Camila C. Francolin, D. A. Benson, William W. Hager, and Anil V. Rao

17 Pages·2014·0.53 MB·English

OPTIMALCONTROLAPPLICATIONSANDMETHODS
Optim.ControlAppl.Meth.(2014)
PublishedonlineinWileyOnlineLibrary(wileyonlinelibrary.com).DOI:10.1002/oca.2112
Costate approximation in optimal control using integral Gaussian
quadrature orthogonal collocation methods(cid:2)
CamilaC.Françolin1,DavidA.Benson2,WilliamW.Hager3 andAnilV.Rao1,*,†
1DepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida,Gainesville,FL32611,USA
2TheCharlesStarkDraperLaboratory,Inc.,Cambridge,MA02139,USA
3DepartmentofMathematics,UniversityofFlorida,Gainesville,FL32611,USA
SUMMARY
Twomethodsarepresentedforapproximatingthecostateofoptimalcontrolproblemsinintegralformusing
orthogonalcollocationatLegendre–Gauss(LG)andLegendre–Gauss–Radau(LGR)points.Itisshownthat
thederivativeofthecostateofthecontinuous-timeoptimalcontrolproblemisequaltothenegativeofthe
costate of the integral form of the continuous-time optimal control problem. Using this continuous-time
relationshipbetweenthedifferentialandintegralcostate,itisshownthatthediscreteapproximationsofthe
differentialcostateusingLGandLGRcollocationarerelatedtothecorrespondingdiscreteapproximations
of the integral costate via integration matrices. The approach developed in this paper provides a way to
approximatethecostateoftheoriginaloptimalcontrolproblemusingtheLagrangemultipliersoftheintegral
formoftheLGandLGRcollocationmethods.Themethodsaredemonstratedontwoexampleswhereitis
shownthatboththedifferentialandintegralcostateconvergeexponentiallyasafunctionofthenumberof
LGorLGRpoints.Copyright©2014JohnWiley&Sons,Ltd.
Received31October2013; Revised19January2014; Accepted24January2014
KEYWORDS: optimalcontrol;Gaussianquadrature;orthogonalcollocation;directtranscription
1. INTRODUCTION
Over the past two decades, direct collocation methods have become popular in the numerical
solution of nonlinear optimal control problems. In a direct collocation method, the state and
control are discretized at a set of appropriately chosen points in the time interval of interest.
The continuous-time optimal control problem is then transcribed to a finite-dimensional nonlinear
programming problem (NLP), and the NLP is solved using well known software [1, 2]. Recently,
agreatdealofresearchhasbeendoneontheclassofGaussianquadratureorthogonalcollocation
methods [3–25]. In a Gaussian quadrature orthogonal collocation method, the state is approxi-
matedusingabasisofeitherLagrangeorChebyshevpolynomials,andthedynamicsarecollocated
at points associated with a Gaussian quadrature. The most common Gaussian quadrature collo-
cation points are Legendre–Gauss (LG) [3, 4, 6–9, 11], Legendre–Gauss–Radau (LGR) [9–14],
and Legendre–Gauss–Lobatto [15–25]. All three types of Gaussian quadrature points are defined
on the domain Œ(cid:2)1;1(cid:3) but differ in that the LG points include neither of the endpoints, the LGR
points include one of the endpoints, and the LGL points include both of the endpoints. The use
of global polynomials together with Gaussian quadrature collocation points is known to provide
accurateapproximationsthatconvergeexponentiallyfastforproblemswhosesolutionsaresmooth
*Correspondence to: Anil V. Rao, Department of Mechanical and Aerospace Engineering, University of Florida,
Gainesville,FL32611,USA.
†E-mail:anilvrao@ufl.edu.
‡Thisarticlewaspublishedonlineon27February2014.Errorsweresubsequentlyidentified.Thisnoticeisincludedin
theonlineandprintversionstoindicatethatbothhavebeencorrectedon22May2014.
Copyright©2014JohnWiley&Sons,Ltd.
C.C.FRANÇOLINETAL.
[6, 9–11, 14]. An advantage of these methods is that by computing the solution of the control
problem accurately at a small number of carefully chosen points, one obtains an accurate global
approximation. Because the problem solution is approximated in a small dimensional space, the
numericalalgorithmscanbeveryefficient.
