Table Of ContentSpringer Optimization and Its Applications 97
Chongyang Liu
Zhaohua Gong
Optimal Control
of Switched
Systems Arising
in Fermentation
Processes
Springer Optimization and Its Applications
VOLUME 97
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor–CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(PrincetonUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(LehighUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthe
lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped,
thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge
ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof
themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe
interdisciplinarynatureofthefield.Optimizationhasbeenabasictoolinallareas
ofappliedmathematics,engineering,medicine,economics,andothersciences.
The series SpringerOptimization and Its Applicationspublishesundergraduate
and graduate textbooks, monographs and state-of-the-art expository work that
focusonalgorithmsforsolvingoptimizationproblemsandalsostudyapplications
involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization
(convexand nonconvex),networkflow problems,stochastic optimization,optimal
control,discreteoptimization,multiobjectiveprogramming,descriptionofsoftware
packages,approximationtechniquesandheuristicapproaches.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393
Chongyang Liu • Zhaohua Gong
Optimal Control of Switched
Systems Arising in
Fermentation Processes
ChongyangLiu
ZhaohuaGong
MathematicsandInformationScience
ShandongInstituteofBusiness
andTechnology
Yantai,Shandong,China
ISSN1931-6828 ISSN1931-6836(electronic)
ISBN978-3-662-43792-6 ISBN978-3-662-43793-3(eBook)
DOI10.1007/978-3-662-43793-3
SpringerHeidelbergNewYorkDordrechtLondon
JointlypublishedwithTsinghuaUniversityPress,Beijing
ISBN:978-7-302-37332-2TsinghuaUniversityPress,Beijing
LibraryofCongressControlNumber:2014949499
MathematicsSubjectClassification:49J15,49J21,65K10,49M37,92C42
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Preface
Switchedsystemshaveattractedmuchinterestfromthecontrolcommunity,notonly
becauseoftheirinherentcomplexitybutalsoduetotheirpracticalimportancewith
a wide range of applications in engineering, nature, and social sciences. Optimal
controlofswitchedsystems,whichrequiresdeterminingboththeoptimalswitching
sequenceandtheoptimalcontinuousinput,hasattractedmanyresearchersrecently.
This phenomenonis due to the problem’ssignificance in theory and applications.
This book is not intended to compete with the many existing excellent books on
optimal control theory and switched systems. We simply cannot write a better
one! Our intention is to supplement them from the viewpoints of applications in
fermentationprocesses.
The modern fermentation industry, which is largely a product of the twentieth
century, is dominated by aerobic/anaerobiccultivations intended to make a range
of high-value products. However, since most fermentation processes create very
diluteandimpureproducts,thereisagreatneedtoincreasevolumetricproductivity
and to increase the product concentration. As a result, significant work is needed
to optimize the operation and design of bioreactors to make production more
efficientandmoreeconomical.Itisobviousthata model-basedefficientapproach
is necessary to ensure maximum productivity with the lowest possible cost in
fermentation processes, without requiring a human operator. Nevertheless, the
mathematicaldeterminationofoptimalcontrolinafermentationprocesscanbevery
difficult and open-ended due to the presence of nonlinearities in process models,
inequalityconstraintsonprocessvariables,andimplicitprocessdiscontinuities.
In this book, we present some mathematical models arising in fermenta-
tion processes. They are in the form of nonlinear multistage system, switched
autonomous system, time-dependent switched system, state-dependent switched
system,multistagetime-delaysystem,andswitchedtime-delaysystem.Onthebasis
of these dynamicalsystems, we consider the optimization problemsincluding the
v
vi Preface
optimalcontrolproblemsandtheoptimalparameterselectionproblems.Wediscuss
some important theories, such as existence of optimal controls and optimization
algorithmsfortheoptimizationproblemsmentionedabove.
