofthe continuous-timeoptimalcontrolproblem.Itisnotedthatt hetransformedKKTconditions inEqs.( 7–112 )-( 7–116 )arethediscretizedformsofthecontinuous-timerst-ord er optimalityconditionsinEq.( 7–16 ),Eq.( 7–14 ),Eq.( 7–19 ),Eq.( 7–20 ),andEq.( 7–21 ), respectively.Furthermore,itisshowninTheorem 2 thatthesystem( 7–119 )isa pseudospectralschemeforthecostatedynamics,i.e. D yk ~ = ~ k ,1 k N (7–120) Therefore,thelefthandsideofEq.( 7–119 )isanapproximationofthecostatedynamics atthe k th collocationpoint, k =(2,..., N ) .Asaresult,Eq.( 7–119 )representsthe discretizedversionofthecostatedynamicsinEq.( 7–15 )at k =(2,..., N ) .Next,itis notedthatatthenalpoint,thediscreteequivalentofcont inuousboundaryconditions ( 7–18 )isthesameasthediscretecostateatthenalpointin( 7–118 ).However,atthe initialpoint,thediscreteequivalentofcontinuousbound arycondition( 7–17 )iscoupled inthediscretecostatedynamicsattheinitialpointin( 7–117 ). TheequivalenceofthetransformedKKTconditionoftheNLPatt heinitialboundary ( 7–117 )tothediscretizedformofcontinuousrst-orderoptimali tyconditionin( 7–17 )is nowestablishedbymanipulating( 7–108 ). Returningtothedenitionof ~ N +1 in( 7–108 ),weobtain ~ N +1 = D TN +1 = N X i =1 i D i N +1 = N X i =1 N X j =1 i D ij (7–121) = 1 w 1 + N X i =1 N X j =1 i D y ji w j w i = 1 w 1 + N X i =1 N X j =1 j D y ij w i w j (7–122) = ~ 1 + N X i =1 N X j =1 w i ~ j D y ij = ~ 1 + N X i =1 w i D yi ~ (7–123) = ~ N X i =1 T ( i ) w i r Y i ( g i + h ~ i f i ih ~ r i C i i ), (7–124) 196

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where( 7–121 )followsfromtheidentity( 4–64 )giveninProposition 2 ,( 7–122 )isthe denition( 7–102 )and( 7–103 )of D y ,( 7–123 )isthedenition( 7–109 )of ~ i ,and( 7–124 ) istherst-orderoptimalitycondition( 7–119 )and( 7–117 ).Rearranging( 7–124 )such that ~ = ~ N +1 + N X i =1 T ( i ) w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (7–125) Next,thecontinuouscostatedynamicsinEq.( 7–15 )are ( )= T ( ) r y H = T ( ) r y ( g + h f ih r C i ). (7–126) Integratingthecontinuouscostatedynamicsin( 7–126 )usingtheRadauquadrature, ~ 1 = ~ N +1 + N X i =1 T ( i ) w i r Y i ( g i + h ~ i f i ih ~ r i C i i ). (7–127) Comparing( 7–125 )with( 7–127 )gives ~ 1 = ~ (7–128) Eq.( 7–128 )isthemissingboundaryconditionattheinitialpointthat wascoupled withthediscretecostatedynamicsattheinitialpointin( 7–117 ).Itisalsoimpliedby Eq.( 7–128 )thattheextratermin( 7–117 )isinfact zero ,thereby,making( 7–117 ) consistentwithdiscretecostatedynamicsattheinitialpo int.Hence,thesystem oftransformedKKTconditionsoftheNLPisequivalenttother st-orderoptimality conditionsofthecontinuous-timeoptimalcontrolproblem andaccuratecostate estimatesareobtainedfromtheKKTmultipliersusingtherela tionshipgivenin 197

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Eqs.( 7–108 )-( 7–109 )as N +1 ~ N +1 = D TN +1 (7–129) k ~ k = k w k ,1 k N (7–130) r k ~ r k = k w k ,1 k N (7–131) ~ = (7–132) 7.3.3EquivalentImplicitIntegrationScheme ItisnowshownthattheLGRpseudospectraldiscretizationo fthestateequationhas anequivalentintegratedformulation.Let p beanypolynomialofdegreeatmost N .By theconstructionofthe N ( N +1) matrix D ,wehave Dp =_ p where p k = p ( k ),1 k N +1, (7–133) p k =_ p ( k ),1 k N (7–134) Thentheidentity p = Dp canbewrittenas p = D 1 p 1 + D 2: N +1 p 2: N +1 (7–135) Multiplyingby D 1 2: N +1 andutilizingProposition 7 gives p k = p 1 + D 1 2: N +1 p k ,2 k N +1. (7–136) Next,adifferentexpressionfor p k p 1 basedontheLagrangepolynomialapproximation ofthederivativeisobtained.Let L yi ( ) betheLagrangeinterpolationpolynomials associatedwiththecollocationpoints: L yi = N Y j =1 j 6 = i j i j ,1 i N (7–137) NoticethattheLagrangepolynomials L i denedin( 7–82 )aredegree N whilethe Lagrangepolynomials L yi aredegree N 1 .Thenbecause p isapolynomialofdegreeat 198