The costate variable in an optimal control problem is related to the sensitivity of the objective
function to perturbations in the system dynamics. The costate is essentially the derivative of the
objective function with respect to a perturbation in the system dynamics. For example, see [26]
and the references therein. Previous research on costate approximation using Gaussian quadrature
collocation has focused on the use of the differential form of the collocation methods. However,
recentresearchstronglyindicatesthattheremaybecomputationaladvantagestousingtheintegral
formofLGandLGRcollocationoverthedifferentialform.Infact,themostcurrentimplementation
ofLGRcollocationistheMATLABoptimalcontrolsoftwareGPOPS(cid:2)II [27],whichusesthe
integralformofLGRcollocationbydefaultbecauseithasbeenfoundthroughavarietyofexamples
thattheintegralformprovidesmoreconsistentresults.TheimplicitintegralformsofLGandLGR
collocation are consistent with the implementations used by established optimal control software
packagessuchasSOCS[28], DIRCOL[29],OTIS[30],ICLOCS[31],andACADO[32].
It is important to note that while the differential and integral forms of LG and LGR collocation
produce equivalent primal solutions (i.e., state and control), these two formulations produce com-
pletely different dual variables. Moreover, the discretized versions of the integral and differential
dynamics have much different numerical characteristics. For example, when refining the mesh in
ordertoachieveaspecifiederrortolerance,theerrorestimatesfortheintegraldynamicsaremuch
more stable and reliable than the error estimates derived from the differential dynamics [27, 33].
Onthebasisofthecomputationalimportanceoftheintegralformofthecollocationmethods,this
paperwillanalyzetherelationshipbetweentheLagrangemultipliersassociatedwiththediscretized
integralforms,andthecostateofthecontinuousoptimalcontrolproblem.Ourearlierwork[9–11]
analyzedtherelationshipsbetweentheLagrangemultipliersarisinginthediscreteandcontinuous
differentialformulationsoftheoptimalcontrolproblem.
The approach developed in this paper provides a way to approximate the costate of the original
optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR
collocation methods. Transformations are derived that relate the Lagrange multipliers of the inte-
gral 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 con-
trol problem in integral form. A new continuous-time dual variable called the integral costate is
thenintroduced,wheretheintegralcostateistheLagrangemultiplieroftheintegraldynamiccon-
straint.Thefirst-orderoptimalityconditionsoftheintegralformoftheoptimalcontrolproblemare
derived in terms of the integral costate. The integral form of the optimal control problem is then
discretizedusingtheintegralLGandLGRcollocationmethodsandrelationshipsbetweenthedis-
creteformoftheintegralcostateandthecostateoftheoriginaldifferentialoptimalcontrolproblem
aredeveloped.ItisshownthattheLGRintegrationmatrixthatrelatesthedifferentialcostatetothe
integral costate is singular while the corresponding LG integration matrix is full rank. These rela-
tionshipsleadtoawaytoapproximatethecostateoftheoriginaloptimalcontrolproblemusingthe
LagrangemultipliersoftheintegralformoftheLGandLGRcollocationmethods.Thetwomethods
developed in this paper are demonstrated on two examples where it is found that the costate con-
verges exponentially, consistent with the analysis in [34] for unconstrained control problems with
smoothsolutions.Althoughwefocusonunconstrainedcontrolproblemsinthispaper,therelations
we establish between the continuous costate and the Lagrange multipliers for the discrete integral
formsarealsoapplicableforproblemswithcontrolconstraintsorendpointconstraints.Ontheother
hand, when state constraints are present, the relationship between the continuous costate and the
multipliersinthediscreteproblemismorecomplexasshownin[35]and[36].
Thispaperisorganizedasfollows.InSection2,weintroducetheconventionsandnotationused
intheremainderofthispaper.InSection3,weformulatethecontinuous-timeoptimalcontrolprob-
lemwiththedynamicconstraintsformulatedinbothdifferentialandintegralform,andwepresent
the first-order optimality conditions for each form. In Sections 4 and 5, we present the LG and
LGR collocation methods in both differential and integral forms, derive the first-order optimality
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
COSTATEAPPROXIMATIONINOPTIMALCONTROL
conditionsineachform,developthetransformedadjointsystem,andderiveacostateapproximation
in terms of the Lagrange multipliers of the integral forms. In Section 6, we provide two examples
that demonstrate the accuracy of the LG and LGR costate approximation methods derived in this
paper.Finally,inSection7,weprovideconclusionsonourwork.
