Table Of ContentLecture Notes in Electrical Engineering 257
Wojciech Mitkowski
Janusz Kacprzyk
Jerzy Baranowski
Editors
Advances in the Theory and
Applications of Non-integer
Order Systems
5th Conference on Non-integer Order Calculus
and Its Applications, Cracow, Poland
Lecture Notes in Electrical Engineering
Volume 257
Forfurthervolumes:
http://www.springer.com/series/7818
·
Wojciech Mitkowski Janusz Kacprzyk
Jerzy Baranowski
Editors
Advances in the Theory and
Applications of Non-integer
Order Systems
5th Conference on Non-integer Order Calculus
and Its Applications, Cracow, Poland
ABC
Editors
WojciechMitkowski JerzyBaranowski
DepartmentofAutomaticsand DepartmentofAutomaticsand
BiomedicalEngineering BiomedicalEngineering
FacultyofElectricalEngineering, FacultyofElectricalEngineering,
Automatics,ComputerScience Automatics,ComputerScience
andBiomedicalEngineering andBiomedicalEngineering
AGHUniversityofScienceandTechnology AGHUniversityofScienceandTechnology
Cracow Cracow
Poland Poland
JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
Warsaw
Poland
and
PIAP–IndustrialInstituteofAutomationand
Measurements
Warsaw
Poland
ISSN1876-1100 ISSN1876-1119 (electronic)
ISBN978-3-319-00932-2 ISBN978-3-319-00933-9 (eBook)
DOI10.1007/978-3-319-00933-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013939744
(cid:2)c SpringerInternationalPublishingSwitzerland2013
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Preface
It can be said without exaggeration that modeling is what science has been all about
sinceitsearlydevelopmentstagesintheantiquity.Whenproperformaltoolsandtech-
niqueshavebecomeknownandoperational,alogicalconsequencehasbeentheemer-
gence of broadly perceived mathematical modeling that have dealt with all kinds of
systemsandprocesses,fromsimpleonesconcerningjustanaspectortwotothemost
complex ones exemplified by socio-economicor biologicalsystems and processes. It
hasbecomeclearthatmathematicalmodelingcanbeveryusefulforsolvingvirtuallyall
realworldproblemsprovidedthatanadequatemathematicalmodelcanbedeveloped,
identifiedandfinallyimplemented.A questforbetterandbettermodelshastherefore
occurred in all fields of science and technology, and a wide array of various model
classeshasbeenproposedbyscholarsandresearchersallovertheworld,followedby
deeptheoreticalanalysesandapplications.
Thisvolumeisconcernedwithvariousaspectsofaveryinterestingclassofmodels,
the so-called non-integer systems, also known as fractional systems, which have re-
centlyattractedanincreasingattentioninthescientificcommunityofsystemsscience,
appliedmathematics,controltheory,etc.whohasshownthatsuchmodelscanbeboth
theoreticallychallengingandrelevantformanyfieldsofscienceandtechnologyexem-
plified by the modeling of signal transmission, electric noise, dielectric polarization,
heattransfer,electrochemicalreactions,thermalprocesses,acoustics,etc.
As with many relevant concepts in modern science, the very roots of the concept
of a non-integerorder system, otherwise known as a fractionalsystem, can be traced
at least to the 1960s. One of the earliest remarkson the idea of a non-integerderiva-
tive, which is a basic underlyingelement of a non-integersystem, obviouslytogether
with the related concept of a non-integer integration, can be found in a letter from
Leibniz to de l'Hôpital (also spelled as de l’Hospital) dated August 3,1695 in which
Leibnizgaveafirstanswertoaquestionposedbydel'Hôpitalaboutthemeaningofa
non-integerderivative,especiallyinthecaseof1/2.Sincethenfractionalcalculushad
beenafieldofactiveresearchofsuchgreatthinkersandmathematiciansasBernoulli,
Euler,Lagrange,tonamethemostprominentonesonly.Probablythefirstdetaileddef-
inition of a fractional derivative was stated in 1812 by Laplace in his book “Théorie
analytiquedesprobabilités”.Ingeneral,onecancertainlyclaimthatmostofthegreat
VI Preface
mathematicians have had some contact with this area at one time or another as they
had well been aware of theoretical challenges and importance for the modeling of a
multitudeofrealworldproblems.
