Control Theory for Linear Systems

403 Pages·2001·1.673 MB·English

by  H. L. Trentelman, Anton Stoorvogel, Malo Hautus

Control theory for linear systems
Harry L. Trentelman
Research Instituteof Mathematics
andComputer Science
Universityof Groningen
P.O. Box 800,9700AV Groningen
The Netherlands
Tel. [phone]
Fax. [phone]
E-mail. [e-mail]
AntonA. Stoorvogel
Dept. ofMathematicsand ComputingScience
EindhovenUniv. of Technology
P.O.Box 513, 5600MB Eindhoven
The Netherlands
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Fax. [phone]
E-mail. [e-mail]
MaloHautus
Dept. ofMathematicsand ComputingScience
EindhovenUniv. of Technology
P.O.Box 513, 5600MB Eindhoven
The Netherlands
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Fax. [phone]
E-mail. [e-mail]
2
Preface
Thisbookoriginatesfromseveraleditionsoflecturenotesthat wereusedas teach-
ing material for the course ‘Control Theory for Linear Systems’, given within the
frameworkofthenationalDutchgraduateschoolofsystemsandcontrol,inthepe-
riodfrom1987to1999. Theaimofthiscourseistoprovideanextensivetreatment
ofthetheoryoffeedbackcontroldesignforlinear,finite-dimensional,time-invariant
statespacesystemswithinputsandoutputs.
One of the important themes of control is the design of controllers that, while
achievinganinternallystableclosedsystem,maketheinfluenceofcertainexogenous
disturbanceinputsongivento-be-controlledoutputvariablesassmallaspossible.In-
deed,intheappropriatesensethisthemeiscoveredbytheclassicallinearquadratic
regulator problem and the linear quadratic Gaussian problem, as well as, more re-
cently,bytheH2andH∞controlproblems.Mostoftheresearcheffortsonthelinear
quadraticregulatorproblemandthelinearquadraticGaussianproblemtookplacein
theperiodupto1975,whereasinparticularH∞controlhasbeentheimportantissue
inthemostrecentperiod,startingaround1985.
In,roughly,theintermediateperiod,from1970to1985,muchattentionwasat-
tractedbycontroldesignproblemsthat requiretomakethe influenceoftheexoge-
nousdisturbancesontheto-be-controlledoutputsequaltozero.Thestaticstatefeed-
backversionsofthesecontroldesignproblems,oftencalleddisturbancedecoupling,
or disturbancelocalization, problems were treated in the classical textbook‘Linear
MultivariableControl: A Geometric Approach’,by W.M. Wonham. Around1980,
acompletetheoryonthedisturbancedecouplingproblembydynamicmeasurement
feedback became available. A central role in this theory is played by the geomet-
ric(i.e.,linearalgebraic)propertiesofthecoefficientmatricesappearinginthesys-
tem equations. In particular, the notions of (A,B)-invariantsubspace and (C,A)-
invariantsubspaceplayanimportantrole. Thesenotions,andtheirgeneralizations,
alsoturnedoutto becentralin understandingandclassifyingthe‘finestructure’of
the system under consideration. For example, important dynamic properties such
assysteminvertibility,strongobservability,strongdetectability,theminimumphase
property,outputstabilizability,etc.,canbecharacterizedintermsofthesegeometric
concepts. Thenotionsof(A,B)-invarianceand(C,A)-invariancealsoturnedoutto
beinstrumentalinothersynthesisproblems,likeobserverdesign,problemsoftrack-
ingandregulation,etc.
vi Preface
Inthisbook,wewilltreatboththe‘pre-1975’approachrepresentedbythelinear
quadraticregulatorproblemandthe H controlproblem,as wellasthe‘post-1985’
2
approachrepresentedbythe H∞ controlproblemanditsapplicationstorobustcon-
trol.However,wefeelthatatextbookdedicatedtocontroltheoryforlinearstatespace
systemsshouldalsocontainthecentralissuesofthe‘geometricapproach’,namelya
treatmentofthedisturbancedecouplingproblembydynamicmeasurementfeedback,
andthegeometricconceptsaroundthissynthesisproblem.Ourmotivationforthisis
three-fold.