2. CONVENTIONSANDNOTATION
The following notation and conventions are used throughout this paper. Except where explicitly
noted,vectorsinthispaperarerowvector.Inparticular,ify.(cid:4)/ 2 Rn isthestatevectorattime(cid:4),
theny.(cid:4)/DŒy .(cid:4)/;(cid:3)(cid:3)(cid:3) ;y .(cid:4)/(cid:3).Generally,ifYisamatrix,thenY isthei-throwofY,whileY
1 n i iWj
denotesthesubmatrixformedbyrowsi throughj.TwoexceptionsarethedifferentiationmatrixD
and the integration matrix A, in which case D and A refer to the ith column of D or A. Finally,
i i
D> denotesthetransposeofmatrixD,andD> denotesthetransposeoftheith columnofD.Given
i
vectors x and y 2 Rn, the notation hx;yi is used to denote the standard Euclidean inner product
between x and y. Furthermore, if f W Rn (cid:2)! Rm and Y is N by n, then f.Y/ is the matrix whose
i-throwisf.Y /.Ify 2 Rn,thenrf.y/denotestheJacobianoffevaluateaty;theJacobianisan
i
m(cid:4)n matrix whose i-th row is rf .y/. In particular, the gradient of a scalar-valued function is a
i
rowvector.Finally,theKroneckerdeltafunctionisdefinedbyı D1andı D0ifi ¤j.
ii ij
3. CONTINUOUS-TIMEBOLZAOPTIMALCONTROLPROBLEM
In this section, we state the differential and integral forms of the continuous-time Bolza optimal
controlproblemunderconsiderationinthispaper.Inaddition,weprovidethefirst-orderoptimality
conditionsofeachformoftheproblemandexplainhowthesetwosetsofoptimalityconditionsare
relatedtooneanother.
3.1. Differentialandintegralformsofoptimalcontrolproblem
Consider the following continuous-time optimal control problem defined on the interval
(cid:4) 2Œ(cid:2)1;C1(cid:3). Determine the state y.(cid:4)/ 2 Rn and the control u.(cid:4)/ 2 Rm that minimize the
costfunctional
Z C1
J Dˆ.y.C1//C g.y.(cid:4)/;u.(cid:4)//d(cid:4); (1)
(cid:2)1
subjecttothedynamicconstraint
yP.(cid:4)/(cid:2)f.y.(cid:4)/;u.(cid:4)//D0; (cid:4) 2Œ(cid:2)1;C1(cid:3); (2)
andtheboundarycondition
y.(cid:2)1/Dy : (3)
0
Itisnotedthatthetimeinterval(cid:4) 2Œ(cid:2)1;C1(cid:3)canbetransformedtotheintervalŒt ;t (cid:3)viatheaffine
0 f
transformation
t (cid:2)t t Ct
f 0 f 0
t D (cid:4) C :
2 2
Henceforth,(1)–(3)willbereferredtoasthedifferentialoptimalcontrolproblem.
The differential optimal control problem given in (1)–(3) can be re-written in the following
integralform.Inparticular,integratingthedynamicsgivenin(2),wehave
Z (cid:2)
y.(cid:4)/Dy.(cid:2)1/C f.y.(cid:4)/;u.(cid:4)//d(cid:4):
(cid:2)1
Theoptimalcontrolprobleminintegralformisthenstatedasfollows.Determinethestatey.(cid:4)/ 2
Rn andthecontrolu.(cid:4)/2Rm thatminimizethecostfunctional
Z C1
J Dˆ.y.C1//C g.y.(cid:4)/;u.(cid:4)//d(cid:4); (4)
(cid:2)1
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
C.C.FRANÇOLINETAL.
subjecttotheintegralconstraint
Z (cid:2)
y.(cid:4)/(cid:2)y.(cid:2)1/(cid:2) f.y.(cid:4)/;u.(cid:4)//dt D0; (cid:4) 2Œ(cid:2)1;C1(cid:3); (5)
(cid:2)1
andtheboundarycondition
y.(cid:2)1/Dy : (6)
0
Henceforth,(4)–(6)willbereferredtoastheintegraloptimalcontrolproblem.
3.2. First-orderoptimalityconditionsofdifferentialandintegralforms
The first-order optimality conditions for the differential optimal control problem, given by the
Pontryaginminimumprinciple,are[37]
yP.(cid:4)/Df.y.(cid:4)/;u.(cid:4)//; (cid:4) 2Œ(cid:2)1;C1(cid:3); (7)
y.(cid:2)1/Dy ; (8)
0
0Dr H.y.(cid:4)/;u.(cid:4)/;(cid:2).(cid:4)//; (cid:4) 2Œ(cid:2)1;C1(cid:3); (9)
u
(cid:2)(cid:2)P.(cid:4)/Dr H.y.(cid:4)/;u.(cid:4)/;(cid:2).(cid:4)//; (cid:4) 2Œ(cid:2)1;C1(cid:3) (10)
y
(cid:2).C1/Drˆ.y.C1//: (11)
HereHistheHamiltoniandefinedby
H.y;u;(cid:2)/Dg.y;u/Ch(cid:2);f.y;u/i; (12)
and(cid:2)istheLagrangemultiplierassociatedwiththedifferentialdynamicsgivenin(7).
