Modern Control Systems with LabVIEW

104 Pages·2011·15.87 MB·English

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.
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Thereaderassumesallriskofuseofthisbookandofallinformation,theories,andprogramscontainedordescribedinit. Thisbookmaycontaintechnical
inaccuracies,typographicalerrors,othererrorsandomissions,andout-of-dateinformation. Neithertheauthornorthepublisherassumesanyresponsibilityor
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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