Mathematical theory of adaptive control

490 Pages·2006·3.215 MB·English

by  Vladimir G. Sragovich, Jan Spalinski

Mathematical Theory of
t
Adaptive Control
INTERDISCIPLINARY MATHEMATICAL SCIENCES
Series Editor: Jinqiao Duan (Illinois Inst. of Tech., USA)
Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin,
Charles Doering, Paul Fisher, Andrei V. Fursikov, Fred R. McMorris,
Daniel Schertzer, Bjorn Schmalfuss, Xiangdong Ye, and
Jerzy Zabczyk
Published
Vol. 1: Global Attractors of Nonautonomous Dissipative Dynamical Systems
David N. Cheban
Vol. 4: Mathematical Theory of Adaptive Control
Vladimir G. Sragovich
Vol. 5: The Hilbert–Huang Transform and Its Applications
Norden E. Huang & Samuel S. P. Shen
Forthcoming
Mathematica in Finance
Michael Kelly
Interdisciplinary Mathematical sciences–vol.4
Mathematical Theory of
Adaptive Control
Vladimir G. Sragovich
Russian Academy of science, Russia
Translator
I. A. Sinitzin
Russian Academy of science, Russia
Editor
J. Spalinski
warsaw University of Technonlogy, Poland
Assistant Editors
l. Stettner and J. Zabczyk
polish Academy of sciences, poland
World scientific
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Interdisciplinary Mathematical Sciences — Vol. 4
MATHEMATICAL THEORY OF ADAPTIVE CONTROL
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November14,2005 17:16 WSPC/SPI-B324-MathematicalTheoryofAdaptiveControl(RokTing) fm
to my teachers,
...
Professors of Moscow University,
Aleksandr Khintchine
and
Abram Plesner
November14,2005 17:16 WSPC/SPI-B324-MathematicalTheoryofAdaptiveControl(RokTing) fm
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
November14,2005 17:16 WSPC/SPI-B324-MathematicalTheoryofAdaptiveControl(RokTing) fm
PREFACE
The primary notions of control theory are those of a controlled object, a control
aim and a control algorithm (strategy). Both a Markov chain and an ordinary or
stochastic differential equation with controls entering into its description can be
considered as control problems.
We choose the controls so that the controlled object has certain desired prop-
erties, called the control aims. For example, a functional defined on the states of a
Markovchainmaybe requiredto be extremeorthe solutionsofthe givenequation
shouldbe stable in some sense.Solving a controlproblemmeans finding a strategy
(an algorithm) giving the choice rules of the controls to achieve the control aim
given beforehand.
For many decades control theory was based on the assumption that the con-
trolledobjectwasknownexactlywithintheframeworkofitsmathematicaldescrip-
tion (model). For example, if the mathematical model of the considered object is
the linear difference equation of order n
xt+a1xt−1+a2xt−2+···+anxt−n =b1ut−1+···+bmut−m+ψ(t)
wherex is the stateoftheobject, uisthe control,ψ(t)isthe externaldisturbance
t
(or noise), then the values of the coefficients (a ,b ) are supposed to be known and
i i
the states x to be observedateachmomentt. Moreover,either anexplicit formof
t
the function ψ(t) or the probabilistic characteristicsof the noise ψ(t) are supposed
tobeknowninthedeterministicorstochasticcasesrespectively.Wecallthetheory
of control based on these assumptions classical control theory.
However, in many applied engineering problems a priori we do not have this
information about the controlled object. This has led to the creation of adaptive
control theory. There are three possible approaches.
Thefirstconsistsofemployingthemissingdataassoonastheyarriveduringthe
controlprocess.Thesecondapproachisbasedoncontrollingtheobjectgivenincom-
pletely and searching missing information simultaneously. This approach gives the
identification method connecting the estimation procedures of the unknown char-
acteristics of the object with the control methods of classical theory. This method
has a wide use. The third approach consists of constructing algorithms of control
not requiring detailed knowledge about the object.
Due to successfuldevelopment,especiallyofthe lastapproach,adaptivecontrol
theory may be regarded as an independent discipline. According to the general
concept of adaptive control, instead of working with the incomplete mathematical
modelofthecontrolledobject,weneedtofindaclass(acollection)ofmathematical
modelscontainingthemodelthatweareinterestedin.Hence,thecontrolaimstated
vii
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viii Preface
