分析案例課件discrete-time system

Chapter7 ysisandDesignofLinearDiscrete-TimeSystem（Sampled-dataSystem）TheSamplingProcessandSamplingSignalRecoveryandZero-Order ysisofDiscrete-TimeDigitalControlDesignforDiscrete-Time1Designfordiscrete-timesystemscanbedoneins- andw- ,respectively.Designfordiscrete-timeDesignbydiscreteequivalentDiscreteDesign：Digital

TheImpulseTransferFunctionfortheDigital(z)

GD(z)G(z)1GD(z)G(z)e(z)

1GD(z)G(z)

1(z)G(z)G(z)

(z) (z)e 1(z) e(z)

e(z)1(z)，E(z)e(z)R(z)GD(z)

e(z)G(z) DeadbeatControlDesign最少拍DeadbeatControlSystems:Matchingaparticulartestinputwithinanumberofsteps.——Nosteady-stateerroronthesamplingpoint.）1.Aunifieddescriptionoftypicaltest

11Tz(1z1T2z1(1z12(1z11(t z 1 (1r(t)

R(z)

(z1)2

t2

T2z(z2(z

T2z1(1z1 3GD(z)

(z)e(z)G(z)DesignIdea：ObtainGD(z)byconstructing(z)sothattheoutputcanthetypicaltestsignalwithintheminimum ZerosofG(z)onor theunitcircle,exceptfor(1,j0)R(z)

(1z1E(z)e(z)R(z)，e(z)1(z)e()lim(z1)e(z)R(z)

(z)(1z1)vF(z1eTomaketheGD(z)simplestandofthelowest-order,wecanchooseF(z-1)as1.e(z)1(z)1-(1-z-1) Fromthedesignidea,weknow e(T)E(z)(z)R(z) (z) (1z1) e(T)lim(1z1

(1z1)F(z

e(z)e(z)(1z1)F(z) (1(z)1(z)1(1z1)bz1bz2bG(z) G(z) D(z)ebz1bz

Theruletoconstruct(z):Allpolesof(z)arelocatedontheoriginofz-5(z)fortypicaltest forr(t)TheC.L.impulsetransferEE(z)TheThesystemcantracktheinputby1step6

r(t)tTheC.L.impulsetransfer (z)2z1z2EE(z)Tz1TheThesystemcantracktheinputby2step7forr(t)1t22TheC.L.impulsetransfer

(z)3z13z2z3EE(z)1T2z11T2z22TheThesystemcantracktheinputby3step8DeadbeatControlDesign（1r(t R(z) （1v

G(z)G(z)D(z)(z)e1e(z)

GD(zGD(z)1(t)

1z1

1z1

z (1z1)G(z) Tz (1z1

(1z1

2z1

z1(2z1(1z1)2G(z t T2z1(1z1 2(1z1

(1

1

3z1z3

z1(33z1z2(1z1 9AlgorithmforDeadbeatControl①ObtainG(z)—Supposetherearenopolesandzerosofon theunit②Determinee(z)fortheparticulartestr(t

R(z)

(1z1

e(z)(1z1)e(z)(1z1)③Obtain(z)1e(z)④AchieveG(z)

(z)e (z)G(z)eExample1.ConsiderthesystemshownintheabovefigureT=1DesigndeadbeatcontrollersGD(z)forr(t)=1(t)、t1eTs

1

C0

C2 Solution.G(z)Z

(s1)(s2)

)Z s s22z1Z11

1

s

sz1

12(z1)

z z

z

ze2T

z

ze2T(1e2T2eT)z

2e2TT

(zeT)(ze2T0.4(z(z0.368)(zReferringtotheresultforr(t)1(t)intheDesignR(z)

z

e(z)1e(z)1(z)eGD(z)

(z)e(z)G(z)

1z

(z0.368)(z0.4(z2.5(z0.368)(z(z1)(zC(z)(z)R(z)z1

1z1z1[1z1z2]z1z2z3eE(z)(z)R(z)(1z1)e

1z1 Forr(t)R(z)

Tz

ee

e(z)(1z1(1z1

(z)1(z)2z1GD(z)

(z)e(z)G(z)

2z1(1z1

(z0.368)(z0.4(z5(z0.5)(z0.368)(z0.136)(z1)2(z0.365)E(z)e(z)R(z)C(z)(z)R(z)(2z1z2)(1z1 Althoughthedeadbeatcontrolsystemtracksaparticulartestinputaccura ywithinanumberofsteps,ithasthefollowingdisadvantages:Itisdesignedonlyforaparticular Theoutputhasripplesalthoughtherearenoerrorsonthesamplingpoints.ThecontrolinputchangesLG(z)haspolesorzerosonor theunitcircleL

zv(1ziz1nG(z) niziisthezeroofpiisthepoleofn (z)

z

(1pz1i (z) i i (z)G(z) (1zi )e(z)in (z) n

z

(1piz1)(z)GD(z)

(z)G(z) )e①Ifthere zvinGD(z)，GD(z)isun-Thus,wehavetoensurethatthere inwhichpromisesGD(z)is②Ifthereiszionor theunitcircle,GD(z)isThen,thoseziwillbedesignedasthezerosof③Note(z)GD(z)G(z)e(z)Iftherearepion theunit(z)willbeThenthosepiwillbedesignedasthezerosofExampleGiventhediscretesystemdescribedasintheFollowingfigure,where

G0(s)s(0.1s1)(0.05s1)T

Gh(s)

Designadeadbeatcontroller (tSolution：theO.L.impulsetransferfunction(zGhFor

，wecane(z)1(z)z1

Becausethereexistsz=－1.065 theunitThus,zshouldalsobethezeroofThereexistz1inG(z)，z1shouldbein(z),(z)

Because

(z)1

from(3),(z)isnowapolynomial z1oforderTosatisfy（4），e(z)mustbeapolynomialonoforder2，thusbasedon(1),wee(z)(1z1)(1Wherea1isaconstanttobechosen

Thusmultipliedbyaconstant tobedesignedlater,we(z)1z1(11.065z1)

From（5）and（6），we

a1

b1e(z)(1z1)(10.516z1)(z)0.484z1(11.065z1)ThenthedeadbeatcontrollerD(z)1

1(1z1)(1(1z1)(10.135z1)(10.631(10.05z1)(10.516z1)ThentheZ-transformz41zz41z0.484z1z2z3Systemcanfollowtheinputatthe2ndstep,whichisonesteplater.Ripple-freedeadbeatcontrolRipple：thoughthesystemoutputsarestableatthesamplingtime,theyarevaryingbetweentwosamplingtime,見p251,圖7-50。Objective:Notonlytrackingtheinputatthesamplingtime,theoutputsareripple-free.NecessaryCondition:Givenainputr(t) t t2 (q1)! qThenG0(s)exists(q-1)z

tqCC ETE*2T-E2(z)D(z)E(z)solution：ensureE2(z)beingapolynomialonz-1ofafiniteorder。Condition: