Time Delay Systems: Methods, Applications and New Trends

442 Pages·2012·6.97 MB·english

by  Vladimir L. Kharitonov (auth.)

Lecture Notes
in Control and Information Sciences 423
Editors:M.Thoma,F.Allgöwer,M.Morari
Rifat Sipahi, Tomáš Vyhlídal,
Silviu-Iulian Niculescu,
and Pierdomenico Pepe (Eds.)
Time Delay Systems:
Methods, Applications
and New Trends
ABC
SeriesAdvisoryBoard
P.Fleming,P.Kokotovic,
A.B.Kurzhanski,H.Kwakernaak,
A.Rantzer,J.N.Tsitsiklis
Editors
Prof.RifatSipahi Prof.Silviu-IulianNiculescu
NortheasternUniversity L2S(UMRCNRS8506)
DepartmentofMechanicaland CNRSSupélec
IndustrialEngineering Gif-sur-Yvette
Boston,USA France
Prof.TomášVyhlídal Prof.PierdomenicoPepe
CzechTechnicalUniversity UniversityofL’Aquila
FacultyofMechanicalEngineering DepartmentofElectricaland
DepartmentofInstrumentationand InformationEngineering
ControlEngineering L’Aquila,Italy
Prague,CzechRepublic
ISBN978-3-642-25220-4 e-ISBN978-3-642-25221-1
DOI10.1007/978-3-642-25221-1
LectureNotesinControlandInformationSciences ISSN0170-8643
LibraryofCongressControlNumber:2011940774
(cid:2)c 2012Springer-VerlagBerlinHeidelberg
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Preface
Inmanyproblemsinengineering,physics,andbiology,variousprocessestakeeffect
onlyafteracertainamountoftimeelapsesaftertheonsetofastimulus,aninput,and
anymeansofcause.Thisperiodoftime,whichisapplicationspecific,canarisedue
tomanyreasonsincluding,amongothers,transmittinginformationacrossawireless
networkinnetworkcontrolsystems,shippingproductsfromonelocationtoanother
insupplychains,producingadecisionincardrivinguponreceivingastimulus.
Duetothepresenceofdelays,instantaneousinformationcannotbeavailableand
therefore many control actions are produced based on the historical evolution of
the governing dynamics. This represents a major source of instability, especially
in cases whenthe controllersare designedbyneglectingdelays.In otherwords,a
controllerdesignedinaconventionalwaybyassumingthatthesystemathanddid
notpresentdelays, maynotnecessarilyguaranteethe stability ofthe same system
thatactuallyincludesdelays.
Since 1950s, many studies have been devoted to understanding the effects of
delaysandexploringcontroltheoreticapproachestocircumventthedetrimentalef-
fects of delays to dynamicalbehavior. One of the first tools to dealwith delaysis
the celebratedSmith predictor,dated 1957.On the otherhand,an interestingphe-
nomenondiscoveredinthesestudiesistheso-calledstabilizingeffectofdelays:in-
ducingdelaysinthecontrolloopmayleadtostableclosed-loopsystemwhichcan
belostwhenthedelaybecomeszero.Suchastabilizingfeatureofdelayindynami-
calsystemsworksagainstthehumanintuitionandthereforenecessitatesanalytical
studies. Consequently,this featurehas becomethe motivationof manytheoretical
andexperimentalstudiesintheliterature,especiallyinthepasttwodecades.
Duringthisperiod,manyresultshavebeenpresentedatthemaincontrolconfer-
ences(IEEECDC, ACC, IFACsymposia),inspecializedworkshops(IFACTime-
DelaySystemsseries),andpublishedintheleadingjournalsofcontrolengineering,
manufacturingengineering,operationsresearch,mathematicalbiology,systemsand
controltheory,appliedandnumericalmathematics.Furthermore,numerousbooks
onthetopicappeared,andtworeviewarticleswerepublishedquiterecentlyinIEEE
ControlSystemsMagazine.Inthecontrolarea,theresultsareprimarilybasedonthe
analysisofsystemswithdelays(suchasstability,controllability,performance),and
VI Preface
on the synthesisof controllers(designingan appropriatecontrollaw thatachieves
certaincontrolobjectives).Forvariousspecificclassesofmodels,includinglinear,
nonlinear,andstochasticsystems,manyresultshavebeenobtained,whileinthenu-
mericalanalysisareatheresearchhasfocusedontime-stepping,discretizationand
semi-discretization,computationofstabilityandcorrespondingstabilitycharts,and
bifurcationanalysis.