in advance refers to no concrete object but to all objects from the specified class.
The strategy(the controlalgorithm)being designedmustapply to allobjects from
the given class. For this reason the algorithms appearing in adaptive theory are
more difficult than those in classical theory.
We would like to emphasize three distinctive features of this book in compar-
ison with other books dedicated to the same topic. First, it is the wide range of
objects studied (in order of increasing complexity): discrete processes of automata
type (inertia-free), the process generated by recurrent procedures, minimax prob-
lems,finite Markovchains(withbothobservableandunobservablestates),Markov
and semi-Markov processes, discrete time stationary processes, linear difference
stochasticequations,ordinarydifferentialequations(wemaycallthisdeterministic
adaptive theory) and, finally, stochastic Ito equations.The controlledobjects listed
above are mainly stochastic, and hence belong to controlled stochastic processes.
The second feature of this book is the detailed description of the research of
the Eastern School of adaptive control which has not been easily accessible to the
western reader.
The third feature is the formal definition of adaptive control strategy which has
been given for the first time. This notion is used throughout the present volume.
This can be stated as follows.
Let K be a class of controlled objects (controlled random processes) and let
Z denote a control aim defined for all objects from K. Finally, let Σ be a set of
strategies which apply to all objects from K.
Then a strategy from Σ that secures the attainment of the aim Z for every
objectfromK is calledanadaptive strategy.The goalofadaptive theory(probably
unreachable)is to obtainnecessaryandsufficientconditionsfor the existance ofan
adaptive strategy for every collection K,Z and Σ above.
The purpose of the present volume is twofold. On the one hand, for the math-
ematically well-trainedstudents of the appropriate specialities the book may serve
for a text-book on adaptive control theory. On the other hand, the author hopes
that even the specialists will find an inspiration here for their own research.
Manyresultsdeservingattentioncouldnothavebeenincludedinthe maintext
of the book due to constraints on the book’s volume. Therefore, to the author’s
regret, some significant results have been put into appendix — Comments and
Supplements.
Thereadersshouldhaveagoodknowledgeofundergraduatemathematics.Nev-
ertheless, most chapters begin with sections containing all necessary information
(without proofs) to be used.
Bibliography is divided into two parts. The first one (General References)
contains the list of the auxiliary citations. The second part (Special References)
presents the original scientific works which form the basis for our consideration.
Thispartissupplementedbysomeinterestingworksbut,unfortunately,theauthor
hadno possibility to reviewthem indetail. As mentionedabovethe briefsurveyof
themisgivenintheCommentsandSupplements.Whilecomposingthebibliography
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Preface ix
the following rule was used. If the results obtained by some authors are cited in a
monograph then the readers will be referred to this monograph only.
Toreferencethetextthe followingschemeisused.Ineverychapterthesections
are numbered successively by two digits. The first of them denotes the chapter
number, for example, (3.2) refers to Sec. 2 in Chap. 3. Each section has a separate
numerationofequations,theorems,lemmasandsoonconsistingofonenumberonly.
The references to an item from another chapter (or section) are given completely
(for example, Theorem 1 from Sec. 1, Chap. 2).
Asubstantialpartofthebookhasbeenwritteninclosecontactwiththeauthors
of the appropriate results. Whether they are post-graduates, colleagues or friends
of author is mentioned in the comments to chapters. Their advice was very useful.
Here, the author would like to especially mention and to express many thanks to
Professor Vladimir A. Brusin and Professor Aleksandr S. Poznyak as well as to
Dr. Eugenij S. Usachev.
The author is grateful to the Committee of Scientific Research in Warsaw
(Poland)forprovidingthe financialsupporttocomplete this workandto translate
themanuscriptfromRussianintoEnglish.Iexpressoncemoremysinceregratitude
to Professor L(cid:4)ukasz Stettner.
Vladimir G. Sragovich