Firstly, in a context of making the influence of the exogenous disturbances on
theto-be-controlledoutputsassmallaspossible,itisnaturaltoaskfirstunderwhat
conditionsontheplantthisinfluencecanactuallybemadetovanish,i.e.,underwhat
conditionstheclosedlooptransfermatrixcanmadezerobychoosinganappropriate
controller.
Secondly,asalsomentionedabove,thenotionsofcontrolledinvarianceandcon-
ditionedinvariance,andtheirgeneralizationsof weaklyunobservablesubspaceand
stronglyreachablesubspace,playaveryimportantroleinstudyingthedynamicprop-
ertiesofthesystem.Asanexample,thesystempropertyofstrongobservabilityholds
ifandonlyifthesystemcoefficientmatriceshavethegeometricpropertythattheas-
sociated weakly unobservable subspace is equal to zero. As another example, the
systempropertyofleft-invertibilityholdsifandonlyiftheintersectionoftheweakly
unobservablesubspace and the stronglyreachable subspace is equal to zero. Also,
the importantnotionsofsystem transmissionpolynomialsand system zeros can be
givenaninterpretationin termsoftheweaklyunobservablesubspace, etc. Inother
words,agoodunderstandingofthefine,structural,dynamicpropertiesofthesystem
goeshandinhandwithanunderstandingofthebasicgeometricpropertiesassociated
withthesystemparametermatrices.
Thirdly,alsointhelinearquadraticregulatorproblem,intheH controlproblem,
2
and in the H∞ control problem, the idea of disturbance decoupling and its associ-
ated geometricconceptsplay an importantrole. For example, the notionof output
stabilizability,andtheassociatedoutputstabilizablesubspaceofthesystem,turnout
toberelevantinestablishingnecessaryandsufficientconditionsfortheexistenceof
a positive semi-definite solution of the LQ algebraicRiccati equation. Also, by an
appropriatetransformationofthesystemparametermatrices,theH controlproblem
2
can be transformed into a disturbance decoupling problem. In fact, any controller
that achieves disturbancedecouplingfor the transformedsystem turns out to be an
optimal controller for the original H2 problem. The same holds for the H∞ con-
trolproblem:byanappropriatetransformationofthesystemparametermatrices,the
originalproblemofmakingthe H∞ normoftheclosedlooptransfermatrixstrictly
less than a given tolerance, is transformed into a disturbance decoupling problem.
Anycontrollerthatachievesdisturbancedecouplingforthetransformedsystemturns
outtoachievetherequiredstrictupperboundonH∞-normoftheclosedlooptransfer
matrix.
Theoutlineofthisbookisasfollows. Afterageneralintroductioninchapter1,
andasummaryofthemathematicalprerequisitesinchapter2,chapter3ofthisbook
Preface vii
dealswiththebasicmaterialonlinearstatespacesystems.Wereviewcontrollability
andobservability,thenotionsofcontrollableeigenvaluesandobservableeigenvalues,
andbasistransformationsinstatespace. Thenwetreattheproblemofstabilization
bydynamicmeasurementfeedback.Asintermediatestepsinthissynthesisproblem,
wediscussstateobservers,detectability,theproblemofpoleplacementbystaticstate
feedback,andthenotionofstabilizability.
Thecentralissueofchapters4to6istheproblemofdisturbancedecouplingby
dynamicmeasurementfeedback. First,inchapter4,weintroducethenotionofcon-
trolled invariance,or (A,B)-invariance. As an immediateapplication, we treat the
problemofdisturbancedecouplingbystaticstatefeedback.Next,weintroducecon-
trollabilitysubspaces,andstabilizabilitysubspaces. Theseareusedtotreatthestatic
statefeedbackversionsofthedisturbancedecouplingproblemwithinternalstability,
andtheproblemofexternalstabilization.Inchapter5,weintroducethecentralnotion
ofconditionedinvariance,or(C,A)-invariance. Next,wediscussdetectabilitysub-
spaces,andtheirapplicationtotheproblemofdesigningestimatorsinthepresence
ofexternaldisturbances. Inchapter6,wecombinethenotionsofcontrolledinvari-
anceandconditionedinvarianceintothenotionof(C,A,B)-pairofsubspaces.Asan
immediate,straightforward,applicationwetreatthedynamicmeasurementfeedback
version of the disturbance decoupling problem. Next, we take stability issues into
consideration,andconsider(C,A,B)-pairsofsubspacesconsistingofadetectability
subspaceanda stabilizabilitysubspace. This structureis appliedto resolvethedy-
namicmeasurementfeedbackversionoftheproblemofdisturbancedecouplingwith
internalstability. Thefinalsubjectofchapter6istheapplicationoftheideaofpairs
of(C,A,B)-pairstotheproblemofexternalstabilizationbydynamicmeasurement
feedback.
