Table Of ContentM. A. Krasnosel'skiI A. V. PokrovskiI
Systems
with Hysteresis
Translated from the Russian by
Marek Niezg6dka
With 81 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Mark A. Krasnosel'skii
Aleksei V. Pokrovskii
Institute of Control Sciences
ul. Profsojuznaja 65
117806 Moscow GSP-7, USSR
Marek Niezg6dka
Systems Research Institute, Polish Academy of Sciences
Newelska 6
01-447 Warsaw, Poland
Title of the Russian original edition:
Sistemy s gisteresisom
Publisher Nauka, Moscow 1983
Mathematics Subject Classification (1980): 47HI5,34A60, 49ElO, 49E15
ISBN-13: 978-3-642-64782-6 e-ISBN-13: 978-3-642-61302-9
DOl: 10.1007/978-3-642-61302-9
Library of Congress Cataloging-in-Publication Data
Krasnosel'skiI, M. A. (Mark Aleksandrovich), 1920- . [Sistemy s gisterezisom. English]
Systems with hysteresis I Mark A. Krasnosel'skii, Aleksei
V. Pokrovskii ; translated from the Russian by Marek Niezg6dka. p. cm.
Translation of: Sistemy s gisterezisom.
Bibliography: p.
Includes index.
ISBN-13: 978-3-642-64782-6
1. Hysteresis. I. Pokrovskii, A. V. (Aleksei Vadimovich) II. Title.
QC754.2.H9K7313 1989 538'.3-dcl9 88-39156 CIP
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Preface
Fo~~d6, onee mo~e - told me my ~~t~net
And led me w~-6ely, M an old -6eholaJt,
T~ough a v~~g~nai, nonpM-6able ~eed
On ~med ~ee-6, Ulae-6 and 4JUineM hOMM-6 ...
B. PMtUnak
... The mO-6t ~mpo~tant ~-6 that evuyth~ng
develop-6 -6y-6temat~eaiiy. Evuyth~ng -6hould
eomp~ehend a -6Y-6tem.
- A -6y-6temat~e -6Y-6tem, - -6~d -6e~o~ -6~~be
Vane~ w~th a -6eept~e -6m~le.
- Ye-6 ... a -6y-6tema~zed -6y-6temat~e -6Y-6tem ••.
J.
f{aJe~
Nonlinearities of hysteresis type are common for various branches
of science and technology, including physics, mechanics, biology and
civil engineering, in particular. Effects like magnetic hysteresis,
dielectric hysteresis or plastic hysteresis are all well-known phenome
na. This book discloses general methods which are applicable to de
scribing and studying a large class of systems with hysteresis non
linearities. New mathematical techniques based on resolving the systems
into elementary hysteresis carriers - hysterons - have been developed
for this purpose. The hysterons are treated as transducers with
well-nefined state space, input-output relations, and input-state rela
tions.
The way we interpret hysteresis phenomena in this book is similar
to the standard approach within continuum mechanics (related to the
sixth Hilbert problem), developed by Coleman, Noll, Truesdell et al.
Models of hysteresis phenomena, dating back to Maxwell, Boltzmann,
Madelung, Prandtl, Masing, Volterra, von Mises, Saint-Venant, Tresca,
Ishlinskii and others, are also made use of in this book. Techniques
of the linear systems theory, recently developed by Zadeh, Desoer,
Arbib, Falb, Kalman and others, play an important role as well.
In the study of any system, two essentially different situations
have to be distinguished.
VI PREFACE
In the first case, at given external actions (inputs) the problem
consists in determining the response (output) of a system. For this
purpose, an algorithm should be set up which permits a numerical or
analytic construction of the input-output and input-state relations
(for example, the correspondences between velocity and position in me
chanics, external field - magnetic induction in magnetism theory,
deformation - stress in plasticity theory, etc.). In this construc
tion, it is often sufficient to admit only inputs with the simplest
5tructure: piecewise linear, piecewise smooth, splines, etc.
