MATLABFOR ENGINEERS APPLICATIONS IN CONTROL, ELECTRICAL ENGINEERING,IT AND ROBOTICS Edited by Karel Perutka MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics Edited by Karel Perutka Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright 2011 InTech All chapters are Open Access articles distributed under the Creative CommonsNon Commercial Share Alike Attribution 3.0 license, which permits to copy,distribute, transmit, and adapt the work in any medium, so long as the originalwork is properly cited. After this work has been published by InTech, authorshave the right to republish it, in whole or part, in any publication of which theyare the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is acceptedfor the accuracy of information contained in the published articles. The publisherassumes no responsibility for any damage or injury to persons or property arising outof the use of any materials, instructions, methods or ideas contained in the book.

Publishing Process Manager Davor Vidic Technical Editor Teodora Smiljanic Cover Designer Jan Hyrat Image Copyright teacept, 2011. Used under license from Shutterstock.com MATLAB (Matlab logo and Simulink) is a registered trademark of The MathWorks, Inc. First published September, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics, Edited by Karel Perutka p.cm.ISBN 978-953-307-914-1 free online editions of InTech Books and Journals can be found atwww.intechopen.com Contents PrefaceIX Part 1Theory1 Chapter 1Implementation of a New Quasi-Optimal ControllerTuning Algorithm for Time-Delay Systems3 Libor Peka and Roman Prokop Chapter 2Control of Distributed Parameter Systems - Engineering Methods and Software Support in the MATLAB &Simulink Programming Environment27 Gabriel Hulk, Cyril Belav, Gergely Takcs,Pavol Buek and Peter Zajek Chapter 3Numerical Inverse Laplace Transforms forElectrical Engineering Simulation51 Lubomr Brank Chapter 4Linear Variable Differential TransformerDesign and Verification Using MATLABand Finite Element Analysis75 Lutfi Al-Sharif, Mohammad Kilani, Sinan Taifour,Abdullah Jamal Issa, Eyas Al-Qaisi, Fadi Awni Eleiwiand Omar Nabil Kamal Part 2Hardware and Photonics Applications95 Chapter 5Computational Models Designed in MATLAB toImprove Parameters and Cost of Modern Chips97 Peter Malk Chapter 6Results Processing in MATLAB forPhotonics Applications119 I.V. Guryev, I.A. Sukhoivanov, N.S. Gurieva,J.A. Andrade Lucio and O. Ibarra-Manzano VIContents Part 3Power Systems Applications153 Chapter 7MATLAB Co-Simulation Tools forPower Supply Systems Design155 Valeria Boscaino and Giuseppe Capponi Chapter 8High Accuracy Modelling of Hybrid Power Supplies189 Valeria Boscaino and Giuseppe Capponi Chapter 9Calculating Radiation from Power Lines forPower Line Communications223 Cornelis Jan Kikkert Chapter 10Automatic Modelling Approach for Power Electronics Converters: Code Generation (C S Function, Modelica,VHDL-AMS) and MATLAB/Simulink Simulation247 Asma Merdassi, Laurent Gerbaud and Seddik Bacha Chapter 11PV Curves for Steady-State SecurityAssessment with MATLAB267 Ricardo Vargas, M.A Arjona and Manuel Carrillo Chapter 12Application of Modern Optimal Control inPower System: Damping DetrimentalSub-Synchronous Oscillations301 Iman Mohammad Hoseiny Naveh and Javad Sadeh Chapter 13A New Approach of Control SystemDesign for LLC Resonant Converter321 Peter Drgoa, Michal Frivaldsk and Anna Simonov Part 4Motor Applications339 Chapter 14Wavelet Fault Diagnosis of Induction Motor341 Khalaf Salloum Gaeid and Hew Wooi Ping Chapter 15Implementation of Induction Motor Drive ControlSchemes in MATLAB/Simulink/dSPACE Environmentfor Educational Purpose365 Christophe Versle, Olivier Deblecker and Jacques Lobry Chapter 16Linearization of Permanent Magnet SynchronousMotor Using MATLAB and Simulink387 A. K. Parvathy and R. Devanathan Part 5Vehicle Applications407 Chapter 17Automatic Guided Vehicle Simulationin MATLAB by Using Genetic Algorithm409 Anibal Azevedo ContentsVII Chapter 18Robust Control of Active Vehicle Suspension Systems Using Sliding Modes and Differential Flatness with MATLAB425 Esteban Chvez Conde, Francisco Beltrn Carbajal, Antonio Valderrbano Gonzlez andRamn Chvez Bracamontes Chapter 19Thermal Behavior of IGBTModule for EV (Electric Vehicle)443 Mohamed Amine Fakhfakh, Moez Ayadi,Ibrahim Ben Salah and Rafik Neji Part 6Robot Applications457 Chapter 20Design and Simulation of Legged WalkingRobots in MATLAB Environment459 Conghui Liang, Marco Ceccarelli and Giuseppe Carbone Chapter 21Modeling, Simulation and Control of a PowerAssist Robot for Manipulating Objects Basedon Operators Weight Perception493 S. M. Mizanoor Rahman, Ryojun Ikeura and Haoyong Yu Preface MATLABisapowerfulsoftwarepackagedevelopedbytheMathWorks,Inc.,the multi-national corporation with the companys headquarters in Natick, Massachusetts, United States of America. The software is a member of the family of the mathematical computingsoftwaretogetherwithMaple,Mathematica,Mathcadetc.anditbecame thestandardforsimulationsinacademiaandpractice.Itofferseasy-to-understand programminglanguage,sharingsourcecodeandtoolboxeswhichsolvetheselected area from practice.The software is ideal for light scientific computing, data processing and math work. Its strength lies in toolboxes for Control and Electrical Engineering. Thisbookpresentsinterestingtopicsfromtheareaofcontroltheory,robotics,power systems,motorsandvehicles,forwhichtheMATLABsoftwarewasused.Thebook consists of six parts. Firstpartofthebookdealswithcontroltheory.Itprovidesinformationabout numericalinverseLaplacetransform,controloftime-delaysystemsanddistributed parameters systems. Therearetwochaptersonlyinthesecondpartofthebook.Oneisaboutthe applicationofMATLABformodernchipsimprovement,andtheotheronedescribes results of MATLAB usage for photonics applications. Next part of the book consists of chapters which have something in common with the powersystemsapplications,forexampletwochaptersareaboutpowersupply systems and one is about application of optimal control in power systems. Thispartisfollowedbythe partaboutMATLABapplicationsusedinfaultdiagnosis of induction motor, implementation of induction motor drive control and linearization of permanent magnet synchronous motors. Thelastbutonepartofthebookprovidestheapplicationforvehicles,namelythe guidedvehiclesimulation,newconfigurationofmachine,behaviorofmodulefor electric vehicle and control of vehicle suspension system. ThelastpartdealswithMATLABusageinrobotics,withthemodeling,simulation and control of power assist robot and legged walking robot. XPreface ThisbookprovidespracticalexamplesofMATLABusagefromdifferentareasof engineeringandwillbeusefulforstudentsofControlEngineeringorElectrical Engineeringtofindthenecessaryenlargementoftheirtheoreticalknowledgeand several models on which theory can be verified. It helps with the future orientation to solve the practical problems. Finally, I would like to thank everybody who has contributed to this book. The results ofyourworkareveryinterestingandinspiring,Iamsurethebookwillfindalotof readers who will find the results very useful. Karel Perutka Tomas Bata University in Zln Czech Republic Part 1 Theory 1 Implementation of a New Quasi-Optimal Controller Tuning Algorithm forTime-Delay Systems Libor Peka and Roman Prokop Tomas Bata University in Zln Czech Republic 1. IntroductionSystemsandmodelswithdeadtimeoraftereffect,alsocalledhereditary,anisochronicor time-delay systems (TDS), belonging to the class of infinite dimensional systems have been largelystudiedduringlastdecadesduetotheirinterestingandimportanttheoreticaland practicalfeatures.Awidespectrumofsystemsinnaturalsciences,economics,pure informaticsetc.,bothreal-lifeandtheoretical, isaffectedbydelayswhichcanhavevarious forms;tonamejustafewthereaderisreferrede.g.to(Greckietal.,1989;Marshalletal., 1992; Kolmanovskii & Myshkis, 1999; Richard, 2003; Michiels & Niculescu, 2008; Peka et al., 2009)andreferencesherein.Lineartime-invariantdynamicsystemswithdistributedor lumped delays (LTI-TDS) in a single-input single-output (SISO) case can be represented by a setoffunctionaldifferentialequations(Hale&VerduynLunel,1993)orbytheLaplace transferfunctionasaratioofso-calledquasipolynomials(Elsgolts &Norkin,1973)inone complexvariables,ratherthanpolynomialswhichareusualinsystemandcontroltheory. Quasipolynomialsareformedaslinearcombinationsofproductsofs-powersand exponentialterms.Hence,theLaplacetransformofLTI-TDSisnolongerrationalandso-called meromorphic functions have to be introduced. A significant feature of LTI-TDS is (in contrasttoundelayedsystems)itsinfinitespectrumandtransferfunctionpolesdecide- exceptsomecasesofdistributeddelays,seee.g.(Loiseau,2000)-abouttheasymptotic stability as in the case of polynomials. It is a well-known fact that delay can significantly deteriorate the quality of feedback control performance,namelystabilityandperiodicity.Therefore,designasuitablecontrollawfor such systems is a challenging task solved by various techniques and approaches; a plentiful enumeration of them can be found e.g. in (Richard, 2003). Every controller design naturally requiresandpresumesacontrolledplantmodelinanappropriateform.Ahugesetof approachesusestheLaplacetransferfunction;however,itisinconvenienttoutilizearatio of quasipolynomials especially while natural requirements of internal (impulse-free modes) andasymptoticstabilityofthefeedbackloopandthefeasibilityandcausalityofthe controller are to be fulfilled.Themeromorphicdescriptioncanbeextendedtothefractionaldescription,tosatisfy requirementsabove,sothatquasipolynomialsarefactorizedintoproperandstable meromorphicfunctions.Theringofstableandproperquasipolynomial(RQ) MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 4 meromorphic functions (RMS) is hence introduced (Ztek & Kuera, 2003; Peka & Prokop, 2010).Althoughtheringcanbeusedforadescriptionofevenneutralsystems(Hale& VerduynLunel,1993),onlysystemswithso-calledretardedstructureareconsideredas theadmissibleclassofsystemsinthiscontribution.Incontrasttomanyotheralgebraic approaches,theringenablestohandlesystemswithnon-commensuratedelays,i.e.itis notnecessary thatall systemdelays can beexpressedas integermultiplesofthe smallest one. Algebraic control philosophy in this ring then exploits the Bzout identity, to obtain stable and proper controllers, along with the Youla-Kuera parameterization for reference tracking and disturbance rejection. Theclosed-loopstabilityisgiven,asfordelaylesssystems,bythesolutionsofthe characteristicequationwhichcontainsaquasipolynomialinsteadofapolynomial.These infinitemanysolutionsrepresentclosed-loopsystempolesdecidingaboutthecontrol systemstability.Sinceacontrollercanhaveafinitenumberofcoefficientsrepresenting selectableparameters,thesehavetobesettodistributetheinfinitespectrumsothatthe closed-loop system is stable and that other control requirements are satisfied. Theaimofthischapteristodescribe,demonstrateandimplementanewquasi-optimal pole placement algorithm for SISO LTI-TDS based on the quasi-continuous pole shifting themainideaofwhichwaspresentedin(Michielsetal.,2002)-totheprescribed positions.Thedesiredpositionsareobtainedbyovershootanalysisofthestepresponse foradominantpairofcomplexconjugatepoles.Acontrollerstructureisinitially calculatedbyalgebraiccontrollerdesigninRMS.Notethatthemaximumnumberof prescribedpoles(includingtheirmultiplicities)equalsthenumberofunknown parameters.Iftheprescribedrootslocationscannotbereached,theoptimizingofan objectivefunctioninvolvingthedistanceofshiftingpolestotheprescribedonesandthe rootsdominancyisutilized.TheoptimizationismadeviaSelf-OrganizingMigration Algorithm (SOMA), see e.g. (Zelinka, 2004). Matlab m-file environment is utilized for the algorithmimplementationand,consequently,resultsaretestedinSimulinkonan attractive example of unstable SISO LTI-TDS. The chapter is organized as follows. In Section 2 a brief general description of LTI-TDS is introducedtogetherwiththecoprimefactorizationfortheRMSringrepresentation.Basic ideas of algebraic controller design in RMS with a simple control feedback are presented in Section3.Themainandoriginalpartofthechapterpole-placementshiftingbased tuningalgorithmisdescribedinSection4.Section5focusesSOMAanditsutilization whensolvingthetuningproblem.Anillustrativebenchmarkexampleispresentedin Section 6. 