<%BANNER%>

Costate Estimation in Optimal Control Problems Using Orthogonal Collocation at Gaussian Quadrature Points

MISSING IMAGE

Material Information

Title:
Costate Estimation in Optimal Control Problems Using Orthogonal Collocation at Gaussian Quadrature Points
Physical Description:
1 online resource (197 p.)
Language:
english
Creator:
Francolin, Camila C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Rao, Anil
Committee Members:
Dixon, Warren E
Lind Jr, Richard C
Hager, William Ward

Subjects

Subjects / Keywords:
control -- costate -- estimation -- gaussian -- optimal -- quaerature
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Costate estimation is an important step in the numerical solution of optimal control problems as it provides a reliable way of verifying the optimality of the approximated solution. In this dissertation the problem of estimating the costate in an optimal control problem using orthogonal collocation at Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) points is presented. First, methods are presented for estimating the costate using orthogonal collocation at the LG or LGR points when the dynamic constraints of the optimal control problem are formulated in integral form. Specifically, transformations are derived that relate the Lagrange multipliers of the integral forms of the LG and LGR collocation methods to the costate of the original optimal control problem. These transformations are derived by writing the original continuous-time optimal control problem in integral form. A new continuous-time dual variable called the integral costate is then introduced, where the integral costate is the Lagrange multiplier of the integral dynamic constraint. The first-order optimality conditions of the integral form of the optimal control problem are derived in terms of the integral costate. The integral form of the optimal control problem is then discretized using the integral LG and LGR collocation methods and relationship between the discrete form of the integral costate and the costate of the original differential optimal control problem are developed. It is shown that the LGR integration matrix that relates the differential costate to the integral costate is singular while the corresponding LG integration matrix is full rank. The approach developed in this research then provides a way to estimate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. Furthermore, the costate estimates presented in this research result in a set of Karuhn-Kush-Tucker conditions of the nonlinear program which are a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem both in differential and integral forms. The second part of this research focuses on state inequality path constrained optimal control problems. Problems with active state-inequality path constraints are difficult to solve due to the high-index differential-algebraic equations (DAE) that result from the constraint activity. This DAE index fluctuation in the solution domain results in possible discontinuities in the dual variables which are hard to approximate numerically. Due to these discontinuities, previous costate estimates for direct transcription methods using collocation at LG or LGR points resulted in a transformed adjoint system which was not a discrete approximation to the first-order optimality conditions in the presence of state inequality path constraints. In this research a different set of costate estimates are developed which result in a transformed adjoint system that is a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem. Specifically, a costate estimate using the methods of direct adjoining, indirect adjoining, and indirect adjoining with continuous multipliers is derived. The equivalence between the first-order optimality conditions of the finite-dimensional nonlinear program and the first-order optimality conditions of the continuous-time optimal control problem ensures convergence of the discrete problem to a local minimum which satisfies the optimality conditions of the original problem. This costate estimate can thus be used to verify the optimality of the approximated solution.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Camila C Francolin.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Rao, Anil.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045808:00001

MISSING IMAGE

Material Information

Title:
Costate Estimation in Optimal Control Problems Using Orthogonal Collocation at Gaussian Quadrature Points
Physical Description:
1 online resource (197 p.)
Language:
english
Creator:
Francolin, Camila C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Rao, Anil
Committee Members:
Dixon, Warren E
Lind Jr, Richard C
Hager, William Ward

Subjects

Subjects / Keywords:
control -- costate -- estimation -- gaussian -- optimal -- quaerature
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Costate estimation is an important step in the numerical solution of optimal control problems as it provides a reliable way of verifying the optimality of the approximated solution. In this dissertation the problem of estimating the costate in an optimal control problem using orthogonal collocation at Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) points is presented. First, methods are presented for estimating the costate using orthogonal collocation at the LG or LGR points when the dynamic constraints of the optimal control problem are formulated in integral form. Specifically, transformations are derived that relate the Lagrange multipliers of the integral forms of the LG and LGR collocation methods to the costate of the original optimal control problem. These transformations are derived by writing the original continuous-time optimal control problem in integral form. A new continuous-time dual variable called the integral costate is then introduced, where the integral costate is the Lagrange multiplier of the integral dynamic constraint. The first-order optimality conditions of the integral form of the optimal control problem are derived in terms of the integral costate. The integral form of the optimal control problem is then discretized using the integral LG and LGR collocation methods and relationship between the discrete form of the integral costate and the costate of the original differential optimal control problem are developed. It is shown that the LGR integration matrix that relates the differential costate to the integral costate is singular while the corresponding LG integration matrix is full rank. The approach developed in this research then provides a way to estimate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. Furthermore, the costate estimates presented in this research result in a set of Karuhn-Kush-Tucker conditions of the nonlinear program which are a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem both in differential and integral forms. The second part of this research focuses on state inequality path constrained optimal control problems. Problems with active state-inequality path constraints are difficult to solve due to the high-index differential-algebraic equations (DAE) that result from the constraint activity. This DAE index fluctuation in the solution domain results in possible discontinuities in the dual variables which are hard to approximate numerically. Due to these discontinuities, previous costate estimates for direct transcription methods using collocation at LG or LGR points resulted in a transformed adjoint system which was not a discrete approximation to the first-order optimality conditions in the presence of state inequality path constraints. In this research a different set of costate estimates are developed which result in a transformed adjoint system that is a discrete approximation of the first-order optimality conditions of the continuous-time optimal control problem. Specifically, a costate estimate using the methods of direct adjoining, indirect adjoining, and indirect adjoining with continuous multipliers is derived. The equivalence between the first-order optimality conditions of the finite-dimensional nonlinear program and the first-order optimality conditions of the continuous-time optimal control problem ensures convergence of the discrete problem to a local minimum which satisfies the optimality conditions of the original problem. This costate estimate can thus be used to verify the optimality of the approximated solution.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Camila C Francolin.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Rao, Anil.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045808:00001


This item has the following downloads:


Full Text

PAGE 1

COSTATEESTIMATIONFOROPTIMALCONTROLPROBLEMSUSING ORTHOGONALCOLLOCATIONATGAUSSIANQUADRATUREPOINTS By CAMILACLEMENTEFRANCOLIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

PAGE 2

c r 2013CamilaClementeFrancolin 2

PAGE 3

Totheoneswhosupportedme: Mam aeandCarol Andtotheoneswhoinspiredme: PapaiandNicolas 3

PAGE 4

ACKNOWLEDGMENTS GettingaPhDhasbeen,handsdown,thehardestthingIhaveac complishedthus farinmylife.ItgoeswithoutsayingIdidnoneofitbymyself ,andIowemanythanksto thepeoplewho,inmanyways,helpedmethroughthisprocess. Forobviousreasons, noneofthiswouldhavebeenpossiblewithoutmyfacultyadvi sor,Dr.AnilRao.Ithank youforalwaysholdingmetothehigheststandards.Youweret oughwhenyouneeded tobe,andencouragingtherestofthetime.Youhelpedmeevol veandgaincondence asaresearcher,andIamamuchbetterscientistforit.Iwoul dalsoliketothankmy committeemembersforhelpingmethroughthedoctoralproce ss:Dr.WilliamHager, Dr.RichardLind,andDr.WarrenDixon.Iamdeeplygratefult oDr.WilliamHagerfor patientlytakingthetimetomeetanddiscussmyresearch,an dkindlycorrectingme whenIwaswrong. IwouldalsoliketothanktheOfceofNavalResearch,especi allyDr.Maria MedeirosandDr.DavidDrumheller,fortheirnancialsuppo rt.ThetimeIspentworking attheNavalUnderseaWarfareCenterwashighlyinstructive .ThankyoutoChrisDuarte andGerryMartelfortheirmentorshipduringthetimeIspent there. APhDisnotjustaboutacademicgrowth,butalsoaboutperson aldevelopment.I havealotofpeopletothankforthelatterpartofmyformatio n.Irstthankthosewho, throughtheirloveofscienceanddiscovery,inspiredmetog odownthispath,asitis notaneasyonetopick.MyrstinspirationwasmyDad,whomIw atchedgothrough thisprocesssomanyyearsago.Hewasinaforeigncountrywit htwosmallchildren andhestillmadeitlookeasy.Hetaughtmebyexampleatavery youngagetoalways questionthings,andtoneverloseasenseofcuriosity;itis stillthissensecuriositythat propelsmetokeeplearning.Nick,youweremysecondsourceo finspiration.Iwould neverhavehadthecouragetotakethispathifyouhadn'tbeen there,forgingaheadwith nofearandshowingmetheway.Youshowedmethisprocesswasp ossiblebytaking onestepatatime,andIthankyouforyouinspiringmewithyou renthusiasmandlove 4

PAGE 5

ofscience.Next,Iwanttothankallthosewhoencouragedmea ndhelpedmetokeep goingwhenquittingwouldhavebeensomucheasier.Greg,you alwaysbelievedinme, evenwhenIdidn't.Ed,thankyouforkeepingmemotivatedtow ardgraduationbyasking mewhenIwouldbedoneeachandeverytimeyousawme.I'msoluc kytohaveyou inmylife.Thomas,thankyouformovingmyfutonalltheeight timesittooktogetme graduated.ThankstoallmyfamilyinBrazil:tiaTerezinha, Thais,Luluca,Erika,Celma, JoaoMatheus,MariaAlice;everytimeIwenttoseeyouoverth esummersIcameback renewed.Maxie,youwerealwayssittingatmyfeetthroughth eupsanddownsofthe researchprocess.Andalwayshadhealinglickswhenthingsd idn'tgoaccordingtoplan (which,asitturnedout,happenedquitealot). IwouldliketothankallthemembersofVDOL.EspeciallyChri sandDivyainthe beginning,andBegum,Matt,andBriantowardtheend(yesBri an,youareanhonorary VDOLmember).Matt,howcouldIeverthankyouforallyoursup port.Thankyoufor patientlylisteningwhenIoverindulgedintellingyouever ygorydetailofmyresearch, kindlytellingmeitwouldbeokaywhenmyresultsdidn'tturn outasexpected,and sharinginmyexcitementwhenitnallydidturnoutasexpect ed.Thankyouforputting afenceupinmybackyardjustsoIcouldwritethisdissertati onwithnodistractionsfrom mydog.You'remylifeboat. Finally,tomyMomandmySister.GettingaPhDisjustoneofth ethingsthatIcould neverhaveaccomplishedwithoutyoubothbymyside.Yousupp ortedmeemotionally, nancially,andanyotherwaythatispossible.Mom,thankyo uforeachandeverytime youhelpedmemove,cleanedmyhouse,orranmyerrandsjustso Icouldhavemore timetonishapieceofwork.Ihopetoalwaysmakeyouproud.C arol,youshowed methatitwaspossibletosucceed,anditwasokaytofail,bec ausetherstfollowsthe latter.YouarealwaystherewhenIneedsomeonetotalkto(or whenIhavesomeoneto sue).Thankyou. 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................9 LISTOFFIGURES .....................................10 ABSTRACT .........................................13 CHAPTER 1INTRODUCTION ...................................15 2MATHEMATICALBACKGROUND .........................25 2.1Continuous-TimeBolzaOptimalControlProblem ..............25 2.1.1First-OrderOptimalityConditions ...................26 2.1.2First-OrderOptimalityConditionsofIntegralFormu lation ......27 2.1.3ControlInequalityPathConstraints ..................29 2.1.4StateInequalityPathConstraints ...................31 2.2Differential-AlgebraicEquations ........................32 2.2.1Index-ReductionforDifferential-AlgebraicEquati ons ........34 2.2.2SolutionsofHigh-IndexDifferential-AlgebraicEqu ations ......37 2.3StateInequalityPathConstrainedOptimalControlProb lems .......40 2.3.1IndirectAdjoining ............................41 2.3.2DirectAdjoining .............................45 2.3.3IndirectAdjoiningWithContinuousMultipliers ............47 2.4NumericalPropertiesofOrthogonalCollocationMethod s .........49 2.4.1FunctionApproximationandInterpolation ..............49 2.4.1.1FamilyofLegendre-Gausspoints .............51 2.4.2NumericalIntegration ..........................54 2.4.2.1Low-orderintegrators ....................54 2.4.2.2Gaussianquadrature ....................57 2.5OrthogonalCollocationfortheSolutionofOptimalCont rolProblems ...59 2.5.1GlobalCollocationatLGPoints ....................61 2.5.2GlobalCollocationatLGRPoints ...................63 2.5.3GlobalCollocationatFlippedLGRPoints ..............65 2.5.4Variable-OrderCollocationatLGPoints ...............66 2.5.5Variable-OrderCollocationatLGRPoints ..............68 2.5.6Variable-OrderCollocationatFlippedLGRPoints ..........70 3COSTATEESTIMATIONUSINGTHEINTEGRALFORMULATION .......72 3.1Continuous-TimeBolzaOptimalControlProblem ..............73 3.1.1DifferentialandIntegralFormsofOptimalControlPr oblem .....74 3.1.2First-OrderOptimalityConditionsofDifferentiala ndIntegralForms 75 6

PAGE 7

3.2CostateEstimationUsingIntegralLegendre-GaussColl ocation ......76 3.2.1DifferentialFormofLGCollocation ..................76 3.2.2KKTConditionsUsingDifferentialLGCollocation ..........78 3.2.3IntegralFormofLGCollocation ....................80 3.2.4KKTConditionsUsingIntegralLGCollocation ............82 3.2.5ARelationshipBetweenIntegralandDifferentialCos tateEstimates 85 3.3CostateEstimationUsingIntegralLegendre-Gauss-Rad auCollocation ..86 3.3.1DifferentialFormofLGRCollocation .................87 3.3.2KKTConditionsUsingDifferentialLGRCollocation .........88 3.3.3IntegralFormofLGRCollocation ...................91 3.3.4KKTConditionsUsingIntegralLGRCollocation ...........93 3.3.5ARelationshipBetweenIntegralandDifferentialCos tateEstimates 97 3.4Discussion ...................................97 3.5ConcludingRemarks ..............................98 4MOTIVATIONFORNEWCOSTATEESTIMATE ..................100 4.1Continuous-TimeBolzaOptimalControlProblem ..............101 4.1.1First-OrderOptimalityConditionsofContinuousPro blem ......102 4.2Variable-OrderCollocationatLegendre-GaussPoints ...........103 4.2.1KKTConditionsofVariable-OrderLGCollocationMeth od .....104 4.2.2CostateEstimateandTransformedAdjointSystem .........105 4.3Variable-OrderCollocationatFlippedLegendre-Gauss -RadauPoints ...108 4.3.1KKTConditionsofVariable-OrderFlippedLGRColloca tionMethod 109 4.3.2CostateEstimateandTransformedAdjointSystem .........110 4.4Discussion ...................................113 5COSTATEESTIMATIONFORSTATECONSTRAINEDPROBLEMS ......116 5.1Continuous-TimeState-ConstrainedOptimalControlPr oblem .......117 5.1.1First-OrderOptimalityConditions ...................119 5.2CostateEstimationUsingLegendre-GaussCollocation ...........120 5.2.1Variable-OrderCollocationatFlippedLGPoints ...........120 5.2.2CostateEstimateandTransformedAdjointSystem .........122 5.3CostateEstimationUsingFlippedLegendre-Gauss-Rada uCollocation ..127 5.3.1Variable-OrderCollocationatFlippedLGRPoints ..........127 5.3.2CostateEstimateandTransformedAdjointSystem .........129 5.4Discussion ...................................137 6EXAMPLES ......................................139 6.1Example1:MayerOptimalControlProblem .................140 6.1.1SolutionUsingCollocationatLGPoints ...............140 6.1.2SolutionUsingCollocationatLGRPoints ..............144 6.2Example2:LagrangeOptimalControlProblem ...............147 6.2.1SolutionUsingCollocationatLGPoints ...............148 6.2.2SolutionUsingCollocationatLGRPoints ..............151 7

PAGE 8

6.3Example3:First-OrderStateInequalityPathConstrain tProblem .....154 6.3.1SolutionUsingCollocationatLGPoints ...............154 6.3.1.1Previouslyderivedcostateestimate ............157 6.3.1.2Indirectadjoiningwithcontinuousmultipliers .......160 6.3.2SolutionUsingCollocationatFlippedLGRPoints ..........162 6.3.2.1Previouslyderivedcostateestimate ............162 6.3.2.2Indirectadjoiningwithcontinuousmultipliers .......164 6.4Example4:Second-OrderStateInequalityPathConstrai ntExample ...168 6.4.1SolutionUsingCollocationatLGPoints ...............169 6.4.1.1Previouslyderivedcostateestimate ............172 6.4.1.2Indirectadjoiningwithcontinuousmultipliers .......175 6.4.2SolutionUsingCollocationatFlippedLGRPoints ..........178 6.4.2.1Previouslyderivedcostateestimate ............181 6.4.2.2Indirectadjoiningwithcontinuousmultipliers .......184 7CONCLUSIONS ...................................187 REFERENCES .......................................191 BIOGRAPHICALSKETCH ................................197 8

PAGE 9

LISTOFTABLES Table page 2-1Absolutemaximumerrorin v ( t ) and u ( t ) forproblems A and B ........40 9

PAGE 10

LISTOFFIGURES Figure page 2-1ErrorinthesolutionofDAEsystem .........................39 2-2Functionapproximationusinguniformlyspacedpoints ..............52 2-3DistributionofGaussianquadraturepoints .....................53 2-4FunctionapproximationusingLGpoints ......................55 2-5FunctionapproximationusingLGRpoints .....................56 2-6Errorassociatedwithfunctionapproximationusinguni form,LG,andLGRpoints 57 2-7ApproximationofintegralusingTrapezoidrule. ..................58 2-8ErrorinapproximationofintegralusingTrapezoidrule asafunctionofN ....58 2-9ErrorinapproximationofintegralusingGaussianquadr atureasafunctionofN 60 2-10DistributionofLGpointsforglobalcollocation ...................62 2-11DistributionofLGRpointsforglobalcollocation ..................64 2-12DistributionofippedLGRpointsforglobalcollocati on ..............65 2-13DistributionofLGpointsforvariable-ordercollocat ion ..............67 2-14DistributionofLGRpointsforvariable-ordercolloca tion .............69 2-15DistributionofippedLGRpointsforvariable-orderc ollocation .........71 4-1RelationshipBetweentheDirectandIndirectMethods ..............115 5-1EquivalenceoftheDirectandIndirectMethods ..................138 6-1PrimalsolutionforExample1obtainedusingintegralco llocationatLGpoints. .141 6-2CostatesolutionsforExample1obtainedusingcollocat ionatLGpoints. ....142 6-3CostateerrorsforExample1obtainedusingcollocation atLGpoints. .....143 6-4PrimalsolutionforExample1obtainedusingintegralco llocationatLGRpoints. 145 6-5CostatesolutionsforExample1obtainedusingcollocat ionatLGRpoints. ...146 6-6CostateerrorsforExample1obtainedusingcollocation atLGRpoints. ....146 6-7StateandcontrolforExample2obtainedusingintegralL Gcollocation. ....148 6-8CostatesolutionsforExample2obtainedusingcollocat ionatLGpoints. ....149 10

PAGE 11

6-9CostateerrorsforExample2obtainedusingcollocation atLGpoints. .....150 6-10StateandcontrolforExample2obtainedusingintegral LGR. ..........152 6-11CostatesolutionsforExample2obtainedusingcolloca tionatLGRpoints. ...153 6-12CostateerrorsforExample2obtainedusingcollocatio natLGRpoints. ....153 6-13PrimalsolutionforExample3obtainedusingcollocati onatLGpoints. .....155 6-14ErrorsinstateandcontrolforExample3obtainedusing LGcollocation. ....156 6-15CostateestimateasderivedbyRef.[1]forExample3. ..............158 6-16CostateerrorsforestimatederivedinRef.[1]forExam ple3. ...........159 6-17DualvariablesforExample3obtainedusingcollocatio natLGpoints. .....161 6-18CostateerrorsforExample3obtainedusingcollocatio natLGpoints. .....161 6-19PrimalsolutionforExample3obtainedusingcollocati onatLGRpoints. ....163 6-20ErrorsforExample3obtainedusingcollocationatLGRp oints. .........164 6-21CostateEstimateasderivedbyRef.[1]forExample3. ..............165 6-22CostateerrorsforestimatederivedinRef.[1]forExam ple3. ...........166 6-23CostateestimateforExample3obtainedusingcollocat ionatLGRpoints. ...167 6-24CostateerrorsforExample3obtainedusingcollocatio natLGRpoints. ....167 6-25PrimalsolutionforExample4obtainedusingLGcolloca tion. ..........170 6-26StateandcontrolerrorsforExample4usingcollocatio natLGpoints. .....171 6-27CostateEstimateasderivedbyRef.[1]forExample4. ..............173 6-28CostateerrorsforestimatederivedinRef.[1]forExam ple4. ...........174 6-29CostateEstimateforExample4obtainedusingcollocat ionatLGpoints. ....176 6-30CostateerrorsforExample4obtainedusingcollocatio natLGpoints. .....177 6-31PrimalsolutionforExample4obtainedusingLGRcolloc ation. .........179 6-32StateandcontrolerrorsforExample4usingcollocatio natLGRpoints. ....180 6-33CostateestimateasderivedbyRef.[1]forExample4. ..............182 6-34CostateErrorsusingestimatederivedinRef.[1]forEx ample4. .........183 6-35CostateEstimateforExample4obtainedusingcollocat ionatLGRpoints. ...185 11

PAGE 12

6-36CostateerrorsforExample4obtainedusingcollocatio natLGRpoints. ....186 12

PAGE 13

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy COSTATEESTIMATIONFOROPTIMALCONTROLPROBLEMSUSING ORTHOGONALCOLLOCATIONATGAUSSIANQUADRATUREPOINTS By CamilaClementeFrancolin August2013 Chair:AnilV.RaoMajor:AerospaceEngineering Computingthecostateinanoptimalcontrolproblemisimpor tantforverifying theoptimalityofthesolutionandperformingsensitivitya nalysis.Thisdissertationis concernedwiththeproblemofestimatingthecostateinanop timalcontrolproblem usingorthogonalcollocationatLegendre-Gauss(LG)andLe gendre-Gauss-Radau (LGR)points.First,methodsarepresentedforestimatingt hecostateusingorthogonal collocationattheLGorLGRpointswhenthedynamicconstrai ntsoftheoptimalcontrol problemareformulatedinintegralform.Anewcontinuous-t imedualvariablecalled the integralcostate isintroduced,wheretheintegralcostateistheLagrangemu ltiplier oftheintegraldynamicconstraint.Therst-orderoptimal ityconditionsoftheintegral formoftheoptimalcontrolproblemarederivedintermsofth eintegralcostate.The integralformoftheoptimalcontrolproblemisthendiscret izedusingtheintegralLG andLGRcollocationmethodsandrelationshipbetweenthedi screteformoftheintegral costateandthecostateoftheoriginaldifferentialoptima lcontrolproblemaredeveloped. ItisshownthattheLGRintegrationmatrixthatrelatesthed ifferentialcostatetothe integralcostateissingularwhilethecorrespondingLGint egrationmatrixisfullrank.The approachdevelopedinthisresearchthenprovidesawaytoes timatethecostateofthe originaloptimalcontrolproblemusingtheLagrangemultip liersoftheintegralformofthe LGandLGRcollocationmethods.Furthermore,thecostatees timatespresentedinthis researchresultinasetofKaruhn-Kush-Tuckerconditionso fthenonlinearprogramming 13

PAGE 14

problemwhichareadiscreteapproximationoftherst-orde roptimalityconditionsofthe continuous-timeoptimalcontrolproblembothindifferent ialandintegralforms. Thesecondpartofthisresearchfocusesonstateinequality pathconstrained optimalcontrolproblems.Problemswithactivestate-ineq ualitypathconstraintsare difculttosolveduetothehigh-indexdifferential-algeb raicequations(DAE)thatresult fromtheconstraintactivity.ThisDAEindexuctuationint hesolutiondomainresultsin possiblediscontinuitiesinthedualvariableswhichareha rdtoapproximatenumerically. Duetothesediscontinuities,previouscostateestimatesf ordirecttranscriptionmethods usingcollocationatLGorLGRpointsresultedinatransform edadjointsystem whichwas not adiscreteapproximationtotherst-orderoptimalitycond itionsinthe presenceofstateinequalitypathconstraints.Inthisrese archadifferentsetofcostate estimatesaredevelopedwhichresultinatransformedadjoi ntsystemthatisadiscrete approximationoftherst-orderoptimalityconditionsoft hecontinuous-timeoptimal controlproblem.Specically,acostateestimateusingthe methodof indirectadjoining withcontinuousmultipliers isderived.Theequivalencebetweentherst-orderoptimal ity conditionsofthenite-dimensionalnonlinearprogramand therst-orderoptimality conditionsofthecontinuous-timeoptimalcontrolproblem ensuresconvergenceofthe discreteproblemtoalocalminimumwhichsatisestheoptim alityconditionsofthe originalproblem.Thiscostateestimatecanthusbeusedtov erifytheextremalityofthe approximatedsolution. 14

PAGE 15

CHAPTER1 INTRODUCTION Manyproblemsinengineering,economics,andbiologycanbe modeledas differential-algebraicsystems.Inaddition,itisoftend esiredtooptimizetheperformance ofsuchsystems.Thegoalofanoptimalcontrolproblemistod eterminethestateand controlthatoptimizeagivenperformanceindexsubjecttoa setofdifferential-algebraic constraints.Inaerospaceengineering,optimalcontrolap plicationsincludetrajectory optimization,parameterestimation,andvehicleguidance .Asalludedtoearlier,the constraintsinanoptimalcontrolproblemincludedifferen tialequationsthatdescribethe motionofthedynamicalsystem,pathconstraintsthatdene limitsontheprocess,and eventconstraintsthatdenewaypointsthatmustbemetduri ngthemotion. Optimalcontrolproblemsthatinvolveinequalitypathcons traintsarecommonin aerospaceengineering.Suchconstraintscanbepurelyafun ctionofthecontrol(for example,controllimitssuchasmaximumallowablethrust), purelyafunctionofthe state(forexample,no-yzoneconstraints),ormoregenera llyafunctionofboththe controlandthestate(forexample,maximumheatingratecon straints).QuotingRef.[ 2 ], “Solvinganoptimalcontrolorestimationproblemisnoteas y”.Optimalcontrolproblems withinequalitypathconstraintsareparticularlychallen gingtosolvebecausetheoptimal trajectorymaycontainregionswheretheinequalityconstr aintisactive.Evenmore challengingareproblemswithinequalitypathconstraints thatarepurelyafunctionof thestate,leadingto high-index differential-algebraicequation(DAE)constraints[ 3 – 6 ]. Systemswithstateinequalitypathconstraintsofindexone orlesscangenerallybe solvednumericallyusingnumericalintegrators.Systemsw ithstateinequalitypath constraintsofindexgreaterthanone,however,posecomput ationalchallengesfor numericalintegrationmethods[ 3 ].Inthecontextofanoptimalcontrolproblem,astate inequalitypathconstrainedhigh-indexdifferential-alg ebraicsystemhaveanon-smooth stateandpossiblyadiscontinuouscostate,whileacontrol inequalityconstrained 15

PAGE 16

problemcanhaveadiscontinuousoptimalcontrol[ 7 8 ].Suchdiscontinuitiescanbe difculttoapproximateaccuratelyusingnumericalmethod s. Methodsforapproximatingsolutionstooptimalcontrolpro blemsfallintotwobroad categories:indirectanddirectmethods.Inanindirectmet hodtherst-orderoptimality conditionsarederivedusingthecalculusofvariations,re sultinginaHamiltonian boundary-valueproblem(HBVP)[ 9 ].Inthecasewhentheinequalitypathconstraints areinactiveontheoptimalsolution,theHBVPisa two-point boundaryvalueproblem. Whenthesolutiondomaincontainsactive/inactiveswitche sinstateinequalitypath constraintactivity,however,theHBVPwillhaveinteriorpointconstraints,resultingina multi-point boundaryvalueproblem[ 7 ]. Agreatdealofresearchhasbeendoneonsolvingoptimalcont rolproblemswith stateinequalitypathconstraintsusingindirectmethods[ 8 10 – 13 ].Thisresearchhas yieldedanumberofdifferentwaystoderivethenecessaryco nditionsforoptimality, eachresultinginadifferentsetofconditions.Inthemetho dof directadjoining ,thestate inequalitypathconstraintisaugmentedtotheHamiltonian ,andtherst-orderoptimality conditionsarederivedusingthecalculusofvariations.Th ismethodresultsonaset of“jumpconditions”ontheoptimalcostatewhichmustbeapp liedattheentranceand exitoftheconstrainedarc.Intheaerospaceengineeringli terature,stateinequality constraintshavehistoricallybeenhandledthroughindexreductionofthehigh-index differential-algebraicequation(DAE)systemthatresult sfromthestateconstraintactivity [ 2 ].Thenecessaryconditionsforoptimalityarederivedfrom thecalculusofvariations usinganapproachtermed indirectadjoining inwhichthestateinequalityconstraintis differentiatedbeforebeingadjoinedtotheLagrangian[ 7 ].Usingthisapproach,Ref.[ 10 ] developsasetoftangencyconditionsthatareenforcedatth eentranceofaconstrained arc,oftenleadingtodiscontinuitiesinthecostate.Theco ntrolalongtheconstrained arcisthendenedbysettingtozerothelowestderivativeof theinequalityconstraint thatisanexplicitfunctionofthecontrolvariable.Thecos tatediscontinuitiesthatarise 16

PAGE 17

fromthenecessaryconditionsforoptimalitythenbecomeaf unctionofthetangency conditions.InRef.[ 13 ]amodiedproblemisposedwheretheoriginalpathconstrai nt isaugmentedtothecostfunctionalandthetangencyconditi onsareappliedat both theentranceandexitoftheconstraintactivity.Theformul ationofRef.[ 13 ]leadstoa reductioninthedimensionofthestatespaceintheregionof activeconstraintactivity.In [ 14 ]anumericaltechniquefordealingwiththeseproblemsisde velopedusingsteepest descent. Anothertechniqueforsolvingstateinequalitypathconstr ainedoptimalcontrol problemsisthemethodof indirectadjoiningapproachwithcontinuousmultipliers [ 15 ].Inthismethod,thediscontinuityinthecostateis“subtr actedout,”leadingto asetofoptimalityconditionsthatyieldacontinuouscosta teevenififthesolution liesonaconstrainedarc[ 16 – 19 ].Reference[ 15 ]summarizesthemethodsof direct adjoining indirectadjoining ,and indirectadjoiningwithcontinuousmultipliers used inthederivationofthenecessaryconditionsforoptimalit yofastateinequalitypath constrainedoptimalcontrolproblem. Indirectmethodsareattractivebecausethesolutionofthe HBVPisanextremal andthusmustsatisfytherst-orderoptimalityconditions fromthecalculusofvariations. Consequently,asolutionobtainedusinganindirectmethod canbeaccurate.The HBVP,however,generallydoesnothaveananalyticsolution .Therefore,numerical methodsmustbeemployed.Commonnumericalapproachesfors olvingtheHBVPare shooting,multipleshooting,andcollocation[ 20 ].NumericalimplementationsofIndirect methodsposeanumberofcomputationalchallenges.First,t herst-orderoptimality conditionsareoftendifculttoderive.Second,theradius ofconvergenceoftheresulting Hamiltonianboundaryvalueproblemcanbenotablysmalldue toinstabilitiesinthe Hamiltoniandynamics[ 21 ].Asaresult,anindirectmethodoftenrequiresagoodiniti al guessforboththestateandthecostate[ 2 7 9 ].However,providinganinitialguessfor thecostateisoftendifcultbecausethecostatehasnophys icalinterpretation.Finally,in 17

PAGE 18

thecasewhentheoptimalsolutionhasconstrainedanduncon strainedarcs,itbecomes necessarytoestimatetheconstrainedarcsequence[ 2 ].Estimatingswitchesinpath constraintactivityisoftendifcultwhennoa-prioriknow ledgeofthesolutionstructureis available. Thesecondclassofnumericalmethodsinoptimalcontrolare directmethods. Differentfromindirectmethods,directmethodsparametri zethecontroland/orthestate, andthecontinuous-timeproblemisdiscretizedandtranscr ibedintoanite-dimensional nonlinearprogrammingproblem(NLP).TheresultingNLPcan thenbesolvedusing welldevelopedoptimizationsoftware[ 22 – 25 ].Directmethodshavegainedagreatdeal ofpopularityastheyavoidanumberofthepitfallsassociat edwithindirectmethods. Specically,becauseadirectmethoddirectlytranscribes theoptimalcontrolproblem intoaNLP,thelengthyderivationsoftherst-orderoptima lityconditionsareavoided. Also,directmethodsdonotrequireaninitialguessforthec ostate,andtheproblem canbemodiedrelativelyeasilywithouthavingtore-deriv etheoptimalityconditions [ 2 26 27 ].Manydirectmethods,however,arenotasaccurateasindir ectmethodsand theyrequirefurtheranalysistoverifyoptimalityonceaso lutionisachieved. Directmethodscanemployeitherasequentialorasimultane ousoptimization approach.Inasequentialapproachthecontrolisparametri zedandthedynamics areintegratedoverthetrajectorydomain.Oneexampleofas equentialoptimization methodisthedirectshootingmethod[ 28 – 30 ].Inadirectshootingmethodthe controlisparametrizedandthedynamicsareintegratedusi ngnumericalintegration methods.Directshootingmethodsareusefulwhenthecontro lcanbeparametrized usingfewparameters,keepingtheproblemsizesmall.Asthe numberofvariables neededtoparametrizethecontrolincreases,however,conv ergencetoasolution usingdirectshootingmethodsbecomesdifcult.Directmul tiple-shootingmethods improveconvergencebysubdividingthesolutiondomainint omultipleintervals[ 28 ].The shootingmethodisthenappliedineachinterval,andcontin uityofthestateisenforced 18

PAGE 19

attheintervalboundaries.Multiple-shootingmethodshav ebetterconvergencethan shootingmethodsbecausetheintegrationofthestatedynam icsisdoneovershorter intervals.Bothdirectshootinganddirectmultipleshooti ngmethods,however,arenot computationallyefcientduetothesequentialnumericali ntegrationtechniqueusedto integratethedynamics.Furthermore,convergencestillde pendsona-prioriknowledge oftheconstrainedandunconstrainedarcsequence. Aparticulardirectmethodsknownasacollocationmethod,e mployasimultaneous optimizationapproach[ 2 30 – 36 ].Collocationmethodsparametrizeboththecontroland thestate,andthedifferential-algebraicequationsareen forcedatasetofdiscretepoints inthedomain[ 2 26 37 ].Directcollocationmethodsareattractivebecausetheyr equire noaprioriknowledgeofthesolutionstructure[ 38 ].Furthermore,directcollocation methodsarelesssensitivetotheinitialguessthanthesequ entialapproachofshooting methods[ 2 ].Well-knownsoftwareimplementationsofdirectcollocat ionmethodsinclude SOCS DIDO DIRCOL ,and GPOPS [ 39 – 42 ]. Directcollocationmethodscanemploylocal h -methodcollocation,orglobal p -methodcollocation.OftentheclassofRunge-Kuttamethod sisusedtocollocate andintegratethesystemdynamics[ 2 33 43 – 45 ].Runge-Kuttamethodsareusually employedas h -methodsinwhichthesolutiondomainissubdividedintoman yintervals andaxedlow-degreeapproximationisusedineachinterval .Thistypeofscheme iscomputationallyefcientasithasasparsestructuretha tcanbeexploitedbyNLP solvers[ 46 ].Convergenceofthenumericaldiscretizationusing h -methodsisthen achievedbyincreasingthenumberofintervalsinthedomain .Duetothepolynomial convergencerateofthiskindofscheme,however, h -methodscanleadtoextremely largeNLP's[ 45 47 48 ]. Incontrasttolocal h -methods,global p -methodsuseasinglepolynomialto approximatethestateovertheentiredomain[ 26 27 49 ].Convergenceina p -method isthenobtainedbyincreasingthedegreeoftheapproximati ngpolynomial.A p -method 19

PAGE 20

hastheadvantagethatitconvergesexponentiallyforprobl emsforproblemswhose solutionsaresmooth.Inthecasewhenthesolutionisnotsmo oth(asoftenhappensin thepresenceofactiveinequalityconstraints)theconverg encerateissignicantlyslower. Furthermore,theNLParisingfroma p -methodislesssparsethantheNLParisingfrom an h -method. Thisresearchwillemployan hp -methodusingcollocationattheLGandLGRpoints [ 50 51 ].Inan hp method,orvariable-ordermethod,thesolutiondomainisdi vided intoamesh,andthedegreeoftheapproximatingpolynomial( thatis,thenumberofLG orLGRcollocationpoints)ineachintervalisallowedtovar y.Usingan hp -methoditis possibletodividetheproblemintointervalssuchthatthes olutionineachintervalis smooth.Thusconvergenceisachievedbyincreasingthedegr eeoftheapproximating polynomialineachinterval.Inthismanneritispossibleto achieveahighaccuracy solutionsolutionwhilekeepingtheNLPsmallerthanwhatmi ghtbepossibleusingan h -method. Overthelastdecade,oneclassofdirectcollocationmethod swhichhasrisen toprominenceinthenumericalsolutionofoptimalcontrolp roblemsistheclassof orthogonalcollocationmethods[ 26 27 34 – 36 42 49 52 – 65 ].Orthogonalcollocation methodsparametrizethestateusingglobalpolynomialsand collocatethedifferential-algebraic equationsusingnodesobtainedfromaGaussianquadrature. Thethreemostcommonly usedsetsofcollocationpointsare Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR),and Legendre-Gauss-Lobatto (LGL)points.Thesethreesetsofpointsare obtainedfromtherootsofaLegendrepolynomialand/orline arcombinationsofa Legendrepolynomialanditsderivatives.Allthreesetsofp ointsaredenedonthe domain [ 1,1] ,butdiffersignicantlyinthattheLGpointsinclude neither ofthe endpoints,theLGRpointsinclude one oftheendpoints,andtheLGLpointsinclude both oftheendpoints.Inaddition,theLGRpointsareasymmetric relativetothe originandarenotuniqueinthattheycanbedenedusingeith ertheinitialpointor 20

PAGE 21

theterminalpoint.AlthoughcollocationattheLGLpointsp rovidesstateandcontrol approximationsattheendpoints,itwasshownbyRefs.[ 64 65 ]thatthecontroland costateapproximationsusingLGLpointstendstobeinnacur ateduetoarank-decient differentiationmatrix.FurthermoreRef.[ 36 ]alsoshowsthatusingcollocationatthe LGandLGRpointsyieldsahighlyaccurateapproximationtot heoptimalstate,control, andcostate.BecausecollocationattheLGandLGRpointspro videsimilaraccuracy whereascollocationatLGLpointscanprovideerroneoussol utions,thisresearchwill focusonusingcollocationattheLGandLGRpoints. Whenapproximatingthesolutiontoanoptimalcontrolprobl emusinganynumerical method,itisimportanttoanalyzethesolutioninanattempt toverifytheconvergenceof thediscreteproblemtoalocalminimaofthecontinuous-tim eproblem[ 4 5 66 67 ].One keyadvantageofanorthogonalcollocationmethodistheele ganttransformationsof theKKTconditionsoftheNLPtotherst-orderoptimalityco nditionsderivedanalytically fromthecalculusofvariations[ 36 64 65 68 ].Suchtransformationshavepreviously beenderivedforoptimalcontrolproblemswithnoactivesta teinequalitypathconstraints andwhenthedynamicconstraintsareformulatedintheirdif ferentialform.When available,suchtransformationsshowthattherst-ordero ptimalityconditionsofthe discreteNLPareequivalenttothediscreteformoftherstorderoptimalityconditions ofthecontinuous-timeoptimalcontrolproblemderivedfro mthecalculusofvariations. Therefore,inthisresearchagapofcostateestimationtheo ryisclosedusingcollocation atLGandLGRpointsbyderivingamappingforthecostateesti mateforthecasewhen thedynamicconstraintsareexpressedinintegralformandi nthepresenceofstate inequalitypathconstraints. WhiletheLGandLGRmethodsareequivalentregardlessofwhe thercollocation isperformedineitherdifferentialorintegralform,thedi fferentialformofeithermethod hasbeenpredominantlyused.Recently,however,morepract icalworkhasbeendone inimplementingboththedifferentialandintegralformsof LGandLGRcollocationusing 21