The first-order optimality conditions for the integral optimal control problem, derived in the
Appendix,arethefollowing:
Z (cid:2)
y.(cid:4)/Dy.(cid:2)1/C f.y.t/;u.t//dt; (cid:4) 2Œ(cid:2)1;C1(cid:3); (13)
(cid:2)1
y.(cid:2)1/Dy ; (14)
0
0Dr H.y.(cid:4)/;u.(cid:4)/;(cid:2).(cid:4)//; (cid:4) 2Œ(cid:2)1;C1(cid:3); (15)
u
r.(cid:4)/Dr H.y.(cid:4)/;u.(cid:4)/;(cid:2).(cid:4)//; (cid:4) 2Œ(cid:2)1;C1(cid:3); (16)
y
where
Z C1
(cid:2).(cid:4)/Drˆ.y.C1//C r.t/dt; (17)
(cid:2)
andristhemultiplierassociatedwiththeintegraldynamicsof(13).Thus,(17)givestherelation-
ship between the multipliers in the differential and integral formulations. Differentiating (17), we
seethatr.(cid:4)/D(cid:2)(cid:2)P.(cid:4)/;i.e.,themultiplierfortheintegraldynamicsisthenegativederivativeofthe
multiplierforthedifferentialdynamics.Theremainderofthispaperisdevotedtoderivingtwodis-
creteapproximationsofthedifferentialcostate,(cid:2).(cid:4)/,usingdiscreteapproximationsoftheintegral
costate,r.(cid:4)/.
4. COSTATEAPPROXIMATIONUSINGINTEGRALLEGENDRE–GAUSSCOLLOCATION
Inthissection,wepresenttheLGcollocationmethodandestablishtherelationbetweentheintegral
anddifferentialdiscretizedproblems.
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
COSTATEAPPROXIMATIONINOPTIMALCONTROL
4.1. DifferentialformofLegendre–Gausscollocation
The differential optimal control problem is now approximated using collocation at LG points (see
Refs. [6, 9–11]). The LG points are denoted .(cid:4) ;:::;(cid:4) / and are defined on the open interval
1 N
.(cid:2)1;C1/.Thestateisapproximatedbythepolynomial
N N
X Y (cid:4) (cid:2)(cid:4)j
y.(cid:4)/(cid:5)Y.(cid:4)/D Y L .(cid:4)/; L .(cid:4)/D ; (18)
i i i
(cid:4) (cid:2)(cid:4)
i j
iD0 jD0
j¤i
where(cid:4) D (cid:2)1isanadditionalpointwhereweapproximatethestate,andL .(cid:4)/,i D 0;:::;N is
0 i
abasisofLagrangepolynomialsofdegreeN withsupportpoints.(cid:4) ;:::;(cid:4) /.Thetimederivative
0 N
ofthestateapproximationat(cid:4) D(cid:4) ,16i 6N is
i
N N
X X
yP.(cid:4) /(cid:5)YP.(cid:4) /D Y LP .(cid:4) /D Y D DŒDY (cid:3) ; (19)
i i j j i j ij 0WN i
jD0 jD0
whereY D Y.(cid:4) /andDistheN (cid:4).N C1/LGdifferentiationmatrixwhoseelementsaregiven
j j
byD DLP .(cid:4) /.Notethatweonlycollocatethedynamicsatthequadraturepoints(cid:4) ,16i 6N,
ij j i i
notattheinitialtime(cid:4) D (cid:2)1.Ifw D .w ;:::;w /istherowvectorofLGquadratureweights
0 1 N
and(cid:4) DC1istheterminaltime,thenthediscretizedcontrolproblemis
NC1
N
X
min J Dˆ.Y /C w g.Y ;U /; (20)
NC1 j j j
jD1
subjecttothecollocateddynamics
DY (cid:2)f.Y ;U /D0; (21)
0WN 1WN 1WN
Y (cid:2)Y (cid:2)wf.Y ;U /D0; (22)
NC1 0 1WN 1WN
Y Dy ; (23)
0 0
It is noted for LG collocation that (22) provides an LG quadrature approximation, Y , of the
NC1
stateatthefinalnoncollocatedpoint(cid:4) DC1.TheNLPdescribedby(20)–(23)willbereferred
NC1
toasthedifferentialLGcollocationmethod.
