Table Of ContentSpringer Tracts in Mechanical Engineering
Zhengzhi Han
Xiushan Cai
Jun Huang
Theory of
Control Systems
Described by
Diff erential
Inclusions
Springer Tracts in Mechanical Engineering
Boardofeditors
Seung-BokChoi,InhaUniversity,Incheon,SouthKorea
HaibinDuan,BeijingUniversityofAeronauticsandAstronautics,Beijing,
P.R.China
YiliFu,HarbinInstituteofTechnology,Harbin,P.R.China
CarlosGuardiola,UniversitatPolitècnicadeValència,València,Spain
Jian-QiaoSun,UniversityofCalifornia,Merced,USA
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Zhengzhi Han (cid:129) Xiushan Cai (cid:129) Jun Huang
Theory of Control Systems
Described by Differential
Inclusions
123
ZhengzhiHan XiushanCai
ShanghaiJiaoTongUniversity ZhejiangNormalUniversity
Shanghai,China Jinhua,China
JunHuang
SoochowUniversity
Suzhou,China
ISSN2195-9862 ISSN2195-9870 (electronic)
SpringerTractsinMechanicalEngineering
ISBN978-3-662-49243-7 ISBN978-3-662-49245-1 (eBook)
DOI10.1007/978-3-662-49245-1
LibraryofCongressControlNumber:2015960942
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Preface
When I give lecture to my students, I preferto introducethe historyof inventions
in engineering development rather than to state only theoretically significance. It
seemsthatmystudentsveryliketohearthesestorieswhichmakethembeinterested
in the teaching materials presented in the classes and understand the engineering
background.Hence,Ispentlotsoftimetoreadandcollectthesestories;ofcourse,
mostofthemwerewrittenbyChinese.
Ithinkthedevelopmentofcontroltheorymaybedividedintothreephases.
From the book of Joseph T. M. Needham, south-pointed cart designed and
manufacturedbyChinesepeoplebeforeChristwasanautomaticmechanicdevice.
Onthecart,therewasapuppetmadeofwood,onearmofthepuppetalwayspointed
to the south no matter how to turn the cart. Needham said it might be the first
automaticdeviceintheworld.
However, the south-pointed cart was a legend. Nobody has seen the design or
sketch drawing and heard any information from archaeological experts. A great
inventionofautomaticdevicewasthecentrifugalgovernorforthesteammachine.
JamesWattheldthreeinventionsforsteammachines;thecentrifugalgovernorwas
the last one by which the Industrial Revolution solved the problem of power and
thenhappened.Manyresearchersagreedthatthecentrifugalgovernorwasthefirst
automatic control device and used until now. Until the invention of centrifugal
governorbyWatt,therehadbeenseveralcontrolleddevices,butnocontroltheory.
After the invention, lots of physicians investigated the reason why the small
device could make the speed stable for a huge steam machine. It was said that
J. Maxwell also considered the problem, but I did not find any valuable results
abouthisinvestigationinthisarea.Theproblemwassolvedbymathematicianswho
workedinthefieldofdifferentialequations;theyestablishedthetheoryofstability
and found the conditions for the stability. Using their words, the governor made
the plant stable. Among these mathematicians, two people, Jules H. Poincaré and
AleksandrM. Lyapunov,haveto be mentioned.Theyindependentlypresentedthe
stability theory for differentialequationsand gave vital effects for the subsequent
investigationofthestabilityofdifferentialequations.
v
vi Preface
Inallfairness,theresearchofstabilityofdifferentialequationsisapartofcontrol
theory. The governor is a controller, the controlled steam machine is a control
system,andthecontrollerstabilizestheplantbythewordsofthetwentiethcentury.
Tillnow,everyclassicaltextbookofcontroltheoryhastostatethestabilityofcontrol
systems. However, at the years of Poincaré and Lyapunov, there was no such a
subjectofcontroltheory.Ontheotherhand,thetheoryofdifferentialequationliked
thesunatnoonsothatitsbrightwouldmakeotherslosetheircolors.Hence,Icalled
thephasethePregnancyPeriodofControlTheory.Thereexistedcontrollerdesign
and the researches of stability and stabilization, but there was no terminology of
controltheory.Itisremarkablethatthemathematicalmodelusedinthispregnancy
phasewasdifferentialequations.