PAGE 22

socalledvariable-ordermethodswherethetimeintervalis partitionedintoameshand meshrenementtechniquesareusedtodetermineanappropri atemeshthatmeetsa speciedsolutionaccuracytolerance[ 69 70 ].Thisresearchindicatesstronglythattheir maybecomputationaladvantagestousingtheintegralformo fLGandLGRcollocation overthedifferentialform.Infact,themostcurrentimplem entationofLGRcollocationis theMATLABoptimalcontrolsoftware GPOPS II [ 69 ]. GPOPS II usestheintegral formofLGRcollocationasthedefaultbecauseithasbeenfou ndthroughavariety ofexamplesthattheintegralformprovidesmoreconsistent results.Moreover,the useoftheimplicitintegralformofLGandLGRcollocationis mostconsistentwiththe implementationsusedbymanyestablishedoptimalcontrols oftwarepackagessuchas SOCS [ 39 ], DIRCOL [ 41 ], OTIS [ 71 ], ICLOCS [ 72 ],and ACADO [ 73 ]. WhilethedifferentialandintegralformsoftheLGandLGRme thodsaremathematically equivalentwithregardtotheprimalvariables(thatis,the stateandcontrol),the twoformulationsproducecompletelydifferentdualvariab les.Inparticular,the relationshipbetweentheLagrangemultipliersofthecollo cationconditionsofthe dynamicconstraintsandthecostateoftheoptimalcontrolp roblemhasbeenwell documented[ 27 36 64 65 ].Ontheotherhand,thecorrespondingrelationship betweentheLagrangemultipliersassociatedwiththeinteg ralformsofLGandLGR collocationandthecostateoftheoptimalcontrolproblemh asnotbeenestablished. WhenemployingtheintegralformsofLGandLGRcollocation, however,itmaybeof interesttoeitherverifyoptimalityorperformsensitivit yanalysisinamannerconsistent withthatwhichwouldbeperformedwhenusingvariationalme thods.Insuchcasesit isusefultoobtainacostateestimatewhenusingtheintegra lformsoftheLGandLGR methods. Inthisresearchamethodsforestimatingtheoptimalcontro lcostateusingthe integralformsofLGandLGRcollocationisdeveloped.Speci cally,transformations arederivedthatrelatetheLagrangemultipliersoftheinte gralformsoftheLGand 22

PAGE 23

LGRcollocationmethodstothecostateoftheoriginaloptim alcontrolproblem.These transformationsarederivedbywritingtheoriginalcontin uous-timeoptimalcontrol probleminintegralform.Anewcontinuous-timedualvariab lecalledthe integralcostate isthenintroduced,wheretheintegralcostateistheLagran gemultiplieroftheintegral dynamicconstraint.Therst-orderoptimalityconditions oftheintegralformofthe optimalcontrolproblemarederivedintermsoftheintegral costate.Theintegralformof theoptimalcontrolproblemisthendiscretizedusingthein tegralLGandLGRcollocation methodsandtherelationshipsbetweenthediscreteformoft heintegralcostateandthe costateoftheoriginaldifferentialoptimalcontrolprobl emaredeveloped.Itisshown thattheLGRintegrationmatrixthatrelatesthedifferenti alcostatetotheintegralcostate issingularwhilethecorrespondingLGintegrationmatrixi sfullrank.Theapproach developedinthisresearchthenprovidesawaytoestimateth ecostateoftheoriginal optimalcontrolproblemusingtheLagrangemultipliersoft heintegralformoftheLGand LGRcollocationmethods. Next,inequalitypathconstrainedoptimalcontrolproblem sareanalyzed.Although previousresearchhassuccessfullyderivedahigh-accurac yestimateofthecostatefrom theKKTmultipliersoftheNLPforthecaseofaproblemwithno activestateinequality pathconstraints,Ref.[ 1 ]subsequentlyshowedthatinthecasewhenthecostateis discontinuous(asisthecaseinthepresenceofactivestate inequalitypathconstraints), thiscostateestimateleadstoasetofrst-orderoptimalit yconditionsoftheNLPthat arenotequivalenttothediscreteformofthevariationalop timalityconditions.Thislack ofequivalenceleadstoaninaccurateapproximationofthec ostate.Therefore,inthis researchthisinaccuracyisrectiedbydevelopinganewapp roachforcostateestimation usingthemethodof indirectadjoiningwithcontinuousmultipliers [ 15 19 ].Thecostate estimatederivedinthisresearchleadstoatransformedadj ointsystemwhichisa discreteapproximationoftherst-orderoptimalitycondi tionsofthecontinuous-time problem. 23

PAGE 24

Thecontributionsofthisresearchareasfollows.First,co stateestimatesare derivedusingcollocationatLegendre-GaussandLegendreGauss-Radaupointsforthe casewhenthedynamicconstraintsoftheoptimalcontrolpro blemareformulated inintegralform.Second,itisdemonstratedthatthecostat emappingderivedfor collocationattheLGandLGRpointsleadstoasetoftransfor medoptimalityconditions oftheNLPthatareshowntobeadiscreterepresentationofth enecessaryconditions foroptimalityofthecontinuous-timeproblem.Third,arel ationshipbetweentheintegral andthedifferentialformsofthecostateestimateisgivena nditisshownthatthe twosetsofoptimalityconditionsareequivalent.Fourth,a newcostateestimatefor collocationatLGandLGRpointsisderivedforproblemswith activestateinequality pathconstraints.Thiscostateestimateisshowntoleadtoa transformedadjointsystem oftheNLPwhichisadiscreteapproximationofthenecessary conditionsforoptimality ofthecontinuous-timeoptimalcontrolproblem.Finally,e xamplesarepresentedthat characterizetheaccuracyofthecostateestimatespresent edinthisresearch. 24

PAGE 25

CHAPTER2 MATHEMATICALBACKGROUND Inthischapterthemathematicalbackgroundnecessarytoun derstandthescopeof theresearchisprovided.First,ageneralcontinuous-time Bolzaoptimalcontrolproblem isdenedandtherst-orderoptimalityconditionsofthisc ontinuous-timeBolzaoptimal controlproblemarisingfromthecalculusofvariationsare derived.Second,anoverview ofmethodsforsolvingdifferential-algebraicequations( DAE)ispresented.Inparticular, itisshownthatoptimalcontrolproblemswithactivestatei nequalitypathconstraintslead tohigh-indexDAEswhichareinherentlydifculttosolveus ingnumericalmethods.A methodof index-reduction isthenpresentedtoovercomethenumericaldifcultiestha t arisefromhigh-indexDAEsystems.Third,variousmethodsa representedtoderivethe necessaryconditionsforoptimalityofstateinequalitypa thconstrainedcontinuous-time optimalcontrolproblems.Fourth,methodsfortranscribin gageneralcontinuous-time optimalcontrolproblemtoanonlinearprogram(NLP)usingo rthogonalcollocationat Legendre-GaussandLegendre-Gauss-Radaupointsaredescr ibed.Finally,inorder toexplainandlegitimizetheuseoforthogonalcollocation methodstosolveoptimal controlproblems,abriefbackgroundisprovidedinfunctio ninterpolationandnumerical integration. 2.1Continuous-TimeBolzaOptimalControlProblem Withoutlossofgenerality,considerthefollowingoptimal controlprobleminBolza form.Determinethestate, y ( t ) 2 R n ,andthecontrol, u ( t ) 2 R m ,thatminimizethecost functional J =( y ( t f ))+ Z t f t 0 g ( y ( t ), u ( t )) dt (2–1) subjecttothedynamicconstraint d y dt =_ y ( t )= f ( y ( t ), u ( t )) 2 R n (2–2) 25

PAGE 26

theboundarycondition ( y ( t 0 ))= 0 2 R q (2–3) andthestateandcontrolinequalitypathconstraint C ( y ( t ), u ( t )) 0 2 R c (2–4) ThecostfunctionalofEq.( 2–1 )consistsofa Mayer cost,whichisevaluatedpurelyat theendpointsofthedomain,anda Lagrange ,orintegral,cost.Theoptimalcontrol problemofEqs.( 2–1 )–( 2–4 )willbereferredtoasthe continuousBolzaproblem 2.1.1First-OrderOptimalityConditions Therst-ordernecessaryconditionsforanextremalsoluti onofthecontinuous Bolzaproblemcanbederivedusingthecalculusofvariation s[ 9 ].First,usingLagrange multipliers,theconstraintsoftheoptimalcontrolproble mareaugmentedtothecost functionaltogeneratetheaugmentedcostfunctional J a =( y ( t f )) > ( y ( t 0 )) (2–5) + Z t f t 0 g ( y ( t ), u ( t )) > ( t )(_ y ( t ) f ( y ( t ), u ( t ))) > ( t ) C ( y ( t ), u ( t )) dt where ,and aretheLagrangemultipliersassociated,respectively,wi ththe boundaryconditionsofEq.( 2–3 ),thedynamicconstraintsofEq.( 2–2 ),andthe inequalitypathconstraintsofEq.( 2–4 ). Next,takingtherstvariationoftheaugmentedcostwithre specttoallfree variables(i.e., y ( t ) u ( t ) , ( t ) ,and ( t ) ),weobtain J a = @ @ y ( t f ) y f > > @ @ y ( t 0 ) y 0 + Z t f t 0 @ g @ y y + @ g @ u u > (_ y f )+ > f y y + f u u y > C > C y y + C u u dt (2–6) 26

PAGE 27

Theterm > y inEq.( 2–6 )canbeintegratedbypartsasfollows Z t f t 0 > y dt = > ( t f ) y ( t f ) > ( t 0 ) y ( t 0 )+ Z t f t 0 > y dt (2–7) ApplyingtherelationshipofEq.( 2–7 )toEq.( 2–6 ),therstvariationoftheaugmented costcanthenberewrittenasafunctionoftheaugmentedHami ltonian H ( y u , )= g + > f > C (2–8) as J a = > + > @ @ y ( t 0 ) + > ( t 0 ) y ( t 0 )+ @ @ y ( t f ) > ( t f ) y ( t f ) + Z t f t 0 @ H @ y + y + @ H @ u u > (_ y f ) > C dt Anextremalsolutionwillsatisfythecondition J a =0 .Becausethevariationsofthe freevariablesarenotzero,theonlywaytoobtainanextrema lsolutionistosatisfythe followingsetofrst-orderoptimalityconditions: y = f ( y u ), 0 = ( y ( t 0 )), (2–9) > ( t 0 )= > @ @ y ( t 0 ) (2–10) > ( t f )= @ @ y ( t f ) (2–11) @ H @ u = @ g @ u + > @ f @ u > @ C @ u =0, (2–12) @ H @ y = @ g @ y + > @ f @ y > @ C @ y = > (2–13) 0 > S ( y )= 0 (2–14) 2.1.2First-OrderOptimalityConditionsofIntegralFormu lation ThecontinuousBolzaproblemgivenbyEqs.( 2–1 )–( 2–4 )canbereformulatedsuch thatthedynamicconstraintofEq.( 2–2 )arewritteninintegralform.Reformulatingthe probleminthiswaywillbeofinteresttothisresearchsotha tarelationshipbetweenthe 27

PAGE 28

Lagrangemultipliersoftheintegralformcanberelatedtot heLagrangemultipliersofthe originaldifferentialform.Inintegralform,theoptimalc ontrolproblemistodeterminethe state, y ( t ) 2 R n ,andthecontrol, u ( t ) 2 R m ,thatminimizethecostfunctional J =( y ( t f ))+ Z t f t 0 g ( y ( t ), u ( t )) dt (2–15) subjecttotheintegralconstraint y ( t )= y ( t 0 )+ Z t t 0 f ( y ( t ), u ( t )) dt (2–16) andtheboundarycondition ( y ( t 0 ))= 0 2 R q (2–17) TheoptimalcontrolproblemgivenbyEq.( 2–15 ),alongwiththeconstraintsof Eqs.( 2–16 )and( 2–17 ),willbereferredtoasthe integralBolzaProblem Therst-ordernecessaryconditionsforanextremalsoluti onoftheintegralBolza problemcanagainbederivedthroughthecalculusofvariati ons.First,theconstraintsof Eqs.( 2–16 )and( 2–17 )areaugmentedtothecostsuchthat J a =( t f )+ > t 0 + Z t f t 0 g ( y u ) p > y y ( t 0 ) Z t 0 f ( y u ) dt d (2–18) where p ( t ) and aretheLagrangemultipliersassociatedwiththedynamicco nstraints ofEq.( 2–16 )andtheboundaryconditionsofEq.( 2–17 ),respectively.Next,therst variationistakenwithrespecttoallfreevariables( y u p ,and ),suchthat J a = @ @ y ( t f ) y f > @ @ y ( t 0 ) y 0 > + Z t f t 0 @ g ( y u ) @ y y + @ g ( y u ) @ u u p > y y ( t 0 ) Z t t 0 f ( y u ) d p > y + p > y ( t 0 )+ Z t f t p > d @ f ( y u ) @ y y + Z t f t p > d @ f ( y u ) @ u u dt (2–19) 28

PAGE 29

ItisnotedthatthefollowingrelationshipwasusedinEq.( 2–19 ): Z t f t 0 q ( t ) Z t t 0 p ( ) d dt = Z t f t 0 p ( t ) Z t f t q ( ) d dt (2–20) Furthermore,notethatthevariationofthenalstateisnot independent,butdependson thevariationsoftheinitialstatesandthestateandcontro latintermediatepoints.Thus, thevariationofthenalstateisgivenas y f = y 0 + Z +1 1 @ f @ y y + @ f @ u u dt (2–21) Eq. 2–21 canbesubstitutedintoEq.( 2–19 )toobtainanexpressionfortherstvariation ofthecostwithrespecttotheindependentvariables. Anextremalsolutionwillsatisfythecondition J a =0 .Becausethevariationsofthe freevariablesarenotzero,theonlywaytoobtainanextrema lsolutionisbysatisfying thefollowingsetofrst-orderoptimalityconditions: y = y ( t 0 )+ Z t t 0 f ( y u ) dt ( y ( t 0 ))= 0 (2–22) 0 = @ g @ u + Z t f t p > d + @ y ( t f ) @ f ( y u ) @ u (2–23) p > = @ g y + Z t f t p > d + @ @ y @ f ( y u ) @ y (2–24) > @ y ( t 0 ) = Z t f t 0 p > dt + @ @ y ( t f ) (2–25) 2.1.3ControlInequalityPathConstraints Nowconsideranoptimalcontrolproblemwithanactivecontr olinequalitypath constraint.Whenacontrolconstraintisinactive,theopti malcontrolisdeterminedusing the strongform ofPontryagin'sMinimumPrinciple,givenbyEq.( 2–12 )[ 7 9 ].Whenthe inequalityconstraintisactive,however,asubsetoftheop timalcontrolisdeterminedby therelation C k ( y ( t ), u ( t ))= 0 t 2 [ t 1 t 2 ], (2–26) 29

PAGE 30

wherethesubscript k denotesthesubsetofactiveconstraintsinthetimeinterva l [ t 1 t 2 ] [ t 0 t f ] Aspecialcaseofanactivecontrolinequalityconstraintis onethatresultsina bangbang control.IfthecontrolappearslinearlyintheHamiltonian denedbyEq.( 2–8 ), thestrongformofPontryagin'sMinimumPrinciplegivenbyE q.( 2–12 )providesno informationabouttheoptimalcontrol,andthe weakform ofPontryagin'sMinimum Principlemustbeusedinstead[ 9 ].Denoting u asthecontrolthatminimizesthecost functionalofEq.( 2–1 ),bydenitionthefollowingmustholdtrue: J ( u ) J ( u ) 0, foralltrajectorieswiththeadmissiblecontrol u sufcientlycloseto u .Furthermore,for smallvariationsaroundtheoptimaltrajectory J ( u ) J ( u )= J ( u u ) 0. ItisknownthatthevariationinthecostisrelatedtotheHam iltonianby J ( u u )= Z t f t 0 @ H @ u u u dt J ( u u )= Z t f t 0 [ H ( y u + u ) H ( y u ) ] dt Therefore, H ( y u + u ) H ( y u ) 0, H ( y u + u ) H ( y u ), foralladmissiblevariationsin u .Fromthisdiscussionitcanbeconcludedthatthe optimalcontrolminimizestheHamiltoniangiveninEq.( 2–8 ).Thisoptimalityconditionis calledthe weakform ofPontryagin'sMinimumPrincipleandisgivenas u = arg min u H ( y u ). (2–27) 30

PAGE 31

Activecontrolinequalitypathconstraints may causenitediscontinuitiesinthe controlattheenteringandexitcornersoftheconstraintac tivity.Activityinthesecontrol constraints,however,onlyproducesdiscontinuitiesinth etimederivativeofthestateand costate;andingeneralthestateandcostatethemselvesare continuousacrosssuch corners.Furthermore,theHamiltonianwillalsobecontinu ousinthepresenceofthese constraints.2.1.4StateInequalityPathConstraints Ofprimaryinterestinthisresearchisthecasewhentheineq ualitypathconstraint ofEq.( 2–4 )ispurelyafunctionofthestateandtheindependentvariab le(thatis,the inequalitypathconstraintisnotafunctionofthecontrol) suchas S ( y ( t )) 0 t 2 [ t 0 t f ]. (2–28) Forthosetimeintervalswhereanextremalsolutionliesont heconstraintboundary(and contrarytothecaseofacontrolinequalitypathconstraint ),theoptimalcontrolcannotbe determinedbytheactiveconstraintbecauseitisnotanexpl icitfunctionofthecontrol. Whileontheconstraintboundary,theDAEsdescribingthesy stemwillthenhavethe form y ( t )= f ( y ( t ), u ( t )), (2–29) 0= S k ( y ( t )), t 2 [ t 1 t 2 ], (2–30) where k denotestheactiveconstraintinthetimeinterval [ t 1 t 2 ] [ t 0 t f ] .Equation( 2–30 ) isanalgebraicconstraintthat,whensatised,removesade greeoffreedomfromthe differentialequationsdenedbyEq.( 2–29 ).Removalofthisdegreeoffreedomresults inwhattheDAEliteraturereferstoasa high-index differential-algebraicequation.In general,high-indexDAEaredifculttosolvenumerically. Methodsforsolvinghigh-index DAEswillbediscussedinthefollowingsection. 31

PAGE 32

Activestateinequalitypathconstraintwillcausetrouble notonlywhensolvingan optimalcontrolproblemnumerically,butalsowhensolving anoptimalcontrolproblem analytically.Theconstraintactivityintroducesadditio nalunknownsthatcannotbe determinedbyapplyingtherst-orderoptimalityconditio nsofEqs.( 2–9 )–( 2–13 ). Specically,theadditionalunknownsincludethetimesoft heentranceandexitofthe constrainedarcandthepathconstraintmultiplier ( t ) .Therefore,whensolvingthese problemanalytically,therst-orderoptimalityconditio nsderivedintheprevioussection mustbemodiedtoaccountfortheconstrainedarcs.Actives tateinequalitypath constraintswilloftenproducediscontinuitiesinthecost ateandtheHamiltonianatthe entranceand/orexitoftheconstraintactivity.Furthermo re,itisoftenthecasethatthe timederivativeofthestatewillbediscontinuousatthecor nersofaconstrainedarc.The stateandcontrol,however,willgenerallynotbediscontin uouseveninthepresenceof activestateinequalitypathconstraints. 2.2Differential-AlgebraicEquations ThenumericalsolutionofDAEscanbefarmorecomplicatedth anthenumerical solutionofordinarydifferentialequations(ODEs).Theac curacyofanumerical methodforthesolutionofaDAEdependsuponontheDAE'ssolv ability,index,and theconsistencyofinitialconditions[ 3 4 ].ThemostgeneralrepresentationofaDAEis giveninthenonlinearimplicitform F (_ y ( t ), y ( t ), u ( t ))= 0 (2–31) TheDAEissaidtobe solvable ifafamilyofuniquesolutionsexistslocally.Furthermore theindexoftheDAEisdenedastheminimumnumberoftimesth atallorsubsetsof theDAEgivenbyEq.( 2–31 )needtobedifferentiatedwithrespecttotimeinorderto determine y ( t ) asacontinuousfunctionofthestateandcontrol.Itisnoted thattheDAE indexcanvaryalongasolutiontrajectoryofanonlinearDAE .Finally,theconsistency 32

PAGE 33

ofinitialconditionsforaDAEsystemisdenedbyasetofini tialconditions( y 0 y 0 )that satisestheextendedsystem( 2–31 )attime t 0 InordertobetterunderstandthedifferencebetweenaDAEan danODE,suppose theDAEgivenbyEq.( 2–31 )hastheform F (_ y y u )= 0 = A ( t ) _y + B ( t ) y + D ( t ) u + e ( t ). (2–32) TheJacobianofEq.( 2–32 )isdenedas @ F @ y = A ( t ). (2–33) Ifthematrix A ( t ) isfullrank,itispossibletosolveEq.( 2–32 )forthestateas [ A ( t )] 1 0 =[ A ( t )] 1 ( A ( t ) _y + B ( t ) y + D ( t ) u + e ( t ) ) y = [ A ( t )] 1 B ( t ) y [ A ( t )] 1 D ( t ) u [ A ( t )] 1 e ( t ), (2–34) Equation( 2–34 )isanordinarydifferentialequation.Therefore,ingener al,ifthesystem JacobiangiveninEq.( 2–33 )isinvertible,Eq.( 2–31 )canbetransformedintoanODEof theform y = f ( y u ) .If,however,theJacobianofEq.( 2–33 )issingular,thenEq.( 2–31 ) isadifferential-algebraicsystemwhichcanbewrittenint hesemi-explicitform y ( t )= f ( y ( t ), u ( t )), (2–35) 0= C ( y ( t ), u ( t )). (2–36) Furthermore,whentheDAEsystemisinthesemi-explicitfor mofEqs.( 2–35 )–( 2–36 ) andthematrix G = @ C @ u issingular,thenthesystemissaidtobea high-indexDAEsystem OnewaytounderstandthesolutionofaDAEsystem,andwhyhig h-indexsystems canbeproblematic,isbyviewingthealgebraicEq.( 2–36 )asawayof“eliminating”the controlvariablesuchthatastandardODEintegrationmetho dcanbeusedtoobtain 33

PAGE 34

anumericalsolution.Inthatcaseonedegreeoffreedomisre movedfromtheODEs foreachalgebraicequationthatissatised.Forinstance, usingNewton'smethod, Eq.( 2–36 )canbesolvediterativelyforthecontrol, u ,as u k = u k G 1 C (2–37) where u k isthenewapproximationatthe k th iterativestep.Theresultingcontrolcould thenbesubstitutedintotheODEofEq.( 2–35 )suchthat y ( t )= f ( y u ). ThislastODEcouldthenbesolvedbyemployingavailablenum ericalODEsolvers. Althoughthisapproachwouldbetimeconsumingandisnotapr acticalwayofobtaining asolution,itdoesillustratethecomputationalchallenge sassociatedwithhigh-index constraints.Namely,if C 1 u isrankdecientthennotonlycantheoperationgivenby Eq.( 2–37 )notbeperformed,butthealgebraicconstraintswillalson otuniquelyspecify allofthedegreesoffreedomofthesystem.2.2.1Index-ReductionforDifferential-AlgebraicEquati ons DifferentialAlgebraicEquationsofindexatmostonecange nerallybesolved numericallyusingmethodsdevelopedforthesolutionofODE s.Forsystemsofindex higherthanone,however,suchmethodsmayhavepoorconverg ence,mayconverge tothewrongsolution,ormaynotconvergeatall[ 3 ].Twogeneralapproachesaccepted intheDAEliteratureexistforobtainingnumericalsolutio nstothesetypesofhigh-index systems.Therstisthroughtheuseofpresentlyavailablen umericalmethodsand codesthataremodiedODEsolversdesignedspecicallyfor high-indexDAEsystems suchasbackward-differentiationformulas(BDF).Theseco ndisthroughindexreduction ofthesystembysymbolicmanipulationoftheDAEequations. Inthescopeofthisresearch,itisdesiredtonotonlyndanu mericalsolution totheDAEsystembutalsoperformtheoptimizationdescribe dbyEqs.( 2–1 )–( 2–4 ). 34

PAGE 35

Thisoptimizationisdonebytranscribingtheoptimalcontr olproblemintoanonlinear programmingproblem(NLP).Therefore,specializedcodesf orsolvinghigh-index systemsarenotdesirableastheywouldresultinacomputati onallyinefcientNLP structure.Furthermore,thesolutionstructureoftheresu ltingoptimalcontrolproblem willoftenconsistofbothconstrainedandunconstrainedar cs,makingtheDAEindex uctuatethroughoutthesolution.Itisdesirable,however ,tohaveoneDAEsolverfor bothconstrainedandunconstrainedsegmentsofthetraject ory.Forthesereasons,the methodofindexreductionmightbepreferredoverspecializ edapproachesforsolving thehigh-indexDAEthatresultfromstateinequalityconstr ainedproblems. Themethodofindexreductionisformulatedasfollows.Give ntheDAE y ( t )= f ( y ( t ), u ( t )), (2–38) 0= S ( y ( t )), (2–39) itisclearthatthematrix G = @ S @ u isrankdecient.ThereforethesystemgivenbyEqs.( 2–38 )–( 2–39 )isahigh-indexDAE system.Nowif S = 0 ,thenitmustalsobetruethatallitstimederivativesareze ro. Therefore,Eq.( 2–39 )canbedifferentiatedwithrespecttotimesuchthat 0 = @ S @ y y + @ S @ u u 0 = @ S @ y f ( y u )+ @ S @ u u (2–40) Thisdifferentiationandback-substitutionprocedurecan berepeateduntilthecontrol appearsexplicitlyinthealgebraicconstraints.If r timederivativesareneededforthe derivativeoftheconstraintwithrespecttothecontrol G ( r ) = @ r S @ u r 35

PAGE 36

tobefullrank,thentheconstraintsaresaidtohave indexr .Once G ( r ) isfullrank,the DAEsystemofEqs.( 2–38 )–( 2–39 )canberewrittenequivalentlyas y ( t )= f ( y ( t ), u ( t )), u ( t )=[ G ( r ) ] 1 @ S @ y f ( y u ) (2–41) Equation( 2–41 )isanODEsystemthatcanbesolvedusingwell-knownalgorit hms. Therefore,ingeneral,theDAEindexistheminimumnumberof timesthattheoriginal constraintsmustbedifferentiatedwithrespecttotimeino rdertoobtainanODE. Inthecontextofdynamicoptimization,indexreductionisu sedonlytothepoint wheretheoptimalcontrolcanbeexplicitlydenedbytheact ivealgebraicconstraint. Therefore,onelesstimederivativeneedbetaken,suchthat theresultingDAEisdened by y ( t )= f ( y ( t ), u ( t )), 0= d q dt q S k ( y ( t ), u ( t )), (2–42) where k denotestheactiveconstraint.Inthedynamicoptimization literature,the algebraicconstraintgiveninEq.( 2–42 )istermeda q th orderconstraint.Therefore,in general,theorderofastateinequalitypathconstraintist heminimumnumberoftimes thatconstraintmustbedifferentiatedwithrespecttotime beforethecontrolappears explicitlyintheexpression.Furthermore,the order ofaconstraintandthe index ofa DAEarerelatedbytheexpression q = r 1 .Finally,asthealgebraicconstraintsare differentiated,theintermediate q 1 timederivativesarenotdiscarded.Instead,they areevaluatedattheinitialtime t 0 andusedasasetof consistentinitialconditions forthe newDAEgivenbyEq.( 2–42 ): 2666666664 S ( y ( t )) d dt S ( y ( t )) ... d ( q 1) dt ( q 1) S ( y ( t )) 3777777775 t = t 0 = 0 (2–43) 36

PAGE 37

2.2.2GaussianQuadratureCollocationMethodforSolution sofHigh-IndexDAEs Supposethestate y fromEqs.( 2–35 )–( 2–36 )isapproximatedas y ( ) Y ( )= N +1 X i =1 Y i L i ( ), (2–44) where i ( i =1,..., N +1) arethediscretizationpointsinthedomain 2 [ 1,+1] Y i 2 R n isa rowvector ofthestateapproximatedat i ,andtheLagrangepolynomials L i ( ) aredenedas L i ( )= N +1 Y j =1 j 6 = i j i j (2–45) Itisnotedthatthedomain t 2 [ t 0 t f ] canbetransformedto 2 [ 1,+1] throughthe afnetransformation t = t f t 0 2 + t f + t 0 2 (2–46) ThetimederivativeofthestateapproximationgiveninEq.( 2–44 )isthengivenas y ( ) Y ( )= N +1 X i =1 L i ( ) Y i (2–47) Whileanysetofpoints i canbeusedassupportpointsforthestateapproximation, ithasbeenshownthatnon-uniformdiscretizationpointsob tainedfromtherootsof orthogonalpolynomialssuchasChebyshevorLegendrepolyn omialswillminimize theinterpolationerrorassociatedwithRungephenomenon. Inthisresearch,the Legendre-Gauss-Radau(LGR)pointsplustheinitialpoint 1 areusedasthesupport points.ApplyingthetimederivativeofEq.( 2–47 )atthe N LGRpoints, ( 1 ,..., N ) ,gives y ( j ) Y ( j )= N +1 X i =1 L j ( i ) Y i = N +1 X i =1 D ji Y i ,( j =1,..., N ), (2–48) where D ji ( j =1,..., N );( i =1,..., N +1) isa ( N ) ( N +1) matrixofknowncoefcients knownasthe LGRdifferentiationmatrix 37

PAGE 38

TheDAEgivenbyEqs.( 2–35 )–( 2–36 )canthenbeapproximatedbythesetof algebraicequations N +1 X i =1 D ji Y i t f t 0 2 f ( Y j U j )= 0 ,( j =1,..., N ), (2–49) C ( Y j U j )=0,( j =1,..., N ), (2–50) where U j 2 R m isa rowvector ofthecontrolapproximationat j .Eqs.( 2–49 )–( 2–50 )are asetofnonlinearequationsthatcanbesolvedfortheunknow ns Y i ,( i =1,..., N +1) U j ,( j =1,..., N ) usingknownnumericalmethods. Example: Inordertoillustratethebenetsofindexreduction,consi derthefollowing simpleexamplewhereadoubleintegratorhasitsstateconst rained: Problem A 8>>>>><>>>>>: x ( t )= v ( t ), v ( t )= u ( t ), x ( t )= ` (2–51) where ( x v ) isthestate, u isthecontroland t 2 [ 1,+1] .Becausethealgebraic constraintmustbedifferentiatedtwicewithrespecttotim ebeforethecontrolappears explicitlyintheexpression,Eq.( 2–51 )isanexampleofan index-3 DAEanda secondorder constraint.Indexreductionresultsinthesystem Problem B 8>>>>><>>>>>: x ( t )= v ( t ), x ( 1)= ` v ( t )= u ( t ), v ( 1)=0, u ( t )=0, (2–52) whichisequivalenttotheoriginalsystemgiveninEq.( 2–51 ).Theanalyticsolutionto theDAEinEq.( 2–51 )isgivenas x ( t )= ` v ( t )=0, u ( t )=0. (2–53) 38

PAGE 39

BecausetheanalyticsolutiontoEq.( 2–51 )isconstantintheentiredomain,itis reasonabletoexpectsmallerrorsinthesolutionusinglowdegreepolynomialapproximations (thatis,asmallnumberofcollocationpoints). ag 24 6 8 10 1214 16 18 -2 -4 -6-8 -10 -12 -14 -16-18 NumberofCollocationPointslog 10 AbsoluteErrorin x ( t )Problem A Problem B Figure2-1.Basetenlogarithmofthemaximumabsoluteerror in x ( t ) asafunctionof thenumberofLGRcollocationpointsforproblems A and B Bothproblems A and B ,denedbyEqs.( 2–51 )and( 2–52 ),respectively,were solvedusingthemethoddescribedbyEqs.( 2–49 )–( 2–50 ).Figure 2-1 showsthe basetenlogarithmofthemaximumabsoluteerrorinthe x ( t ) componentofthestate asafunctionofthenumberofLGRcollocationpoints.Itcanb eseenthatusingthe formulationgiveninEq.( 2–52 )resultsinmuchsmallererrorsascomparedwith theformulationinEq.( 2–51 ).Furthermore,Eq.( 2–51 )requiresalargenumberof collocationpointstoachieveareasonableaccuracytolera nceof 10 8 ,whereasthe numericalsolutionofEq.( 2–52 )hasamuchsmallererrorofapproximately 10 16 regardlessofthedegreeoftheapproximatingpolynomial.T heerrorsfoundforthe secondcomponentofthestate, v ( t ) ,andthecontrol, u ( t ) ,areshowninTable 2-1 .From theseresultsitcanbeseenthatthedisparityintheaccurac yofthesolutionbetween 39

PAGE 40

theformulationsofEqs.( 2–51 )and( 2–52 )isevenlargerforthevariablesthatare notexplicitlydenedbythealgebraicconstraintintheori ginalproblemformulation. Furthermore,itcanbeseenthatanaccuracyof O (10 5 ) canstillbeattainedwithout performingindex-reduction.Thereforeitcanbeconcluded thatindexreductiongreatly reducesthenumericalerrorwhensolvinghigh-indexDAEsys tems,butwhenno index-reductionisperformeditisstillpossibletoachiev eareasonablelevelofaccuracy whensolvinghigh-indexDAEoforderthree. Table2-1.Absolutemaximumerrorin v ( t ) and u ( t ) forproblems A and B NumberofLGRPoints v ( t ) MaxAbsoluteError u ( t ) MaxAbsoluteError Problem A Problem B Problem A Problem B 3 3.38 10 3 07.97 10 3 0 6 1.29 10 4 09.79 10 4 0 9 1.80 10 5 02.57 10 4 0 12 5.49 10 6 08.08 10 5 0 15 2.57 10 6 03.98 10 5 0 18 1.13 10 6 02.57 10 5 0 2.3StateInequalityPathConstrainedOptimalControlProb lems FromthediscussionofSection 2.2 ,itisclearthatoptimalcontrolproblemswith activestateinequalityconstraintsleadtohigh-indexDAE sandaredifculttosolve numerically.Furthermore,itwasshownthatwhensolvingth eseproblemsanalytically therst-orderoptimalityconditionsofSection 2.1 mustbemodiedinordertoaccount fortheextraunknownsintroducedbytheconstraintactivit y.Manymethodsareavailable intheliteratureforderivingthenecessaryconditionsfor optimalityofastate-inequality constrainedproblem.Ofthesemethods,threeofthemwillbe discussedhere[ 7 8 15 ]. Therstmethodiscalled indirectadjoining [ 15 ].Inindirectadjoiningtheindex-reduction ofthehigh-indexsystemofDAEsistakenintoaccount,asrs tderivedby[ 7 ].Indirect adjoiningresultsinacostatethathasdiscontinuitiesatt heentranceoftheconstraint activity.Thesecondmethodpresentedforsolvingstateine qualitypathconstrained problemsiscalled directadjoining [ 15 ].Indirectadjoining,thestateinequalitypath constraintisdirectlyaugmentedtothecostandtherst-or deroptimalityconditionsare 40

PAGE 41

derived.Usingdirectadjoiningthecostatemaybedisconti nuousattheentranceor exitofaconstrainedarcbecauseofjumpsinthestateconstr aintmultiplier.Thethird methodpresentediscalled indirectadjoiningwithcontinuousmultipliers [ 15 ].Inindirect adjoiningwithcontinuousmultiplers,thecostatediscont inuityis“subtractedout”from thecostatedynamics,yieldingacostatethatiscontinuous eveninthepresenceofstate constraintactivity. ConsideragaintheoptimalcontrolproblemofSection 2.1 ,restatedheresuch thattheinequalitypathconstraintisafunctionofonlythe state.Determinethestate, y ( t ) 2 R n ,andthecontrol u ( t ) 2 R m ,thatminimizethecostfunctional ( y ( t f ))+ Z t f t 0 g ( y ( t ), u ( t )) dt (2–54) subjecttothedynamicconstraint y ( t )= f ( y ( t ), u ( t )), (2–55) theboundarycondition ( y ( t 0 ))= 0 (2–56) andthestateinequalitypathconstraint S ( y ( t )) 0. (2–57) 2.3.1IndirectAdjoining Therst-orderoptimalityconditionsofthestateinequali typathconstrainedproblem ofEqs.( 2–54 )–( 2–57 )arenowderivedbyapplyingthemethodofindirectadjoinin g. ItwaspreviouslyshowninSection 2.2.1 thathigh-indexDAEaredifculttosolve numerically.Thus,intheindirectadjoiningmethodofderi vingtherst-orderoptimality conditions,indexreductionisperformedonthehigh-index DAEsystemthatresultsfrom thestateconstraintactivity,resultinginamodiedoptim alcontrolproblemformulation. Inordertosimplifytheanalysispresentedhere,ascalarin equalitypathconstraintis 41

PAGE 42

considered.Generalityisnotlost,however,becauseindex reductioncanbeapplied toavectorinequalitypathconstraintbyconsideringeachc omponentindividually. Furthermore,itisassumedthestateconstraintisactivein aninterval [ t 1 t 2 ] suchthat S ( y ( t ))=0 2 [ t 1 t 2 ] [ t 0 t f ] .Ontheconstrainedarc,thestateconstraintmustbe satisedsuchthat S ( y ( t ))=0, t 2 [ t 1 t 2 ]. (2–58) BecauseEq.( 2–58 )mustbesatisedontheoptimalsolution,allderivativeso fthe pathconstraintin [ t 1 t 2 ] mustalsobezero.Performingindex-reductionasdescribed in Section 2.2.1 ,Eq.( 2–58 )isdifferentiated q times,where q isthe lowest derivativeof S thatisanexplicitfunctionofthecontrol.Theintermediat etimederivativesarethen denedas ( y ( t )) 2666666664 S ( y ( t )) d dt S ( y ( t )) ... d ( q 1) dt ( q 1) S ( y ( t )) 3777777775 t 2 [ t 1 t 2 ]. (2–59) Theseintermediatetimederivativesareevaluatedattheen tranceoftheconstrained arc, t 1 ,andactasconsistentinitialconditionsforthemodiedDA E.Reference[ 7 ] denotestheseconstraintsas tangencyconditions .Physically,theinterpretationofthese conditionsisthatsincetheconstraintfunctioniscontrol lableonlybychangingits q th timederivative,theredoesnotexistanitecontrolforwhi chthesystemwillremainon theconstraintboundaryunlessthetangencyconditionsare alsosatised[ 10 ]. Consequently,thestateinequalitypathconstraintgivenb yEq.( 2–57 )canbe replacedbythetangencyconditionsalongwiththefollowin gstateandcontrolinequality pathconstraint: d ( q ) dt ( q ) S ( y ( t ), u ( t ))= C ( y ( t ), u ( t )) 0, t 2 [ t 1 t 2 ]. (2–60) 42

PAGE 43

TheoptimalcontrolproblemofEqs.( 2–54 )–( 2–57 )canthenbemodiedasfollowsto accountfortheDAEindexreductionasfollows.Minimizethe costfunctional ( y ( t f ))+ Z t f t 0 g ( y ( t ), u ( t )) dt (2–61) subjecttothedynamicconstraint y ( t )= f ( y ( t ), u ( t )), (2–62) theboundarycondition ( y ( t 0 ))= 0 (2–63) thetangencyconditions ( y ( t 1 ))= 0 (2–64) andthestateandcontrolinequalitypathconstraint C ( y ( t ), u ( t )) 0, t 2 [ t 1 t 2 ]. (2–65) Therst-orderoptimalityconditionsofthemodiedproble mofEqs.( 2–61 )–( 2–65 )can beobtainedusingthecalculusofvariationsaspreviouslyd escribedinSection 2.1 .They aregivenastheoriginalconstraintsofEqs.( 2–62 )–( 2–65 )alongwiththeconditions 0 = @ H ( y u ~ ,~ ) @ u (2–66) ~ > = @ H ( y u ~ ,~ ) @ y (2–67) ~ > ( t 0 )= > @ @ y ( t 0 ) (2–68) ~ > ( t + 1 )= ~ > ( t 1 )+ ~ > ( t 1 ) @ @ y ( t 1 ) (2–69) ~ ( t + 2 )= ~ ( t 2 ), (2–70) ~ > ( t f )= @ @ y ( t f ) (2–71) S ( y ) 0,~ 0,~ > S ( y )=0. (2–72) 43