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most N 1 ,itcanbeapproximatedexactlybytheLagrangepolynomials L yi : p = N X i =1 p i L yi ( ). (7–138) Integrating p from 1 to k ,followingrelationshipisobtained p ( k )= p ( 1)+ N X i =1 p i A ki A ki = Z k 1 L yi ( ) d ,2 k N +1. (7–139) Utilizingthenotation( 7–133 )and( 7–134 ),wehave p k = p 1 + ( A p ) k ,2 k N +1. (7–140) Therelations( 7–136 )and( 7–140 )aresatisedforanypolynomialofdegreeatmost N Equating( 7–136 )and( 7–140 )toobtain A p = D 1 2: N +1 p Choose p fromthecolumnsoftheidentitymatrixtodeducethat A = D 1 2: N +1 .Rewriting Eq.( 7–86 )suchthat D 1 Y 1 + D 2: N +1 266664 Y 2 ... Y N +1 377775 = 266664 T ( 1 ) f ( Y 1 U 1 ) ... T ( N ) f ( Y N U N ) 377775 (7–141) Multiplyingby A = D 1 2: N +1 andutilizingProposition 7 toobtain Y k = Y 1 + N X i =1 A ki T ( i ) f ( Y i U i ),2 k N +1. (7–142) Hence,theEq.( 7–142 )istheequivalentimplicitintegralformofthestateequat ionin ( 7–86 ).Theelementsof A aretheintegralsoftheLagrangebasis L yi ,whiletheelements of D arethederivativesoftheLagrangebasis L i denedin( 7–82 ).Itisnotedthatthe stateatthenalpointisalsoobtainedintheimplicitinteg rationscheme. 199

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7.4InapplicabilityofLobattoPseudospectralMethod ConsidertheLGLcollocationpoints ( 1 ,..., N ) ontheinterval [ 1,1] where 1 = 1 and N =1 (theterminaltime,correspondingto t =+ 1 ).Thestateisapproximatedby apolynomialofdegreeatmost N 1 as y ( ) Y ( )= N X i =1 Y i L i ( ), (7–143) where Y i 2 R n and L i ( ) isaLagrangepolynomialofdegree N 1 denedas L i ( )= N Y j =1 j 6 = i j i j ,( i =1,..., N ). (7–144) DifferentiatingtheseriesofEq.( 7–143 )withrespectto ,anapproximationtothe derivativeofthestatein domainisobtainedas y ( ) Y ( )= N X i =1 Y i L i ( ). (7–145) Thecollocationconditionsarethenformedbyequatingthed erivativeofthestate approximationinEq.( 7–145 )totheright-handsideofthestatedynamicconstraintsin Eq.( 7–11 )atthe N LGLpoints, ( 1 ,..., N ) N X i =1 Y i L i ( k )= T ( k ) f ( Y k U k ),( k =1,..., N ), (7–146) N X i =1 D ki Y i = T ( k ) f ( Y k U k ), D ki = L i ( k ). (7–147) Hence,thediscreteapproximationtothesystemdynamicsis D k Y LGL = T ( k ) f ( Y k U k ),1 k N (7–148) 200

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where D k isthe k th rowofdifferentiationmatrix D =[ D ki ],(1 k N ),(1 i N ) calledthe Lobattopseudospectraldifferentiationmatrix and Y LGL isdenedas Y LGL = 266664 Y 1 ... Y N 377775 Thus,thediscreteapproximationtothesystemdynamics y ( )= T ( ) f ( y ( ), u ( )) is obtainedbyevaluatingthesystemdynamicsateachcollocat ionpointandreplacing y ( k ) byitsdiscreteapproximation D k Y LGL TheobjectivefunctioninEq.( 7–10 )isapproximatedbytheLegendre-Gauss-Lobatto quadratureas J = Z +1 1 T ( ) g ( y ( ), u ( )) d N X k =1 T ( k ) w k g ( Y k U k ), (7–149) where w k isthequadratureweightassociatedwith k .Lastly,thepathconstraintsin Eq.( 7–13 )areenforcedatthe N LGLcollocationpointsas T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ). (7–150) Thecontinuous-timenonlinearinnite-horizonoptimalco ntrolproblemofEqs.( 7–10 )-( 7–13 ) isthenapproximatedbythefollowingNLP:Minimizethecost J = N X k =1 T ( k ) w k g ( Y k U k ), (7–151) subjecttothefollowingequalityandinequalityconstrain ts D k Y LGL T ( k ) f ( Y k U k )= 0 ,( k =1,..., N ), (7–152) Y 1 y 0 = 0 (7–153) T ( k ) C ( Y k U k ) 0 ,( k =1,..., N ). (7–154) 201

