Control of Indefinite Nonlinear Dynamic Systems: Induced Internal Feedback

210 Pages·1998·3.954 MB·English

by  Professor Stanislav V. Emel'yanov, Professor Isaak Burovoi, Dr Fedor Levada (auth.)

Lecture Notes
in Control and Information Sciences 132
Editor: .M Thoma
S.V. Emel'yanov, I.A. Burovoi and .FI Yu Levada I
lortnoC of
Indefinite raenilnoN
cimanyD smetsyS
decudnI Internal kcabdeeF
Translated from the Russian by P.S. Ivanov
r e g ~ n i r p S
Series Advisory Board
A. Bensoussan M.J. ' Grimble • P. Kokotovic • H. Kwakernaak
I.L. Massey Y.Z. • Tsypkin
Authors
Professor Stanislav V. Emel'yanov
The Academy of Sciences of Russia, Institute for System Analysis,
9, Prospect 60 let Octyabrya, Moscow 117312, Russia
Professor Isaak Burovoi
Moscow Steel and Alloys Institute, 4, Leninsky Prospect, Moscow 117936, Russia
[Dr Fedor Levada ]
ISBN 3-540-76245-0 Springer-Verlag Berlin Heidelberg New York
British Library Cataloguing in Publication Data
Emel'yanov, .S V.
Controlofindefinite nonlinear dynamic systems : induced
internal feedback
1.Nonlinear control theory 2.Feedback control systems
Dynamics
I.Title Burovoi, II. I.A. III.Levada, .F Yu
629.8'36
3540762450 ISBN
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
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Contents
Preface xi
Chapter 1. Introduction 1
Section 1. Processes, Systems, Actions: General Issues ................ 1
1.1. What Do We Want? .......................................... 1
1.2. what Can the Interrelatiobne tween Processes Afford? .......... 1
1.3. Language of the General Theory of Systems ................... 2
1.4. A System and an Object ...................................... 3
1.5. Transfer of Actions Through the Chain of Systems ............. 4
1.6. Direct Actions: A Small Trap ................................. 5
1.7. Sources of Actions ............................................ 6
1.8. Is There a Need for an Intermediate System ................... 8
Section 2. Practice: Survey of Examples ................................ 9
2.1. Biology: The "Predator-Victim" System ....................... 9
2.2. Power Engineering: An Atomic Power Plant .................. 10
2.3. Economics: Nationed Market Economy and State Budget ..... 10
2.4. Production Engineering:
Chemical Interaction Between Two Reagents ................. 11
2.5. Medicine: Hormonotherapy for Diabetes ..................... 12
2.6. Mechanics: Irregular Rectilinear Motion ...................... 12
Section 3. Mathematical Tools and the Subject of Study ............... 13
3.1. How Can We Turn from Words to Deeds? 14
. . . . . . . . . . . . . . . . . . . .
3.2. Why Did We Turn to Differential Equations? ................. 15
3.3. Indeterminate Differential Equations 16
. . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Language of the Theory ofA utomatic Control ................ 17
3.5. What is a Dynamic System? ................................. 17
3.6. Differential Equations as Models of Dynamic Systems ......... 18
3.7. Model of a Controllable Object .............................. 20
3.8. Control Problems: How to Handle Them ..................... 21
Chapter 2. Control of Elementary Dynamic Systems 32
Section 4. Control of a One-Dimensional Object 32
.......................