Therecentyears,orbettertosaydecades,havewitnessedahugeincreaseofinterest
and research activities in the area of non-integer order systems. The main reason is
obviouslythe fact that they have provedtheir power as an effective and efficient tool
for the modeling of various systems processes and systems in the areas of physics,
chemistry,biology,electricalengineering,etc.However,inadditiontothatapplication
motivatedinterestthathasoccurredmostlyinsciencesandengineering,muchprogress
hasalsobeendoneintheoreticalandnumericalanalyses,mostlyinthefieldofapplied
andnumericalmathematics.
Thisvolumeisconcernedwithnoveladvancesinboththetheoryandapplicationsof
non-integerorder systems. The main focusis on controland biomedicalengineering,
andsystemstheory.Thesefieldsarebroadlyperceivedandthereforetheresultpresented
inthisvolumecanbeofusetomanyfieldsofscienceandtechnology.
Thebookisdividedintosixpartscoveringtheareasthatmaycertainlybeconsidered
tobeofamaininterestofthescientificcommunity,andoftoprelevancebothfromthe
pointofviewoftheoreticalchallengesandapplications.
Part 1 is concerned with the problem of realization. This problem, which is well
knownfromthetheoryofstandarddynamicalsystems,becomesfascinatingandchal-
lenginginthecontextoffractionalsystems.
TadeuszKaczorek(Realizationproblemfordescriptorpositivefractionalcontinuous-
timelinearsystems)considerstherealizationproblemfordescriptorpositivefractional
continuoustimelinearsystemswithregularpencils.Hepresentsconditionsfortheexis-
tenceofpositiverealizationsofthedescriptorfractionalsystemsalongwithprocedures
forthecomputationoftherealizationsofimpropertransfermatrices.Hethendemon-
strateseffectivenessoftheproposedproceduresonnumericalexamples.
ŁukaszSajewski(Positivestableminimalrealizationoffractionaldiscretetimelinear
systems)analysesthepositivestableminimalrealizationproblemforfractionaldiscrete
timelinearsystemsandproposesamethodforfindingapositivestableminimalrealiza-
tionofa givenpropertransfermatrix.Healso establishessufficientconditionsforthe
existenceofapositivestableminimalrealizationofthisclassoflinearsystems.
Part2concernstheproblemofstabilityofthenon-integerordersystems.Thisarea
isalsoveryinterestingsinceeveninsimplesystemsconditionsontheeigenvaluesand
thespeedofreachingzeroissubstantiallydifferentthanwiththeintegerordersystems.
Mikołaj Busłowicz (Frequency domain method for stability analysis of linear
continuous-time state-space systems with double fractional orders) considers the sta-
bility problemof continuous-timelinearsystems describedby the state equationwith
doublefractionalorders.Hepresentsafrequencydomainmethodforstabilitychecking
ofthesystemwithcommensurateornon-commensurateorders.Themethodproposed
is based on the Argument Principle. The considerations are illustrated by numerical
examples.
Malgorzata Wyrwas, Ewa Girejko, Dorota Mozyrska, Ewa Pawluszewicz (Stability
offractionaldifferencesystemswithtwoorders)studythestabilityofnonlinearsystems
with the Caputo fractional difference with two orders. They use the Lyapunov direct
Preface VII
methodtoanalyzethestabilityofthesystem,andpresentsufficientconditionsforthe
uniformstabilityandtheuniformasymptoticstability.
Andrzej Ruszewski (Stability conditions of fractional discrete-time scalar systems
withtwodelays)considersthestabilityproblemsoffractionaldiscrete-timelinearscalar
systemswithtwodelays.UsingtheclassicalD-partitionmethod,theauthordetermines
the boundaries of the stability regions in the parameter space. Based on the stability
regions, new conditions for the practical stability and for the asymptotic stability are
given.
Part 3 is focused on more advanced concepts such as controllability,observability
andoptimalcontrol,andtheirrelatedtoolsandtechniques.
EwaPawłuszewiczandDorotaMozyrska(Constrainedcontrollabilityofh-difference
linear systems with two fractional orders ) study the problem of controllability in a
finitenumberofstepswithcontrolconstrainsofh-differencelinearcontrolsystemswith
twofractionalorders.TheauthorsconsidersystemswiththeCaputotypeh-difference
operatorsandwithcontrolsthevaluesofwhicharefromagivenconvexandbounded
subsetofthecontrolspace.Thenecessaryandsufficientconditionsfortheconstrained
controllabilityinafinitenumberofstepsaregiven.
