CISM International Centre for Mechanical Sciences 604 Courses and Lectures Dimitri Breda   Editor Controlling Delayed Dynamics Advances in Theory, Methods and Applications International Centre for Mechanical Sciences CISM International Centre for Mechanical Sciences Courses and Lectures Volume 604 ManagingEditor PaoloSerafini,CISM—InternationalCentreforMechanicalSciences,Udine,Italy SeriesEditors ElisabethGuazzelli,LaboratoireMatièreetSystèmesComplexes,UniversitéParis Diderot,Paris,France AlfredoSoldati,InstituteofFluidMechanicsandHeatTransfer, TechnischeUniversitätWien,Vienna,Austria WolfgangA.Wall,InstituteforComputationalMechanics,TechnischeUniversität München,Munich,Germany AntonioDeSimone,BioRoboticsInstitute,Sant’AnnaSchoolofAdvancedStudies, Pisa,Italy For more than 40 years the book series edited by CISM, “International Centre for Mechanical Sciences: Courses and Lectures”, has presented groundbreaking developments in mechanics and computational engineering methods. It covers such fields as solid and fluid mechanics, mechanics of materials, micro- and nanomechanics, biomechanics, and mechatronics. The papers are written by internationalauthoritiesinthefield.Thebooksareatgraduatelevelbutmayinclude someintroductorymaterial. Dimitri Breda Editor Controlling Delayed Dynamics Advances in Theory, Methods and Applications Editor DimitriBreda CDLab-ComputationalDynamics Laboratory UniversityofUdine Udine,Italy ISSN 0254-1971 ISSN 2309-3706 (electronic) CISMInternationalCentreforMechanicalSciences ISBN 978-3-031-00981-5 ISBN 978-3-031-01129-0 (eBook) https://doi.org/10.1007/978-3-031-01129-0 ©CISMInternationalCentreforMechanicalSciences2023 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Delaysareubiquitousinengineeringandnaturalsciences:communicationdelaysin controldevicesortheincubationperiodduringanepidemicspreadarejustacouple of noteworthy examples. The inclusion of past history in the time evolution adds nontrivial complexities with respect to ordinary systems, balancing the advantage ofdealingwithmorerealisticmodels.Equationsinvolvingdelaysgenerateinfinite- dimensionaldynamicalsystems,askingforadvancedtoolsandmethodsintheback- groundmathematicalanalysis,thenumericaltreatmentandthedevelopment,design andoptimizationofcontrolstrategies.Eventually,understandingfundamentalissues likestabilityiscrucial,especiallyforvaryingoruncertainparameters. ThesepremisesmotivatedtheorganizationofaninternationalcourseatCISMin 2019,andthisbookcollectscontributionsofthelecturersaboutanalytical,numerical andapplicationaspectsoftime-delaysystems,undertheparadigmofcontroltheory. Theaimisatdiscussingrecentadvancesinthesedifferentcontexts,alsohighlighting theinterdisciplinaryconnections. Chapter“TheTwinSemigroupApproachTowardsPeriodicNeutralDelayEqua- tions” deals with twin semigroups and norming dual pairs for neutral delay equa- tions, including time-dependent perturbations in view of periodic problems. Then inchapter“CharacteristicMatrixFunctionsandPeriodicDelayEquations”,charac- teristic matrix functions are introduced to analyze spectral properties, focusing on monodromyoperatorsofneutralperiodicdelayequations. Chapters “Pseudospectral Methods for the Stability Analysis of Delay Equa- tions.PartI:TheInfinitesimalGeneratorApproach and“PseudospectralMethods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach” concern the use of pseudospectral collocation techniques to reduce to finite dimension the dynamical analysis of both delay differential and renewal equations. Discretizations of the infinitesimal generator of the relevant semigroup (Chapter “Pseudospectral Methods for the Stability Analysis of Delay Equations. PartI:TheInfinitesimalGeneratorApproach”)andofthesemigroupitself(Chapter “PseudospectralMethodsfortheStabilityAnalysisofDelayEquations.PartII:The SolutionOperatorApproach”)aredescribedinviewofanalyzinglocalstabilityand performingbifurcationanalysis. v vi Preface The focus moves then to the characteristic roots of linear time-invariant time- delay systems in view of stability. Frequency-sweeping techniques are illustrated in chapter “Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications”, while frequency- domainapproachesarepresentedinchapter“CountingCharacteristicRootsofLinear DelayDifferentialEquations.PartII:FromArgumentPrincipletoRightmostRoot AssignmentMethods”,linkingmaximalmultiplicitytodominancy,alsoinviewof low-complexitycontrollers. Chapter“BifurcationAnalysisofSystemsWithDelays:MethodsandTheirUse inApplications”presentsadynamicalsystemspointofviewtostudyproblemswith possiblystate-dependentdelays.ByusingthemostrecentreleaseofDDE-BIFTOOL, thenumericalcontinuationofsteadystatesandperiodicorbits,theirbifurcationsand relevant normal forms are addressed, also through the analysis of two longer case studies. Chapters“DesignofStructuredControllersforLinearTime-DelaySystems”gives an overview of control design methods, grounded in