系统的能控性和能观性英文版.docx

系统的能控性和能观性英文版.docx

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系统的能控性和能观性英文版

Unit13ControllabilityandObservability

Asystemissaidtobecontrollableattime

ifitispossiblebymeansofanunconstrained

controlvectortotransferthesystemfromanyinitialstate

toanyotherstateinafinite

intervaloftime.Asystemissaidtobeobservableattime

if,withthesysteminstate

itispossibletodeterminethisstatefromtheobservationoftheoutputoverafinitetime

interval.

TheconceptsofthecontrollabilityandobservabilitywereintroducedbyKalman.Theyplayan

importantroleinthedesignofcontrolsystemsinstatespace.Infact,theconditionsof

controllabilityandobservabilitymaygoverntheexistenceofacompletesolutionofthecontrol

systemdesignproblem.Thesolutiontothisproblemmaynotexistofthesystemconsideredis

notcontrollable.Althoughmostphysicalsystemsarecontrollableandobservable,

correspondingmathematicalmodelsmaynotpossessthepropertyofcontrollabilityand

observability.

CompleteStateControllabilityofContinuous-TimeSystems

Considerthecontinuous-timesystem

（13.1）

whereX=statevector（n-vector）

u=controlsignal（scalar）

A=

matrix

B=

matrix

ThesystemdescribedbyEquation（13.1）issaidtobestatecontrollableat

ifitis

possibletoconstructanunconstrainedcontrolsignalthatwilltransferaninitialstatetoany

finalstateinafinitetimeinterval

.Ifeverystateiscontrollable,thenthesystemis

saidtobecompletelystatecontrollable.

Weshallnowderivetheconditionforcompletestateofcontrollability.Withoutlossof

generality,wecanassumethatthefinalstateistheoriginofthestatespaceandthattheinitial

timeiszero,or

.

ThesolutionofEquation（13.1）is

Applyingthedefinitionofcompletestatecontrollabilityjustgiven,wehave

or

（13.2）

And

canbewritten

（13.3）

SubstitutingEquation（13.3）intoEquation（13.2）gives

（13.4）

Letusput

ThenEquation（13.4）becomes

（13.5）

Ifthesystemoscompletelystatecontrollable,then,givenanyinitialstateX（0）,Equation

（13.5）mustbesatisfied.Thisrequiresthattherankofthe

matrix

ben.

Fromthisanalysis,wecanstatetheconditionforcompletestatecontrollabilityasfollows.

ThesystemgivenbyEquation（13.5）iscompletelystatecontrollableifandonlyifthevectors

arelinearlyindependent,orthe

matrix

istherankn.

TheresultjustobtainedcanbeextendedtothecasewherethecontrolvectorUis

r-dimensional.Ifthesystemisdescribedby

WhereUisanr-vector,thenitcanbeprovedthattheconditionofforcompletestate

controllabilityisthatthe

matrix

beofrankn,orcontainnlinearlyindependentcolumnvectors.Thematrix

iscommonlycalledthecontrollabilitymatrix.

CompleteObservabilityofContinuous-TimeSystem

Inthissectionwediscusstheobservabilityoflinearsystems.Considertheunforced

systemdescribedbythefollowingequations

（13.6）

（13.7）

whereX=statevector（n-vector）

Y=outputvector（m-vector）

A=

matrix

C=

matrix

Thesystemissaidtobecompletelyobservableifeverystate

canbedeterminedfrom

theobservationofY（t）overafinitetimeinterval,

.Thesystemis,therefore,

completelyobservableifeverytransitionofthestateeventuallyaffectseveryelementofthe

outputvector.Theconceptofobservabilityosusefulinsolvingtheproblemorreconstructing

unmeasurablestatevariablefrommeasurablevariablesintheminimumpossiblelengthoftime.

Inthissectionwetreatonlylinear,time-invariantsystems.Therefore,withoutlossof

generality,wecanassumethat

.

Theconceptofobservabilityisveryimportantbecause,inpractice,thedifficulty

encounteredwithstatefeedbackcontrolisthatsomeofthestatevariablesarenotaccessiblefor

directmeasurement,withtheresultthatitbecomesnecessarytoestimatetheunmeasurable

statevariablesinordertoconstructthecontrolsignals.Suchestimatesofstatevariablesare

possibleofandonlyifthesystemiscompletelyobservable.

Indiscussionobservabilityconditions,weconsidertheunforcedsystemasgivenby

Equation（13.6）and（13.7）.Thereasonsforthisareasfollows,Ifthesystemisdescribedby

then

AndY（t）is

SincethematricesA,B,C,andDareknownandu（t）isalsoknown,thelasttermson

theright-handsideofthislastequationareknownquantities.Therefore,theymaybe

subtractedfromtheobservedvalueofY（t）.Hence,forinvestigatinganecessaryandsufficient

conditionforcompleteobservability,itsufficestoconsiderthesystemdescribedbyEquations

（13.6）and（13.7）.

ConsiderthesystemdescribedbyEquations（13.6）and（13.7）.TheoutputvectorY（t）is

And

canbewrittenas

Hence,weobtain

or

（13.8）

Ifthesystemiscompletelyobservable,then,giventheoutputY（t）overatimeinterval

X（0）isuniquelydeterminedfromEquation（13.8）.Itcanbeshownthatthisrequiresthe

rankofthe

matrix

toben.