2. Description of LTI-TDS TheaimofthissectionistopresentpossiblemodelsofLTI-TDS;first,thatintimedomain using functional differential equations, second, the transfer function (matrix) via the Laplace transform.Then,thelatterconceptisextendedsothatanalgebraicdescriptioninaspecial ringisintroduced.Notethatforthefurtherpurposeofthischapterthestate-space functional description is useless. 2.1 State-space model A LTI-TDS system with both input-output and internal (state) delays, which can have point (lumped) or distributed form, can be expressed by a set of functional differential equations Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 5 ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )0 01 110 0d dd dH ABN Nii i ii iL L Ni iit tt t tt tt t d t dt t = === + + ++ + + = } }x xH Ax A x B uB u A x B uy Cx(1) where x n isavectorofstatevariables, u mstandsforavectorofinputs, y l representsavectorofoutputs,Ai,A(),Bi,B(),C,Hiarematricesofappropriate dimensions,0iL arelumped(point)delaysandconvolutionintegralsexpress distributed delays (Hale & Verduyn Lunel, 1993; Richard, 2003; Vyhldal, 2003). If i H 0for any i = 1,2,...NH, model (1) is called neutral; on the other hand, if i = H 0for every i = 1,2,...NH,so-calledretardedmodelisobtained.Itshouldbenotedthatthestateofmodel (1)isgivennotonlybyavectorofstatevariablesinthecurrenttimeinstant,butalsobyasegmentofthelastmodelhistory(infunctionalBanachspace)ofstateandinput variables ( ) ( ) , , , 0 t t L + + x u (2) Convolutionintegralsin(1)canbenumericallyapproximatebysummationsfordigital implementation;however,thiscandestabilizeevenastablesystem.Alternatively,onecan integrate (1) and add a new state variable to obtain derivations on the right-hand side only. Inthecontrary,themodelcanalsobeexpressedinmoreconsistentfunctionalformusing Riemann-Stieltjesintegralssothatbothlumpedanddistributeddelaysareunderone convolution.Forfurtherdetailsandotherstate-spaceTDSmodelsthereaderisreferredto (Richard, 2003). 2.2 Input-output model Thiscontributionisconcernedwithretardeddelayedsystemsintheinput-output formulationgovernedbytheLaplacetransferfunctionmatrix(consideringzeroinitial conditions)asin(3).Hence,intheSISOcase(weareconcerningabouthere),thetransfer functionisnolongerrational,asforconventionaldelaylesssystems,andameromorphic function as a ratio of retarded quasipolynomials (RQ) is obtained instead. ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )1 2101011 01012 010det exp exp dadj exp exp dexp exp dAABL Ni iiL Ni iiL Ni iis s s ss s ss s ss s +=+=+== = ( = ( ( (= ( ( = + + }}} Y G U UI A A A C I A A A B B B(3) A (retarded or neutral) quasipolynomial of degree n has the generic form MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 6 ( )( )0 1exp , 0n hn iij ij iji jqs s xs s = == + (4) where0njx in the neutral case for some j, whereas a RQ owns0njx =for all j. However, the transfer function representation in the form of a ratio of two quasipolynomials isnotsuitableinordertosatisfycontrollerfeasibility,causalityandclosed-loop(Hurwitz) stability (Loiseau 2000; Ztek & Kuera, 2003). Rather more general approaches utilize a field offractionswhereatransferfunctionisexpressedasaratiooftwocoprimeelementsofa suitablering. Aringisaset closedforadditionandmultiplication,with aunitelement for additionandmultiplication andaninverseelementforaddition.Thisimpliesthatdivision is not generally allowed. 2.3 Plant description in RMS ring ApowerfulalgebraictoolensuringrequirementsaboveisaringofstableandproperRQ-meromorphic functions (RMS). Since the original definition of RMS in (Ztek & Kuera, 2003) doesnotconstitutearing,someminorchangesinthedefinitionwasmadein(Peka& Prokop,2009).Namely,althoughtheretardedstructureofTDSisconsideredonly,the minimalringconditionsrequiretheuseofneutralquasipolynomialsatleastinthe numerator as well. An element( )MSTs Ris represented by a proper ratio of two quasipolynomials ( )( )( )ysTsxs=(5) whereadenominator( ) xs isaquasipolynomialofdegreenandanumeratorcanbe factorized as ( ) ( ) ( ) exp ys ys s = (6) where( ) ys isaquasipolynomialofdegreeland 0.( ) Ts isstable,whichmeansthat there is no pole s0 such that{ }0Re 0 s ; in other words, all roots of( ) xswith{ }0Re 0 s are those of( ) ys . Moreover, the ratio is proper, i.e. l n.Thus,( ) Tsis analytic and bounded in the open right half-plane, i.e. { }( )Re 0supsTs< (7) Asmentionedabove,inthiscontributiononlyretardedsystemsareconsidered,i.e. ( ) xs , ( ) ysare RQs. Let the plant be initially described as ( )( )( )bsGsas= (8) where( ) as ,( ) bsare RQs. Hence, using a coprime factorization, a plant model has the form ( )( )( )BsGsAs= (10) Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 7 where( ) ( ) ,MSAs Bs R arecoprime,i.e.theredoesnotexistanon-trivial(non-unit) commonfactorofbothelements.Notethatasystemofneutraltypecaninduceproblem sincetherecanexistacoprimepair( ) ( ) , As Bs whichisnot,however,Bzoutcoprime which implies that the system can not be stabilized by any feedback controller admitting the Laplace transform, see details in (Loiseau et al., 2002). 3. Controller design in RMS ThissectionoutlinescontrollerdesignbasedonthealgebraicapproachintheRMSring satisfying the inner Hurwitz (Bounded Input Bounded Output - BIBO) stability of the closed loop, controller feasibility, reference tracking and disturbance rejection. ForalgebraiccontrollerdesigninRMSitisinitiallysupposedthatnotonlytheplantis expressed by the transfer function over RMS but a controller and all system signals are over the ring. As a control system, the common negative feedback loop as in Fig. 1 is chosen for the simplicity, where( ) Wsis the Laplace transform of the reference signal,( ) Dsstands for thatoftheloaddisturbance,( ) Es istransformedcontrolerror,( )0U s expressesthe controlleroutput(controlaction),( ) Us representstheplantinput,and( ) Ys istheplant outputcontrolledsignalintheLaplacetransform.Theplanttransferfunctionisdepicted as ( ) Gs , and( )RG sstands for a controller in the scheme. Fig. 1. Simple control feedback loop Control system external inputs have forms ( )( )( )( )( )( ),W DW DH s H sWs DsF s F s= = (11) where( )WH s ,( )DH s ,( )WF s ,( )DF s RMS. The following basic transfer functions can be derived in the control system in general ( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( ),,WY DYWE DEYs BsQs Ys Bs PsG s G sWs Ms Ds MsEs As Ps Es Bs PsG s G sWs Ms Ds Ms= = = == = = = (12) where MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 8 ( )( )( )RQsG sPs= (13) ( ) ( ) ( ) ( ) ( ) Ms As Ps BsQs = + (14) and( ) Qs , ( ) Psare from RMS and the fraction (13) is (Bzout) coprime (or relatively prime). Thenumeratorof( ) Ms RMSagreestothecharacteristicquasipolynomialoftheclosed loop. Following subsections describes briefly how to provide the basic control requirements. 3.1 Stabilization Accordingtoe.g.(Kuera,1993;Ztek&Kuera,2003),theclosed-loopsystemisstableif and only if there exists a pair( ) ( ) ,MSPs Qs Rsatisfying the Bzout identity ( ) ( ) ( ) ( ) 1 As Ps BsQs + = (15) a particular stabilizing solution of which,( ) ( )0 0, P s Q s , can be then parameterized as ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )00,MSPs P s BsTsQs Q s AsTs Ts= = R (16) Parameterization (16) is used to satisfy remaining control and performance requirements. 3.2 Reference tracking and disturbance rejection Thequestionishowtoselect( )MSTs R in(16)sothattasksofreferencetrackingand disturbance rejection are accomplished. The key lies in the form of( )WEG sand( )DYG sin (12). Consider the limits ( ) ( ) ( ) ( )( ) ( )( )( )0 00lim lim limlimt D s D s DYDsDy t sY s sG s DsH ssBs PsF s = ==(17) ( ) ( ) ( ) ( )( ) ( )( )( )0 00lim lim limlimt W s W s WEWsWe t sE s sG s WsH ssAs PsF s = ==(18) where D meansthattheoutputisinfluencedonlybythedisturbance,andsymbol Wexpresses that the signal is a response to the reference. Limit (17) is zero if( )0lims DY s< and( )DY s isanalyticintheopenrighthalf-plane.Moreover,forthefeasibilityof( )Dy t , ( )DY s mustbeproper.Thisimpliesthatthedisturbanceisasymptoticallyrejectedif ( )DY s RMS. Similarly, the reference is tracked if( )WE s MSR . Inotherwords,( )DF s mustdividetheproduct( ) ( ) Bs Ps in MSR ,and( ) ( ) As Ps mustbe divisible by( )WF sin MSR . Details about divisibility inMSRcan be found e.g. in (Peka & Prokop,2009).Thus,ifneither( ) Bs hasanycommonunstablezerowith( )DF s nor( ) As Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 9 hasanycommon unstablezerowith( )WF s ,one hastoset all unstable zeros of( )DF s and ( )WF s (withcorrespondingmultiplicities)aszerosof( ) Ps .Notethatzerosmeanzero pointsofawholetermin MSR ,notonlyofaquasipolynomialnumerator.Unstablezeros agrees with those with{ } Re 0 s . 4. Pole-placement shifting based controller tuning algorithm Inthiscrucialsection,theideaofanewpole-placementshiftingbasedcontrollertuning algorithm(PPSA)ispresented. Althoughsomestepsof PPSAare taken over someexisting pole-shiftingalgorithms,theideaofconnectionwithpoleplacementandtheSOMA optimization is original. 4.1 Overview of PSSA WefirstgiveanoverviewofallstepsofPPSAand,consequently,describeeachinmore details.Theprocedurestartswithcontrollerdesignin MSR introducedintheprevious section. The next steps are as follows: 1.Calculatetheclosed-loopreference-to-outputtransferfunction( )WYG s .Let numl anddenl ,respectively,benumbersofunknown(free,selectable)realparametersofthe numerator and denominator, respectively. Sign num denl l l = + . 