Table Of Content

Modern Control Systems with ™ LabVIEW Robert H. Bishop Modern Control Systems with LabVIEWTM Robert H. Bishop Marquette University ISBN-13:978-1-934891-18-6 ISBN-10:1-934891-18-5 10 9 8 7 6 5 4 3 2 Publisher:TomRobbins GeneralManager:ErikLuther TechnicalOversight:AndyChang DevelopmentEditor:CatherinePeacock Compositor:PaulMailhot,PreTeXInc. (cid:2)c2012NationalTechnologyandSciencePress. Allrightsreserved.Neitherthisbook,noranyportionofit,maybecopiedorreproducedinanyformorbyanymeanswithoutwrittenpermissionofthepublisher. NTSPressrespectstheintellectualpropertyofothers,andweaskourreaderstodothesame. Thisbookisprotectedbycopyrightandotherintellectualproperty laws. Wherethesoftwarereferredtointhisbookmaybeusedtoreproducesoftwareorothermaterialsbelongingtoothers,youshouldusesuchsoftwareonlyto reproducematerialsthatyoumayreproduceinaccordancewiththetermsofanyapplicablelicenseorotherlegalrestriction. LabVIEW,USRP,UniversalSoftwareRadioPeripheral,andNationalInstrumentsaretrademarksofNationalInstruments. Allothertrademarksorproductnamesarethepropertyoftheirrespectiveowners. AdditionalDisclaimers: Thereaderassumesallriskofuseofthisbookandofallinformation,theories,andprogramscontainedordescribedinit. Thisbookmaycontaintechnical inaccuracies,typographicalerrors,othererrorsandomissions,andout-of-dateinformation. Neithertheauthornorthepublisherassumesanyresponsibilityor liabilityforanyerrorsoromissionsofanykind,toupdateanyinformation,orforanyinfringementofanypatentorotherintellectualpropertyright. Neithertheauthornorthepublishermakesanywarrantiesofanykind,includingwithoutlimitationanywarrantyastothesufficiencyofthebookorofany information,theories,orprogramscontainedordescribedinit,andanywarrantythatuseofanyinformation,theories,orprogramscontainedordescribedin thebookwillnotinfringeanypatentorotherintellectualpropertyright. THISBOOKISPROVIDED“ASIS.”ALLWARRANTIES,EITHEREXPRESSOR IMPLIED,INCLUDING,BUTNOTLIMITEDTO,ANYANDALLIMPLIEDWARRANTIESOFMERCHANTABILITY,FITNESSFORAPARTICULAR PURPOSE,ANDNON-INFRINGEMENTOFINTELLECTUALPROPERTYRIGHTS,AREDISCLAIMED. Norightorlicenseisgrantedbypublisherorauthorunderanypatentorotherintellectualpropertyright,expressly,orbyimplicationorestoppel. INNOEVENTSHALLTHEPUBLISHERORTHEAUTHORBELIABLEFORANYDIRECT,INDIRECT,SPECIAL,INCIDENTAL,COVER,ECONOMIC, ORCONSEQUENTIALDAMAGESARISINGOUTOFTHISBOOKORANYINFORMATION,THEORIES,ORPROGRAMSCONTAINEDORDESCRIBED INIT,EVENIFADVISEDOFTHEPOSSIBILITYOFSUCHDAMAGES,ANDEVENIFCAUSEDORCONTRIBUTEDTOBYTHENEGLIGENCEOF THEPUBLISHER,THEAUTHOR,OROTHERS.Applicablelawmaynotallowtheexclusionorlimitationofincidentalorconsequentialdamages,sotheabove limitationorexclusionmaynotapplytoyou. C O N T E N T S Preface v 1 MathematicalModelsofSystems 1 2 StateVariableModels 18 3 FeedbackControlSystemCharacteristics 22 4 PerformanceofFeedbackControlSystems 28 5 StabilityofLinearFeedbackSystems 39 6 RootLocusMethod 48 7 FrequencyResponseMethods 53 8 StabilityintheFrequencyDomain 59 9 DesignofFeedbackControlSystems 70 10 DesignofStateVariableFeedbackSystems 77 11 RobustControlSystems 88 12 DigitalControlSystems 92 iii P R E F A C E WelcometoControlSystemDesignwithLabVIEWTM As a companion to the textbook, Modern Control Systems by Richard C. Dorf and Robert H. Bishop, this supplement provides a set of comprehensive tutorials and exercises utilizing the LabVIEW Control Design and Simulation