Table Of ContentMathematical Engineering
Gordon Blower
Linear
Systems
Mathematical Engineering
SeriesEditors
Jörg Schröder, Institute of Mechanics, University of Duisburg-Essen, Essen,
Germany
Bernhard Weigand, Institute of Aerospace Thermodynamics, University of
Stuttgart,Stuttgart,Germany
Jan-PhilipSchmidt,UniversitätofHeidelberg,Heidelberg,Germany
AdvisoryEditors
GünterBrenn,InstitutfürStrömungslehreundWärmeübertragung,TUGraz,Graz,
Austria
DavidKatoshevski,Ben-GurionUniversityoftheNegev,Beer-Sheva,Israel
JeanLevine,CAS- MathematiquesetSystemes,MINES-ParsTech,Fontainebleau,
France
Gabriel Wittum, Goethe-University Frankfurt am Main, Frankfurt am Main,
Germany
Bassam Younis, Civil and Environmental Engineering, University of California,
Davis,Davis,CA,USA
Today,thedevelopmentofhigh-techsystemsisunthinkablewithoutmathematical
modeling and analysis of system behavior. As such, many fields in the mod-
ern engineering sciences (e.g. control engineering, communications engineering,
mechanicalengineering,androbotics)callforsophisticatedmathematicalmethods
inordertosolvethetasksathand.
The series Mathematical Engineering presents new or heretofore little-known
methods to support engineers in finding suitable answers to their questions,
presentingthosemethodsinsuchmannerastomakethemideallycomprehensible
andapplicableinpractice.
Therefore,theprimaryfocusis—withoutneglectingmathematicalaccuracy—on
comprehensibilityandreal-worldapplicability.
Tosubmitaproposalorrequestfurtherinformation,pleaseusethePDFProposal
Formorcontactdirectly:Dr.ThomasDitzinger([email protected])
IndexedbySCOPUS,zbMATH,SCImago.
Gordon Blower
Linear Systems
GordonBlower
DepartmentofMathematicsandStatistics
LancasterUniversity
Lancaster,UK
ISSN2192-4732 ISSN2192-4740 (electronic)
MathematicalEngineering
ISBN978-3-031-21239-0 ISBN978-3-031-21240-6 (eBook)
https://doi.org/10.1007/978-3-031-21240-6
MathematicsSubjectClassification:93C05,93B52,15A16,34D08
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To Jian
Preface
Thebookisaboutthemathematicsoflinearsystems, particularlycontinuoustime
autonomous linear systems with finite-dimensional state space. These cover an
interesting range of applications in science and engineering and are basic to the
study of more complex systems. The material is at a level suitable for a third or
fourthyearundergraduatestudentinaUKuniversity.Weassumethatthereaderis
familiarwithcalculus,linearalgebraandbasiccomplexanalysisanddevelopthese
ideasfurtherwithintheparticularcontextoflinearsystems.
Chapter 1 is about how to describe linear systems by differential equation,
block diagramsand linear algebra.One of the attractionsof linear systems is that
they can be discussed in several different ways. Chapter 2 begins the systematic
development of the theory, where methods of linear algebra are used to describe
linear systems and compute transfer functions. The purpose of transfer functions
is deferred until Chap.4, where they are introduced in parallel via the Laplace
transform.InChap.2,weuseonlybasiclinearalgebra,whileinchapterthreemore
advancedtechniquesareintroduced.Inmostcases,theresultsareprovedindetail,
and thereare indicationsabouthow the mathematicalquestionscan be posedin a
formsuitableforcalculationviaMATLAB.Acrucialaspectoflinearsystemstheory
isitsadaptabilitytotreatsystemsofveryhighdimension,forwhichcomputersare
essential. Some readers may wish to defer the final few sections of Chap.3 until
later.Moregenerally,somesectionsofthebookaremorechallengingthanothers,
andreaderscanpassoversomeresultsiftheyfindthemdifficult.
In Chap.4, we give a conventional discussion of the Laplace transform for
functionson (0,∞) with basic applicationsto differential equations. Inversionof
Laplace transformsis a complicated topic, and in this book we give some special
casessuchasHeaviside’sexpansiontheorembeforeaddressingthegeneralcasein
Chap.6;thisisconsistentwiththehistoricaldevelopmentofthesubject.Methodsof
vii
viii Preface
complexanalysisaremainlydeferreduntilChap.5.WealsocovertheFouriercosine
transformandobtainaninversionformulafortheL1cosinetransform.TheFourier
transform is essential for applications to signal processing, which are pursued in
Chap.10.
In Chaps.5, 6, and 7, we consider three approaches to the stability problem.