4.1. Model of an Object and Control Capabilities .................. 23
vi Contents
4,2. What Can a Majorant Do? ................................... 24
4.3. Problem of Correctness
and Completion of Control Synthesis ......................... 25
4.4. Character of Transient Processes ............................. 28
4.5. Extension of the Problem:
An Indeterminate Coefficient Involved with a Control Action.. 30
4.6. Extension of the Problem: Additional State Parameters ........ 30
4.7. Extension of the Problem: Variable Control Accuracy ......... 31
4.8. Prospects of Further Extensions .............................. 32
Section 5. Control of a Nonlinear Multidimensional Object ............. 32
5.1. Model of the Object and Discussion of the Problem ........... 32
5.2. Lyapunov Function .......................................... 33
5.3. What Norm Must We Select? ................................ 34
5.4. Control Synthesis ........................................... 35
5.5. Character of Transient Processes ............................. 36
5.6. Possibilities for Extension of the Problem ..................... 37
5.7. Comments on the Control Law .............................. 37
Chapter 3. Control Problem for an Intermediate Link 39
Section 6. Results -- Predecessors ................................... 39
6.1. Statement of the Problem:
Remote and Immediate Objectives of Control ................. 40
6.2. Linear Control .............................................. 41
6.3. Strong Feedback ............................................ 42
6.4. Bounded Actions ............................................ 43
6.5, Varying Structure System .................................... 45
6.6. Dynamic Binary Control .................................... 46
Section 7. New Properties of Two-Dimensional Problems .............. 48
7.1. Discussion of Statement of the Problem ...................... 48
7.2. Control Synthesis ........................................... 49
7.3. Correctness Problem ........................................ 50
7.4. Control in the General Case ................................. 50
7.5. Behavior of Solutions ........................................ 50
7.6. Some Remarks .............................................. 52
Section 8. Internal Feedback .......................................... 53
8.1. Natural Control Loops ...................................... 53
8.2. Internal Feedback and Control Problems ..................... 54
8.3. Induced Internal Feedback .................................. 57
8.4. Induction with Error ........................................ 58
Contents vii
8.5. Chains of Induction Problems ................................ 60
8.6. Clusters of Induction Problems ............................... 61
8.7. What is the Need for so Many Problems? ..................... 63
Section 9. Synthesis of the Induction Control 64
..........................
9.1. Statement of the Induction Problem .......................... 64
9.2. Concept of Control Synthesis ................................ 67
9.3. Control Law ................................................ 68
9.4. What Else Is to Be Done? ................................... 69
Section 10. Correctness of the Closed System .......................... 70
10.1. Continuity of the Control .................................... 70
10.2. The Lipschitz Condition ..................................... 72
10.3. Summary of the Results ..................................... 74
Section 11. The System with Induced Feedback:
General Properties of Trajectories .......................... 74
ll.l. Variation of an Induction Error .............................. 74
11.2. Induction Conditions for Desired Feedback ................... 76
11.3. Do We Need Exceptional Trajectories? ....................... 78
11.4. Hyperbolic Asymptotics 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5. Exponential Asymptotics ..................................... 81
11.6. Summary of the Results ..................................... 82
Section 12. Control Synthesis Errors and their Aftereffects ............. 83
12.1. Errors Localized in a Certain Domain ........................ 83
12.2. Underestimation of Model Parameters ........................ 84
12.3. It There Any Need to Find our Errors? ....................... 86
Section 13. Values of the Control Actions in the Induction System ...... 87
13.1. What Do We Need to Study? ................................ 87
13.2. The One-Dimensional Linear Example ....................... 88
13.3. A High Coefficient in the Induction Control .................. 89
13.4. A Varying High Coefficient 9l
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5. What Is to Be Done
to Effect Savings in Control from the Outset? ................. 93
13.6. A Fly in the Ointment ....................................... 94
13.7. Basic Result ................................................. 95
Section 14. Potential Functions of Induction Control .................... 96
14.1. Potential Systems. Levels of Clarity of Representation .......... 96
14.2. Construction of the Potential Function of Control ............ 97
14.3. Interpretation of Common Properties of Trajectories .......... 99
