Functional fractional calculus for system identification and controls

250 Pages·2008·2.225 MB·English

by  Shantanu Das

Functional Fractional Calculus for System
Identification and Controls
Shantanu Das
Functional Fractional
Calculus for System
Identification and Controls
With68Figuresand11Tables
Author
ShantanuDas
Scientist
ReactorControlDivision,
BARC,Mumbai–400085
[e-mail]
LibraryofCongressControlNumber:2007934030
ISBN 978-3-540-72702-6 SpringerBerlinHeidelbergNewYork
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This workisdedicatedto myblindfatherSri
SoumendraKumar Das,mymotherPurabi,
mywifeNita,my sonSankalan,my sister
Shantasree,brother-in-lawHemant,and to
mylittlenieceIshita
Preface
This work is inspired by thought to have an overall fuel-efficient nuclear plant
control system. I picked up the topic in 2002 while deriving the reactor control
laws, which aimed at fuel efficiency. Controlling the nuclear reactor close to its
natural behavior by concept of exponentshape governor,ratio control and use of
logarithmiclogic,aimsatthefuelefficiency.Thepower-maneuveringtrajectoryis
obtained by shaped-normalized-periodfunction, and this defines the road map on
which the reactor should be governed. The experience of this concept governing
theAtomicPowerPlantofTarapurAtomicPowerStationgiveslesseroverallgains
comparedtotheolderplants,whereconventionalproportionalintegralandderiva-
tive type (PID) scheme is employed. Therefore, this motivation led to design the
scheme for control system than the conventional schemes to aim at overall plant
efficiency.Thus,IfelttheneedtolookbeyondPIDandobtainedtheanswerinfrac-
tional order control system, requiringfractionalcalculus (a 300-year-oldsubject).
Thisworkistakenfromalargenumberofstudiesonfractionalcalculusandhereit
isaimedatgivinganapplication-orientedtreatment,tounderstandthisbeautifulold
new subject. The contribution in having fractionaldivergenceconceptto describe
neutron flux profile in nuclear reactors and to make efficient controllers based on
fractionalcalculusisaminorcontributioninthisvast(hidden)areaofscience.This
work is aimed at to make this subject popular and acceptable to engineering and
sciencecommunitytoappreciatetheuniverseofwonderfulmathematics,whichlies
betweentheclassicalintegerorderdifferentiationandintegration,whichtillnowis
notmuchacknowledged,andishiddenfromscientistsandengineers.
ShantanuDas
vii
Acknowledgments
I’m inspired by the encouragements from the director of BARC, Dr. Srikumar
Banerjee, for his guidance in doing this curiosity-drivenresearch and apply them
for technological achievements. I acknowledge the encouragement received from
Prof.Dr.ManojKumarMitra,DeanFacultyofEngineering,andProf.Dr.Amitava
Gupta of the Jadavpur University (Power EngineeringDepartment), for accepting
theconceptoffractionalcalculusforpowerplantcontrolsystem,andhiseffortalong
with his PhD and ME studentsto developthe controlsystem fornuclearindustry,
aimed at increasing the efficiency and robustness of total plant. I also acknowl-
edgetheencouragementsreceivedfromProf.Dr.SiddhartaSen,IndianInstituteof
TechnologyKharagpur(ElectricalEngineeringDepartment)andDr.KarabiBiswas
(Jadavpur University), to make this book for ME and PhD students who will
carrythisknowledgeforresearchininstrumentationandcontrolscience.Fromthe
Department of Atomic Energy, I acknowledge the encouragementsreceived from
Dr.M.S.Bhatia(LPTDBARCandHBNIFaculty,BombayUniversityPhDguide),
Dr.AbhijitBhattacharya(CnID),andDr.Aulluck(CnID)forrecognizingtherich-
ness and potential for research and development in physical science and control
systems, especially Dr. M.S. Bhatia for his guidance to carry forward this new
work for efficient nuclear power plant controls. I’m obliged to Sri A.K. Chandra,
AD-R&D-ESNPCIL,forrecognizingthepotentialofthistopicandtohaveinvited
me to present the control concepts at NPCIL R&D and to have this new control
schemedevelopedforNPCILplants.I’malsoobligedtoSriG.P.Srivatava(Direc-
tor EIG-BARC) and Sri B.B. Biswas (Head of RCnD-BARC) for their guidance
andencouragementin expandingthescope ofthisresearchanddevelopmentwith
variousuniversitiesandcolleges,andtowritethisbook.LastlyIthankSriSubrata
Dutta(RCS-RSD-BARC) andDrD Datta(HPD-BARC), mybatchmatesof1984
BARCTrainingSchool,tohaveappreciatedandrespectedthelogicinthisconcept
of fractionalcalculusand to have givenme immense moralsupportwith valuable
suggestionstocompletethiswork.