PAGE 44

IntheconditionsofEqs.( 2–66 )–( 2–72 ), ~ ( t ) 2 R n isthecostate, ~ ( t ) 2 R isthe Lagrangemultiplierassociatedwiththepathconstraintof Eq.( 2–65 ), 2 R p isthe Lagrangemultiplierassociatedwiththeboundaryconditio nofEq.( 2–63 ), ~ 2 R q is theLagrangemultiplierassociatedwiththetangencycondi tionsofEq.( 2–64 ),andthe augmentedHamiltonianisdenedas H = g ( y u )+ ~ > f ( y u ) ~ > C ( y u ). (2–73) Theaforementionedapproachiscalled indirectadjoining becausethe q th derivativeof thestateconstraintisadjoinedtotheHamiltonianrathert hanthestateconstraintitself. Itcanbeseenthatbyapplyingtheseconditions,activestat einequalitypath constraintsmayleadtoadiscontinuouscostateattheentra nce,butnotattheexit,of theconstrainedarc,ascanbeseenbyEqs.( 2–95 )and( 2–96 ).Furthermore,notethat thecostatediscontinuitiesattheentranceoftheconstrai nedarcarequantiedbythe costatejumpconditionsofEq.( 2–95 ).Thus,Eq.( 2–95 )actsasterminalconditionsfor theintervalprecedingtheconstrainedarc,andtheoptimal controlproblemcanbeseen asathree-pointboundaryvalueproblem. Althoughthemethodofindirectadjoiningofferstheobviou sbenetsofindex-reduction, asstatedinSection 2.2.1 ,inpracticethistechniquemaybecumbersometoapply. Inparticular,index-reductionrequiresareformulationo ftheprobleminwhichthe derivativesofthestateinequalitypathconstraintmustbe takenanalytically.Furthermore, thesolutionstructureoftheproblem(thatis,thesequence ofconstrainedand unconstrainedarcs)mustbeknownapriori.Becausesolutio nstooptimalcontrol problemsareoftenobtainedthroughanautomatedprocess(s uchasamesh-renement technique),indirectadjoiningisanimpracticalsolution method. ItwasseeninSection 2.2.1 thatitisstillpossibletoobtainanaccuracyof O (10 5 ) whenestimatingsolutionstohigh-orderDAEofuptoorderth ree.Becausemost aerospaceengineeringapplicationsgenerallydonotformu latestateinequalitypath 44

PAGE 45

constraintsoforderhigherthantwo,itisreasonabletoexp lorealternatemethodsof solvingstateinequalitypathconstrainedoptimalcontrol problemsthatdonotinvolve index-reduction.InSections 2.3.2 and 2.3.3 twosuchmethodswillbediscussed: themethodof directadjoining andthemethodof indirectadjoiningwithcontinuous multipliers 2.3.2DirectAdjoining Usingthemethodofdirectadjoining,therst-orderoptima lityconditionsofthestate inequalitypathconstrainedproblemofEqs.( 2–54 )–( 2–57 )canbederivedinthesame manneraswasdoneinSection 2.1 .Theseconditionsaregivenas y = f ( y u ), 0 = ( y ( t 0 )), (2–74) 0 = @ H ( y u , ) @ u (2–75) > = @ H ( y u , ) @ y (2–76) > ( t 0 )= > @ @ y ( t 0 ) (2–77) > ( t f )= @ @ y ( t f ) (2–78) S ( y ) 0 0 > S ( y )= 0 (2–79) wheretheaugmentedHamiltonianisgivenas H ( y u , )= g ( y u )+ > f ( y u ) > S ( y ). (2–80) Therst-orderoptimalityconditionsofEqs.( 2–74 )–( 2–79 )arenecessaryconditions foroptimalityofapurestateinequalitypathconstrainedp roblem.Theyare,however, insufcienttofullydetermineanextremalsolution.Inpar ticular,thestateconstraint multipliersgivenbyEq.( 2–79 )maybediscontinuousattheentranceandexitof aconstrainedarc.Becausethestateconstraintmultiplier sandthecostateare relatedthroughthecostatedynamicsofEq.( 2–76 ),thediscontinuitiesinthestate constraintmultiplierwillinturncausediscontinuitiesi nthecostate.Toquantifythese 45

PAGE 46

discontinuities,thecostatedynamicsofEq.( 2–76 )canbeintegratedas > ( t + 1 )= > ( t 0 ) Z t 1 t 0 @ g ( y u ) @ y + > ( t ) @ f ( y u ) @ y dt + Z ( t 0 t 1 ] > ( t ) @ S ( y ) @ y dt (2–81) where t 0 t 1 t f .Now,denethevariable ( t )= ( t ) .TheintegralofEq.( 2–81 ) canthenbere-writtenas > ( t + 1 )= > ( t 0 ) Z t 1 t 0 @ g ( y u ) @ y + > ( t ) @ f ( y u ) @ y dt Z ( t 0 t 1 ] @ S > ( y ) @ y d ( t ). (2–82) Because ( t ) isafunctionofaboundedvariation,itcanbedecomposeduni quelyas ( t )= 1 ( t )+ 2 ( t ), (2–83) where 1 ( t ) isabsolutelycontinuouswithrespectto t and 2 ( t ) issingularwith respectto t .Therefore,thesecondintegralontheright-handsideofEq .( 2–82 )can beexpressedas Z [ t 0 t 1 ] @ S > ( y ) @ y d ( t )= Z t 1 t 0 @ S > ( y ) @ y 1 ( t ) dt + Z ( t 0 t 1 ] @ S > ( y ) @ y d 2 ( t ). (2–84) Also, ( t ) ismonotonesothatitcanhaveatmostcountablymanyjumps.M oreover, itisreasonabletoassumethat ( t ) issufcientlywellbehavedtohaveapiecewise continuousderivative.Inthatcasethesecondintegralont heright-handsideof Eq.( 2–84 )canbeexpressedas Z [ t 0 t 1 ] @ S > ( y ) @ y d 2 ( t )= X i 2 ( t 0 t 1 ] @ S > ( y ( i )) @ y [ ( + 1 ) ( i )], (2–85) where i arethepointsofdiscontinuityof ( t ) .Next,deningthecostatediscontinuity as ( )= ( ) ( + ), (2–86) theintegralofEq.( 2–81 )canbeexpressedas > ( t + 1 )= > ( t 0 ) Z t 1 t 0 @ g ( y u ) @ y + > ( t ) @ f ( y u ) @ y > ( t ) @ S ( y ) @ y dt + X i 2 ( t 0 t 1 ] ( ) > @ S ( y ( i )) @ y 46

PAGE 47

Therefore,thecostatedynamicscanbeexpressedas > = @ g @ y + > @ f @ y > @ S @ y (2–87) Furthermore,let t = denoteanentrytimeintoaconstrainedarc,anexittimefrom aconstrainedarc,orasinglecontactpointinwhich S ( y ( ))= 0 .Thenthecostate trajectorymayhaveadiscontinuitygivenbythefollowingj umpconditions > ( + )= > ( )+ ( ) > @ S ( y ( i )) @ y (2–88) Because ( t ) 0 ,thefunction ( t ) isnon-decreasinginthesolutiondomain, andthus ( k ) 0 .Furthermore,thecondition h ( k ) S ( Y ( T k )) i = 0 musthold true.Thus,usingdirectadjoining,therst-orderoptimal ityconditionsforthestate inequalitypathconstrainedoptimalcontrolproblemofEqs .( 2–54 )–( 2–57 )aregivenby Eqs.( 2–74 )–( 2–76 )alongwiththecostatediscontinuityconditionsofEq.( 2–88 ). Therst-orderoptimalityconditionsderivedherearevery similartotheconditions derivedinSection 2.3.1 .However,itisnotedthatintheconditionsderivedinSecti on 2.3.1 the q th derivativeofthestateinequalitypathconstraintisenfor cedonthe constrainedarc,whereastheoriginalundifferentiatedpa thconstraintwasenforced intheproblemformulationofthissection.Moreover,theop timalityconditionsstatedin Section 2.3.1 normalizethejumpconditionssuchthatthecostateisdisco ntinuousonly attheentrytimeoftheconstrainedarc,whilethecondition sstatedinthissectionmake nosuchdistinction.2.3.3IndirectAdjoiningWithContinuousMultipliers Thethirdmethodforderivingthenecessaryconditionsforo ptimalityofastate inequalitypathconstrainedoptimalcontrolproblemisthe methodofindirectadjoining withcontinuousmultipliers.Usingthemethodofindirecta djoiningwithcontinuous multipliersyieldsacostatethatiscontinuouseveninthep resenceofstateinequality pathconstraints.Becausediscontinuitiesaredifcultto approximatenumerically,the 47

PAGE 48

methodofindirectadjoiningwithcontinuousmultiplierso ffersanadvantageoverthe methodsofdirectandindirectadjoiningwhichapproximate adiscontinuouscostate. Therst-orderoptimalityconditionsforthestateinequal itypathconstrainedoptimal controlproblemgivenbyEqs.( 2–54 )–( 2–57 )arenowderivedbyusingthemethodof indirectadjoiningwithcontinuousmultipliers.Similart otheapproachusedinSection 2.3.2 ,thecostatedynamicsofEq.( 2–76 )canbeintegratedtogive > ( t + 1 )= > ( t 0 ) Z t 1 t 0 @ g ( y u ) @ y + > ( t ) @ f ( y u ) @ y dt + Z ( t 0 t 1 ] > ( t ) @ S ( y ) @ y dt where t 0 t 1 t f .Now,let ( t )= ( t ) .Furthermore,because ( t ) 0 ( t ) must beanon-decreasingfunctionoftime.Thecostatediscontin uitycannowbe“subtracted” bydeninganewcostate p ( t ) suchthat p > ( t )= > ( t )+ > ( t ) @ S ( y ) @ y (2–89) TheHamiltoniancanbedenedintermsofthecontinuouscost ate p ( t ) suchthat H ( y u p )= g ( y u )+ p > f ( y u ) > S ( y u ). (2–90) ItcanbeseenthatthisexpressioncanbedecomposedtotheHa miltonianbysubstituting therelationship S ( y ) @ S ( y ) @ y y = @ S ( y ) @ y f ( y u ). intoEq.( 2–90 )suchthat ~ H ( y u )= g ( y u )+ > f ( y u ) H ( y u p ). (2–91) 48

PAGE 49

Therst-orderoptimalityconditionsarethengiveninterm softhecontinuous costateasEqs.( 2–55 )–( 2–57 )alongwiththeconditions 0 = @ H ( y u p ) @ u (2–92) p > = @ H ( y u p ) @ y (2–93) p > ( t 0 )= > @ @ y ( t 0 ) + ( t 0 ) > @ S ( y ( t 0 )) @ y (2–94) > ( t f )= @ @ y ( t f ) + ( t f ) > @ S ( y ( t f )) @ y (2–95) ( t f ) 0 ,_ 0 C ( y ) 2N ( ), (2–96) Furthermore,let C ( R q ) denotethespaceofcontinuousfunctionsmapping [ t 0 t f ] to R q Assuming isLipschitzcontinuousandnon-decreasingwith ( t f ) 0 ,theset-valued map N ( ) isdenedas N ( )= f z 2C ( R q ): z 0 ,_ > z = 0 > ( t f ) z ( t f )=0 g 2.4NumericalPropertiesofOrthogonalCollocationMethod s Numericallysolvinganoptimalcontrolproblemsrequirekn owledgeofanumberofdifferentconcepts.Inparticular,twoconceptsareimportant inconstructingadiscretizednitedimensionaloptimizationproblemfromacontinuous-timeo ptimalcontrolproblem:polynomial approximationandnumericalintegration.Polynomialappr oximationisimportantbecausethe innite-dimensionalcontinuousfunctions(thatis,thest atecomponents)oftheoptimalcontrol problemareapproximatedbyanite-dimensionalLagrangep olynomialbasis.Numericalintegrationmethodsareimportantbecausethedynamicconstraints andthecostmustbeintegratedas partoftheoptimization.Inthissection,theseimportantm athematicalconceptsthatareusedto transcribeacontinuous-timeoptimalcontrolproblemtoan onlinearprogrammingproblem(NLP) usingorthogonalcollocationatGaussianquadraturepoint sarereviewed. 2.4.1FunctionApproximationandInterpolation Collocationmethodsforsolvingoptimalcontrolproblemsa pproximatethecontinuous functionsoftimeatasetofsupportpoints.Inthisresearch ,Lagrangepolynomialsareusedto 49

PAGE 50

interpolatethestateandthecostate.Specically,givena continuousfunction y ( t ) ,thereexists auniquepolynomial Y ( t ) ofdegree N 1 whichuses N arbitrarysupportpoints ( t 1 ..., t N ) 2 [ t 0 t f ] ,suchthat Y ( t i )= y ( t i ),( i =1,..., N ). (2–97) Furthermore,theuniquepolynomialcanbedescribedbyLagr angeinterpolationsuchthat Y ( t )= N X i =1 y i L i ( ), (2–98) where y i = y ( t i ) and L i ( t ) aretheLagrangepolynomials L i ( t )= N Y j =1 j 6 = i j i j (2–99) OneimportantpropertyofLagrangeinterpolatingpolynomi alsisthattheysatisfytheisolation property L i ( t j )= 8><>: 1 for i = j 0 for i 6 = j (2–100) Eq.( 2–100 )isimportantforthisresearchbecauseitleadstoasparsen onlinearprogram transcriptionoftheoptimalcontrolproblembeingsolved. Thustheisolationpropertyleadstoa transcriptionwhichcanbeefcientlysolvedbyanonlinear programsolver.Theerrorassociated withtheLagrangeapproximationofa N th timesdifferentiablefunctionisgivenby y ( t ) Y ( t )= ( t t 1 )...( t t N ) N y N ( ), (2–101) where y N ( ) isthe N th derivativeofthefunction y ( t ) evaluatedatapoint 2 [ t 0 t f ] .Itis seenfromthiserrorformulathattheerrorisexactlyzeroat anyofthesupportpointsofthe interpolatingpolynomial.Furthermore,sincetheerroris adirectfunctionofthe N th derivative of y ( t ) ,theLagrangeinterpolationapproximationusing N supportpointswillbeexactfor polynomialsofdegreeatmost N 1 Althoughsmoothfunctionscanbeaccuratelyapproximateda sstatedabove,thebehavior oftheinterpolationerroras N approachesinnityfornon-smoothfunctionsbecomeserrat ic,a behaviorcalled Runge phenomenon.Rungephenomenonischaracterizedbylargeamp litude 50

PAGE 51

oscillationsintheinterpolatingpolynomialsnearthedom ainboundarieswhenthesupportpoints areuniformlydistributedand N becomeslarge.InordertounderstandRungephenomenon better,considerthefollowingfunctiondenedinthedomai n 2 [ 1,+1] y ( )= 1 1+50 2 (2–102) ThefunctioninEq.( 2–102 )wasapproximatedusing N uniformlydistributedsupportpointsfor abasisofLagrangeinterpolatingpolynomialsandtheresul tfor N =11 and N =41 isshown inFig.( 2-2 ).Theapproximationsfor N =11 and N =41 pointscorrespondtopolynomials ofdegree 10 and 40 .Itcanbeseenthatasthenumberofsupportpointsisincreas ed,the errorintheinterpolationbecomeslargerneartheendpoint sduetothelargeoscillationsin thepolynomialapproximations.Thustheinterpolationdoe snotconvergetothefunctionbeing approximatedas N isincreased. 2.4.1.1FamilyofLegendre-Gausspoints OnewaytorectifytheeffectofRungephenomenonistouseano n-equallyspacedset supportpoints.Inthisresearchthesupportpointsusedare thepointsobtainedfromtherootsof aLegendrepolynomialand/orlinearcombinationsofaLegen drepolynomialanditsderivatives. Thesepointsareknowntohaveadistributionthatminimizes Rungephenomenon.Inparticular, twosetsofpointsareofinterest:theLedendre-Gauss(LG)p oints,andtheLegende-GaussRadau(LGR)points.Boththesesetsofpointsaredenedonth edomain 2 [ 1,+1] ,but differsignicantlyinthattheLGpointsincludeneitherof theendpointswhereastheLGRpoints includeoneoftheendpoints.Inaddition,theLGRpointsare asymmetricrelativetotheorigin andarenotuniquebecausetheycanbedenedusingeitherthe initialortheterminalpoint.The LGRpointsthatincludetheterminalpointareoftencalledt he ipped LGRpoints.Theipped LGRpointsareamirrorimageoftheLGRpointsthatincludeth einitialpointinthedomain. Thesepointscanbecalculatedasfollows.Denotingthe N th degreeLegendrepolynomialby P N ( )= 1 2 N N d N d N h ( 2 1) N i 51

PAGE 52

0 0 0.2 0.4 0.5 0.6 0.8 1 1 -0.8 -0.6 -0.4-0.2 -0.5 -1 -1 2 3 2.5 3.5 1.5 y ( )y ( )Y ( ) (A)ApproximationofthefunctiongivenbyEq.( 2–102 )using11 uniformlyspacedsupportpoints. 0 0 0.2 0.4 0.5 0.6 0.8 1 -0.8-0.6-0.4-0.2 -0.5-1.5 -2 -1 -1 -2.5 6 x10 y ( )y ( )Y ( ) (B)ApproximationofthefunctiongivenbyEq.( 2–102 )using41 uniformlyspacedsupportpoints. Figure2-2.ApproximationofthefunctiongivenbyEq.( 2–102 )using11and41 uniformlyspacedsupportpoints. 52

PAGE 53

0 0.2 0.4 0.6 0.8 1 -0.8-0.6-0.4-0.2 -1 LGR LGR-f LG Figure2-3.DistributionofLegendre-Gauss,Legendre-Gau ss-Radau,andFlipped Legendre-Gauss-RadauPointsinthedomain 2 [ 1,+1] the LG pointsaredenedastherootsof P N ( ) andtheLGRpointsaredenedastheroots obtainedfrom P N 1 ( )+ P N ( ) .Fig.( 2-3 )isaschematicrepresentationoftheLG,LGR,and ippedLGRpointsinthedomain [ 1,+1] for N =5 ThefunctiongivenbyEq.( 2–102 )isnowapproximatedusingabasisofLagrangeinterpolatingpolynomialwithLGandLGRsupportpoints.Afunction y ( ) canbeapproximatedinthe domain [ 1,+1] usingabasisofLagrangeinterpolatingpolynomialsandthe LGpointsas Y ( )= N +1 X i =0 y ( i ) L ) i ( ) (2–103) where ( 1 ,..., N ) arethe N LGpoints, 0 = 1 ,and N +1 =+1 .Similarly,thesamefunction y ( ) canbeapproximatedinthedomain [ 1,+1] usingabasisofLagrangeinterpolating polynomialsandtheLGRpointsas Y ( )= N +1 X i =1 y ( i ) L ) i ( ) (2–104) where ( 1 ,..., N ) arethe N LGpoints,and N +1 =+1 .Fig.( 2-4 )showstheresultsofthe functionapproximationwhen N =11 and N =41 LGsupportpointsareused.Furthermore, Fig.( 2-5 )showstheresultsofthefunctionapproximationwhen N =11 and N =41 LGRsupport 53

PAGE 54

pointsareused.Itcanbeseenthatasthenumberofsupportpo intsisincreasedthepolynomial approximationconvergestothetruefunction.Inordertoun derstandthebehavioroftheerror, Fig.( 2-6 )plotsthebasetenlogarithmofthemaximumabsoluteerrord enedas E y =log 10 jj Y ( ) y ( ) jj 1 asafunctionof N foranapproximationusingLagrangeinterpolatingpolynom ialswithuniformlyspaced,LG,andLGRsupportpoints.Itcanbeseenthat asthenumberofsupport pointsincreases,thepolynomialapproximationwhichuses uniformlyspacedsupportpoints divergesfromthefunctionbeingapproximated,whereasusi ngLGandLGRsupportpointsthe approximationconvergestothetruefunction.2.4.2NumericalIntegration Numericalintegrationplaysakeyrolewhennumericallyapp roximatingthesolutiontoa continuous-timeoptimalcontrolproblem.Inparticular,a noptimalcontrolproblemrequiresthata costbeminimized(ormaximized),andthiscostisintegrate dacrossthetimedomainofinterest. Furthermore,thedynamicconstraintsmustbeintegrated.T herefore,anoptimalcontrolproblem optimizesandintegratessimultaneously.Asummaryofnume ricalintegrationmethodswillnow begiveninordertobetterunderstandthemethodsusedinthi sresearch. 2.4.2.1Low-orderintegrators Acommontechniqueusedtoapproximatetheintegralofafunc tionistouselow-degree polynomialapproximations.Theapproximationoftheinteg ralisthenfoundbysummingthelowordermethodintegralapproximationsofeachsubinterval. Onecommonlyusedtechniquethat useslow-ordermethodsiscalledthecompositetrapezoidru le[ 2 ].Thecompositetrapezoidrule dividesthedomainofinterestintomanyuniformlydistribu tedsub-intervalsandapproximates thefunctiontobeintegratedwithastraightlinethatpasse sthroughthefunctionattheendpoints ofthesubinterval.Therefore,for N approximatingsubintervals,thecompositetrapezoidrule is givenby Z t f t 0 f ( t ) dt t f t 0 2 N [ f ( t 0 )+2 f ( t 1 )+2 f ( t 2 )+...+2 f ( t N 1 )+ f ( t N ) ] (2–105) 54

PAGE 55

1.2 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 -0.8 -0.6 -0.4 -0.2 -0.2 -1 y ( )y ( )Y ( ) (A)ApproximationofthefunctiongivenbyEq.( 2–102 )using11 Legendre-Gausspoints. 0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.6 0.7 0.8 0.8 0.9 1 1 -0.8-0.6-0.4-0.2 -1 y ( )y ( )Y ( ) (B)ApproximationofthefunctiongivenbyEq.( 2–102 )using41 Legendre-Gausspoints. Figure2-4.ApproximationofthefunctiongivenbyEq.( 2–102 )using11and41 Legendre-Gausspoints. 55

PAGE 56

1.2 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 -0.8 -0.6 -0.4 -0.2 -0.2 -1 y ( )y ( )Y ( ) (A)ApproximationofthefunctiongivenbyEq.( 2–102 )using11 Legendre-Gauss-Radaupoints. 0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.6 0.7 0.8 0.8 0.9 1 1 -0.8-0.6-0.4-0.2 -1 y ( )y ( )Y ( ) (B)ApproximationofthefunctiongivenbyEq.( 2–102 )using41 Legendre-Gauss-Radaupoints. Figure2-5.ApproximationofthefunctiongivenbyEq.( 2–102 )using11and41 Legendre-Gauss-Radaupoints. 56

PAGE 57

0 0 30 40 5060708090100 -5 10 10 -10 20 20 15 5 NE yUniform LG LGR Figure2-6.Basetenlogarithmofinnitynormerrorasafunc tionofnumberofsupport points, N ,forapproximatingthefunctiongivenbyEq.( 2–102 ). where ( t 0 ,..., t N ) arethesubintervalboundaries,orgridpoints,atwhichthe functionisbeing evaluated.Inordertodemonstratetheuseofthecompositet rapezoidrule,considerapproximatingtheintegralofthefunction f ( ) inthedomain 2 [ 1,+1] : f ( )= Z +1 1 exp( ) d (2–106) Figure( 2-7 )showsagraphicalrepresentationofthetrapezoidruleapp roximationusingthree intervals.Furthermore,Fig.( 2-8 )showsthebasetenlogarithmerrorofthisintegrationasa functionofthebasetenlogarithmnumberofapproximatingi ntervals.Itcanbeseenthatthe convergenceofthismethodisslowas 10 5 subintervalsarenecessarytoreachanerrorof O (10 10 ) 2.4.2.2Gaussianquadrature Incontrastwithlow-orderintegratorssuchasthecomposit etrapezoidrule,aGaussian quadratureisahighaccuracyintegratorwhichdisplaysexp onentialconvergencewhenapproximatingtheintegralofsmoothfunctions.Gaussianquadratu rerulesapproximatetheintegralofa 57

PAGE 58

0 0 0.2 0.4 0.5 0.6 0.8 1 1 -0.8-0.6-0.4-0.2 -1 2 3 2.5 1.5 f ( )f ( )Approx. Figure2-7.Approximationoftheintegralofthefunctiongi venbyEq.( 2–106 )usinga4 intervalTrapezoidrule. -4 2 4 4.55 -5 3 2.5 3.5 -6 -7 -8 -9 -10 -11 log 10 Errorlog 10 NumberofApproximatingIntervals Figure2-8.Basetenlogarithmerroroftheintegralofthefu nctiongivenbyEq.( 2–106 ) asafunctionofthebasetenlogarithmnumberofapproximati ngintervals. 58

PAGE 59

functionbyevaluatingtheexpression Z t f t 0 f ( t ) dt N X i =1 w i f ( i ), (2–107) where w i arethequadrature weights associatedwiththesetofpointschosentoapproximatethe integration.ThethreesetsofpointsdenedbyGaussianqua draturearetheLegendre-GaussLobatto(LGL)points,theLegendre-Gauss-Radau(LGR)poin ts,andtheLegendre-Gauss(LG) points.TheLGL,LGR,andLGquadraturerulesareexactforpo lynomialsofdegreeatmost 2 N 3 2 N 2 ,and 2 N 1 ,respectively.InthisresearchtheLGandLGRpointsareuse d. The N LGpointsaretherootsofthe N th degreeLegendrepolynomial P N ( ) ,andthe correspondingLGquadratureweightsaregivenas w i = 2 1 2 i [ P N ( i )] 2 ,( i =1,..., N ). Similarly,the N LGRpointsarecomputedfromtherootsof P N ( )+ P N 1 ( ) ,andthecorrespondingLGRquadratureweightsaregivenas w 1 = 2 N 2 w i = 1 (1 i )[ ( P N 1 ( i )] 2 ,( i =2,..., N ). Finally,the ipped LGRpointsandweightsaresimplythenegativeoftheLGRpoin ts. TheaccuracyoftheLGandLGRquadraturemethodscanbeseenf romthefunction f ( )=exp( ) inthedomain 2 [ 1,+1] .Figure 2-9 showsthebasetenlogoftheerror fortheapproximationoftheintegralinEq.( 2–106 )asafunctionof N .Itcanbeseenthatthe convergencerateusingGaussianquadratureisexponential .Furthermore,itisseenthat N =6 LGorLGRpointsresultsinanerroroflessthan O (10 10 ) .Forcomparison,thissameerror of O (10 10 ) required 10 5 subintervalsusingthecompositeTrapezoidrule.Thus,the benets ofusingGaussianquadratureoverlow-orderintegratorswh enapproximatingtheintegralofa smoothfunctionisevident. 2.5OrthogonalCollocationfortheSolutionofOptimalCont rolProblems ItisnowpossibletocombinetheconceptsdescribedinSecti on 2.4 inordertodevelopa methodtoapproximatethesolutionofthecontinuous-timeo ptimalcontrolproblemofSection 59

PAGE 60

6 7 89 10 -16 -14 -4 -2 4 5 3 -6 -8 -10 -12 N LGPoints LGRPoints log 10 Error Figure2-9.Basetenlogarithmoftheerrorintheintegratio nofthefunctiongivenby Eq.( 2–106 )asafunctionofnumberofLGandLGRPointsUsed. 2.1 .Inthissectionthemethodsoforthogonalcollocationatbo thLegendre-Gauss(LG)and Legendre-Gauss-Radau(LGR)pointsaredescribed.Boththe semethodsforapproximating solutionstooptimalcontrolproblemscanbeusedaseitherg lobalcollocationmethodsor variable-ordercollocationmethods.Globalcollocationm ethodsuseonesinglepolynomial approximationtocollocatethedifferential-algebraiceq uationsovertheentiredomain.Global collocationatLGandLGRpointsisadvantageouswhensolvin gproblemswhosesolutionsare smooth,becausetheLGandLGRmethodsconvergeexponential ly.Whenthesolutionisnot smooth,however,theconvergencerateissignicantlylowe r.Inthecasewheretheoptimal solutionliesonaconstrainedarcforasubsetofthesolutio ndomain,non-smoothfeaturesinthe solutionstateand/orcontrolcanoccur.Forsuchproblems, itwillthusbebenecialtoemploy variable-orderLGorLGRcollocation.Inavariable-orderc ollocationschemethesolutiondomain isdividedintoamesh,andthedegreeoftheapproximatingpo lynomial(thatis,thenumberof LGorLGRcollocationpoints)ineachmeshintervalisallowe dtovary.Thismethodisuseful becauseitallowsforcapturingnon-smoothnessinthesolut iondomainatintervalboundaries. BecauseboththeLGandtheLGRsetofpointsaredenedonthed omain 2 [ 1,+1] ,the followingafnetransformationwillbeusedtomapthetimed omain t 2 [ t 0 t f ] to 2 [ 1,+1] 60

PAGE 61

whenusingglobalcollocation: t = t f t 0 2 + t f + t 0 2 (2–108) Furthermore,itisnotedthat dt d = t f t 0 2 h 2 h = t f t 0 Whenusingvariable-ordercollocation,thetimedomain t 2 [ t 0 t f ] isdividedintoamesh consistingof K meshintervals wherethemeshpointsare t 0 = T 0 < T 1 < < T K 1 < T K = t f ,andthecorrespondingmeshintervalsare [ T k 1 T k ],( k =1,..., K ) .Thereforeeachmesh intervalcanbemappedtothedomain 2 [ 1,+1] throughtheafnetransformation t = t k t k 1 2 + t k + t k 1 2 ,( k =1,..., K ). Itisalsonotedthat dt d = t k t k 1 2 h ( k ) 2 h ( k ) = t k t k 1 Thefollowingnotationandconventionswillbeusedinthedi scussionthatensues.First,all vectorfunctionsoftimearedenotedas row vectors,thatis,if y ( ) 2 R n isavectorfunctionofthe scalarvariable ,then y ( )=[ y 1 ( ), y n ( )] .Next,anycapitalboldfacecharacter, Y ,denotes amatrixofsize M n ,whereeachrowof Y i correspondstotheevaluationofafunction y ( ) at aparticularvalue = i .Next,thenotation Y i : j denotesrows i through j ofthematrix Y ,except whenreferringtoadifferentiationmatrix D ,inwhichcase D i referstothe i th column of D .Finally, D > denotesthetransposeofmatrix D ,and D >i denotesthetransposeofthe i th columnof D 2.5.1GlobalCollocationatLegendre-GaussPoints Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingglobalorthogonal collocationatLegendre-Gauss(LG)pointsisnowdescribed [ 27 ].TheLGpointsaredened inthedomain ( 1,+1) whichdoesnotincludeeitheroftheendpoints.However,the method derivedhereforcollocationattheLGpointsstill approximates ,butdoesnot collocate ,thestate atbothendpoints 0 = 1 and N +1 =+1 .Figure 2-10 showstheLGcollocationpointsaswell astheendpointsatwhichthestateis approximatedbutnotcollocated forvariousvaluesof N 61

PAGE 62

0 0.2 0.4 0.6 0.8 1 -0.8-0.6-0.4-0.2 -1 3 4 5 6 7 8 9 DiscretizationPoints CollocationPointsNumberofLGPoints, NFigure2-10.DistributionofLegendre-Gaussdiscretizati onandcollocationpointsinthe domain 2 [ 1,+1] ThestateisthenapproximatedusingaLagrangepolynomialw ithsupportpointsatthe N LGpointsplusthe noncollocated point 0 = 1 ,suchthat y ( ) Y ( )= N X i =0 Y i L i ( ), (2–109) wheretheLagrangepolynomials L i ( ) aredenedas L i ( )= N Y j =0 j 6 = i j i j ;( i =0,..., N ). (2–110) Thestateapproximationisthendifferentiatedat = j ( j =1,..., N ) as Y ( j ) N X i =0 Y i L i ( j )=[ DY 0: N ] j (2–111) where D ij = L i ( j ) ,( i =1,..., N j =0,..., N ) arethecomponentsofthe N ( N +1) Legendre-Gauss(LG)differentiationmatrix 62

PAGE 63

Next,thecostfunctionalofEq.( 2–1 )isapproximatedwithaGaussianquadrature.The nite-dimensionaltranscriptionofthecontinuous-timeo ptimalcontrolproblemofEqs.( 2–1 )– ( 2–4 )thenbecomestominimizethecostfunction J =( Y N +1 )+ h 2 N X j =1 w j g ( Y j U j ) (2–112) subjecttothealgebraicconstraints DY 0: N = h 2 f ( Y 1: N U 1: N ), (2–113) Y N +1 = Y 0 + h 2 N X j =1 w j f ( Y j U j ), (2–114) ( Y 0 )= 0 (2–115) C ( Y 1: N U 1: N ) 0 (2–116) where w =( w 1 ,..., w N ) aretheLG quadratureweights .ItisnotedforLGcollocationthat Eq.( 3–24 )providesanLGquadratureapproximationofthestate, Y N +1 ,atthenalnoncollocatedpoint N +1 =+1 2.5.2GlobalCollocationatLegendre-Gauss-RadauPoints Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingglobalorthogonal collocationatLegendre-Gauss-Radau(LGR)pointsisnowde scribed[ 64 ].TheLGRpointsare denedinthedomain [ 1,+1) suchthat 1 = 1 isaLGRcollocationpointbut N +1 =+1 is anoncollocatedpoint.However,themethodderivedherefor collocationattheLGRpointsstill approximates ,butdoesnot collocate ,thestateattheterminalpoint N +1 =+1 .Figure 2-11 showstheLGRcollocationpointsaswellasthenoncollocate dterminalpointforvariousvalues of N ThestateisthenapproximatedusingaLagrangepolynomialw ithsupportpointsatthe N LGRpointsplusthenoncollocatedpoint N +1 =+1 suchthat y ( ) Y ( )= N +1 X i =1 Y i L i ( ), (2–117) 63

PAGE 64

0 0.2 0.4 0.6 0.8 1 -0.8-0.6-0.4-0.2 -1 3 4 5 6 7 8 9 DiscretizationPoints CollocationPointsNumberofLGRPoints, NFigure2-11.DistributionofLegendre-Gaussdiscretizati onandcollocationpointsinthe domain 2 [ 1,+1] wheretheLagrangepolynomials L i ( ) aredenedas L i ( )= N +1 Y j =1 j 6 = i j i j ;( i =1,..., N +1). (2–118) Thestateapproximationisthendifferentiatedat = j ( j =1,..., N ) as Y ( j ) N +1 X i =1 Y i L i ( j )=[ DY 1: N +1 ] j (2–119) where D ij = L i ( j ) ,( i =1,..., N j =1,..., N +1) arethecomponentsofthe N ( N +1) Legendre-Gauss-Radau(LGR)differentiationmatrix .Next,thecostfunctionalofEq.( 2–1 )is approximatedbyanLGRquadrature.Thenite-dimensionala pproximationofthecontinuoustimeoptimalcontrolproblemofEqs.( 2–1 )–( 2–4 )isthengivenasfollows.Minimizethecost function J =( Y N +1 )+ h 2 N X j =1 w j g ( Y j U j ) (2–120) 64

PAGE 65

0 0.2 0.4 0.6 0.8 1 -0.8-0.6-0.4-0.2 -1 3 4 5 6 7 8 9 DiscretizationPoints CollocationPointsNumberofFlippedLGRPoints, NFigure2-12.DistributionofLegendre-Gaussdiscretizati onandcollocationpointsinthe domain 2 [ 1,+1] subjecttothealgebraicconstraints DY 1: N +1 = h 2 f ( Y 1: N U 1: N ), (2–121) ( Y 1 )= 0 (2–122) C ( Y 1: N U 1: N ) 0 (2–123) where w =( w 1 ,..., w N ) aretheLGRquadratureweights. 2.5.3GlobalCollocationatFlippedLegendre-Gauss-Radau Points Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingglobalorthogonal collocationatthe ipped Legendre-Gauss-Radau(LGR)pointsisnowdescribed[ 64 ].TheLGR pointsaredenedinthedomain ( 1,+1] suchthat N =+1 isaLGRcollocationpointbut 0 = 1 isanoncollocatedpoint.However,themethodderivedheref orcollocationattheLGR pointsstill approximates ,butdoesnot collocate ,thestateattheinitialpoint 0 = 1 .Figure ( 2-12 )showstheippedLGRcollocationpointsaswellastheiniti alpointatwhichthestateis approximatedbutnotcollocated forvariousvaluesof N 65

PAGE 66

ThestateisthenapproximatedusingaLagrangepolynomialw ithsupportpointsatthe N LGRpointsplusthenoncollocatedpoint 0 = 1 ,suchthat y ( ) Y ( )= N X i =0 Y i L i ( ), (2–124) wheretheLagrangepolynomials L i ( ) aredenedas L i ( )= N Y j =0 j 6 = i j i j ;( i =0,..., N ). (2–125) Thestateapproximationisthendifferentiatedat = j ( j =1,..., N ) as Y ( j ) N X i =0 Y i L i ( j )=[ DY 0: N ] j (2–126) where D ij = L i ( j ) ,( i =1,..., N j =0,..., N ) arethecomponentsofthe N ( N +1) Legendre-Gauss-Radau(LGR)differentiationmatrix Next,thecostfunctionalofEq.( 2–1 )isapproximatedbyanLGRquadrature.Thenitedimensionalapproximationofthecontinuous-timeoptimal controlproblemofEqs.( 2–1 )–( 2–4 )is thengivenasfollows.Minimizethecostfunction J =( Y N )+ h 2 N X j =1 w j g ( Y j U j ) (2–127) subjecttothealgebraicconstraints DY 0: N = h 2 f ( Y 1: N U 1: N ), (2–128) ( Y 0 )= 0 (2–129) C ( Y 1: N U 1: N ) 0 (2–130) where w =( w 1 ,..., w N ) aretheippedLGRquadratureweights. 2.5.4Variable-OrderCollocationatLegendre-GaussPoint s Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingvariable-order collocationattheLegendre-Gauss(LG)pointsisnowdescri bed.Whenimplementingthe variable-orderLGmethod,asinglevariableisusedforthev alueofthestateattheendofmesh interval k andthestartofmeshinterval k +1 ,thatis, Y ( k ) N k +1 Y ( k +1) 0 ,1 k K 1 suchthat 66

PAGE 67

3 4 5 6 7 8 9 t T 0 T 1 T 2 T 3 DiscretizationPoints CollocationPointsNumberofLGpointsPerInterval, N kFigure2-13.Distributionofmultiple-intervalLegendreGaussdiscretizationand collocationpointsforvariousvaluesof N k .Thedomain [ t 0 t f ] issplitintoK=3intervals suchthat t 0 = T 0 and t f = T 3 continuityinthestateisenforced.Hence,redundantvaria blesdeningthestateattheinterior meshpointsareeliminated.Figure 2-13 showstheLGcollocationpointsaswellasthemesh pointsatwhichthestateis approximatedbutnotcollocated forwhen K =3 andforvarious valuesof N Inmultiple-intervalLGcollocation,thestateisapproxim atedineachmeshinterval k as y ( k ) ( ) Y ( k ) ( )= N k X i =0 Y ( k ) i L ( k ) i ( ), L ( k ) i ( )= N k Y j =0 i 6 = j ( k ) j i ( k ) j (2–131) Differentiating Y ( k ) ( ) inEq.( 2–131 )withrespectto ,yields Y ( k ) ( j ) N k X i =0 Y ( k ) i L ( k ) i ( j )=[ D ( k ) Y ( k ) 0: N k ] j (2–132) where D ( k ) ij = L ( k ) i ( ) j ( i =1,..., N k j =0,..., N k ) arethecomponentsofthe N k ( N k +1) Legendre-Gauss(LG)differentiationmatrix inthe k th meshinterval. Next,thecostfunctionalofEq.( 2–1 )isapproximatedusingamultiple-intervalLGquadrature.Thenite-dimensionalapproximationofthecontinuo us-timeoptimalcontrolproblemof 67

PAGE 68

Eqs.( 2–1 )–( 2–4 )isthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K +1 )+ K Xk =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (2–133) subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (2–134) Y ( k +1) 0 = Y ( k ) 0 + h ( k ) 2 N k X j =1 w ( k ) j f ( Y ( k ) j U ( k ) j ),( k =1,..., K 1), (2–135) Y ( K ) N +1 = Y ( K ) 0 + h ( K ) 2 N k X j =1 w ( K ) j f ( Y ( K ) j U ( K ) j ), (2–136) ( Y (1) 0 )= 0 (2–137) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ,( k =1,..., K ), (2–138) where w ( k ) =( w ( k ) 1 ,..., w ( k ) N k ) aretheLG quadratureweights ininterval k .ItisnotedforLG collocationthatEq.( 2–135 )providesanLGquadratureapproximation, Y ( k ) 0 ,ofthestateatthe nalnoncollocatedpoint ( k ) N +1 =+1 ininterval ( k =1,..., K 1) ,whileEq.( 2–136 )providesan LGquadratureapproximation, Y ( K ) N +1 ,ofthestateatthenalnoncollocatedpointofthedomain, t f = ( k ) N +1 =+1 2.5.5Variable-OrderCollocationatLegendre-Gauss-Rada uPoints Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingvariable-order collocationattheLegendre-Gauss-Radau(LGR)pointsisno wdescribed.Whenimplementing thevariable-orderLGRmethod,asinglevariableisusedfor thevalueofthestateattheendof meshinterval k andthestartofmeshinterval k +1 ,thatis, Y ( k ) N k +1 Y ( k +1) 1 ,1 k K 1 suchthatcontinuityinthestateisenforced.Hence,redund antvariablesdeningthestateatthe interiormeshpointsareeliminated.ItisnotedthattheLGR pointsareparticularlyconducive tothistypeofcollocation;becauseonlyoneofthedomainen dpointsiscollocated,thereisno “doublecollocation”attheboundaries.Also,theonlynonc ollocatedpointisthelastpointofthe nalinterval, t f = ( K ) N K +1 =+1 .Figure( 2-14 )showstheLGRcollocationpointsaswellasthe terminalpointatwhichthestateis approximatedbutnotcollocated forwhen N =3 andfor variousvaluesof K 68

