Random Processes in Nonlinear Control Systems (Mathematics in Science and Engineering, Volume 15)

RANDOM PROCESSES IN NONLINEAR CONTROL SYSTEMS A. A . Pervozvanskii MOSCOW, U.S.S.R.

Translated by SCRIPTA TECHNICA, INC, Translation Editor : Ivo Herzer COLUMBIA UNIVERSITY, NEW YORK

@)

1965

W ACADEMIC PRESS New York London

COPYRIGHT 0 1965,

BY

ACADEMICPRESSINC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 1 1 1 Fifth Avenue, New York, New York lo003 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.1

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-17383

PRINTED IN THE UNITED STATES OF AMERICA

RANDOM PROCESSES IN NONLINEAR CONTROL SYSTEMS THIS BOOK WAS ORIGINALLY PUBLISHED AS: SLUCHAYNYYE PROTSESSY V NELINEYNYKH AVTOMATICHESKIKH SISTEMAKH BY STATE PRESS FOR PHYSICAL AND MATHEMATICAL LITERATURE, MOSCOW, 1962

As the theory of control processes has become of greater interest and importance, its domain has widened considerably. On one hand, we see an awareness of deterministic processes subject to realistic constraints; on the other hand, we see a recognition of the necessity for a determined study of stochastic and adaptive processes. T h e many years during which deterministic processes were the sole objects of research has bequeathed us a most important library of special processes and particular methods for their treatment. Furthermore, over time, a number of quite important approximation techniques have been developed. No such catalog of problems and methods exists for stochastic processes, and certainly not for nonlinedr stochastic processes. T h e purpose of the present book is to supply a set of methods which can be effectively used, together with detailed applications. I t should, for this reason, prove extremely useful to those engaged in research in the general theory of stochastic processes, as well as those specifically concerned with control. We have taken the liberty of changing the title slightly, replacing the older adjective “automatic” by “control,” and adding the term “nonlinear” to indicate the principal contribution of the book.

RICHARDBELLMAN

Santa Monica, California February 1965

V

T h e designer of control systems invariably has to deal with nonlinear phenomena. Indeed, only over a limited range can linear relations describe the real elements of such a system. Backlash and damping interfere with the linear performance of small signals, whereas mechanical and energy limitations often prevent the use of high-power signals. On the other hand, it has been shown in recent years that the dynamic properties of control systems can be considerably improved by the introduction of nonlinear techniques. Similarly, the increased utilization of self-adjusting and, in particular, of extremal systems, points up the significance of nonlinear relations. Furthermore, statistical methods are beginning to be used intensively in computations for complicated modern control systems. Today, the well-read engineer, specializing in control systems, knows very well that statistical methods, especially methods in the theory of random variables, make it possible to study and construct systems which, in the first place, successfully combat interference and which, in the second place, work reasonably well, not only for several common fixed signals but for a whole spectrum of possible factors that may arise under real conditions. Hence, the combination of the two subjects indicated above-the linearity in dynamic idealizations of real systems and the statistical nature of the input signals-is of vital interest in the theory of control systems. T h e monograph by V. S. Pugachev [65], which is the ,most comprehensive work on the theory of random functions and its application to problems in control systems, deals at great length with nonlinear problems. However, the broad range of questions covered in this work precludes the possibility of developing practical approximation techniques and of giving sufficient attention to the physical aspects of nonlinear phenomena. T h e elegant exposition of the approximation techniques of statistical analysis given by Ye. P. Popov and I. P. Pal’tov [64] fills this gap only to a limited extent. At the same time, a large number of articles have been published dealing with important practical topics related to random processes in nonvii

...

Vlll

Preface

linear control systems. This book attempts to give a systematic representation of these publications and to submit new material, previously unpublished. For practical reasons, and because of the personal preference of the author, this book pays special attention to simple techniques of approximation. However, to a large extent, it is concerned with exact methods because the application to many special problems allows a more precise understanding of nonlinear phenomena. From a computational point of view, this is also often a simpler method. This is illustrated in great detail in the development of several problems concerning statistical theory for extremal systems, which may be of special interest to the reader, both as an independent topic and as an example of the application of the various techniques. T h e book consists of an introduction, four chapters, and appendices. T h e introduction formulates the basic problem of statistical theory in control systems ; it introduces and discusses several general methods of calculation and, finally, it reviews elementary propositions in the theory of random functions and the characteristics of transformations which are used in dynamic idealizations of real control systems. Chapter 1 gives methods of analysis and synthesis of nonlinear transformations without feedback. In developing the problems of analysis, particular attention is paid to the qualitative effects that take place when random signals pass through nonlinear devices which are most frequently found in practice, as well as to the formal research techniques. Essentially this chapter describes the methods of obtaining exact solutions. Although this involves a rather cumbersome exposition, it shows nonetheless that, for a number of typical nonlinearities, the necessary computations have already been carried out. Thus, an engineer not interested in the technique itself can use the prepared formulas that are given in the appendices. T h e problem of synthesis by the criterion of the mean-square deviation is examined in a quite general form. An application of this criterion, several means of constructing optimal nonlinear filters, are studied ; the problem of statistical linearization of noninertial nonlinearities is also investigated. Chapter 2 contains a short general survey of statistical methods applicable to nonlinear systems with feedback and an examination of the stationary states of such systems. T h e application of the