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Itisnotedthat,thesingularityin T ( )= 0 ( ) at =+1 ispresentintheNLP formulation.Because =+1 isacollocationpoint, T ( )= 0 ( ) isevaluated atthesingularityinEqs.( 7–151 )-( 7–154 ).Thiscausesasingularitytooccurin theformulationoftheNLP.Therefore,theLobattopseudosp ectralmethodinwhich collocationisperformedattheterminalpointoftheinterv alcannotbeusedforsolving aninnite-horizonoptimalcontrolproblem. 7.5Examples Inthissectiontwoexamplesoftheinnite-horizonGaussan dRadaupseudospectral methodsdevelopedinthischapterareconsidered.Therste xampleisaone-dimensional nonlinearinnite-horizonoptimalcontrolproblem.Ther stexamplewassolvedusing boththeinnite-horizonGaussandRadaupseudospectralme thodsdevelopedinthis chapterandthemethodofRef.[ 65 ]usingallthreetimetransformations ( ) .Anerror analysisisperformedanderrorcomparisonbetweendiffere nttimetransformationsis done.Anerrorcomparisonbetweentheerrorsfromthemethods ofthisdissertationand themethodofRef.[ 65 ]isalsoperformed.Thesecondexampleisthetwo-dimension al innite-horizonLQRproblem.Thesecondexamplewassolved usingboththe innite-horizonGaussandRadaupseudospectralmethodsde velopedinthischapter usingallthreetimetransformations ( ) .Anerrorcomparisonforboththemethods usingallthreetransformationsisperformed.7.5.1Example1:Innite-HorizonOne-DimensionalNonlinea rProblem Considerthefollowingnonlinearinnite-horizonoptimal controlproblem[ 64 ]. Minimizethecostfunctional J = 1 2 Z 1 0 log 2 y ( t )+ u ( t ) 2 dt (7–155) subjecttothedynamicconstraint y ( t )= y ( t )log y ( t )+ y ( t ) u ( t ), (7–156) 202

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withtheinitialcondition y (0)=2. (7–157) Theexactsolutiontothisproblemis y ( t )=exp( x ( t )), u ( t )= (1+ p 2) x ( t ), ( t )=(1+ p 2)exp( x ( t )) x ( t ), x ( t )=log2exp( t p 2). (7–158) TheexampleofEqs.( 7–155 )–( 7–157 )wassolvedfor N =(5,10,15,20,25,30) using boththeinnite-horizonpseudospectralmethodsdescribe dinthischapterandthe approachofRef.[ 65 ]withthethreestrictlymonotonictransformationsofthed omain 2 [ 1,+1) giveninEqs.( 7–5 )–( 7–7 ).AllsolutionswereobtainedusingtheNLPsolver SNOPTwithoptimalityandfeasibilitytolerancesof 1 10 10 and 2 10 10 ,respectively. Furthermore,thefollowinginitialguessofthesolutionwa sused: y ( )= y 0 u ( )= (7–159) wherewerecallthat 2 [ 1,1] .Figs. 7-3 – 7-8 showthesolutionsobtainedfromthe innite-horizonGaussandRadaupseudospectralmethodsal ongsidetheexactsolution usingthethreetransformationsofthedomain 2 [ 1,+1) giveninEqs.( 7–5 )–( 7–7 ) for N =30 collocationpoints.ItisseenthattheGPMandtheRPMsolution are indistinguishablefromtheexactsolutionforallthreequa ntities(state,control,and costate).Inparticular,itisseenthattheinnitehorizon GPMandRPMsolvethe problemontheentireinnitetimedomain. 203

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1.2 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.2 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-3.SolutionobtainedfromtheGausspseudospectral methodusing a ( ) Next,themaximumbasetenlogarithmofthestate,control,a ndcostateerrorsare denedas E y =max k log 10 jj y ( k ) y ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E =max k log 10 y ( k ) y ( k ) 1 (7–160) InEq.( 7–160 )theindex k spanstheapproximationpointsinthecaseofeitherthe stateandcostateandspansonlythecollocationpointsinth ecaseofthecontrol.Itis 204