Although many results are mature, the existing work has so far remained valid
for stand-alonesystems thathave relativelysmall dimensionsand a small number
ofdelays.Ontheotherhand,thechallengesfacedbytheresearchersisincreasing,
in parallel with the rapidly developing technology and higher expectations from
researchersandcontrolengineerpractitioners.Themainchallengesareduetohan-
dling large-scale systems, nonlinear systems, discrete-continuoustype mixed sys-
tems,interconnectedsystemsovernetworks,multi-inputmulti-outputsystemswith
delays of various features such as time-varying, uncertain, and stochastic delays.
In the analysis and controller synthesis of such systems, there still remains many
unchartedresearchterritorieswherenewresultsandmajorimpactsareneeded.
The arising complexities in the control problems are also in parallel with the
versatilityofsystemsalignedwiththeemergenceofmultipledisciplines,including
engineering,physics,chemistry,biology,operationsresearch,andeconomics.This
trendalsocallsforresearchersfromdifferentdisciplinestoworktogether,inorder
to better understand and address control problems. Therefore, it is of no surprise
todaythatcontrolengineers,medicaldoctors,chemicalengineers,andmathemati-
cians work together in defining new frameworks for dealing with the complexity
of the systems to be handled. In one sense, the research on time delay systems
is expandingand becomingmore pervasive,and the publicationsin this field will
growevenfurtherandwillbeaccessedbymanyresearchersincludingthoseoutside
engineering.
This bookpresentsthemostrecenttrendsas wellas newdirectionsin the field
of control and dynamics of time delay systems. The field is extremely active and
the book captures the most recent snapshot of the research results. The book is
collectedunderfiveparts:(i)Methodology:FromRetardedtoNeutralContinuous
DelayModels,(ii)Systems,SignalsandApplications,(iii)NumericalMethods,(iv)
Predictor-based Control and Compensation, and (v) Networked Control Systems
andMulti-agentSystems.Thethemesofthesechapterscloselytiewiththediscus-
sionsandmotivationsabove.Thecontributionsinthechaptersareawell-balanced
combinationoftwomainresources;invitedpapers,andtheworkpresentedat2009
IFAC Workshop on Time Delay Systems held in Sinaia, Romania and 2010 IFAC
WorkshoponTime DelaySystemsheldinPrague,CzechRepublic.Thisworkshop
represents the main specialized meeting venue in the field, and thus captures the
mostupdatedresearchtrends.Intheselectionofthetopicsandthecontributors,the
editorshavenotonlyaimedatmaintainingthehighesttechnicalqualityofthepre-
sentations,butalsoatachievinganappropriatebalanceacrossthe chapters(i)-(v).
It is worthy to note that the book proposal does not have significant overlap with
thecontentsof2009-2010IFACWorkshopsonTimeDelaySystems(IFACTDS).
Attendees of the workshopcontributingto this book proposalalso improvedtheir
Preface VII
contributionsbeyondwhattheypresentedattheworkshopandtheeditorscomfort-
ablyclaimthatatleast50%ofthebookcontent-wisecomprisesnewcontributions
thatarecloselyrelevanttothemostrecenttrendsdiscussedabove.
StructureoftheVolume
Asmentionedearlier,thisvolumeisdividedintofiveparts,whereeachpartiscom-
posedoffivetoeightchapters.TheyarerespectivelydevotedtoPartI:Methodology:
FromRetardedtoNeutralContinuousDelayModels,PartII:Systems,Signalsand
Applications, Part III: Numerical Methods, Part IV: Predictor-based Control and
Compensation, and Part V: Networked Control Systems and Multi-agent Systems.