2.ChooseasimplemodelofastableLTIsystemintheformofthetransferfunction ( ), WYmG s withanumeratorofdegree numn andthedenominatorofdegree denn . Calculate step response maximum overshoots of the model for a suitable range of its numnzerosand denn poles(includingtheirmultiplicities).If num numn n and den denn n , respectively,arenumbersofallrealzeros(poles)andpairsofcomplexconjugatezeros (poles) of the model, it must hold that num numn l and den denn l , respectively. 3.Prescribeallpolesandzerosofthemodelwithrespecttocalculatedmaximum overshoots (and maximal overshoot times). If the poles and zeros are dominant (i.e. the rightmost), the procedure is finished. Otherwise do following steps. 4.Shifttherightmost(orthenearest)zerosandpolestotheprescribedlocations successively.Ifthenumberofcurrentlyshiftedpolesandconjugatepairs den sp denn n l ishigherthen denn ,trytomove therestof dominant(rightmost)poles to the left. The same rule holds for shifted zeros, analogously. 5.Ifallprescribedpolesandzerosaredominant,theprocedureisfinished.Otherwise, selectasuitablecostfunctionreflectingthedistanceofdominantpoles(zeros)from prescribedpositionsanddistancesofspectralabscissasofboth,prescribedand dominant poles (zeros). 6.Minimize the cost function, e.g. via SOMA. Now look at these steps of the algorithm at great length. 4.2 Characteristic quasipolynomial and characteristic entire function Algebraiccontrollerdesigninthe MSR ringintroducedinSection3resultsinacontroller owning the transfer function ( )RG scontaining a finite number of unknown (free, selectable) parameters.ThetaskofPPSAistosettheseparameterssothatthepossiblyinfinite spectrumoftheclosedloophasdominant(rightmost)poleslocatedin(ornearby)the prescribedpositions.Ifpossibly,onecanprescribeandplacedominantzeros as well.Note MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 10thatcontrollerdesignin MSR usingthefeedbacksystemasinFig.1resultsininfinite spectrum of the feedback if the controlled plant is unstable. If the (quasi)polynomial numerator and denominator of( ) Gshave no common roots in the openright-halfplane,theclosed-loopspectrumisgiven entirely by rootsof thenumerator ( ) ms of( ) Ms ,thesocalledcharacteristicquasipolynomial.Inthecaseofdistributed delays,( ) Gs hassomecommonrootswith{ } Re 0 s inboth,thenumeratorand denominator,andtheserootsdonotaffectthesystemdynamicssincetheycanceleach other. In this case, the spectrum is given by zeros of the entire function( ) ( ) /Ums m s , i.e. the characteristicentirefunction,where( )Um s isa(quasi)polynomialtheonlyrootsofwhich are the common unstable roots.The(quasi)polynomialdenominatorof( )WYG s agreeswith( ) ms .Itsroleismuchmore importantthantheroleofthenumeratorof( )WYG s sincetheclosed-loopzerosdoesnot influencethestability.Inthelightofthisfact,thesettingofclosed-looppoleshasthe priority.Therefore,onehastoset denl freedenominatorparametersfirst.Free(selectable) parameters in the numerator of( )WYG sare to be set only if there exist those which are not contained in the denominator. The number of such additional parameters is numl . 4.3 Closed-loop model and step response overshoots Thetasknowishowtoprescribetheclosed-looppolesappropriately.Wechooseasimple finite-dimensional model of the reference-to-output transfer function and find its maximum overshoots and overshoot times for a suitable range of the model poles. Let the prescribed (desired) closed-loop model be of the transfer function ( )( )( )1 0 1, 1 221 1 1 0WYmb s b s zG s k ks s s s s a s a+ = = + +(19) where 1 2 1 0 1 0, , , , , k k b b a a aremodelparameters 1z -standsforamodelzeroand1s - is a model stable pole where 1sexpresses its complex conjugate. To obtain the unit static gain of( ), WYmG sit must hold true 21 01 20 1,s ak kb z= = (20) Sign 1j, 0, 0 s = + < and calculate the impulse function( ), WYmg tof( ), WYmG susing the Matlab function ilaplace as ( ) ( ) ( ) ( )1, 2exp cos sinWYmzg t k t t t (= ( (21) Since( ) ( ), , WYm WYmi t h t = ,where( ), WYmh t isthestepresponsefunction,thenecessary condition for the existence ofa step response overshoot at time tO is ( ),0, 0WYm O Oi t t = > (22) The condition (22) yields these two solutions: either Ot (which is trivial) or Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 11 ( )12211arccosOztz | | |= | | +\ .(23) whenconsidering( ) [ ] arccos 0, x .Obviously,(23)hasinfinitelymanysolutions.If 10, 0 z < < , the maximum overshoot occurs at time ( )maxminOt t = (24) One can further calculate the step response function( ), WYmh tas ( ) ( ) ( ) ( )21 1 2, 1 121exp cos sinWYmz s kh t t z t t zs ( | | (|= | (\ . (25) Define now the maximum relative overshoot as ( ) ( )( ), max ,, ,max,:WYm WYmWYmWYmh t hhh =(26) see Fig. 2. Fig. 2. Reference-to-output step response characteristics and the maximum overshootUsing definition (26) one can obtain ( )( )( )( )max21 1 1, ,max1cos sinexpWYmt tz t z s th tz =| | + | = | |\ .(27) MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 12Obviously, , ,max WYmh is afunctionofthreeparameters,i.e. 1, , n ,whichisnotsuitable for a general formulation of the maximal overshoot. Hence, let us introduce new parameters ,z as 1,zz = = (28) which give rise from (23), (24) and (27) to ( ) ( ) ( ) ( )( )( )2, ,max max, max, max,max, max21exp cos 1 sinmin arccos1WYm norm z norm z normznormh t t tt t = + + | | | | ||= = || || +\ . \ .(29) where max,normtrepresents the normalized maximal overshoot time. We can successfully use Matlab to display function( ), ,max,WYm zh and( )max,,norm zt graphically, for suitable ranges of,z as can be seen from Fig. 3 Fig. 7.Recall that model(19) gives rise to1, 2, 1, 1num den num denn n n n = = = = . Fig. 3. Maximum overshoots( ), ,max,WYm zh (a) and normalized maximal overshoot times( )max,,norm zt (b) for[ ] 0.1, 2 = ,{ } 0.2, 0.4, 0.6, 0.8, 1z = . Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 13 Fig. 4. Maximum overshoots( ), ,max,WYm zh (a) and normalized maximal overshoot times( )max,,norm zt (b) for [ ] 0.1, 2 = ,{ } 2, 3, 4, 5, 10z = . Fig. 5. Maximum overshoots( ), ,max,WYm zh (a) and normalized maximal overshoot times( )max,,norm zt (b) for [ ] 2, 10 = ,{ } 0.2, 0.4, 0.6, 0.8, 1z = . Fig. 6. Maximum overshoots( ), ,max,WYm zh (a) and normalized maximal overshoot times( )max,,norm zt (b) for[ ] 2, 10 = ,{ } 2, 3, 4, 5, 10z = . MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 14

Fig. 7. Maximum overshoots( ), ,max,WYm zh (a) and normalized maximal overshoot times( )max,,norm zt (b) for[ ] 1, 5, 4.5 = ,{ } 2.8, 3, 3.2, 3.4, 3.6z =- A detailed view on small overshoots. Theprocedureofsearchingsuitableprescribedpolescanbedonee.g.asinthefollowing way. A user requires , ,max0.03WYmh = (i.e. the maximal overshoot equals 3 %),4 =(i.e. thequarterdumping)and max5 t = s.Fig.7givesapproximately2.9z = whichyields max,1.2normt .Thesetwovaluestogetherwith(28)and(29)resultin 10.96 0.24j, s = +10.7 z = . 4.4 Direct pole placement Thissubsectionextendsstep3ofPPSAfromSubsection4.1.Thegoalistoprescribepoles andzerosoftheclosed-loopatonce.Thedrawbackhereisthattheprescribedpoles (zeros)mightnotbedominant(i.e.therightmost).TheprocedurewasutilizedtoLTI-TDS e.g. in (Ztek & Hlava, 2001). Givenquasipolynomial( ) ms withavector [ ]1 2, ,...,Tlv v v = v loflfreeparameters,the assignment of n prescribed single roots i , i = 1...n, can be done via the solution of the set of algebraic equations in the form [ ] ( , ) 0, 1...iism i n== = v (30) Inthecaseofcomplexconjugatepoles,onehastotaketherealandimaginarypart separately as [ ]{ }[ ]{ }Re ( , ) 0, Im ( , ) 0i ii is sm m = == = v v (31) for every pair of roots. If a root ihas the multiplicity p, it must be calculated d( , ) 0, 0... 1dijijsm j ps= (= = ( ( v (32) or Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 15 d dRe ( , ) 0, Im ( , ) 0, 0... 1d di ij ji ij js sm m j ps s = = ((= = = (( (( v v (33) Note that if( ) msis nonlinear with respect tov , one can solve a set on non-linear algebraic equations directly, or to use an expansion 0001( , )( , ) ( , )iki i js j jm sm m vv = == ( + ( ( v vvv v (34) where 0v meansapointinwhichtheexpansionismadeoraninitialestimationofthe solutionand[ ]1 2, ,...,Tlv v v = v isavectorofparametersincrements.Equations(34) shouldbesolvediteratively,e.g.viathewell-knownNewtonmethod.Note,furthermore, that the algebraic controller design inMSRfor LTI-TDS results in the linear set (30)-(34) with respect to selectable parameters both, in the numerator and denominator of( )WYG s . It is clear that a unique solution is obtained only if the set ofn l =independent equations is given.Ifn l < ,equations(30)-(34)canbesolvedusingtheMoore-Penrose(pseudo)inverse minimizing the norm 221kiiv== v , see (Ben Israel & Greville, 1966). Contrariwise, whenever n l > ,itisnotpossibletoplacerootsexactlyandthepseudoinverseprovidesthe minimization of squares of the left-hand sides of (30)-(34). Themethodologydescribedinthissubsectionisutilizedonboth,thenumeratorand denominator. 4.5 Continuous poles (zeros) shifting Oncethepoles(zeros)areprescribed,itoughttobecheckedwhethertheserootsarethe rightmost.Ifyes,thePPSAalgorithmstops;ifnot,onemaytrytoshiftpolessothatthe prescribedonesbecomedominant.Therearetwopossibilities.First,thedominantroots move to the prescribed ones; second, roots nearest to the prescribed ones are shifted while therestofthespectrum(orzeros)issimultaneouslypushedtotheleft.Thefollowing describes it in more details. Wedescribetheprocedurefortheclosed-loopdenominatoranditsroots(poles);the numeratorisservedanalogouslyforallitsfreeparameterswhicharenotincludedinthe denominator. Recall that denlis the number of unknown (selectable) parameters, dennstands forthenumberofmodel(prescribed)poles(includingtheirmultiplicities), denn represents thenumberofrealpolesandconjugatepairsofprescribedpolesand spn isthenumberof currently shifted real poles and conjugate pairs. Generally, it holds that den sp denn n l (35) Theideaofcontinuouspolesshiftingdescribedbelowwasintroducedin(Michielsetal., 2002).Similarprocedurewhich,however,enablestoshiftlessnumberofpolessince sp denn l includes every single complex pole instead of a conjugate pair, was investigated in (Vyhldal,2003).Roughlyspeaking,thelatterisbasedonsolutionof(30)-(34)where 0vrepresentsthevectorofactualcontrollerparameters, 0= + v v v arenewcontroller MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 16parametersand i meansprescribedpoles(inthevicinityoftheactualones)here.Now look at the former methodology in more details. The approach (Michiels et al., 2002) is based on the extrapolation 00( , ) ( , )( , ) ( , ) 0, 1... , 1...iii i i j sp denss jm s m sm m v i n j ls v === == ( ( + + = = =( ( ( 00v vv vv vv v(36) yielding 1( , ) ( , )iiiss j jm s m sv s v ==== ( ( ( ( ( 00v vv vv v(37) where 0vrepresents the vector of actual controller parameters, imeans actual poles and i and jv are increments of poles and controller parameters, respectively. In case of a p-multiple pole, the following term is inserted in (36) and (37) instead of( ) ms( )ddpp mss(38) However,(38)canbeusedonlyifthepoleincludingallmultiplicitiesismoved.If,onthe other hand, the intention is to shift a part of poles within the multiplicity to the one location andtherestofthemultiplicitytoanother(orother)location(s),itisbettertoconsidera multiple pole as a nest of close single poles. Then a matrix ijv (= ( ( S den spxl n (39) is called the sensitivity matrix satisfying + = v S (40) where 1 2, ,...,spTn ( = and +Smeans the pseudoinverse.It holds that { } ReRei ij jv v = ` )(41) thus, if poles are shifted in a real axis only, it can be calculated {} { } Re Re+ = v S (42) Otherwise, the following approximation ought to be used { }Re+ v S (43) Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 17 Thecontinuousshiftingstartswith sp denn n = .Then,onecantakethenumberof dennrightmostpolesandmovethemtotheprescribedones.Therightmostclosed-looppole movestotherightmostprescribedpoleetc.Alternatively,thesamenumberofdominant poles(orconjugatedpairs)canbeconsidered;however,thenearestpolescanbeshiftedto theprescribedones.Iftwoormoreprescribedpolesownthesamedominantpole,itis assignedtotherightmostprescribedpoleandremovedfromthelistofmovedpoles.The number{ } ,sp den denn n l is incremented whenever the approaching starts to fail for any pole. If sp denn n > , the rest of dominant poles is pushed to the left. More precisely, shifting to the prescribed poles is described by the following formula p sp s =(44) and pushing to the left agrees with = (45) whereis a discretization step in the space of poles, e.g.0.001 = , pis a prescribed pole and smeans a pole moved to the prescribed one. If sp denn l = and all prescribed poles become the rightmost (dominant) ones, PPSA is finished. Otherwise, do the last step of PPSA introduced in the following subsection. 5. Minimization of a cost function via SOMA This step is implemented whenever the exact pole assignment even via shifting fails. In the firstpartofthissubsectionwearrangethecostfunctiontobeminimized.Then,SOMA algorithm (Zelinka, 2004) belonging to the wide family of evolution algorithms is introduced andbrieflydescribed.Again,theprocedureisgivenforthepole-optimization;thezero-optimization dealing with the closed-loop numerator is done analogously. 5.1 Cost function Thegoalnowistorearrangefeedbackpoles(zeros)sothattheyaresufficientlycloseto theprescribedonesand,concurrently,theyareasthemostdominantaspossible.This requirement can be satisfied by the minimizing of the following cost function ( ) ( ) ( ){ }, , , ,1RedennR s i p i d i p iiF d d == + = + v v v (46) where( ) dv isthedistanceofprescribedpoles , p i fromthenearestones , s i ,( )Rd vexpressesthesumofdistancesofdominantpolesfromtheprescribedonesand0 >representsarealweightingparameter.Thehigher is,thepoledominancyofismore important in( ) Fv . Recall that (when the dominant poles were moved) ,1 ,2 , ,1 ,2 , ,1 ,2 ,, ,den den dens s s n p p p n d d d n (47) Alternatively, one can include both, the zeros and poles, in (46), not separately. MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 18Poles can be found e.g. by the quasipolynomial mapping root finder (QPMR) implemented in Matlab, see (Vyhldal & Ztek, 2003). Hence, the aim is to solve the problem ( ) arg minopt F = v v(48) We use SOMA algorithm based on genetic operations with a population of found solutions andmovingofpopulationspecimenstoeachother.Abriefdescriptionofthealgorithm follows. 5.2 SOMA SOMAisrankedamongevolutionalgorithms,morepreciselygeneticalgorithms,dealing withpopulationssimilarlyasdifferentialevolutiondoes.Thealgorithmisbasedonvector operationsoverthespaceoffeasiblesolutions(parameters)inwhichthepopulationis defined. Population specimens cooperate when searching the best solution (the minimum of the cost function) and, simultaneously, each of them tries to be a leader. They move to each other and the searching is finished when all specimens are localized on a small area. InSOMA,everysinglegeneration,inwhichanewpopulationisgenerated,iscalledamigrationround.Thenotionofspecificcontrolandterminationparameters,whichhavetobesetbeforethealgorithmstarts,willbeexplainedineverystepofamigrationround below. First, population { }1 2, ,...,PopSizeP = v v vmust be generated based on a prototypal specimen. ForPPSA,thisspecimenisavectorofcontrollerfreeparameters,v ,ofdimensiondenD l = . TheprototypalspecimenequalsthebestsolutionfromSubsection4.5.Onecanchoosean initialradius(Rad)ofthepopulationinwhichotherspecimensaregenerated.Thesizeof population (PopSize), i.e. the number of specimens in the population, is chosen by the user. Each specimen is then evaluated by the cost function (46).The simplest strategy called All to One implemented here then selects the best specimen - leader, i.e. that with the minimal value of the cost function ( )argminmr mrL iiF = v v(49) where L denotes the leader, i is i-th of specimen in the population and mr means the current migrationround.Thenallotherspecimenaremovedtowardstheleaderduringthe migrationround.Themovingisgivenbythreecontrolparameters:PathLength,Step,PRT. PathLength should be within the interval[1.1, 5] and it expresses the length of the path when approaching the leader. PathLength = 1 means that the specimen stops its moving exactly at thepositionoftheleader. Steprepresentsthesamplingofthepathandoughttobevalued [0.11, ] PathLength .E.g.apairPathLength=1andStep=0.2agreeswiththatthespecimen makes5stepsuntilitreachestheleader. [ ] 0, 1 PRT enablestocalculatetheperturbation vector PRTVector which indicates whether the active specimen moves to the leader directly or not. PRTVector is defined as { }1 2, ,..., 0, 11 if0 elsedendenTlli iiPRTVector p p pp rnd PRTp (= = ,kW,kD.StabilizationviatheBzoutidentity(15)resultse.g.inthe following particular solution ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( )( ) ( )( )43 23 2 1 0 002 2 3 2 3 22 1 0 3 2 1 043 22 1 0 002 2 3 2 3 22 1 0 3 2 1 0exp expexp expq s qs q s q s mQ ss s a s s p s p s p b s q s qs q s qs p s p s p s mP ss s a s s p s p s p b s q s qs q s q + + + += + + + + + + + ++ + + += + + + + + + + +(56) usingthegeneralizedEuclideanalgorithm,see(Peka&Prokop,2009),wherep2,p1,p0,q3, q2, q1, q0 are free parameters. In order to provide reference tracking and load disturbance rejection,useparameterization(16)whileboth,( )WF s and( )DF s ,divide( ) Ps ;inother words, the numerator of( ) Psmust satisfy ( ) 0 0 P = . If we take ( )( )( )( )( ) ( )( )40 02 2 3 2 3 22 1 0 3 2 1 0exp expt s mTs s a s s p s p s p b s q s qs q s q += + + + + + + + +(57) ( ) Psis obtained in a quite simple form with a real parameter t0 which must be set as 40 00p mtb= (58) Finally, the controller numerator and denominator in MSR , respectively, have forms Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 21 ( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) ( ) ( )( )( )( )( ) ( )( )43 2 4 2 23 2 1 0 0 0 02 2 3 2 3 22 1 0 3 2 1 043 2 42 1 0 0 0 02 2 3 2 3 22 1 0 3 2 1 0expexp expexpexp expbq s qs q s q s m p ms s a sQsb s s a s s p s p s p b s q s qs q s qs p s p s p s m p m sPss s a s s p s p s p b s q s qs q s q + + + + + = ( + + + + + + + + + + + + += + + + + + + + +(59) Hence, the controller has the transfer function ( )( )( ) ( )( )( )( ) ( ) ( )43 2 4 2 23 2 1 0 0 0 043 2 42 1 0 0 0 0expexpRbq s qs q s q s m p ms s a sG sb s p s p s p s m p m s + + + + + = (+ + + + + ( (60) and the reference-to-output function reads ( )( )( ) ( )( )( ) ( )( ) ( )( )( )( ) ( )( )43 2 4 2 23 2 1 0 0 0 042 2 3 2 3 20 2 1 0 3 2 1 0exp expexp expWYG sb bq s qs q s q s m p ms s a s ss m s s a s s p s p s p b s q s qs q s q (+ + + + + + ( = (+ + + + + + + + + (61) which gives rise to the characteristic quasipolynomial ( )( ) ( )( )( )( ) ( )( )42 2 3 2 3 20 2 1 0 3 2 1 0exp expmss m s s a s s p s p s p b s q s qs q s q (= + + + + + + + + + (62) Obviously, the numerator of ( )WYG sdoes not have any free parameter not included in the denominator, i.e. lnum = 0. Moreover, the factor( )40s m +has a quadruple real pole; to cancel it,itmustholdthat{ }0 1Re m s >> = .Hencelden=7.Now,therearetwopossibilities eithersetzeroexactlytoobtainconstrainedcontrollerparameter(thenlden=6)ortodeal withthenumeratoranddenominatorof(61)togetherin(46)wedecidedtoutilizethe former one. Generally, one can obtain e.g. ( )( )( )( )43 21 0 3 1 2 1 1 1 004 2 20 1 1 1expbz m qz qz qz qpmz z a z + + + += (63) from (61). Choose , ,max0.5WYmh = ,0.5 = and max10 t = s.FromFig.3wehave0.9z = , max,2normt whichgives 10.2, 0.18, 0.1 z = = = .Thentakee.g. 05 m= .Insertingplant parameters in (63) yields ( )0 0 1 2 35.4078 0.18 0.0324 0.005832 p q q q q = + (64) The concrete quasipolynomial which roots are being set, thus, reads ( ) ( )( )( )( )( )( )2 2 3 21 2 1 0 1 2 33 23 2 1 0exp 0.1 5.4078 0.18 0.0324 0.0058320.2exp 0.4m s s s s s p s p s q q q qs q s qs q s q= + + + + + + + + (65) MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 22Initial direct pole placement results in controller parameters as 3 2 1 0 2 1 01.0051, 0.9506, 1.2582, 0.2127, 1.1179, 0.4418, 0.0603 q q q q p p p = = = = = = = (66) and poles locations in the vicinity of the origin are displayed in Fig. 9. Fig. 9. Initial poles locations Theprocessofcontinuousrootsshiftingisdescribedbytheevolutionofcontroller parameters,thespectralabscissa(i.e.therealpartoftherightmostpole ,1 d )andthe distance of the dominant pole from the prescribed one ,1 ,1 d p , as can be seen in Fig. 10 Fig. 12, respectively. Note that p0 is related to shifted parameters according to (64). Fig. 10. Shifted parameters evolution Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 23 Fig. 11. Spectral abscissa evolution Fig. 12. Distance of the rightmost pole from the prescribed one When shifting, it is suggested to continue in doing this even if the desired poles locations are reached since one can obtain a better poles distribution i.e. non-dominant poles are placed moreleftinthecomplexspace.Moreover,onecandecreasethenumberofshiftedpoles during the algorithm whenever the real part of a shifted pole becomes too different from a group of currently moved poles. The final controller parameters from the continuous shifting are 3 2 1 0 2 1 04.7587, 2.1164, 2.6252, 0.4482, 0.4636, 0.529, 4.6164 q q q q p p p = = = = = = = (67) and the poles location is pictured in Fig. 13. MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 24 Fig. 13. Final poles locations As can be seen, the desired prescribed pole is reached