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1.2 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.2 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-4.SolutionobtainedfromtheRadaupseudospectral methodusing a ( ) notedthatthestateandcostateareobtainedonthe entirehorizon withtheindex N +1 correspondingtothestateandcostateat =+1 ,orequivalently,at t =+ 1 Thestate,controlandcostateerrorsobtainedusingtheGau ssandRadaumethods ofthispaperareshown,respectively,inFigs. 7-9 7-10 and 7-11 alongsidetheerror obtainedusingthemethodofRef.[ 65 ]withthetransformationgiveninEqs.( 7–5 )and ( 7–7 ).Itisseenforallthreetransformationsandforbothmetho dsofthispaper,the state,control,andcostateerrorsdecreaseinessentially alinearmanneruntil N =30 205

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-5.SolutionobtainedfromtheGausspseudospectral methodusing b ( ) demonstratinganapproximatelyexponentialconvergencer ate.Furthermore,itis observedthateithertheGaussorRadaumethodofthispapery ieldsapproximately thesameerrorforaparticularvalueof N andchoiceoftransformation.Moreover,itis seenthattheerrorsarelargestandsmallest,respectively ,usingthetransformations ofEqs.( 7–6 )and( 7–7 ).Infact,thetransformationofEq.( 7–7 )isatleastoneorderof magnitudemoreaccuratethaneitheroftheothertwotransfo rmations.Finally,itisseen thattheerrorsfromthetwomethodsofthispaperusingthetr ansformationofEq.( 7–7 ) 206

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1.2 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.2 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-6.SolutionobtainedfromtheRadaupseudospectral methodusing b ( ) aresignicantlysmallerthanthoseobtainedusingthemeth odofRef.[ 65 ](wherethe transformationofEq.( 7–5 )areused).WhenthetransformationofEq.( 7–7 )isused, however,thestateerrorsfromthemethodofRef.[ 65 ]arenearlythesameasthose obtainedusingtheGaussandRadaumethods,whilethecontro landcostateerrorsare approximatelyoneorderofmagnitudelargerusingthemetho dofRef.[ 65 ]. ThedifferentbehaviorofthefunctionsgiveninEqs.( 7–5 )–( 7–7 )isunderstoodif weapplythechangeofvariablestothecontinuoussolution. Theoptimalstateinthe 207

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-7.SolutionobtainedfromtheGausspseudospectral methodusing c ( ) transformedcoordinatesisasfollows: y a ( )=exp log2exp p 2 1+ 1 (7–161) y b ( )=exp log2 1 2 p 2 (7–162) y c ( )=exp log2 1 2 2 p 2 (7–163) 208

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2 1.8 1.6 1.4 1.2 1 1 0.8 0.5 0 0.5 1 State y ( t ) y ( t ) AStatesolution. 1 0.5 0.5 0 0 0.5 0.5 1 1 1.5 2 Control u ( t ) u ( t ) BControlsolution. 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0.5 1 Costate y ( t ) y ( t ) CCostatesolution. Figure7-8.SolutionobtainedfromtheRadaupseudospectral methodusing c ( ) Herethesubscripts a b ,and c correspondtothethreechoicesof giveninEqs.( 7–5 )–( 7–7 ). Anadvantageofusingalogarithmicchangeofvariablesgiven inEqs.( 7–6 )or( 7–7 ), ascomparedtothefunctiongiveninEq.( 7–5 ),isthatlogarithmicfunctionsessentially movecollocationpointsassociatedwithlargevaluesof t totheleft.Becausetheexact solutionchangesslowlywhen t islarge,thisleftwardmovementofthecollocationpoints isbenecialsincemorecollocationpointsaresituatedwhe rethesolutionischanging mostrapidly.Thedisadvantageofalogarithmicchangeofva riablesisseeninthe 209

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30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 NE y a ( ) (Gauss) b ( ) (Gauss) c ( ) (Gauss) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) AGausserror. 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 N E y a ( ) (Radau) b ( ) (Radau) c ( ) (Radau) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) BRadauerror. Figure7-9.MaximumstateerrorsforexampleusinginnitehorizonGaussandRadau pseudospectralmethodalongsideerrorsobtainedusingthe Radaumethod ofRef.[ 65 ]. 210

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30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 NE u a ( ) (Gauss) b ( ) (Gauss) c ( ) (Gauss) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) AGausserror. 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 N E u a ( ) (Radau) b ( ) (Radau) c ( ) (Radau) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) BRadauerror. Figure7-10.Maximumcontrolerrorsforexampleusinginni te-horizonGaussand Radaupseudospectralmethodalongsideerrorsobtainedusi ngtheRadau methodofRef.[ 65 ]. 211