It was commonly recognized the birth of control theory was in World War II.
By1938,twoexcellentexpertsdisappearedfromthepublicfieldofvision.Herdrik
W. Bode and Norbert Weiner were recruited by the army to improve the control
of the radar and anti-aircraft fire. From the film of Eagles over London, one may
understand the contribution of the control of anti-aircraft systems. After the war,
MITopenedacourseofPrincipleofServoMotors.Thecourserevealedthetheory
used in anti-aircraftsystem control. It was also remarkable that the research used
thetransferfunctionasmathematicalmodeltoanalyzeanddesigncontrollers.The
control theory established at that time was called by Classical Control Theory in
China.Italsowascalledas2Ptheorysincethedesigncouldbecompletedbyusing
PencilCPaper.2Phasgreatsignificancebecauseitsolvedthecontrolsystemdesign
inthecomputerlessera.
Mostpeoplelovethenewandhatetheold.Whenanewtheorywasborn,people
were happy and celebrated its birth, and then they found lots shortcomings of
the theory, and then started to complain. The classical control theory can handle
single-input-single-output (SISO) systems very effectively, but it is hard to treat
the multi-inputand multi-output(MIMO) systems. By 1960, Rudolph E. Kalman
appliedstatespacedescriptiontoresearchthepropertiesofcontrolsystems,sothat
the MIMO systemscould be treatedas the same asSISO systems. Two important
concepts, controllability and observability were defined. When Kalman proposed
his paper, his inventionwas not attached enoughimportance.About 2 years later,
the Kalman filter was successfully applied in aerospace technology, people then
lookedthestatespacemethodwithneweyesandcalleditModernControlTheory.
In1982,KalmanvisitedChina,afterhispresentation,onepersonaskedaquestion
about the difference between the Kalman filter and Wiener filter. After thinking
a while, Kalman answered the question with a new question, he said: Do you
know the reason why the Soviet Union launched manned spacecraft before USA,
but USA arrived in the Moon first? With one more minutes silence, Kalman
answered the question himself, “It was because the Soviet Union people did not
understand Kalman filter.” As an evidence, I knew in the aerospace engineering
Chinese scientists are applyingKalman filter to processthe informationsent back
byChang’e.Itisnodoubt,themoderncontroltheoryusesstatespacemodelinits
investigation.
Preface vii
What can we conclude from the history of control theory? I draw two conclu-
sions: The mathematical model is the mark for the developing phases of control
theory;Thebirthofnewtheoryneedsanengineeringbackground.
Since 1980s, I have heard many times that the third-generation control theory
has been born, such as the theory of large-scale systems, the theory of intelligent
control systems, the theory for complex networks, etc. Except the mathematical
models have been renewed, the birth of a new theory was not accepted by most
people.
In the turn of this century, the word of uncertainty appeared on the desks of
manyscientistswhowereworkingondifferentresearchareas.Itisveryusualthat
thesameactionwillleadtodifferentresults.Forexample,therearetenresistances
made in a productionline at almost the same time. They are all labeled 1 K, and
set on ten baseboards.After working10 h, their resistance values may be similar,
after 10 days, difference starts to happen, after 10 months, it is certain that these
resistances may have more than seven values. It is necessary for us to use a set
to describe these resistance values. The value of a simple resistance is a variable
what can be said for a natural phenomenon? When we deal with the design of
:
controlsystem,thedifferentialequationxDf .x;u/isappliedasthedescriptionof
controlsystems,wherexisthestateanduthecontrol.Theabovediscussionasserts,
fora determinedcontrolu,youcannotexpectthestate isdeterminedalthoughthe
initialconditionisfixed.Theinnerparameters,theenvironmentconditionsandother
unknowncauses lead to the uncertainty of function f(x,u), and the change shows
irregularly.Therefore, it is much suitable to treat f(x,u) as a set-valued mapping,
:
andthedynamicmodelbecomesx 2f .x;u/.Thisisadifferentialinclusion,andis
anewmathematicalmodelofcontrolsystems.Doestheintroductionofdifferential
inclusion model result in a new era of the development of control theory? My
answer is “no, I do not know.” The present research results cannot support such
a conclusion, but it probably leads to a new generation if there is a vital social
necessary.