14.4. Aftereffects of Errors and Results of Preventive Measures ...... 99
viii Contents
Section 15. Structure of the noitcudnI System ......................... 100
15.1. Language of Block Diagrams ................................ 100
15.2. Block Diagrams of Controllable Systems ..................... 102
15.3. Structure of the Control Law ............................... 105
15.4. What is a Binary System? ................................... 106
Chapter 4. Construction of Tracking Systems 901
Section 16. gnikcarT Structures with Binary Multilevel Systems
........ 109
16.1. Induction Control in the Tracking System ................... 110
16.2. How to State the Induction Problem? ....................... 111
16.3. Generation of Inducing Feedback ........................... 112
16.4. Constructiono f the Control System ......................... 114
16.5. Multilevel Binary Structure ................................. 116
16.6. Operation of the Multilevel System .......................... 117
Section 17. Ways of gnidnetxE the Capabilities fo the Approach
....... 120
17.1 • Oscillations in Tracking Systems ............................ 121
17.2. Oscillations in Induction Systems ........................... 122
17.3. Ways of Eliminating Oscillations ............................ 124
17.4. Termwise Induction and a New Class of Problems ........... 127
Section 18. Termwise noitcudnI ni Systems with gnihcnarB Structures. 127
18.1. General Design Considerations ............................. 128
18.2. Statement of the Termwise Induction Problem ............... 129
t8.3. Synthesis and Properties of the Control Algorithm ........... 130
18.4. Structure Representation ................................... 131
18.5. Termwise Induction in Tracking Systems .................... 132
18.6. Intact Clusters ............................................. 135
Chapter 5. Induction Control. Practical Examples 137
Section 19. :ygoloiB of Control Predator-Victim Systems
............... 138
19.1. Discussion of Practical Issues ............................... 138
19.2. Model of the Predator-Victim Biocenosis .................... 139
19.3. Reproduction of Predators for Annihilation of Victims ....... 141
19.4. Where Is it Possible to Stabilize Predator-Victim Systems? ... 144
19.5. Stabilization of Systems
Involving Reproduction of Predators ........................ 145
19.6. Stabilization of Systems Involving Removal of Predators ...... 147
19.7. Stabilization of Systems InvolvingR emoval of Victims ....... 148
19.8. Stabilization of Systems
by Removal of Predators and Victims ....................... 150
Contents ix
Section 20. Power Engineering: Control of a Nuclear Reactor
.......... 152
20.1. Why Does the Nuclear Reactor Heat up? .................... 152
20.2. Chain Reaction ............................................ 153
20.3. Existence of a Neutron ..................................... 154
20.4. Populations of Neutrons .................................... 156
20.5. Mathematical Model of the Reactor ......................... 158
20.6. Statement of the Problem and Properties of the Model ....... 159
20.7. Why Must the Control be Nonlinear? ....................... 161
20.8. Construction of the Control Law ............................ 162
20.9. Properties of the Closed System ............................. 163
Section 21. Economy: Stabilization of the Trajectories of Growth
...... 165
21.1 • On Modeling of Economic Systems ......................... 165
21.2. Essence of Processes Under Study .......................... 166
21•3. Equations of the Model .................................... 167
21.4. Labor Force and Constraints on Variables ................... 169
21.5. Economic Growth and Statement of the Problem ............ 170
21.6. Linear Control and the Uncertainty Problem ................ 172
21.7. Control of Unemployment as Induction ..................... 172
21•8. Induction Control .......................................... 173
21.9. Comments on the Results .................................. 174
Section 22. Technology: Control of an Exothermic Reaction ............ 175
22,1. Production Process and the Control Problem ................ 175
22,2. Mathematical Model of the Process ......................... 176
22.3. Statement of the Control Problem .......................... 178
22.4. Statement of the Induction Problem ......................... 179
22.5. Synthesis of the Induction Control .......................... 181
22.6. Constraints and Drawbacks ................................. 182
Section 23. Medicine:
Control of Carbohydrate Metabolism for Diabetes
......... 183
23.1. Carbohydrate Metabolism and Insulin-Dependent Diabetes.,. 183
23.2. Mathematical Model of Carbohydrate Metabolism ........... 184
23.3. Constraints on Variables and Statement of the Problem ...... 186
23.4. Is Diabetes Doomed? ....................................... 187
23.5• Synthesis of the Control .................................... 188
23.6. Properties of the Closed System ............................. 190
Conclusion 1-91
Bibliography 193
Preface
This book is written for a wide circle of readers who take interest in concepts of
basic sciences, specifically in concepts of cybernetics, automation, and control.
