Impulsive Control Theory

355 Pages·2001·4.3 MB·English

by  Dr. Tao Yang (auth.)

Lecture Notes
in Control and Information Sciences 272
Editors:M.Thoma · M.Morari
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TaoYang
Impulsive Control Theory
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Dr.TaoYang
UniversityofCaliforniaatBerkeley
DepartmentofElectricalEngineeringandComputerSciences
Berkeley,CA94720
USA
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This book is dedicated to my parents
and to
my wife
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Preface
Theconceptofimpulsive controlanditsmathematicalfoundationcalledim-
pulsive differential equations,ordifferential equations with impulse effects,or
differential equations with discontinuous righthand sides have a long history.
In fact, in mechanical systems impulsive phenomena had been studied for a
long time under different names such as: mechanical systems with impacts.
Thestudyofimpulsivecontrolsystems(controlsystemswithimpulseeffects)
has also a long history that can be traced back to the beginning of modern
controltheory. Many impulsive controlmethods were successfully developed
underthe frameworkofoptimalcontrolandwere occasionallycalledimpulse
control. The so called impulse control is not exactly the impulsive control as
willbedefinedinthisbook.The readershouldnotmixupthesetwokindsof
control methods though in many papers they were treated as the same. Re-
cently,thereisatendencyofintegratingimpulsivecontrolintohybrid control
systems. However, this effort does not have much help to the development
of impulsive control theory because impulsive systems can only be studied
bytheverymathematicaltoolbasedonimpulsivedifferentialequations.The
effort to invent avery generalframeworkofhybrid controlsystem forstudy-
ing impulsive control and other hybrid control problems will contribute no
essential knowledge to impulsive control.
On the other hand, the long history of impulsive differential equations
and impulsive controlsystems did not mean that we alreadyhad a good un-
derstandingofimpulsivecontrolsystems.Thisisbecauseformanyyears,the
study of impulsive control problems had been restricted to only a few kinds
ofspecialproblemssuchasmechanicalsystemswithimpactsandtheoptimal
control of spacecraft. Another fact contributed to the slow development of
impulsive control is that the early research activities were reported as Rus-
sian literature and therefore was not well-known to the English community.
Only within the last two decades impulsive differential equations had been
intensively studied in English literature and at least 7 English books had
been published on impulsive differential equations.
EvenafterthepublicationofmanyEnglishbooksonimpulsivedifferential
equations during the period of 1982 to 1995, the control community still
sawnothingexcitingaboutthesemathematicaltoolsbecausethewell-known
plantsthatcanbestudiedbythesemathematicaltoolsseemtobetoolimited.
VIII Preface
Forexample,mechanicalsystemswithimpactsarenotamainfocusofcontrol
community,predator-preysystemscannotattractseriousattentionofcontrol
engineers. Unfortunately, mathematicians only know the above few kinds
of real examples that fall into the scope of impulsive differential equations.
And tomakethingsworse,theexistingmonographsonimpulsivedifferential
equations target mainly mathematicians as potential readers.
Fortunately, this slow developing pace of impulsive control system had
been changed at the end of last century because of the following facts:
♣ thetheoryofimpulsivedifferentialequationshadbeengraduallydiffused
into control community;
♣ much more new plants, which can be modeled by impulsive differential
equations,werefoundsuchasnanoelectronicdevicesandchaoticspread-
spectrum communication systems . More important, these plants are of
great interests to electrical engineers because of their industrial applica-
tions.
Thisisthefirst one oftwobooksthat arededicatedtoimpulsivesystems
andcontrol.Thesecondoneentitled“ImpulsiveSystemsandControl:Theory
andApplications”willbepublishedbyNova Science Publishers, Inc.[44].In
thisbook,theemphasisis put onthetheoreticalaspectsofimpulsivecontrol
systems.Therefore,theexistenceandstabilityofimpulsivecontrolstrategies
are studied in a very detailed manner. In the second book, the emphasis
will be put on the applications of impulsive control theory. Both books will
benefit three parties:
1. give mathematicians the real applications of impulsive differential equa-
tions and evoke new activities in pure and applied mathematical re-
searches on impulsive differential equations;
2. provide engineers with a tool box and a well organized mathematical
theory for impulsive control problems.
3. providephysicistsandengineerswithanewframeworkofmodelingmany
impulsive effects caused by quantum effects of nanodevices.
Special Styles of the Book
In this book some special styles are designed to help the reader understand
the text quickly and clearly. For theorems, lemmas, definitions and etc., the
symbols are used to terminate statements. For remarks, examples and
proofs, th(cid:2)e symbols , and are used to show where are the ends of
statements. Since ma(cid:7)nyFkinds o(cid:4)f impulsive control systems were studied in
this book, black boxes were used to highlight the control systems at the
beginnings of correspondingsections. For readerswho do not bother to read
details ofproofs, the highlightedcontrol systemmodels serveas anindex for
quick check of conditions for impulsive controller design for different plants.
Aspecialhypertextinterface,calledreasoning flow chart isusedtorepresent
Preface IX
