Invariance Entropy for Deterministic Control Systems: An Introduction

290 Pages·2013·1.98 MB·English

by  Christoph Kawan (auth.)

Lecture Notes in Mathematics 2089
Christoph Kawan
Invariance
Entropy for
Deterministic
Control Systems
An Introduction
Lecture Notes in Mathematics 2089
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304

Christoph Kawan
Invariance Entropy
for Deterministic
Control Systems
An Introduction
123
ChristophKawan
InstituteofMathematics
UniversityofAugsburg
Augsburg,Germany
ISBN978-3-319-01287-2 ISBN978-3-319-01288-9(eBook)
DOI10.1007/978-3-319-01288-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013948764
MathematicsSubjectClassification(2010):94A17,93C10,93C15
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Foreword
Thetopicofmathematicalcontroltheoryistheexplorationofthepossibilitiesand
thelimitationsofchangesindynamicalsystemsduetoinputs.Henceconnectionsto
thetheoryofdynamicalsystemsareimmanent.InthismonographChristophKawan
adds a new angle to this connection by bringing to bear concepts and techniques
fromtheglobaltheoryoftopologicalanddifferentiabledynamicalsystemsuponthe
problemtodetermineminimaldatarates,averytimelysubjectincontrol.Herethe
combinationwithargumentsfromnonlinearcontrolseemsparticularlynoteworthy.
Thisfieldhasdevelopedoverthelastfewyearsandthepresenttextshowsthatit
hasreachedacertainmaturity.Atthesametime,Ihopethatthemanyopenproblems
willleadtofurtherfruitfulinvestigations.
Augsburg,Germany FritzColonius
June2013
vii

Preface
Thisbookissupposedtoserveasanintroductiontothetheoryofinvarianceentropy
whichisrelatedtothecontroltaskofmakingasubsetinthestatespaceofacontrol
systeminvariant.InspiredbytheseminalworkofNair etal.[85]aboutnotionsof
entropymeasuringthecomplexityofcertaincontroltasks,andtheBowen–Dinaburg
characterizationsoftopologicalentropyinmetricspaces,FritzColoniuscreatedthe
conceptofinvarianceentropyin2007.Atthattime,IstartedtowritemyPh.D.thesis
under his supervision at the Mathematical Institute of the Universityof Augsburg
and had the pleasure and great opportunity to work on this new topic in the field
ofinformation-basedcontrol.Thetextathandpresentsthetheoryobtainedinfive
fruitfulyearsof research in Augsburgand duringtwo research stays in Campinas
(Brazil)inAugust2010andintheperiodfromSeptembertoNovember2011.There
I had the chance to work with Luiz San Martin who showed great interest in our
researchandcontributedseveralimportantideas.Inthistext,thetheoryaspresented
sofarinthearticles[23,63–65]andinthethesis[62]isalsoputonanewlevelof
generality. We work with a fairly general definition of control systems which is
basicallytheonethatcanbefoundinSontag’sbook[102].Despitethefactthatthis
definitiontreatsdiscrete-andcontinuous-timesystemssimultaneously,theemphasis
inthistextclearlyliesoncontinuous-timesystemsgivenbydifferentialequations.
However,whereitisnogreatdealtoprovearesultalsoindiscretetimeand/orina
purelytopologicalsetting,wedonothesitatetodoso.
The central motivation behind the theory presented in this book comes from
the need to deal with communication constraints in digitally networked control
systems. Here the assumption of classical control theory that information can
be transmitted within control loops instantaneously, lossless, and with arbitrary
precisionis no longer satisfied. Realistic mathematicalmodelsof many important
real-worldcommunicationandcontrolnetworkshaveto take intoaccountgeneral
data-rate constraints in the communication channels, time delays, partial loss of
information, and variable network topologies. This raises the question about the
smallestpossibleinformationrateabovewhichagivencontroltaskcanbesolved.
Though networked control systems can have a complicated topology, consisting
of multiple sensors, controllers, and actuators, a first step towards understanding
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