Chapter5usesmethodsofcomplexanalysisandgeometricfunctiontheorytohelp
us visualize the transfer function in terms of its Nyquist contour. In Chap.6, we
presentalgebraicapproacheswhicharealgorithmicandcanbecarriedoutinexact
arithmeticwithoutapproximation.TheninChap.7weuselinearalgebrainamanner
thatisespeciallysuitedtolargematrices.Alloftheseapproachesaremosteffective
when they are implemented with the aid of computers, and for large systems,
computers are essential. Although these approaches are separated into distinct
chapters,thedifferencebetweenthemshouldnotbeoverstated.Theyaredifferent
routes towardsthe same goal and the same problemcan be expressed in different
butequivalentwaysintermsofLaplacetransforms,polynomialsormatrices.Also,
some families of transcendental functions such as the Bessel functions can be
convenientlydescribedintermsofalgebraicdifferentialrings.
In Chap.8, we consider orthogonal polynomials. This topic is often taught
alongsidenumericalanalysis,asexamplesinthetheoryofdifferentialequationsor
asanapplicationofHilbertspacetheory.Inthisbook,weemphasizethatsequences
of orthogonal polynomials can be generated efficiently using discrete time linear
systems, which make use of the three term recurrence relation. In applications to
signalprocessing,itiscommontouseexamplessuchastheChebyshevpolynomials
tocreatefilters,andtheseareparticularlywellsuitedtoourapproach.Thechapter
covers some of the other classical orthogonal polynomials, such as the Laguerre
system,whichwelaterusetoprovefundamentalresultsaboutFourierintegrals.
Chapter9isconcernedwithGreen’sfunctions,inthesenseofCauchytransforms
ofanintegrableweightonaboundedrealinterval.Thishasobviousapplicationsto
orthogonalpolynomialsandmoments,andsomelessobviousapplicationtorandom
linear processes. Some of these results describe fundamentalexamples in random
matrixtheorysuchasthesemicirclelaw.Therearediverseapplications,suchasthe
famousMay-Wigner law in mathematicalbiology,which demonstrate how useful
linearsystemsareintreatingcomplexproblemsinmanybranchesofscience.
The results of the first nine chapters mainly apply to linear systems with finite
dimensionalstate spacesandthefinitematricesthatoperateonthem.InChap.10,
we introduce Hilbert space with a view to describing some infinite-dimensional
linear systems. As soon as one attained the generality of Hilbert space, one
realizes the need to have suitably adapted tools with which to carry out explicit
computations. For this reason, we introduce Hardy space on the right half-plane
with its orthonormal basis derived from the Laguerre functions. The intended
Preface ix
application of these is to signal processing, and in Chap.11 we include famous
results such as Shannon’ssampling theorem for band limited functions.Although
the historical development had an interlude of several decades, it is now natural
tofollowthisdirectlywiththeShannonwaveletbasisforL2.Thebookconcludes
withadiscussionofTelatar’smodelofwirelesstransmission,whichisoftencalled
‘single user MIMO’. This is an important instance where random matrix theory
entersintolinearsystemsandsuggestsareasforfurtherstudy.
Chapters 1 and 2 are essential for understanding the rest of the book. Then
readerswhoaremainlyinterestinginlinearalgebracanprogresstoChaps.3and7.
Chapters 4 and 5 feature Laplace transforms and do not depend on the more
advanced tools from linear algebra. Chapters 8 and 9 are best read together, and
thesefeednaturallyintoChap.10.ThefinalChap.11requiresChaps.9and10.The
followingdiagramindicatesthislogicaldependence.
There are many aspects of control theory that we do not discuss in this
book. Reliability, cost of manufacture,usability and tuning of componentsare all
importanttopicsthatweleavetobooksthatemphasizeengineering.
This book is based on lectures for a third-yearmodule at Lancaster University
formathematicsstudentsandseveralprojectsforstudentsinmathematics,physics
andenvironmentalscience.Iamgratefultothesestudentsforhelpfulcommentson
the course materials which progressively improved the module. The module also
benefitedfromahelpfulreviewbymycolleaguesNadiaMazzaandDavidTowers,
whopersuadedmetoincorporateprojectsintothemoduleassessmentandenabled
students to work on more extended exercises. Lucinda Hadley’s PhD thesis on
wirelesscommunicationsuggestedsomeofthecontemporarytopics.Iamgrateful
toRemiLodhofSpringerwhohelpedguidetheproject.AformerstudentYufeiLi
proofreadthemanuscriptanderadicatedseveralerrors.
x Preface
Linearalgebra Transforms
2Solvingbymatrices 1Linearsystemsdescription 4Laplace transforms
3Eigenvalues,blockmatrices 6Algebraicstability 5Frequencyresponse, stability
7Stabilityvialinearalgebra 8Discrete-time
9Green’sfunctions 10Hilbert spaces
11Wireless and wavelets
After studying this book, we hope that the reader is confidently prepared to
pursuethetopicviaIEEE journalsorapplicationstoengineeringproblems.
Lancaster,UK GordonBlower