Preface

ix

concept of statistical linearization to the analysis of closed systems in the presence of wide-band normally distributed random inputs is described in detail ; techniques are developed that make for a rigorous method of computing the distortion in the form of a correlation function of the signal that passes through a nonlinear element. These methods also give approximate solutions for special cases when the spectral density of the input signal has a narrow bandwidth or when its distribution is somewhat different from the normal. Furthermore, this chapter deals with the problem of synthesis of optimal linear correctional chains with some given nonlinearity. Special attention is paid to the case when the nonlinearity imposes a limitation on the magnitude of the output signal. T h e conclusion gives an exact method of analysis for several nonlinear problems by using the theory of Markov processes. I n Chapter 3 we develop techniques for studying nonstationary operation in nonlinear closed systems, mostly applicable to problems in which the nonstationary aspect has a periodic character. These problems have great significance both in the analysis of the effect of an impulse on a system with harmonic and stationary random signals and also in connection with the practical problem of guaranteeing stability with respect to random interference. T h e solution makes extensive use of the concepts of the frequency characteristics of input signals ; this introduces several approximative devices, which are based on the obvious combination of the ideas of statistical and harmonic linearization. T h e last section of this chapter presents an exact method for studying periodic operation in relay systems with small random disturbances. T h e solution is based on the solution lacing technique commonly used in the theory of relay systems. Chapter 4 is devoted to an examination of extremal systems. In view of the relative novelty of this subject, the exposition begins with a survey of the basic methods of constructing extremal systems and their classification, and, proceeds in the order dictated by this classification, which does not, however, correspond to methods used in analysis. I t covers systems with a time separation between operation and testing of the on-off and proportional type systems with a frequency separation of these operations, and finally oscillatory systems where operation and testing are completely coincident. T h e main objective of this study is to obtain an estimate of the qualitative operation of extremal systems and some idea of how to choose the parameters.

X

Preface

T h e development of most of the methods described in the text is supplemented by examples. Familiarity with these illustrative examples is necessary for the reader who wants to acquire greater skill in computational methods and, particularly, a better understanding of the qualitative nature of nonlinear phenomena. T h e appendices contain the basic information necessary for practical calculations ; they also discuss additional mathematical theories, such as an introduction to the theory of Markov chains and processes. T h e book is written for the design engineer as well as for the research scientist whose work is concerned with control systems. I t is assumed that, in addition to the usual fundamental mathematics taught at university level, the reader is acquainted with the elements of the theory of probability and of random functions, for example, with the material covered by J. Kh. Laning and R. G. Battin in Chapters2-5 and 7 of ref. [51] or by V. V. Solodovnikov in Chapters 2-7 of ref. [80]. A reader who is beginning the study of the theory of random functions given in the monograph by Pugachev [65] should be familiar with sections on linear problems, which are summarized in the preface to that monograph. I n spite of its extreme brevity, this book covers all the necessary elements of the general questions concerning random variables. T h e reader must also understand the general theory of control systems, and, in particular, the approximation techniques of solving nonlinear problems, at least on the level of the course of Ye. P. Popov [62]. Altogether, the demands on the preliminary preparation of the reader are somewhat advanced, even though this is primarily a technical and not a mathematical book. Borrowing an appropriate expression from one of his teachers, A. I. Lurie, the author would remark that the book is written “by an engineer for engineers.” Therefore, wherever a strict mathematical explanation was found to be either too difficult or too cumbersome, it has been replaced by an explanation based on simple physical concepts, which are founded on an analogy or on a practical experiment. I n particular, the discussion is simplified in the introduction and in Sections 1.5, 2.2, 2.3, 3.1, 3.2, and 3.3, where approximation methods for closed systems are described. Most of these sections (with the exception of that part of the text that is in small print) are

Preface

xi

independent of the rest of the book. An engineer who wishes to master the practical side of the subject as quickly as possible may wish to concentrate on these sections. T h e author is deeply grateful to Ye. P. Popov by whose initiative this work on nonlinear problems was begun, and who encouraged the publication of this book and, also, to A. A. Fel’dbaum who gave valuable advice on its structure. T h e author expresses gratitude to his colleagues in the department headed by Professor A. I. Lurie at the Leningrad Polytechnic Institute, whose kind attention and help was invaluable in allowing the book to go to press. T h e author would also like to sincerely thank 0. K. Sobolev for his diligent editing of the manuscript.

