Process control and optimization using a modified Carleman linearization

129 Pages·1997·3.5 MB·English

by  AlTuwaim, Mohammad Saleh, 1960-

PROCESSCONTROLANDOPTIMIZATIONUSINGAMODIFIEDCARLEMAN
LINEARIZATION
1
By
MOHAMMADSALEHALTUWAIM
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1997
.
ACKNOWLEDGEMENTS
Iwouldliketoexpressmygreatestappreciationtomydissertationadvisor,Dr.
SpyrosSvoronos,forsharinghisexperience,consistentsupportandencouragement. I
would like alsoto express mythanksand appreciationto my co-advisor, Dr. Oscar
Crisalle,forhisconsistentsupportandencouragement. IwouldliketothankDr. Gar
Hoflund,Dr.BenKoopman,andDr.HaniphLatchmanfortheirsupportandforserving
onmysupervisorycommittee.
HighonmylistofacknowledgementsforfinancialsupportmustbethePublic
AuthorityforAppliedEducationandTraining(PAAET),Kuwait,especiallytheCollege
ofTechnologicalStudies. Inaddition,Iwouldliketoexpressmygratitudetothestaffof
theCulturalDivisionattheEmbassyoftheStateofKuwait.
Iamgreatlyindebtedtomyfriendsfortheirhelpandcontinuedsupport. Iowe
special thanks to all my colleagues inthe Chemical Engineering Department at the
UniversityofFlorida, especiallyWilburWoo,Pi-HsinLiu, JoseLorenzo, and Sunjay
Gupta
Last,butbynomeansleast,IwouldliketosaythatIamindebtedtomyfamily,
especially my parents, for their love, patience, support, motivation, and continuing
encouragement and prayers, which made this work possible. I owe thanks to the
inspirationofmywife,Muna,myson,Saleh,andmydaughter,Shareefa.
5
TABLEOFCONTENTS
page
ACKNOLEDGEMENTS
ii
NOMENCLATURE v
ABSTRACT
viii
CHAPTERS
1 INTRODUCTION 1
2 NEWDISCRETIZATIONMETHOD 5
2.1 ModificationtotheCarlemanMethod 5
2.2 PerformanceMeasure 9
2.3 StructureoftheApproximatingMatrixFunction 11
2.4 PropertiesoftheDiscrete-TimeRepresentation 13
2.5 Examples 19
3 NONLINEAROPTIMALCONTROL 33
3.1StatementoftheProblem 33
3.2ControlLawStructure 34
3.3FormulationoftheSuboptimalProblem 37
3.4Example 39
4 CONCLUSIONS 50
APPENDICES
A MATLABCODESFOREXAMPLEONESECTION2.5
(FIRSTORDER) 52
B MATLABCODESFOREXAMPLEONESECTION2.5
(SECONDORDER) 64
C MATLABCODESFOREXAMPLETWOSECTION2.
(FIRSTANDSECONDORDER) 80
iii
D MATLABCODESFOREXAMPLEONECHAPTER3
(SECOND,FOURTHANDSIXTHORDER) 95
E CALCULATIONSRELATEDTOLEMMA1(CHAPTER2) 110
REFERENCES 113
BIOGRAPHICALSKETCH 118
iv
s
NOMENCLATURE
a parameterdefinedbyeq.(2.36)
ai,&2 parametersdefinedbyeq.(2.34)
an,ai2 parametersdefinedbyeq.(2.38)
A parameterdefinedbyeq.(2.37)
Aij matrixdefinedbyeq.(2.2)
b parameterdefinedbyeq.(2.36)
c parameterdefinedbyeq.(2.36)
c heatcapacityofthemixture,kJ/kgK
Ca concentrationofspeciesAinthereactor,kgmole/m
CAin concentrationofspeciesAinthefeed,kgmole/m
Ca steadystatevalueofCa
d parameterdefinedbyeq.(2.36)
d, integerexponentdefinedbyeq.(2.16)
relative-errorvectordefinedbyeq.(2.13)
E activationenergy,kJ/kgmole
f scalardefinedbyeq.(2.32)
F volumetricfloerate,m3/s
Fk matrixdefinedbyeq.(2.9)
gk vectordefinedbyeq.(2.6)
gk vectordefinedbyeq.(2.4)
hji elementsofthematrixfunction,H(y,ur)
H matrixfunctiondefinedeq.(2.8)
-AH heatofreaction,kJ/kgmole
J performancemeasuredefinedbyeq.(2.14)
k truncationorder
v
ki reactionrateconstant
l numberofcoefficientsoftheythrowwhichiszero
L, scalardefinedbyeq.(2.33)
n sizeofthestatevector
U reactionrate
R universalgasconstant,kJ/kgmoleK
s scalardefinedbyeq.(2.32)
Si,S2 scalarsdefinedbyeq.(2.34)
Sk matrixdefinedbyeq.(2.6)
Sk' matrixdefinedbyeq.(2.4)
Sk matrixdefinedbyeq.(2.6)
Sk' matrixdefinedbyeq.(2.4)
t time
tr timeatthebeginningofrthsamplinginterval
T samplingperiod
Tr therthsamplingperiod
u scalarofcontrolvariable
vectorofinputsequencedefinedbyeq.(2.14)
Umax vectorofmaximumvaluesofinput
Umin vectorofminimumvaluesofinput
Ur inputvectorattherthsamplinginterval
V reactorvolume,m3
x statevector
X estimateofstatevector
Xjn dimensionlessvariabledefinedeq.(2.30)
Xli dimensionlessvariabledefinedeq.(2.36)
X2i dimensionlessvariabledefinedeq.(2.36)
vectordefinedbyeq.(2.3)withoutrepeatingterms
vectordefinedbyeq.(2.3)
vi
x Athcomponentofx
k
xk estimateofxk
x* estimateofxk usingproductsofestimatesoffirstorder
terms
x[kl vectoroftheAthKroneckerproductwithoutrepetitions
x'[k] vectoroftheAthKronieckerproduct,definedbyeq.(2.2)
x[k]* estimateofx(klusingproductsofestimatesoffirstorder
terms
Z frequencyfactor,s'1
Greekletters
a dimensionlessparameterdefinedbyeq.(2.30)
y vectorofcoefficientsdefinedbyeq.(2.8)
8 vectordefinedbyeq.(2.10)
8, vectorthatconsistsofthetoprowsof8
(D matrixdefinedbyeq.(2.10)
<Dn submatrixof<Dconsistsoftopnrowsandcolumns
correspondtox[kr
r* vectorsdefinedbyeq.(2.19)
Xi eigenvalueofF*
e reactor’stemperature,K
0m temperatureoffeedingstream,K
0s steadystatevalueof0
X dimensionlessparameterdefinedbyeq.(2.30)
Si polynomicexpansiondefinedbyeq.(2.19)
vii
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
PROCESSCONTROLANDOPTIMIZATIONUSINGMODIFIEDCARLEMAN
LINEARIZATION
By
MohammadSalehAlTuwaim
December,1997
Chairman:Dr.SpyrosA.Svoronos
Cochairman:Dr.OscarD.Crisalle
MajorDepartment:ChemicalEngineering
Theeffectivedesignofdigitalcontrolsystemsoftenrequirestheavailabilityof
highlyaccuratediscreterepresentationsofcontinuous-timesystems. Thediscretizationis
donephysicallyviaasample-and-holdmechanism;hence,thechallengefortheengineer
istoproduceagooddiscretemodelofthesampledsystem.
RecentsuccessfuldiscretizationmethodsarebasedonhighorderTaylorseries
expansions or lower order Taylor series expansions combined with the Carleman
linearization procedure. In this work we introduce a new approach based on a
modificationoftheBellman-RichardsonextensionoftheCarlemanlinearization. Akey
idea is to approximate truncated terms as linear combinations ofkept terms with
coefficientsthatarepolynomialfunctionsofthemanipulatedinput. Thesecoefficients
viii
aredeterminedbysolvinganumericaloptimizationproblem.Theresultinglinearsystem
is then integrated overthe sampling period toyield the discrete-time model after a
straightforwardsubstitutionofvariables.
Numericalstudiesareconductedtoevaluatethenewmethodusingtwotestcases
availableintheliterature,namelyaone-dimensionalmodelofanisothermalcontinuous
stirredtankreactor(CSTR)andatwo-dimensionalmodelofanonisothermalCSTR. The
resultsdemonstratethatthenewmethodgivesexcellentdiscretizationswithverylow
orderexpansions. ThenewmethodoutperformstheCarlemandiscretizationmethod;in
fact,thefirstorderimplementationofthenewmethodyieldsdiscretizationscomparable
tothosefromasecondorderCarleman.
Inaddition,anewmethodfordeterminingsuboptimalfeedbackcontrollawsfor
nonlineardynamicalsystemsisdeveloped. Thenewapproachintroducesastructurefor
thecontrollawsimilartothatusedinneuralnetworks. Thenonlineardynamicalsystem
andtheperformancemeasurewithconstraintsontheinputsarethenlinearizedusingthe
Calemanmethod. Thus,thedynamicoptimizationproblemistransformedtoanalgebraic
expressionthatisoptimizedtoyieldtheparametersofthecontrollaw. Thenewmethod
isappliedtoaone-dimensionalexampleavailableintheliterature. Theperformanceof
thesynthesizedfeedbackcontrollerusingtheproposedmethodisveryclosetothatofthe
optimalcontrollaw.
IX
CHAPTER
1
INTRODUCTION
The development of a discretization method for nonlinear continuous-time
systems is motivated by the computer-based implementation of advanced control
systems. Therefore,discretizationschemesthatleadtoaccurateandsimplediscrete-time
modelsareessentialtoolsforthedesignofeffectivedigitalcontrolsystems.
Thelackofgeneralexactdiscretizationmethodsfornonlinearsystemsforcesthe
controldesignertochooseoneoftwoalternativeapproaches. Thefirstalternativeisto
designacontinuous-timecontrollerbased onthecontinuousmodel and implement it
digitally through a sample-and-hold element. This approach is adequate for small
samplingperiods. Thesecondalternativeistodesignadiscretecontrollerbasedonan
approximatediscretemodel. Suchamodelcanbeobtainedusingclassicalmethodssuch
as the Euler, Heun, and Runge-Kutta techniques (Chapra and Canale, 1988). A
disadvantageofthesemethodsisthattheiraccuracyrapidlydeterioratesasthesampling
periodincreases. Thisdisadvantageisovercomeinarecentlydevelopedmethodbased
ontheCarlemanlinearization(Svoronosetal, 1994). Thismethodyieldsanonlinear
discrete-timemodelbyintegratingaCarlemanapproximationoftheoriginalcontinuous-
timemodel. KazantzisandKravaris(1993)showedthataccuraciescomparabletothose
ofthe Carleman-linearization-based method can be achieved by using Taylor series
l