4.2. Karush–Kuhn–TuckerconditionsusingdifferentialLegendre–Gausscollocation
In [10], it is shown that the Karush–Kuhn–Tucker (KKT) conditions for the differential LG
collocationmethodassociatedwith(20)–(23)canbewritteninthefollowingform:
DY Df.Y ;U /; Y Dy ; (24)
0WN 1WN 1WN 0 0
Y DY Cwf.Y ;U /; (25)
NC1 0 1WN 1WN
0Dr H.Y ;U ;(cid:2) /; (26)
u i i i
.D(cid:3)(cid:2) / D(cid:2)r H.Y ;U ;(cid:2) /; (27)
1WNC1 i y i i i
(cid:2) Drˆ.Y /; (28)
NC1 NC1
i D1;2;:::;N,whereD(cid:3) isdefinedby
N
D(cid:3) D(cid:2)wjD ; 16i;j 6N; and D(cid:3) D(cid:2)XD(cid:3): (29)
ij w ji i;NC1 ij
i
jD1
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
C.C.FRANÇOLINETAL.
It was shown in Theorem 1 of [9] that D(cid:3) is a differentiation matrix for the space of polynomials
ofdegreeN.Moreprecisely,ifb isapolynomialofdegreeatmostN andb 2 RNC1 isthevector
whoseithelementisb Db.(cid:4) /,16i 6N C1,then.D(cid:3)b/ DbP.(cid:4) /.Inthetranscriptionof(24)–
i i i i
(28), the state is differentiated by a matrix D [given by (19)], which is based on the derivatives of
polynomials of degree N with coefficients at the N LG points plus the initial noncollocated point
(cid:4) D (cid:2)1, whereas the costate is differentiated by a matrix D(cid:3) [given by (29)], which is based on
0
the derivatives of polynomials of degree N with coefficients at the N LG points plus the terminal
noncollocatedpoint(cid:4) DC1.
NC1
4.3. IntegralformofLegendre–Gausscollocation
TheintegraloptimalcontrolproblemisnowdiscretizedusingtheintegralformofLGcollocation.
It has been shown in [10] that the LG differentiation matrix D given by (19) has the property that
thesquarematrixD obtainedbyremovingthefirstcolumnofDisfull-rankand(cid:2)D(cid:2)1 D D1.
1WN 1WN 0
Wemultiplythedifferentialdynamicsin(24)byADD(cid:2)1 toobtain
1WN
Y D1Y CAf.Y ;U /; (30)
1WN 0 1WN 1WN
where 1 is an N (cid:4) 1 column vector of all ones. Combining (30) with (25) gives the discretized
dynamicsfortheintegralformulation.Theintegraloptimalcontrolproblemof(4)–(6)canthenbe
approximatedviathefollowingfinite-dimensionalnonlinearprogrammingproblem:Minimizethe
costfunctionof(20)subjecttothealgebraicconstraints
Y D1y CAf.Y ;U /; (31)
1WN 0 1WN 1WN
Y Dy Cwf.Y ;U /: (32)
NC1 0 1WN 1WN
The NLP described by the objective function of (20) and the dynamics of (31) and (32) will be
referredtoastheintegralLGcollocationmethod.
4.4. Karush–Kuhn–TuckerconditionsusingintegralLegendre–Gausscollocation
TheKKTconditionsoftheintegralLGcollocationmethodarefoundbytakingthepartialderivatives
oftheLagrangianLoftheNLPwithrespecttoeveryfreevariableandsettingtheresulttozero.The
Lagrangianis
N
X
LDˆ.Y /C w g.Y ;U /ChR ;1y CAf.Y ;U /(cid:2)Y i
NC1 i i i 1WN 0 1WN 1WN 1WN
iD1
ChR ;y Cwf.Y ;U /(cid:2)Y i;
NC1 0 1WN 1WN NC1
whereRistheN C1bynmatrixofmultipliersassociatedwiththediscretedynamics.Thepartial
derivativeswithrespecttothecontrolandthestateyield
0Dw r g.Y ;U /Cr h.ATR / ;f.Y ;U /iCw r hR ;f.Y ;U /i; 16i 6N;
i u i i u 1WN i i i i u NC1 i i
R Dw r g.Y ;U /Cr h.ATR / ;f.Y ;U /iCw r hR ;f.Y ;U /i; 16i 6N;
i i y i i y 1WN i i i i y NC1 i i
R Drˆ.Y /:
NC1 NC1
Wemakethechangeofvariables
w
r DR =w ; 16i 6N; r DR ; A D i A(cid:3): (33)
i i i NC1 NC1 ji w ij
j
Afterthesesubstitutionsandafterdividingtheith equationbyw ,weobtain
i
0Dr H.Y ;U ;(cid:3) /; (34)
u i i i
r Dr H.Y ;U ;(cid:3) /; (35)
i y i i i
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
COSTATEAPPROXIMATIONINOPTIMALCONTROL
r Drˆ.Y /; (36)
NC1 NC1
i D1;2;:::;N,where
(cid:3) Dr C.A(cid:3)r/ : (37)
i NC1 i
Because(cid:2) Dr by(28),weseethat(34)–(35)and(26)–(27)areequivalentifthefollowing
NC1 NC1
conditionshold:
(a) (cid:2) D1r CA(cid:3)r and (b) r D(cid:2)D(cid:3)(cid:2) :
1WN NC1 1WN 1WN 1WNC1
We will now show that conditions (a) and (b) are equivalent. This equivalence is based on the
(cid:2) (cid:3)(cid:2)1
following key property: D(cid:3) D (cid:2)A(cid:3). For example, if (b) holds, then by the definition of
1WN
D(cid:3) in(29),wehave
NC1
(cid:2)r DD(cid:3)(cid:2) DD(cid:3) (cid:2) CD(cid:3) (cid:2) DD(cid:3) (cid:2) (cid:2)D(cid:3) 1(cid:2) :
1WN 1WNC1 1WN 1WN NC1 NC1 1WN 1WN 1WN NC1
(cid:2) (cid:3)(cid:2)1 (cid:2) (cid:3)(cid:2)1
Wemultiplyby D(cid:3) D(cid:2)A(cid:3)toobtain(a).Theidentity D(cid:3) D(cid:2)A(cid:3)isnowestablished.