PAGE 69

3 4 5 6 7 8 9 t T 0 T 1 T 2 T 3 DiscretizationPoints CollocationPointsNumberofLGRPointsPerInterval, N kFigure2-14.Distributionofmultiple-intervalLegendreGauss-Radaudiscretizationand collocationpointsforvariousvaluesof N k .Thedomain [ t 0 t f ] issplitintoK=3intervals suchthat t 0 = T 0 and t f = T 3 Inmultiple-intervalLGRcollocation,thestateisapproxi matedineachmeshinterval k as y ( k ) ( ) Y ( k ) ( )= N k +1 X i =1 Y ( k ) i L ( k ) i ( ), L ( k ) i ( )= N k +1 Y j =1 i 6 = j ( k ) j i ( k ) j (2–139) Differentiating Y ( k ) ( ) inEq.( 2–139 )withrespectto ,yields Y ( k ) ( j ) N k +1 X i =1 Y ( k ) i L ( k ) i ( j )=[ D ( k ) Y ( k ) 1: N k +1 ] j (2–140) where D ( k ) ij = L ( k ) i ( ) j ( i =1,..., N k j =1,..., N k +1) arethecomponentsofthe N k ( N k +1) Legendre-Gauss-Radau(LGR)differentiationmatrix inthe k th meshinterval. Next,thecostfunctionalofEq.( 2–1 )isapproximatedusingamultiple-intervalLGquadrature.Thenite-dimensionalapproximationofthecontinuo us-timeoptimalcontrolproblemof Eqs.( 2–1 )–( 2–4 )isthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K +1 )+ K Xk =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (2–141) 69

PAGE 70

subjecttothealgebraicconstraints D ( k ) Y ( k ) 1: N k +1 = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (2–142) ( Y (1) 1 )= 0 (2–143) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ,( k =1,..., K ), (2–144) where w ( k ) =( w ( k ) 1 ,..., w ( k ) N k ) aretheLGR quadratureweights ininterval k 2.5.6Variable-OrderCollocationatFlippedLegendre-Gau ss-RadauPoints Themethodforapproximatingsolutionstooptimalcontrolp roblemsusingvariable-order collocationattheippedLegendre-Gauss-Radau(LGR)poin tsisnowdescribed.When implementingtheippedvariable-orderLGRmethod,asingl evariableisusedforthevalue ofthestateattheendofmeshinterval k andthestartofmeshinterval k +1 ,thatis, Y ( k ) N k Y ( k +1) 0 ,1 k K 1 suchthatcontinuityinthestateisenforced.Hence,redund ant variablesdeningthestateattheinteriormeshpointsaree liminated.Itisnotedthattheipped LGRpointsareparticularlyconducivetothistypeofcolloc ation;sinceonlyoneofthedomain endpointsarecollocated,thereisno“doublecollocation” attheboundaries.Also,theonly noncollocatedpointistherstpointoftherstinterval, t 0 = (1) 0 = 1 .Figure( 2-15 )showsthe ippedLGRcollocationpointsaswellastheinitialpointat whichthestateis approximatedbut notcollocated forwhen N =3 andforvariousvaluesof K Inmultiple-intervalippedLGRcollocation,thestateisa pproximatedineachmeshinterval k as y ( k ) ( ) Y ( k ) ( )= N k X i =0 Y ( k ) i L ( k ) i ( ), L ( k ) i ( )= N k Y j =0 i 6 = j ( k ) j i ( k ) j (2–145) Differentiating Y ( k ) ( ) inEq.( 4–46 )withrespectto ,yields Y ( k ) ( j ) N k X i =0 Y ( k ) i L ( k ) i ( j )=[ D ( k ) Y ( k ) 0: N k ] j (2–146) where D ( k ) ij = L ( k ) i ( ) j ( i =1,..., N k j =0,..., N k ) arethecomponentsofthe N k ( N k +1) ippedLegendre-Gauss-Radau(LGR)differentiationmatri x inthe k th meshinterval. 70

PAGE 71

3 4 5 6 7 8 9 t T 0 T 1 T 2 T 3 DiscretizationPoints CollocationPointsNumberofFlippedLGRPointsPerInterval, N kFigure2-15.Distributionofmultiple-intervalFlippedLe gendre-Gauss-Radau discretizationandcollocationpointsforvariousvalueso f N k .Thedomain [ t 0 t f ] issplit intoK=3intervalssuchthat t 0 = T 0 and t f = T 3 Next,thecostfunctionalofEq.( 2–1 )isapproximatedusingamultiple-intervalLGquadrature.Thenite-dimensionalapproximationofthecontinuo us-timeoptimalcontrolproblemof Eqs.( 2–1 )–( 2–4 )isthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K )+ K Xk =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (2–147) subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (2–148) ( Y (1) 0 )= 0 (2–149) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ,( k =1,..., K ), (2–150) where w ( k ) =( w ( k ) 1 ,..., w ( k ) N k ) aretheippedLGR quadratureweights ininterval k 71

PAGE 72

CHAPTER3 COSTATEESTIMATIONUSINGTHEINTEGRALFORMULATION Aswaspreviouslydiscussed,theLegendre-Gauss(LG)andLe gendre-Gauss-Radau (LGR)methodsforapproximatingsolutionstooptimalcontr olproblemsareequivalent regardlessofwhetherthecollocationisperformedindiffe rentialorintegralform. Typically,however,thedifferentialformofeithermethod hasbeenused.Therefore,the relationshipbetweentheLagrangemultipliersofthediffe rentialformofthecollocation methodsandthecostateoftheoptimalcontrolproblemhasbe enwelldocumented. Ontheotherhand,thecorrespondingrelationshipbetweent heLagrangemultipliers associatedwiththeintegralformsofLGandLGRcollocation andthecostateofthe optimalcontrolproblemhasnotbeenestablished.Inthisch aptermethodsforestimating theoptimalcontrolcostateusingtheintegralformsofLGan dLGRcollocationare developed.Specically,transformationsarederivedthat relatetheLagrangemultipliers oftheintegralformsoftheLGandLGRcollocationmethodsto thecostateofthe originaloptimalcontrolproblem.Anewcontinuous-timedu alvariablecalledthe integralcostate isintroduced,wheretheintegralcostateistheLagrangemu ltiplierofthe integraldynamicconstraint.Therst-orderoptimalityco nditionsoftheintegralformof theoptimalcontrolproblemarederivedintermsoftheinteg ralcostate.Theintegral formoftheoptimalcontrolproblemisthendiscretizedusin gtheintegralLGandLGR collocationmethodsandrelationshipbetweenthediscrete formoftheintegralcostate andthecostateoftheoriginaldifferentialoptimalcontro lproblemaredeveloped.The approachdevelopedinthisresearchthenprovidesawaytoes timatethecostateofthe originaloptimalcontrolproblemusingtheLagrangemultip liersoftheintegralformofthe LGandLGRcollocationmethods. Thefollowingnotationandconventionswillbeusedthrough outthischapter.First, allvectorfunctionsoftimearedenotedas row vectors,thatis,if y ( ) 2 R n isavector functionofthescalarvariable ,then y ( )=[ y 1 ( ), y n ( )] .Next,anycapital 72

PAGE 73

boldfacecharacter, Y ,denotesamatrixofsize M n ,whereeachrowof Y i corresponds totheevaluationofafunction y ( ) ataparticularvalue = i .Next,thenotation Y i : j denotesrows i through j ofthematrix Y ,exceptwhenreferringtoadifferentiationmatrix D ortheintegrationmatrix A ,inwhichcase D i and A i referstothe i th column of D and A .Finally, D > denotesthetransposeofmatrix D ,and D >i denotesthetransposeofthe i th columnof D .Givenvectors x and y 2 R n ,thenotation h x y i isusedtodenotethe standardinnerproductbetween x and y .Furthermore,if f : R n R m ,then r f isthe m by n Jacobianmatrixwhose i th rowis r f i .Inparticular,thegradientofascalar-valued functionisarowvector.If : R m n R and Y isan m by n matrix,then r denotes the m by n matrixwhose ( i j ) elementis ( r ( Y )) ij = @ ( Y ) =@ Y ij Thisremainderofthischapterisorganizedasfollows.InSe ction 3.1 thecontinuous-time optimalcontrolproblemispresentedwiththedynamicconst raintsformulatedinboth differentialandintegralform,andtherst-orderoptimal ityconditionsofeachformulation aregiven.InSections 3.2 and 3.3 ,theLegendre-GaussandLegendre-Gauss-Radau collocationmethodsinbothdifferentialandintegralform sarepresented,andthe rst-orderoptimalityconditionsofeachformarederived. Furthermore,thetransformed adjointsystemoftheintegralformisderived,andacostate estimateispresentedin termsoftheLagrangemultipliersoftheintegralforms.Fin ally,Section 3.4 providesa discussionofthedifferencesbetweenthecollocationsche mesatLGandLGRpoints. 3.1Continuous-TimeBolzaOptimalControlProblem Inordertomakethisexpositionclearer,inthischapterthe Bolzacontinuous-time optimalcontrolproblemofSection 2.1 isformulatedinthedomain 2 [ 1,+1] .Itis notedthatthetimeinterval 2 [ 1,+1] canbetransformedtotheinterval [ t 0 t f ] viathe afnetransformation t = t f t 0 2 + t f + t 0 2 (3–1) Furthermore,thisproblemformulationisstatedwithnoine qualitypathconstraints. Inequalitypathconstrainedoptimalcontrolproblemswill bediscussedinChapter 5 73

PAGE 74

3.1.1DifferentialandIntegralFormsofOptimalControlPr oblem ConsideragaintheBolzacontinuous-timeoptimalcontrolp roblemfromSection 2.1 denedontheinterval 2 [ 1,+1] .Determinethestate y ( ) 2 R n andthecontrol u ( ) 2 R m thatminimizethecostfunctional J =( y (+1))+ Z +1 1 g ( y u ) d (3–2) subjecttothedynamicconstraint y ( ) f ( y u )= 0 (3–3) andtheboundarycondition ( y ( 1))= 0 (3–4) Henceforth,Eqs.( 3–2 )–( 3–4 )willbereferredtoasthe differentialoptimalcontrol problem ThedifferentialoptimalcontrolproblemgiveninEqs.( 3–2 )–( 3–4 )canbere-written inthefollowingintegralform.Inparticular,integrating thedynamicsgiveninEq.( 3–3 ), yields y ( )= y ( 1)+ Z 1 f ( y u ) d Theoptimalcontrolprobleminintegralformisthenstateda sfollows.Determinethe state, y ( ) 2 R n ,andthecontrol, u ( ) 2 R m ,thatminimizethecostfunctional J =( y (+1))+ Z +1 1 g ( y ( ), u ( )) d (3–5) subjecttotheintegralconstraint y ( ) y ( 1) Z 1 f ( y u ) dt = 0 (3–6) andtheboundarycondition ( y ( 1))= 0 (3–7) 74

PAGE 75

Henceforth,Eqs.( 3–5 )–( 3–7 )willbereferredtoasthe integraloptimalcontrolproblem 3.1.2First-OrderOptimalityConditionsofDifferentiala ndIntegralForms Therst-orderoptimalityconditionsofthedifferentialo ptimalcontrolproblem obtainedfromthecalculusofvariationswerederivedinSec tion 2.1.1 andaregivenas y ( )= f ( y u ), ( y ( 1))= 0 (3–8) 0 = r u H ( y u ), (3–9) = r y H ( y u ), (3–10) ( 1)= r y h ( 1) i (3–11) (+1)= r y (+1), (3–12) Next,therst-orderoptimalityconditionsoftheintegral optimalcontrolproblem obtainedfromapplyingthecalculusofvariationswerederi vedinSection 2.1.2 aregiven as y = y ( 1)+ Z 1 f ( y u ) d ( y ( 1))= 0 (3–13) 0 = r u g + h Z +1 p dt + r y (+1) r u f ( y u ) i (3–14) p = r y g + h Z +1 p dt + r y (+1) r y f ( y u ) i (3–15) r y h ( 1) i = Z +1 1 p d + r y (+1), (3–16) where H ( y u )= g ( y u )+ h f ( y u ) i (3–17) istheHamiltonian, ( ) isthecostateofthedifferentialoptimalcontrolproblem( and willbereferredtohenceforthasthe differentialcostate ), p ( ) isthecostateofintegral optimalproblem(andwillbereferredtohenceforthasthe integralcostate ),and isthe Lagrangemultiplierassociatedwiththeboundaryconditio nofEq.( 3–4 ).Itisnotedthat thedifferentialcostateandtheintegralcostatearediffe rentinthat ( ) istheLagrange 75

PAGE 76

multiplierassociatedwiththedifferentialequationcons traintofEq.( 3–3 )while p ( ) is theLagrangemultiplierassociatedwiththeintegralequat ionconstraintofEq.( 3–6 ). Thedifferentialandintegralcostatearerelatedas ( )= r y (+1)+ Z +1 p dt (3–18) Inparticular,substitutingEq.( 3–18 )togetherwiththerelationship ( )= p ( ) (3–19) intotherst-orderoptimalityconditionsoftheintegralo ptimalproblemasgivenin Eqs.( 3–13 )–( 3–16 )yieldsthetherst-orderoptimalityconditionsofthedif ferential optimalcontrolproblemasgiveninEqs.( 3–8 )–( 3–12 ).Theremainderofthischapteris devotedtoderivingtwodiscreteapproximationsofthediff erentialcostate, ( ) ,using discreteapproximationsoftheintegralcostate, p ( ) 3.2CostateEstimationUsingIntegralLegendre-GaussColl ocation Inthissectionacostateestimateforthedifferentialopti malcontrolproblemis developedviaanestimateoftheintegralcostateobtainedu singtheintegralformof theLegendre-Gaussorthogonalcollocationmethod.InSect ion 3.2.1 thedifferential formoftheLegendre-Gausscollocationmethodisdescribed .InSection 3.2.2 the rst-orderoptimalityconditionsofthenonlinearprogram mingproblemdescribedin Section 3.2.1 areprovided.InSection 3.2.3 theintegralformoftheLegendre-Gauss collocationmethodisdescribed.InSection 3.2.4 therst-orderoptimalityconditionsof thenonlinearprogrammingproblemdescribedinSection 3.2.1 areprovided.Finally,in Section 3.2.5 adifferentialcostateestimateusingtheintegralcostate estimatederivedin Section 3.2.4 isdeveloped. 3.2.1DifferentialFormofLegendre-GaussCollocation ThedifferentialoptimalcontrolproblemofEqs.( 3–2 )–( 3–4 )cannowbeapproximated usingcollocationatLegendre-Gauss(LG)pointsaswasdesc ribedinSection 2.5.1 .The 76

PAGE 77

LGpointsaredenedinthedomain ( 1,+1) whichdoesnotincludeeitherofthe endpoints.ThestateisapproximatedusingaLagrangepolyn omialwithsupportpoints atthe N LGpointsplusthenoncollocatedpoint 0 = 1 ,suchthat y ( ) Y ( )= N X i =0 Y i L i ( ), L i ( )= N Y j =0 j 6 = i j i j ;( i =0,..., N ). (3–20) where L i ( ),( i =0,..., N ) isabasisofLagrangepolynomialsofdegree N withsupport points ( 0 ,..., N ) .Thetimederivativeofthestateat = j isthenapproximatedas Y ( j ) N X i =0 Y i L i ( j )= [ DY 0: N ] j (3–21) where D isthe N ( N +1) Legendre-Gaussdifferentiationmatrix whoseelementsare givenas D ij = L i ( j ) ,( i =1,..., N j =0,..., N ) .Furthermore,thecostfunctionalof Eq.( 3–2 )isapproximatedusingaLegendre-Gaussquadratureas J =( Y N +1 )+ N X j =1 w j g ( Y j U j ). (3–22) Thedifferentialoptimalcontrolproblemisthenapproxima tedviathefollowingnite-dimensional nonlinearprogrammingproblem.Minimizethecostfunction ofEq.( 3–22 )subjecttothe algebraicconstraints DY 0: N = f ( Y 1: N U 1: N ), (3–23) Y N +1 = Y 0 + N X j =1 w j f ( Y j U j ), (3–24) ( Y 0 )= 0 (3–25) where w =( w 1 ,..., w N ) isavectorofLegendre-Gaussquadratureweights.Itis notedforLGcollocationthatEq.( 3–24 )providesanLGquadratureapproximation, Y N +1 ,ofthestateatthenalnoncollocatedpoint N +1 =+1 .TheNLPdescribedby Eqs.( 3–22 )–( 3–25 )willbereferredtoasthe differentialLegendre-Gausscollocation method 77

PAGE 78

3.2.2KKTConditionsUsingDifferentialLegendre-GaussCo llocation TheKarush-Kuhn-Tucker(KKT)rst-orderoptimalitycondi tionsofthedifferential Legendre-Gausscollocationmethodarenowderived[ 27 64 ].FirsttheLagrangianis formedsuchthat L =( Y N +1 )+ h w g ( Y 1: N U 1: N ) ih ( Y 0 ) i h 1: N DY 0: N f ( Y 1: N U 1: N ) i h N +1 Y N +1 Y 0 w > f ( Y 1: N U 1: N ) i where ( 1: N N +1 ) aretheKKTmultipliersassociatedwiththeconstraintsof Eqs.( 3–25 ),( 3–23 ),and( 3–24 ),respectively.Next,theKKTconditionsareobtainedby differentiatingtheLagrangianwithrespecttoallvariabl esintheNLP.Theyaregivenas DY 0: N = f ( Y 1: N U 1: N ), ( Y 0 )= 0 (3–26) Y N +1 = Y 0 + w > f ( Y 1: N U 1: N ), (3–27) 0 = r U H G ( Y 1: N U 1: N 1: N +1 ), (3–28) W 1 D >1: N 1: N = r Y H G ( Y 1: N U 1: N 1: N +1 ) (3–29) D >0 1: N = N +1 r Y h ( Y 0 ) i (3–30) N +1 = r Y ( Y N +1 ). (3–31) where H G ( Y 1: N U 1: N 1: N +1 )= 1 > g ( Y 1: N U 1: N )+ h W 1 1: N + 1 N +1 f ( Y 1: N U 1: N ) i (3–32) isthediscreteHamiltonian.Furthermore, 1 isan N 1 columnvectorofallones,and W isa N N diagonalmatrixofLGweights.Supposenowthatthefollowin gtransformed 78

PAGE 79

dualvariablesareintroduced: 1: N = W 1 1: N + 1 N +1 (3–33) N +1 = N +1 (3–34) 0 = N +1 D >0 1: N (3–35) Inaddition,considerthe N ( N +1) matrix D y D y ij = w j w i D ji ,and D y i N +1 = N X j =1 D y ij (3–36) for ( i j =1,..., N ) .ItwasshowninRef.[ 36 ]that D y isadifferentiationmatrixforthe spaceofpolynomialsofdegree N .Inotherwords,if b isapolynomialofdegreeatmost N and b 2 R N +1 isthevectorwhose i th elementis b i = b ( i ) ,then ( D y b ) i = b ( i ). (3–37) UsingthetransformationsdescribedinEqs.( 3–33 )–( 3–35 )alongwithEq.( 3–36 ),the KKTconditionsofthedifferentialLegendre-Gausscolloca tionmethodofEqs.( 3–26 )–( 3–31 ) canbewrittenas[ 64 ] DY 0: N = f ( Y 1: N U 1: N ), ( Y 0 )= 0 (3–38) Y N +1 = Y 0 + w T f ( Y 1: N U 1: N ), (3–39) 0 = r U H ( Y 1: N U 1: N 1: N ), (3–40) D y 1: N +1 = r Y H ( Y 1: N U 1: N 1: N ), (3–41) 0 = r Y h ( Y 0 ) i (3–42) N +1 = r Y ( Y N +1 ), (3–43) where H isadiscreteformoftheHamiltoniangivenbyEq.( 3–17 ).Itisseenby examinationthatEqs.( 3–38 )–( 3–43 )isadiscreteformofEqs.( 3–8 )–( 3–12 ). 79

PAGE 80

Itisnotedinthistranscriptionthatthestateisbeingdiff erentiatedbyamatrix D [givenbyEq.( 3–21 )]whichisbasedonthederivativesofpolynomialsofdegree N with coefcientsatthe N LGpointsplustheinitialnoncollocatedpoint 0 = 1 ,whereasthe costateisbeingdifferentiatedbyamatrix D y [givenbyEq.( 3–36 )]whichisbasedon thederivativesofpolynomialsofdegree N withcoefcientsatthe N LGpointsplusthe terminalnoncollocatedpoint N +1 =+1 3.2.3IntegralFormofLegendre-GaussCollocation Theintegraloptimalcontrolproblemisnowdiscretizedusi ngtheintegralformof Legendre-Gausscollocation.IthasbeenshowninRef.[ 64 ]thatthedifferentialLG transcriptionmethodgiveninSection 3.2.1 canequivalentlybeexpressedasanimplicit integrationscheme.Inparticular,let p beanypolynomialofdegreeatmost N .Bythe constructionofthe N ( N +1) matrix D ,then Dp =_ p where p i = p ( i ),( i =0,..., N ), p i =_ p ( i ),( i =0,..., N ). (3–44) Now,let 1 bea N 1 columnvectorcomposedofones.since D isadifferentiation matrix,itfollowsthatthecomponentsofthevector D1 arethederivativesatthe collocationpointsoftheconstantpolynomial p ( )=1 .Therefore, D1 = 0 ,which impliesthat D1 = D 0 + D 1: N 1 = 0 .Rearranging, D 0 = D 1: N 1 (3–45) IthasbeenshownbyRef.[ 64 ]thatthematrix D 1: N isfullrank.Therefore,multiplyingby D 1 1: N givestherelationship D 1 1: N D 0 = 1 (3–46) Furthermore,theexpression p canequivalentlybewrittenas p = Dp = D 0 p 0 + D 1: N p 1: N (3–47) 80

PAGE 81

premultiplyingby D 1 1: N andutilizingrelationshipgivenbyEq.( 3–46 ),thefollowingis obtained p i = p 0 + D 1 1: N p i ,( i =1,..., N ). (3–48) Now,toshowthatthe N N matrix D 1 1: N isanintegrationmatrix,adifferent expressionfor p i p 0 canbeobtainedbasedontheintegrationoftheinterpolanto fthe derivative.Let L yj ( ) betheLagrangepolynomialbasisgivenby L yj = N Y i =1 i 6 = j i j i ,( j =1,..., N ). (3–49) NoticethattheLagrangepolynomials L i denedinthedifferentialproblemformulation, givenbyEq.( 3–20 )aredegree N whiletheLagrangepolynomials L yj aredegree N 1 Thenbecause p isapolynomialofdegreeatmost N 1 ,itcanbeinterpolatedexactly bytheLagrangepolynomials L yj : p = N X j =1 p j L yj ( ) (3–50) Integrating p from 1 to i ,thefollowingrelationshipisobtained p ( i )= p ( 1)+ N X j =1 p j A ij A ij = Z i 1 L yj ( ) d ,( i =1,..., N ), (3–51) whichcanequivalentlybewrittenas p i = p 0 + ( A p ) i ,( i =1,..., N ). (3–52) Therelations( 3–48 )and( 3–52 )aresatisedforanypolynomialofdegreeatmost N .By equating( 3–48 )and( 3–52 )thefollowingbecomestrue A p = D 1 1: N p 81

PAGE 82

Thus,thedynamicconstraintsofEq.( 3–6 )canbeapproximatedas Y 1: N = 1 Y 0 + Af ( Y 1: N U 1: N ), (3–53) where 1 denotesa N 1 columnvectoroftheconstant 1 foreverycomponent,and A = D 1 1: N isthe N N Legendre-Gaussintegrationmatrix denedbyEq.( 3–51 ). ThusthestateateachLegendre-Gausspointisapproximated viaquadratureusingthe Legendre-Gaussintegrationmatrix.Next,thestateat =+1 isapproximatedusinga Legendre-Gaussquadratureas Y N +1 = Y 0 + N X i =1 w i f ( Y i U i ). UsingthesepropertiesofLGcollocation,theintegralopti malcontrolproblemof Eqs.( 3–5 )–( 3–7 )canbeapproximatedviathefollowingnite-dimensionaln onlinear programmingproblem.MinimizethecostfunctionofEq.( 3–22 )subjecttothealgebraic constraints Y 1: N = 1 Y 0 + Af ( Y 1: N U 1: N ), (3–54) Y N +1 = Y 0 + w T f ( Y 1: N U 1: N ), (3–55) ( Y 0 )= 0 (3–56) TheNLPdescribedbyEqs.( 3–22 )and( 3–54 )–( 3–56 )willbereferredtoasthe integral Legendre-Gausscollocationmethod 3.2.4KKTConditionsUsingIntegralLegendre-GaussColloc ation Similarlyaswasdoneforthedifferentialformofthediffer entialLGtranscription,the Karush-Kuhn-Tucker(KKT)rst-orderoptimalityconditio nsoftheintegralLegendre-Gauss 82

PAGE 83

collocationmethodarenowderived.First,theLagrangiani sdenedas L =( Y N +1 )+ h w g ( Y 1: N U 1: N ) ih ( Y 0 ) i h P 1: N Y 1: N 1 Y 0 Af ( Y 1: N U 1: N ) i h N +1 Y N +1 Y 0 w > f ( Y 1: N U 1: N ) i where ( P 1: N N +1 ) aretheKKTmultipliersassociatedwiththeconstraintsof Eqs.( 3–56 ),( 3–54 ),and( 3–55 ),respectively.Next,theKKTconditionsareobtainedby differentiatingtheLagrangianwithrespecttoallfreevar iablesintheNLP.Theyaregiven as Y 1: N = 1 Y 0 + Af ( Y 1: N U 1: N ), ( Y 0 )= 0 Y N +1 = Y 0 + w > f ( Y 1: N U 1: N ), (3–57) 0 = r U w > g ( Y 1: N U 1: N )+ h A > P f ( Y 1: N U 1: N ) i + h N +1 w > f ( Y 1: N U 1: N ) i (3–58) P 1: N = r Y w > g ( Y 1: N U 1: N )+ h A > P f ( Y 1: N U 1: N ) i + h N +1 w > f ( Y 1: N U 1: N ) i (3–59) N +1 = r Y h ( Y 0 ) i 1 > P 1: N (3–60) N +1 = r Y ( Y N +1 ), (3–61) Supposenowthatthefollowingtransformeddualvariablesa reintroduced: p 1: N = W 1 P 1: N N +1 = N +1 (3–62) Inaddition,considerthe N N matrix A y A yij = w j w i A ji (3–63) for ( i j =1,..., N ) .Itwillnowbeproventhat A y isabackwardintegrationmatrixforthe spaceofpolynomialsofdegree N 1 Theorem1. Thematrix A y describedbyEq.( 3–63 )isabackwardsintegrationmatrixfor thespaceofpolynomialsofdegree N 1 .Morespecically,if p isapolynomialofdegree 83

PAGE 84

atmost N 1 and p 2 R N isthevectorwith i th component p i = p ( i ),( i =1,..., N ) ,then ( A y p ) i = Z +1 i p ( t ) dt (3–64) Proof. Let p q denotepolynomialsofdegreeatmost N 1 suchthat p j = p ( j ) and q j = q ( j ) ( j =1,..., N ) .Theorderofintegrationinadoubleintegralcanbeswitche dby changingthelimitsofintegrationsuchthat Z +1 1 q ( ) Z 1 p ( t ) dt d = Z +1 1 p ( ) Z +1 q ( t ) dt d (3–65) Now,since p and q arepolynomialsofdegreeatmost N 1 ,then p R +1 qdt and q R 1 pdt arepolynomialsofdegreeatmost 2 N 1 .SinceGaussquadratureisexact forpolynomialsofdegree 2 N 1 ,theintegralsinEq.( 3–65 )canbereplacedbytheir quadratureequivalentstoobtain N X j =1 w j q j Z 1 p ( t ) dt = N X j =1 w j p j Z +1 q ( t ) dt (3–66) Substituting R 1 p ( t ) dt = Ap and R +1 q ( t ) dt = A y q thefollowingexpressionisobtained ( Wq ) > Ap =( A y q ) > Wp q > ( A > W WA y ) p =0. (3–67) Since p and q arearbitraryvectors,itmustbetruethat A > W WA y = 0 (3–68) whichimpliesthat A yij = w j w i A ji ,( i j =1,..., N ). Furthermore,apolynomialofdegree N 1 isuniquelydenedbyitsvalueat N points, andcanthusbeexactlyinterpolatedbyaLagrangeinterpola tingpolynomialofdegree 84

PAGE 85

N 1 .Therefore: Z +1 i q ( ) d = N X j =1 A yij q j A yij = Z +1 i L yj ( ) d (3–69) where L y isthebasisofinterpolatingpolynomialsofdegree N 1 denedinEq.( 3–51 ). UsingthetransformationsdescribedinEq.( 3–62 )alongwiththedenitionof A y in Eq.( 3–63 ),theKKTconditionsoftheintegralformoftheLegendre-Ga usscollocation methodasshowninEqs.( 3–57 )–( 3–61 )canbewrittenas Y 1: N = 1 Y 0 + Af ( Y 1: N U 1: N ), ( Y 0 )= 0 Y N +1 = Y 0 + w > f ( Y 1: N U 1: N ), (3–70) 0 = r U g ( Y 1: N U 1: N )+ h A y p 1: N + r Y ( Y N +1 ) r U f ( Y 1: N U 1: N ) i (3–71) p 1: N = r Y g ( Y 1: N U 1: N )+ h A y p 1: N + r Y ( Y N +1 ) r Y f ( Y 1: N U 1: N ) i (3–72) w > p 1: N = r Y h ( Y 0 ) ir Y ( Y N +1 ), (3–73) N +1 = r Y ( Y N +1 ). (3–74) ItisseenbyexaminationthatEqs.( 3–70 )–( 3–74 )areadiscreteformofthenecessary conditionsforoptimalityoftheintegraloptimalcontrolp roblemgiveninEqs.( 3–13 )–( 3–16 ). ItmustbenotedthatintheintegralLegendre-Gausscolloca tionmethodthis transcriptionthatthestate, y ( ) ,andthedualvariable, p ( ) ,areapproximatedusing anintegrationmatrixforthespaceofpolynomialsofdegree N 1 ,thedifferencebeing thatthestateisapproximatedusingaforwardquadratureus ingthematrix A givenin Eq.( 3–51 )whiletheintegralcostate, p ( ) ,isapproximatedusingabackwardquadrature usingthematrix A y giveninEq.( 3–69 ). 3.2.5DifferentialCostateEstimateUsingIntegralLegend re-GaussCollocation TheresultsofSections 3.2.1 – 3.2.4 cannowbeusedtodeneanestimate forthedifferentialcostateusingtheestimateoftheinteg ralcostate.Inparticular, thetransformednecessaryconditionsofEq.( 3–38 )–( 3–43 )areequivalenttothe 85

PAGE 86

transformednecessaryconditionsofEqs.( 3–70 )–( 3–73 )ifthediscreteapproximations of ( ) and p ( ) arerelatedas D y 1: N +1 = p 1: N (3–75) where D y isasdenedinEq.( 3–36 ). IthasbeenshownbyRef.[ 64 ]thatthematrix D y hasthefollowingproperties thataresimilartothepropertiesthematrix D :(a)thesquarematrix D y1: N obtainedby removingthelastcolumnof D y isfull-rank,and(b) [ D y1: N ] 1 D yN +1 = 1 .Usingthese properties,Eq.( 3–75 )canberewrittenas 1: N = 1 N +1 [ D y1: N ] 1 p 1: N 1: N = 1 N +1 + A y p 1: N (3–76) wherethematrix A y istheintegrationmatrixdenedbyEq.( 3–69 ).Fromthisrelationship, itcanbeseenthatthedifferentialandintegraldualvariab leestimatesarerelated by [ D y1: N ] 1 = A y .Furthermore,thecostateattheinitialnoncollocatedpoi nt 0 is approximatedthroughaGaussianquadraturesuchthat 0 = N +1 + w > p 1: N (3–77) ItcanbeseenthatapplyingthetransformationsofEqs.( 3–75 )–( 3–77 )tothetransformed rst-orderoptimalityconditionsgiveninEqs.( 3–70 )–( 3–74 )willresultinthetransformed rst-orderoptimalityconditionsgiveninEqs.( 3–38 )–( 3–43 ). 3.3CostateEstimationUsingIntegralLegendre-Gauss-Rad auCollocation Inthissectionacostateestimateforthedifferentialopti malcontrolproblemis developedviaanestimateoftheintegralcostateobtainedu singtheintegralform oftheLegendre-Gauss-Radauorthogonalcollocationmetho d.InSection 3.3.1 the differentialformoftheLegendre-Gausscollocationmetho disrevisited.InSection 3.3.2 therst-orderoptimalityconditionsofthenonlinearprog rammingproblemdescribed 86

PAGE 87

inSection 3.3.1 arederived.InSection 3.3.3 theintegralformoftheLegendre-Gauss collocationmethodisdescribed.InSection 3.3.4 therst-orderoptimalityconditionsof thenonlinearprogrammingproblemdescribedinSection 3.3.1 arederived.Finally,in Section 3.3.5 adifferentialcostateestimateusingtheintegralcostate estimatederivedin Section 3.3.4 ispresented. 3.3.1DifferentialFormofLegendre-Gauss-RadauCollocat ion ThedifferentialoptimalcontrolproblemofEqs.( 3–2 )–( 3–4 )cannowbeapproximated usingcollocationatLegendre-Gauss-Radau(LGR)pointsas wasdescribedinSection 2.5.2 .TheLGRpointsaredenedinthedomain [ 1,+1) suchthat 1 = 1 isaLGR collocationpointbut N +1 =+1 isanoncollocatedpoint.Thestateisapproximated usingaLagrangepolynomialwithsupportpointsatthe N LGRpointsplusthe noncollocatedpoint N +1 =+1 ,suchthat y ( ) Y ( )= N +1 X i =1 Y i L i ( ), L i ( )= N +1 Y j =1 j 6 = i j i j ;( i =1,..., N +1). (3–78) where L i ( ),( i =1,..., N +1) isabasisofLagrangepolynomialsofdegree N withsupportpoints ( 1 ,..., N +1 ) .Thetimederivativeofthestateat = j isthen approximatedas Y ( j ) N +1 X i =1 Y i L i ( j )= [ DY 1: N +1 ] j (3–79) where D isthe N ( N +1) Legendre-Gauss-Radaudifferentiationmatrix whose elementsaregivenas D ij = L i ( j ) ,( i =1,..., N j =1,..., N +1) .Furthermore,the costfunctionalofEq.( 3–2 )isapproximatedusingaLegendre-Gauss-Radauquadrature as J =( Y N +1 )+ N X j =1 w j g ( Y j U j ). (3–80) Thedifferentialoptimalcontrolproblemisthenapproxima tedviathefollowingnite-dimensional nonlinearprogrammingproblem.Minimizethecostfunction ofEq.( 3–80 )subjecttothe 87

PAGE 88

algebraicconstraints DY 1: N +1 = f ( Y 1: N U 1: N ), (3–81) ( Y 1 )= 0 (3–82) where w =( w 1 ,..., w N ) isavectorofLegendre-Gauss-Radauquadratureweights.Th e NLPdescribedbyEqs.( 3–80 )–( 3–82 )willbereferredtoasthe differentialLegendreGauss-Radaumethod 3.3.2KKTConditionsUsingDifferentialLegendre-Gauss-R adauCollocation SimilarlyaswasdoneforcollocationattheLGpoints,theKa rush-Kuhn-Tucker (KKT)rst-orderoptimalityconditionsofthedifferentia lLegendre-Gauss-Radau collocationmethodarenowderived[ 64 ].FirsttheLagrangianisformedsuchthat L =( Y N +1 )+ h w g ( Y 1: N U 1: N ) ih ( Y 1 ) i h 1: N DY 1: N +1 f ( Y 1: N U 1: N ) i where ( 1: N ) aretheKKTmultipliersassociatedwiththeconstraintsofE qs.( 3–82 ) and( 3–81 ),respectively.Next,theKKTconditionsareobtainedbydi fferentiatingthe LagrangianwithrespecttoallvariablesintheNLP.Theyare givenas DY 1: N +1 = f ( Y 1: N U 1: N ), ( Y 1 )= 0 (3–83) 0 = r U H R ( Y 1: N U 1: N 1: N ), (3–84) D >1: N 1: N = r Y H R ( Y 1: N U 1: N 1: N ) e 1 r Y h ( Y 1 ) i (3–85) D >N +1 1: N = r Y ( Y N +1 ), (3–86) where H R ( Y 1: N U 1: N 1: N )= w > g ( Y 1: N U 1: N )+ h 1: N f ( Y 1: N U 1: N ) i (3–87) 88

PAGE 89

isthediscreteHamiltonian,and e 1 istherstcolumnoftheidentitymatrix.Supposenow thatthefollowingtransformeddualvariablesareintroduc ed: 1: N = W 1 1: N (3–88) N +1 = D >N +1 1: N (3–89) Inaddition,considerthe N N matrix D y D y 11 = D 11 1 w 1 and D y ij = w j w i D ji otherwise (3–90) for ( i j =1,..., N ) .ItwasshowninRef.[ 36 ]that D y isadifferentiationmatrixforthe spaceofpolynomialsofdegree N 1 .Inotherwords,if b isapolynomialofdegreeat most N 1 and b 2 R N isthevector i th element b i = b ( i ) ,then ( D y b ) i = b ( i ). (3–91) UsingthetransformationsdescribedinEqs.( 3–88 )–( 3–89 )alongwithEq.( 3–90 ), theKKTconditionsofdifferentialLegendre-Gauss-Radauc ollocationmethodof Eqs.( 3–83 )–( 3–86 )canbewrittenas[ 36 ] DY 1: N +1 = f ( Y 1: N U 1: N ), ( Y 1 )= 0 (3–92) 0 = r U H ( Y 1: N U 1: N 1: N ), (3–93) D y 1: N = r Y H ( Y 1: N U 1: N 1: N )+ e 1 w 1 ( r Y h ( Y 1 ) i 1 ) (3–94) N +1 = r Y ( Y N +1 ), (3–95) Theseequationsareincompletebecauseanewvariable N +1 wasintroducedwithout addinganewequation.Anequationforthisnewvariablecanb edevelopedby manipulatingthematrix D .Because D isadifferentiationmatrix,ithastheproperty that D 1 = 0 ,where 1 isavectorwhosecomponentsareallconstantandequalto1.T his 89

PAGE 90

impliesthat D N +1 = N X j =1 D 1: N j D >N +1 = N X i =1 D i N +1 i = N X i =1 N X j =1 D i j i N +1 = 1 + N X i =1 N X j =1 w i j D yi j = 1 + N X j =1 w j ( D y ) j (3–96) wheretherelationshipsin( 3–88 )and( 3–91 )wereusedtoobtainEq.( 4–70 ).Itcanbe seenthatthisrelationshipapproximatestheintegralofth ecostatedynamicsacrossthe domainviaaRadauquadrature.CombiningEqs.( 3–92 )–( 3–95 )withEq.( 4–70 ),the completetransformedadjointsystemcanthenbewrittenas DY 1: N +1 = f ( Y 1: N U 1: N ), ( Y 1 )= 0 (3–97) 0 = r U H ( Y 1: N U 1: N 1: N ), (3–98) D y 1: N = r Y H ( Y 1: N U 1: N 1: N )+ e 1 w 1 ( r Y h ( Y 1 ) i 1 ) (3–99) N +1 = r Y h ( Y 0 ) i N X i =1 w i r Y H ( Y i U i i ), (3–100) N +1 = r Y ( Y N +1 ), (3–101) where H isadiscreteformoftheHamiltoniangivenbyEq.( 3–17 ).Itisseenin Eq.( 3–100 )thatthecostateatthenoncollocatednalpoint =+1 isapproximated viaaLegendre-Gauss-Radauquadratureofthecostatedynam icsacrossthesolution domain.Consequently,Eq.( 3–100 )isasubtlewayofenforcingtherelationship 1 = r Y h ( Y 1 ) i anditisexpectedthatthelasttermofEq.( 3–99 )willbesmallwhilethe remainingtermsinEq.( 3–99 )areacollocationcollocationschemeforthecontinuous adjointequation.Thetransformedoptimalityconditionso fEqs.( 3–97 )–( 3–101 )are, thus,adiscreteformofthenecessaryconditionsforoptima lityofthedifferentialoptimal controlproblemdescribedbyEqs.( 3–8 )–( 3–12 ).Itisnotedintheseconditionsthat 90

