Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland PDF
Preview 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 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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown orhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnectionwithreviews orscholarly analysis ormaterial suppliedspecifically forthepurposeofbeingentered andexecuted ona computersystem,forexclusive usebythepurchaser ofthework.Duplication ofthis publication orparts thereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation,initscur- rentversion,andpermissionforusemustalways beobtained fromSpringer. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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.
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