Without acknowledging the work of several scientists dealing to renew and
enrichthisparticularsubjectallovertheglobe,theworkwillremainincomplete,—
especially Dr. Ivo, Petras Department of Informatics and process control BERG
facility Technical University Kosice Slovak Republic, to have helped me to deal
with doubtsin digitizedcontrollerin fractionaldomain.I tookthe inspirationand
ix
x Acknowledgments
learnt the subject from several presentations and works of Dr. Alain Oustaloup,
CNRS-University Bordeaux, Dr. Francesco Mainardi, University Bologna Italy,
Dr. Stefan G Samko University do Algarve Portugal, Dr. Katsuyuki Nishimoto,
InstituteofAppliedMathematicsJapan,Dr.IgorPodlubnyKosiceSlovakRepublic,
Dr.Kiran,MKolwankar,andDr.AnilD.Gangal,DepartmentofPhysics,University
ofPune,India.TheeffortsofDr.CarlF.Lorenzo,GlenResearchCenterCleveland
Ohio, Dr. Tom T Hartley, University of Akron, Ohio, who has popularized this
old(new)subjectof fractionalcalculusis worthacknowledging.Appliedworkon
anomalousdiffusionbyProff.R.K.Saxena(JaiNarainVyasUniversityRajasthan)
Dr. Santanu Saha Ray, and Proff. Dr. Rasajit Kumar Bera (Heritage Institute of
TechnologyKolkata),whichisasourceofinspiration,isalsoacknowledged.Ihave
beeninspiredbytheworkonmodernfractionalcalculusinthefieldofappliedmath-
ematicsandappliedsciencebyDr.M.Caputo,Dr.RudolfGorenflo,Dr.R. Hifler,
Dr. W.G. Glockle, Dr. T.F. Nonnenmacher, Dr. R.L. Bagley, Dr. R.A. Calico,
Dr. H.M. Srivastava, Dr. R.P Agarwal, Dr. P.J. Torvik, Dr. G.E.Carlson,
Dr. C.A. Halijak, Dr. A. Carpinteri, Dr. K. Diethelm, Dr. A.M.A El-Sayed,
Dr. Yu. Luchko, Dr. S. Rogosin, Dr. K.B. Oldham, Dr. V. Kiryakova,
Dr.B.Mandelbrot,Dr.J.Spanier,Dr.Yu.N.Robotnov,Dr.K.S.Miller,Dr.B.Ross,
Dr. A. Tustin, Dr. Al-Alouni,Dr. H.W. Bode, Dr. S. Manabe, Dr. S.C. Dutta Roy,
Dr. W. Wyss; their work are stepping stone for applications of fractional calculus
forthiscentury.Iconsiderthesescientistsasfathersofmodernfractionalcalculus
ofthetwenty-firstcenturyandsalutethem.
ShantanuDas
ScientistReactorControlDivisionBARC
About the Contents of This Book
The book is organized into 10 chapters. The book aims at giving a feel of this
beautiful subject of fractional calculus to scientists and engineers and should be
taken as a start point for research in application of fractional calculus. The book
is aimed for appreciation of this fractional calculus and thus is made as applica-
tion oriented, from various science and engineering fields. Therefore, the use of
tooformalmathematicalsymbolismandmathematicalformaltheoremstatinglan-
guagearerestricted.Chapters3 and4 givean overviewoftheapplicationoffrac-
tionalcalculus,beforedealingindetailtheissuesaboutfractionaldifferintegrations
andinitialization.Thesetwochaptersdealwithalltypesofdifferentialoperations,
includingfractionaldivergenceapplicationandusageoffractionalcurl.Chapter1is
thebasicintroduction,dealingwithdevelopmentofthefractionalcalculus.Several
definitionsoffractionaldifferintegrationsandthemostpopularonesareintroduced
here;thechaptergivesthefeeloffractionaldifferentiationofsomefunctions,i.e.,
howtheylook.Toaidtheunderstanding,diagramsaregiven.Chapter2dealswith
theimportantfunctionsrelevanttofractionalcalculusbasis.Laplacetransformation
is given for each function, which are important in analytical solution. Chapter 3
givestheobservationoffractionalcalculusinphysicalsystems(likeelectrical,ther-
mal, controlsystem, etc.)description.Thischapteris madeso thatreadersgetthe
feelofreality.Chapter4isanextensionofChap.3,wheretheconceptoffractional
divergenceand curl operator is elucidated with application in nuclear reactor and
electromagnetism.Withthis,thereadergetsabroadfeelingaboutthesubject’swide
applicabilityinthefieldofscienceandengineering.Chapter5isdedicatedtoinsight
of fractionalintegrationfractionaldifferentiationand fractionaldifferintegralwith
physical and geometric meaning for these processes. In this chapter, the concept
of generating function is presented, which gives the transfer function realization
fordigitalrealizationinrealtimeapplicationofcontrols.Chapter6triestogener-