Thefirstpartis concernedwiththe newresultsonretardedandneutraltypedelay
differentialequations. The second part is devoted to problems with flows, signals
andthe correspondingreal-worldapplications.Thethirdpartisrelatedto numeri-
calmethods,whichare indispensablepartofthe researchmomentuminthisfield.
The fourth and the fifth parts are concerned with the recent trends, which cover
respectively predictor-based control/compensationand networked control systems
withmultiagents.Inwhatfollows,wepresentthedifferentchaptersandstreamsof
thedifferentparts.
PartIMethodology:FromRetardedtoNeutralContinuousDelayModels
Almost all continuous dynamical systems with delays are modeled by either re-
tardedorneutral-typedelaydifferentialequations.Itisthereforeextremelyimpor-
tantto developappropriateanalysistoolsand methods,and synthesisschemesfor
suchdifferentialequations.Inthisresearchdirection,manyproblemsarestillopen,
including,amongothers,improvementsoncomputationalefficiency,controlofnon-
linear systems using observers, new ideas for control system design, connections
between frequency and time-domain tools, new ways in constructing Lyapunov
quadratic functionalsthat are not necessarily continuous.In this part of the book,
someoftheseimportantproblemsarediscussedandaddressed.
ThefirstchapterinthispartiscontributedbyVLADIMIR L. KHARITONOV and
presentsnewresultsincomputingquadraticfunctionalsforlinearneutral-typesys-
tems. Particularemphasishere is on the Lyapunovmatrices, which are definedby
special matrix valued functions. The focus on such matrices is important as this
efforttieswithstabilityanalysisandrobuststabilityanalysisoftimedelaysystems.
ThesecondchapteriscontributedbyMICHAELDILORETOandJEANJACQUES
LOISEAUanditdiscussesthestabilityofpositivedifferenceequations,whicharise
inmanystabilityproblemsassociatedwithneutral-typedelaydifferentialequations.
Thechaptershowsthatthestabilityoflineardifferenceequationswithpositiveco-
efficientsisrobustwithrespecttodelays.Thisresultiscrucialsinceotherwisethe
stabilityanalysisofdifferenceequationswithmultipledelayscanbecomputation-
allycumbersome.Theresultsupplementsacriticalstabilityanalysisstepinneutral
typesystems.
ThethirdchaptercontributedbyKEQIN GU,YASHUN ZHANG,andMATTHEW
PEET concerns the positivity of quadratic functionals, which are key components
VIII Preface
intime-domainbasedstabilityanalysis.Undersingleanddoubleintegralpositivity
conditions, the chapter utilizes appropriate notions from operator theory to show
the positivity of quadratic functionals, and studies coupled differential-difference
equationsusingefficientsum-of-squarestools.
The fourth chapter, which is contributed by MILENA ANGUELOVA and
MIROSLAV HALA´S, presents a class of retarded-type nonlinear systems that ad-
mitaninput-outputrepresentationofneutraltype.Interestingly,therepresentation
becomespossibleonlywithnonlinearitiespresentinthedynamicalsystem,andthe
authorspresentconditionsunderwhichsucharepresentationexists.
The fifth chapter is contributedby GILBERTO OCHOA, SABINE MONDIE´, and
VLADIMIR L. KHARITONOV. The chapterstudiesan importantproblemin linear
neutral-typesystems,wherethecomputationofcriticaleigenvalueshasdirectlinks
withsystemstability.Differentfromexistingresults,theauthorsconsiderLyapunov
matrixinordertocomputesucheigenvalues,whicharethenusedtofindthedelay
intervalsinwhichthesystemmaintainsitsasymptoticstability.