PAGE 91

thetimederivativeofthestateisbeingapproximatedusing thedifferentiationmatrix D forthespaceofpolynomialsofdegree N [seeEq.( 3–79 )],whilethecostateisbeing differentiatedbyadifferentiationmatrix D y forthespaceofpolynomialsofdegree N 1 [seeEq.( 3–90 )]. 3.3.3IntegralFormofLegendre-Gauss-RadauCollocation Theintegraloptimalcontrolproblemisnowdiscretizedusi ngtheintegralformof Legendre-Gauss-Radaucollocation.IthasbeenshowninRef .[ 64 ]thatthedifferential LGtranscriptionmethodgiveninSection 3.2.1 canequivalentlybeexpressedasan implicitintegrationscheme.Inparticular,supposethat p isanypolynomialofdegree atmost N .Then,bytheconstructionofthe N ( N +1) differentiationmatrix D ,then Dp = _p where p i = p ( i ),( i =1,..., N +1), i =_ p ( i ),( i =1,..., N +1). (3–102)Now,let 1 bea N 1 columnvectorcomposed ofones.since D isadifferentiationmatrix,itfollowsthatthecomponents ofthevector D1 arethederivativesatthecollocationpointsoftheconstan tpolynomial p ( )=1 Therefore, D1 = 0 ,whichimpliesthat D1 = D 1: N 1 + D N +1 = 0 .Rearranging, D N +1 = D 1: N 1 (3–103) IthasbeenshownbyRef.[ 64 ]thatthematrix D 1: N isfullrank.Therefore,pre-multiplying by D 1 1: N givestherelationship D 1 1: N D N +1 = 1 (3–104) Furthermore,theexpression p canequivalentlybewrittenas p = Dp = D 1: N p 1: N + D N +1 p N +1 (3–105) premultiplyingby D 1 1: N andutilizingtherelationshipgivenbyEq.( 3–104 ),thefollowingis obtained p i = p N +1 + D 1 1: N p i ,( i =1,..., N ). (3–106) 91

PAGE 92

Next,let ~ L j ( ) beLagrangepolynomialbasisgivenby ~ L j ( )= N Y i =1 i 6 = j i j i ,( j =1,..., N ). Then p canbeinterpolatedexactlyas p ( )= N X j =1 p j ~ L j ( ). Integratingfrom +1 to i p i = p N +1 +( A_p ) i ,( i =1,..., N ), (3–107) A ij = Z i +1 ~ L j ( ) d ,( i j =1,..., N ). (3–108) By( 3–106 )and( 3–107 ),itcanbeseenthat A = D 1 1: N .Thus,thedynamicconstraintsof Eq.( 3–6 )canbeapproximatedas Y 1: N = 1 Y N +1 + Af ( Y 1: N U 1: N ), (3–109) where 1 denotesa N 1 columnvectoroftheconstant 1 foreverycomponent,and A = D 1 1: N isthe N N Legendre-Gauss-Radauintegrationmatrix givenbyEq.( 3–108 ). Theintegraloptimalcontrolproblemcanthenbeapproximat edviathefollowing nite-dimensionalnonlinearprogrammingproblem.Minimi zethecostfunctionof Eq.( 3–80 )subjecttothealgebraicconstraints Y 1: N = 1 Y N +1 + Af ( Y 1: N U 1: N ), (3–110) ( Y 1 )= 0 (3–111) TheNLPdescribedbyEqs.( 3–80 ),( 3–110 ),and( 3–111 )willbereferredtoasthe integralLegendre-Gauss-Radaucollocationmethod 92

PAGE 93

3.3.4KKTConditionsUsingIntegralLegendre-Gauss-Radau Collocation TheKarush-Kuhn-Tucker(KKT)rst-orderoptimalitycondi tionsoftheintegral Legendre-Gauss-Radaucollocationmethodaregivenas Y 1: N = 1 Y N +1 + Af ( Y 1: N U 1: N ), ( Y 1 )= 0 (3–112) 0 = r U w > g ( Y 1: N U 1: N )+ h A > P 1: N f ( Y 1: N U 1: N ) i (3–113) P 1: N = r Y w > g ( Y 1: N U 1: N )+ h A > P 1: N f ( Y 1: N U 1: N ) i e 1 r Y h ( Y 1 ) i (3–114) N X i =1 P i = r Y ( Y N +1 ). (3–115) where ( P 1: N ) aretheKKTmultipliersassociatedwiththeconstraintsofE qs.( 3–110 ) and( 3–111 ),respectively.Supposenowthatweintroducethefollowin gtransformed dualvariables: p 2: N = W 1 2: N ,2: N P 2: N N +1 = N X i =1 P i (3–116) Inaddition,supposeweintroducethe N N matrix A y A yij = w j w i A ji + w j (3–117) for ( i j =1,..., N ) .Wewillnowprovethat A y isabackwardintegrationmatrixforthe spaceofpolynomialsofdegree N 2 Theorem2. Thematrix A y describedbyEq.( 3–117 )isabackwardsintegrationmatrix forthespaceofpolynomialsofdegree N 2 .Morespecically,let p beapolynomialof degreeatmost N 2 and p 2 R N 1 bethevectorwith i th component p i = p ( i ),( i = 2,..., N ) .Sinceapolynomialofdegree N 2 isuniquelydenedbyitsvaluesat N 1 points, p canbeexpressedexactlyas p ( )= N X j =2 p i L yj ( ), L yj ( )= N Y i =2 j 6 = i i j i (3–118) 93

PAGE 94

Furthermore,let p 1 = p ( 1 ) betheextrapolatedvalueof p ( ) evaluatedat 1 suchthat p ( 1 ) p 1 = N X j =2 p j L yj ( 1 ). (3–119) Thenthematrix A y approximatestheintegral ( A y p ) i = Z +1 i p ( t ) dt (3–120) Proof. Let p denotethepolynomialsofdegreeatmost N 2 denedinthestatement ofthetheorem.Furthermore,let q denoteapolynomialofdegree N 1 whichsatises q j = q ( j ) ( j =1,..., N ) .If p and q aresmooth,real-valuedfunctionsthenintegrationby partsgives Z +1 1 p ( ) Z +1 q ( t ) dt d = Z 1 +1 p ( ) d Z 1 +1 q ( ) d + Z +1 1 q ( ) Z +1 p ( t ) dt d (3–121) Now,since p isapolynomialofdegreeatmost N 2 and q isapolynomialofdegreeat most N 1 ,then q R +1 pdt and p R +1 qdt arepolynomialsofdegreeatmost 2 N 2 SinceRadauquadratureisexactforpolynomialsofdegree 2 N 2 ,theintegralsin Eq.( 3–121 )canbereplacedbytheirquadratureequivalentstoobtain N X j =1 w j p j Z +1 q ( t ) dt = N X i =1 w i p i N X j =1 w j p j + N X j =1 w j q j Z +1 p ( t ) dt (3–122) Substituting R +1 q ( t ) dt = Aq and R +1 p ( t ) dt = A y p withtherstcolumnof A y consisting ofa N 1 columnvectorofzeros,thefollowingexpressionisobtaine d ( Aq ) > Wp = ( Wq ) > 1 N N Wp +( Wq ) > A y p q > ( A > W + W 1 N N W WA y ) p =0. (3–123) Since p and q arearbitraryvectors,then A > W + W 1 N N W WA y = 0 (3–124) 94

PAGE 95

whichimpliesthat A yij = w j w i A ji + w j ,( i j =1,..., N ). Ithaspreviouslybeenshownthatthematrix ~ A denedas ~ A ij = w j w i A ji ,( i j =1,..., N ) (3–125) hastheform ~ A ij = Z i 1 L yj ( ) d + w 1 L yj ( 1 ),( i j =2,..., N ), ~ A i 1 = w 1 ,( i =1,..., N ), ~ A 1 j = w 1 L yj ( 1 ),( j =2,..., N ). (3–126) Since A yij = ~ A ij + w j ,for ( i j =1,..., N ) ,then A y hastheform A yij = Z i 1 L yj ( ) d + w 1 L yj ( 1 )+ w j ,( i j =2,..., N ), A yi 1 =0,( i =1,..., N ), A y1 j = w 1 L yj ( 1 )+ w j ,( j =2,..., N ). (3–127) Tobetterunderstandthestructureofthematrix A y ,supposethe N 1 vector p is multipliedby A y where p i = p ( i ) and p ( ) isthepolynomialofdegree N 2 denedin Eq.( 3–118 ).Theresultingoperationyields ( A y p ) i = N X j =2 w 1 p j L yj ( 1 )+ N X j =2 w j p j N X j =2 p j Z i 1 L yj ( ). (3–128) Thesummation P Nj =2 p j L yj ( 1 ) extrapolatesthe ( N 2) th -degreepolynomial p ( ) to theinitialpoint 1 .ThereforethesumoftherstandsecondtermsinEq.( 3–128 )is aLegendre-Gauss-Radauquadratureapproximationof R +1 1 p ( ) d .Furthermore, because p ( ) isapolynomialofdegree N 2 itcanbeinterpolatedexactlyasdened inEq.( 3–118 ).ThereforethenaltermofEq.( 3–128 )approximates R 1 p ( ) d .This showsthat A y isthusanintegrationmatrixforthespaceofpolynomialsof degree N 2 95

PAGE 96

whichapproximates ( A y p ) i = Z +1 i p ( ) d ( ). Recallthattherstcolumnof A y isallzeros(thatis, A y1 = 0 ),andthat p isbeing approximatedbyapolynomialofdegree N 2 whichisuniquelydenedbyitsvaluesat N 1 points.UsingthetransformationsdescribedinEq.( 3–116 )alongwithEq.( 3–117 ), theKKTconditionsoftheintegralLegendre-Gauss-Radauco llocationmethodasshown inEqs.( 3–112 )–( 3–115 )canbewrittenas Y 1: N = 1 Y 1 + Af ( Y 1: N U 1: N ), ( Y 1 )= 0 (3–129) 0 = r U g ( Y 1: N U 1: N )+ h A y2: N p 2: N + r Y ( Y N +1 ) r U f ( Y 1: N U 1: N ) i (3–130) p 2: N = r Y g ( Y 2: N U 2: N )+ h A y2: N ,2: N p 2: N + r Y ( Y N +1 ) r Y f ( Y 2: N U 2: N ) i (3–131) 1 w 1 ( P 1 + r Y h ( Y 1 ) i ) = r Y g ( Y 1 U 1 )+ h A y1,2: N p 2: N + r Y ( Y N +1 ) r Y f ( Y 1 U 1 ) i (3–132) P 1 = N X i =2 w i p i r Y ( Y N +1 ). (3–133) Substitutingthevaluesof p i ( i =2,..., N ) ,and P 1 obtainedfromEqs.( 3–131 )and ( 3–132 ),respectively,intoEq.( 3–133 )yields N X i =1 w i r Y g ( Y i U i )+ h A yi ,2: N p 2: N + r Y ( Y N +1 ), r Y f ( Y i U i ) i = r Y h ( Y 1 ) ir Y ( Y N +1 ), (3–134) Equation( 3–134 )isaLegendre-Gauss-Radauquadratureapproximationtoth e continuous-timeconditionofEq.( 3–16 ).ItcanbeseenthatEqs.( 3–130 )and( 3–131 ) areapproximationstothecontinuous-timeoptimalitycond itionsofEqs.( 3–14 )and ( 3–15 ),respectively.Therefore,ithasbeenshownthatEqs.( 3–129 )–( 3–131 )along withEq.( 3–134 )formasetofnecessaryconditionsforoptimalityandaread iscrete 96

PAGE 97

approximationtothenecessaryconditionsforoptimalityo ftheintegralformofthe optimalcontrolproblemasdenedinEqs.( 3–5 )–( 3–7 ). 3.3.5DifferentialCostateEstimateUsingIntegralLegend re-Gauss-Radau Collocation TheresultsofSections 3.3.1 – 3.3.4 cannowbeusedtodeneanestimateforthe differentialcostateusingtheintegralcostateestimatef romtheLegendre-Gauss-Radau collocationmethod.Inparticular,thetransformednecess aryconditionsofEq.( 3–97 )–( 3–101 ) areequivalenttothetransformednecessaryconditionsofE q.( 3–129 )–( 3–131 )along withEq.( 3–134 )ifthediscreteapproximationsof ( ) and p ( ) arerelatedas 1: N = N +1 + A y p 2: N (3–135) wherethematrix A y istheintegrationmatrixforthespaceofpolynomialsofdeg ree N 2 denedinEq.( 3–117 ).Equation( 3–135 )canequivalentlybewrittenindifferentialform suchthat D y 1: N = p 1: N (3–136) wherethematrix D y isadifferentiationmatrixforthespaceofpolynomialsofd egree N 1 denedbyEq.( 3–90 ).ItcanbeseenthatapplyingthetransformationofEq.( 3–135 ) totheoptimalityconditionsoftheintegralLegendre-Gaus s-Radaucollocationmethod asgiveninEqs.( 3–129 )–( 3–131 )alongwithEq.( 3–134 )willresultintherst-order optimalityconditionsofproblemthedifferentialLegendr e-Gauss-Radaucollocation methodasgiveninEqs.( 3–97 )–( 3–101 ). 3.4Discussion Whileitmayappearatrstglanceasifthecostateestimateu singeithertheintegral Legendre-GaussorintegralLegendre-Gauss-Radaumethodi sthesame,theestimate obtainedusingeitherofthemethodshasnuancesthatdistin guishitfromtheestimate obtainedusingtheothermethod.First,the N N Legendre-Gauss-Radaudifferentiation matrixofEq.( 3–90 )issingularforthespaceofpolynomialsofdegree N 1 ,whilethe 97

PAGE 98

N ( N +1) Legendre-GaussdifferentiationmatrixofEq.( 3–36 )isfullrankforthespace ofpolynomialsofdegree N .Second,the N N matrix A y inEq.( 3–117 )associatedwith theintegralLegendre-Gauss-Radaumethodissingularands imultaneouslyintegrates andinterpolatesapolynomialofdegree N 2 ,whilethematrix N N matrix A y in Eq.( 3–69 )isanintegrationmatrixforthespaceofpolynomialsofdeg ree N 1 .Finally, althoughbothintegralcollocationmethodsprovideestima tesofthedifferentialcostate, ( ) ,atalldiscretizationpointsinthedomain(includingboth endpoints),theintegral Legendre-Gausscollocationmethodproducesanapproximat ionofintegralcostate, p ( ) ,atonlythe N Legendre-GausspointswhiletheintegralLegendre-GaussRadau collocationmethodprovidesestimatesof p ( ) at N 1 interiorLegendre-Gauss-Radau pointsand extrapolates thevalueof p ( ) totheinitial( = 1 )Legendre-Gauss-Radau point. 3.5ConcludingRemarks Amethodwaspresentedforcostateestimationofanoptimalc ontrolproblem usingorthogonalcollocationatLegendre-GaussandLegend re-Gauss-Radaupoints whenthedynamicconstraintsarepresentedinintegralform .Akeyfeatureofthese collocationschemesisthattheinverseofthematrixassoci atedwithanimplicitLG(or LGR)integrationschemeistheLG(orLGR)differentiationm atrix.Hence,themethods presentedinthischaptercanbethoughtofaseitheranimpli citintegrationmethodor adifferentialmethod.ItwasshownthattheKKTmultipliers stemmingfromtheimplicit integrationtranscriptioncanberelatedtothecostateoft hedifferentialformofthe problemviaanintegrationmatrix.TheLGcollocationschem eyieldedacostateestimate whichwasapproximatedbyapolynomialofdegree N ,whereasthecostateestimatefor theLGRcollocationschemewasapproximatedbyapolynomial ofdegree N 1 .The relationshipbetweenthecostateofthedifferentialformu lationandthedualmultipliers oftheimplicitintegralformulationprovidedanequivalen cebetweentherst-order 98

PAGE 99

optimalityconditionsoftheoptimalcontrolproblemwhenp osedwiththeconstraintsin eitherform. 99

PAGE 100

CHAPTER4 MOTIVATIONFORNEWCOSTATEESTIMATE Inthischapteramotivationisgivenfordevelopingnewcost ateestimatesfor variable-ordercollocationattheLegendre-Gauss(LG)and ippedLegendre-Gauss-Radau (LGR)pointswhensolvingproblemswithactivestateinequa litypathconstraints.In particular,apreviouslyderivedcostateestimateforvari able-ordercollocationattheLG andippedLGRpointswillbepresented[ 1 ].Itwillbeshownthatinthecasewhenthe costateisdiscontinuous(asisthecaseinthepresenceofac tivestateinequalitypath constraints),thiscostateestimateleadstoasetofrst-o rderoptimalityconditionsofthe NLPthatarenotequivalenttothediscreteformofthevariat ionaloptimalityconditions. Thislackofequivalenceleadstoaninaccurateapproximati onofthecostate. Thefollowingnotationandconventionswillbeusedthrough outthischapterin ordertomaketheexpositionmoreclear.First,allvectorfu nctionsoftimearedenoted as row vectors,thatis,if y ( ) 2 R n isavectorfunctionofthescalarvariable ,then y ( )=[ y 1 ( ), y n ( )] .Next,anycapitalboldfacecharacter, Y ,denotesamatrixof size M n ,whereeachrowof Y i correspondstotheevaluationofafunction y ( ) ata particularvalue = i .Next,thenotation Y i : j denotesrows i through j ofthematrix Y exceptwhenreferringtoadifferentiationmatrix D ortheintegrationmatrix A ,inwhich case D i and A i referstothe i th column of D and A .Finally, D > denotesthetranspose ofmatrix D ,and D >i denotesthetransposeofthe i th columnof D .Givenvectors x and y 2 R n ,thenotation h x y i isusedtodenotethestandardinnerproductbetween x and y .Furthermore,if f : R n R m ,then r f isthe m by n Jacobianmatrixwhose i th rowis r f i .Inparticular,thegradientofascalar-valuedfunctionis arowvector.If : R m n R and Y isan m by n matrix,then r denotesthe m by n matrixwhose ( i j ) elementis ( r ( Y )) ij = @ ( Y ) =@ Y ij Theremainderofthischapterisorganizedasfollows.Secti on 4.1 reformulatesthe continuous-timeBolzaoptimalcontrolproblemofSection 2.1 suchthatthedomainis 100

PAGE 101

dividedintoameshandeachmeshintervalisdenedinthedom ain 2 [ 1,+1] .The transformationofvariablesisdonetofacilitatecomparis onoftherst-orderoptimality conditionsofthecontinuous-timeproblemwiththemodied optimalityconditions ofthevariable-ordercollocationmethods.Next,Section 4.2 presentsapreviously derivedcostateestimateforvariable-ordercollocationa ttheLGpoints,developsthe transformedadjointsystem,andshowsthatthecostateesti mateisonlyvalidforthe casewhenthecostateiscontinuousacrossintervalboundar iesinthedomain.Similarly, inSection 4.3 apreviouslyderivedcostateestimateforvariable-orderc ollocationat theippedLGRpointsispresented,thetransformedadjoint systemisdeveloped,and itisshownthatthecostateestimatedoesnotleadtoanaccur ateapproximationof thecontinuous-timeoptimalcostateinthecasewhenthecos tateisdiscontinuous. Finally,Section 4.4 discussestheimplicationsoftheinaccuracyinthecostate estimates presented. 4.1Continuous-TimeBolzaOptimalControlProblem ConsideragainthecontinuousBolzaproblemthatwaspresen tedinSection 2.1 .To simplifycomparisonswiththetransformedadjointsystem, thedomain t 2 [ t 0 t f ]= I isnowdividedinto K sub-intervals S k =[ T k 1 T k ] [ t 0 t f ],( k =1,..., K ) ,where T 0 = t 0 T K = t f T k 1 < T k ,( k =1,..., K ) ,and S Kk =1 S k = I .Furthermore, withoutlossofgeneralitytheoptimalcontrolproblemcanb escaledbytransformingthe independentvariableineachsub-intervalfrom t 2 [ T k 1 T k ] to ( k ) 2 [ 1,+1] viathe afnetransformation t = T k T k 1 2 ( k ) + T k + T k 1 2 (4–1) suchthat dt = T k T k 1 2 d ( k ) h ( k ) 2 where h ( k ) T k T k 1 (4–2) Theoptimalcontrolproblemproblemthenbecomestodetermi nethestate y ( k ) ( ) 2 R n andthecontrol u ( k ) ( ) 2 R m ineachsub-interval ( k =1,..., K ) ,tominimizethecost 101

PAGE 102

functional J =( y ( K ) (+1))+ K X k =1 h ( k ) 2 Z +1 1 g ( y ( k ) ( ), u ( k ) ( )) d (4–3) subjecttothedynamicconstraints y ( k ) ( )= h ( k ) 2 f ( y ( k ) ( ), u ( k ) ( )),( k =1,..., K ), (4–4) theboundaryconditions ( y (1) ( 1))= 0 (4–5) andthestateandcontrolinequalitypathconstraint C ( y ( k ) ( ), u ( k ) ( )) 0 ,( k =1,..., K ). (4–6) 4.1.1First-OrderOptimalityConditionsofContinuousPro blem Therst-orderoptimalityconditionsoftheoptimalcontro lproblemgivenby Eqs.( 4–3 )–( 4–6 )canbederivedfromthecalculusofvariationsinthemanner described bySection 2.1 .Theyaregivenas y ( k ) = f ( y ( k ) u ( k ) ), ( y (1) ( 1))= 0 ,( k =1,..., K ), (4–7) 0 = r u H ( y ( k ) u ( k ) ( k ) ( k ) ),( k =1,..., K ), (4–8) ( k ) = h ( k ) 2 r y H ( y ( k ) u ( k ) ( k ) ( k ) ),( k =1,..., K ), (4–9) (1) ( 1)= r y h ( y (1) ) ij = 1 (4–10) ( K ) (+1)= r y ( y ( K ) ) j =+1 (4–11) C ( y ( k ) u ( k ) ) 0 ( k ) ( ) 0 h ( k ) C ( y ( k ) u ( k ) ) i = 0 ,( k =1,..., K ), (4–12) where ( ( k ) ( k ) ) aretheLagrangemultipliersassociatedwiththeconstrain tsof Eqs.( 4–4 )and( 4–6 )ininterval k ,and aretheLagrangemultipliersassociatedwith theboundaryconditionsofEq.( 4–5 ).Furthermore,theHamiltonianininterval k isgiven as H ( y ( k ) u ( k ) ( k ) ( k ) )= g ( k ) + h ( k ) f ( k ) ih ( k ) C ( k ) i (4–13) 102

PAGE 103

4.2Variable-OrderCollocationatLegendre-GaussPoints TheoptimalcontrolproblemofEqs.( 4–3 )–( 4–6 )isnowdiscretizedusingvariable-order collocationattheLegendre-Gauss(LG)pointsasdescribed inSection 2.5.4 .First,recall thattheLGpoints ( 0 ,..., N +1 ) aredenedinthedomain 2 ( 1,+1) suchthat 0 = 1 and N +1 =+1 arenoncollocatedpoints.Thestateisthenapproximatedin eachmeshinterval k as y ( k ) ( ) Y ( k ) ( )= N k X i =0 Y ( k ) i L ( k ) i ( ), L ( k ) i ( )= N k Y j =0 i 6 = j ( k ) j i ( k ) j (4–14) Differentiating Y ( k ) ( ) inEq.( 4–14 )withrespectto ,yields Y ( k ) ( j ) N k X i =0 Y ( k ) i L ( k ) i ( j )=[ D ( k ) Y ( k ) 0: N k ] j (4–15) where D ( k ) ij = L ( k ) i ( ) j ( i =1,..., N k j =0,..., N k ) arethecomponentsofthe N k ( N k +1) Legendre-Gauss(LG)differentiationmatrix inthe k th meshinterval.Itis notedthatwhenimplementingthevariable-orderLGmethod, asinglevariableisused forthevalueofthestateattheendofmeshinterval k andthestartofmeshinterval k +1 ,thatis, Y ( k ) N k +1 Y ( k +1) 0 ,1 k K 1 suchthatcontinuityinthestateisenforced. Next,thecostfunctionalofEq.( 2–1 )isapproximatedusingamultiple-intervalLG quadrature.Thenite-dimensionalapproximationoftheco ntinuous-timeoptimalcontrol problemofEqs.( 2–1 )–( 2–4 )isthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K +1 )+ K X k =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (4–16) 103

PAGE 104

subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ) (4–17) Y ( k +1) 0 = Y ( k ) 0 + h ( k ) 2 N k X j =1 w ( k ) j f ( Y ( k ) j U ( k ) j ),( k =1,..., K 1) (4–18) Y ( K ) N +1 = Y ( K ) 0 + h ( K ) 2 N k X j =1 w ( K ) j f ( Y ( K ) j U ( K ) j ), (4–19) ( Y (1) 0 )= 0 (4–20) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ,( k =1,..., K ), (4–21) where w ( k ) =( w ( k ) 1 ,..., w ( k ) N k ) aretheLG quadratureweights ininterval k .Itisnoted forLGcollocationthatEq.( 4–18 )providesanLGquadratureapproximation, Y ( k ) 0 ,of thestateatthenalnoncollocatedpoint ( k ) N +1 =+1 ininterval ( k =1,..., K 1) ,while Eq.( 4–19 )providesanLGquadratureapproximation, Y ( K ) N +1 ,ofthestateatthenal noncollocatedpointofthedomain, t f = ( k ) N +1 =+1 4.2.1KKTConditionsofVariable-OrderLegendre-GaussCol locationMethod Therst-orderoptimalityconditionsofthediscreteprobl emgivenbyEqs.( 4–16 )–( 4–21 ), alsocalledtheKKTconditionsoftheNLP,arenowderived.Fi rst,theLagrangianis denedas L = ( Y ( K ) N K +1 ) h ( Y (1)0 ) i + K X k =1 N k X i =1 h ( k ) 2 w ( k ) i g ( k ) i h ( k ) i C ( k ) i i K X k =1 N k X i =1 h ( k ) i D ( k ) i ,0: N k Y ( k ) 0: N k h ( k ) 2 f ( k ) i i K 1 X k =1 N k X i =1 h ( k ) N k +1 Y k +1 0 Y ( k ) 0 h ( k ) 2 w ( k ) i f ( k ) i i N K X i =1 h ( K ) N K +1 Y ( K ) N K +1 Y ( K ) 0 h ( K ) 2 w ( K ) i f ( K ) i i (4–22) where ( ( k ) i ( k ) N k +1 ( k ) i ) aretheLagrangemultipliersassociatedwiththedynamic constraintsofEq.( 4–17 ),thequadratureconstraintsofEqs.( 4–18 )–( 4–19 ),andthe 104

PAGE 105

inequalitypathconstraintofEq.( 4–21 ),respectively,ininterval k attheLGRpoint i .Furthermore denotestheLagrangemultipliersassociatedwiththebound ary conditionsofEq.( 4–20 ).Notethatfunctiondependencieshavebeenomittedforcla rity, suchthat g ( k ) i g ( Y ( k ) i U ( k ) i ) ,andsimilarly f ( k ) i f ( Y ( k ) i U ( k ) i ) and C ( k ) i C ( Y ( k ) i U ( k ) i ) TheKKTconditionsoftheNLParethengivenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (4–23) Y ( k +1) 0 = Y ( k ) 0 + N k X i =1 w ( k ) i f ( Y ( k ) i U ( k ) i ),( k =1,..., K 1), (4–24) Y ( k ) N K +1 = Y ( K ) 0 + N K X i =1 w ( K ) i f ( Y ( K ) i U ( K ) i ), (4–25) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k +1 ( k ) 1: N k ),( k =1,..., K ), (4–26) D ( k ) > 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k +1 ( k ) 1: N k ),( k =1,..., K ), (4–27) D (1) > 0 (1)1: N 1 = (1)N 1 +1 r Y h ( Y (1) 0 ) i (4–28) D ( k ) > 0 ( k ) 1: N 1 = ( k ) N k +1 ( k 1) N k +1 ,( k =2,..., K ), (4–29) ( K ) N K +1 = r Y ( Y ( K ) N K +1 ), (4–30) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k C ( Y ( k ) 1: N k U ( k ) 1: N k ) i = 0 ,( k =1,..., K ). (4–31) ThediscreteHamiltonianininterval k isgivenas H ( Y ( k ) 1: N k U 1: N k ( k ) 1: N k +1 ( k ) 1: N k )= w ( k ) > g ( k ) 1: N k + h ( k ) 1: N k + W ( k ) 1 ( k ) N k +1 f ( k ) 1: N k ih 2 h ( k ) ( k ) 1: N k C ( k ) 1: N k i where W ( k ) isa N N diagonalmatrixofLGquadratureweightsininterval k ,and 1 isa N 1 columnvectorofones. 4.2.2CostateEstimateandTransformedAdjointSystem Considerthefollowingcostateestimate,rstderivedbyRe f.[ 1 ],thatrelatesthe KKTconditionsmultipliersofEqs.( 4–23 )–( 4–31 )tothedualvariablesoftherst-order 105

PAGE 106

optimality[givenbyEqs.( 4–7 )–( 4–12 )]ofthecontinuous-timeoptimalcontrolproblem: ( k ) i = 2 h ( k ) ( k ) i w ( k ) i ,( i =1,..., N k ),( k =1,..., K ), (4–32) ( k ) i = ( k ) N k +1 + ( k ) i w ( k ) i ,( i =1,..., N k ),( k =1,..., K ), (4–33) ( k ) 0 = ( k ) N k +1 D ( k ) > 0 ( k ) 1: N k ,( k =1,..., K ), (4–34) ( k ) N k +1 = ( k ) N k +1 ( k =1,..., K ). (4–35) Next,let D y bean N k ( N k +1) matrixdenedasfollows: D y ij = w j w i D ji ,and D y i N +1 = N X j =1 D y ij (4–36) for i =1,..., N .Basedonthetheorydevelopedin[ 36 ], D y isadifferentiationmatrixfor thespaceofpolynomialsofdegree N .Thatis,if b isapolynomialofdegreeatmost N and b 2 R N isthevectorwith i -thelement b i = b ( i ) ,then ( D y b ) i = b ( i ). 106

PAGE 107

UsingtheadjointdifferentiationmatrixdenedinEq.( 4–36 )alongwiththetransformations givenbyEqs.( 4–32 )–( 4–35 ),theKKTsystemofEqs.( 4–23 )–( 4–31 )canberewrittenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (4–37) Y ( k +1) 0 = Y ( k ) 0 + N k X i =1 w ( k ) i f ( Y ( k ) i U ( k ) i ),( k =1,..., K 1), (4–38) Y ( K ) N K +1 = Y ( K ) 0 + N K X i =1 w ( K ) i f ( Y ( K ) i U ( K ) i ), (4–39) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (4–40) D y ( k ) 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (4–41) (1)0 = r Y h ( Y (1) 0 ) i (4–42) ( k ) 0 = ( k 1) N k +1 ( k =2,..., K ), (4–43) ( K ) N K +1 = r Y ( Y ( K ) N K +1 ), (4–44) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k C ( Y ( k ) 1: N k U ( k ) 1: N k ) i = 0 ,( k =1,..., K ). (4–45) where H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k +1 ( k ) 1: N k ) isadiscreteformoftheHamiltoniangivenby Eq.( 4–13 ).Itcanbeseenthatthetransformedoptimalityconditions ofEqs.( 4–37 )–( 4–42 ) areadiscreteformoftherst-orderoptimalityconditions ofthecontinuous-timeoptimal controlproblemgivenbyEqs.( 4–7 )–( 4–10 ).Furthermore,thecostateterminalcondition givenbyEq.( 4–11 )issatisedinthediscretetransformedadjointsystembyE q.( 4–44 ), andthecomplementaryslacknessconditionsofEq.( 4–12 )aresatisedbythediscrete conditionofEq.( 4–45 ).Fromtherelationshipgiveninthetransformedadjointsy stem byEq.( 4–43 ),however,itisseenthatinthediscreteproblemthecostat eacross intervalboundariesmustbecontinuous.Itwaspreviouslys howninSection 2.3 thatdiscontinuitiesinthecostatestemfrominequalitypa thconstraintactivityinthe solutiondomain.Thereforeinthepresenceofstateinequal ityconstraints,thecostate becomesdiscontinuous,andthetransformedadjointsystem oftheNLPisnotadiscrete approximationoftherst-orderoptimalityconditionsoft hecontinuous-timeproblem. 107

PAGE 108

4.3Variable-OrderCollocationatFlippedLegendre-Gauss -RadauPoints TheoptimalcontrolproblemofEqs.( 4–3 )–( 4–6 )isnowdiscretizedusingvariable-order collocationatthe ipped Legendre-Gauss-RadaupointsasdescribedinSection 2.5.6 .First,recallthattheippedLGRpoints ( 0 ,..., N ) aredenedinthedomain 2 ( 1,+1] suchthat 0 = 1 isanoncollocatedpoint.Thestateineachmeshinterval k isapproximatedas y ( k ) ( ) Y ( k ) ( )= N k X i =0 Y ( k ) i L ( k ) i ( ), L ( k ) i ( )= N k Y j =0 i 6 = j ( k ) j ( k ) i ( k ) j (4–46) Differentiating Y ( k ) ( ) inEq.( 4–46 )withrespectto ,yields Y ( k ) ( j ) N k X i =0 Y ( k ) i L ( k ) i ( j )=[ D ( k ) Y ( k ) 0: N k ] j (4–47) where D ( k ) ij = L ( k ) i ( ) j ( i =1,..., N k j =0,..., N k ) arethecomponentsofthe N k ( N k +1) ippedLegendre-Gauss-Radau(LGR)differentiationmatri x inthe k th mesh interval. Theoptimalcontrolproblemthenbecomestominimizethecos t J ( Y ( K ) N K )+ K X k =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (4–48) subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (4–49) ( Y (1) 0 )= 0 (4–50) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ,( k =1,..., K ), (4–51) where w ( k ) =( w ( k ) 1 ,..., w ( k ) N k ) aretheippedLGR quadratureweights ininterval k Whenimplementingtheippedvariable-orderLGRmethod,as inglevariableisusedfor thevalueofthestateattheendofmeshinterval k andthestartofmeshinterval k +1 thatis, Y ( k 1) N k Y ( k ) 0 ,2 k K suchthatcontinuityinthestateisenforced. 108

PAGE 109

4.3.1KKTConditionsofVariable-OrderFlippedLegendre-G auss-RadauCollocationMethod Therst-orderoptimalityconditionsofthediscreteprobl emgivenbyEqs.( 4–48 )–( 4–51 ), alsocalledtheKKTconditionsoftheNLP,arenowderived.Fi rst,theLagrangianis denedas L = ( Y ( K ) N K ) h ( Y (1)0 ) i + K X k =1 N k X i =1 h ( k ) 2 w ( k ) i g ( k ) i h ( k ) i C ( k ) i i N 1 X i =1 h (1)i D (1) i ,0 Y (1)0 + D (1)i ,1: N 1 Y (1)1: N 1 h ( k ) 2 f (1)i i K 1 X k =1 N k X i =1 h ( k ) i D ( k ) i ,0 Y ( k 1) N + D ( k ) i ,1: N k Y ( k ) 1: N k h ( k ) 2 f ( k ) i i (4–52) where ( ( k ) i ( k ) i ) aretheLagrangemultipliersassociatedwiththedynamicco nstraints ofEq.( 4–49 )andtheinequalitypathconstraintofEq.( 4–51 )ininterval k attheLGR point i .Furthermore denotestheLagrangemultipliersassociatedwiththebound ary conditionsofEq.( 4–50 ).Notethatfunctiondependencieshavebeenomittedforcla rity, suchthat g ( k ) i g ( Y ( k ) i U ( k ) i ) ,andsimilarly f ( k ) i f ( Y ( k ) i U ( k ) i ) and C ( k ) i C ( Y ( k ) i U ( k ) i ) TheKKTconditionsoftheNLParethengivenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (4–53) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (4–54) D ( k ) > 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ) e N k D ( k +1) 0 ( k +1) 1: N k ,( k =1,..., K 1), (4–55) D ( K ) > 1: N K ( K ) 1: N K = h ( k ) 2 r Y H ( Y ( K ) 1: N K U ( K ) 1: N K ( K ) 1: N K ( K ) 1: N K )+ e N K r Y ( Y ( K ) N K ), (4–56) D (1) > 0 (1)1: N 1 = r Y h ( Y (1) 0 ) i (4–57) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k C ( Y ( k ) 1: N k U ( k ) 1: N k ) i = 0 ,( k =1,..., K ), (4–58) 109

PAGE 110

where e N denotesthe N th columnoftheidentitymatrixandthediscreteHamiltoniani n interval k is H ( Y ( k ) 1: N k U 1: N k ( k ) 1: N k )= w ( k ) > g ( k ) 1: N k + h ( k ) 1: N k f ( k ) 1: N k ih 2 h ( k ) ( k ) 1: N k C ( k ) 1: N k i (4–59) 4.3.2CostateEstimateandTransformedAdjointSystem Considerthefollowingcostateestimate,rstderivedbyRe f.[ 1 ],thatrelatesthe KKTconditionsmultipliersofEqs.( 4–53 )–( 4–58 )tothedualvariablesoftherst-order optimality[givenbyEqs.( 4–7 )–( 4–12 )]ofthecontinuous-timeoptimalcontrolproblem: ( k ) i = 2 h ( k ) ( k ) i w ( k ) i ,( i =1,..., N k ),( k =1,..., K ), (4–60) ( k ) i = ( k ) i w ( k ) i ,( i =1,..., N k ),( k =1,..., K ), (4–61) ( k ) 0 = D ( k ) > 0 ( k ) 1: N k ,( k =1,..., K ). (4–62) Next,let D y bean N k N k matrixdenedasfollows: D y ij = 8><>: D NN + 1 w N i = j = N w j w i D ji i j =2,..., N (4–63) Basedonthetheorydevelopedin[ 36 ], D y isadifferentiationmatrixforthespaceof polynomialsofdegree N 1 .Thatis,if b isapolynomialofdegreeatmost N 1 and b 2 R N isthevectorwith i -thelement b i = b ( i ) ,then ( D y b ) i = b ( i ). 110

PAGE 111

UsingtheadjointdifferentiationmatrixdenedinEq.( 4–63 )alongwiththetransformations givenbyEqs.( 4–60 )–( 4–62 ),theKKTsystemofEqs.( 4–53 )–( 4–58 )canberewrittenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (4–64) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (4–65) D y ( k ) 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ) e N k w N k ( k +1) 0 ( k ) N ,( k =1,..., K 1), (4–66) D y ( K ) 1: N K ( K ) 1: N K = h ( k ) 2 r Y H ( Y ( K ) 1: N K U ( K ) 1: N K ( K ) 1: N K ( K ) 1: N K ) e N K w N K r Y ( Y ( K ) N ) ( K ) N (4–67) (1)0 = r Y h ( Y (1) 0 ) i (4–68) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k C ( Y ( k ) 1: N k U ( k ) 1: N k ) i = 0 ,( k =1,..., K ), (4–69) where H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ) isadiscreteformoftheHamiltoniangivenby Eq.( 4–13 ).Theseequationsareincompletebecauseanewvariable ( k ) 0 was introducedwithoutaddinganewequation.Anequationforth isnewvariablecanbe developedbymanipulatingthematrix D ( k ) .Considernowa ( N +1) 1 columnvector composedofones.Thecomponentsofthevector D ( k ) 1 arethederivativesatthe collocationpointsofthepolynomialwhosevalueis1at i ( i =1,..., N +1) ininterval k .Thederivativeoftheconstantpolynomialiszeroeverywhe re.Thus, D ( k ) 1 = 0 ,which impliesthat D ( k ) 0 = N k X j =1 D ( k ) 1: N k j D ( k ) > 0 ( k ) = N k X i =1 D ( k ) i ,0 ( k ) i = N k X i =1 N k X j =1 D ( k ) i j ( k ) i ( k ) 0 = ( k ) N k + N k X i =1 N k X j =1 w ( k ) i ( k ) j D y ( k ) i j = ( k ) N k + N k X j =1 w ( k ) j [ D y ( k ) ( k ) ] j (4–70) 111