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30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 NE a ( ) (Gauss) b ( ) (Gauss) c ( ) (Gauss) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) AGausserror. 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 N E a ( ) (Radau) b ( ) (Radau) c ( ) (Radau) a ( ) (Ref.[ 65 ]) c ( ) (Ref.[ 65 ]) BRadauerror. Figure7-11.Maximumcostateerrorsforexampleusinginni te-horizonGaussand Radaupseudospectralmethodalongsideerrorsobtainedusi ngtheRadau methodofRef.[ 65 ]. 212

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function log(1 ) wherethegrowthissoslownear =+1 thatthetransformedsolution possessesasingularityinaderivativeat =+1 .Inotherwords,the j th derivativeofa functionoftheform (1 ) ,where > 0 isnotaninteger,issingularat =+1 for j > .Inparticular, y b ( ) hasonederivativeat =+1 butnottwo,while y c ( ) hastwo derivativesat =+1 butnotthree.Toachieveexponentialconvergence, y ( ) should beinnitelysmooth.Forthisparticularproblem,thechoic eofEq.( 7–7 )hasthefollowing niceproperties: y c ( ) isrelativelysmoothwithtwoderivatives,althoughnotin nitely smooth,andcollocationpointscorrespondingtolargevalu esof t ,wherethesolution changesslowly,aremovedtotheleft[whencomparedto t =(1+ ) = (1 ) ]where thesolutionchangesmorerapidly.Asaresult,for 5 N 30 ,thefunctionofEq.( 7–7 ) yieldsasolutionthatisoftentwoormoreordersofmagnitud emoreaccuratethanthe otherchoicesfor 7.5.2Example2:Innite-HorizonLQRProblem ConsiderthefollowingoptimalcontrolproblemtakenfromR ef.[ 65 ].Denoting x ( t )=[ x 1 ( t ) x 2 ( t )] T 2 R 2 asthestateand u ( t ) 2 R asthecontrol,minimizethecost functional J = 1 2 Z 1 0 x T Qx + u T Ru dt (7–164) subjecttothedynamicconstraint x = Ax + Bu (7–165) andtheinitialcondition x (0)= 264 4 4 375 (7–166) Thematrices A B Q ,and R forthisproblemaregivenas A = 264 012 1 375 B = 264 01 375 Q = 264 2001 375 R = 1 2 (7–167) 213

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Theexactsolutiontothisproblemis x ( t )=exp( [ A BK ] t ) x (0) u ( t )= Kx ( t ) ( t )= Sx ( t ) (7–168) where K istheoptimalfeedbackgainand S isthesolutiontothealgebraicRiccati equation.Inthiscase K and S aregiven,respectively,as K = 4.8284271247461932.557647291327851 S = 264 6.0312730495357522.4142135623730972.4142135623730971.278823645663925 375 (7–169) TheoptimalcontrolproblemofEqs.( 7–164 )–( 7–166 )wassolvedusingthe innite-horizonGaussandRadaupseudospectralmethodsus ingtheNLPsolver SNOPTwithdefaultoptimalityandfeasibilitytolerancesof 10 6 and 2 10 6 respectively,for N =5 to N =40 bystepsof5.Figs. 7-12 7-17 showthesolutions obtainedfromtheinnite-horizonGaussandRadaupseudosp ectralmethodsalongside theexactsolutionusingthethreetransformationsofthedo main 2 [ 1,+1) given inEqs.( 7–5 )–( 7–7 )for N =40 collocationpoints.ItisseenthattheGPMandthe RPMsolutionareindistinguishablefromtheexactsolutionf orallthreequantities(state, control,andcostate).Inparticular,itisseenthatthein nitehorizonGPMandRPM solveforthestateandthecostateontheentireinnitetime domainwhereasthecontrol isobtainedonlyatthecollocationpoints,whichdonotincl udethenalpoint. Supposenowthatwedenethefollowingmaximumabsoluteerro rsbetweenthe solutionobtainedfromtheNLPsolverandtheexactsolution : E x =max k log 10 jj x ( k ) x ( k ) jj 1 E u =max k log 10 jj u ( k ) u ( k ) jj 1 E =max k log 10 jj ( k ) ( k ) jj 1 (7–170) 214

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 2 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-12.SolutionobtainedfromtheGausspseudospectra lmethodusing a ( ) Thevaluesof E x E u ,and E areshowninFigs. 7-18 – 7-22 .Itisseenthatallerrors decreaselinearlyuntilapproximately N =40 ,againdemonstratinganexponential convergencerate.Furthermore,itisobservedthateithert heGaussortheRadau methodofthisdissertationyieldsapproximatelythesamee rrorforaparticularvalueof N andchoiceoftransformation.Moreover,itisseenthatforl ownumberofcollocation points,i.e., N 30 theerrorsarethelargestandthesmallest,respectively,u sing thetransformationsofEqs.( 7–6 )and( 7–7 ).Infact,thetransformationofEq.( 7–7 ), 215