Thelastchapter,contributedbyROB H. GIELEN, MIRCEA LAZARand SORIN
OLARU, focusses on the relation between stability of delay difference equations
(DDEs) and the existenceof D-contractivesets, which providea regionof attrac-
tion. First, it is established that a DDE admits a D-contractive set if and only if
it admits a Lyapunov-Razumikhinfunction.This and subsequently derivedneces-
saryconditionsprovideafirststeptowardsthederivationofanotionofasymptotic
stabilityforDDEswhichisequivalenttotheexistenceofaD-contractiveset.
PartIISystems,SignalsandApplications
This part presents an array of contributions that encompass system-level applica-
tions to various real-world control problems, in which time delays affect the in-
volvedsignals.Theorganizationofthischapterisasfollows.
ThefirstchapterinthispartiscontributedbySERGEI AVDONIN andLUCIANO
PANDOLFI.Theauthorsfocusontherelationbetweentemperatureandfluxforheat
equationswithmemory.First,itisprovedthattherelationbetweentemperatureand
heat governingequationsare strict for small times, but these quantities are essen-
tiallyindependentforlargetimes.Intheproblemanalysis,theobservationthatthe
independencecanbeinterpretedasakindofcontrollabilityisutilized.
Thesecondchapter,contributedbyLOTFI BELKOURA,investigatestheidentifi-
abilityandalgebraicidentificationoftime-delaysystems.Itisshownthattheiden-
tifiabilitypropertyofageneralclassofsystemsdescribedbyconvolutionequations
can be formulatedin terms of approximate controllabilityor weak controllability,
dependingontheavailablemodels.Consequently,analgebraicmethodfortheiden-
tificationofdelaysystemsbasedonbothstructuredinputsandarbitraryinput-output
trajectoriesispresented.Theproposedon-linealgorithmsforbothparametersand
delayestimationarevalidatedonboththesimulationandexperimentalstudieswith
noisydata.
ThethirdchapteriscontributedbyRUDY CEPEDA-GOMEZandNEJATOLGAC.
Thestudyaddressestheconsensusproblemforagroupofautonomousagentswith
secondorderdynamicsandtime-delayedcommunications.Aconsensusprotocolfor
Preface IX
asystemofagentswithsecond-orderdynamicsisproposed,undertheassumptions
thatallagentscancommunicatewitheachotherandthereisaconstantcommunica-
tiondelayforall.UsingtheClusterTreatmentofCharacteristicRoots(CTCR)pro-
cedure,acompletestabilitypictureisobtained,takingintoaccountthevariationsin
thecontrolparametersandthecommunicationdelay.Casestudiesandsimulations
arepresentedtoillustratetheanalyticalderivations.
The fourth chapter,contributed by ERIK I. VERRIEST, focuseson the analysis
of state-space construction for systems with time-varying delay. The fundamental
conditionforthe existenceofthe state-spaceisthatthedelayderivativeshouldbe
boundedbyone.Besides,usingadiscretizationapproach,simplederivationofthe
spectralreachabilityconditionforlineartime-invariant(LTI)delaysystemsispro-
vided.Itisalsoshownthatwhenasystemwithfixeddelayismodeledasoneina
classwithlargerdelay,reachabilitycannolongerbepreserved.
The fifth chapter of the part is contributedby VLADIMIR RAˇSVAN, which dis-
cusses some dynamical models in automatic control that are connected with dis-
tributed parameters in one dimension. These models are described by boundary
valueproblemsforhyperbolicpartialdifferentialequations(PDEs).Thefunctional
equationsassociatedtotheseproblemsareconsideredbyusingtheintegrationofthe
Riemanninvariantsalongthecharacteristics.Thechaptercontainssomeillustrating
applicationsfromvariousfields:nuclearreactorswithcirculatingfuel,canalflows
control, overheadcrane, drilling devices, without forgettingthe standard classical
exampleof the nonhomogeneoustransmission lines for distortionlessand lossless
propagation. Specific features of the control models are discussed in connection
withthecontrolapproachwhereveritapplies.
