自适应作业1系统辨识.docx
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自适应作业1系统辨识.docx
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自适应作业1系统辨识
AdaptiveControl
Assignment1
SystemIdentification
姓名:
****
学号:
*************
班级:
***********
Answers:
1.a)Obtainthesystemmodelequationandwriteitinlinearregressionform.
Thesystemmodelequation:
It’sautoregressiveform:
b)Simulatethesystembygenerating1000datapoints.Plotu(t)andy(t).
c)Obtaintheleastsquaresestimatorforthissystem.
Theleastsquaresestimatorfortheparametervectoris:
Theestimatedvalueofsystemparametersare:
2.
a)Generateanyinputandgettheresponse.Plotu(t)andy(t).Ignorethesystemnoise
TheARXmodels:
It’sautoregressiveform:
Wheninputisastepfunction,theoutputis:
Wheninputisasinwave,theoutputis:
b)Writearecursiveleastsquaresprogramtoidentifythismodelandtestyourprogram.
Theleastsquaresestimate
canbeobtainedfrom:
Theestimatedvalueofsystemparametersare:
Testmyrecursiveleastsquaresprogram:
Clearly,theresponsewiththeleastsquaresestimateisalmostassameastheoriginalsystemresponse.
c)Testtheresponseandtherecursiveleastsquaresprogramifawhitenoiseisadded.
Obviously,theresponsewiththeleastsquaresestimateisalmostassameastheoriginalsystemresponse.SoIthinkitispredictingthecorrectsystemparameters.
d)Commentonhowdifferenttypesofinputs,initialLN,andlengthofdataaffectthefinalestimation.
Caseone:
recursiveleastsquares:
recursiveleastsquares
value
AStepfunctionsignal
errorllEll2
Asinwavesignal
errorllEll2
The
Estimated
value
ofsystem
parameters
LN=1e+6*I
W=
-1.5363
0.8607
0.0416
0.0395
0.00148
0.00454
The
Estimated
value
ofsystem
parameters
LN=1e+5*I
W=
-1.5363
0.8607
0.0416
0.0395
0.00155
0.044297
The
Estimated
value
ofsystem
parameters
LN=1e+4*I
W=
-1.5363
0.8607
0.0416
0.0395
0.00478
0.356367
The
Estimated
value
ofsystem
parameters
LN=1e+3*I
W=
-1.5363
0.8607
0.0416
0.0395
0.04446
1.211363
Conclusion:
1)TheerrorofparameterestimationwillbesmallerwithbiggerinitialLN.Sothesystemidentificationwillbemoreaccurate.
2)Differenttypesofinputscanaffectthefinalestimation,inthiscase,aStepfunctionsignalisbetterthenAsinwavesignal.
Casetwo:
recursiveleastsquareswithaforgettingfactor
RLS-withaforgettingfactor
Asinwavesignal
errorllEll2
RLS-withaforgettingfactor
Asinwavesignal
errorllEll2
The
Estimated
value
ofsystem
parameters
c=0.4
LN=1e+6*I
The
Estimated
value
ofsystem
parameters
c=0.4
LN=1e+5*I
The
Estimated
value
ofsystem
parameters
c=0.5
LN=1e+6*I
4.704492e-009
The
Estimated
value
ofsystem
parameters
c=0.5
LN=1e+5*I
5.474138e-009
The
Estimated
value
ofsystem
parameters
c=0.8
LN=1e+6*I
2.211398e-008
The
Estimated
value
ofsystem
parameters
c=0.8
LN=1e+5*I
1.090623e-007
The
Estimated
value
ofsystem
parameters
c=0.9
LN=1e+6*I
0.001754
The
Estimated
value
ofsystem
parameters
c=0.9
LN=1e+5*I
0.014448
Conclusion:
1)whencissmaller,theestimatedvaluesaremoreprecise,butthesmallerccouldmakeSystemIdentificationinstability.Forsinusoidalsignals,whenc<0.5,theestimatedvaluesbecomethedivergence.Therefore,thegeneralrangeofcis0.95to0.98.
2)Forthissystem,RLS-withaforgettingfactorismoreaccuratethenrecursiveleastsquares.
3)TheerrorofparameterestimationwillbesmallerwithbiggerinitialLN.Sothesystemidentificationwillbemoreaccurate.
e)ShowhowthesystemparametersintheθtracktowardsthetruevaluesAandBaseachnewiterationoccurs.
Caseone:
RLS,withoutinterrupt
Casetwo:
RLS-withinterrupt
Conclusion:
1.whentheinputistherandomsignal,thespeedofidentificationisthefastest.
2.Thespeedofidentificationisfasterandmoreaccuratewhenthereisn’tinterrupt.
3.Findtheorderofthefollowinginputsignals:
•Toobtainestimatesofaparametricmodel,theinputsignalhastobe“rich”enoughtoexciteallmodesofthesystem.
•Aninputsignalissaidtobepersistentlyexciting(P.E.)ofordernifthefollowinglimitexists:
andthematrixispositivesemi-definite(non-singular).
•Thesignaluwiththepropertyc(k)ispersistentlyexcitingofordernifandonlyif
a)Astepfunctionsignal;
Letu(t)=1fort>0andzerootherwise.Itfollowsthat
AstepcanbethusatmostbePEoforder1.Since
So:
ItfollowsthatitisPEoforder1.
b)Apulsefunctionsignal;
ItfollowsfromEq
thatCn
0forallnifuisapulse.ApulsethusisnotPEforanyn.
c)Asinusoidfunctionsignal;
Letu(t)=sinwt.Itfollowsthat
AsinusoidcanthusatmostbePEoforder2.Since
itfollowsthatasinusoidisactuallyPEoforder2.
d)Arandomsignal.
Considerthestochasticprocess
U(t)=H(q)e(t)
Wheree(t)iswhitenoiseandH(q)isapulsetransferfunction.ItfollowsfromthedefinitionofwhitenoisethatEq
issatisfiedforthesignaleforanynonzeropolynomialA(q).Thispropertyalsoholdsforthesignalu.SothesignaluisthusPEofanyorder.
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