Theobjectiveofthisbookistopresent,inasystematicmanner,theoptimalcon-
trols under different mathematical models in fermentation processes. By bringing
forward fresh novel methods and innovative tools, we are to provide a state-of-
the-artandcomprehensivesystematictreatmentofoptimalcontrolproblemsarising
infermentationprocesses.Thiscannotonlydevelopnonlineardynamicalsystem,
optimal control theory, and optimization algorithms but also increase process
productivity of product and serve as a reference for commercial fermentation
processes.
Acknowledgments
Forthecompletionofthebook,weareindebtedtomanydistinguishedindividuals
in our community. We would like to thank Prof. Enmin Feng and Prof. Zhilong
Xiu,DalianUniversityofTechnology,China,forbringingourattentiontothisarea.
Almostallthematerialspresentedinthisbookareextractedfromworkdonejointly
withthem.ItisourpleasuretoexpressourgratitudetoProf.KokLayTeo,Dr.Ryan
Loxton,andDr.QunLin,CurtinUniversity,Australia,fortheirvaluablecomments
duringourvisitingatCurtinUniversityfromJanuary2013toJuly2014.
We gratefully acknowledge the unreserved support, constructive comments,
and fruitful discussions from Dr. Lei Wang, Dr. Yaqin Sun, and Dr. Qingrui
Zhang,Dalian Universityof Technology,China; Dr. Jianxiong Ye, Fujian Normal
University, China; Dr. Bangyu Shen, Huaiyin Normal University, China; and
Dr.Jin’gangZhai,LudongUniversity,China.
We are also gratefulto Prof. Yuliang Han and Prof. Guang’aiSong, Shandong
InstituteofBusinessandTechnology,China,fortheirkindinvitationsinpublishing
thebook.
FinancialSupport
We acknowledgethefinancialsupportfromthe NationalNaturalScienceFounda-
tionofChinaunderGrants11201267,11001153,and11126077,fromtheShandong
Province Natural Science Foundation of China under Grant ZR2010AQ016, and
fromShandongInstituteofBusinessandTechnologyunderGrantY2012JQ02.
Yantai,Shandong,China ChongyangLiu
January2014 ZhaohuaGong
Contents
1 Introduction................................................................. 1
1.1 SwitchedSystem ..................................................... 1
1.2 OptimalControl...................................................... 2
1.2.1 StandardOptimalControl.................................. 2
1.2.2 OptimalSwitchingControl ................................ 4
1.3 FermentationProcess ................................................ 5
1.3.1 GenericFermentationProcess ............................. 5
1.3.2 1,3-PropanediolFermentation ............................. 7
1.3.3 KineticsandPhysiologicalModeling...................... 8
1.4 OutlineoftheBook .................................................. 9
2 MathematicalPreliminaries............................................... 13
2.1 LebesgueMeasureandIntegration.................................. 13
2.2 NormedSpaces....................................................... 17
2.3 LinearFunctionalsandDualSpaces ................................ 20
2.4 BoundedVariation.................................................... 22
3 ConstrainedMathematicalProgramming............................... 25
3.1 Introduction........................................................... 25
3.2 Gradient-BasedAlgorithms.......................................... 26
3.2.1 OptimalityConditions...................................... 27
3.2.2 TheQuadraticPenaltyMethod ............................ 28
3.2.3 AugmentedLagrangianMethod........................... 30
3.2.4 SequentialQuadraticProgramming ....................... 32
3.3 EvolutionaryAlgorithms............................................. 35
3.3.1 ParticleSwarmOptimization .............................. 35
3.3.2 DifferentialEvolution ...................................... 36
3.3.3 Constraint-HandlingTechniques........................... 38
vii
viii Contents
4 ElementsofOptimalControlTheory .................................... 41
4.1 Introduction........................................................... 41
4.2 DynamicalSystems .................................................. 41
4.2.1 OrdinaryDifferentialSystem .............................. 41
4.2.2 Delay-DifferentialSystem ................................. 44
4.2.3 SwitchedSystem ........................................... 48
4.3 OptimalControlProblems........................................... 49
4.3.1 StandardOptimalControlProblem........................ 49
4.3.2 OptimalMultiprocessControlProblem................... 50
4.4 NecessaryOptimalityConditions ................................... 52
4.4.1 NecessaryConditionsforStandardOptimal
ControlProblem............................................ 52
4.4.2 NecessaryConditionsforOptimalMultiprocesses ....... 54
5 OptimalControlofNonlinearMultistageSystems..................... 59
5.1 Introduction........................................................... 59
5.2 ControlledMultistageSystems...................................... 60
5.3 PropertiesoftheControlledMultistageSystems................... 63
5.4 OptimalControlModels............................................. 66
5.5 ComputationalApproaches.......................................... 68
5.6 NumericalResults.................................................... 73
5.7 Conclusion............................................................ 76
6 OptimalControlofSwitchedAutonomousSystems.................... 77
6.1 Introduction........................................................... 77
6.2 SwitchedAutonomousSystems..................................... 78
6.3 OptimalControlModels............................................. 80
6.4 ComputationalApproaches.......................................... 82
6.5 NumericalResults.................................................... 85
6.6 Conclusion............................................................ 86
7 OptimalControlofTime-DependentSwitchedSystems............... 89
7.1 Introduction........................................................... 89
7.2 Time-DependentSwitchedSystems................................. 90
7.3 ConstrainedOptimalControlProblems............................. 93
7.4 ComputationalApproaches.......................................... 94
7.4.1 ApproximateProblem...................................... 94
7.4.2 ContinuousStateConstraints .............................. 96
7.4.3 OptimizationAlgorithms................................... 98
7.5 NumericalResults.................................................... 99
7.6 Conclusion............................................................ 103
8 OptimalControlofState-DependentSwitchedSystems............... 105
8.1 Introduction........................................................... 105
8.2 State-DependentSwitchedSystems................................. 106
8.3 OptimalControlModels............................................. 110
Contents ix
8.4 SolutionMethodsfortheInnerOptimizationProblem............. 113
8.5 NumericalResults.................................................... 118
8.6 Conclusion............................................................ 119
9 OptimalParameterSelectionofMultistageTime-DelaySystems..... 123
9.1 Introduction........................................................... 123
9.2 ProblemFormulation................................................. 124
9.2.1 MultistageTime-DelaySystems........................... 124
9.2.2 PropertiesoftheMultistageTime-DelaySystems........ 126
9.3 ParametricSensitivityAnalysis...................................... 128
9.3.1 SensitivityFunctions ....................................... 128
9.3.2 NumericalSimulationResults ............................. 132
9.4 OptimalParameterSelectionProblems ............................. 135
9.4.1 OptimalParameterSelectionModels...................... 135
9.4.2 AComputationalProcedure................................ 136
9.4.3 NumericalResults.......................................... 139
9.5 Conclusion............................................................ 142
10 OptimalControlofMultistageTime-DelaySystems ................... 143
10.1 Introduction........................................................... 143
10.2 ControlledMultistageTime-DelaySystems........................ 144
10.3 ConstrainedOptimalControlProblems............................. 148
10.4 ComputationalApproaches.......................................... 149
10.5 NumericalResults.................................................... 155
10.6 Conclusion............................................................ 158
11 OptimalControlofSwitchedTime-DelaySystems..................... 159
11.1 Introduction........................................................... 159
11.2 SwitchedTime-DelaySystems...................................... 160
11.3 OptimalControlProblems........................................... 163
11.3.1 FreeTimeDelayedOptimalControlProblem ............ 163
11.3.2 TheEquivalentOptimalControlProblem................. 164
11.4 NumericalSolutionMethods........................................ 166
11.4.1 ApproximationProblem.................................... 166
11.4.2 AComputationalProcedure................................ 167
11.5 NumericalResults.................................................... 173
11.6 Conclusion............................................................ 174
References......................................................................... 177
