Introduction to Mathematical Systems Theory: Discrete time Linear Systems, Control and Identification

199 Pages·2021·2.755 MB·English

by  Christiaan Heij, André C. M. Ran, Frederik van Schagen

Christiaan Heij
André C. M. Ran
Frederik van Schagen
Introduction to
Mathematical
Systems Theory
Discrete Time Linear Systems,
Control and Identifi cation
Second Edition
Introduction to Mathematical Systems Theory
Christiaan Heij • André C. M. Ran (cid:129)
Frederik van Schagen
Introduction
to Mathematical
Systems Theory
Discrete Time Linear Systems,
Control and Identification
Second Edition
ChristiaanHeij AndréC.M.Ran
DepartmentofEconometrics DepartmentofMathematics
ErasmusUniversityRotterdam VrijeUniversiteit
Rotterdam,TheNetherlands Amsterdam,TheNetherlands
FrederikvanSchagen
DepartmentofMathematics
VrijeUniversiteit
Amsterdam,TheNetherlands
ISBN978-3-030-59652-1 ISBN978-3-030-59654-5 (eBook)
https://doi.org/10.1007/978-3-030-59654-5
MathematicsSubjectClassification:93-XX
1stedition:©BirkhäuserBasel2007
2ndedition:©SpringerNatureSwitzerlandAG2021
©SpringerNatureSwitzerlandAG2021
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Preface
This book has grown out of more than 15years of lecturing an introductory course in
systemtheory,controlandidentificationforstudentsintheareasofBusinessMathematics
and Computer Science, Econometrics and Mathematics at the Vrije Universiteit in
Amsterdam. The interests and mathematical background of our students motivated our
choicetofocusonsystemsindiscretetimeonly,becausethetopicscanthenbestudiedand
understood without preliminary knowledge of (deterministic and stochastic) differential
equations.Thisbookdoesrequiresomepreliminaryknowledgeofcalculus,linearalgebra,
probabilityandstatistics,andsomepartsusetheelementaryresultsonFourierseries.
The book treats the standard topics of introductory courses in linear systems and
control theory. Deterministic systems are discussed in the first five chapters, with the
following main topics: realization theory, observability and controllability, stability and
stabilization by feedback, and linear quadratic optimal control. Stochastic systems are
treated in Chaps.6 to 8, with main topics:realization,filteringand prediction(including
the Kalman filter), and linear quadratic Gaussian optimal control. Chapters 9 and 10
discuss system identification and modelling from data, and Chap.11 concludes with a
briefoverviewoffurthertopics.
Exercises form an essential ingredient of any successful course in this area. The
exercises are not printed in the book and are instead incorporatedon the accompanying
CD-ROM. Theexercisesare oftwo types:thatis, theoryexercisesto trainmathematical
skillsinsystemtheoryandpracticalexercisesapplyingsystemandcontrolmethodstodata
setsthatarealsoincludedontheCD-ROM.ManyexercisesrequiretheuseofMatlabora
similarsoftwarepackage.
We did benefit greatly from the comments of many colleagues who, over the years,
participatedinteachingfromthisbook.Inparticular,weliketomentionthecontributions
of(inalphabeticalorder)SanneterHorst,RienKaashoek,DerkPik,JanH.vanSchuppen
and Alistair Vardy. We thank them for their comments, which have improved the text
considerably. In addition, many students helped us in improving the text by asking
questionsandpointingoutmisprints.
v
vi Preface
PrefacetotheSecondEdition
Thesecondeditionofthebookisaslightlyalteredversionofthefirstedition.Wecorrected
manymisprintsand mis-statementsthat were presentin the first edition.In addition,we
mademanysmallchangestothetexttoimprovereadability.
There are several more extensive changes as well. Chapter 3 has been expanded to
incorporate two sections dealing with the subspace identification algorithm, providing
a construction of a minimal realization directly from sequences of inputs and outputs.
In Sect.6.5, material was added to make the discussion of spectra of several types of
processesmorecomplete.Chapter11hasbeenexpandedwithseveralpartsdealingwith
modern developments.In connectionwith this, the list of referenceshas been expanded
considerably.Wehopethiswillbeusefulbyprovidingpointersforfurtherstudy.
Changingtechnologyhaschangedthewayinwhichtheexercisesarepresented.Atthe
timeofwritingofthefirstedition,aCD-ROMwasstateoftheart;however,present-day
laptopsare not equippedwith a CD drive anymore.For that reason, we have decided to
maketheexercisesavailableasaseparatefileontheSpringerLink’sbookwebsite.
Rotterdam,TheNetherlands ChristiaanHeij
Amsterdam,TheNetherlands AndréC.M.Ran
Amsterdam,TheNetherlands FrederikvanSchagen
ElectronicSupplementaryMaterialTheonlineversionofthisbook(https://doi.org/10.1007/978-
3-030-59654-5_11)containssupplementarymaterial,whichisavailabletoauthorizedusers.
Contents
1 DynamicalSystems................................................................ 1
1.1 Introduction.................................................................. 1
1.2 SystemsandLaws........................................................... 5
1.3 StateRepresentations ....................................................... 7
1.4 Illustration ................................................................... 8
2 Input-OutputSystems............................................................. 11
2.1 InputsandOutputsintheTimeDomain ................................... 11
2.2 FrequencyDomainandTransferFunctions................................ 14
2.3 StateSpaceModels.......................................................... 16
2.4 EquivalentandMinimalRealizations...................................... 22
2.5 TheRestrictedShiftRealization............................................ 23
3 StateSpaceModels................................................................ 27
3.1 Controllability............................................................... 27
3.2 Observability ................................................................ 30
3.3 StructureTheoryofRealizations........................................... 34
3.4 AnAlgorithmforMinimalRealizations................................... 39
3.5 TheSubspaceIdentificationAlgorithm.................................... 41
3.6 AnExample ................................................................. 45