Fromthisanalysiswecanstatetheconditionforcompleteobservabilityasfollows.

ThesystemdescribedbyEquation（13.6）and（13.7）iscompletelyobservableofandonly

isthe

matrix

isofranknorhasnlinearlyindependentcolumnvectors.Thismatrixiscalledthe

observabilitymatrix.

KeyWordsandTerms

1.controllabilityn.可控性

2.observabilityn.可观测性

3.controllableadj.可控的

4.observableadj.可观测的

5.mathematicalmodel数学模型

6.propertyn.性质，属性

7.continuous-timesystem连续时间系统

8.generalityn.一般性，普遍性

9.rankn.秩

10.linearlyindependent线性无关

11.time-invariantsystem时变系统

12.sufficev.满足

Notes

Althoughmostphysicalsystemsarecontrollableandobservable,corresponding

mathematicalmodelsmaynotpossessthepropertyofcontrollabilityandobservability.

尽管大多数的物理系统都是可控的和可观测的，它们所对应的数学模型并不一定具有可控性和可观测性。

Thesystemossaidtobecompletelyobservableifeverystate

canbedetermined

fromtheobservationofY（t）overafinitetimeinterval,

.

如果在有限的时刻t，

，从系统的输出Y（t）的观测中能确定每一个状态向量的初值

，则称系统是完全可观测的。

Theconceptofobservabilityisveryimportantbecause,inpractice,thedifficulty

encounteredwithstatefeedbackcontrolisthatsomeofthestatevariablesarenotaccessiblefor

directmeasurement,withtheresultthatitbecomesnecessarytoestimatetheunmeasurable

statevariablesinordertoconstructthecontrolsignals.

可观测性的概念非常重要，在实际中，状态反馈控制中所遇到的困难在于，一些状态变量是不能够直接测量的，因此有必要估计不可测量的状态变量来构成控制信号。

encounteredwithstatefeedbackcontrol为过去分词作定语，修饰thedifficulty

thatsomeofthestatevariablesarenotaccessiblefordirectmeasurement为表语从句。

thatitbecomesnecessarytoestimatetheunmeasurablestatevariablesinordertoconstruct

thecontrolsignal.为同位语从句，解释theresult。

Exercises

1.Considerthesystemdefinedby

Isthesystemcompletelystatecontrollable?

2.Considerthesystem

Theoutputisgivenby

Showthatthesystemisnotcompletelyobservable.

3.PleasetranslatethefollowingparagraphintoChinese.

Asystemissaidtobecontrollableattime

ifitispossiblebymeansofanunconstrained

controlvectortotransferthesystemfromanyinitialstate

toanyotherstateinafinite

intervaloftime.Asystemissaidtobeobservableattime

if,withthesysteminstate

itispossibletodeterminethisstatefromobservationoftheoutputoverafinitetime

interval.

Unit14InternalModelControl

InthelastchapterwepresentedseveralmethodsfortuningPIDcontrollersanddeveloped

amodel-basedprocedure（directsynthesis）tosynthesizeacontrollerthatyieldsadesired

closed-loopresponsetrajectory.Inthischapter,wefirstdevelopan"open-loopcontrol"design

procedurethatthenleadstothedevelopmentofaninternalmodelcontrol（IMC）structure.

ThereareanumberofadvantagestotheIMCstructure（andcontrollerdesignprocedure）,

comparedwiththeclassicalfeedbackcontrolstructure.Oneisthatitbecomesveryclearhow

processcharacteristicssuchastimedelaysandRHPzerosaffecttheinherentcontrollabilityof

theprocess.IMCsaremucheasiertotunethanarecontrollersinastandardfeedbackcontrol

structure.

Afterstudyingthischapter,thereadershouldbeableto:

●Designinternalmodelcontrollersforstableprocess（eitherminimumornon-minimum

phase）;

●Sketchtheclosed-loopresponseofthemodelisperfect;

●Derivetheclosed-looptransferfunctionsforIMC;

●DesignIMCimproveddisturbancesforIMC.

IntroductiontoModel-BasedControl

InthepreviouschapterswefocusedontechniquestotunePIDcontrollers.Theclosed-loop

oscillationtechniquedevelopedbyZieglerandNicholsdidnotrequireamodeoftheprocess.

Directsynthesis,however,wasbasedtheuseofaprocessmodelandadesiredclosed-loop

responsetosynthesizeacontrollaw;oftenthisresultedinacontrollerwithaPIDstructure.

Inthischapterwedevelopamodel-basedprocedure,whereaprocessmodelis"embedded"in

thecontroller.Byexplicitlyusingprocessknowledge,byvirtueoftheprocessmodel,

improvedperformancescanbeobtained.

Considerthestirred-tankheatercontrolproblemshowninFigure14.1.Wecanusea

modeloftheprocesstodecidetheheatflow（Q）thatneedstobeaddedtotheprocesstoobtain

adesiredtemperature（T）trajectory,specifiedbytheset-point（

）.Asimplesteady-state

energybalanceprovidesthesteady-stateheatflowneededtoobtainanewsteady-state

temperature,forexample.Byusingadynamicmodel,wecanfindthetime-dependentheat

profileneededtoyieldaparticulartime-dependenttemperatureprofile.

Assumethatthechemicalprocessisrepresentedbyalineartransferfunctionmodel,and

thatitisopen-loopstable.Theinput-outputrelationshipisshowninFigu