thederivingprocessesofmanyproofs.Atypicalexampleisshownin(3.181).
The following table lists some special symbols used in reasoning flow charts.
symbols meanings symbols meanings
A⇐B from B we have A; A⇒B from A we have B;
B
(cid:1)A (cid:2)(cid:3) B(cid:4) A B is due to C; ⇑ A from A we have B;
! !
(cid:5)C
A
⇒C from A and B we have C; (cid:1)(cid:2)B(cid:3)(cid:4) from A we have B.
B
A
where the symbol denotes any of <,>,≤,≥,≺,(cid:8),(cid:9),(cid:10),etc.
!
Organization of the Book
This book is organized in a highly self-contained and reader-friendly way.
Manyimportanttheoremsareaccompaniedby detailed proofsbecausethese
proofs can show the reader which assumption leads to which conclusion and
therefore make it easy to understand, to apply and to improve these results.
More important, the detailed proofs, which in many cases are constructive,
canguidethedesignofimpulsivecontrollers.Althoughthereadercanbrowser
papers and books to find many of these proofs, different symbols, jargons
and typos contributed by different authors and publishers will make it a
time-consuming and discouragingtask.
In Chapter 1 the definitions of different kinds of impulsive control sys-
tems arepresented.Somebasicknowledgeofimpulsivedifferential equations
such as the existence and continuations of solutions are introduced briefly.
Explicit forms of solutions for different kinds of impulsive differential equa-
tions are presented. Some conditions for avoiding beating phenomena are
also presented. Some extensively used definitions and mathematical results
are summarized in this chapter.
In Chapter 2 we study time-invariant and time-varying linear impulsive
control systems. The stability and controllability are presented.
In Chapter 3 the stability of impulsive control systems are studied based
on comparison methods. We present the results based on single comparison
systemandmulticomparisonsystemswhichcanalsobecalledasvectorcom-
parison system. The applications to impulsive control of chaotic systems are
presented.
In Chapter 4 different methods for designing impulsive controllers with
fixed-timeimpulsesarepresented.Inthischaptertheplantsarenonlinearand
in manycaseswe assume that they canbe decomposed intolinear partsand
nonlinear parts. We also use Lyapunovsecond method to study the stability
of impulsive control systems. The stability of sets and the stability in terms
of two measures are also studied.
InChapter5theimpulsivecontrolsystemswithimpulsesatvariabletime
arestudiedbyusinglineardecompositionmethodsandmethodsbasedontwo
X Preface
measures.The stability of prescribedtrajectoriesor control strategiesis also
studied.
InChapter6thepracticalstabilityofimpulsivecontrolsystemsarestud-
iedbyusingdifferentmethodsbasedonsinglecomparisonsystem,multicom-
parisonsystemandtwomeasures.Thecontrollabilityinpracticalsenseisalso
studied. We also study the practical stability of linear systems. Applications
to impulsive control of nonautonomous chaotic systems are presented.
In Chapter 7 the partial stability of impulsive control systems with im-
pulsesatfixed-timeandvariabletimeisstudied.Wealsostudythestabilityof
integro-differential impulsive control systems based on comparison methods,
methods in terms of two measures and practical stability concept.
In Chapter 8 the principle of impulsive verb control are presented. Since
verbcontrolisabrand mewcontrolparadigm,somebasicknowledgeofverb
controlispresentedatthebeginningofthischapter.Thenthebasicprinciples
of verb impulsive control are presented. This chapter may of great interest
to those readers who are familiar with fuzzy control because verb control is
a natural extension of fuzzy control.
InChapter 9westudy the impulsivecontrolof periodicmotionsin linear
periodic autonomous and nonautonomous systems. We also use parameter
perturbation methods to control periodic motions in impulsive control sys-
tems.Applicationstosteppingmotorcontrolandimpulsivecontrolofchaotic
systems to periodic motions are presented.
InChapter10wepresenttheimpulsivecontrolofalmostperiodicmotions.
We study two kinds of plants; namely, almost periodic plants and periodic
plants driven by almost periodic control signals. The results can be used to
control chaotic systems and design nanoelectronic circuits.
In Chapter 11 we present the applications of impulsive control theory
to nanoelectronics which is an emerging discipline in electrical engineering.
Although nanoelectronics is an extension of classical microelectronics, many
device models in nanoelectronics are entirely different from those used in
microelectronics because of the quantum mechanical effects of nanodevices.
In this chapter we first present elementary impulsive device models that can
be used to model different kinds of devices used in nanoelectronics. Some
examplesofnanodevicesarethenmodeledbyimpulsivedevicemodels.Since
one promisingmethod to implement Booleanlogic in nanoelectronic circuits
istoencodetwodigitbits;namely,0or1byusingdifferentphaseinformation
of a periodic solution, the existence and stability of periodic solutions are of
greatinteresttothedesignofnanoelectroniccircuits.Inthischapter,wewill
study the stability of periodic solutions of different kinds of nanoelectronic
circuits.
Sincetherestrictiontothevolumeofthisbook,itisimpossibletoinclude
all aspects of impulsive systems and control. Therefore, some theoretical as-
pects such as global stability, absolute stability, optimal impulsive control
Preface XI
and some application aspects such as impulsive secure and spread spectrum
communications will be put in the second one of the two sister books [44].
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Berkeley, California, June 2000 Tao Yang