Module. NI LabVIEW is a graphicalandtextuallanguageforprototypingrealcontrolandsignalprocessingsystemsenablingaseamlessflowfromsimulation to deployment on real-time control hardware. This text guides students through the process of modeling a system, analyzing the modeltobuildacontroller,andvalidatingtherobustnessofthecontrolsystemthoroughsimulation. Itscontentsandorganization mirrorthecorrespondingsectionsinMCSmakingitanidealteachingcompanion,especiallywhenLabVIEWisbeingutilizedina correspondingcontrolslaboratory. ItisassumedreaderswillhaveaccesstoLabVIEW2010orlater,theControlDesignandSimulationModule,andMathscript RT.Withthesetoolsthereadercaneasilybuild,simulate,andanalyzetheexamplesandproblemsincludedinthetext. SolutionVIs formanyoftheproblemsareincludedforuseasaprogrammingreferenceandcanalsobeusedasastartingpointforsolvingmore advanceddesignproblems. AlloftheLabVIEWexamplesweredevelopedandtestedonaPCcompatiblewithLabVIEWExpress 2010. TheavailableVIscanbedownloadedfromhttp://www.ntspress.com/publications/modern-control-systems-with-labview. ForstudentsunfamiliarwithLabVIEW,itwillbeveryhelpfultohaveaccesstotheLearningwithLabVIEW textbookbyRobert H. Bishop, available from Prentice Hall. For readers new to LabVIEW control, a wealth of documentation exists. Please see www.ni.com/academic/controls.htm. WewishtoexpressappreciationtoAndyChangatNationalInstrumentsandtoJorgeAlvarezatTheUniversityofTexasatAustin fortheirworkincreatingtheLabVIEWVIsandthescreencapturesusedinthemakingofthiscompaniontext. Specialthanksalso tothefolksatNationalInstruments,especiallyErikLutherandDr. JeannieFalcon,fortheircontinuedsupport. RobertH.Bishop v C H A P T E R 1 Mathematical Models of Systems Applicationofthemanyclassicalandmoderncontrolsystemdesignandanalysistoolsisbasedonmathematicalmodels. LabVIEW canbeusedwithsystemsgivenintheformoftransferfunctiondescriptions. WebeginthischapterbyshowinghowtouseLabVIEW to assist in the analysis of a typical spring-mass-damper mathematical model of a mechanical system. Using a LabVIEW virtual instrument(orVI)wecandevelopaninteractiveanalysiscapabilitytoanalyzetheeffectsofnaturalfrequencyanddampingonthe unforcedresponseofthemassdisplacement. Wealsodiscusstransferfunctionsandblockdiagrams. Inparticular,weareinterested inhowLabVIEWcanassistusinmanipulatingpolynomials,computingpolesandzerosoftransferfunctions,computingclosed-loop transferfunctions,computingblockdiagramreductions,andcomputingtheresponseofasystemtoaunitstepinput. Thechapter concludeswithadesignexampleforanelectrictractionmotorcontroldesign. 1.1 Simulating Spring-Mass-Damper Systems A spring-mass-damper mechanical system is shown in Fig. 1.1. The motion of the mass, denoted by y(t), is described by the differentialequation My¨(t)+by˙(t)+ky(t)=r(t). Theunforceddynamicresponseofthespring-mass-dampermechanicalsystemis y(0) (cid:3) (cid:2) (cid:4) y(t)= (cid:2) e−ζωntsin ωn 1−ζ2 t +θ , 1−ζ2 whereθ =cos−1ζ,ωn2 =k/M and2ζωn =b/M. Theinitialdisplacementisy(0)andy˙(0)=0. Thetransientsystemresponseis underdampedwhenζ <1,overdampedwhenζ >1,andcriticallydampedwhenζ =1. Example1.1 Spring-Mass-DamperSimulation WecanuseLabVIEWtovisualizetheunforcedtimeresponseofthemassdisplacementfollowinganinitialdisplacementofy(0). Considertheunderdampedcase,where √ 1 y(0)=0.15m, ωn = 2rad/sec, ζ = √ , (k/M =2, b/M =1). 