4 Stability............................................................................. 49
4.1 InternalStability............................................................. 49
4.2 Input-OutputStability....................................................... 54
4.3 StabilizationbyStateFeedback ............................................ 57
4.4 StabilizationbyOutputFeedback.......................................... 60
5 OptimalControl ................................................................... 65
5.1 ProblemStatement.......................................................... 65
5.2 DynamicProgramming..................................................... 68
5.3 LinearQuadraticControl ................................................... 72
vii
viii Contents
6 StochasticSystems................................................................. 81
6.1 Modelling.................................................................... 81
6.2 StationaryProcesses......................................................... 82
6.3 ARMAProcesses............................................................ 86
6.4 StateSpaceModels.......................................................... 89
6.5 SpectraandtheFrequencyDomain ........................................ 94
6.6 StochasticInput-OutputSystems........................................... 99
7 FilteringandPrediction........................................................... 101
7.1 TheFilteringProblem....................................................... 101
7.2 SpectralFiltering............................................................ 104
7.3 TheKalmanFilter........................................................... 108
7.4 TheSteadyStateFilter...................................................... 116
8 StochasticControl................................................................. 121
8.1 Introduction.................................................................. 121
8.2 StochasticDynamicProgramming......................................... 122
8.3 LQGControlwithStateFeedback ......................................... 125
8.4 LQGControlwithOutputFeedback....................................... 129
9 SystemIdentification.............................................................. 137
9.1 Identification................................................................. 137
9.2 RegressionModels.......................................................... 138
9.3 MaximumLikelihood....................................................... 141
9.4 EstimationofAutoregressiveModels...................................... 143
9.5 EstimationofARMAXModels ............................................ 147
9.6 ModelValidation............................................................ 150
9.6.1 LagOrders......................................................... 150
9.6.2 ResidualTests...................................................... 151
9.6.3 InputsandOutputs................................................. 153
9.6.4 ModelSelection ................................................... 155
10 CyclesandTrends ................................................................. 157
10.1 ThePeriodogram............................................................ 157
10.2 SpectralIdentification....................................................... 163
10.3 Trends........................................................................ 170
10.4 SeasonalityandNonlinearities.............................................. 172
11 FurtherDevelopments ............................................................ 177
11.1 ContinuousTimeSystems .................................................. 177
11.2 OptimalControl............................................................. 178
Contents ix
11.3 NonlinearSystems .......................................................... 180
11.3.1 ApplicationsinLifeSciences..................................... 182
11.4 InfiniteDimensionalSystems............................................... 182
11.5 RobustandAdaptiveControl............................................... 183
11.6 StochasticSystems.......................................................... 186
11.7 NetworkedSystems......................................................... 186
11.8 HybridSystems.............................................................. 187
11.9 SystemIdentification........................................................ 187
Bibliography............................................................................. 189
Index...................................................................................... 193
1
Dynamical Systems
1.1 Introduction
Many phenomena investigated in such diverse areas as physics, biology, engineering,
and economics show a dynamical evolution over time. Examples are thermodynamics
and electromagnetism in physics, chemical processes and adaptation in biology, control
systems in engineering,and decisionmakingin macroeconomics,finance,and business
economics.Themainquestionsanalysedinthisbookarethefollowing.
(cid:129) Whattypeofmathematicalmodelscanbeusedtostudysuchdynamicalprocesses?
(cid:129) Onceamodelclassisselectedandweknowtheparametersinthemodel,howcanwe
achievespecificobjectivessuchasstability,uncertaintyreductionandoptimaldecision
making?
(cid:129) Ifwedonotknowtheparametersinthemodelexactly,howcanweestimatethemfrom
availabledataandhowreliableistheobtainedmodel?
The first question is the topic of Chaps. 2, 3 and 6, the second one of Chaps. 4, 5, 7,
8 and 9, and the third one of Chaps. 9 and 10. The answers to these questions will in
general depend on accidental particularities of the problem at hand. However, there are
important common characteristics of these problems which can be expressed in terms
of mathematical models. We first give some examples to illustrate the main ideas in
modelling,estimation,forecastingandcontrol.
Example1.1.1 Suppose that for a certain good the market functions as follows. The
quantity currently produced will be supplied to the market in the next period. Supply
and demand determine the market price. Let D denote the quantity demanded, S the
ˆ
quantity supplied, P the market price, P the anticipated price used by the suppliers in
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