Chapters7 and8ofthis bookdealwithsystem structure. Inchapter7, we first
giveareviewofsomebasicmaterialonpolynomialmatrices,elementaryoperations,
Smith form, and left- and right-unimodularity. Then we introduce the notions of
transmission polynomialsand zeros, in terms of the system matrix associated with
thesystem. Wethendiscusstheweaklyunobservablesubspace,andtherelatedno-
tion of strong observability, and finally give a characterization of the transmission
polynomialsandzerosintermsofalinearmapassociatedwiththeweaklyunobserv-
ablesubspace. Inchapter8wediscusstheideaofdistributionsasinputs. Allowing
distributions(insteadofjustfunctions)asinputsgivesrisetosomenewconceptsin
statespace,suchasthestronglyreachablesubspaceandthedistributionallyweakly
unobservablesubspace. Thenotionsofsystemleft-andright-invertibilityareintro-
duced, and characterized in terms of these new subspaces. The basic material on
distributionsthatisusedinchapter8istreatedinappendixAofthisbook.
In chapter 9 we treat the problem of tracking and regulation. In this problem,
certainvariablesoftheplantarerequiredtotrackanapriorigivensignal,regardless
ofthedisturbanceinputandtheinitialstateoftheplant.Boththesignaltobetracked
as well as the disturbance input are modeled as being generated by an additional
finite-dimensionallinearsystem,calledtheexosystem. Conditionsfortheexistence
of a regulatorare givenin terms of the transmissionpolynomialsof certainsystem
matricesassociatedwiththeinterconnectionoftheplantandtheexosystem.Wealso
viii Preface
address the issue of well-posedness of the regulator problem, and characterize this
propertyintermsofright-invertibilityoftheplant,andtherelationbetweenthezeros
oftheplantandthepolesoftheexosystem.
Inchapter10we givea detailedtreatmentofthelinearquadraticregulatorpro-
blem.First,weexplainhowtotransformthegeneralproblemtoaso-calledstandard
problem. Then we treat the finite-horizon problem in terms of the solution of the
Riccati differential equation. Next, we discuss the infinite-horizon problem, both
the free-endpoint as well as the zero-endpoint problem, and characterize the opti-
malcostandoptimalcontrollawsfortheseproblemsintermsofcertainsolutionsof
thealgebraicRiccati equation. Finally, theresults arereformulatedforthegeneral,
non-standardcase.
Chapter11is aboutthe H controlproblem. First, we explainhowthe original
2
stochasticlinearquadraticGaussianproblemcanbereformulatedasthedeterminis-
ticproblemofminimizingthe L normoftheclosedloopimpulseresponsematrix,
2
equivalently, the H -normof the closed loop transfer matrix. Then we discuss the
2
problemsofminimizingthisH -normovertheclassofallinternallystabilizingstatic
2
state feedback controllers, and over the class of all internally stabilizing dynamic
measurementfeedbackcontrollers. Inbothcases,theoriginalproblemisreducedto
a disturbancedecouplingproblembymeansoftransformationsinvolvingrealsym-
metricsolutionsoftherelevantalgebraicRiccatiequations.
Chapters12,13,14,and15dealwiththe H∞ controlproblem,andits applica-
tion to problems of robust stabilization. In chapter 12, the H∞ control problem is
introduced,and it is explainedhow it can be applied, via the celebratedsmall gain
theorem,toproblemsofrobuststabilization.Next,chapter13givesacompletetreat-
ment of the static state feedback version of the H∞ control problem, both for the
finite-horizonaswellastheinfinitehorizoncase.Then,inchapter14,thegeneraldy-
namicmeasurementfeedbackversionofthe H∞ controlproblemistreated. Again,
both the finite, as well as the infinite horizonproblemare discussed. In particular,
the celebratedresult onthe existenceof H∞ suboptimalcontrollersin terms of the
existenceofsolutionsoftwoRiccatiequations,togetherwithacouplingcondition,is
treated. Finally,inchapter15,theresultsofchapter14areappliedtotheproblemof
robuststabilizationintroducedinchapter12. Thechaptercloseswithsomeremarks
onthesingular H∞controlproblem,andwithadiscussionontheminimumentropy
H∞controlproblem.