The second case arises when the system under consideration is mere
ly an element of a larger structure and therefore it cannot be treated
in isolation. Complex systems of such an origin are typical for control
engineering, mechanics and physiCS. If an element with hysteresis
is included in a complex system which is subject to external influences
(controls) and inevitable noises (non-controllable), then the actual
inputs of this element are unknown (and their nature may be rather com
plicated). Any description of hysteresis effects in such a case must
thus allow"for inputs of a general form (for example, arbitrary contin
uous funct~ons) and must be applicable to an analYSis of the dynamics
of the whole system. For this purpose, it must be possible to consider
'the appropriate input-output and input-state relations as operators
with as many "good" properties as possible on various classes of the
inputs. The classic problem of trajectory determination can serve
as a trivial example of such a situation: to perform necessary compu
tations, it is sufficient to use integral sums or mechanical quadra
tures. In order to derive and study equations that describe a certain
process, an integral notion is needed together with information con
cerning some its properties (mean value theorem, integration by parts,
linearity, etc.). Another example of the same situation is given by
problems where, instead of difference operations, differentiation must
be performed.
Throughout the book, the hY.6tVton - a deterministic, static trans
ducer with scalar input and scalar output, controllable and stable
with respect to small-amplitude noises at the input (vibro-correct) -
will be considered as an elementary carrier of hysteresis effects. A
special construction based on passing to some limit will be used to de
scribe the dynamics of the hysteron. First, piecewise monotone inputs
are assumed and analysis techniques standard for nonlinearities such as
PREFACE VII
those in models of a play (backlash) or stop are used. Then, the con
struction is extended onto arbitrary continuous inputs (similarly as at
passing from finite integral sums to an appropriate integral).
A characterization of the hysteron is given and its properties are
studied in Part 1. Plays and stops, extended to all continuous inputs,
are considered as simple examples of hysterons. Part 2 contains for
mulations and proofs of several identification theorems.
If an elasto-plastic element (fibre) has time-independent mechanical
properties, it can often be treated as a hysteron with variable defor
mation and variable stress as the input and output, respectively. If,
due to ageing, temperature changes, etc., the properties are vari
able in time, it proves convenient to introduce a variable hysteron
as a suitable model. A general theory of variable hysterons is devel
oped in Part 3. That theory is based on an analysis of so-called
vibro-correct differential equations.
Hysterons with vector inputs and outputs are considered in Part 4
(for the models due to von Mises, Saint-Venant, Tresca and others).
In Part 5, a study of transducers with discontinuous input-output
relations is developed. Static elements described by discontinuous
superposition operators (of special importance in the theory of systems
with variable structure), as well as ideal and non-ideal relays (on
-off elements) are studied there in detail. Special attention is given
to the role of small noises at the input; taking them into account
simplifies an analysis of the dynamics of transducers.
In Part 6, the notion of hysteron is modified sO.as to cover phe-
nomena of self-magnetization type. The first of those modifications
refers to the classical Madelung's model. The others employ notions
of stochastic integrals. All the constructions are based on the use of
vibro-correct differential equations. Relationships between solutions
of the vibro-correct equations and solutions of some stochastic equa
tions (understood in the Ito or Stratonovich sense) are established.
In Part 7, a study of complex non-deterministic hysteresis non
linearities is developed. By extending the constructions. originally
due to Masing, Ishlinskii, Preisach, Giltay and others, some nonlinear
systems composed of hysterons and relays are considered.
The techniques developed in this book turn out useful at studying
closed-loop systems Which contain hysteresis elements. The tools we use
include functional analysis and topology methods applied to problems
VIII PREFACE
such as stability analysis (including absolute stability questions),
an analysis of forced periodic oscillations and self-oscillations,
a study of singular perturbations, construction of averaged equations,
the selection of special functioning conditions and their study, nu
merical analysis, etc. Concerning all of the problems mentioned above,
numerous results have been obtained by many authors. This, however,
would have to be the subject of another book !
My personal interest in hysteresis phenomena arose in connection
with the theory of nonlinear oscillations. A small seminar had been
organized at the Voronezh University; besides myself, there were also
two young mathematicians, B.t-1. Darinskii and P.P. Zabreiko, as well as
three students, I.V. Emelin, E.A. Lifshitz and A.V. Pokrovskii, who
took part in the seminar. With extraordinary patience, B.M. Darinskii
acquainted us with various hysteresis models in plasticity theory.
More than ten years ago I began working at the Institute of Control
Problems (Institute of Automatic Control and Remote Mechanics) in
Moscow. It was at this time that I started dealing with various prob
lems of automatic control and general systems theory. It turned out
that, despite their importance in many concrete problems, systems with
hysteresis represented an area within the general systems theory which
had hardly been researched. 'From then on, hysteresis itself, physical
phenomena with hysteresis effects and mathematical techniques applica
ble to an analysis of systems with hysteresis nonlinearities have re
mained of interest to me as well as to several students and young col
leagues of mine.