ThesixthchapteriscontributedbyTAMASINSPERGER,RICHARD WOHLFART,
JANOS TURI and GABOR STEPAN. The authors study the dynamics of the stick
balancing control where the output for the feedback controller is provided by an
accelerometerattachedtothestick.Intheanalysis,differentmodelsareconsidered
intheloopwithproportional-derivativecontroller.Thestabilityofthefeedbacksys-
temisstudiedforthecaseswithandwithoutfeedbackdelays.Besides,theeffectof
controller discretization is studied. It is shown that if the accelerometer dynamics
isconsideredinthemodelwithfeedbackdelay,theadvancedclosedloopdynamics
(withinfinitelymanyunstablepoles)isobtaiend.Oncethecontrollerisdiscretized,
thesystemcanbestabilizeddespiteitsadvancednature.
The seventh chapter, contributed by LOUAY SALEH, PHILIPPE CHEVREL and
JEAN-FRANC¸OIS LAFAY, is dedicated to the study the characteristics of the op-
timalpreviewcontrolforlateralsteeringofapassengervehicle,providedthenear
futurecurvatureoftheroadisknown.Thesynthesisisperformedincontinuoustime
andleadstoatwo-degreesoffreedomfeedbackandfeedforwardcontroller,whose
feedforwardpartisafiniteimpulseresponsefilter.Boththetheoreticalandexperi-
mentalresultsshowthatthepreviewcontrolactionenablesthevehicletotrackthe
centerofthelanewithasmallertrackingerror.
In the last, the eighth chapter of this part, HITAY O¨ZBAY, CATHERINE BON-
NET, HOUDA BENJELLOUN and JEAN CLAIRAMBAULT study local asymptotic
stability conditions for the positive equilibrium of a system modeling cell
X Preface
dynamicsinleukemia.Localasymptoticstabilityconditionsarederivedforthepos-
itive equilibriumpoint of this nonlinear model. As the main result, guidelines for
developmentof therapeutic actions are proposed based on the stability conditions
derivedintermsofinequalities.
PartIIINumericalMethods
Roughlyspeaking,duetothemanycomputationalchallengesinvolvedintimedelay
systems,itisofnosurprisetodaythattheneedandthedemandforreliablenumer-
ical tools is always a focal point, with more and more versatility expected from
these tools. In this line of research,this partpresentsoptimalcontroldesign tech-
niques,discretizationofsolutionoperators,applicationstodistributeddelays,tools
fordelay-independentstabilitytest,analysisofnon-uniformlysampledsystems,nu-
mericalmethodsforconstructingLyapunovmatricesandanalyzingthestabilityof
systemsfoundinbiology.Thedetailsofthechaptersareasfollows.
ThefirstchapterofthispartcontributedbyWIMMICHIELSprovidesanoverview
oftheeigenvalue-basedrobustcontroldesignmethodsforlineartimedelaysystems
with the fixed-order controllers. The analysis concerns the computation of stabil-
ity determiningcharacteristic rootsand the computationof H2 and H∞ type cost
functions.Thecontrolsynthesisisperformedusingdirectoptimizationalgorithms
applied to minimizing either the spectral abscissa (stabilization problem), or H ,
2
or H∞ norms(robustcontroldesign),which are, in general,non-convexandeven
non-smooth objective functions. As mentioned in the chapter, the author and his
colleagues have recently implemented the methods into the freely available soft-
waretools.
In the second chapter,contributedby DIMITRI BREDA, STEFANO MASET and
ROSSANA VERMIGLIO,anumericalschemetodiscretizethesolutionoperatorsof
lineartime-invarianttime-delaysystemsinHilbertspacesisproposedandanalyzed.
Combining polynomialcollocation and Fourier projection, a numerical scheme is
proposedfor discretizing the solution operator of the system under consideration.
Next, step-by-step analysis of the discretization approaches results in the matri-
ces necessary for code implementations, and detailed convergenceanalysis of the
methodisperformed.