1WN 1WN
Theorem1
The matrix A(cid:3) defined in (33) is a backwards integration matrix for the space of polynomials of
degreeN (cid:2)1.Thatis,ifp isapolynomialofdegreeatmostN (cid:2)1andp2RN isthevectorwith
ith componentp Dp.(cid:4) /,then
i i
Z C1
.A(cid:3)p/ D p.t/dt: (38)
i
(cid:2)i
(cid:2) (cid:3)(cid:2)1
Moreover,(cid:2)A(cid:3) D D(cid:3) .
1WN
Proof
LetpandqdenotepolynomialsofdegreeatmostN (cid:2)1suchthatp Dp.(cid:4) /andq Dq.(cid:4) /for
j j j j
j D1;:::;N.Changingtheorderofintegration,wehave
Z C1(cid:4) Z (cid:2) (cid:5) Z C1(cid:4) Z C1 (cid:5)
q.(cid:4)/ p.t/dt d(cid:4) D p.(cid:4)/ q.t/dt d(cid:4): (39)
(cid:2)1 (cid:2)1 (cid:2)1 (cid:2)
Becausepandq arepolynomialsofdegreeatmostN (cid:2)1,itfollowsthat
Z C1 Z (cid:2)
p.(cid:4)/ q.t/dt and q.(cid:4)/ p.t/dt
(cid:2) (cid:2)1
arepolynomialsofdegreeatmost2N(cid:2)1.BecauseLGquadratureisexactforpolynomialsofdegree
atmost2N (cid:2)1,theintegralsin(39)canbereplacedbytheirLGquadratureequivalentstoobtain
XN Z (cid:2)j XN Z C1
w q p.t/dt D w p q.t/dt: (40)
j j i i
jD1 (cid:2)1 iD1 (cid:2)i
In[10],itisshownthat
Z (cid:2)j
p.t/dt D.Ap/ ; 16j 6N; (41)
j
(cid:2)1
wherepisacolumnvector.LetL(cid:3) denotetheLagrangebasisfunctionsdefinedby
j
N
L(cid:3) D Y (cid:4) (cid:2)(cid:4)i ; j D1;:::;N;
j (cid:4) (cid:2)(cid:4)
j i
iD1
j¤i
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
C.C.FRANÇOLINETAL.
anddefinetheN (cid:4)N matrixBby
Z C1
b D L(cid:3).t/dt:
ij j
(cid:2)i
BythedefinitionofB,itfollowsthat
Z C1
q.t/dt D.Bq/ : (42)
i
(cid:2)i
Combining(42)with(40)and(41),weobtain
N N N N
XX XX
w q A p D w p B q :
j j ji i i i ij j
jD1iD1 iD1jD1
Rearrangingthislastexpressiongives
N N
XX
q Œw A (cid:2)w B (cid:3)p D0:
j j ji i ij i
jD1iD1
Because this last resultmusthold forall p and q, we conclude thatthe bracketed expressionmust
vanish.Therefore,
w
j
B D A ;
ij ji
w
i
whichshowsthatBDA(cid:3).Consequently,(42)yields(38).