alize the concept of initialization function, which actually embeds hereditary and
historyofthefunction.Here,attemptismadetogivesomelighttodecomposition
properties of the fractional differintegration. Generalization is called as the frac-
tional calculus theory, with the initialization function which becomes the general
theoryanddoescovertheintegerorderclassical calculus.Inthischapter,the fun-
damentalfractionaldifferentialequationis taken and the impulse response to that
is obtained.Chapter7 givesthe Laplace transformtheory—a generaltreatmentto
xi
xii AbouttheContentsofThisBook
cover initialization aspects. In this chapter, the concept of w-plane on which the
fractional controlsystem properties are studied is described. In Chap. 7 elaborate
dealing is carried on for scalar initialization and vector initialization problems. In
Chaps. 6 and 7 elaborate block diagrams are given to aid the understanding of
these concepts. Chapter 8 gives the application of fractional calculus in electrical
circuits and electronic circuits. Chapter 9 deals with the application of fractional
calculusinotherfieldsofscienceandengineeringforsystemmodelingandcontrol.
Inthischapter,themodernaspectsofmultivariatecontrolsaretouchedtoshowthe
applicabilityinfractionalfeedbackcontrollersandstateobserverissues.Chapter10
givesa detailed treatmentof the orderof a system and its identificationapproach,
with concepts of fractional resonance, and ultra-damped and hyper-damped sys-
tems.Alsoabriefispresentedonfutureformalizationofresearchanddevelopment
forvariableorderdifferintegrationsandcontinuousordercontrollerthatgeneralizes
conventional control system. Bibliography gives list of important and few recent
publications,ofseveralworksonthisold(new)subject.Itisnotpossibletoinclude
alltheworkdoneonthissubjectsincepast300years.Undoubtedly,thisisanemerg-
ingareaorresearch(notsopopularatpresentinIndia),butthenextdecadewillsee
theplethoraofapplicationsbasedonthisfield.Maybethetwenty-firstcenturywill
speakthelanguageofnature,thatis,fractionalcalculus.
ShantanuDas
Contents
1 IntroductiontoFractionalCalculus ............................... 1
1.1 Introduction.................................................. 1
1.2 BirthofFractionalCalculus .................................... 1
1.3 FractionalCalculusaGeneralizationofIntegerOrderCalculus ....... 2
1.4 HistoricalDevelopmentofFractionalCalculus..................... 3
1.4.1 ThePopularDefinitionsofFractionalDerivatives/Integralsin
FractionalCalculus ..................................... 7
1.5 AboutFractionalIntegrationDerivativesandDifferintegration ....... 9
1.5.1 FractionalIntegrationRiemann–Liouville(RL).............. 9
1.5.2 FractionalDerivativesRiemann–Liouville(RL)LeftHand
Definition(LHD)....................................... 10
1.5.3 FractionalDerivativesCaputoRightHandDefinition(RHD) .. 10
1.5.4 FractionalDifferintegralsGrunwaldLetnikov(GL) .......... 12
1.5.5 CompositionandProperty ............................... 14
1.5.6 FractionalDerivativeforSomeStandardFunction ........... 15
1.6 SolutionofFractionalDifferentialEquations ...................... 16
1.7 AThoughtExperiment......................................... 16
1.8 QuotableQuotesAboutFractionalCalculus ....................... 17
1.9 ConcludingComments......................................... 18
2 FunctionsUsedinFractionalCalculus............................. 19
2.1 Introduction.................................................. 19
2.2 FunctionsfortheFractionalCalculus............................. 19
2.2.1 GammaFunction....................................... 19
2.2.2 Mittag-LefflerFunction.................................. 22
2.2.3 AgarwalFunction ...................................... 27
2.2.4 Erdelyi’sFunction ...................................... 27
2.2.5 Robotnov–HartleyFunction.............................. 27
2.2.6 Miller–RossFunction ................................... 27
2.2.7 GeneralizedRFunctionandGFunction.................... 28
2.3 ListofLaplaceandInverseLaplaceTransformsRelatedtoFractional
Calculus..................................................... 30
2.4 ConcludingComments......................................... 33
xiii