PAGE 112

wheretherelationshipsin( 4–61 )–( 4–62 )and( 4–63 )wereusedtoobtainEq.( 4–70 ). Itcanbeseenthatthisrelationshipapproximatestheinteg ralofthecostatedynamics acrosstheinterval k viaaRadauquadrature.Thatis,itapproximatestherelatio nship ( 1)= (+1)+ Z +1 1 ( ) d CombiningEqs.( 4–66 )–( 4–67 )withEq.( 4–70 ),thecompletetransformedadjoint systemcanthenbewrittenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (4–71) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (4–72) D y ( k ) 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ) e N k w N k ( k +1) 0 ( k ) N ,( k =1,..., K 1), (4–73) ( k +1) 0 = ( k ) 0 h ( k ) 2 N k X j =1 w ( k ) j r Y H ( Y ( k ) j U ( k ) j ( k ) j ( k ) j ),( k =1,..., K 1), (4–74) D y ( K ) 1: N K ( K ) 1: N K = h ( k ) 2 r Y H ( Y ( K ) 1: N K U ( K ) 1: N K ( K ) 1: N K ( K ) 1: N K ) e N K w N K r Y ( Y ( K ) N ) ( K ) N (4–75) r Y ( Y ( K ) N )= ( K ) 0 h ( K ) 2 N K X j =1 w ( K ) j r Y H ( Y ( K ) j U ( K ) j ( K ) j ( K ) j ), (4–76) (1)0 = r Y h ( Y (1) 0 ) i (4–77) C ( Y ( k ) 1: N k U ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k C ( Y ( k ) 1: N k U ( k ) 1: N k ) i = 0 ,( k =1,..., K ), (4–78) ItisseenthatEq.( 4–76 )isaLegendre-Gauss-Radauquadratureofthecostate dynamicsacrossinterval K .Consequently,theright-handsideofEq.( 4–76 )approximates thecostateatthenalpointinthedomain.Eq.( 4–76 )isthusasubtlewayofenforcing therelationship ( K ) N = r Y ( Y N k ) anditisexpectedthatthelasttermofEq.( 4–75 ) willbesmallwhiletheremainingtermsinEq.( 4–75 )areacollocationcollocation schemeforthecontinuousadjointequationinthenalinter val K .Similarly,the 112

PAGE 113

right-handsideofEq.( 4–74 )approximatesthecostate, ( k ) N ,attheterminalpoint ininterval k viaaLegendre-Gauss-Radauquadratureofthecostatedynam icsfor ( k =1,..., K 1) .Equation( 4–74 )isthereforeasubtlewayofenforcingtherelationship ( k ) N = ( k +1) 0 ( k =1,..., K 1) .Ifthecostateiscontinuousacrossaninterval boundary,therelationship ( k ) N = ( k +1) 0 holdstrueandthelasttermofEq.( 4–73 ) willbesmallwhiletheremainingtermsinEq.( 4–73 )areacollocationschemeforthe continuousadjointequation.Thereforeincaseswhentheco stateiscontinuousacross anintervalboundary,thetransformedoptimalityconditio nsofEqs.( 4–71 )–( 4–78 )are adiscreteformoftherst-orderoptimalityconditions[gi venbyEqs.( 4–7 )–( 4–12 )]of thecontinuous-timeoptimalcontrolproblem.However,ith aspreviouslybeenshownin Section 2.3 thatthepresenceofactivestateinequalitypathconstrain tsinthesolution domainmaycausediscontinuitiesinthecostate.Therefore ,inthepresenceofactive stateinequalitypathconstraintsinthesolutiondomain, ( k ) N 6 = ( k +1) 0 attheentrance orexitofaconstrainedarc,andthelasttermofEq.( 4–73 )willnotbesmall.Therefore Eq.( 4–73 )willnotbeacollocationschemeforthecontinuousadjoint equationusingthis costateestimate. 4.4Discussion InthischapteramethodrstderivedbyRef.[ 1 ]forobtainingcostateestimates fromtheKKTmultipliersoftheNLPwaspresented.Thisderiv ationshowedthatif thecostateiscontinuous,variable-ordercollocationatt heLGandLGRpointsyields asetoftransformedoptimalityconditionsoftheKKTsystem whichareadiscrete representationofthecontinuous-timerst-ordernecessa ryconditionsoftheoptimal controlproblem,ascanbeseeninFig.( 4-1 ).Ifthecostateisdiscontinuous,however, variable-ordercollocationattheLGandLGRpointsyieldsa setoftransformed optimalityconditionsoftheKKTsystemwhichareaninexact discreterepresentation ofthecontinuous-timerst-ordernecessaryconditions.T hisresultwasrstshownby 113

PAGE 114

Ref.[ 74 ],whosuggestedthatthecostateestimatemustbemodiedin ordertoaccount forthecostatediscontinuities: “...high-accuracyapproximationsareachievedbytheprop osed hp-methodifthecostateiscontinuous.Ifmeshpointsareat thelocation ofdiscontinuityinthecostate,thetransformedadjointsy stemisaninexact discreterepresentationofthecontinuous-timerst-orde rnecessary conditions.Foradynamicrenementalgorithmthatwillexa ctlylocate theswitchinactivityofinequalitypathconstraints,itis likelythatthecostate maybediscontinuousatmeshpoints.Itisnecessarytodeter minehow tomakethetransformedadjointsystemadiscreterepresent ationofthe continuous-timerst-ordernecessaryconditionsforadis continuouscostate solutionifmeshpointsareatthelocationofdiscontinuity inthecostate”. ItwaspreviouslyshowninSection 2.3 thatdiscontinuitiesinthecostatestemfrom inequalitypathconstraintactivityinthesolutiondomain .Thereforeinthepresence ofstateinequalityconstraints,thecostatebecomesdisco ntinuous,andtherst-order optimalityconditionsoftheNLParenotadiscreteapproxim ationoftherst-order optimalityconditionsofthecontinuous-timeproblem.The refore,inthisresearchanew methodofcostateestimationforvariable-ordercollocati onatLGandLGRpointswillbe derivedusingthreedifferentmethods.Specically,acost ateestimateusingthemethod indirectadjoiningwithcontinuousmultipliers willbepresented.Itwillbeshownthat thismethodforcostateestimationusingvariable-orderco llocationattheLGandLGR pointsleadstoatransformedadjointsystemwhichisadiscr eterepresentationofthe continuous-timerst-ordernecessaryconditionsevenint hepresenceofstateinequality pathconstraints. 114

PAGE 115

Figure4-1.Relationshipbetweenthedirectandindirectme thodsforsolvinganoptimal controlproblem.Intheindirectmethod,theproblemisrst optimizedthroughthe calculusofvariations,leadingtoasetofconditionswhich canthenbediscretizedand solved.Inthedirectmethod,theproblemisrstdiscretize dandtranscribedtoanNLP, thenitisoptimizedbysolvingtheKKTsystem.Thetwosystem sareequivalentonly whenthecostateiscontinuousinthesolutiondomain. 115

PAGE 116

CHAPTER5 COSTATEESTIMATIONFORSTATECONSTRAINEDPROBLEMS AswasshowninChapter 4 ,previousresearchhassuccessfullyderiveda high-accuracyestimateofthecostateusingcollocationat theLegendre-Gaussand Legendre-Gauss-Radaupointsforthecaseofaproblemwithn oactivestateinequality pathconstraints.However,Ref.[ 1 ]showedthatinthecasewhenthecostateis discontinuous(asisthecaseinthepresenceofactivestate inequalitypathconstraints), thispreviouslyderivedcostateestimateleadstoasetofr st-orderoptimalityconditions oftheNLPthatarenotequivalenttothediscreteformofthev ariationaloptimality conditions.Thisnon-equivalenceleadstoaninaccurateap proximationofthecostate.In ordertorectifythisinacuracy,inthischapteramethodfor estimatingthecostateofstate inequalitypathconstrainedoptimalcontrolproblemsusin gcollocationatLGandLGR pointsisdevelopedusingthemethodof indirectadjoiningwithcontinuousmultipliers Themethodofindirectadjoiningwithcontinuousmultiplie rswaschosento developacostateestimateoverthedirectandindirectadjo iningmethodsfortwo reasons.First,themethodofindirectadjoiningrequiresa modicationoftheoriginal problemformulationthroughindex-reductionofthediffer ential-algebraicequations.The reformulationoftheproblemmustbedoneanalytically,and requirespriorknowledge ofthesolutionstructure.Thus,whenusinganautomatedsol utionprocess(suchas ameshrenementtechnique),thisproceduremightbecumber sometoimplement. Second,boththemethodsofindirectanddirectadjoiningre sultinadiscontinuous costate.Becausediscontinuitiesaredifculttoapproxim atenumerically,boththese methodsmayyieldlargeerrorsinthecostateestimateifthe locationofthediscontinuity isnotexact.Thus,becausethemethodofindirectadjoining withcontinuousmultipliers yieldsacontinuouscostate,itoffersanadvantageoverthe methodsofdirectand indirectadjoiningwhichapproximateadiscontinuouscost ate. 116

PAGE 117

SimilartotheapproachofChapter 4 ,thefollowingnotationandconventionswill beusedthroughoutthischaptertomaketheexpositionmorec lear.First,allvector functionsoftimearedenotedas row vectors,thatis,if y ( ) 2 R n isavectorfunctionof thescalarvariable ,then y ( )=[ y 1 ( ), y n ( )] .Next,anycapitalboldfacecharacter, Y ,denotesamatrixofsize M n ,whereeachrowof Y i correspondstotheevaluation ofafunction y ( ) ataparticularvalue = i .Next,thenotation Y i : j denotesrows i through j ofthematrix Y ,exceptwhenreferringtoadifferentiationmatrix D orthe integrationmatrix A ,inwhichcase D i and A i referstothe i th column of D and A .Finally, D > denotesthetransposeofmatrix D ,and D >i denotesthetransposeofthe i th column of D .Givenvectors x and y 2 R n ,thenotation h x y i isusedtodenotethestandardinner productbetween x and y .Furthermore,if f : R n R m ,then r f isthe m by n Jacobian matrixwhose i th rowis r f i .Inparticular,thegradientofascalar-valuedfunctionis arow vector.If : R m n R and Y isan m by n matrix,then r denotesthe m by n matrix whose ( i j ) elementis ( r ( Y )) ij = @ ( Y ) =@ Y ij Theremainderofthischapterisorganizedasfollows.First ,Section 5.1 formulates thecontinuous-timestateinequalitypathconstrainedopt imalcontrolproblemand statestherstorderoptimalityconditionsofthecontinuo usproblem.Next,inSections 5.2 and 5.3 anewcostateestimateisderivedusingvariable-ordercoll ocationatthe Legendre-GaussandippedLegendre-Gauss-Radaupoints,r espectively,throughthe methodofindirectadjoiningwithcontinuousmultipliers. Itisshownforeachofthese derivedcostateestimatesthatthetransformedrst-order optimalityconditionsofthe NLPareadiscreteformoftherst-orderoptimalityconditi onsofthecontinuous-time optimalcontrolproblem.Finally,inSection 5.4 thederivedcostateestimatesare discussed,andconclusionsaregiven. 5.1Continuous-TimeStateInequalityPathConstrainedOpt imalControlProblem Thestateinequalitypathconstrainedoptimalcontrolprob lemtobestudiedin theremainderofthischapterisnowpresented.Tosimplifyc omparisonswiththe 117

PAGE 118

transformedadjointsystem,thedomain t 2 [ t 0 t f ]= I isdividedinto K intervals S k =[ T k 1 T k ] [ t 0 t f ],( k =1,..., K ) ,where T 0 = t 0 T K = t f T k 1 < T k ,( k = 1,..., K ) ,and S Kk =1 S k = I .Furthermore,withoutlossofgeneralitytheoptimalcontr ol problemcanbescaledbytransformingtheindependentvaria bleineachintervalfrom t 2 [ T k 1 T k ] to ( k ) 2 [ 1,+1] viatheafnetransformation t = T k T k 1 2 ( k ) + T k + T k 1 2 (5–1) suchthat dt = T k T k 1 2 d ( k ) h ( k ) 2 where h ( k ) T k T k 1 (5–2) Thestateinequalityconstrainedoptimalcontrolproblemp roblemisstatedas follows.Determinethestate y ( k ) ( ) 2 R n ,andthecontrol u ( k ) ( ) 2 R m ineachinterval ( k =1,..., K ) ,tominimizethecostfunctional J =( y ( K ) (+1))+ K X k =1 h ( k ) 2 Z +1 1 g ( y ( k ) ( ), u ( k ) ( )) d (5–3) subjecttothedynamicconstraints y ( k ) ( )= h ( k ) 2 f ( y ( k ) ( ), u ( k ) ( )),( k =1,..., K ), (5–4) theboundaryconditions ( y (1) ( 1))= 0 (5–5) andthestateinequalitypathconstraint S ( y ( k ) ( )) 0 ,( k =1,..., K ). (5–6) Thecontinous-timeoptimalcontrolproblemofEqs.( 5–3 )–( 5–6 )willbethetopicofthe remainderofthischapter. 118

PAGE 119

5.1.1First-OrderOptimalityConditionsUsingMethodofIn directAdjoiningwith ContinuousMultipliers Therst-orderoptimalityconditionsofthestateinequali typathconstrainedoptimal problemgivenbyEqs.( 5–3 )–( 5–6 )werederivedinSection 2.3.3 usingthemethodof indirectadjoiningwithcontinuousmultipliers.Thesecon ditionsarerepeatedhereas y ( k ) = f ( y ( k ) u ( k ) ), ( y (1) ( 1))= 0 ,( k =1,..., K ), (5–7) 0 = r u H ( y ( k ) u ( k ) p ( k ) ( k ) ),( k =1,..., K ), (5–8) p ( k ) = h ( k ) 2 r y H ( y ( k ) u ( k ) p ( k ) ( k ) ),( k =1,..., K ), (5–9) p (1) ( 1)= r y ( h ( y (1) ) i + h ( k ) S ( y ( k ) ) i ) = 1 (5–10) p ( K ) (+1)= r y (( y ( K ) )+ h ( K ) S ( y ( K ) ) i ) =+1 (5–11) ( K ) (+1) 0 ,_ ( k ) 0 S ( y ( k ) ) 2N ( ( k ) ),( k =1,..., K ), (5–12) where p ( k ) ( ) and ( k ) ( ) aretheLagrangemultipliersassociatedwiththedynamic constraintsofEq.( 5–4 )andthestateinequalitypathconstraintsofEq.( 5–6 ),respectively, ininterval k .Furthermore, istheLagrangemultiplierassociatedwiththeboundary conditionsofEq.( 5–5 ).TheHamiltonianininterval k H ( y ( k ) u ( k ) p ( k ) ( k ) ) ,isdened as H ( y ( k ) u ( k ) p ( k ) ( k ) )= g ( y ( k ) u ( k ) )+ h p ( k ) f ( y ( k ) u ( k ) ) ih ( k ) S ( y ( k ) ) i (5–13) where S ( k ) r S ( y ( k ) ) f ( y ( k ) u ( k ) ) .Let S ( R q ) denotethespaceofcontinuousfunctions mapping [ t 0 t f ] to R q .Assuming ( k ) isLipschitzcontinuousandnondecreasingwith ( K ) (+1) 0 ,theset-valuedmap N ( ( k ) ) isdenedas N ( ( k ) )= f z ( k ) 2S ( R q ): z ( k ) 0 h ( k ) z ( k ) i = 0 h ( K ) (+1), z ( K ) (+1) i =0 g for ( k =1,..., K ) 119

PAGE 120

5.2CostateEstimationUsingLegendre-GaussCollocation TheoptimalcontrolproblemofEqs.( 5–3 )–( 5–6 )isnowdiscretizedusingvariable-order collocationattheLegendre-GausspointsasdescribedinSe ction 2.5.4 .Unlikeprevious implementationsoftheLGcollocationmethod,thestateine qualitypathconstraintis enforcedatallLGpoints and allinteriormeshpoints ( T 1 ,..., T K 1 ) butis not enforced attheendpoints T 0 = 1 and T K =+1 .LGcollocationprovidesanapproximationto thestate,butnotthecontrol,atthemeshpoints.Therefore itisimpossibletoenforce inequalitypathconstraintsthatareafunctionofthecontr olatanyofthemeshpoints. However,becausethisresearchisconcernedwithinequalit ypathconstraintsthatare afunctionofpurelythestate,itispossibletoenforcethei nequalitypathconstraintat allinteriormeshpoints.Indeed,itwillbeseenthatanaccu rateapproximationofthe costateusingvariable-ordercollocationattheLGpointsc anonlybeachievedwhenthe stateinequalitypathconstraintisenforcedatalltheinte riormeshpoints. 5.2.1Variable-OrderCollocationatLegendre-GaussPoint s RecallthattheLGpoints ( 1 ,..., N ) aredenedinthedomain 2 ( 1,+1) suchthat 0 = 1 and N +1 =+1 arenoncollocatedpoints.Whenimplementingthe variable-orderLGmethod,asinglevariableisusedforthev alueofthestateattheend ofmeshinterval k andthestartofmeshinterval k +1 ,thatis, Y ( k ) N k +1 Y ( k +1) 0 ,( k = 1,..., K 1) suchthatcontinuityinthestateisenforced. TheNLPisthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K +1 )+ K X k =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (5–14) 120

PAGE 121

subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (5–15) Y ( k +1) 0 = Y ( k ) 0 + h ( k ) 2 N k X j =1 w ( k ) j f ( Y ( k ) j U ( k ) j ),( k =1,..., K 1), (5–16) Y ( K ) N +1 = Y ( K ) 0 + h ( K ) 2 N k X j =1 w ( K ) j f ( Y ( K ) j U ( K ) j ), (5–17) ( Y (1) 0 )= 0 (5–18) S ( Y ( k ) 0: N k ) 0 ( k =2,..., K ), (5–19) S ( Y (1) 1: N 1 ) 0 (5–20) ItisseenthatthequadratureconstraintsofEqs.( 5–16 )and( 5–17 )providean approximationtothenalstateinintervals ( k =1,..., K 1) and K ,respectively, throughaGaussianquadratureapproximationtotheintegra lofthestatedynamics acrossthatinterval. Therst-orderoptimality(KKT)conditionsofthediscrete problemgivenby Eqs.( 5–14 )–( 5–19 )arederivedinthesamemannerofSection 4.2 .First,theLagrangian isdenedas L = ( Y ( K ) N K +1 ) h ( Y (1)0 ) i + K X k =1 N k X i =1 h ( k ) 2 w ( k ) i g ( k ) i h ( k ) i S ( k ) i i K X k =1 N k X i =1 h ( k ) i D ( k ) i ,0: N k Y ( k ) 0: N k h ( k ) 2 f ( k ) i i K 1 X k =1 h ( k ) N k +1 Y k +1 0 Y ( k ) 0 h ( k ) 2 w ( k ) i f ( k ) i ih ( k ) N k +1 S ( k +1) 0 i N K X i =1 h ( K ) N K +1 Y K N K +1 Y ( K ) 0 h ( K ) 2 w ( K ) i f ( K ) i i (5–21) where ( k ) 1: N k +1 and ( k ) 1: N k +1 aretheLagrangemultipliersassociatedwiththedynamic constraintsofEq.( 5–15 ),thequadratureconstraintsofEqs.( 5–16 )–( 5–17 ),andthe inequalitypathconstraintofEq.( 5–19 ),respectively,ininterval k .Furthermore 121

PAGE 122

denotestheLagrangemultipliersassociatedwiththebound aryconditionsofEq.( 5–18 ). TheKKTconditionsoftheNLParethengivenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (5–22) Y ( k +1) 0 = Y ( k ) 0 + N k X i =1 w ( k ) i f ( Y ( k ) i U ( k ) i ),( k =1,..., K 1), (5–23) Y ( k ) N K +1 = Y ( K ) 0 + N K X i =1 w ( K ) i f ( Y ( K ) i U ( K ) i ), (5–24) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k +1 ( k ) 1: N k ),( k =1,..., K ), (5–25) D ( k ) > 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k +1 ( k ) 1: N k ),( k =1,..., K ), (5–26) D (1) > 0 (1)1: N 1 = (1)N 1 +1 r Y h ( Y (1) 0 ) i (5–27) D ( k ) > 0 ( k ) 1: N 1 = ( k ) N k +1 ( k 1) N k +1 r Y h ( k 1) N +1 S ( Y ( k ) 0 ) i ,( k =2,..., K ), (5–28) ( K ) N K +1 = r Y ( Y ( K ) N K +1 ), (5–29) S ( Y ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k S ( Y ( k ) 1: N k ) i = 0 ,( k =1,..., K ), (5–30) S ( Y ( k ) 0 ) 0 ( k 1) N +1 0 h ( k 1) N +1 S ( Y ( k ) 0 ) i = 0 ,( k =2,..., K ). (5–31) ThediscreteHamiltonianininterval k isgivenas H ( Y ( k ) 1: N k U 1: N k ( k ) 1: N k +1 )= w ( k ) > g ( k ) 1: N k + h ( k ) 1: N k + W ( k ) 1 ( k ) N k +1 f ( k ) 1: N k ih 2 h ( k ) ( k ) 1: N k S ( k ) 1: N k i where W ( k ) isa N N diagonalmatrixofLGquadratureweightsininterval k ,and 1 isa N 1 columnvectorofones. 5.2.2CostateEstimateandTransformedAdjointSystem Inordertorelatethenecessaryconditionsforoptimalityo fthecontinuousproblem [givenbyEqs.( 5–7 )–( 5–12 )]tothediscreteKKTconditionsoftheNLP[givenby 122

PAGE 123

Eqs.( 5–22 )–( 5–31 )],considerthefollowingtransformeddualvariables p ( k ) 1: N k =[ W ( k ) ] 1 ( k ) 1: N k + 1 ( k ) N +1 + r Y h ( k ) 1: N k S ( Y ( k ) 1: N k ) i ,( k =1,..., K ), (5–32) p ( k ) N k +1 = ( k ) N k +1 + r Y h ( k ) N k +1 S ( Y ( k ) N k +1 ) i ,( k =1,..., K ), (5–33) ( k ) 1: N = W ( k ) D y ( k ) 1: N +1 ,( k =1,..., K ), (5–34) ( K ) N K +1 = ( K ) N K +1 = 0 (5–35) ( k ) N k +1 = ( k ) N k +1 + ( k +1) 0 ( k =1,..., K ), (5–36) p ( k ) 0 = p ( k ) N k +1 D ( k ) > 0 W ( k ) [ p ( k ) 1: N k 1 p ( k ) N k +1 ],( k =1,..., K ), (5–37) ( k ) 0 = ( k ) N k +1 + N k X i =1 ( k ) i ,( k =1,..., K ). (5–38) where D y isa N ( N +1) matrixdenedby D y ij = w j w i D ji ,and D y i N +1 = N X j =1 D ij (5–39) for i =1,..., N .Basedonthetheorydevelopedin[ 36 ], D y isadifferentiationmatrixfor thespaceofpolynomialsofdegree N .Thatis,if b isapolynomialofdegreeatmost N and b 2 R N +1 isthevectorwith i -thelement b i = b ( i ) ,then ( D y b ) i = b ( i ). (5–40) Furthermore,itcanbeshownthat D y hassimilarpropertiestotheGaussdifferentiation matrix D .Specically,aswasseeninChapter 3 ,thefollowingholdtrue:(a)the squarematrix D y1: N obtainedbyremovingthelastcolumnof D y isfull-rank,and(b) ( D y1: N ) 1 D yN +1 = 1 .Usingtheseproperties,Eq.( 5–34 )canberewrittenas ( k ) 1: N k = ( k ) N +1 [ W ( k ) D y ( k ) 1: N ] 1 ( k ) 1: N k = ( k ) N k +1 A y ( k ) [ W ( k ) ] 1 ( k ) 1: N k (5–41) wherethematrix A y isa backward integrationmatrixforthespaceofpolynomialsof degree N 1 .Specically,let L yi ( ) beabasisofLagrangeinterpolatingpolynomialsof 123

PAGE 124

degree N 1 L yi ( )= N Y j =1 j 6 = i j i j (5–42) Thenif q isapolynomialofdegreeatmost N 1 with q ( i )=_ q i ,itcanbeinterpolated exactlybytheLagrangepolynomials L yi suchthat q ( )= N X i =1 q i L yi ( ). (5–43) Integratingthisexpressionbackwardsyields q ( j )= q (+1)+ N X i =1 A yji q i A yji = Z j +1 L yi ( ) d (5–44) Furthermore,aconstantofintegration(inthiscaseatermi nalcondition)isneededfor theintegrationoftheKKTmultipliers .Sincetheconstraint S ( Y ( k ) 1: N ) 0 isnotenforced atthenalpoint ( K ) N K +1 =+1 inthenalinterval K ,bydenitionitsassociatedmultiplier iszero.Thus, ( K ) N K +1 = ( K ) N K +1 = 0 .Theterminalconditionfortheremainderofthe intervals, ( k =1,..., K 1) ,isdenedbyEq.( 5–36 ). ExpressionsarenowdevelopedthatjustifyEqs.( 5–37 )and( 5–38 )asapproximations ofthestateconstraintmultiplierandthecostate,respect ively,at ( k ) = 1 ( k = 1,..., K ) .First,let ( k ) ( ) bethepolynomialofdegree N thatsatises ( k ) ( i )= ( k ) i for ( i =1,..., N +1) ininterval k ( k =1,..., K ) .SincetheLegendre-Gaussquadratureis exactforapolynomialofdegree N 1 ,then ( k ) 0 = ( k ) N +1 Z +1 1 ( k ) ( ) d = ( k ) N +1 N X i =1 w ( k ) j ( k ) ( i ). (5–45) Furthermore, D y isadifferentiationmatrixforthespaceofpolynomialsofd egree N ,as seenbyEq.( 5–40 ).Therefore,Eq.( 5–45 )becomes ( k ) 0 = ( k ) N +1 N k X i =1 w ( k ) j D y ( k ) i ,1: N +1 ( k ) 1: N +1 = ( k ) N +1 + N k X i =1 i ( k ) whereEq.( 5–34 )wasusedinthelastsubstitution. 124

PAGE 125

Next,itisshownthatthecostateapproximationgivenbyEq. ( 5–37 )isequivalentto applyingtheFundamentalTheoremofCalculus.Let 1 2 R N +1 denotethevectorwhose elementsareallunity.Becausethecomponentsofthevector D ( k ) 1 arethederivatives oftheconstantpolynomial q ( )=1 attheLGpoints, D ( k ) 1 = 0 ,whichimpliesthat D ( k ) 0 = N k X j =1 D ( k ) j (5–46) TakingthetransposeofEq.( 5–46 )andsubstitutingtherowsof D y ( k ) fortherowsof D ( k ) thefollowingexpressionisobtained: D ( k ) 0 = N X i =1 N X j =1 w i w j D y ( k ) ij (5–47) Finally,post-multiplyingtheresultby W ( k ) [ p ( k ) 1: N 1 p ( k ) N +1 ] ,andsubtracting p ( k ) N +1 fromboth sidesyields p ( k ) N +1 + D ( k ) > 0 W ( k ) [ p ( k ) 1: N 1 p ( k ) N +1 ] = p ( k ) N +1 + N k X j =1 w ( k ) j D y ( k ) j ,1: N +1 p ( k ) 1: N +1 (5–48) foreachinterval ( k =1,..., K ) .Nowlet p ( k ) ( ) bethepolynomialofdegree N that satises p ( k ) ( i )= p ( k ) i for ( i =1,..., N +1) ( k =1,..., K ) ,thenusingthesamelogicas wasdoneforEq.( 5–45 ) p ( k ) 0 = p ( k ) N +1 + Z +1 1 p ( k ) ( ) d = p ( k ) N +1 + N k X i =1 w ( k ) i p ( k ) ( i ). ComparingthisexpressionwithEq.( 5–48 ),itisseenthat p ( k ) 0 ( k =1,..., K ) givenby Eq.( 5–37 )isconsistentwiththeFundamentalTheoremofCalculus. 125

PAGE 126

UsingthetransformationsdescribedinEqs.( 5–32 )–( 5–38 )alongwithEq.( 5–39 ), theKKTconditionsoftheNLPgivenbyEqs.( 5–22 )–( 5–31 )canbewrittenas D ( k ) Y ( k ) 0: N = f ( Y ( k ) 1: N U ( k ) 1: N ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (5–49) Y ( k +1) 0 = Y ( k ) 0 + w ( k ) > f ( Y ( k ) 1: N U ( k ) 1: N ),( k =1,..., K 1), (5–50) Y ( K ) N +1 = Y ( K ) 0 + w ( k ) > f ( Y ( k ) 1: N U ( k ) 1: N ), (5–51) 0 = r U H ( Y ( k ) 1: N U ( k ) 1: N p ( k ) 1: N ( k ) 1: N ),( k =1,..., K ), (5–52) D y ( k ) p ( k ) 1: N +1 = r Y H ( Y ( k ) 1: N U ( k ) 1: N p ( k ) 1: N ( k ) 1: N ),( k =1,..., K ), (5–53) p ( k ) 0 = r Y h ( Y (1) 0 ) i + h (1)0 S ( Y (1) 0 ) i (5–54) p ( k +1) 0 = p ( k ) N +1 ,( k =1,..., K 1), (5–55) p ( K ) N +1 = r Y ( Y ( K ) N +1 )+ h ( K ) N +1 S ( Y N +1 ) i (5–56) 0= h D y ( k ) ( k ) 1: N +1 S ( Y ( k ) 1: N ) i ,( k =1,..., K ), (5–57) S ( Y ( k ) 1: N ) 0 D y ( k ) ( k ) 1: N +1 0 ( K ) N +1 = 0 ,( k =1,..., K ), (5–58) where H ( k ) isadiscreteformoftheHamiltonianininterval k givenbyEq.( 5–13 ).Furthermore, thepartialdifferentialsoftheHamiltonianininterval k giveninEqs.( 5–52 )and( 5–53 ) aregivenas r U H ( k ) = r U g ( Y ( k ) i U ( k ) i )+ h p ( k ) i r U f > ( Y ( k ) i U ( k ) i ) ih ( k ) i r Y S > ( Y ( k ) i ) r U f > ( Y ( k ) i U ( k ) i ) i and r Y H ( k ) = r Y g ( Y ( k ) i U ( k ) i )+ h p ( k ) i r Y f > ( Y ( k ) i U ( k ) i ) ih ( k ) i r Y S > ( Y ( k ) i ) r Y f > ( Y ( k ) i U ( k ) i ) i 2 h ( k ) D y ( k ) i ,1: N +1 r Y h ( k ) 1: N +1 S ( Y ( k ) 1: N +1 ) i + h 2 h ( k ) D y ( k ) i ,1: N +1 ( k ) 1: N +1 r Y S ( Y ( k ) i ) i for i =1,..., N .Notethattheproductrulewasusedtodifferentiatethesta teinequality constraint,thatis,thefollowingidentitywasused: h d dt r y S > ( y ) i = d dt h r y S > ( y ) ih d dt r y S > ( y ) i 126

PAGE 127

Carefulcomparisonofthenecessaryconditionsforoptimal ityofthediscreteand continuousproblems[givenbyEqs.( 5–49 )–( 5–58 )andEqs.( 5–7 )–( 5–12 ),respectively] revealstheirequivalence.Notethecostateiscontinuousa crossintervalboundaries,as givenbyEq.( 5–55 ).Furthermore,itisreinforcedthatthestateisbeingdiff erentiatedby amatrix D whichisbasedonthederivativesofpolynomialsofdegree N withcoefcients atthe N LGpointsplustheinitialuncollocatedpoint 0 = 1 ,whereasthecostateand thestateconstraintmultipliersarebeingdifferentiated byamatrix D y whichisbased onthederivativesofpolynomialsofdegree N withcoefcientsatthe N LGpointsplus theterminaluncollocatedpoint N +1 =+1 .Finally,notethattheintegrationmatrix A associatedwiththestatedynamicsintegratesthestatefor wardinthedomain,and requiresaninitialcondition,whereastheintegrationmat rix A y associatedwiththe costateandthestateconstraintmultipliersisabackwardi ntegrationmatrixwhich requiresaterminalcondition. 5.3CostateEstimationUsingFlippedLegendre-Gauss-Rada uCollocation TheoptimalcontrolproblemofEqs.( 5–3 )–( 5–6 )isnowdiscretizedusingvariable-order collocationattheippedLegendre-Gauss-Radaupointsasd escribedinSection 2.5.6 ItisnotedthattheippedLGRpointsareparticularlycondu civetovariable-order collocation;sinceonlyoneofthedomainendpointsarecoll ocated,thereisno“double collocation”attheboundaries.Also,theonlynoncollocat edpointistherstpointofthe rstinterval, t 0 = (1) 0 = 1 .Thus,theippedLGRpointsareanimprovementover theLGpoints,whichprovidednoinformationontheoptimalc ontrolatanyofthemesh points.5.3.1Variable-OrderCollocationatFlippedLegendre-Gau ss-RadauPoints RecallthattheippedLGRpointsaredenedonthedomain ( 1,+1] suchthat N =+1 isaLGRcollocationpointbut 0 = 1 isanoncollocatedpoint.When implementingtheippedvariable-orderLGRmethod,asingl evariableisusedforthe valueofthestateattheendofmeshinterval k andthestartofmeshinterval k +1 ,that 127

PAGE 128

is, Y ( k ) N k Y ( k +1) 0 ,1 k K 1 suchthatcontinuityinthestateisenforced.Hence, redundantvariablesdeningthestateattheinteriormeshp ointsareeliminated. TheoptimalcontrolproblemofEqs.( 5–3 )–( 5–6 )isnowdiscretizedusingvariable-order collocationatthe ipped Legendre-Gauss-RadaupointsasdescribedinSection 2.5.6 TheNLPisthengivenasfollows.Minimizethecostfunction J ( Y ( K ) N K )+ K X k =1 N k X j =1 h ( k ) 2 w ( k ) j g ( Y ( k ) j U ( k ) j ), (5–59) subjecttothealgebraicconstraints D ( k ) Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ),( k =1,..., K ), (5–60) ( Y (1) 0 )= 0 (5–61) S ( Y ( k ) 1: N k ) 0 ,( k =1,..., K ). (5–62) Therst-orderoptimality(KKT)conditionsofthediscrete problemgivenby Eqs.( 5–59 )–( 5–62 )arederivedinthesamemannerofChapter 4 .First,theLagrangian isdenedas L =( Y ( K ) N K ) h ( Y (1)0 ) i + K X k =1 N k X i =1 h ( k ) 2 w ( k ) i g ( k ) i h ( k ) i S ( k ) i i N 1 X i =1 h (1)i D (1) i ,0 Y (1)0 + D (1)i ,1: N 1 Y (1)1: N 1 h ( k ) 2 f (1)i i K 1 X k =1 N k X i =1 h ( k ) i D ( k ) i ,0 Y ( k 1) N + D ( k ) i ,1: N k Y ( k ) 1: N k h ( k ) 2 f ( k ) i i (5–63) where ( k ) i and ( k ) i aretheLagrangemultipliersassociatedwiththedynamicco nstraints ofEq.( 5–60 )andtheinequalitypathconstraintofEq.( 5–62 ),respectively,ininterval k attheLGRpoint i .Furthermore istheLagrangemultipliersassociatedwiththe boundaryconditionsofEq.( 5–61 ). 128

PAGE 129

TheKKTconditionsoftheNLParethengivenas D ( k ) 0: N k Y ( k ) 0: N k = h ( k ) 2 f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (5–64) 0 = r U H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ),( k =1,..., K ), (5–65) D ( k ) > 1: N k ( k ) 1: N k = h ( k ) 2 r Y H ( Y ( k ) 1: N k U ( k ) 1: N k ( k ) 1: N k ( k ) 1: N k ) e N k D ( k +1) 0 ( k +1) 1: N k ,( k =1,..., K 1), (5–66) D ( K ) > 1: N K ( K ) 1: N K = h ( k ) 2 r Y H ( Y ( K ) 1: N K U ( K ) 1: N K ( K ) 1: N K ( K ) 1: N K )+ e N K r Y ( Y ( K ) N K ), (5–67) D (1) > 0 (1)1: N 1 = r Y h ( Y (1) 0 ) i (5–68) S ( Y ( k ) 1: N k ) 0 ( k ) 1: N k 0 h ( k ) 1: N k S ( Y ( k ) 1: N k ) i = 0 ,( k =1,..., K ), (5–69) where e N denotesthe N -thcolumnoftheidentitymatrixandthediscreteHamiltoni anin interval k is H ( Y ( k ) 1: N k U 1: N k ( k ) 1: N k )= w ( k ) > g ( k ) 1: N k + h ( k ) 1: N k f ( k ) 1: N k ih h ( k ) 2 ( k ) 1: N k S ( k ) 1: N k i (5–70) 5.3.2CostateEstimateandTransformedAdjointSystem Inordertorelatethenecessaryconditionsforoptimalityo fthecontinuousproblem [givenbyEqs.( 5–7 )–( 5–12 )]tothediscreteKKTconditionsoftheNLP[givenby Eqs.( 5–64 )–( 5–69 )],considerthefollowingtransformeddualvariables p ( k ) 1: N =[ W ( k ) ] 1 ( k ) + r Y h ( k ) 1: N S ( Y ( k ) 1: N ) i ,( k =1,..., K ), (5–71) ( K ) 1: N = W ( K ) ~ D ( K ) ( K ) 1: N (5–72) p ( k ) 0 = D ( k ) > 0 W ( k ) p ( k ) 1: N ,( k =1,..., K ), (5–73) ( k ) 0 = D ( k ) > 0 W ( k ) ( k ) 1: N ,( k =1,..., K ), (5–74) where W ( k ) isthediagonalmatrixwiththequadratureweights w onthediagonalin interval k and ~ D isdenedby ~ D = W 1 D T1: N W (5–75) 129

PAGE 130

Next,let D y bean N k N k matrixdenedasfollows: D y ij = 8><>: D NN + 1 w N i = j = N w j w i D ji i j =2,..., N (5–76) Basedonthetheorydevelopedin[ 36 ], D y isadifferentiationmatrixforthespaceof polynomialsofdegree N 1 .Thatis,if b isapolynomialofdegreeatmost N 1 and b 2 R N isthevectorwith i -thelement b i = b ( i ) ,then ( D y b ) i = b ( i ). Thematrix ~ D isidenticaltothedifferentiationmatrix D y introducedinEq.( 5–76 )except forthe ( N N ) element: ~ D NN = D NN = D y NN 1 w N (5–77) Because ~ D equals D y exceptforthesingleelementinrow N giveninEq.( 5–77 ),it followsthat ( ~ Db ) i = 8><>: b ( i ),1 i N 1, b ( N ) b ( N ) = w N i = N (5–78) Hence, ~ D behaveslikeadifferentiationmatrixexceptforthelastro wthatboth differentiatesandevaluates.Notethat D y issingularwhile ~ D isinvertiblesince D 1: N isinvertible.Inparticular,byEq.( 5–75 ), ~ D canbeinvertedas ~ D 1 = W 1 D T 1: N W Nowlet ~ A denote ~ D 1 ,byEq.( 5–72 ),thefollowingrepresentationfor ( K ) intermsof ( K ) inthelastmeshinterval K isobtained: ( K ) 1: N = [ W ( K ) ~ D ( K ) ] 1 ( K ) 1: N = ~ A ( K ) [ W ( K ) ] 1 ( K ) 1: N (5–79) 130

PAGE 131

Itcanbeshownthat ~ A isanintegrationmatrixwhichalsoextrapolatesthenalva lueof thevectoritoperateson.Specically,theelementsof ~ A aregivenas ~ A ij = Z i +1 ~ L j ( ) d + w N ~ L j ( N ),( i j =1,..., N 1), ~ A iN = w N ,( i =1,..., N ), ~ A Nj = w N ~ L j ( N ),( j =1,..., N 1), (5–80) wherethe N 1 Lagrangeinterpolatingpolynomials ~ L j ( ) aredenedas ~ L j ( ) = N 1 Y i =1 j 6 = i i j i j =1,..., N 1. (5–81) ItisknownthattheKKTmultiplier isrelatedtothedualvariablesofthecontinuous-time problemasfollows ( k ) i = ( k ) i w ( k ) i ,( i =1,..., N 1), ( k ) N = ( k ) N + ( k ) w ( k ) N (5–82) whereitisrecalledfromChapter 2 that ( k ) =_ ( k ) ,and isthemultiplierassociated withthe“jump”,ordiscontinuityofthestateconstraintmu ltiplier.Next,letthecontinuous-time stateconstraintmultiplier beapolynomialofdegreeatmost N 2 suchthat i = ( i ) for ( i =1,..., N 1) .ThispolynomialcanbedescribedexactlyusingtheLagrang e interpolatingbasesofEq.( 5–81 )suchthat ( )= N 1 X j =1 j ~ L j (5–83) SubstitutingtheexpressionsfromEq.( 5–82 )intoEq.( 5–80 )itisseenthat ~ A ( k ) i ( k ) 1: N = 8>>>>><>>>>>: Z i +1 ( k ) d + N 1 X j =1 j ~ L ( k ) j ( N ) w ( k ) N ( k ) N ,( i =1,..., N 1), N 1 X j =1 j ~ L ( k ) j ( N ) w ( k ) N ( k ) N ,( i = N ). (5–84) 131