and it is also the dominant one. Thus, optimizationcanbeomitted.However,trytoperformSOMAtofindtheminimalcost function (16) with this setting: Rad = 2, PopSize = 10, D = 6, PathLength = 3, Step = 0.21, PRT = 0.6, Migration = 10, MinDiv = 10-6, . Yet, the minimum of the cost function remains in thebestsolutionfromcontinuousshifting,i.e.accordingto(67),withthevalueofthecost function as( )42.93 10 F= v. 7. Conclusion This chapter has introduced a novel controller design approach for SISO LTI-TDS based on algebraicapproachfollowedbypole-placement-likecontrollertuningandanoptimization procedure.ThemethodologyhasbeenimplementedinMatlab-Simulinkenvironmentto verify the results. Theinitialcontrollerstructuredesignhasbeenmadeovertheringofstableandproper meromorphicfunctions,RMS,whichofferstosatisfypropernessofthecontroller,reference tracking and rejection of the load disturbance (of a nominal model). The obtained controller has owned some free (unset) parameters which must have been set properly. In the crucial part of the work, we have chosen a simple finite-dimension model, calculated its step-response maximum overshoots and times to the overshoots. Then, using a static pole placement followed by continuous pole shifting dominant poles have been attempted to be placed to the desired prescribed positions. Finally,optimizationofdistancesofdominant(therightmost)polesfromtheprescribed ones has been utilized via SOMA algorithm. The whole methodology has been tested on an attractive example of a skater on a swaying bow described by an unstable LTI TDS model. Theprocedureissimilartothealgorithmintroducedin(Michielsetal.,2010);however, therearesomesubstantialdifferencesbetweenthem.Firstly,thepresentedapproachis made in input-output space of meromorphic Laplace transfer functions, whereas the one in (Michiels et al., 2010) deals purely with state space. Second, in the cited literature, a number Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems 25 poleslessthenanumberoffreecontrollerparametersissetexactlyandtherestofthe spectrum is pushed to the left as much as possible. If it is possible it is necessary to choose otherprescribedpoles.Weinitiallyplacethepolesexactly;however,theycanleavetheir positionsduringtheshifting.Anyway,ouralgorithmdoesnotrequireresetofselection assigned poles. Moreover, we suggest unambiguously how the prescribed poles (and zeros) positions are to be chosen based on model overshoots. Last but not least, in (Michiels et al., 2010), the gradient sampling algorithm (Burke et al., 2005) on the spectral abscissa was used while SOMA together with more complex cost function is considered in this chapter. ThepresentedapproachislimitedtoretardedSISOLTI-TDSwithoutdistributeddelays only.Itsextensiontoneutralsystemsrequiressomeadditionalconditionsonstabilityand existence of a stabilizing controller. Systems with distributed delays can be served in similar wayasitisdonehere,yetwiththecharacteristicmeromorphicfunctioninsteadof quasipolynomial.Multivariablesystemswouldrequiredeepertheoreticanalysisofthe controllerstructuredesign.Themethodologyisalsotime-comsuptingandthususelessfor online controller design (e.g. for selftuners). Inthefutureresearch,onecansolvetheproblemsspecifiedabove,chooseotherreference-to-output models and control system structures. There is a space to improve and modify the optimization algorithm. 8. Acknowledgment The authors kindly appreciate the financial support which was provided by the Ministry of Education, Youth and Sports of the Czech Republic, in the grant No. MSM 708 835 2102 and bytheEuropeanRegionalDevelopmentFundundertheprojectCEBIA-TechNo. CZ.1.05/2.1.00/03.0089.9. References Ben-Israel,A.&Greville,T.N.E(1966).GeneralizedInverses:TheoryandApplications.Wiley, New York Burke,J.;Lewis,M&Overton,M.(2005).ARobustGradientSamplingAlgorithmfor NonsmoothNonconvexOptimization.SIAMJournalofOptimization,Vol.15,Issue 3, pp. 751-779, ISSN 1052-6234 Elsgolts,L.E.&Norkin,S.B.(1973).IntroductiontotheTheoryandApplicationofDifferential Equations with Deviated Arguments. Academic Press, New York Grecki, H.; Fuksa, S.; Grabowski, P. & Korytowski, A. (1989). Analysis and Synthesis of Time Delay Systems, John Wiley & Sons, ISBN 978-047-1276-22-7, New York Hale, J.K., & Verduyn Lunel, S.M. (1993). Introduction to Functional Differential Equations. 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Loiseau,J.-J.(2000).AlgebraicToolsfortheControlandStabilizationofTime-Delay Systems. Annual Reviews in Control, Vol. 24, pp. 135-149, ISSN 1367-5788 Loiseau,J.-J.(2002).Neutral-TypeTime-Delay SystemsthatarenotFormallyStablearenot BIBOStabilizable.IMAJournalofMathematicalControlandInformation,Vol.19,No. 1-2, pp. 217-227, ISSN 0265-0754 Michiels,W.&Niculescu,S.(2008).StabilityandStabilizationofTimeDelaySystems:An eigenvaluebasedapproach.AdvancesinDesignandControl,SIAM,ISBN978-089-8716-32-0, Philadelphia Peka,L.&Prokop,R.(2009).SomeobservationsabouttheRMSringfordelayedsystems, Proceedingsofthe17thInternationalConferenceonProcessControl09, pp.28-36, ISBN 978-80-227-3081-5, trbsk Pleso, Slovakia, June 9-12, 2009. Peka,L.;Prokop,R.&Dostlek,P.(2009).CircuitHeatingPlantModelwithInternal Delays.WSEASTransactiononSystems,Vol.8,Issue9,pp.1093-1104,ISSN1109-2777 Peka, L. & Prokop, R. (2010). 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Kybernetika, Vol. 44, No. 5, pp. 633-648, ISSN 0023-5954 2 Control of Distributed ParameterSystems - Engineering Methods andSoftware Support in the MATLAB &Simulink Programming Environment Gabriel Hulk, Cyril Belav, Gergely Takcs,Pavol Buek and Peter Zajek Institute of Automation, Measurement and AppliedInformatics Faculty of Mechanical EngineeringCenter for Control of Distributed Parameter SystemsSlovak University of Technology Bratislava Slovak Republic 1. Introduction Distributedparametersystems(DPS)aresystemswithstate/outputquantitiesX(x,t) /Y(x,t)parameterswhicharedefinedasquantityfieldsorinfinitedimensional quantitiesdistributedthroughgeometricspace,wherexingeneralisavectorofthe three dimensional Euclidean space. Thanks to the development of information technology andnumericalmethods,engineeringpracticeislatelymodellingawiderangeof phenomenaandprocessesinvirtualsoftwareenvironmentsfornumericaldynamical analysispurposessuchasANSYS-www.ansys.com,FLUENT(ANSYSPolyflow)- www.fluent.com,ProCASTwww.esi-group.com/products/casting/,COMPUPLAST www.compuplast.com,SYSWELDwww.esi-group.com/products/welding,COMSOL Multiphysics-www.comsol.com,MODFLOW,MODPATH,...www.modflow.com, STAR-CD-www.cd-adapco.com,MOLDFLOW-www.moldflow.com,...Basedonthe numericalsolutionoftheunderlyingpartialdifferentialequations(PDE)thesevirtual software environments offer colorful, animated results in 3D. Numerical dynamic analysis problemsaresolvedbothfortechnicalandnon-technicaldisciplinesgivenbynumerical models defined in complex 3D shapes. From the viewpoint of systems and control theory thesedynamicalmodelsrepresentDPS.A newchallengeemergesforthecontrol engineeringpractice,whichistheobjectivetoformulatecontrolproblemsfordynamical systemsdefinedasDPSthroughnumericalstructuresovercomplexspatialstructuresin 3D. Themainemphasisofthischapteristopresentaphilosophyoftheengineeringapproach for the control of DPS - given by numerical structures, which opens a wide space for novel applicationsofthetoolboxesandblocksetsoftheMATLAB&Simulinksoftware environment presented here. MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 28The first monographs in the field of DPS control have been published in the second half of the last century, where mathematical foundations of DPS control have been established. This mathematicaltheoryisbasedonanalyticalsolutionsoftheunderlyingPDE(Butkovskij, 1965;Lions,1971;Wang,1964).Thatisthe decomposition ofdynamicsintotimeand space componentsbasedontheeigenfunctionsofthePDE.Recentlyinthefieldofmathematical control theory of DPS,publications on control ofPDE have appeared (Lasiecka & Triggiani, 2000; ). AnengineeringapproachforthecontrolofDPSisbeingdevelopedsincetheeightiesofthelastcentury(Hulketal.,1981-2010).Inthefieldoflumpedparameterssystem (LPS) control, where the state/output quantities x(t)/y(t) parameters are given as finite dimensional vectors, the actuator together with the controlled plant make up a controlled LPS.Inthissensetheactuatorsandthecontrolledplantasa DPScreatea controlled lumped-input and distributed-parameter-output system (LDS). In this chapter the general decompositionofdynamicsofcontrolledLDSintotimeandspacecomponentsisintroduced,whichisbasedonnumericallycomputeddistributedparametertransient andimpulsecharacteristicsgivenoncomplexshapedefinitiondomainsin3D.Basedonthisdecompositionamethodicalframeworkofcontrolsynthesisdecompositionintospaceandtimetaskswillbepresentedwhereinspacedomainapproximation problemsaresolvedandintimedomainsynthesisofcontrolisrealizedbylumped parametercontrolloops.Forthesoftwaresupportofmodelling,controlanddesignofDPS,theDistributedParameterSystemsBlocksetforMATLAB&Simulink(DPSBlockset)-www.mathworks.com/products/connections/hasbeendeveloped withintheCONNECTIONSprogramframeworkofTheMathWorks,asaThird-Party ProductofTheMathWorksCompany(Hulketal.,2003-2010).Whensolvingproblemsinthetimedomain,toolboxesandblocksetsoftheMATLAB&Simulinksoftware environment such as for example the Control Systems Toolbox, Simulink Control Design, SystemIdentificationToolbox,etc.areutilized.Inthespacerelationtheapproximation taskisformulatedasanoptimizationproblem,wheretheOptimizationToolboxismadeuseof.A webportalnamedDistributedParameterSystemsControl- www.dpscontrol.skhasbeencreatedforthoseinterestedinsolvingproblemsofDPScontrol(Hulketal.,2003-2010).Thiswebportalfeaturesapplicationexamplesfromdifferentareasofengineeringpracticesuchasthecontroloftechnologicalandmanufacturingprocesses,mechatronicstructures,groundwaterremediationetc.MoreoverthiswebportaloffersthedemoversionoftheDPSBlocksetwiththeTutorial,Show,DemosandDPSWizardfordownload,alongwiththeInteractiveControl serviceforthe interactive solutionof model control problemsvia the Internet. 2. DPS DDS LDS Generally in the control of lumped parameter systems the actuator and the controlled plant createa lumpedparametercontrolledsystem.InthefieldofDPScontroltheactuators togetherwiththecontrolledplant-generallybeinga distributed-inputanddistributed-parameter-outputsystem(DDS)createa controlledlumped-inputanddistributed-parameter-output system (LDS). Fig. 1.-3. and Fig. 6., 11., 14.Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 29 Fig. 1. Controlled DPS as LDS - heating of metal body of complex-shape { }iiSA (s) ,{ }iiSG (s) ,{ }iiT (x, y, z)- models of actuatorsDDS- distributed-input and distributed-parameter-output system { }iiU - lumped actuating quantities /{ }ii - complex-shape definition domain in 3D / actuation subdomains Y( x,y,z,t) temperature field distributed output quantity Fig. 2. Lumped-input and distributed-parameter-output system LDS ( ) { }iiU t lumped input quantities ( ) ( ) Y x, y, z, t Y ,t = x