Before 1980, several authors applied differentialinclusion to deal with control
systems. At this period, they tried to extend the maximum principle to the
differential inclusion systems. Similar conclusion has been established. By Year
2000, lots authors were interested in the controllability of such systems. In the
twenty-first century, researchers applied differential inclusions to deal with Luré
system and polytope system. This is the simple history of differential inclusion
control theory. In the past almost four decades, the development of differential
inclusion control theory is very slow. There are two excuses for the researchers
workinginthearea.Oneisthedevelopmentofdifferentialinclusiontheoryisquite
slow related to other fields of mathematics. The another reason may be the lack
of engineering needs. In China, I could only find two books for the theory. One
is Differential Inclusions – Set-Valued Maps and Viability Theory written by J-P
AubinandA. Cellina,andthe anotheris Introductionofthe Theoryofdifferential
Inclusions by G. V. Smirnov. In Amazon, I did not find other more. These facts
motivatedauthorstospendalmost5yearstowritethisbook,wereallywishitcan
generalizetheinvestigationofdifferentialinclusioncontrolsystems.
viii Preface
The arrangementof the book is as follows. The first chapter provides prelimi-
naries,andfocusesonconvexanalysiswhichiselementalforfurtherinvestigation.
Chapter 2 introducesset-valuedmappingsand differentialinclusions.The authors
required themselves that the chapter can contain basic knowledge of differential
inclusions so that readers can carry out farther research with a massy base. To
understandtheconclusionspresentedinthischapterneedsknowledgeoffunctional
analysis and differentialequations. The third chapter deals with convexprocesses
which can be treated as an extension of linear mappings. The fourth chapter
considerspolytopecontrolsystemswhichcanbelookedasaconvexhullofseveral
linearsystems. The last chapterisaboutLuré differentialinclusionsystems. Most
ofthechapterdealswiththedesignofobserversforthesystems.
BothMs.XiushanCaiandMr.JunHuangweremystudentswhostudiedunder
mysupervisionfortheirPh.D.degrees.WhentheystudiedinShanghaiJiaoTong
University,theymademanystudiesforthecontrolofdifferentialinclusionsystems.
NowtheyjoinedZhejiangNormalUniversityandSoochowUniversity,respectively,
andstillareinterestedinresearchofsuchsystems.MystudentsWeiZhang,Leipo
Liu,JunfengZhang,HaiWu,ShaojieShenandPeiquanWangjoinedseminarsand
discussionswhentheystudiedinmylab.Wearegratefultotheircontributions.
Shanghai,China ZhengzhiHan
18August2015
Contents
1 ConvexSetsandConvexFunctions........................................ 1
1.1 NormedSpacesandInnerProductSpaces............................. 1
1.1.1 SetsandMappings.............................................. 1
1.1.2 NormedSpaces ................................................. 3
1.1.3 ElementaryTopology........................................... 7
1.1.4 LimitationTheorems ........................................... 9
1.1.5 InnerProductSpaces ........................................... 12
1.2 ConvexSets ............................................................. 14
1.2.1 ConvexSetsandTheirProperties.............................. 15
1.2.2 ConstructionofConvexSets................................... 17
1.2.3 SeparationTheorems........................................... 23
1.3 ConvexFunctions....................................................... 28
1.3.1 ConvexFunctionsandTheirElementaryProperties.......... 28
1.3.2 ExamplesofConvexFunctions................................ 31
1.3.3 DerivativeofConvexFunctions................................ 33
1.4 Semi-continuousFunctions............................................. 41
1.4.1 Semi-continuousFunctionsandTheirElementary
Properties........................................................ 41
1.4.2 ExamplesofSemi-continuousFunctions...................... 45
References..................................................................... 51
2 Set-ValuedMappingsandDifferentialInclusions........................ 53
2.1 Set-ValuedMappings................................................... 53
2.1.1 DefinitionoftheSet-ValuedMappings........................ 53
2.1.2 ContinuitiesofSet-ValuedMappings.......................... 55
2.1.3 TangentConesandNormalCones............................. 62
2.1.4 DerivativeofSet-ValuedMappings............................ 67
2.2 SelectionofSet-ValuedMappings..................................... 72
2.2.1 TheMinimalSelection ......................................... 73
2.2.2 MichaelSelectionTheorem.................................... 76
2.2.3 LipschitzianApproximation.................................... 80
2.2.4 TheoremsforFixedPoints ..................................... 87
ix