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 2 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-13.SolutionobtainedfromtheRadaupseudospectra lmethodusing a ( ) c ( ) ,isatleastoneorderofmagnitudemoreaccuratethaneither oftheothertwo transformations.For N 30 ,however,thetransformationofEq.( 7–5 ), a ( ) ,seems tobethemostaccurateofallthethreetransformations.Mor eprecisely,therateof converganceforthetransformation a ( ) isthehighestandtherateofconvergence forthetransformation b ( ) istheleast.For N =5 ,theerrorsaretheleastandthe greatestforthetransformations c ( ) and a ( ) ,respectively.Duetothehigherrate ofconvergencefortransformation a ( ) ,at N =15 ,theerrorsfromthetransformation 216

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-14.SolutionobtainedfromtheGausspseudospectra lmethodusing b ( ) a ( ) becomesmallerthantheerrorsfromthetransformation b ( ) andnally,fromthe errosfromthetransformation c ( ) at N =35 217

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-15.SolutionobtainedfromtheRadaupseudospectra lmethodusing b ( ) 218

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 2 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-16.SolutionobtainedfromtheGausspseudospectra lmethodusing c ( ) 219

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4 3 21 1 0.5 0 0 0.5 1 1 2 3 4 State x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) AStatesolution. 10 8 6 4 2 1 0.5 0 0 0.5 1 2 Control u ( t ) u ( t ) BControlsolution. 5 1 0.5 0 0 0.5 1 5 10 15 Costate x 1 ( t ) x 2 ( t ) x 1 ( t ) x 2 ( t ) CCostatesolution. Figure7-17.SolutionobtainedfromtheRadaupseudospectra lmethodusing c ( ) 220

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40 30 20 10 0 0 1 2 3 4 5 6 7 NE x 1 (GPM) a ( ) b ( ) c ( ) AGausserror. 40 30 20 10 0 0 1 2 3 4 5 6 7 N E x 1 (RPM) a ( ) b ( ) c ( ) BRadauerror. Figure7-18.Maximumrstcomponentofstateerrorsforexam pleusinginnite-horizon GaussandRadaupseudospectralmethod 221

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40 30 20 10 0 1 2 3 4 5 6 7 NE x 2 (GPM) a ( ) b ( ) c ( ) AGausserror. 40 30 20 10 0 0 1 2 3 4 5 6 7 N E x 2 (RPM) a ( ) b ( ) c ( ) BRadauerror. Figure7-19.Maximumsecondcomponentofstateerrorsforex ampleusing innite-horizonGaussandRadaupseudospectralmethod 222

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40 30 20 10 0 1 2 3 4 5 6 NE u (GPM) a ( ) b ( ) c ( ) AGausserror. 40 30 20 10 0 0 1 2 3 4 5 6 N E u (RPM) a ( ) b ( ) c ( ) BRadauerror. Figure7-20.Controlerrorsforexampleusinginnite-hori zonGaussandRadau pseudospectralmethod 223

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replacements 40 30 20 10 0 0 1 2 3 4 5 6 7 8 NE 1 (GPM) a ( ) b ( ) c ( ) AGausserror. 40 30 20 10 0 0 1 2 3 4 5 6 7 8 N E 1 (RPM) a ( ) b ( ) c ( ) BRadauerror. Figure7-21.Maximumrstcomponentofcostateerrorsforex ampleusing innite-horizonGaussandRadaupseudospectralmethod 224

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replacements 40 30 20 10 0 1 2 3 4 5 6 7 8 NE 2 (GPM) a ( ) b ( ) c ( ) AGausserror. 40 30 20 10 0 0 1 2 3 4 5 6 7 8 N E 2 (RPM) a ( ) b ( ) c ( ) BRadauerror. Figure7-22.Maximumsecondcomponentofcostateerrorsfor exampleusing innite-horizonGaussandRadaupseudospectralmethod 225

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7.6Summary TwomethodsbasedontheLGandtheLGRcollocation,calledre spectively, theinnite-horizonGausspseudospectralmethodandthein nite-horizonRadau pseudospectralmethodhavebeenpresentedinthischapter. Theproposedmethods yieldapproximationstothestateandthecostateontheenti rehorizon,including approximationsat t =+ 1 .Furthermore,ithasbeenshownthatthetransformed KKTconditionsforboththemethodsareequivalenttothediscr etizedversionofthe continuous-timerst-orderoptimalityconditions.Bothof themethodscanbewritten equivalentlyineitheradifferentialoranimplicitintegr alform.Lastly,ithasbeenshown thatcollocationbasedontheLGLpointsisnotsuitablefors olvinginnite-horizon optimalcontrolproblems.Ithasbeenshownthatthemap :[ 1,+1) [0,+ 1 ) can betunedtoimprovethequalityofthediscreteapproximatio n.Innumericalexamples, thediscretesolutionexhibitsexponentialconvergenceas afunctionofthenumberof collocationpoints. 226