The third chapter of this part is contributed by ELIAS JARLEBRING, WIM
MICHIELS and KARL MEERBERGEN.TheauthorsshowthattheArnoldimethod,
which is a well-established numericalmethodfor standardand generalizedeigen-
value problems, can also be used for infinite-dimensionalproblems. First, the au-
thorsintroducetheinfiniteArnoldimethodandprovideastep-by-stepalgorithmfor
implementation. Consequently, the method is adapted for time-delay system with
distributeddelays.AfteroutliningtheconnectionwiththeFouriercosinetransform,
theconvergencepropertiesintheeigenvaluecomputationareillustratedontwoex-
amples.
The fourth chapter, contributed by ALI FUAT ERGENC, presents an original
method for determining delay-independentstability zones of the general LTI dy-
namics with multiple delays againstparametric uncertainties. The method utilizes
Preface XI
theextendedKroneckersummationanduniquepropertiesofself-inversivepolyno-
mials. Themain resultof the chapteris representedbythe sufficientconditionfor
delay-independentstability. Several examplesdemonstrate the applicability of the
theoreticalresults.
Inthefifthchapter,LAURENTIUHETEL,ALEXANDREKRUSZEWSKIandJEAN-
PIERRERICHARDproposeamethodforcomputingtheLyapunovexponentofsam-
pleddatasystemswithsamplingjitter.Theproposedmethodishybrid,inthesense
thatitcombinescontinuous-timemodels(basedontime-delaysystems)withpoly-
topic embedding methods (specific to discrete-time approaches). A lower bound
of the Lyapunovexponentcan be expressed as a generalized eigenvalueproblem.
Numerical examples complete the presentation and illustrate the improvement in
comparisonwithotherclassicalapproaches.
Inthesixthchapter,OLGA N. LETYAGINAandALEXEY P. ZHABKO proposea
numericalprocedurefortheconstructionofLyapunovfunctionalswithaprescribed
time-derivativefor the case of delaysystemswith periodiccoefficients.Similar to
thecaseoftime-invariantsystems,thefunctionalsaredefinedbyusingspecialLya-
punovmatrices.Next,theissuesofexistence,exponentialestimation,robuststabil-
ityandcomputationalissuesarestudied.
The seventh chapter by WARODY LOMBARDI, SORIN OLARU and SILVIU-
IULIAN NICULESCU proposes a generic approach to obtain polytopic models for
systemswith inputsaffectedbytime-delays.Threedifferentcasesarestudiedthat
ariseduringthediscretizationprocesswithrespecttothestructureofthecontinuous-
statetransitionmatrix.Thegoalistomodelthevariableinputdelaysasapolytopic
uncertaintyin order to preservea linear differenceinclusion framework.The pro-
posedapproachguaranteesafixedcomplexitysimplex-typeglobalembedding.
PartIVPredictor-BasedControlandCompensation
Predictor-based control goes back to 1950s with the results of Smith predictor,
whichhasbecomeoneofthemajorcontributionsincontrolsystemsfieldintermsof
controllingaclassofsystemswithdelay.Manyotherpredictive-basedapproaches
currentlyexistintheliterature,includingapopularonecalledmodelpredictivecon-
trol,whichhasfoundbroadapplicationinindustry.Presenceofdelayposesmajor
problemsindesigningpredictorsandusingthemascompensationschemes.
ThefirstchapterbyMIROSLAVKRSTICprovidesatutorialintroductiontometh-
odsforstabilizationofsystemswithlonginputdelays,theso-calledpredictorfeed-
back techniques. The methods are based on techniques originally developed for
boundarycontrolofpartialdifferentialequationsusingthebacksteppingapproach.
Severaladaptiveandcompensationschemesareconsideredforbothlinearandnon-
linear systems with delaysof differentnature.Primarily,the chapterdemonstrates
that the construction of backstepping transformations allow one to deal with de-
laysandPDEdynamicsattheinput,aswellasinthemainlineofapplyingcontrol
action.Itisshownthatusingdirectandinversebacksteppingtransformations,Lya-
punovfunctionalsandexplicitstabilityestimatescanbeconstructed.
The second chapter, contributed by ION NECOARA, IOAN DUMITRACHE and
JOHAN A.K.SUYKENS,proposestwomethodsforsolvingdistributivelyseparable