PAGE 132

Thus,itisseenthatthematrix ~ A integratesandextrapolates themultiplier for ( i = 1,..., N 1) .Furthermore,theexpressionfor ( i = N ) amountstoanapproximationof thejumpmultiplier ( k ) .Consequently,therighthandsideofEq.( 5–79 )isequivalentto integratingthestateinequalityconstraintmultipliersb ackwardfrom ( K ) =+1 inthelast meshinterval K .Furthermore,because ( k ) isbeingintegratedbackwardacrosseach meshinterval,thevalueoftheintegrationininterval k +1 mustbeaddedasaninitial conditionoftheintegrationininterval k ,yielding: ( k ) 1: N = 1 ( k +1) 0 ~ A ( k ) [ W ( k ) ] 1 ( k ) 1: N ,( k =1,..., K 1), (5–85) where 1 isa N 1 columnvectorcomposedofones. TojustifythatEq.( 5–74 )isanapproximationofthestateconstraintmultiplier at ( k ) = 1 for ( k =1,..., K ) ,itisnowshownthatthisdenitionisequivalentto applyingtheFundamentalTheoremofCalculus.Let 1 2 R N +1 denotethevectorwhose elementsareallunity.Becausethecomponentsofthevector D 1 arethederivativesof theconstantpolynomial q ( )=1 attheLGRpoints, D 1 = 0 ,whichimpliesthat D 0 = N X j =1 D j (5–86) TakingthetransposeofEq.( 5–86 ),substitutingtherowsof ~ D fortherowsof D T ,and multiplyingtheresultontherightby W ( K ) ( K ) 1: N yields D ( K ) 0 > W ( K ) ( K ) 1: N = ( K ) 0 = N X j =1 w ( K ) j ~ D ( K ) j ,1: N ( K ) 1: N (5–87) Next,let ( ) bethepolynomialofdegree N 1 thatsatises ( i )= i for 1 i N UsingEq.( 5–78 )togetherwiththefactthattheLegendre-Gauss-Radauquad ratureis exactforapolynomialofdegree N 2 ,then 0 = N + N X i =1 w j ( i )= N + Z +1 1 ( ) d 132

PAGE 133

Because 0 istheapproximationtothestateconstraintmultiplierat = 1 ,while N istheapproximationat =+1 ,itisseenthat 0 inEq.( 5–74 )isconsistentwiththe FundamentalTheoremofCalculus.Usingthesamelogicasabo ve,let 1 2 R N denote thevectorwhoseelementsareallunity.Becausethecompone ntsofthevector D y 1 are thederivativesoftheconstantpolynomial q ( )=1 attheLGRpoints, D y 1 = 0 ,which implies: N X j =1 D yj = 0 (5–88) Because ~ D isidenticaltothedifferentiationmatrix D y introducedinEq.( 5–76 )exceptfor the ( N N ) element,substitutingthevaluesof ~ D intoEq.( 5–88 )resultsin [ ~ D 1 ] i = 8><>: 0for( i =1,..., N 1), 1 w N for( i = N ). (5–89) Thus,substitutingthevaluesof ( k ) 1: N obtainedfromEq.( 5–85 )intotheright-handsideof Eq.( 5–87 )resultsintherelationship D ( k ) 0 > W ( k ) ( k ) 1: N = ( k ) 0 = ( k +1) 0 + N X j =1 w ( k ) j [ W 1 ( k ) 1: N ] j ,( k =1,..., K 1). (5–90) Furthermore,substitutingtheexpressionsgivenbyEq.( 5–82 )intoEq.( 5–90 )resultsin theexpression ( k ) 0 = ( k +1) 0 ( k ) + N X j =1 w ( k ) j j ,( k =1,..., K 1). (5–91) whichsaysthatthevalueofthestateconstraintmutliplier attherstpointofaninterval isgivenbyaRadauquadraturewhichapproximatesthebackwa rdintegralofthestate constraintdynamicsacrossthatintervalsummedwithaterm inalconditiongivenby ( k ) N = ( k +1) 0 + ( k ) .Finally,inasimilarfashion,itcanbeshownthatEq.( 5–73 )implies 133

PAGE 134

thefollowingrelation: D 0 Wp 1: N = p 0 = p N + N X j =1 w j D yj ,1: N p 1: N (5–92) Because D y isadifferentiationmatrix,itisseenthatEq.( 5–73 )isalsoconsistentwith theFundamentalTheoremofCalculus. Now,usingthetransformationsdescribedinEqs.( 5–71 )–( 5–74 )alongwith Eqs.( 5–75 )and( 5–77 ),theKKTconditionsoftheNLPgivenbyEqs.( 5–64 )–( 5–69 ) canbewrittenas D ( k ) Y ( k ) 0: N k = f ( Y ( k ) 1: N k U ( k ) 1: N k ), ( Y (1) 0 )= 0 ,( k =1,..., K ), (5–93) 0 = r U H ( Y ( k ) 1: N U ( k ) 1: N p ( k ) 1: N ( k ) 1: N ),( k =1,..., K ), (5–94) D y ( k ) 1: N k p ( k ) 1: N k = r Y H ( Y ( k ) 1: N U ( k ) 1: N p ( k ) 1: N ( k ) 1: N ) (5–95) + e N k w N k h p ( k ) N k p ( k +1) 0 i ,( k =1,..., K 1), (5–96) D y ( K ) 1: N K p ( K ) 1: N K = r Y H ( Y ( K ) 1: N U ( K ) 1: N p ( K ) 1: N ( K ) 1: N ) + e N K w N K h p ( K ) N r Y ( Y ( K ) N ) h ( K ) N S ( Y ( K ) N ) i i (5–97) p (1)0 = r Y h ( Y (1) 0 ) i + h 0 S ( Y 0 ) i (5–98) S ( Y ( k ) 1: N ) 0 ~ D ( k ) ( k ) 1: N 0 h ~ D ( k ) ( k ) 1: N S ( Y ( k ) 1: N ) i =0,( k =1,..., K ), (5–99) where H isthecontinuousHamiltoniandenedinEq.( 5–13 ).Next,substituting Eq.( 5–97 )intoEq.( 5–92 )yields p ( K ) 0 = r Y ( Y ( K ) N ) h ( K ) N S ( Y ( K ) N ) i N K X i =1 w ( K ) i r Y H ( Y ( K ) i U ( K ) i p ( K ) i ( K ) i ). (5–100) If p ( K ) i werethecontinuouscostateevaluatedat i ininterval K ,thenbythecontinuous adjointequation,thesuminEq.( 5–100 )approximatestheintegralof p ( K ) between 1 134

PAGE 135

and +1 .Eq.( 5–100 )amountstoanapproximationtotherelation p ( K ) N = r Y ( Y ( K ) N )+ h ( K ) N S ( Y ( K ) N ) i (5–101) whereEq.( 5–101 )isadiscreteformofthecontinuousoptimalityconditiong ivenin Eq.( 5–11 ).Consequently,itisexpectedthatthe e N terminEq.( 5–97 )shouldbe small,whiletheremainingtermsinEq.( 5–97 )amounttoacollocationschemeforthe continuousadjointequation.Next,itisshownthatthelast terminthebracketsonthe right-handsideofEq.( 5–96 )willbesmall.SubstitutingEq.( 5–96 )intoEq.( 5–92 )yields p ( k +1) 0 = p ( k ) 0 N k X i =1 w ( k ) i r Y H ( Y ( k ) i U ( k ) i p ( k ) i ( k ) i ),( k =1,..., K 1). (5–102) If p ( k ) i werethecontinuouscostateevaluatedat i ininterval k ,thenbythecontinuous adjointequation,thesuminEq.( 5–102 )approximatestheintegralof p ( K ) between 1 and +1 .Eq.( 5–102 )amountstoanapproximationtotherelation p ( k ) N k = p ( k +1) 0 (5–103) Thisconditionshowsthatthecostatewillbecontinuousacr ossmeshintervalboundaries. Consequently,itisexpectedthatthe e N terminEq.( 5–96 )shouldbesmall,whilethe remainingtermsinEq.( 5–96 )amounttoacollocationschemeforthecontinuousadjoint equation. Theconnectionbetweenthetransformedoptimalityconditi onsandtheoriginal continuousoptimalityconditionsisquitesubtle.Forexam ple,thenonnegativity conditionsforthederivativeofthestatemultiplier ( k ) andthecomplementaryslackness conditionsinEq.( 5–12 )areembeddedinaveryunusualwayinthediscreteoptimalit y conditions.AspointedoutinEq.( 5–78 ),ifthediscretemultiplier k ) 1: N associatedwith thestateconstraintisinterpolatedbyapolynomial ( ) ofdegree N 1 ,thenthe nonnegativityconditionsinEq.( 5–98 )onlyensurenonnegativityofthepolynomial 135

PAGE 136

derivativeat 1 through N 1 .At N ,thediscretepositivityconditionamountsto ( K ) ( N ) ( K ) ( N ) w ( K ) N 0, (5–104) ( K ) ( N )+ ( k ) ( N )+ ( k +1) ( 0 ) w ( k ) N 0,( k =1,..., K ). (5–105) Toillustratehowtheseconditionswork,supposethatthest ateconstraintisinactiveover theentirenalinterval [ 1,+1] forthediscreteproblem.Thatis, S ( Y ( K ) i ) < 0 forall i .In thiscase,complementaryslacknessimpliesthat ( K ) ( i )=0 for 1 i N 1. (5–106) Becausethederivativeofapolynomialofdegree N 1 is N 2 ,the N 1 conditions Eq.( 5–106 )implythatthederivativeisidenticallyzero.Hence, ( K ) ( N )= 0 in Eq.( 5–104 ),anditisconcludedthat ( K ) ( N )= (+1) 0 .Finally,fromthe complementaryslacknesscondition, 0 = S ( Y ( K ) N ) T (_ ( K ) ( N ) ( K ) ( N ) = w N ) = S ( Y ( K ) N ) T ( K ) ( N ) = w N whichimpliesthat ( K ) ( N )= ( K ) N = 0 when S ( Y ( K ) N ) < 0 .Hence,thecontinuous optimalityconditions ( K ) (+1) 0 and h ( K ) (+1), S ( y ( K ) (+1)) i = 0 aresatised inthediscreteproblem.Furthermore,ifcomplementarysla cknessholdsininterval K ,then ( K ) isanon-decreasingfunctionininterval K ,and ( K ) 0 ( K 1) N ,thusthe secondterminEq.( 5–105 )willbegreaterthanzero.Asimilarargumentcanbemade overeachinterval,suchthatthecondition ( k ) 0 issatisedfor ( k =1,..., K ) Thus,ithasbeenshownthattheconditionsofthetransforme dadjointsystemgiven byEqs.( 5–93 )–( 5–99 )areadiscreteformoftherst-orderoptimalitycondition softhe continuous-timeoptimalcontrolproblemgivenbyEqs.( 5–7 )–( 5–12 ). Thetransformedadjointsystemforvariable-ordercolloca tionattheLGRpointsis complex.Forinstance,thedifferentiationmatrixassocia tedwiththestatedynamics, D 136

PAGE 137

isa N ( N +1) full-rankdifferentiationmatrixassociatedwiththespac eofpolynomials ofdegreeatmost N .Conversely,thedifferentiationmatrixassociatedwitht hecostate dynamics, D y isarank-defcient N N differentiationmatrixassociatedwiththespace ofpolynomialsofdegreeatmost N 1 .Finally,thematrixassociatedwiththestate constraintmultiplier, ~ D ,isafull-rank N N matrixthatdifferentiatesandevaluatesthe terminalconditionsimultaneously. 5.4Discussion Inthischaptercostateestimateswerederivedforestimati ngthecostateofstate inequalitypathconstrainedoptimalcontrolproblemsusin gorthogonalcollocation attheLegendre-GaussandtheippedLegengre-Gauss-Radau Points.These conditionsresultinacontinuousapproximationtothecost ateeveninthepresence ofstateinequalitypathconstraints.Furthermore,thecos tateestimatederivedhere reducestothecostateestimategivenbyRef.[ 1 ],presentedinChapter 4 ,whenno stateinequalitypathconstraintsarepresentintheoptima lcontrolproblem.Finally, Itwasshownthatthecostateestimateusingthemethodofind irectadjoiningwith continuousmultipliersresultedinatransformedadjoints ystemthatisadiscreteformof therst-orderoptimalityconditionsofthecontinuous-ti meproblem.Fig.( 5-1 )illustrates theequivalencebetweenthetransformedadjointsystemder ivedfromtheNLPandthe rst-orderoptimalityconditionsofthecontinuous-timep roblemderivedfromthecalculus ofvariations. 137

PAGE 138

Figure5-1.Relationshipbetweenthedirectandindirectme thodsforsolvinganoptimal controlproblem.Intheindirectmethod,theproblemisrst optimizedthroughthe calculusofvariations,leadingtoasetofconditionswhich canthenbediscretized andsolved.Inthedirectmethod,theproblemisrstdiscret izedandtranscribedto anNLP,thenitisoptimizedbysolvingtheKKTsystem.Thetwo systemsareshownto beequivalenteveninthepresenceofadiscontinuouscostat e. 138

PAGE 139

CHAPTER6 EXAMPLES Inthischapter,fourexamplesarestudiedusingthemethods developedinChapters 3 and 5 .Thersttwoexamplesdemonstratetheeffectivenessofthe costateestimation methodsderivedinChapter 3 usingtheintegralformofLGandLGRcollocation. TherstexampleisasinglestatenonlinearMayeroptimalco ntrolproblemwhile thesecondexampleisasinglestatenonlinearLagrangeopti malcontrolproblem. Next,twostateinequalitypathconstrainedoptimalcontro lproblemsaresolvedusing variable-orderLGandLGRcollocationasdescribedbyChapt er 5 .Therststate inequalityconstrainedexamplecontainsarst-orderstat einequalitypathconstraint, whilethesecondstateinequalityconstrainedexamplecont ainsasecond-orderstate inequalitypathconstraint.TheLGandLGRcostateestimate sderivedinRef.[ 1 ] areshowntoproduceinaccurateestimatesofthedualvariab lesforbothexamples, whiletheLGandLGRcostateestimatesusingthemethodofind irectadjoiningwith continuousmultipliers(asdescribedinChapter 5 )areshowntoproduceaccurate approximationsofthedualvariables. Threemainobservationsaremadefromtheexamplessolvedin thischapter.First, itisshownthatvariable-ordercollocationattheLGandLGR pointsproducesaccurate approximationstostateinequalitypathconstrainedoptim alcontrolproblems.Because collocationattheLGpointsdoesnotprovideanapproximati ontotheoptimalcontrolat anyofthemeshpointswhereascollocationattheLGRpointsp rovidesanapproximation oftheoptimalcontrolatallinteriormeshpoints,collocat ionattheLGRmethodisfound tobethepreferredmethodofsolution.Second,itisshownth atforstateinequalitypath constraintsofatmostordertwo,itisnotnecessarytorefor mulatetheoptimalcontrol problembyreducingtheindexoftheDAEinordertoobtainana ccurateapproximation. Index-reductionrequiresananalyticreformulationofthe optimalcontrolproblemwhich maybecumbersome,ifnotimpossible,toimplementwhenusin gmeshrenement. 139

PAGE 140

Third,becausethemethodofindirectadjoiningwithcontin uousmultipliersproduces acostateestimatethatiscontinuouseveninthepresenceof stateinequalitypath constraint,highlyaccurateestimatesofthecostatecanbe obtainedevenwhenusing low-orderpolynomialapproximationsinthestate(thatis, asmallnumberofcollocation points). 6.1Example1:MayerOptimalControlProblem Therstexampleconsideredisanonlinearone-dimensional Mayeroptimalcontrol problem[ 64 ].Itisstatedasfollows: Minimize J = y (2) subjectto 8><>: y = 5 2 ( y + yu u 2 ), y (0)=1. Theoptimalsolutionisgivenas y ( t )= 4 a ( t ) u ( t )= y ( t ) 2 p y ( t )= (15exp(5 t = 2) 1)( 3exp(5 t = 2) 1) (2 b exp(5 t = 2)) y ( t )= exp(2ln( a ( t )) 5 t = 2) b where a ( t )=1+3exp(5 t = 2) ,and b =exp( 5)+6+9exp(5) .Thisexamplewassolved usingtheintegralLGandLGRcollocationmethodsusingtheN LPsolverSNOPT[ 22 ], whereSNOPTwasimplementedusingoptimalityandfeasibili tytolerancesof 1 10 8 and 2 10 8 ,respectively.Theinitialguessusedforthestateandcont rolwasalinear interpolationfromtheinitialstatevaluetozero.Forcoll ocationateithersetofpoints, theintegralcostate, p y ( ) ,wasestimatedfromtheKKTmultipliersoftheNLP,and thedifferentialcostate, y ( ) ,wassubsequentlycomputedfromtheintegralcostate approximation.6.1.1SolutionUsingCollocationatLegendre-GaussPoints Example1wassolvedusingintegralcollocationatLGpoints asdescribedin Chapter 3 .Figure 6-1 showsthestateandcontrolapproximationobtainedusing N =20 140

PAGE 141

0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.6 0.7 0.8 0.8 0.9 1.2 1.4 1.61.8 1 1 2 t State y ( t ) y ( t ) (A)State. 0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.8 1.2 1.4 1.61.8 0.05 0.15 0.25 0.35 0.45 12 t Control u ( t ) u ( t ) (B)Control. Figure6-1.PrimalsolutionforExample1obtainedusingint egralcollocationatLG points.LGcollocationpoints.Itisseenthatintegralcollocation attheLGpointsprovidesa highlyaccurateapproximationtotheoptimalsolution. Next,theintegralcostate, p y ( ) ,wascomputedattheLGpointsusingEq.( 3–62 ), andthedifferentialcostate, y ( ) ,wasestimatedattheLGpointsplusthenoncollocated endpoints 0 and N +1 usingtheresultsofSection 3.2.5 .Figure 6-2 showsboththe 141

PAGE 142

Costate -0.5 0 0 0.5 0.5 1.5 1.5 2.5 -1 1 1 2 2 t p y y p y y Figure6-2.Integralanddifferentialcostatesolutionsfo rExample1obtainedusingLG collocation.integralandthedifferentialcostatesobtainedusing N =20 LGcollocationpoints. Itisseenthatthecostateestimateisindistinguishablefr omtheoptimalcostate. Furthermore,Fig. 6-3 showsthebasetenlogarithmofthe L 1 -normerrorforthe integralanddifferentialcostateswhenapproximatedusin g ( N =2,4,6,...,20) LG collocationpoints.Itisinterestingtonotethatthediffe rentialcostateestimateconverges exponentiallyasafunctionof N andreachesanaccuracyof O (10 12 ) for N =20 whereastheintegralcostateestimateachievesanaccuracy ofapproximately O (10 11 ) for N =20 142

PAGE 143

0 0 -8 -6 -4 -2 4 8 12 -12 16 20 -10 Nlog 10 Innity-NormErrorrr y y rr 1 rr p y p y rr 1 Figure6-3.IntegralanddifferentialcostateerrorsforEx ample1obtainedusingLG collocation. 143

PAGE 144

6.1.2SolutionUsingCollocationatLegendre-Gauss-Radau Points Next,Example1wassolvedusingintegralLGRcollocationas describedinChapter 3 .Figure 6-4 showsthestateandcontrolapproximationobtainedusing N =20 LGRcollocationpoints.Itisseenthat,similartocollocat ionattheLGpoints,integral collocationattheLGRpointsprovideshighlyaccurateappr oximationtotheoptimal solution.However,unlikecollocationattheLGpoints,col locationattheLGRpoints providesanapproximationofthecontrolattheterminalbou ndarypoint,making collocationattheLGRpointsmoredesirablethancollocati onattheLGpoints. Next,theintegralcostate, p y ( ) ,wascomputedattheLGRpointsusingEq.( 3–116 ), andthedifferentialcostate, y ( ) ,wasestimatedattheLGRpointsplusthenoncollocated endpoint N +1 usingtheresultsofSection 3.3.5 .Notethatthevalue p y ( 1 ) (where 1 = 1 fortheintegralLGRcollocationmethod)wasfoundbyextrap olatingthe LagrangeinterpolatingpolynomialasdescribedbyEqs.( 3–118 )and( 3–119 ).Figure 6-5 showsboththeintegralandthedifferentialcostatesobtai nedusing N =20 LGRcollocationpoints.Itisseenthatthecostateestimate isindistinguishablefrom theoptimalsolution.Furthermore,Fig. 6-6 showsthebasetenlogarithmofthe L 1 -normerrorfortheintegralanddifferentialcostateswhen approximatedusing ( N =2,4,6,...,20) LGRcollocationpoints.Itisseenthatthedifferentialand integral costateestimatesconvergesexponentiallyasafunctionof N untiltheerrorreaches approximately O (10 10 ) for N =20 144

PAGE 145

0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.6 0.7 0.8 0.8 0.9 1.2 1.4 1.61.8 1 1 2 t State y ( t ) y ( t ) (A)State. 0 0 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.6 0.8 1.2 1.4 1.61.8 0.05 0.15 0.25 0.35 0.45 1 2 t Control u ( t ) u ( t ) (B)Control. Figure6-4.PrimalsolutionforExample1obtainedusingint egralcollocationatLGR points. 145

PAGE 146

Costate -0.5 0 0 0.5 0.5 1.5 1.5 2.5 -1 -1.5 1 1 2 2 t p y y p y y Figure6-5.Integralanddifferentialcostatesolutionsfo rExample1obtainedusingLGR collocation. 0 0 -8 -6 -4 -2 4 8 12 16 20 -10 -12 Nlog 10 Innity-NormErrorrr y y rr 1 rr p y p y rr 1 Figure6-6.IntegralanddifferentialcostateerrorsforEx ample1obtainedusingLGR collocation. 146

PAGE 147

6.2Example2:LagrangeOptimalControlProblem Thissecondexampleconsideredisanonlinearone-dimensio nalLagrangeoptimal controlproblemgivenasfollows. Minimize J = 1 2 Z tf 0 (log 2 y + u 2 ) dt subjectto 8>>>><>>>>: y = y log y + yu y (0)=5, y ( tf )=3. Theoptimalsolutiontothisexampleisgivenas y ( t )=exp( x ( t )), y ( t )= x ( t ) = y ( t ), p y ( t )= x ( t ) y ( t ) x ( t )_ y ( t ) ( y ( t )) 2 (6–1) where x ( t )= c 1 exp( t p 2)+ c 2 exp( t p 2), x ( t )= c 1 (1+ p 2)exp( t p 2)+ c 2 (1 p 2)exp( t p 2), (6–2) and 264 c 1 c 2 375 = 264 11 exp( t f p 2)exp( t f p 2) 375 264 log y 0 log y f 375 (6–3) TheexamplewassolvedusingtheintegralLGandLGRcollocat ionmethodsusingthe NLPsolverSNOPT[ 22 ],whereSNOPTwasimplementedusingoptimalityandfeasibi lity tolerancesof 1 10 8 and 2 10 8 ,respectively,withtheexactstateandcontrol evaluatedatthediscretizationpointsastheinitialguess .Forcollocationateithersetof points,theintegralcostate, p y ( t ) ,wasestimatedfromtheKKTmultipliersoftheNLP, andthedifferentialcostate, y ( t ) ,wassubsequentlycomputedfromtheintegralcostate approximation. 147

PAGE 148

3.5 3.5 4.5 4.5 5.5 0 0.5 1.5 1.5 2.5 2.5 1 1 2 2 3 3 4 4 5 5 t State y ( t ) y ( t ) (A)State. -3.5 -2.5 -1.5 3.5 4.5 -0.5 0 0 0.5 0.5 1.5 2.5 -4 -3 -2 -1 12 3 4 5 t Control u ( t ) u ( t ) (B)Control. Figure6-7.StateandcontrolforExample2obtainedusingin tegralLGcollocation. 6.2.1SolutionUsingCollocationatLegendre-GaussPoints Example2wassolvedusingintegralcollocationatLGpoints ,asdescribedin Chapter 3 .Figure 6-7 showstheprimalsolution(thatis,thestateandcontrol)ob tained using N =32 LGcollocationpoints.Itisseenthatintegralcollocation attheLGpoints provideshighlyaccurateapproximationtotheoptimalsolu tion. 148

PAGE 149

-0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 t Costatep y y p y y Figure6-8.IntegralanddifferentialcostateforExample2 obtainedusingLGcollocation. Next,theintegralcostate, p y ( ) ,wasestimatedattheLGpointsusingEq.( 3–62 ), andthedifferentialcostate, y ( ) ,wasestimatedattheLGpointsplusthenoncollocated endpoints 0 and N +1 usingtheresultsofSection 3.2.5 .Figure 6-8 showsboththe integralandthedifferentialcostatesobtainedusing N =32 LGcollocationpoints.Itis seenthatboththedifferentialandintegralcostateestima tesareindistinguishablefrom theoptimalcostates.Figure 6-9 showsthebasetenlogarithmofthe L 1 -normerrorfor theintegralanddifferentialcostatesfor ( N =4,8,12,...,32) LGcollocationpoints.Both thedifferentialandintegralcostateestimatesconverges exponentiallyasafunctionof N untiltheerrorreachesapproximately O (10 12 ) for N =32 149

PAGE 150

32 28 24 -12 0 0 -8 -6 -4 -2 4 8 12 16 20 -10 Nlog 10 Innity-NormErrorrr y y rr 1 rr p y p y rr 1 Figure6-9.IntegralanddifferentialcostateerrorsforEx ample2obtainedusingintegral LGcollocation. 150

PAGE 151

6.2.2SolutionUsingCollocationatLegendre-Gauss-Radau Points Next,Example2wassolvedusingintegralcollocationatLGR points,asdescribed inChapter 3 .Figure 6-10 showsthestateandcontrolapproximationsobtainedusing N =32 LGRcollocationpoints.Itisseenthat,similartocollocat ionattheLGpoints, integralcollocationattheLGRpointsprovideshighlyaccu rateapproximationtothe optimalsolution.However,unlikecollocationattheLGpoi nts,collocationattheLGR pointsprovidesanapproximationofthecontrolatthetermi nalboundarypoint,making collocationattheLGRpointsmoredesirablethancollocati onattheLGpoints. Theintegralcostate, p y ( ) ,wascomputedattheLGRpointsusingEq.( 3–116 ),and thedifferentialcostate, y ( ) ,wasestimatedattheLGRpointsplusthenoncollocated endpoint N +1 usingtheresultsofSection 3.3.5 .Notethatthevalue p y ( 1 ) (where 1 = 1 fortheintegralLGRcollocationmethod)wasfoundbyextrap olatingthe LagrangeinterpolatingpolynomialasdescribedbyEqs.( 3–118 )and( 3–119 ).Figure 6-11 showsboththeintegralandthedifferentialcostatesobtai nedusing N =32 LGRcollocationpoints.Itisseenthatthecostateestimate isindistinguishablefrom theoptimalsolution.Furthermore,Fig. 6-12 showsthebasetenlogarithmofthe L 1 -normerrorfortheintegralanddifferentialcostateswhen approximatedusing ( N =4,8,12,...,32) LGRcollocationpoints.Itisseenthatthedifferentialand integral costateestimatesconvergesexponentiallyasafunctionof N untiltheerrorreaches approximately O (10 12 ) for N =32 151

PAGE 152

3.5 3.5 4.5 4.5 5.5 0 0.5 1.5 1.5 2.5 2.5 1 1 2 2 3 3 4 4 5 5 t State y ( t ) y ( t ) (A)State. -3.5 -2.5 -1.5 3.54.5 -0.5 0 0 0.5 0.5 1.5 2.5 -4 -3 -2 -1 1 2 3 4 5 t Control u ( t ) u ( t ) (B)Control. Figure6-10.StateandcontrolforExample2obtainedusingi ntegralLGR. 152

PAGE 153

-0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 t Costatep y y p y y Figure6-11.IntegralanddifferentialcostateforExample 2obtainedusingintegralLGR collocationwith N =32 32 28 24 -12 0 0 -8 -6 -4 -2 4 8 12 16 20 -10 Nlog 10 Innity-NormErrorrr y y rr 1 rr p y p y rr 1 Figure6-12.IntegralanddifferentialcostateerrorsforE xample2obtainedusingintegral LGRcollocation. 153

PAGE 154

6.3Example3:First-OrderStateInequalityPathConstrain tProblem Considerthefollowingstateinequalitypathconstrainedo ptimalcontrolproblem: Minimize Z 3 0 e t udt subjectto 8>>>>><>>>>>: y = u y (0)=0, y 1+( t 2) 2 0, 0 u 3. (6–4) Theoptimalstateandcontrolforthisexamplearegivenas y = 8>>>><>>>>: 0, t 2 [0,1), 1 ( t 2) 2 t 2 [1,2], 1, t 2 (2,3], u = 8>>>><>>>>: 0, t 2 [0,1), 2(2 t ), t 2 [1,2], 0, t 2 (2,3], (6–5) ThisexamplewassolvedusingLGandippedLGRcollocationw iththeNLPsolver SNOPT,whereSNOPTwasimplementedusingdefaultsettings. Thesolutiondomain wasdividedintothreeintervalswith N collocationpointsineachinterval.Theboundaries betweentheintervalswerechosentobethetimeinstantswhe rethestateconstraint changesbetweenactiveandinactive,namely,theintervalb oundarieswereat t =1 and t =2 .Furthermore,astraightlineinitialguessbetweentheini tialstateandunitywas usedforthestateandcontrol.6.3.1SolutionUsingCollocationatLegendre-GaussPoints Example3wassolvedusingvariable-ordercollocationatth eLegendre-Gauss points,asdescribedinChapter 2 .Figure 6-13 showsthestateandcontrolapproximations obtainedusing N =10 collocationpointsperinterval.Itisseenthatvariable-o rderLG collocationprovideshighlyaccurateapproximationstoth eoptimalsolutioneventhough index-reductionofthestateinequalitypathconstraintwa snotperformed.Figure 6-14 154

PAGE 155

0 0 0.2 0.4 0.5 0.6 0.8 1.2 1.5 2.5 1 1 2 3 -0.2 tState y ( t ) y ( t ) (A)State. 0 0 0.5 0.5 1.5 1.5 2.5 2.5 1 1 2 2 3 t Control u ( t ) u ( t ) (B)Control. Figure6-13.PrimalsolutionforExample3obtainedusingco llocationatLGpoints. showsthebasetenlogarithmofthe L 1 -normerrorforthestateandthecontrolfor ( N =2,4,6,...,20) collocationpointsperinterval.Itisinterestingtoseeth attheLG stateandcontrolapproximationsarehighlyaccurateevenu singlow-degreestate approximations(thatis,usingasmallnumberofcollocatio npoints). 155

PAGE 156

-8.5 -9.5 -10.5 -11 -11.5 12 -9 -8 2 4 6 8 10 1214 1618 20 -10 Nlog 10 Innity-NormError k y y k 1 k u u k 1 Figure6-14.StateandcontrolerrorsforExample3obtained usingLGcollocation. 156

PAGE 157

6.3.1.1Previouslyderivedcostateestimate TheaccuracyofthecostateestimateofRef.[ 1 ]isnowcomparedtothecostate estimatederivedinChapter 5 .TheanalyticoptimalcostateforExample3using themethodofdirectadjoiningcanbefoundbyapplyingther st-orderoptimality conditionsderivedinSection 2.3.2 .Thecostateandstateconstraintmultipliersare given,respectively,as = 8>>>><>>>>: e 1 t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3], = 8>>>><>>>>: 0, t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3]. (6–6) Figure 6-15 showstheresultofthecostateapproximationfor N =10 collocation pointsperinterval.Itcanbeseenthatthecostate, ( t ) ,isnotapproximatedcorrectlyat theintervalboundarieswherethediscontinuitiesoccur.F urthermore,theapproximation ofthestateconstraintmultiplier, ,isquitepoor.Figure 6-16 showsthebaseten logarithmofthe L 1 -normerrorforthecostateandthestateconstraintmutlipl iersfor ( N =2,4,6,...,20) collocationpointsperinterval.Itisseenthatthecostate haslarge errorsneartheknowndiscontinuitiesintheoptimalcostat e.Furthermore,thestate constraintmultiplierestimatediverges. 157

PAGE 158

0 0 0.5 1.5 2.5 0.05 1 2 3 -0.4 -0.35 -0.3 -0.25 -0.15 -0.1 -0.05 -0.2 tCostate ( t ) ( t ) (A)Costate. -0.5 0 0 0.5 1.5 2.5 -4 -3 -2 -1 1 2 3 -3.5 -2.5 -1.5 t StateConstraintMultiplier ( t ) ( t ) (B)StateConstraintMultiplier. Figure6-15.CostateEstimateasderivedbyRef.[ 1 ]forExample3obtainedusing collocationatLGpoints. 158

PAGE 159

0 1.2 1 2 4 6 8 10 12 14 16 1820 -0.2 -0.4 -0.6 0.2 0.4 0.6 0.8 Nlog 10 Innity-NormErrork k 1 k k 1 Figure6-16.ErrorsincostateestimatederivedbyRef.[ 1 ]forExample3obtainedusing LGcollocation. 159

PAGE 160

6.3.1.2Costateestimateusingmethodofindirectadjoinin gwithcontinuous multipliers Thecostateestimatederivedusingthemethodofindirectad joiningwithcontinuous multipliersforcollocationattheLGpoints,describedinC hapter 5 ,isnowanalyzed.The optimalcostateforExample3usingthemethodofindirectad joiningwithcontinuous multiplierscanbefoundbyapplyingtherst-orderoptimal ityconditionsderivedin Section 2.3.3 .Thecostateandthestateconstraintmultipliersaregiven ,respectively,as p = 8>>>><>>>>: 0, t 2 [0,1), 0, t 2 [1,2], 0, t 2 (2,3], = 8>>>><>>>>: e 1 t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3]. (6–7) Figure 6-17 showsthecostateapproximationfor N =10 .Itcanbeseenthatthe estimatespresentedinChapter 5 provideanaccuratecostateapproximationofthe continuousoptimalcontrolproblem.Figure 6-18 showsthebasetenlogarithmofthe L 1 -normerrorforthecostateandstateconstraintmultiplier .Itcanbeseenthatthe erroronthestateinequalityconstraintmultiplierdecrea sesasthenumberofcollocation pointsisincreased.Furthermore,theerroronthecostatea pproximationremains approximatelyzero.Therefore,thecostateestimateprodu cesanaccuracyof O (10 12 ) evenwheninacuraciesinthestateconstraintmultiplierar epresentduetotheuseof low-orderpolynomialapproximationsinthestate. 160

PAGE 161

0.5 1 1.5 2 2.5 3 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0 0.05 tDualVariablesp ( t ) ( t ) p ( t ) ( t ) Figure6-17.DualvariablesforExample3obtainedusingcol locationatLGpoints. 0 -2 -4 -6-8 -10 -12 -14 2 4 6810 12 14 161820 Nlog 10 AbsoluteError ( t ) p ( t ) Figure6-18.CostateerrorsforExample3obtainedusingcol locationatLGpoints. 161

PAGE 162

6.3.2SolutionUsingCollocationatFlippedLegendre-Gaus s-RadauPoints Example3isnowsolvedusingvariable-orderippedLGRcoll ocationasdescribed inChapter 2 .Figure 6-19 showsthestateandcontrolapproximationsobtainedusing N =10 collocationpointsperinterval.Itisseenthatvariable-o rdercollocationatthe LGpointsprovideshighlyaccurateapproximationstotheop timalsolutioneventhough index-reductionofthestateinequalitypathconstraintwa snotperformed.Figure 6-20 showsthebasetenlogarithmofthe L 1 -normerrorforthestateandthecontrolusing ( N =2,4,6,...,20) collocationpointsperinterval.Itisseenthatthesolutio nusingLGR collocationislessaccuratethanthesolutionobtainedusi ngcollocationattheLGpoints. Thisdifferenceinaccuracyisexpected,astheLGpointsare knowntohaveahigher accuracyquadraturethantheLGRpoints.Furthermore,itis seenthatthestateand controlreachaccuraciesof O (10 5 ) and O (10 8 ) ,respectively,for N =20 6.3.2.1Previouslyderivedcostateestimate TheaccuracyofthecostateestimateofRef.[ 1 ]usingippedLGRcollocationis nowcomparedagainsttheaccuracyofthecostateestimatede rivedinthisresearch. TheoptimalcostateforExample3usingthemethodofdirecta djoiningcanbefoundby applyingtherst-orderoptimalityconditionsderivedinS ection 2.3.2 .Thecostateand stateconstraintmultipliersaregiven,respectively,as = 8>>>><>>>>: e 1 t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3], = 8>>>><>>>>: 0, t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3]. (6–8) Figure 6-21 showstheresultofthecostateapproximationwhen N =10 collocation pointsperintervalwereused.Itcanbeseenthatalthoughth ecostateisapproximated 162

PAGE 163

0 0 0.2 0.4 0.5 0.6 0.8 1.2 1.5 2.5 1 1 2 3 -0.2 tState y ( t ) y ( t ) (A)State. 0 0 0.2 0.4 0.5 0.6 0.8 1.2 1.4 1.5 1.6 1.8 2.5 1 1 2 2 3 t Control u ( t ) u ( t ) (B)Control. Figure6-19.PrimalsolutionforExample3obtainedusingva riable-ordercollocationat LGRpoints. 163

PAGE 164

-16 -14 12 -8 -6 -4 -2 2 4 6 8 10 12 14 16 1820 -10 Nlog 10 Innity-NormError k y y k 1 k u u k 1 Figure6-20.StateandcontrolerrorsforExample3obtained usingvariable-order collocationatLGRpoints.accurately,thestateconstraintmultiplier, ,isapproximatedverypoorlywherethe costateisdiscontinuous.Figure 6-22 showsthebasetenlogarithmofthe L 1 -norm errorforthecostateandthestateconstraintmutliplierwh enapproximatedusing ( N =2,4,6,...,20) collocationpointsperinterval.Itisseenthatthecostate estimate haslargeerrorsneartheknowndiscontinuitiesintheoptim alcostate.Furthermore,itis seenthatthestateconstraintmultiplierestimatediverge s. 6.3.2.2Costateestimationusingmethodofindirectadjoin ingwithcontinuous multipliers Thecostateestimatederivedusingthemethodofindirectad joiningwithcontinuous multipliersforcollocationattheippedLGRpoints,descr ibedinChapter 5 ,isnow analyzed.TheanalyticoptimalcostateforExample3usingt hemethodofindirect adjoiningwithcontinuousmultiplierscanbefoundbyapply ingtherst-orderoptimality conditionsderivedinSection 2.3.3 .Thecostateandthestateconstraintmultipliersare 164

PAGE 165

Dual Solution 0 0 0.5 1.5 2.5 1 2 3 -0.4 -0.35 -0.3 -0.25 -0.15 -0.1 -0.05 -0.2 t ( t ) ( t ) (A)Costate. -14 -12 -10 -8 -6 Dual Solution 0 0 0.5 1.5 2.5 -4 -2 1 2 3 t ( t ) ( t ) (B)StateConstraintMultiplier. Figure6-21.CostateEstimateasderivedbyRef.[ 1 ]forExample3obtainedusingLGR collocation. 165

PAGE 166

-0.5 0.5 0 1.5 1 2 2 4 6 8 10 12 14 16 1820 Nlog 10 Innity-NormErrork k 1 k k 1 Figure6-22.ErrorsincostateestimatederivedbyRef.[ 1 ]forExample3obtainedusing LGRcollocation.given,respectively,as p = 8>>>><>>>>: 0, t 2 [0,1), 0, t 2 [1,2], 0, t 2 (2,3], = 8>>>><>>>>: e 1 t 2 [0,1), e t t 2 [1,2], 0, t 2 (2,3]. (6–9) Figure 6-23 showsthecostateapproximationfor N =10 obtainedbyusingthe methoddescribedinChapter 5 forcollocationatLGRpoints.Itcanbeseenthatthe estimatepresentedinthisresearchprovidesanaccurateap proximationofthecostate. Figure 6-24 showsthebasetenlogarithmofthe L 1 -normerrorforthecostateandstate constraintmultiplierapproximations.Itcanbeseenthatt heerroronthestateinequality constraintmultiplierdecreasesexponentiallyasthenumb erofcollocationpointsis increased.Furthermore,theerroronthecostateapproxima tionremainsapproximately zero.Therefore,thecostateestimateproducesanaccuracy of O (10 12 ) evenwhen 166

PAGE 167

0.5 1 1.5 2 2.53 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0 0.05 tDualVariablesp ( t ) ( t ) p ( t ) ( t ) Figure6-23.CostateestimateforExample3obtainedusingc ollocationatLGRpoints. 0 -2 -4 -6-8 -10 -12 -14 2 4 6 8 10 1214 161820 Nlog 10 AbsoluteError ( t ) p ( t ) Figure6-24.CostateerrorsforExample3obtainedusingcol locationatLGRpoints. inacuraciesinthestateconstraintmultiplierarepresent duetotheuseoflow-order polynomialapproximationsinthestate. 167

