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.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
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