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CHAPTER8 CONCLUSION 8.1DissertationSummary Itisdifcult,ifnotimpossible,tondthesolutiontoacon tinuous-timeoptimal controlproblemfromtherst-ordernecessaryconditionsd erivedfromthecalculus ofvariationsformostoftheengineeringapplications.Num ericalapproximationsto thesolutionoftheoptimalcontrolproblem,therefore,mus tbeused.Manynumerical methodsexistforapproximatingthesolutionofanoptimalc ontrolproblem.These methodsgenerallyfallintooneoftwocategories:indirect methodsanddirectmethods. Indirectmethodsattempttondasolutiontotheoptimalcon trolproblembyapproximating therst-ordernecessaryconditionsderivedfromthecalcu lusofvariationsandthe Pontryagin'sprinciple.Directmethodsconverttheinnit e-dimensionalcontinuous controlproblemintoanite-dimensionaldiscretenonline arprogrammingproblem (NLP).TheresultingNLPcanthenbesolvedbywell-developed NLPalgorithms. Indirectmethodsgenerallyaremoreaccurate,whiledirect methodshavesimpler,more convenient,formulationsandaremorerobust.Pseudospectr almethodsforsolving optimalcontrolproblemsareaclassofdirecttranscriptio nmethodsthatuseorthogonal collocationforapproximatingthesolutionofdifferentia lequationsandtheGaussian quadraturetoapproximateintegralcost.TheLobattopseud ospectralmethod(LPM) usestheLegendre-Gauss-Lobatto(LGL)pointsintheformul ationwherethecostate estimatesarederiveddirectlyfromtheKarush-Kuhn-Tucker (KKT)multipliersofthe resultingNLP.TheLPM,however,suffersfromadefectinthec ostateestimatesat theboundarypoints.Atthesepointsthecostateestimatesdo notsatisfythecostate boundaryconditionsorthediscretizedcostatedynamics.T hisdefectresultsina relativelypoorestimateofthecostate.TheGausspseudosp ectralmethod(GPM) usestheLegendre-Gauss(LG)pointsanddoesnotsufferfrom adefectincostate estimates.IntheGPM,theKKTconditionsoftheresultingNLPar eadiscreteformof 227

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thecontinuous-timerst-ordernecessaryconditions.Iti snoted,however,thatbecause thedifferentialequationsarecollocatedatonlytheinter iorpoints,thecontrolattheinitial andthenalpointisnotobtainedusingtheGPM. Inthisdissertation,amethod,calledtheRadaupseudospec tralmethod(RPM), hasbeenpresentedfordirecttrajectoryoptimizationandc ostateestimationusing globalcollocationattheLegendre-Gauss-Radau(LGR)poin ts.Atheoreticalfoundation forthemethodhasbeenprovided.Theprimarypropertythatd istinguishestheRPM fromtheLPMandtheGPMisthefactthatthedynamicsarecolloca tedattheinterior pointsplustheinitialpoint.Thisleadstoanelegantequiv alencebetweenthenecessary conditionsoftheNLPandtherst-orderoptimalityconditi onsofthecontinuousproblem whilestillprovidingthecontrolattheinitialpoint.Thef actthatthenecessaryconditions oftheNLPareadiscreteformthecontinuous-timerst-orde rnecessaryconditions allowstheRPMtotakeadvantageoftheconvenientformulatio nsandrobustnessofa directmethod,whilepreservingtheaccuracyofanindirect method.Theresultsofthis researchindicatethattheRPMdescribedherehastheability todetermineaccurate solutionsforthestate,thecontrol,andthecostateofagen eraloptimalcontrolproblem. Next,auniedframeworkhasbeenpresentedbasedontheGPMan dtheRPM.It wasdemonstratedthateachoftheseschemescanbeexpressed ineitheradifferential oranintegralformulationthatareequivalenttoeachother .Itwasalsodemonstrated thattheLPMdoesnotexhibitsuchanequivalence.Thefacttha tthedifferentialforms oftheGPMandtheRPM,asformulatedinthisresearch,canbeexp ressedinan equivalentintegralform,showsthatthedynamicsareinfac tbeingintegratedwhen implementingthesemethods.Eitherformcanbeusedtosolvet heproblem.The differentialform,however,resultsinamoresparseNLPtha tcanbesolvedfaster. Furthermore,boththeGPMandtheRPMprovideanaccuratetrans formation betweentheKKTmultipliersofthediscreteNLPandthecostate ofthecontinuous optimalcontrolproblem.Ithasbeenshownthatthediscrete costatesystemsin 228