Givenp2RN,letp.(cid:4)/denotethepolynomialofdegreeatmostN (cid:2)1thatsatisfiesp.(cid:4) /Dp .
i i
Letq bethepolynomialofdegreeatmostN definedby
Z C1
q.(cid:4)/D p.t/dt: (43)
(cid:2)
Letq 2 RNC1 bethevectorwithcomponentsq D q.(cid:4) /,1 6 i 6 N C1.ByTheorem1in[10]
i i
andby(43),wehave
.D(cid:3) q/ DqP.(cid:4) /D(cid:2)p.(cid:4) /D(cid:2)p ; 16i 6N: (44)
1WNC1 i i i i
By(38),wehaveq DA(cid:3)p.Becauseq D0,itfollowsthat
1WN NC1
D(cid:3) qDD(cid:3) q DD(cid:3) A(cid:3)p: (45)
1WNC1 1WN 1WN 1WN
Combining(44)and(45)yields
D(cid:3) A(cid:3)pD(cid:2)p:
1WN
(cid:2) (cid:3)(cid:2)1
Becausepwasarbitrary,wededucethatD(cid:3) isinvertibleand D(cid:3) D(cid:2)A(cid:3). (cid:2)
1WN 1WN
5. COSTATEAPPROXIMATIONUSINGINTEGRALLEGENDRE–GAUSS–RADAU
COLLOCATION
Inthissection,wedeveloptherelationbetweenthemultipliersarisingintheintegralLGRcolloca-
tionschemeandthecostateassociatedwiththedifferentialLGRcollocationscheme.InSection5.1,
we review the differential form of the LGR collocation method, and in Section 5.2, we review the
first-order optimality conditions for the discrete problem (see Refs. [9–11]). In Section 5.3, we
describetheintegralformoftheLGRcollocationmethod,andweprovidethefirst-orderoptimality
conditions of the nonlinear programming problem described in Section 5.1, and the relationships
betweenthemultipliersintheintegralanddifferentialdiscretizations.
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
COSTATEAPPROXIMATIONINOPTIMALCONTROL
5.1. DifferentialformofLegendre–Gauss–Radaucollocation
We will consider the so-called flipped LGR points located on the half-open interval .(cid:2)1;C1(cid:3), as
in [10], because the algebraic manipulations are somewhat simpler than those for the LGR points
locatedonŒ(cid:2)1;C1/consideredin[9].If.(cid:4) ;:::;(cid:4) /aretheLGRquadraturepointswith(cid:4) DC1,
1 N N
thenthestateisapproximatedas
N N
X Y (cid:4) (cid:2)(cid:4)j
y.(cid:4)/(cid:5)Y.(cid:4)/D Y L .(cid:4)/; L .(cid:4)/D ; (46)
i i i
(cid:4) (cid:2)(cid:4)
i j
iD0 jD0
j¤i
where(cid:4) D(cid:2)1isthestartingtimeandL .(cid:4)/; i D0;:::;N,isabasisofLagrangepolynomialsof
0 i
degreeN withsupportpoints.(cid:4) ;:::;(cid:4) /.Thetimederivativeofthestateapproximationat(cid:4) D(cid:4) ,
0 N i
16i 6N is
N
X
yP.(cid:4) /(cid:5)YP.(cid:4) /D Y LP .(cid:4) /DŒDY (cid:3) ; (47)
i i j j i 0WN i
jD0
whereY DY.(cid:4) /andDistheN (cid:4).N C1/LGRdifferentiationmatrixwhoseelementsaregiven
i i
by D D LP .(cid:4) /. Note that we collocate the dynamics at the quadrature points (cid:4) , 1 6 i 6 N,
ij j i i
whichincludesthefinaltime(cid:4) D C1butnottheinitialtime(cid:4) D (cid:2)1.Ifw D .w ;:::;w /is
N 0 1 N
thevectorofLGRquadratureweights,thenthediscretizedcontrolproblemis
N
X
min J Dˆ.Y /C w g.Y ;U /; (48)
N j j j
jD1
subjecttothecollocateddynamics
DY (cid:2)f.Y ;U /D0; (49)
0WN 1WN 1WN
Y Dy : (50)
0 0
TheNLPdescribedby(48)–(50)willbereferredtoasthe differentialLGRmethod.
5.2. Karush–Kuhn–TuckerconditionsusingdifferentialLegendre–Gauss–Radaucollocation
In[10],itisshownthattheKKTfirst-orderoptimalityconditionsofthedifferentialLGRcollocation
methodcanbewritteninthefollowingform:
DY Df.Y ;U /; Y Dy ; (51)
0WN 1WN 1WN 0 0
0Dr H.Y ;U ;(cid:2) /; (52)
u i i i
1
.D(cid:3)(cid:2) / D(cid:2)r H.Y ;U ;(cid:2) /C ı .(cid:2) (cid:2)rˆ.Y /; (53)
1WN i y i i i iN N N
w
N
where (cid:2) , 1 6 i 6 N is the transformed multiplier associated with the collocated dynamics at
i
(cid:4) , ı is the Kronecker delta, which is zero except for ı D 1, and D(cid:3) is the N (cid:4)N matrix
i iN NN
definedby
1 w
D(cid:3) D(cid:2)D C and D(cid:3) D(cid:2) jD otherwise: (54)
NN NN w ij w ji
N i
ByTheorem1in[10],D(cid:3) isadifferentiationmatrixforthespaceofpolynomialsofdegreeN (cid:2)1.