The text aims to provide a new technique for the solution of a wide class of
automatic control problems. The title of the course reflects the basic systematic
approach the essence of which we will completely clarify in the third section.
For now, we only note that the meaning of the word "induction" stems from the
Latin word "inductio" which implies initiation, demonstration, stimulation, and
so on. That is why mental associations arising in the minds of radio amateurs
(the induced EMF in a conductor) and mathematicians (the induced metric in a
vector space) are far from accidental. We will "induce" coordinates of a control-
lable object to obey a certain specific condition that is sufficient for ensuring the
desired behavior of this object. The text considers simple examples to illustrate
the basic problems, the most important aspects and terms of the theory of au-
tomatic control, and the essential difficulties that everyone has to do with when
he/she faces the problems of automation of actual objects. The book describes
comprehensively and in detail the most popular procedures of control synthesis,
informally discusses their merits and shortcomings, states unresolved problems,
and defines the urgent trends in the progress of the science of automatic control.
First of all, the book distinguishes the most complex and not yet fully inves-
tigated problem of control in the case of a deficiency of both the information on
a controllable object and the characteristics of forces that act on it. The basic
question under discussion lies in the following: is it always too necessary to know
quite exactly the object of interest for its efficient automation or, perhaps, in a
number of cases we can make do with the highly generalized information on its
properties and characteristics?
In this connection, it is pertinent to note that the question raised above is in
essence equivalent to the problem for developing a universal controller intended
to control effectively a wide class of objects. It is also important to stress that
a similar statement of the question contradicts to a certain extent the classical
tradition of control theory by which the information on an object and external
forces is always assumed to be complete.
What is likely to be the most striking fact is that the answer to the ques-
tion put above proves positive in very many situations: there is no need to have
xii Preface
the detailed information on the properties of an object for its high-quality con-
trol because it suffices to possess the information on its belonging to a fairly
wide class of objects displaying some general or similar characteristics. More-
over, it turns out that under nonstrict constraints, the problem for the synthe-
sis of the stabilizing control of a complex indefinite object (multidimensional,
nonlinear, multiple-connected, and so on) can be reduced to a sequence of al-
most trivial operations. The specific character of this method is certainly de-
fined by the decomposition rules which constitute not only the "zest" for the
book, but also the dominant bulk of its subject matter. The basic idea of the
principle successively developed in the book comes to the following: using lo-
cal feedback loops, we strive to induce or, which is the same, to transfer to the
system points, which are beyond the reach of a direct action, the desired con-
trol actions that must change the object dynamics in the requisite direction and
eliminate the effects of uncertainty factors on this dynamics. It stands to rea-
son that this approach leads to quite peculiar control structures which, however,
exhibit a large "safety factor" and ensure a reliable compensation for interfer-
ences. In other words, it is the branching structures of controllers that impart
them the properties of universality and flexibility and the capacity to control
effectively a wide class of objects without the preliminary adjustment of their
parameters.
It is worth noting that the above idea has a certain past history in the theory
of automatic control. The idea that may most closelayg ree with the former one is
the idea of the cascade or subordinate control already developed in the classical
theory and used for the stabilization of power plants. However, a successful
combination of the induced control concept and the principle of duality and
generation of structures relating to the theory of control systems with new types
of feedback loops has made it possible not only to extend appreciably the class of
the objects automatized in this fashion, but also to produce essentially new and
unexpected effects clearly illustrated in the book by examples of the engineering,
physical, economic, and medical types.
Section 1 elucidates the basic notions and sets forth the key idea of the course,
namely, the idea of the transfer of a control action through an intermediate
system that acts as a controllable object subsystem. Practical examples are given in
which this idea appears to be promising. Mathematical tools (ordinary differential
equations) presented in the section serve to formalize the issues discussed in the
lecture course.
Section 2 deals with the control problem for a special object which, in our
opinion, is convenient for use as an intermediate system for conveying control
actions. In this section, we design stabilizing control algorithms that are effective
under the uncertainty conditions of object models.