distributed output quantity Fig. 3. General structure of lumped-input and distributed-parameter-output systems MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 30LDS - lumped-input and distributed-parameter-output system { }iiSA- actuating members of lumped input quantities { }iiGU-generators of distributed input quantities DDS - distributed-input and distributed-parameter-output system { }ii(t) U (t) = U - vector of lumped input quantities of LDS { }iiUA (t)- output quantities of lumped parameter actuators { }iiU ( , t) - distributed output quantities of generators{ }iiGUU( , t) - overall distributed input quantity for DDS Y( , t) Y(x, y, z, t) = x- distributed output quantityInput-output dynamics of these DPS can be described, from zero initial conditions, by ( ) ( )n n nti i i i i0i 1 i 1 i 1Y( , t) Y ( , t) ( , t) U (t) , U t d= = == = = }x x x x G G (1) or in discrete form n n n ki i i i ii 1 i 1 i 1 q 0Y( , k) Y ( , k) H ( , k) U (k) H ( , q)U (k q)= = = == = = x x x x G G(2) where marksconvolutionproductand marksconvolutionsum,Gi(x,t)distributedparameterimpulseresponseofLDStothei-thinput,GHi(x,k)discretetime(DT) distributed parameter impulse response of LDS with zero-order hold units H - HLDS to thei-thinput,Yi(x,t)-distributedoutputquantityofLDStothei-thinput,Yi(x,k)DT distributedoutputquantityofHLDStothei-thinput.Forsimplicityinthischapter distributedquantitiesareconsideredmostlyascontinuousscalarquantityfieldswithunit samplingintervalintimedomain.WhereasDTdistributedparameterstepresponses ( ) { }ii ,kH ,k x H of HLDS can be computed by common analytical or numerical methodsthen DT distributed parameter impulse responses can be obtained as ( ) ( ) ( ) { }i i ii ,kH ,k H ,k H ,k-1 = x x x G H H (3) 3. Decomposition of dynamics The process of dynamics decomposition shall be started from DT distributed parameter step and impulse responses of the analysed LDS. For an illustration, procedure of decomposition ofdynamicsandcontrolsynthesiswillbeshownontheLDSwithzero-orderholdunitsHHLDSdistributedonlyontheinterval[ ] 0, L ,withoutputquantity ( ) ( )nii 1Y x, k Y x, k==discretisedintimerelationandcontinuousinspacerelationonthis interval.Neverthelessthefollowingresultsarevalidingeneralbothforcontinousor discretedistributedquantitiesinspacerelationgivenoncompex-shapedefinitiondomains over 3D as well. Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 31 Fig. 4. i-th discrete distributed parameter impulse response ofHLDS ( )i iH x , k G- partial DT impulse response in time, t - relation to the i-th input considered as response with maximal amplitude in point[ ]ix 0, L ( ) { }ii ,kH x, k G- partial DT impulse responses to the i-th input in space - x - relation ( ) { }ii ,kHR x, k G reduced partial DT impulse responses to the i-th input in space,x relationfor timesteps{ }kkIf the reduced DT partial distributed parameter impulse responses are defined as iii ii ,kH (x, k)HR (x, k)H (x , k) = ` )GGG(4) for( ) { }i ii ,kH x , k 0 G , then the i-th DT distributed output quantiy in (2) can be rewritten by the means of the reduced characteristics as follows ( ) ( ) ( ) ( )i i i i iY x, k H x , k HR x, k U k = G G (5) At fixed ixthe partial DT distributed output quantity in time direction ( )i iY x , kis given as theconvolutionsum( ) ( )i i iH x , k U k G =ki i iq 0H (x , q)U (k q)=G ,incasetherelation ( ) { }i iq 0,kHR x , q 1== Gholds at the fixed point ix . At fixed k, the partial discrete distributed outputquantityinspacedirection( )iY x, k isgivenasalinearcombinationofelements ( ) { }iq 0,kHR x, q=G ,wherethereduceddiscretepartialdistributedcharacteristics ( ) { }iq 0,kHR x, q=G aremultipliedbycorrespondingelementsoftheset ( ) ( ) { }i i iq 0,kH x , q U k q= G ., see Fig. 5. MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 32Thisdecompositionisvalidforallgivenlumpedinput{ }iiU andcorrespondingoutput quantities( ) { }ii ,kY x, k -thusweobtaintimeandspacecomponentsofHLDSdynamicsin the following form:TimeComponentsof Dynamics( ) { }i ii ,kH x ,k G for given i and chosen xi-

variable kSpaceComponentsof Dynamics( ) { }ii,kHR x,k G for given i and chosen k variable x Also reduced components of single distributed output quantities are ( ) { }( )( )iiii ii ,kY x,kYR x,kY x ,k = ` )(6) then( ) { }i ii ,kY x , k 0 canbeconsideredastimecomponentsand( ) { }ii ,kYR x, k asspace components of the output quantities. Whenreducedsteady-statedistributedparametertransientresponsesareintroduced ( ) { } ( ) ( ) { }i i i ii iHR x, H x, / H x , = H H H -for( ) { }i iiH x , 0 H -anddiscretetransfer functions( ) { }i iiSH x ,z areassignedtopartialdistributedparametertransientresponses withmaximalamplitudesatpoints{ }iix ontheinterval[ ] 0, L ,weobtaintimeandspace components of HLDS dynamics for steady-state as: TimeComponentsofDynamics( ) { }i iiSH x ,z - for given i and chosen xi- variable zSpaceComponentsofDynamics( ) { }iiHR x, H - for given i in variable x Fig. 5. Partial distributed output quantities in time and space direction iU- i-th DT lumpedinput quantity ( )i iY x , k- i-th partial DT distributed output quantity intime domain at chosen pointix( )iY x, k / ( )iYR x, k -i-thpartialdistributedoutput/reducedoutputquantityinspace direction Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 33 For the steady-state, whenk ( ) ( ) { }i ii ,kYR x, HR x, H (7) then in the steady-state ( ) ( ) ( )n ni i i i i ii 1 i 1Y(x, ) Y (x , )YR x, Y x , HR x,= == = H (8) Whendistributedquantitiesareusedindiscreteformasfinitesequencesofquantities,the discretizationinspacedomainisusuallyconsideredbythecomputationalnodesofthe numerical model of the controlled DPS over the compex-shape definition domain in 3D. 4. Distributed parameter systems of control BasedondecompositionofHLDSdynamicsintotimeandspacecomponents,possibilities forcontrolsynthesisarealsosuggestedbyananalogousapproach.Inthissectiona methodicalframeworkforthedecompositionofcontrolsynthesisintospaceandtime problemswill bepresentedbyselectdemonstrationcontrolproblems.In thespace domaincontrolsynthesiswillbesolvedasasequenceofapproximationtasksonthesetofspace componentsofcontrolledsystemdynamics,wheredistributedparameterquantitiesin particularsamplingtimesareconsideredascontinuousfunctionsontheinterval[ ] 0, L as elementsofstrictlyconvexnormedlinearspaceXwithquadraticnorm.Itisnecessary to noteasabovethatthefollowingresultsarevalidingeneralforDPSgivenoncompex-shapedefinitiondomainsin3Dbothforcontinousordiscretedistributedquantities,inthe space relation as well. Inthetimedomain,thecontrolsynthesissolutionsarebasedonsynthesismethodsofDT lumped parameter systems of control.4.1 Open-loop control Assumetheopen-loopcontrolofadistributedparametersystem,wheredynamic characteristicsgiveanidealrepresentationofcontrolledsystemdynamicsand( ) V x, t 0 = , thatiswithzeroinitialsteady-state,inwhichallvariablesinvolvedareequaltozerosee seeFig.6forreference.Letusconsiderastepchangeofdistributedreferencequantity- ( ) ( ) W x, k W x, = ,seeFig.7.Forsimplicityletthegoalofthecontrolsynthesisisto generateasequenceofcontrolinputs( ) U k insuchfashionthatinthesteady-state, for k ,thecontrolerror( ) ( ) ( ) E x, k W x, Y x, k = willapproachitsminimalvalue ( ) E x, in the quadratic norm: ( ) ( ) ( ) ( ) min E x, min W x, Y x, E x, = = (9) First, an approximation problem will be solved in the space synthesis (SS) block: ( ) ( ) ( )ni i ii 1min W x, W x , HR x,= H (10) MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 34 Fig. 6. Distributed parameter open-loop system of control LDS - lumped-input and distributed-parameter-output system H - zero-order hold units HLDS - controlled system with zero-order hold units CS - control synthesis TS - time part of control synthesis SS - space part of control synthesis ( ) ( ) ( ) Y x, t /W x, k W x, = - distributed controlled/reference quantity ( ) V x, t

- distributed disturbance quantity W= ( ) { }i iiW x , - vector of lumpedreference quantities ( ) U k - vector oflumped control quantities ( ) ( ) { }i i iiSH x , z / HR x, H - time/space components of controlled system dynamics where( ) { }iiHR x, Hare steady-state reduced distributed parameter transient responses of the controlled system HLDS and( ) { }i iiW x , are parameters of approximation. Functions ( ) { }iiHR x, H formafinite-dimensionalsubspaceofapproximationfunctionsFninthe strictlyconvexnormedlinearspaceofdistributedparameterquantitiesXon[ ] 0, L with quadratic norm, where the approximation problem is to be solved, see Fig. 8. for reference. From approximation theoryin this relation is known the theorem:LetFnbeafinite-dimensionalsubspaceofastrictlyconvexnormedlinearspaceX.Then,foreach f X, there exists a unique element of best approximation. (Shadrin, 2005). So the solution of the approximation problem (10) is guaranteed as a unique bestapproximation( ) ( ) ( )ni i ii 1WO x, W x , HR x,== H ,where( ) { }i iiW W x , = isthe vector of optimal approximation parameters. Hence: Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 35 ( ) ( ) ( )( ) ( ) ( ) ( ) ( )ni i ii 1ni i ii 1min W x, W x , HR x,W x, W x , HR x, W x, WO x,== == = HH(11) Fig. 7. Step change of distributed reference quantity HLDS- controlled system with zero-order hold units { }iiU - lumped control quantities ( ) ( ) W x, k W x, = - step change of distributed reference quantity Fig. 8. Solution ofthe approximation problem HLDS- controlled systemwith zero-order hold units { }iiU - lumped control quantities { } ( ) { }i i ii iW W x , = - optimal approximation parameters, lumped references ( ) { }iiHR x, H - reduced steady-state distributed parameter transientresponses MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 36( ) W x, - distributed reference quantity ( ) WO x, - unique best approximation of reference quantity Let us assume vectorW enters the block oftime synthesis (TS). In this block, there are n single-input /single-output (SISO) lumped parameter control loops:( ) ( ) { }i i iiSH x , z , R z , see Fig.9.forreference.Thecontrolledsystemsoftheseloopsarelumpedparametersystems assigned to HLDS as time components of dynamics:( ) { }i iiSH x , z . Controllers,( ) { }iiR z , are to be chosen such that fork the following relation holds: ( ) ( ) ( ){ } i i i i i ik ki ,klimE x , k lim W x , Y x , k 0 (= = (12) Fig. 9. SISO lumped parameter control loops in the block TS TS - time part of control synthesis ( ) { }i iiSH x , z - time components of HLDS dynamics ( ) { }iiR z- lumped parameter controllers Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 37 ( ) { } } {( ) { }i i i i ii ,kiiY x , k / W W x , = - controlled/reference quantities ( ) { }i ii ,kE x , k- lumped control errors( ) { }ii ,kU k - lumped control quantities ( ) ( ) W k /U k- vector of lumped reference/control quantities When the individual components of the vector( ) { }i iiW W x , = are input in the particular control loops:( ) ( ) { }i i iiSH x , z , R z , the control processes take place. The applied control laws result in the sequences of control inputs:( ) { }ii ,kU k , and respectively the output quantities, fork converging to reference quantities ( ) ( ) ( ) { }i i i i i ii ,kY x , k Y x , W x , = (13) Values ofthese lumped controlled quantities in new steady-state will be further denoted as ( ) ( ) { }i i i iiY x , W x , = (14) see Fig. 9. 