Wojciech Trzasko (Observabilityof positivefractional-orderdiscrete-timesystems)
considers the positive linear discrete-time fractional-order (non-commensurate and
commensurate order) systems described in the state space. He provides a definition
andprovesthenecessaryandsufficientconditionsforthepositivityobservability.The
considerationsareillustratedbyanumericalexample.
AndrzejDzielin´skiandPrzemysławM.Czyronis(Optimalcontrolproblemforfrac-
tional dynamic systems - linear quadratic discrete-time case) formulate and solve an
optimalcontrolproblemwithafixedfinaltimeforthefractionaldiscrete-timesystems
withthequadraticperformanceindex.Theyconsiderthecasesofafreefinalstatewith
afixedfinaltimeandpresentanewmethodforthenumericalcomputationofsolution
oftheoptimalcontrolproblemformulated.
Part4dealtwiththeareaofdistributedsystems.Thesesystemsarechallengingwith
integerorderderivativesbuttheiranalysisinthecontextoffractionalderivativesgives
anadditionalinsight.
Piotr Grabowski (Stabilization of wave equation using standard/fractional deriva-
tiveinboundarydamping)discusestheproblemofstabilizationofawaveequationby
meansofthestandardorfractionalderivativeinboundarydamping.Theproblemisbe-
ingreducedtoaselectionbetweentheproportionalorfractionalintegratoroforder1-α
feedbackcontrollers.Thefractionalintegrationleadstothestrongasymptoticstability
only,whiletheproportionalfeedbackcontrolcanensuretheexponentialstability.This
meansthatexponentialstabilityisnotrobustaroundthevalueα=1.
YuriyPovstenko(FundamentalSolutionstotheCentralSymmetricSpace-TimeFrac-
tional Heat Conduction Equation and Associated Thermal Stresses) investigates the
space-timefractionalheatconductionequationwiththeCaputotimefractionalderiva-
tiveandtheRieszfractionalLaplaceoperator.ThefundamentalsolutionstotheCauchy
and source problems as well as associated thermal stresses are found in the case of
thesphericalsymmetry.Hepresentsnumericalresultsforthetemperatureandstresses
graphicallyforvariousordersofspaceandtimederivatives.
VIII Preface
Tatiana Odzijewicz (Variable Order Fractional Isoperimetric Problem of Several
Variables)studiesthreetypesofpartialvariableorderfractionaloperators.Usinginte-
grationbypartsformulasforvariableorderfractionalintegrals,thenecessaryoptimality
conditionofthe Euler–Lagrangetypefor themultidimensionalisoperimetricproblem
areproved.
BartłomiejDybiec(Mittag-Lefflerpatterninanomalousdiffusion)analyzesvarious
systems described by the bi-fractional Fokker-Planck-Smoluchowskiequation which
displaysomeverygeneralanduniversalproperties.Theseuniversalcharacteristicsorig-
inate in the underlying competition between long jumps (fractional space derivative)
andlongwaitingtimes(fractionaltimederivative).Usingafewselectedmodelexam-
plestheauthordemonstratestheuniversalfeaturesofanomalousdiffusion.
Part 5 concerns the problems of solution and approximation of noninteger order
equationsofcertaintypes.
StefanDomek(Piecewiseaffinerepresentationofdiscreteintime,non-integerorder
systems)considersa multi-modelapproachwhichhasbeenoftenusedforthe model-
ing and controlof physicalprocesses in recent years leading to the class of so-called
switchedsystems.Theirproperties,particularlythestability,observabilityandcontrol-
labilityanalyses,havebecometopicsofintensiveresearchtopicsincontroltheoryand
its applications. He proposes a method of modeling nonlinear, discrete in time, non-
integer order systems by means of piecewise affine multimodels, and then describes
special cases of such models. The discussion is illustrated with results of simulation
tests.