matrix theory and numerical linearalgebraandrelyingonadirectoptimizationofstability,robustnessandperfor- mance indicators as a function of controller or design parameters. Then Chapter “A Scalable Controller Synthesis Method for the Robust Control of Networked Systems”concentratesonascalablecontrollersynthesismethodintheframework ofH∞-normcontrolfornetworkedsystems. Finally, chapters “Regenerative Machine Tool Vibrations” and “Dynamics ofHumanBalancing”discussmodelsof,respectively,machinetoolvibrationsand humanbalancingtasks.Intheformer,thephenomenoncalledsurfaceregeneration isanalyzedintermsofthedelaydifferentialequationsgoverningthevibrations,and stabilitydiagramsareconstructed.Inthelatter,thecentralroleplayedbythereaction timeisaddressedbydiscussingstabilizabilityissuesintermsofthecriticaldelayfor differentfeedbackconcepts. Aftertwoyearsoftheglobalpandemic,thetimehaseventuallycometoputanend tothisvolume:afterall,somedelayisnotcompletelyoutofplacegiventhesubject. Oncemore,letmethankthelecturers:Iamsurethattheirstimulatingcontributions to the course have been much appreciated by the 46 attendees from 13 different countries,towhichIgratefullyaddmyself.Thepricelesshelpandkindpresenceof CISMadministrativestaffarealsoacknowledgedwithtruepleasure. Thisbookistheresultofthelecturers’effort,SjoerdVerduynLunel,Silviu-Iulian Niculescu,BerndKrauskopf,WimMichielsandTamásInsperger(andofthatoftheir co-authors,whomIthankaswell).Iamtremendouslygratefultothem,aswellasto manyothercolleaguesforallIcouldlearnaboutdelaysystems.Sincethelistwould beexcessivelylong,letmejustgivecredittoGaborStépánforhavinginspiredthe coursebehindthisvolume,andtoRossanaVermiglioandStefanoMasetforhaving patientlyintroducedmetothisresearchfield,forwhichItrustthisvolumewillbea validresource. Udine,Italy DimitriBreda Contents TheTwinSemigroupApproachTowardsPeriodicNeutralDelay Equations ......................................................... 1 SjoerdVerduynLunel CharacteristicMatrixFunctionsandPeriodicDelayEquations ........ 37 SjoerdVerduynLunel Pseudospectral Methods for the Stability Analysis of Delay Equations.PartI:TheInfinitesimalGeneratorApproach ............. 65 DimitriBreda Pseudospectral Methods for the Stability Analysis of Delay Equations.PartII:TheSolutionOperatorApproach ................. 95 DimitriBreda Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests andApplications .................................................. 117 Silviu-IulianNiculescu,Xu-GuangLiandArbenÇela Counting Characteristic Roots of Linear Delay Differential Equations.PartII:FromArgumentPrincipletoRightmostRoot AssignmentMethods ............................................... 157 Silviu-IulianNiculescuandIslamBoussaada BifurcationAnalysisofSystemsWithDelays:MethodsandTheir UseinApplications ................................................ 195 BerndKrauskopfandJanSieber DesignofStructuredControllersforLinearTime-DelaySystems ...... 247 WimMichiels AScalableControllerSynthesisMethodfortheRobustControl ofNetworkedSystems ............................................. 289 PieterAppeltansandWimMichiels vii viii Contents RegenerativeMachineToolVibrations .............................. 311 TamásInspergerandGaborStépán DynamicsofHumanBalancing ..................................... 343 TamásInsperger,GaborStépánandJohnMilton The Twin Semigroup Approach Towards Periodic Neutral Delay Equations SjoerdVerduynLunel Abstract Inthefirstpartofthischapterwereviewtherecentlydevelopedtheoryof twinsemigroupsandnormingdualpairsinthelightofneutraldelayequations.Inthe secondpartweextendtheperturbationtheoryfortwinsemigroupstoincludetime- dependent perturbations. Finally we apply this newly developed theory to neutral periodicdelayequations. 1 Introduction Considerafunctionx definedonthehalf-line[0,∞)withvaluesinRn andassume thatthederivativex˙dependsonthehistoryofxandx˙.Moreprecisely,weassumethat thereexistsh >0suchthatx˙(t)dependsonx(τ)andx˙(τ)fort −h ≤τ ≤t.Given theserestrictionswewouldliketoconsideragenerallineardifferentialequation. To formulate precisely what type of equations we consider, we first define the segmentx :[−h,0]→Rn by t x (θ):= x(t +θ), for −h ≤θ ≤0. (1) t Letηandζben×n-matrix-valuedfunctionsofboundedvariationdefinedon[0,∞) suchthatη(0)=ζ(0)=0,η andζ arecontinuousfromtherighton(0,h),η(t)= η(h)andζ(t)=ζ(h)fort ≥h.Wecallsuchfunctionsηandζofnormalizedbounded variation.Furthermoreassumethatη(t)iscontinuousatt =0.(SeeAppendixAfor theprecisedefinitionandbasicpropertiesofsuchfunctions.) Theclassofequationsthatwewillstudycannowbewrittenas (cid:2) (cid:3) (cid:4) (cid:3) d h h x(t)− dη(θ)x(t −θ) = dζ(θ)x(t −θ). (2) dt 0 0 B S.VerduynLunel( ) MathematicalInstitute,UtrechtUniversity,Utrecht,TheNetherlands e-mail:[email protected] ©CISMInternationalCentreforMechanicalSciences2023 1 D.Breda(ed.),ControllingDelayedDynamics,CISMInternationalCentre forMechanicalSciences604,https://doi.org/10.1007/978-3-031-01129-0_1