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boththesemethodsarefullrankwhilethediscretecostates ystemintheLobatto pseudospectralmethodhasanullspace.TheLPMcostateappro ximationwasfound tohaveanerrorthatoscillatesabouttheexactsolution,an dthiserrorwasshown byexampletobeduetothenullspaceintheLobattodiscretec ostatesystem.By comparingtheschemesofthisuniedframeworkwiththeLoba ttopseudospectral method,anefforthasbeenmadetounderstandthedeciencie sintheLobatto pseudospectralmethodthatleadtoaninaccuratesolution. Lastly,theinnite-horizonversionsoftheGaussandtheRa daupseudospectral methodshavebeenderived.Theproposedmethodsyieldappro ximationstothestate andcostateontheentirehorizon,includingapproximation sat t =+ 1 .Theseversions canalsobewrittenequivalentlyineitheradifferentialor animplicitintegralform.Itwas shownthatthemapping :[ 1,+1) [0,+ 1 ) canbetunedtoimprovethequalityof thediscreteapproximation.ItwasalsoshownthattheLobat topseudospectralmethod cannotbeusedtosolvetheinnite-horizonoptimalcontrol problems.Thedynamicsare collocatedatthenalpointintheLobattopseudospectralm ethodandhencecannotbe usedforinnite-horizonproblemsasasingularityexistsa tthenalpointinthemapping :[ 1,+1) [0,+ 1 ) EmpiricalevidencehassuggestedthattheGaussandtheRadau pseudospectral methodsconvergerapidly(exponentially)foralargeclass ofproblemsandgiveabetter costateestimatethantheLobattopseudospectralmethod.T heseadvantageshave beenshownonavarietyofexampleproblems.Ithasbeenshown thatthesemethods, however,arenotwellsuitedforsolvingproblemsthathaved iscontinuitiesinthesolution ordiscontinuitiesinthederivativesofthesolution,andp roblemsthatcontainsingular arcs. 229

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8.2FutureWork 8.2.1ConvergenceProofforGaussandRadauPseudospectralMe thod TheequivalencebetweenthetransformedKKTconditionsofthe NLPandthe discretizedformofthecontinuous-timerst-orderoptima lityconditionsprovidesaway toapproximatethecostateandtheLagrangemultipliersofa noptimalcontrolproblem. Furthermore,theequivalencebetweenthedifferentialand theimplicitintegration schemeestablishesthefactthatinusingthedifferentialf ormofapseudospectral methodtosolveanoptimalcontrolproblem,thestatedynami csareactuallybeing integratednumerically.Thenumericalexamplessuggestth atthesolutionfromthe GaussandtheRadaupseudospectralmethodconvergeatanexp onentialrate.Aformal mathematicalproof,however,thatshowsthatthesolutiont othediscreteNLPconverges totheoptimalsolutionoftheoriginalcontinuous-timeopt imalcontrolproblemisstill missing.8.2.2CostateEstimationUsingLobattoPseudospectralMetho d BoththeGaussandtheRadauschemesprovideanaccuratetrans formationto obtaincostateestimatefromtheKKTmultipliersofthediscre tenonlinearprogramming problem.TheLPMcostateapproximation,however,isfoundto haveanerrorthat oscillatesabouttheexactsolution,andthiserrorisshown byexampletobeduetothe nullspaceintheLobattodiscretecostatesystem.Forther stexampleconsideredin thisdissertation,itwaspossibletondthelinearfactorb ywhichthenullspacecanbe combinedtothesolutionobtainedfromtheNLPtoobtainanac curatecostateestimate. Foramulti-dimensionalproblem,thisapproach,however,i snottrivial.Exploringthenull spaceassociatedwiththeLobattodiscretecostatedynamic sandobtainingaccurate costateestimateusingtheLobattopseudospectralmethodi saninterestingresearch problem. 230

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BIOGRAPHICALSKETCH DivyaGargwasbornin1984inGhaziabad,UttarPradesh,India .Shereceived herBachelorofTechnologydegreeinMechanicalEngineeringf romtheMalaviya NationalInstituteofTechnology,Jaipur,India,inMay200 7.Shethenreceivedher MasterofSciencedegreeinMechanicalEngineeringinDecembe r2008,andherDoctor ofPhilosophydegreeinMechanicalEngineeringinAugust2011 ,attheUniversityof Florida.Herresearchinterestsincludelinearalgebra,nu mericaloptimization,optimal controltheory,andnumericalapproximationstothesoluti onofoptimalcontrolproblems. 238