MarekBłasikandMałgorzataKlimek(ExactSolutionofTwo-TermNonlinearFrac-
tional Differential Equation with Sequential Riemann-Liouville Derivatives) derive a
generalsolutionforaclassofnonlinearsequentialfractionaldifferentialequationswith
theRiemann-Liouvillederivativesofanarbitraryorder.Thesolutionofsuchanequa-
tionexistsinanarbitraryinterval(0;b]providedthenonlineartermobeystherespective
Lipschitzcondition.Theauthorsprovethateachpairofstationaryfunctionsofthecor-
responding Riemann-Liouville derivatives leads to a unique solution in the weighted
continuousfunctionsspace.
PiotrBaniaandJerzyBaranowski(Laguerrepolynomialapproximationoffractional
order linear systems) present a finite dimensional approximation of fractional order
linearsystemsanditsconnectionwiththetransportequation.Theirmainresultsshow
thatthelinearfractionalordersystemcanbeapproximatedbyafinitenumberoflinear
differentialequations.Thediscussionisillustratedonasimpleexampleofafractional
oscillator.
EwaGirejko,DorotaMozyrskaandMałgorzataWyrwas(Solutionsofsystemswith
two-termsfractionaldifferenceoperators)discusssystemswithgeneralizedtwo-terms
fractionaldifferenceoperators.Bythechoiceofacertainkernel,theseoperatorscanbe
reducedtothestandardfractionalintegralsandderivatives.Theystudytheexistenceof
solutionstosuchsystems.
DorotaMozyrska,EwaGirejkoandMałgorzataWyrwas(Comparisonofh-difference
fractionaloperators)comparethreedifferenttypesofh-differencefractionaloperators:
theGrünwald-Letnikov,CaputoandRiemann-Liouvilleones.Theauthorsintroducea
formulaforthefundamentalmatrixofsolutionsforlinearsystemsofthe h-difference
Preface IX
fractionalequationswiththeGrünwald-Letnikovtypeoperatorwhiletheonewiththe
CaputotypeortheRiemann-Liouvilletypeiswellknown.Theypresentnewformulas
forlinearcontrolsystemswiththeoperatorsasmentionedabove.
Part6,thefinalone,isfocusedondifferentkindsofapplications.Variousproblems
fromthefieldsofvariationalcalculus,throughprocessmodelingtocontrolalgorithms
areconsidered.
Małorzata Klimek and Maria Lupa (Reflection Symmetry in Fractional Calculus -
Propertiesand Applications)define the Riesz type derivativesthat are symmetric and
anti-symmetric with respect to the reflection mapping in a finite interval [a; b]. The
authorsprovetherepresentationandintegrationformulasforthefractionalsymmetric
and anti-symmetric integrals and derivatives introduced. It appears that they can be
reducedtooperatorsdeterminedinarbitrarilyshortsubintervals[am;bm].Theauthors
discussfutureapplicationinthereflectionsymmetricfractionalvariationalcalculusand
ageneralizationofpreviousresultsonthelocalisationoftheEuler-Lagrangeequations.
Paweł Skruch (A General Fractional-Order Thermal Model for Buildings and Its
Properties) presents a general model of the temperature dynamics in buildings. His
modeling approach relies on the principles of thermodynamics, in particular of heat
transfer. The model considers heat losses by conduction and ventilation and internal
heatgains.Theparametersofthemodelcanbeuniquelydeterminedfromthegeome-
try of the building and thermalpropertiesof the constructionmaterials. The modelis
describedbyfractional-orderdifferentialequationsandispresentedusingastatespace
notation.Thestabilitypropertyofthemodelisconsideredandprovidesanillustrative
exampleisprovided.
Anna Obra˛czka and Jakub Kowalski (Heat transfer modeling in ceramic materials
usingfractionalorderequations)observethattheuse foclassic numericalmethodsin
themodelingofheattransferinceramicmaterialscausesimpreciseresults.Theirpaper
presents a new way of modeling using the fractional order equations. The numerical
results obtained are compared with the registered heat transfer distribution using an
infraredcamera.Acomparisonshowsthatthepresentedmethodyieldsamuchhigher
accuracy.
Adam Pilat (A comparative study of PIαDμ controller approximationsexemplified
by Active Magnetic Levitation System) examines the PIαDμ discrete fractional order
controllerappliedtotheActiveMagneticLevitationSystem.Hisresearchisbasedon
ProfessorIvoPetras’ Toolboxforfractionalcontrollersynthesis. Thepointofinterest
is the PID controllerconfigurationapplied at the simulation and experimentalstages.