Thebookcloseswithanappendixthatreviewsthebasicmaterialondistribution
theory,asneededinchapter8.
As mentionedin the first paragraphof this preface, the lecturenotes that ledto
this bookwere usedas teachingmaterialforthecourse‘ControlTheoryforLinear
Systems’oftheDutchgraduateschoolofsystemsandcontroloveraperiodofmany
years. During this period, many former and present Ph.D. students taking courses
withtheDutchgraduateschoolcontributedtothecontentsofthisbookthroughtheir
critical remarks and suggestions. Also, most of the problems and exercises in this
bookwereusedasproblemsinthetake-homeexamsthatwerepartofthecourse,so
weretriedouton‘real’students.Wewanttotaketheopportunitytothankallformer
Preface ix
andpresentPh.D.studentsthatfollowedourcoursebetween1987and1999fortheir
constructiveremarksonthecontentsofthisbook.Finally,wewanttothankthoseof
ourcolleaguesthatencouragedustocompletetheprojectofconvertingtheoriginal
setoflecturenotestothisbook.
HarryL.Trentelman
UniversityofGroningen,Groningen,TheNetherlands
AntonA.Stoorvogel
EindhovenUniversityofTechnology,Eindhoven,TheNetherlands
MaloHautus
EindhovenUniversityofTechnology,Eindhoven,TheNetherlands
x Preface
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Controlsystemdesignandmathematicalcontroltheory . . . . . . . 1
1.2 Anexample:instabilityofthegeostationaryorbit . . . . . . . . . . 4
1.3 Linearcontrolsystems . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Example:linearizationaroundthegeostationaryorbit . . . . . . . . 7
1.5 Linearcontrollers . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Example:stabilizingthegeostationaryorbit . . . . . . . . . . . . . 9
1.7 Example:regulationofthesatellite’sposition . . . . . . . . . . . . 10
1.8 Exogenousinputsandoutputstobecontrolled . . . . . . . . . . . . 10
1.9 Example:includingthemoon’sgravitationalfield . . . . . . . . . . 11
1.10 Robuststabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Notesandreferences . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Mathematicalpreliminaries . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Linearspacesandsubspaces . . . . . . . . . . . . . . . . . . . . . 15
2.2 Linearmaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Innerproductspaces . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Quotientspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Differentialequations . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Rationalmatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.9 Laplacetransformation . . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 Notesandreferences . . . . . . . . . . . . . . . . . . . . . . . . . 35
xii Contents
3 Systemswithinputsandoutputs . . . . . . . . . . . . . . . . . . . . . 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Controllability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Basistransformations . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Controllableandobservableeigenvalues . . . . . . . . . . . . . . . 46
3.6 Single-variablesystems . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 Poles,eigenvaluesandstability . . . . . . . . . . . . . . . . . . . . 51
3.8 Liapunovfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.9 Thestabilizationproblem . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Stabilizationbystatefeedback . . . . . . . . . . . . . . . . . . . . 58
3.11 Stateobservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.12 Stabilizationbydynamicmeasurementfeedback. . . . . . . . . . . 65
3.13 Well-posedinterconnection . . . . . . . . . . . . . . . . . . . . . . 66
3.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.15 Notesandreferences . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Controlledinvariantsubspaces . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Controlledinvariance . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Disturbancedecoupling . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Theinvariantsubspacealgorithm . . . . . . . . . . . . . . . . . . . 80
4.4 Controllabilitysubspaces . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Poleplacementunderinvarianceconstraints . . . . . . . . . . . . . 85
4.6 Stabilizabilitysubspaces . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 Disturbancedecouplingwithinternalstability . . . . . . . . . . . . 96
4.8 Externalstabilization . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.10 Notesandreferences . . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Conditionedinvariantsubspaces . . . . . . . . . . . . . . . . . . . . . 107
5.1 Conditionedinvariance . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Detectabilitysubspaces . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Estimationinthepresenceofdisturbances . . . . . . . . . . . . . . 118
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5 Notesandreferences . . . . . . . . . . . . . . . . . . . . . . . . . 124