2 2 . by ky k Wall friction, b Mass y y M M r(t) r Force Figure1.1: Amass-spring-dampersystem. 1 2 CHAPTER1 MATHEMATICALMODELSOFSYSTEMS Figure1.2: VItoanalyzethespring-mass-damper. TheLabVIEWcommandstogeneratetheplotoftheunforcedresponseareshowninFig.1.2. IntheLabVIEWsetup, thevariablesy(0), ωn, ζ, andt areinputtotheuserinterfacepartoftheVI.Thenthe Unforced.viis executed to generate the desired plots. This creates an interactive analysis capability to analyze the effects of natural frequency anddampingontheunforcedresponseofthemassdisplacement. Onecaninvestigatetheeffectsofthenaturalfrequencyandthe dampingonthetimeresponsebysimplyenteringnewvaluesofωnandζ andreruntheUnforced.vi. (cid:2) Forthespring-mass-damperproblem,theunforcedsolutiontothedifferentialequationwasreadilyavailable. Ingeneral,when simulatingclosed-loopfeedbackcontrolsystemssubjecttoavarietyofinputsandinitialconditions,itisdifficulttoobtainthesolution analytically. InthesecaseswecanuseLabVIEWtocomputethesolutionsnumericallyandtodisplaythesolutiongraphically. 1.2 Analyzing Systems Using LabVIEW LabVIEWcanbeusedtoanalyzesystemsdescribedbytransferfunctions. Sincethetransferfunctionisaratioofpolynomials,we begin by investigating how LabVIEW handles polynomials, remembering that working with transfer functions means that both a numeratorpolynomialandadenominatorpolynomialmustbespecified. InLabVIEW,polynomialsarerepresentedbyrowvectors containingthepolynomialcoefficients. Forexample,thepolynomial P(s)=s3+3s2+2s+3 isenteredas[3231]asshowninFig.1.3. CHAPTER1 MATHEMATICALMODELSOFSYSTEMS 3 Figure1.3: EnteringthepolynomialP(s)=s3+3s2+2s+3andcalculatingtherootsofP(s)=0. IfpisarowvectorcontainingthecoefficientsofP(s)inascendingorder,thenPolynomialRoots(p)isarowvectorcontaining the roots of the polynomial. Conversely, if r is a row vector containing the roots of the polynomial, then Create Polynomial from Roots(r) is a row vector with the polynomial coefficients in ascending order. We can compute the roots of the polynomial P(s) = s3+3s2+2s+3withthePolynomialRootsfunction. InFig.1.3,weshowhowtoreassemblethepolynomialwiththe CreatePolynomialfromRootsfunction. Multiplication of polynomials is accomplished with the Multiply Polynomials.vi function. Suppose we want to expand the polynomialN(s),where N(s)=(3s2+2s+1)(s+4). The associated LabVIEW commands using the Multiply Polynomials.vi function are shown in Fig. 1.4. Thus, the expanded polynomialis N(s)=3s3+14s2+9s+4. ThefunctionPolynomialEvaluationisusedtoevaluatethevalueofapolynomialatthegivenvalueofthevariable. Thepolynomial N(s)hasthevalueN(−5)=−66,asshowninFig.1.4. The LabVIEW Control Design & SimulationToolbox treats linear, time-invariant system models as objects, allowing you to manipulate the system models as single entities. In the case of transfer functions, you create the system models using the CD ConstructTransfer Function Model; for state variable models you employ the CD Construct State Space Function Model. Theuseof CDConstructTransferFunctionModelisillustratedinFig.1.5. BasedontheLabVIEWobject-orientedprogrammingcapabilities,thesystemmodelobjectspossessobjectpropertiesthatcan bemodified;likewisefunctionsthatoperateonsystemmodelobjectsarecalledmethods. Forexample,ifyouhavethetwosystem models 10 1 G (s)= and G (s)= , 1 s2+2s+5 2 s+1 youcanaddthemusingtheAddRationalPolynomials.vifunctiontoobtain s2+12s+15 G(s)=G (s)+G (s)= . 1 2 s3+3s2+7s+5