General program frames for a study of hysteresis nonlinearities
from the viewpoint of the systems theory have been clear to me for
a long time. A realization of that program required overcoming diverse
difficulties. Unexpected and unusual associations of hysteresis with
stochastic processes were discovered, a special class of so-called
vibro-correct differential equations were introduced and studied, a
number of non-classical geometric problems had to be solved, properties
of some new types of discontinuous operators required a study, etc.
The first formulations of the fundamental theorems were clumsy and
cumbersome; their first proofs took up dozens of pages; numerous heu
ristically obvious facts, confirmed by numerical experiments, waited
long for rigorous mathematical foundations. So far, basic points
of the general program have been completed, although many questions
PREFACE IX
still remain open.
The very first results were obtained by the participants of the
above-mentioned Voronezh seminar. Further research was, in a diverse
form, developed by T.S. Gil 'man, N.!. Grachev, A.F. Kleptsyn, V.S.
Kozyakin, A.A. Kravchenko, N.P. Panskikh, V.B. Proval ' skii , A.Yu.
Veretennikov and A.A. Vladimirov. During all these years, my co
-author, A. V. Pokrovskii, has played a decisive role in this work:
within a short time he worked his way up from a young assistant to
a collaborator of equal standing. He has obtained a number of the
fundamental results and recently took over, to a large degree, super
vision of colleagues of the next generation.
We have written this book together. A first version of Chapter 19
was worked on and prepared by A.A. V1adimirov. Chapter 39 was pre
pared especially for the English translation of the book by A.A. Krav
chenko.
Taking advantage of this opportunity, I would like to express my
gratitude to N.N. Bogolyubov, A. Yu. Ishlinskii, A.!, Lur'e, Yu.A.
Mitropol'skii, M.A. Rozenblat and Ya.Z. Tsypkin for sometimes short,
sometimes long (but always interesting and profitable) discussions
concerning various aspects of hysteresis phenomena.
Contents
PART 1 STATIC HYSTEROO 1
1. Short-memory transducer 1
1.1 Transducer 1
1.2 States of transducer 2
1.3 Some properties of transducers 3
1.4 Admissible inputs 4
1.5 Vibro-correctness 6
2. Generalized play . 6
2.1 Ordinary play 6
2.2 Generalized play with piecewise monotone inputs 8
2.3 Estimates 10
2.4 Generalized play with continuous inputs 14
2.5 Dependence of outputs upon initial states 15
2.6 Correctness of the definition of the play 16
2.7 Monotonicity 17
2.8 Periodic inputs 19
2.9 Inputs defined on the whole real axis 20
3. Hysteron 22
3.1 Stop 22
3.2 Determining systems of curves 24
3.3 Piecewise monotone inputs 28
3.4 Passage to arbitrary continuous inputs 28
4. Canonical representation of hysteron and proof of
Theorem 3.2 30
4.1 Canonical hysteron 30
XII CONTENTS
4.2 Canonical representation theorem 32
4.3 Proof of Theorem 3.2 35
4.4 Properties of hysteron 35
4.5 Rectification of hysteron 39
5. Distances 40
5.1 Definition of distance 40
5.2 Estimates on differences of output signals 43
6. Various input spaces 47
6.1 Statement of the problem 47
6.2 Spaces of continuously differentiable functions 48
6.3 Play in the space S of absolutely continuous
functions 50
6.4 Hysteron in the space S 51
6.5 Hysterons in spaces Ha 53
6.6 Discontinuous inputs 54
6.7 Hysteron in the space of functions with" bounded
variation 56
6.~ Hysteron in Wiener spaces 57
PART 2 IDENTIFICATION THEOREM 59
7. Identification problem 59
7.1 General identification problem 59
7.2 Prehysteron 61
7.3 Basic identification theorem 63
7.4 Concluding remarks 64
8. Proof of Theorem 7.1 65
8.1 Singular points of the domain n{V) 65
8.2 Construction of curves IT{M) 69
8.3 Construction of curves ~l' ~r 74
8.4 Completion of the proof of Theorem 7.1 75
9. a - identifiability 78
9.1 Statement of the problem 78
9.2 Normal hysteron 79
9.3 Theorem on a- identification 80
9.4 A remark 81
10. Approximate construction of hysteron 81