More precisely, if b is a polynomial of degree at most N (cid:2)1 and b 2 RN is the vector with ith
elementb Db.(cid:4) /for16i 6N,then
i i
.D(cid:3)b/ DbP.(cid:4) /:
i i
Itisnotedthatin(51)–(53),thetimederivativeofthestateisapproximatedusingthedifferentiation
matrixDforthespaceofpolynomialsofdegreeN [(47)],whilethecostateisbeingdifferentiated
byadifferentiationmatrixD(cid:3) forthespaceofpolynomialsofdegreeN (cid:2)1.
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca
C.C.FRANÇOLINETAL.
5.3. IntegralformofLegendre–Gauss–Radaucollocation
It is shown in [10] that the LGR differentiation matrix D given by (47) has the property that the
square matrix D obtained by removing the first column of D is full-rank and (cid:2)D(cid:2)1 D D 1.
1WN 1WN 0
Usingtheseproperties,theintegraldynamicconstraints(5)canbeapproximatedas
Y D1y CAf.Y ;U /; (55)
1WN 0 1WN 1WN
where1asacolumnvectorofallones.TheNLPdescribedby(48)and(55)willbereferredtoas
theLGRcollocationmethod.
TheKKTfirst-orderoptimalityconditionsoftheintegralLGRcollocationmethodarefoundby
taking the partial derivatives of the Lagrangian of the NLP with respect to every free variable and
settingtheresulttozero.TheLagrangianis
N
X
LDˆ.Y /C w g.Y ;U /ChR;1y CAf.Y ;U /(cid:2)Y i;
N i i i 0 1WN 1WN 1WN
iD1
where R , 1 6 i 6 N, is the multiplier associated with the collocated dynamics at (cid:4) . The partial
i i
derivativeswithrespecttothecontrolandthestateyieldtherelations
0Dw r g.Y ;U /Cr h.ATR/ ;f.Y ;U /i; (56)
i u i i u i i i
R Dw r g.Y ;U /Cr h.ATR/ ;f.Y ;U /iCı rˆ.Y /; (57)
i i y i i y i i i Ni N
16i 6N.Next,wemakethechangeofvariables
r DR =w (cid:2).ı =w /rˆ.Y /: (58)
i i i Ni i N
Inaddition,wedefinethematrixA(cid:3) as
w
A(cid:3) D jA : (59)
ij w ji
i
Substitutingtheresultsof(58)and(59)into(56)and(57)anddividingtheith equationbyw ,we
i
obtain
0Dr H.Y ;U ;.A(cid:3)r/ C.A =w /rˆ.Y //;
u i i i Ni i N
r Dr H.Y ;U ;.A(cid:3)r/ C.A =w /rˆ.Y //:
i y i i i Ni i N
In[10],itisshownthat
A DZ (cid:2)i L(cid:3).(cid:4)/; L(cid:3) D YN (cid:4) (cid:2)(cid:4)i :
ij j j (cid:4) (cid:2)(cid:4)
(cid:2)1 j i
iD1
j¤i
Because(cid:4) D C1,wededucethatA D w andA =w D 1.Hence,weobtainthefollowing
N Ni i Ni i
necessaryoptimalityconditions
0Dr H.Y ;U ;(cid:3) /; (60)
u i i i
r Dr H.Y ;U ;(cid:3) /; (61)
i y i i i
16i 6N,where
(cid:3) Drˆ.Y /C.A(cid:3)r/ : (62)
i N i
Comparing(52)and(53)to(60)and(61),weseethattheyareequivalentifthefollowingconditions
hold:
(cid:6) (cid:7)
1 1
(a) (cid:2)D1rˆ.Y /CA(cid:3)r and (b) rD e eT (cid:2)D(cid:3) (cid:2)(cid:2) e rˆ.Y /;
N w N N w N N
N N
wheree isthelastcolumnoftheidentity(thecolumnvectorwhoseentriesareallzeroexceptthe
N
lastentrywhichis1).Theconditions(a)and(b)areequivalentinthatif(a)holds,thensodoes(b)
andif(b)holds,thensodoes(a).ThisequivalenceisbasedonthefollowinganalogofTheorem1,
whichwasestablishedin[34].
Copyright©2014JohnWiley&Sons,Ltd. Optim.ControlAppl.Meth.(2014)
DOI:10.1002/oca