The search for an optimal controller form is dependent upon the quality measure in
the transition phase when the external excitation load is activated. A digital control
experiment was carried out in the MATLAB/Simulink using a USB I/O board. The
controllerrealisationsarecomparedanddiscussed.
Ewa Szymanek(The applicationof fractionalorderdifferentialcalculusforthe de-
scriptionoftemperatureprofilesinagranularlayer)presentsresultsofanactualexper-
imentontheflowofairthroughabulkheadfilledwithagranularmaterial.Theauthor
comparesthedeterminedtemperatureprofilesinthediscussedbulkheadatdifferentex-
ternal and internal temperatures to a numerical description based on fractional order
differentialcalculus.
X Preface
WojciechMitkowskiandKrzysztofOprze˛dkiewicz(Fractional-orderP2Dβ controller
for uncertain parameter DC motor) consider an uncertain parameter DC motor con-
trolledwith theuseofa non-integerorderP2Dβ controllerwithuncertainparameters.
TheauthorsperformananalysisoftheBIBO(BoundedInputBoundedOutput)stability
withrespecttouncertaintyofplantparametersforthissystem.
MikołajBusłowiczandAdamMakarewicz(SynchronizationofthechaoticIkedasys-
tems of fractional order) consider the problem of synchronisation of two fractional
Ikeda delay systems via a master/slave configuration with a linear coupling. The au-
thorsinvestigateeffectsofthefractionalorderandthecouplingrateonsynchronization
usingnumericalsimulationsperformedusingtheNon-integerFractionalControlTool-
boxforMatLab.
Dominik Sierociuk, MichalMacias andWiktor Malesza (Analogmodelingof frac-
tionalswitched-orderderivatives:experimentalapproach)presentexperimentalresults
ofthemodelingofswitched-orderintegratorsbasedondominoladderapproximations
oforder0.5and0.25.Theresultswereobtainedbyincreasinganddecreasingthefrac-
tional order.The quarter-orderimpedancewas implementedusing over 5000 discrete
elements.Theexperimentalcircuitsarebasedonaswitchingschemethatisnumerically
identicaltothesecondordertypeoffractionalvariableorderderivative.Experimental
resultswereanalysedandcomparedwithnumericalresults.
PawełSkruchandWojciechMitkowski(Fractional-OrderModelsoftheUltracapac-
itors) investigate and analyze the dynamic behavior of the ultracapacitors. The ultra-
capacitors are represented by equivalentelectrical circuit models and mathematically
describedbyfractional-orderdifferentialequations.Theauthorsproposeaprocedureto
identifyparametersofthemodels.Theresultsofnumericalsimulationsarecompared
withthosemeasuredexperimentallyinthephysicalsystem.
Waldemar Bauer, Jerzy Baranowski and Wojciech Mitowski (Non-integer order
PIαDμ controlICU-MM) present a dynamicalsystem model that describes glycemia.
Itisbasedonfourdifferentialequationsthatsimulateglucosedynamicsoftraumatised
patient’sblood(atanIntensiveCareUnit).Theauthorspresentthedescriptionofabasic
modelandamethodoftuningthePIαDμ controllerparametersbasedontheintegrated
absoluteerrorastheperformanceindex.
Tomasz Moszkowski and Elzbieta Pociask (Comparisonof Fractional- and Integer-
orderFiltersinFiltrationofMyoelectricActivityAcquiredfromBicepsBrachii)exam-
ine the viability of filtration of a myoelectricsignalusing fractionalorderfilters. The
authorsacquirearawEMGsignalfromm.bicepsbrachiiduringanisometricmaximal
voluntary contraction from ten test subjects, then test the conventionaland fractional
Butterworthfiltersoftwoordergroups,andfinallycomparetheresultsintermsofof-
flinefiltration.
Piotr Duch, Maciej Łaski, Sylwester Błaszczyk, Piotr Ostalczyk (Variable-,
Fractional-OrderDead-Beat Control of a Robot Arm) propose a synthesis method of
the variable-, fractional – order dead–beat controller. It is applied to the control of a
robot arm described as a simple integrating element. They measure and compare the
transient characteristic of a closed-loop system using the proposedcontroller and the
classicalcontrollers.
