现代控制理论课后答案.docx

现代控制理论课后答案.docx

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现代控制理论课后答案

现代控制理论课后答案（Answerstomoderncontroltheory）

Preface

Thisbookiswritteninlinewiththetextbook"moderncontroltheory"writtenbyacademicianZhangSiying.

Thisbookgivesadetailedanswertotheexercisesinthetextbook,whichcanhelpstudentslearnandunderstandthecontentofthetextbook.Duetoalargenumberofexercises,varyingdegreesofdifficulty,althoughthemainobjectistheundergraduatestudentsmajoringinautomationresearchuniversities,butalsocanmakeuseofothertextbooksforundergraduate,graduatestudyandreferencebooks.

Inthebookthefifth,sixth,eighthchapterexercisesbyProfessorGaoLiqunorganizationcompilationandanswer;thefourth,seventhchapterbyProfessorWeiJingYuanorganizationcompilationandanswer,thefirst,secondchapterbyassociateprofessorZhengYantissuecompilationandanswer.

Becauseoftheshorttime,theremaybeerrors,pleasereaderscriticism,correction.Inaddition,somesolutionsandanswersarenotunique,butonlyonesolutionandanswerisgivenhere.

Editor

May2005

Thesecondchapter"thestatespacedescriptionofcontrolsystem"exercisesanswer

2.1thecircuitisshowninfigureP2.1,theinputis,theoutputisthetest,theoptionalstatevariablesarejuxtaposed,andthestatespaceexpressioniswritten.

Fig.P2.1

Tosolvethisproblem,asamplingmechanismanalysismethodisproposed.First,thedifferentialequationsarewrittenaccordingtothecircuitlaw,andthenthestatevariablesarechosentoobtainthecorrespondingstatespaceexpressionofthesystem.Thesystemtransferfunctioncanalsobeobtainedbythecircuitdiagram,andthestatespaceexpressionofthesystemcanbeobtainedbythetransferfunction.Samplingmechanismanalysismethod.

Whenthevoltageatbothendsis,thevoltageatbothendsis

（1）

（2）

Thechoiceofstatevariablesisobtainedbyformula

（1）and

（2）:

Thestatespaceexpressionis:

Thatis:

2.2establishthestatespaceexpressionofthesystemshowninFig.P22.

Fig.P2.2

Thisisaphysicalsystem,anditismoreconvenienttousethemechanismanalysismethodtofindthestatespaceexpression.Theinputistheinputquantity,i.e.,thedisplacementistheoutput,

Selectthestatevariable,=,=,.

AccordingtoNewton'slaw,yes:

Yes,thereis:

Collated:

Equationofstate:

Theoutputequationis:

Writteninmatrixform:

Thestructureofthe2.5systemisshowninfigureP2.5.Thestatespaceexpressionisdeducedbyusingthemarkandthestatevariableinthegraph.Amongthem,theinput,output,andscalarofthesystemarescalarrespectively.

Fig.P2.5systemstructurediagram

Thestructurediagramofthesystemdescribedbyintegrator,amplifierandadderisgiveninfigureP2.5,andtheoutputofeachintegratorinthegraphisstatevariable,whichiscalledthesystemstatevariablediagram.Thestatevariablediagramdescribestherelationshipbetweenthestatevariablesofthesystem,andillustratesthephysicalmeaningofthestatevariable.Thestatevariablescanbedirectlyobtainedbythestatespaceexpressionsystem.

Inthelightofseekingsumpoint,one,two,andthird,thereare

1:

2:

3:

Outputis,get

2.7trytofindthestatespaceexpressionoftheinductance,thebranchcurrentandthestatevariableintheelectricalnetworkshowninthegraph.Hereisthecurrentvalueoftheconstantcurrentsource,andtheoutputistheupperbranchvoltage.

Fig.P2.8RLelectricnetwork

Solutiontostatespaceexpressionbymechanismanalysismethod.Thefollowingdifferentialequationscanbeobtainedfromthecircuitprinciple

Organizethestatespaceexpressionas

2.8differentialequationsofknownsystems

（1）;

（2）;

（3）.

Testcolumnstowritetheirstatespaceexpressions.

（1）ifthestatevariablesarechosen,thenthereare:

Thestatespaceexpressionis:

（2）theLaplacetransformmethodisusedtoobtainthestatespaceexpression.

TheLaplacetransformofdifferentialequation

（2）underzeroinitialconditionisobtained:

Byformula（2.14）and（2.15）,thestatespaceexpressionofthesystemcanbeobtaineddirectly

（3）theLaplacetransformmethodisusedtoobtainthestatespaceexpression.TheLaplacetransformofdifferentialequation（3）underzeroinitialconditionisobtained:

Whenthetransferfunctionisusedtofindthestatespaceexpressionofthesystem,itisnecessarytopayattentiontowhetherthetransferfunctionisstrictlytruerationalfraction,thatis,whetheritislessthanthat,ifnecessary,asfollows

Byformula（2.14）and（2.15）,thestatespaceexpressionofthesystemcanbeobtaineddirectly

2.9thefollowingtransferfunctionsareknown,andthestatespaceexpressionisestablishedbythedirectdecompositionmethod,andthestatevariablediagramisdrawn.