PAGE 168

6.4Example4:Second-OrderStateInequalityPathConstrai ntExample Considerthefollowingsecond-orderstateinequalitycons trainedoptimalcontrol problemfromRef.[ 10 ]: minimize 1 2 Z 1 0 u 2 dt subjectto 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: x = v v = u x (0)=0, x (1)=0, v (0)=1 v (1)= 1, x ( t ) ` Itisknownforthisexamplethattheinequalitypathconstra intisinactivefor `> 1 = 4 isactiveatonlyasinglepointfor 1 = 6 <` 1 = 4 ,andisactivealonganonzeroduration arcfor 0 <` 1 = 6 .Inthecasewhere 0 <` 1 = 6 ,theoptimalstateandcontrolare givenas x ( t )= 8>>>><>>>>: ` h 1 1 t 3 ` 3 i ` ` h 1 1 1 t 3 ` 3 i t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1], v ( t )= 8>>>><>>>>: 1 t 3 ` 2 0, 1 1 t 3 ` 2 t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1], u ( t )= 8>>>><>>>>: 2 3 ` 1 t 3 ` 0, 2 3 ` 1 1 t 3 ` t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1], Avalueof ` =1 = 10 wasusedintheanalysisofthisexample.Thesolutiondomain was dividedintothreeintervalswith N collocationpointsineachinterval.Theboundaries betweentheintervalswerechosentobethetimeinstantswhe rethestateconstraint changesbetweenactiveandinactive,namely, t =3 = 10 and t =7 = 10 .Thesolutionwas 168

PAGE 169

approximatedusing N =5 collocationpoints.AllproblemsweresolvedusingtheNLP solverSNOPTwithdefaultoptimalityandfeasibilitytoler ances.[ 22 ].Theinitialguess usedwastheexactsolution.6.4.1SolutionUsingCollocationatLegendre-GaussPoints Example4wassolvedusingvariable-ordercollocationatth eLegendre-Gauss points,asdescribedinChapter 2 .Figure 6-25 showsthestateandcontrolapproximations obtainedusing N =5 collocationpointsperinterval.Itisseenthatvariable-o rder collocationattheLGpointsprovideshighlyaccurateappro ximationstotheoptimal solutioneventhoughindex-reductionofthestateinequali typathconstraintwasnot performed.Figure 6-26 showsthebase10logarithmofthe L 1 -normerrorforthestate andthecontrolwhenapproximatedusing ( N =2,3,...,10) collocationpointsper interval.Itisinterestingtoseethattheerrorsintheprim alsolutionforthisexampleare largerthantheerrorsobservedfortheprimalsolutionofEx ample3.Thisdifference inaccuracycanbeattributtedtotheincreaseintheorderof thestateinequalitypath constraint.Althoughtheerrorsinthisexamplearelargert hanforExample3,itcanbe seenthatanaccuracyof O (10 6 ) and O (10 5 ) forthestateandcontrol,respectively, canbeobtainedusing N =3 collocationpointsperinterval. 169

PAGE 170

0 0 0.1 0.10.20.3 0.4 0.50.60.70.80.9 1 0.02 0.04 0.06 0.08 0.12 t StateComponent x ( t ) x ( t ) (A) x ( t ) 0 0 0.1 0.2 0.20.3 0.4 0.40.5 0.6 0.60.7 0.8 0.8 0.9 1 1 -0.2 -0.4 -0.6-0.8 -1 t StateComponent v ( t ) v ( t ) (B) v ( t ) 0 00.10.20.30.40.5 0.6 0.70.80.9 -4 -3 -2 1 1 -5-6 -7 -1 t Control u ( t ) u ( t ) (C) u ( t ) Figure6-25.PrimalsolutionforExample4obtainedusingLG collocation. 170

PAGE 171

N 0 -8 -7 -6 -5 -4 -3 -2 -1 2 3 4 56 7 8910 log 10 Innity-NormErrorx ( t ) v ( t ) u ( t ) Figure6-26.StateandcontrolerrorsforExample4usingcol locationatLGpoints. 171

PAGE 172

6.4.1.1Previouslyderivedcostateestimate TheaccuracyofthecostateestimateusingLGcollocationde rivedinRef.[ 1 ]is nowcomparedagainsttheaccuracyofthecostateestimatede rivedinthisresearch. TheoptimalcostateforExample4usingthemethodofdirecta djoiningcanbefoundby applyingtherst-orderoptimalityconditionsderivedinS ection 2.3.2 .Thecostateand stateconstraintmultiplieraregiven,respectively,as x = 8>>>><>>>>: 2 9 ` 2 t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 2 9 ` 2 t 2 [1 3 ` ,1], v = 8>>>><>>>>: 2 3 ` 1 t 3 ` t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 2 3 ` 1 1 t 3 ` t 2 [1 3 ` ,1], = 8>>>><>>>>: 0, t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 0, t 2 [1 3 ` ,1]. (6–10) Figure 6-27 showsthecostateapproximationfor N =5 collocationpointsper interval.Itcanbeseenthatthecostate, ( t ) ,isnotapproximatedcorrectlyatthe intervalboundarieswherethediscontinuitiesoccur.Furt hermore,thestateconstraint multiplier, ,isapproximatedverypoorly.Figure 6-28 showsthebasetenlogarithmof the L 1 -normerrorforthecostateandthestateconstraintmutlipl ierwhenapproximated using ( N =2,3,...,10) collocationpointsperinterval.Itcanbeseenthatlargeer rors aroundthecostatediscontinuitiespreventthecostateest imatefromconvergingtoits optimalsolution. 172

PAGE 173

0 00.10.20.3 0.4 0.50.60.70.8 0.9 1 25 20 15 10 5 -5 -10 -15 -20 -25 t CostateComponent x ( t ) x ( t ) (A)Costate. 0 0 0.1 0.20.30.40.5 0.6 0.70.80.9 6 7 -1 1 1 2 3 4 5 tCostateComponent v ( t ) v ( t ) (B)Costate. 0 00.10.2 0.3 0.40.50.60.70.80.9 1 -50 -100-150-200-250-300-350-400 t ( t ) ( t ) StateConstraintMultiplier (C)StateConstraintMultiplier. Figure6-27.CostateEstimateasderivedbyRef.[ 1 ]forExample4obtainedusing collocationatLGpoints. 173

PAGE 174

1.5 2.5 1 2 2 3 3 4 5 6 7 8 9 10 3.5 0.5 Nlog 10 Innity-NormErrork x x k 1 k v v k 1 k k 1 Figure6-28.ErrorsincostateestimatederivedbyRef.[ 1 ]forExample4obtainedusing LGcollocation. 174

PAGE 175

6.4.1.2CostateEstimationusingmethodofindirectadjoin ingwithcontinuous multipliers Thecostateestimatederivedusingthemethodofindirectad joiningwithcontinuous multipliersforcollocationattheLGpoints,describedinC hapter 5 ,isnowanalyzed.The optimalcostateforExample4usingthemethodofindirectad joiningwithcontinuous multiplierscanbefoundbyapplyingtherst-orderoptimal ityconditionsderivedin Section 2.3.3 .Thecostateandthestateconstraintmultiplieraregiven, respectively,as p x ( t )= 2 9 ` 2 t 2 [0,1], p v ( t )= 8>>>><>>>>: 2 3 ` 1 t 3 ` 0, 2 3 ` 1 1 t 3 ` t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1]. ( t )= 8>>>><>>>>: 4 9 ` 2 2 9 ` 2 0, t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1]. Figure 6-29 showsthedualvariableapproximationsforcollocationatt heLGpoints. ItcanbeseenthatthemappingpresentedinChapter 5 providesanaccurateestimate forthedualvariablesofthecontinuousoptimalcontrolpro blem.Figure 6-30 showsthe basetenlogarithmofthe L 1 -normerrorforthecostateandstateconstraintmultiplier approximationsobtainedusingEqs.( 5–32 )–( 5–38 )andEq.( 5–79 )for N collocation pointsineachofthethreemeshintervals.Itcanbeseenthat theerroronthedual variablesreachanaccuracyof O (10 5 ) forlow-orderpolynomialapproximationsinthe state(thatis,asmallnumberofcollocationnumbers). 175

PAGE 176

-22.5 -22.4 -22.3 -22.2 -22.1 -22 -21.9 -21.8 -21.7 -21.6 -21.5 00.10.2 0.3 0.40.50.60.7 0.8 0.9 1 t p x ( t ) p x ( t ) CostateComponent (A) p x ( t ) 0 0 0.10.20.30.4 0.5 0.60.70.80.9 6 7 -1 1 1 2 3 4 5 t p v ( t ) p v ( t ) CostateComponent (B) p v ( t ) 0 00.1 0.2 0.30.40.50.60.70.80.9 1 -5 -10 -15 -20 -25 -30 -35 -40 -45 t ( t ) ( t ) StateConstraintMultiplier(C) ( t ) Figure6-29.CostateestimateforExample4obtainedusingc ollocationatLGpoints. 176

PAGE 177

0 -6 -5 -4 -3 -2 -1 1 2 2 3 4 56 7 8910 Nlog 10 AbsoluteError ( t ) p x ( t ) p v ( t ) Figure6-30.CostateerrorsforExample4obtainedusingcol locationatLGpoints. 177

PAGE 178

6.4.2SolutionUsingCollocationatFlippedLegendre-Gaus s-RadauPoints Next,Example4wassolvedusingvariable-ordercollocatio nattheipped Legendre-Gauss-Radaupoints,asdescribedinChapter 2 .Figure 6-31 showsthe stateandcontrolapproximationsobtainedusing N =5 collocationpointsperinterval.It isseenthatvariable-ordercollocationattheLGpointspro videsaccurateapproximations totheoptimalsolutioneventhoughindex-reductionofthes tateinequalitypathconstraint wasnotperformed.Figure 6-32 showsthebase10logarithmofthe L 1 -normerrorfor thestateandthecontrolwhenapproximatedusing ( N =2,3,...,10) collocationpoints perinterval.Itisseenthatthestatereachesanaccuracyof O (10 5 ) for N =3 ,whereas thecontrolonlyreachesanaccuracyof O (10 2 ) for N =10 178

PAGE 179

0.01 0.03 0.07 0.09 0 0 0.1 0.10.20.3 0.4 0.50.60.70.80.9 0.05 1 0.02 0.04 0.06 0.08 t StateComponent x ( t ) x ( t ) (A) x ( t ) 0 0 0.1 0.2 0.20.3 0.4 0.40.5 0.6 0.60.7 0.8 0.8 0.9 1 1 -0.2 -0.4 -0.6-0.8 -1 t StateComponent v ( t ) v ( t ) (B) v ( t ) 0 00.10.20.30.40.5 0.6 0.70.80.9 -4 -3 -2 1 -5-6 -7 -1 t Control u ( t ) u ( t ) (C) u ( t ) Figure6-31.PrimalsolutionforExample4obtainedusingLG Rcollocation. 179

PAGE 180

N 0 -5 2 3 4 5 56 7 8910 -10 -20 -25 -30 -35 -15 log 10 Innity-NormErrorx ( t ) v ( t ) u ( t ) Figure6-32.StateandcontrolerrorsforExample4usingcol locationatLGRpoints. 180

PAGE 181

6.4.2.1Previouslyderivedcostateestimate heaccuracyofthecostateestimateusingLGRcollocationde rivedinRef.[ 1 ]isnow comparedagainsttheaccuracyofthecostateestimatederiv edinthisresearch.The optimalcostateforExample4usingthemethodofdirectadjo iningcanbefoundby applyingtherst-orderoptimalityconditionsderivedinS ection 2.3.2 .Thecostateand stateconstraintmultipliersaregiven,respectively,as x = 8>>>><>>>>: 2 9 ` 2 t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 2 9 ` 2 t 2 [1 3 ` ,1], v = 8>>>><>>>>: 2 3 ` 1 t 3 ` t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 2 3 ` 1 1 t 3 ` t 2 [1 3 ` ,1], = 8>>>><>>>>: 0, t 2 [0,3 ` ], 0, t 2 [3 ` ,1 3 ` ], 0, t 2 [1 3 ` ,1]. (6–11) Figure 6-33 showsthecostateapproximationfor N =5 collocationpointsper interval..Itcanbeseenthatalthoughthecostateisapprox imatedaccurately,thestate constraintmultiplier, ,isapproximatedverypoorly.Figure 6-34 showsthebaseten logarithmofthe L 1 -normerrorforthecostateandthestateconstraintmutlipl ierwhen approximatedusing ( N =2,3,...,10) collocationpointsperinterval.Itcanbeseen thatlargeerrorsaroundthecostatediscontinuitiespreve ntthecostateestimatefrom convergingtoitsoptimalsolution. 181

PAGE 182

0 00.10.20.3 0.4 0.50.60.70.8 0.9 1 25 20 15 10 5 -5 -10 -15 -20 -25 t CostateComponent x ( t ) x ( t ) (A)Costate. 0 0 0.1 0.20.30.40.5 0.6 0.70.80.9 6 7 -1 1 1 2 3 4 5 tCostateComponent v ( t ) v ( t ) (B)Costate. -2000 -1500 -1000 -500 500 0 00.10.2 0.3 0.40.50.60.70.80.9 1 t ( t ) ( t ) StateConstraintMultiplier (C)StateConstraintMultiplier. Figure6-33.CostateEstimateasderivedbyRef.[ 1 ]forExample4obtainedusing collocationatLGRpoints. 182

PAGE 183

0 -10 12 -8 -6 -4 -2 2 2 3 4 5 6 7 8 9 10 Nlog 10 Innity-NormErrork x x k 1 k v v k 1 k k 1 Figure6-34.ErrorsincostateestimatederivedbyRef.[ 1 ]forExample4obtainedusing LGRcollocation. 183

PAGE 184

6.4.2.2CostateEstimationusingmethodofindirectadjoin ingwithcontinuous multipliers Thecostateestimatederivedusingthemethodofindirectad joiningwithcontinuous multipliersforcollocationattheippedLGRpoints,descr ibedinChapter 5 ,isnow analyzed.TheanalyticoptimalcostateforExample4usingt hemethodofindirect adjoiningwithcontinuousmultiplierscanbefoundbyapply ingtherst-orderoptimality conditionsderivedinSection 2.3.3 .Thecostateandthestateconstraintmultiplierare given,respectively,as p x ( t )= 2 9 ` 2 t 2 [0,1], p v ( t )= 8>>>><>>>>: 2 3 ` 1 t 3 ` 0, 2 3 ` 1 1 t 3 ` t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1]. ( t )= 8>>>><>>>>: 4 9 ` 2 2 9 ` 2 0, t 2 [0,3 ` ], t 2 [3 ` ,1 3 ` ], t 2 [1 3 ` ,1]. Figure 6-35 showsthedualvariableapproximationsobtainedbyusingth emethod describedinthispaperforcollocationatLGRpoints.Itcan beseenthatthemapping presentedinChapter 5 providesanaccurateestimateforthedualvariablesofthe continuousoptimalcontrolproblem.Figure 6-36 showsthebase10logarithmofthe L 1 -normerrorforthecostateandstateconstraintmultiplier approximationsfor N collocationpointsineachofthethreemeshintervals.Itca nbeseenthattheerror ontheestimateofthestateinequalityconstraintmultipli erandthecostatereachan accuracyof O (10 5 ) for N largerthantwo. 184

PAGE 185

-22.5 -22.4 -22.3 -22.2 -22.1 -22 -21.9 -21.8 -21.7 -21.6 -21.5 00.10.2 0.3 0.40.50.60.7 0.8 0.9 1 t p x ( t ) p x ( t ) CostateComponent (A) p x ( t ) 0 0 0.10.20.30.4 0.5 0.60.70.80.9 6 7 -1 1 1 2 3 4 5 t p v ( t ) p v ( t ) CostateComponent (B) p v ( t ) 0 00.1 0.2 0.30.40.50.60.70.80.9 1 -5 -10 -15 -20 -25 -30 -35 -40 -45 t ( t ) ( t ) StateConstraintMultiplier(C) ( t ) Figure6-35.CostateestimateforExample4obtainedusingc ollocationatLGRpoints. 185

PAGE 186

0 -6 -5 -4 -3 -2 -1 1 2 2 3 4 56 7 8910 Nlog 10 AbsoluteError ( t ) p x ( t ) p v ( t ) Figure6-36.CostateerrorsforExample4obtainedusingcol locationatLGRpoints. 186

PAGE 187

CHAPTER7 CONCLUSIONS Solvinganoptimalcontrolproblemisnoteasy.Formostengi neeringapplications, itisimpossibletoderiveananalyticsolutiontoanoptimal controlproblemusingthe rst-orderoptimalityconditionsderivedfromthecalculu sofvariations.Thus,numerical methodsmustbeusedtoapproximatethesolutiontotheconti nuous-timeproblem. Applyingtherst-orderoptimalityconditionsoftheconti nuous-timeoptimalcontrol problemresultinaHamiltonianboundary-valueproblemwhi chmustbesolved. NumericalmethodsthataproximateasolutiontotheHamilto nianboundary-value problemstemmingfromtherst-orderoptimalitycondition softheoptimalcontrol problemarecalledindirectmethods.Numericalmethodstha temploytheindirect methodtendtoresultinahighlyaccuratesolutionbecauset herst-orderoptimality conditionsoftheoptimalcontrolproblemaresatised.Con vergenceusingindirect methods,however,canbeveryhardtoachieveduetotheunsta blenatureofthe Hamiltonianboundary-valueproblem.Thus,intuitiveinit ialguessesarerequiredto achieveconvergenceusingindirectmethods. Numericalmethodsforsolvingoptimalcontrolproblemstha tdonotformulatethe rst-orderoptimalityconditionsofthecontinous-timepr oblemarecalleddirectmethods. Directmethodsconverttheinnite-dimensionalcontinuou scontrolproblemintoa nite-dimensionaldiscretenonlinearprogrammingproble m(NLP).TheresultingNLP canthenbesolvedbywell-developedNLPalgorithms.Direct methodsareattractive becausetherst-orderoptimalityconditionsneednotbede rived.Furthermore,because theHamiltonianboundary-valueproblemisnotformulated, convergenceusingdirect methodsisusuallyeasiertoobtain.Inthisresearchadirec torthogonalcollocation methodusingcollocationattheLegendre-GaussandLegendr e-Gauss-Radau pointswasanalyzed.Inparticular,anestimateoftheconti nuous-timecostateofthe 187

PAGE 188

continuous-timeoptimalcontrolproblemwasderivedfromt heKKTmultipliersofthe nonlinearprogrammingproblemofthediscreteproblem. Costateestimationisanimportantstepinthenumericalsol utionofoptimalcontrol problems.Mappingthedualvariablesofthenumericalsolut iontothecostateofthe continuous-timeproblemnotonlyallowsforavericationo fthedualsolution,but alsoallowstherst-orderoptimalityconditionsofthenon linearprogrammingproblem (NLP)totakeaformthatisequivalenttotherst-orderopti malityconditionsofthe continuous-timeproblem.Thus,havinganaccuratecostate estimateshowsthatthe KKTconditionssatisedbytheNLPareadiscreteformofthe rst-orderoptimality conditionsofthecontinuousproblemgivenbythecalculuso fvariations,andwill convergetoanoptimalsolutionofthecontinuous-timeprob lemifdiscretizedcorrectly. Costateestimatesfordirectmethodsusingorthogonalcoll ocationattheLegendre-Gauss andLegendre-Gauss-Radaupointshavepreviouslybeenderi vedforoptimalcontrol problemswithnoactivestateinequalitypathconstraintsa ndwhenthedynamic constraintsareformulatedintheirdifferentialform.Int hisresearchagapofcostate estimationtheorywasclosedbyderivingamappingfortheco stateestimateforthecase whenthedynamicconstraintsareexpressedinintegralform andinthepresenceof stateinequalitypathconstraints. Intherstpartofthisresearchacostateestimatewasdevel opedforproblemstated withintegralconstraints.Whilethedifferentialandinte gralformsoftheLGandLGR methodsaremathematicallyequivalentwithregardtothepr imalvariables(thatis,the stateandcontrol),thetwoformulationsproducecompletel ydifferentdualvariables.In particular,therelationshipbetweentheLagrangemultipl iersofthecollocationconditions ofthedynamicconstraintsandthecostateoftheoptimalcon trolproblemhasbeen welldocumented.Ontheotherhand,thecorrespondingrelat ionshipbetweenthe Lagrangemultipliersassociatedwiththeintegralformsof LGandLGRcollocationand thecostateoftheoptimalcontrolproblemhasnotbeenestab lished.Whenemploying 188

PAGE 189

theintegralformsofLGandLGRcollocation,however,itmay beofinteresttoeither verifyoptimalityorperformsensitivityanalysisinamann erconsistentwiththatwhich wouldbeperformedwhenusingvariationalmethods.Insuchc asesitisusefulto obtainacostateestimatewhenusingtheintegralformsofth eLGandLGRmethods. Thus,inthisresearch,acostateestimateforcollocationa tLegendre-Gaussand Legendre-Gauss-Radaupointswasderivedforthecasewhent hedynamicconstraints oftheoptimalcontrolproblemareformulatedinintegralfo rm.Itwasdemonstrated thatthecostatemappingderivedforcollocationattheLGan dLGRpointsleadstoa setoftransformedoptimalityconditionsoftheNLPwhichwe reshowntobeadiscrete representationofthenecessaryconditionsforoptimality ofthecontinuous-timeproblem. Finally,arelationshipbetweentheintegralandthediffer entialformsofthecostate estimatewasgivenanditwasshownthatthetwosetsofoptima lityconditionsare equivalent. Thesecondpartofthisresearchfocusedonproblemswithact ivestateinequality pathconstraints.Althoughpreviousresearchhassuccessf ullyderivedahigh-accuracy estimateofthecostatefromtheKKTmultipliersoftheNLPfo rthecaseofaproblem withnoactivestateinequalitypathconstraints,Ref.[ 1 ]subsequentlyshowedthatinthe casewhenthecostateisdiscontinuous(asisthecaseinthep resenceofactivestate inequalitypathconstraints),thiscostateestimateleads toasetofrst-orderoptimality conditionsoftheNLPthatarenotequivalenttothediscrete formofthevariational optimalityconditions.Thislackofequivalenceleadstoan inaccurateapproximation ofthecostate.Therefore,inthisresearchcostateestimat esforcollocationatLGand LGRpointswerederivedforproblemswithactivestateinequ alitypathconstraints. Thederivedcostateestimatewasshowntoleadtoatransform edadjointsystemof theNLPwhichisadiscreteapproximationofthenecessaryco nditionsforoptimality ofthecontinuous-timeoptimalcontrolproblem.Thisequiv alencewasnotexistentwith priorcostateestimatesusingLGandLGRcollocation.Theco stateestimatesderived 189

PAGE 190

inthisdissertationwereimplementedinfourproblemstoas sesstheiraccuracy.Itwas shownthateachdiscretecostateestimateledtoanaccurate approximationofthe continuous-timecostate. 190

PAGE 191

REFERENCES [1]Darby,C.L.,Garg,D.,andRao,A.V.,“CostateEstimatio nUsingMultiple-Interval PseudospectralMethods,” JournalofSpacecraftandRockets ,Vol.48,No.5, September–October2011,pp.856–866. [2]Betts,J., PracticalMethodsforOptimalControlandEstimationUsing Nonlinear Programming ,SIAMPress,Philadelphia,2nded.,2009. [3]Unger,J.,Kroner,A.,andMarquardt,W.,“Structuralan alysisof differential-algebraicequationsystems-theoryandappl ications,” Computers andChemicalEngineering ,Vol.19,No.8,1995,pp.867–882. [4]Feehery,W.F.andBarton,P.I.,“Dynamicoptimizationw ithstatevariablepath constraints,” ComputersandChemicalEngineering ,Vol.22,No.9,1998,pp.1241 –1256. [5]Maurer,H.andPesch,H.J.,“Directoptimizationmethod sforsolvingacomplex state-constrainedoptimalcontrolprobleminmicroeconom ics,” AppliedMathematics andComputation ,Vol.204,No.2,2008,pp.568–579. [6]B ¨ oskens,Christof,M.H.,“SQP-methodsforsolvingoptimalc ontrolproblemswith controlandstateconstraints:adjointvariables,sensiti vityanalysisandreal-time control,” J.Comput.Appl.Math. ,Vol.120,No.1-2,Aug.2000,pp.85–108. [7]Bryson,A.E.andHo,Y.-C., AppliedOptimalControl ,HemispherePublishing,New York,1975. [8]Speyer,J.L.,“NecessaryConditionsforOptimalityFor PathsLyingonaCorner,” ManagementScience ,Vol.19,No.11,July1973. [9]Kirk,D.E., OptimalControlTheory:AnIntroduction ,DoverPublications,Mineola, NewYork,2004. [10]Bryson,A.E.,Denham,W.F.,andDreyfus,S.E.,“Optima lProgrammingProblems withInequalityConstraintsI:NecessaryConditionsforEx tremalSolutions,” AIAA Journal ,Vol.1,No.11,1962,pp.2544–2550. [11]Dreyfus,S.E., VariationalProblemswithStateVariableInequalityConst raints Ph.D.thesis,HarvardUniversity,Cambridge,Massachusse ts,1962. [12]Jacobson,D.H.,Lele,M.M.,andSpeyer,J.L.,“NewNece ssaryConditionsof OptimalityforControlProblemswithState-VariableInequ alityConstraints,”Tech. rep.,DivisionofEngineeringandAppliedPhysics-Harvard University,Cambridge, Massachussets,1969. [13]Speyer,J.L.andBryson,A.E.,“OptimalProgrammingPr oblemswithaBounded StateSpace,” AIAAJournal ,Vol.6,No.8,August1968,pp.1488–1491. 191

PAGE 192

[14]Denham,W.F.andBryson,A.E.,“OptimalProgrammingPr oblemswithInequality ConstraintsII:SolutionbySteepest-Ascent,” AIAAJournal ,Vol.2,No.1,January 1964,pp.25–34. [15]Hartl,R.F.,Sethi,S.P.,andVickson,R.G.,“ASurveyo ftheMaximumPrinciples forOptimalControlProblemswithStateConstraints,” SIAMReview ,Vol.37,No.2, June1995,pp.181–218. [16]Gollan,B.,“OnOptimalControlProblemswithStateCon straints,” Journalof OptimizationTheoryandApplications ,Vol.32,1980,pp.75–80. [17]Russak,I.B.,“OnProblemswithBoundedStateVariable ,” JournalofOptimization TheoryandApplications ,Vol.5,1970,pp.114–157. [18]Russak,I.B.,“OnGeneralProblemswithBoundedStateV ariable,” Journalof OptimizationTheoryandApplications ,Vol.6,1970,pp.424–452. [19]Vinter,R., OptimalControl(Systems&Control:FoundationsandApplic ations) Birkh ¨ auser,Boston,2000. [20]Russel,R.andShampine,L.,“ACollocationMethodforB oundaryValueProblems,” NumericalMathematics ,Vol.19,1972,pp.1–28. [21]Rao,A.V.andMease,K.D.,“DichotomicBasisApproacht osolving Hyper-SensitiveOptimalControlProblems,” Automatica ,Vol.35,No.4,April 1999,pp.633–642. [22]Gill,P.E.,Murray,W.,andSaunders,M.A.,“SNOPT:AnS QPAlgorithmfor Large-ScaleConstrainedOptimization,” SIAMReview ,Vol.47,No.1,January 2005,pp.99–131. [23]Biegler,L.T.andZavala,V.M.,“Large-ScaleNonlinea rProgrammingUsingIPOPT: AnIntegratingFrameworkforEnterprise-WideOptimizatio n,” Computersand ChemicalEngineering ,Vol.33,No.3,March2008,pp.575–582. [24]W ¨ achter,A., AnInteriorPointAlgorithmforLarge-ScaleNonlinearOpti mization withApplicationsinProcessEngineering ,Ph.D.thesis,DepartmentofChemical Engineering,Carnegie-MellonUniversity,Pittsburgh,PA ,2002. [25]W ¨ achter,A.andBiegler,L.T.,“OntheImplementationofanIn terior-PointFilter Line-SearchAlgorithmforLarge-ScaleNonlinearProgramm ing,” Mathematical Programming ,Vol.106,No.1,2006,pp.25–57. [26]Fahroo,F.andRoss,I.M.,“CostateEstimationbyaLege ndrePseudospectral Method,” JournalofGuidance,Control,andDynamics ,Vol.24,No.2,March–April 2001,pp.270–277. [27]Benson,D.A.,Huntington,G.T.,Thorvaldsen,T.P.,an dRao,A.V.,“Direct TrajectoryOptimizationandCostateEstimationviaanOrth ogonalCollocation 192

PAGE 193

Method,” JournalofGuidance,Control,andDynamics ,Vol.29,No.6, November-December2006,pp.1435–1440. [28]Bock,H.G.andPlitt,K.J.,“AMultipleShootingAlgori thmforDirectSolutionof OptimalControlProblems,” IFAC9thWorldCongress ,Budapest,Hungary,1984. [29]Williamson,W.E.,“UseofPolynomialApproximationst oCalculateSuboptimal Controls,” AIAAJournal ,Vol.9,No.11,1971,pp.2271–2273. [30]Kraft,D., OnConvertingOptimalControlProblemsintoNonlinearProg ramming Codes ,Vol.F15of NATOASISeries, in ComputationalMathematicalProgramming ,Springer-Verlag,Berlin,1985,pp.261–280. [31]Von-Stryk,O.andBulirsch,R.,“DirectandIndirectMe thodsforTrajectory Optimization,” AnnalsofOperationsResearch ,Vol.37,1992,pp.357–373. [32]Hargraves,C.R.andParis,S.W.,“DirectTrajectoryOp timizationUsingNonlinear ProgrammingTechniques,” JournalofGuidance,Control,andDynamics ,Vol.10, No.4,July–August1987,pp.338–342. [33]Enright,P.J.andConway,B.A.,“DiscreteApproximati onstoOptimalTrajectories UsingDirectTranscriptionandNonlinearProgramming,” JournalofGuidance, Control,andDynamics ,Vol.19,No.4,July–August1996,pp.994–1002. [34]Benson,D.A., AGaussPseudospectralTranscriptionforOptimalControl ,Ph.D. thesis,DepartmentofAeronauticsandAstronautics,Massa chusettsInstituteof Technology,Cambridge,Massachusetts,2004. [35]Elnagar,G.,Kazemi,M.,andRazzaghi,M.,“ThePseudos pectralLegendreMethod forDiscretizingOptimalControlProblems,” IEEETransactionsonAutomatic Control ,Vol.40,No.10,1995,pp.1793–1796. [36]Garg,D.,Patterson,M.A.,Darby,C.L.,Francolin,C., Huntington,G.T.,Hager, W.W.,andRao,A.V.,“DirectTrajectoryOptimizationandCo stateEstimation ofFinite-HorizonandInnite-HorizonOptimalControlPro blemsusingaRadau PseudospectralMethod,” ComputationalOptimizationandApplications ,Vol.49, No.2,June2011,pp.335–358. [37]Biegler,L.T.,Ghattas,O.,Heinkenschloss,M.,andva nBloemenWaanders, B.,editors, Large-ScalePDEConstrainedOptimization ,LectureNotesin ComputationalScienceandEngineering,Vol.30,SpringerVerlag,Berlin,2003. [38]Betts,J.T.,Campbell,S.L.,andEngelsone,A.,“Direc tTranscriptionSolutionof OptimalControlProblemswithHigherOrderStateConstrain ts:TheoryvsPractice,” OptimizationandEngineering ,Vol.8,No.1,2007,pp.1–19. [39]Betts,J.T.andHuffman,W.P.,“SparseOptimalControl Software–SOCS,”Tech. Rep.MEA-LR-085,BoeingInformationandSupportServices, Seattle,Washington, July1997. 193

PAGE 194

[40]Ross,I.M.andFahroo,F., User'sManualforDIDO2001 :AMATLABApplication forSolvingOptimalControlProblems ,DepartmentofAeronauticsandAstronautics, NavalPostgraduateSchool,TechnicalReportAAS-01-03,Mo nterey,California, 2001. [41]Von-Stryk,O., User'sGuideforDIRCOLVersion2.1:ADirectCollocationMe thod fortheNumericalSolutionofOptimalControlProblems ,TechnischeUniversitat Darmstadt,Darmstadt,Germany,1999. [42]Rao,A.V.,Benson,D.A.,Darby,C.L.,Francolin,C.,Pa tterson,M.A.,Sanders, I.,andHuntington,G.T.,“Algorithm902:GPOPS,AMatlabSo ftwareforSolving Multiple-PhaseOptimalControlProblemsUsingtheGaussPs eudospectral Method,” ACMTransactionsonMathematicalSoftware ,Vol.37,No.2,April–June 2010,pp.22:1–22:39. [43]J.T.Betts,“SurveyofNumericalMethodsforTrajector yOptimization,” Journalof Guidance,Control,andDynamics ,Vol.21,No.2,March–April1998,pp.193–207. [44]Hager,W.W.,“Runge-KuttaMethodsinOptimalControla ndtheTransformed AdjointSystem,” NumerischeMathematik ,Vol.87,2000,pp.247–282. [45]Dontchev,A.L.,Hager,W.W.,andVeliov,V.M.,“Second -OrderRunge-Kutta ApproximationsInConstrainedOptimalControl,” SIAMJournalonNumerical Analysis ,Vol.38,2000,pp.202–226. [46]Betts,J.T.,“SparseJacobianUpdatesintheCollocati onMethodforOptimal ControlProblems,” JournalofGuidance,Control,andDynamics ,Vol.13,No.3, May–June1990,pp.409–415. [47]Dontchev,A.L.,Hager,W.W.,andMalanowski,K.,“Erro rBoundsfortheEuler ApproximationandControlConstrainedOptimalControlPro blem,” Numerical FunctionalAnalysisandApplications ,Vol.21,2000,pp.653–682. [48]Dontchev,A.,Hager,W.,andVeliov,V.,“UniformConve rgenceandMesh independenceofNewton'smethodforDiscretizedVariation alProblems,” SIAM JournalonControlandOptimization ,Vol.39,2000,pp.961–980. [49]Kameswaran,S.andBiegler,L.T.,“ConvergenceRatesf orDirectTranscription ofOptimalControlProblemsUsingCollocationatRadauPoin ts,” Computational OptimizationandApplications ,Vol.41,No.1,2008,pp.81–126. [50]Darby,C.L.,Hager,W.W.,andRao,A.V.,“Anhp-Adaptiv ePseudospectralMethod forSolvingOptimalControlProblems,” OptimalControlApplicationsandMethods Vol.32,No.4,July–August2011,pp.476–502. [51]Darby,C.L.,Hager,W.W.,andRao,A.V.,“DirectTrajec toryOptimizationUsinga VariableLow-OrderAdaptivePseudospectralMethod,” JournalofSpacecraftand Rockets ,Vol.48,No.3,May–June2011,pp.433–445. 194

PAGE 195

[52]Vlassenbroeck,J.andDooren,R.V.,“AChebyshevTechn iqueforSolving NonlinearOptimalControlProblems,” IEEETransactionsonAutomaticControl Vol.33,No.4,1988,pp.333–340. [53]Vlassenbroeck,J.,“AChebyshevPolynomialMethodfor OptimalControlwithState Constraints,” Automatica ,Vol.24,No.4,1988,pp.499–506. [54]Elnagar,G.andRazzaghi,M.,“ACollocation-TypeMeth odforLinearQuadratic OptimalControlProblems,” OptimalControlApplicationsandMethods ,Vol.18, No.3,1998,pp.227–235. [55]Elnagar,G.andKazemi,M.,“PseudospectralChebyshev OptimalControlof ConstrainedNonlinearDynamicalSystems,” ComputationalOptimizationand Applications ,Vol.11,No.2,1998,pp.195–217. [56]Elnagar,G.N.andRazzaghi,M.A.,“PseudospectralLeg endre-BasedOptimal ComputationofNonlinearConstrainedVariationalProblem s,” JournalofComputationalandAppliedMathematics ,Vol.88,No.2,March1998,pp.363–375. [57]Williams,P.,“JacobiPseudospectralMethodforSolvi ngOptimalControlProblems,” JournalofGuidance,Control,andDynamics ,Vol.27,No.2,March–April2004, pp.293–297. [58]Williams,P.,“ApplicationofPseudospectralmethods forRecedingHorizonControl,” JournalofGuidance,Control,andDynamics ,Vol.27,No.2,2004,pp.310–314. [59]Williams,P.,“Hermite-Legendre-Gauss-LobattoDire ctTranscriptionMethodsin TrajectoryOptimization,” AmericalAstronauticalSociety ,SpaceightMechanics Meeting,August2005. [60]Huntington,G.T., AdvancementandAnalysisofaGaussPseudospectralTranscriptionforOptimalControl ,Ph.D.thesis,MassachusettsInstituteofTechnology, Cambridge,Massachusetts,2007. [61]Huntington,G.T.,Benson,D.A.,andRao,A.V.,“Optima lCongurationof TetrahedralSpacecraftFormations,” TheJournaloftheAstronauticalSciences Vol.55,No.2,April-June2007,pp.141–169. [62]Huntington,G.T.andRao,A.V.,“OptimalRecongurati onofSpacecraft FormationsUsingtheGaussPseudospectralMethod,” JournalofGuidance, Control,andDynamics ,Vol.31,No.3,May-June2008,pp.689–698. [63]Fahroo,F.andRoss,I.M.,“DirectTrajectoryOptimiza tionbyaChebyshev PseudospectralMethod,” JournalofGuidance,Control,andDynamics ,Vol.25, No.1,2002,pp.160–166. [64]Garg,D.,Patterson,M.A.,Hager,W.W.,Rao,A.V.,Bens on,D.A.,andHuntington, G.T.,“AUniedFrameworkfortheNumericalSolutionofOpti malControlProblems 195

PAGE 196

UsingPseudospectralMethods,” Automatica ,Vol.46,No.11,December2010, pp.1843–1851. [65]Garg,D.,Hager,W.W.,andRao,A.V.,“PseudospectralM ethodsforSolving Innite-HorizonOptimalControlProblems,” Automatica ,Vol.47,No.4,April2011, pp.829–837. [66]Dontchev,A.L.andHager,W.W.,“TheEulerApproximati oninStateConstrained OptimalControl,” MathematicsofComputation ,Vol.70,2001,pp.173–203. [67]Maurer,H.andGillessen,W.,“Applicationofmultiple shootingtothenumerical solutionofoptimalcontrolproblemswithboundedstatevar iables,” Computing Vol.15,1975,pp.105–126. [68]Garg,D., AdvancesInGlobalPseudospectralMethodsForOptimalCont rol ,Ph.D. thesis,UniversityofFlorida,Gainesville,Florida,2011 [69]Patterson,M.A.andRao,A.V.,“GPOPS-II:AMATLABSoft wareforSolving Multiple-PhaseOptimalControlProblemsUsinghp-Adaptiv eGaussianQuadrature CollocationMethodsandSparseNonlinearProgramming,” ACMTransactionson MathematicalSoftware ,February2013,Submitted. [70]Patterson,M.A.,Hager,W.W.,andRao,A.V.,“A ph -CollocationSchemefor OptimalControl,” Automatica ,January2013,Submitted. [71]Vlases,W.G.,Paris,S.W.,Lajoie,R.M.,Martens,M.J. ,andHargraves,C.R., “OptimalTrajectoriesbyImplicitSimulation,”Tech.Rep. WRDC-TR-90-3056, BoeingAerospaceandElectronics,Wright-PattersonAirFo rceBase,Ohio,1990. [72]Falugi,P.,Kerrigan,E.,andvanWyk,E., ImperialCollegeLondonOptimalControl SoftwareUserGuide(ICLOCS) ,DepartmentofElectricalEngineering,Imperial CollegeLondon,London,UK,May2010. [73]Houska,B.,Ferreau,H.J.,andDiehl,M.,“ACADOToolki t–AnOpen-Source FrameworkforAutomaticControlandDynamicOptimization, ” OptimalControl ApplicationsandMethods ,Vol.32,No.3,2011,pp.298–312. [74]Darby,C.L., hp-PseudospectralMethodForSolvingContinuous-TimeNon linear OptimalControlProblems ,Ph.D.thesis,UniversityofFlorida,Gainesville,Florid a, 2011. 196

PAGE 197

BIOGRAPHICALSKETCH CamilaFrancolinwasborninRiodeJaneiro,Brazil.Sherece ivedherdualBachelor ofSciencedegreesinaerospaceandmechanicalengineering inDecember2007 fromtheUniversityofFlorida.ShethenreceivedherMaster ofSciencedegreein aerospaceengineeringinMay2010,andherDoctorofPhiloso phyinaerospace engineeringinAugust2013fromtheUniversityofFlorida.H erresearchinterestsinclude numericalapproximationstodifferentialequations,opti malcontroltheory,andnumerical approximationstothesolutionofoptimalcontrolproblems 197