10. for reference. Fig. 10. Quantities of distributed parameter open-loop control in new steady-state HLDS - controlled system with zero-order hold units { }iiU - lumpedcontrol quantities ( ) ( ) { }i i i i iiY x , /W W x , = - controlled/reference quantities in new steady-state ( ) { }i iiE x , - lumped control errors ( ) Y x, - controlled distributed quantity in new steady-state MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 38( ) W x, - distributed reference quantity ( ) WO x, - uniquebest approximation of reference quantity ( ) E x, - distributed control error with minimal norm Thenaccordingtoequations(12-14)forthenewsteady-stateitholds: ( ) ( ) ( ) ( ) { }i i i i i iiY x , HR x, W x , HR x, = H H ,whichimpliesthattheoveralldistributed outputquantityatthetimek :( ) Y x, givestheuniquebestapproximationofthe distributed reference variable:( ) W x, ( ) ( ) ( ) ( ) ( ) ( )n ni i i i i ii 1 i 1Y x, Y x , HR x, W x , HR x, WO x,= = = = H H = (15) Therefore the control error has a unique form as well, with minimal quadratic norm ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )ni i ii 1ni i ii 1E x, W x, Y x, W x, Y x , HR x,W x, W x , HR x, W x, WO x,== = = = = HH(16) Thus the control task, defined at equation (9), is accomplished with( ) { }i iiE x , 0 =- see Fig. 10. for reference. As the conclusion of this section we may state that the control synthesis in open-loop control systemis realized as: Time Tasks of Control Synthesis in lumped parameter control loopsSpace Tasks of Control Synthesis as approximation task. WhenmathematicalmodelscannotprovideanidealrepresentationofcontrolledDPS dynamicsanddisturbancesarepresentwithanoveralleffectontheoutputinsteady-state expressed by( ) EY x, then ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )W x, Y x, EY x, W x, Y x, EY x,W x, WO x, EY x, E x, EY x, + = + = + (17) Finallyatthedesignstageofacontrolsystem,foragivendesiredqualityofcontrol in space domain, it is necessary to choose appropriate number and layout of actuators for the fulfillment of this requirement ( ) ( ) E x, EY x, + (18) 4.2 Closed-loop control with block RHLDS Letusconsidernowa distributedparameterfeedbackcontrolloopwithinitialconditions identicalasthecaseabove,seeFig.11.InblocksSS1a SS2approximationproblemsare solvedwhileinblockRHLDSreduceddistributedoutputquantities( ) { }ii ,kYR x, k are Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 39 generated.BlockTSinFig.12.,containsthecontrollers ( ) { }iiR z designedasthecontrollers forSISOlumpedparametercontrolloops( ) ( ) { }i i iiSH x , z , R z withrespecttotherequest formulated by equation (12). In the k-th step in block SS2 at approximation of( ) Y x, k on the subspace of( ) { }ii ,kYR x, k ( ) ( ) ( )ni i ii 1min Y x, k Y x , k YR x, k=(19) timecomponentsofpartialoutputquantities( ) { }i iiY x , kareobtained,inblockSS1 reference quantities( ) { }i iiW x , are computed. Then on the output of the algebraic block is ( ) ( ) ( ) { }i i i i i iiE x , k W x , Y x , k = . These sequences( ) { }i iiE x , k enter into the TS on ( ) { }iiR zandgive( ) { }iiU k ,whichenterintoHLDSwith( ) { }i iiY x , kontheSS2output-among ( ) { }iiU k and( ) { }i iiY x , k there arerelations( ) { }i iiSH x , z .-Thisanalysisofcontrol synthesisprocessshowsthatsynthesisintimedomainisrealizedonthelevelofone parameter control loops( ) ( ) { }i i iiSH x , z , R z , Fig. 9. Fig. 11. Distributed parameter closed-loop system of control with reduced space components of output quantity MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 40HLDS - controlled system with zero-order hold units RHLDS - model for reduced space components:( ) { }ii ,kYR x, kCS - control synthesis TS/SS1,SS2 - time/space parts of control synthesis K - time/space sampling ( ) Y x, t - distributed output quantity ( ) ( ) W x, , V x, t -reference and disturbance quantities ( ) { } ( ) { }i i i ii ,k i ,kY x , k Y x , k = time components of output quantity ( ) { }i i iiW W x , = / ( ) { }i ii ,kE x , k- reference quantities/control errors ( ) U k - vector of lumped control quantities Fig. 12. The block of time synthesis TS - time part ofcontrol synthesis ( ) { }iiR z- lumped parameter controllers ( ) ( ) { }i ii ,kE k E x , k =- vector of lumped control errors( ) ( ) { }ii ,kU k U k = - vector of lumped control quantities Fork ( ) ( ) { }i ii ,kYR x,k HR x, H ,( ) { } ( ) ( ) { }i i i i i ii ,k iY x , k Y x , W x , = alongwith ( ) ( ) { }i i i ii ,kE x , k E x , 0 = . Thus the control task, defined by equation (9) is accomplished as given by relation (16). In case of the uncertainty of the control process relations similar to (17-18) are also valid.Lets consider now the approximation of( ) W x, in the block SS1 in timestep k, on the set of( ) { }ii ,kYR x, k .Theninthecontrolprocesssequencesofquantities( ) { }i ii ,kx , k Ware obtained,asdesiredquantitiesofSISOcontrolloops( ) ( ) { }i i iiSH x , z , R z whichareclosed throughout the blocks TS, HLDS and SS2, see Fig. 13. Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 41 Fig. 13. Lumped parameter SISO control loops i-th control loop ( )i iSH x , z - i-th time component of HLDS dynamics ( )iR z - i-th lumped parameter controller ( ) ( )i i iY x , k /U k- i-th controlled/control quantity ( ) ( )i i i iW x , k /E x , k - i-th desired quantity/control error Iffork ( ) ( ) { }i ii ,kYR x,k HR x, H , ( ) ( ) { } ( ) ( ) { }i i i i i i i ii ,k iY x , k Y x , k Y x , W x , = = along with( ) ( ) { }i i i ii ,kE x , k E x , 0 = - this actually means that the control task defined in equation (9), is accomplished as given by relation (16).Finallywemaystateasasummary,thatinclosed-loopcontrolwithRHLDSthecontrol synthesis is realized as: Time Tasks of Control Synthesis on the level of lumped parameter control loopsSpace Tasks of Control Synthesis as approximation tasks. AtthesametimethesolutionoftheapproximationprobleminblockSS1onthe approximation set( ) { }ii ,kYR x, k ( ) ( ) ( )ni i ii 1min W x, W x , k YR x, k=(20) in timestep kis obtained ( ) ( ) ( ) ( )ni i ii 1W x, W x , k YR x, k E x, k== + (21) where( ) E x, kistheuniqueelementatthebestapproximationof( ) W x, onthesetof approximatefunctions( ) { }iiYR x, k .Similarybythesolutionofapproximationproblemin the block SS2 - ( ) { }i iiY x , kin the timestep k distributed output quantity( ) Y x, k is given as ( ) ( ) ( )ni i ii 1Y x, k Y x , k YR x, k==(22) MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 424.3 Closed-loop control Let us now consider a distributed parameter feedback loop as featured in Fig. 14. with initial conditions as above, where in timestep k an approximation problem is solved ( ) ( ) ( )ni i ii 1min E x, k E x , k YR x, k=(23) and as a result in timestep k a vector( ) ( ) { }i iiE k E x , k =is obtained. By relations (20-22)the further equations are valid ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )ni i ii 1ni i ii 1ni i i i ii 1ni i i i ii 1min E x, k E x , k YR x, kmin W x, Y x, k E x , k YR x, kmin W x, Y x , k E x , k YR x, kW x, Y x , k E x , k YR x, k==== == = (= + = (= + (24) Theproblemsolution( ) ( ) ( ) ( ) ( )ni i i i ii 1W x, Y x , k E x , k YR x, k E x, k= (= + + isobtainedby approximation. Fig. 14. Distributed parameter closed-loop system of control HLDS- LDSwith zero-order hold units Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 43 CS control synthesis TS /SS time/space control synthesisK time/space sampling ( ) ( ) Y x, t /W x, distributed controlled/reference quantity( ) { }i iiSH x , z - transfer functions - dynamic characteristics of HLDS in timedomain

( ) { }iiYR x, k / ( ) { }iiHR x, H- reduced characteristics in space domain ( ) E x, k distributed control error ( ) V x, t distributed disturbance quantity ( ) E k vector of control errors ( ) U k vector of control quantities Comparisonofrelation(21)andresultoftheapproximationproblem(24)gives ( ) ( ) ( ) ( ) ( ) ( ) ( )n ni i i i i i i ii 1 i 1W x , k YR x, k E x, k Y x , k E x , k YR x, k E x, k= = (+ = + + andthen ( ) ( ) ( ) ( ) ( )n ni i i i i i i ii 1 i 1W x , k YR x, k Y x , k E x , k YR x, k= = (= + ,finally ( ) ( ) ( ) { }i i i i i ii ,kW x , k Y x , k E x , k = + is obtained.Now in vector form this means ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { }i i i i i ii ,kW k Y k E k E k W k Y k E x , k W x , k Y x , k = + = = = (25) Then sequences( ) { }i ii ,kE x , kenter into the block TS on( ) { }iiR z and give( ) { }iiU k , among ( ) { }iiU k and( ) { }i iiY x , k there arerelations( ) { }i iiSH x , z .Finallythisanalysisofcontrol synthesisprocessshowsthatsynthesisintimedomainisrealizedonthelevelofone parametercontrolloops( ) ( ) { }i i iiSH x , z , R z ,Fig.13.-closedthroughoutthestructureof distributedparametercontrolloop,Fig.14.Iffork ( ) ( ) { }i iiYR x,k HR x, H and ( ) { } ( ) ( ) { }i i i i i ii ,k iY x , k Y x , W x , = along with( ) ( ) { }i i i ii ,kE x , k E x , 0 = then ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )n ni i i i i ii 1 i 1n n ni i i i i i i i ii 1 i 1 i 1n ni i i i i i i ii 1 i 1min E x, E x , HR x, W x, Y x, E x , HR x,W x , HR x, E x, Y x , HR x, E x , HR x,W x , Y x , HR x, E x, E x , HR x, E x,= == = == = = == + = (= + = H HH H HH H -(26) isvalid,sotheabovegivencontroltask(9)isaccomplished-whereasinthesteady-state ( ) ( ) ( ) { }i i i i i iiW x , Y x , E x , 0 = = .Byconcludingtheabovepresenteddiscussion,the control synthesis in closed-loop control isrealized as: MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 44Time Tasks of Control Synthesis on the level of lumped parameter control loopsSpace Tasks of Control Synthesis as approximation tasks. When mathematical models cannot provide an ideal description of controlled DPS dynamics and disturbances are present with an overall effect on the output in steady-state, expressed by( ) EY x, then the realtions similar to (17-18) are also valid here.Inpracticemostlyonlyreduceddistributedparametertransientresponsesinsteady-state ( ) { }iiHR x, H areconsideredforthesolutionoftheapproximationtasksintheblockSSof the scheme in Fig. 14. along with robustification of controllers( ) { }iiR z.

ForsimplicityproblemsofDPScontrolhavebeenformulatedhereforthedistributed desiredquantity ( ) W x, .Incaseof( ) W x, k isassumed,thecontrolsynthesisisrealized similarly: -In Space Domain - as problem of approximation in particular sampling intervals -InTimeDomain-ascontrolsynthesisinlumpedparametercontrolloops,closed throughout structures of the distributed parameter control loop.Thesolutionofthepresentedproblemsofcontrolsynthesisanassumptionisused,thatin theframeworkofthechosencontrolsystemstheprescribedcontrolqualitycanbereached bothintheinspaceandtimedomain.Howeverinthedesignofactualcontrolsystemsfor the given distributed parameter systems, usually theoptimization of the number and layout of actuators optimization of dynamical characteristics of lumped/distributed parameter actuatorsoptimization of dynamical characteristics of lumped parameter control loops is required and necessary. 5. Distributed Parameter Systems Blockset for MATLAB & Simulink AsasoftwaresupportforDPSmodelling,controlanddesignofproblemsinMATLAB&SimulinktheprogrammingenvironmentDistributedParameterSystemsBlocksetforMATLAB&Simulink(DPSBlockset)-aThird-PartyProductofTheMathWorkswww.mathworks.com/products/connections/Fig.15.,hasbeendevelopedwithintheprogramCONNECTIONSofTheMathWorksCorporationbytheInstituteof Automation,MeasurementandAppliedInformaticsofMechanicalEngineeringFaculty, SlovakUniversityofTechnologyinBratislava(IAMAI-MEF-STU)(Hulketal.,2003-2010).Fig.16.showsThelibraryofDPSBlockset.TheHLDSandRHLDSblocksmodel controlledDPSdynamicsdescribedbynumericalstructuresasLDSwithzero-orderholdunits-H.DPSControlSynthesisprovidesfeedbacktodistributedparameter controlledsystemsincontrolloopswithblocksfordiscrete-timePID,Algebraic,State-SpaceandRobustSynthesis.TheblockDPSInputgeneratesdistributedquantities, whichcanbeusedasdistributedreferencequantitiesordistributeddisturbances,etc. DPSDisplaypresentsdistributedquantitieswithmanyoptionsincludingexporttoAVI files.TheblockDPSSpaceSynthesisperformsspacesynthesisasanapproximation problem.Asademonstration,someresultsofthediscrete-timePIDcontrolofcomplex-shapemetal body heating by the DPS Blockset are shown in Fig. 17.-19., where the heating process was modelledbyfiniteelementmethodintheCOMSOLMultiphysicsvirtualsoftware environment -www.comsol.com.Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 45 TheblockTutorialpresentsmethodologicalframeworkforformulationandsolutionof control problems for DPS. The block Show contains motivation examples such as: Control of temperature field of 3D metal body (the controlled system was modelled in the virtual software environment COMSOL Multiphysics); Control of 3D beam of smart structure (the controlled systemwasmodelledinthevirtualsoftwareenvironmentANSYS);Adaptivecontrolofglass furnace (the controlled system was modelled by Partial Differential Equations Toolbox of the MATLAB ), and Groundwater remediation control (the controlled system was modelled in the virtual software environment MODFLOW). The block Demos contains examples oriented at the methodology of modelling and control synthesis. The DPS Wizard gives an automatized guideforarrangementandsettingdistributedparametercontrolloopsinstep-by-step operation. Fig. 15. Distributed Parameter Systems Blockset on the web portal of The MathWorks Corporation MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 46 Fig. 16. The library of Distributed Parameter Systems Blockset forMATLAB & Simulink Third-Party Product of The MathWorksControl of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 47 Fig. 17. Distributed parameter control loop for discrete-timePID control ofheating ofmetal body in DPS Blockset environment Fig. 18. Distributed referenceandcontrolledquantities of metal body heating over the numerical net Fig. 19. Quadratic norm of distributed control error and discrete lumped actuating quantities at discrete-timePID control ofheating ofmetal body in DPS Blockset environment MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 486. Interactive control via the Internet For the interactive formulation and solution of DPS demonstration control problems via the Internet,anInteractiveControlservicehasbeenstartedonthewebportalDistributed ParameterSystemsControl-www.dpscontrol.skoftheIAMAI-MEF-STU(Hulk, 2003-2010)seeFig.20.fora screenshotofthesite.Intheframeworkoftheproblem formulation,firstthecomputationalgeometryandmesharechoseninthecomplex3D shapedefinitiondomain,thenDTdistributedtransientresponsesarecomputedinvirtual softwareenvironmentsfornumericaldynamicalanalysisofmachinesandprocesses. Finally, the distributed reference quantity is specified in points of the computational mesh - Fig.18.Representingthesolutiontothoseinterestedanimatedresultsofactuating quantities, quadratic norm of control error, distributed reference and controlled quantity are sent in the form ofDPS Blockset outputs see Fig. 17-19. for illustration. Fig. 20. Web portal Distributed Parameter Systems Control with monograph Modeling, Control and Design of Distributed Parameter Systems with Demonstrations in MATLAB and service Interactive Control 7. Conclusion Theaimofthischapteristopresentaphilosophyoftheengineeringapproachforthe controlofDPSgivenbynumericalstructures,whichopensa widespacefornovel applicationsofthetoolboxesandblocksetsoftheMATLAB&Simulinksoftware environment.Thisapproachisbasedonthegeneraldecompositionintotimeandspace components of controlled DPS dynamics represented bynumerically computed distributed parameter transient and impulse characteristics, given on complex shape definition domains in 3D. Starting out from this dynamics decomposition a methodical framework is presented Control of Distributed Parameter Systems - Engineering Methodsand Software Support in the MATLAB & Simulink Programming Environment 49 fortheanalogousdecompositionofcontrolsynthesisintothespaceandtimesubtasks.In spacedomainapproximationproblemsaresolved,whileinthetimedomaincontrol synthesisisrealizedbylumpedparameterSISOcontrolloops(Hulketal.,1981-2010). BasedonthesedecompositionasoftwareproductnamedDistributedParameterSystems BlocksetforMATLAB&Simulink-aThird-PartysoftwareproductofTheMathWorks- www.mathworks.com/products/connections/hasbeendevelopedwithintheprogram CONNECTIONSofTheMathWorksCorporation,(Hulketal.,2003-2010),wheretime domaintoolboxesandblocksetsofsoftwareenvironmentMATLAB&SimulinkasControl SystemsToolbox,SimulinkControlDesign,SystemIdentificationToolbox,...aremadeuse of.Inthespacedomainapproximationproblemsaresolvedasoptimizationproblemsby means of the Optimization Toolbox.Forthefurthersupportofresearchinthisareaa webportalnamed Distributed Parameter Systems Control - www.dpscontrol.sk was realized (Hulk et al., 2003-2010), see Fig. 20. for anillustration.Ontheabovementionedwebportal,theonlineversionofthemonograph titledModeling,ControlandDesignofDistributedParameterSystemswith Demonstrations in MATLAB - www.mathworks.com/support/books/ (Hulk et al., 1998), ispresentedalongwithapplicationexamplesfromdifferentdisciplinessuchas:controlof technologicalandproductionprocesses,controlanddesignofmechatronicstructures, groundwaterremediationcontrol,etc.Thiswebportalalsooffersforthoseinterestedthe downloadofthedemoversionoftheDistributedParameterSystemsBlocksetfor MATLAB&SimulinkwithTutorial,Show,DemosandDPSWizard.Thisportalalso offerstheInteractiveControlserviceforinteractivesolutionofmodelcontrolproblemsof DPS via the Internet. 8. Acknowledgment ThisworkwassupportedbytheSlovakScientificGrantAgencyVEGAunderthecontract No.1/0138/11forprojectControlofdynamicalsystemsgivenbynumericalstructuresas distributedparametersystemsandtheSlovakResearchandDevelopmentAgencyunderthe contractNo. APVV-0160-07 for project Advanced Methods for Modeling, Control and Design of MechatronicalSystemsasLumped-inputandDistributed-outputSystemsalsotheprojectNo. APVV-0131-10High-techsolutionsfortechnologicalprocessesandmechatroniccomponentsas controlled distributed parameter systems.9. References Butkovskij, A. G. (1965). Optimal control of distributed parameter systems. Nauka, Moscow (in Russian) Hulk, G. et al. (1981). On Adaptive Control of DistributedParameterSystems, Proceedings of 8-th World Congress of IFAC, Kyoto, 1981 Hulk,G.etal.(1987).ControlofDistributedParameterSystemsbymeansofMulti-Input and Multi-Distributed-Output Systems, Proceedings of 10-th World Congress ofIFAC, Munich, 1987 Hulk,G.(1989).IdentificationofLumpedInputandDistributedOutputSystems, Proceedings of 5-th IFAC / IMACS / IFIP Symposium on Control of Distributed Parameter Systems, Perpignan, 1989 MATLAB for Engineers Applications in Control, Electrical Engineering, IT and Robotics 50Hulk,G.etal.(1990).ComputerAidedDesignofDistributedParameterSystemsofControl, Proceedings of 11-th World Congress of IFAC, Tallin, 1990HulkG.(1991).LumpedInputandDistributedOuptutSystemsatthe