（1）

（2）

（1）solution

First,transfertheletter

（1）intostrictlytruerationalform:

Order,thereis

Thatis:

Fromtheupperform,thestatevariablediagramisasfollows:

Thestatespaceexpressionofcontrollabilitycanonicalformcanbeobtaineddirectlybythestatevariablegraphorformula（2.14）and（2.15）

（2）thesolutionisknown:

Order:

Well:

Thestatevariablediagramisasfollows:

Thestateexpressionisasfollows:

Thestatespaceexpressionofthesystemshowninthe2.13columndrawingP2.10.

Fig.P2.10

Desetting

（7）

（8）

Thesystemcanblock

（9）

（10）

TheLaplaceInverseTransformationofthepairisobtained

Thenthestatespaceexpressionofthesystemis

2.14trytoconvertthefollowingequationofstateintodiagonalcanonicalform.

（1）

（2）

（1）solution

Eigenvalue

Solution

Seekingeigenvector

For..:

Yes

Solution

For..:

Yes

Solution

Structure,seeking

Fourth,seek,.

Thediagonalnormalformisobtained

（2）solution

Eigenvalue:

Seekingeigenvector

Yes,yes:

Yes,yes:

Yes,yes:

Structure,seeking.

Fourth,seek,.

Thediagonalnormalformisobtained

Trythefollowing2.15stateequationsintocanonicalform.

Solvingeigenvalue:

Seekingeigenvector

Yes,yes

Thatis

Yes,yes

Thatis

Thatis

Structure,seeking.

Fourth,seek,.

IsJordanStandard

2.16thestatespaceexpressionoftheknownsystemis

Thecorrespondingtransferfunctionisobtained.

solution

，，，

2.19setthedifferenceequationofdiscretesystemas

Findingthestatespaceexpressionofthesystem.

TheZtransformationofthedifferenceequationisobtained:

Thestateequationofdiscretesystemis

Thethirdchapter"solutionofstateequation"problemsolving

3.1calculatethematrixindexofthefollowingmatrices.

（1）solution

（2）solution

（3）solution

（4）solution:

3.2thestateequationandinitialconditionofthesystemareknown

（1）thestatetransitionmatrixisobtainedbytheLaplacetransformmethod;

（2）thestatetransitionmatrixisobtainedbytrialdiagonalcanonicalform;

（3）trytofindthestatetransitionmatrixbythefinitetermmethod;

（4）accordingtothegiveninitialconditions,thesolutionofthehomogeneousequationofstateisobtained.

（1）solution,

Amongthem,

Isthere

And,

Sothestatetransitionmatrixis

（2）solution

For,

For,

（3）theeigenvaluesofthesolutionmatrixare,

Foryes:

Foryes:

Becauseitisadoubleeigenvalue,itisnecessarytoaddtheequation

Thusthesimultaneoussolutionisobtained:

（4）solution:

The3.3matrixistheconstantmatrix,aboutthestateequationofthesystem

When,

When,

Trytodeterminethestatetransitionmatrixandmatrixofthesystem.

Solution:

Becausethezeroinputresponseofthesystemis...

therefore

Combinethem,getthem

Thepropertiesofthestatetransitionmatrixshowthatthestatetransitionmatrixsatisfiesthedifferentialequation

Initialconditions

Therefore,thematrixcanbeobtainedbysubstitutingtheinitialtime:

3.9thetransfermatrixoftheknownsystemis...

Trytodeterminethematrix.

Thesolutionisastatetransitionmatrix,

Sothereis

Bereplacedby..:

3.10theknownstatespaceexpressionofthesystemis

（1）theunitstepresponseofthesystemiscalculated;

（2）theimpulseresponseofthesystemisobtained.

（1）solution,

When,

When,

Theformulaissubstitutedintothesolution:

+

Ifyoutakeit,thenyouhaveit

（2）interpretationby

（1）knowledge

Ifyoutakeit,youwillhaveit

Ifyoutakeit,thenyouhaveit,

3.11thefollowinginputfunctionsareasfollows:

（1）pulsefunction;unitstepfunction;（3）stateresponseunderunitslopefunction.

（1）

（2）

（1）solution

1,

Take,then

Ii,

Ifyoutakeit,thenyouhaveit

3,

Ifyoutakeit,thenyouhaveit

（2）solution

therefore

When,

When,

1,

Ifyoutakeit,youwillhaveit

Ii,

Ifyoutakeit,youwillhaveit

3,

Ifyoutakeit,youwillhaveit

Thecoefficientmatrixofthe3.12lineartime-varyingsystemisasfollows.Trytofindthecorrespondingstatetransitionmatrix

（1）

（2）

（1）solution

because

Descriptionandisnotcommutative,thatis,andisnotcommutative.

Thenthepressedstatetransitionmatrixiscalculated

Forthiscalculation:

Sothestatetransitionmatrixis

（2）solution

Theautonomousstateequationofthecorrespondingsystemis

Solutionobtained

Thetwogroupsoflinearindependentinitialstatevariablesareusedagain:

Twolinearlyindependentsolutionscanbederived:

Thus,afundamentalsolutionmatrixofthesystemisobtained:

Thus,thestatetransitionmatrixcanbedeterminedbyusingthestatetransitionmatrixrelation:

3.14thedifferenceequationsofthelinearsteadydiscretesystemareasfollows:

Ifitissetup,therecurrencemethodisusedtofindout.

solution

Inthesameway,recursionisgiven:

3.15thestateequationofthelinearsteadycontinuoustimesystemisestablished

Takingthesamplingperiod,thestateequationofthecontinuouss