A. A. PERVOZVANSKII

CONTEN TS

•

Eo1Toa 's fouwo•o

vii

PR!rAC.It

INTRODUC TION

Chapter 1 NONLINEAR TRANSFOR MATIONS WITHOUT FEEDBACK Nonlinear Laglnt Transform11tionJ Nonlinur TramformatiO~'I with Laa: The Prob~em of Synthesis. Optimal Condition~ for Variout Clas~s of Tran.sfonnationt App lia~tion of Methods of SynthHit. 1.4 Nonlinear Fihen l.S SlttitliCIIl Linearization

1.1 1.2 1.3

,\e

16 4)

56 68

n

Chapter 1 NONLINEAR TRANSFOR MATIONS WITH FEEDBACK . STATIONAR Y STATES 2.1 A Shore Oetcription of the: Basic Method• of Investigation 2.2 The Application of Stati.sti.cal Linn.riution to the Analysis of Nonline-ar TranafomutioiU with Normally Oittributed Stttionary Signals 2.3 Computation of Frequency Oistortiom lntroduc:ed by Nonlinur E~ts 2.4 Rcstr1ctiont Imposed by the- Requiremtnt That true Input Signal of the Nonlinear System Be Normal 2. , The Synthesi.t o£ Linear Compenution Networks in C latc:d-l..oop Systems with Nonlintt.ritiea 2.6 Application of the Theory o( Markov Procc~Scs in the St·udy of Some Nonlineu Syttemt

xiii

88 91 107

liS 12l

138

Contents

XIV

Chapter 3 NONLINEAR TRANSFORMATIONS WITH FEEDBACK. NONSTATIONARY STATES 3. 1 The Transformation of a Slowly Changing Signal in the Presence of a High- Fr~qucncy Random Interference 3.2 Passage of a S lowly Varying Random S ignal through a Sytrem in a Suue with Periodic Oscillations 3.3 Transformation of the Sum of Wide- Band, Normal, Random Signals, and Harmonic Signals in a Nonlinear System with Feedback (Method of Statistical Lineariution) 3.4 Random Disturbances of Periodic SUI.tes in Relay Systems (Exact Solution by the Method of Alignment)

146 158

167 ISS

Chapter 4 EXTREMAL SYSTEMS 4.1 4.2

Basic Principles of che Operation of EJ(trernal System.-. Extremal Systems with a 1'ime Separation between Testing and Operation; Systems with Proportional Action 4.3 Discrete Extremal Systems with Constant Ste.ps 4.4 Extremal Systems in Which Testing and Operation Are Sepanted by a Frequency Band 4.5 An Automatic Extremal System with Simultaneous Testing and Operation

211 222 248 269

277

Appendix I FUNCTIONS m,(m., a.), h 1(m,, a,), a 1(m,, a,), AND a 1(m,, a, ) FOR SEVERAL TYPICAL NONLINEARITIES I. 2. 3. 4. 5.

6. 7. 8. 9.

The Ideal Relay Y = I sgn X A Relay with a Dead Zone An Element with a Bounded Zone of Linearity An Element with a Dead Zone An Element with a Bounded Zone of Linearity and a Dead Zone An Element with a Characteristic o f the Form Y = Nx2 sgn An Elcmcnt with the Char acteristic Y -= Nx3 A Relay with a H ysteresi5 Loop An Element with 011 Bounded Zone of Linearity with Nonsymmetrical Sounds

293 294 297 297

.~

298 300

300 301 303

Contents

xv

Appendix II REPRESENTATION OF A LAGLESS NONLINEAR TRANSFORMATION IN THE FORM OF AN INTEGRAL TRANSFORMATION IN A COMPLEX REGION. THE THEOREM OF R. PRICE

304

Appendix Ill COMPUTATION OF THE INTEGRALS I.

309

Appendix IV THE COEFFICIENTS OF STATISTICAL LINEARIZATION h 1(a, a) AND h1 (a, a) FOR TYPICAL NONLINEARITIES I.

2. 3. 4. 5. 6.

The Ideal Relay A Relay with a Dead Zone An Element with a Bounded Zone of Linearity An E14!ment with a Bounded Zone of Linearity and De•d Zone An Element with the Characteristicj(X) = NX 1 sgnx E'ements with the Characteristic /(X) .,. NX''>+' where n = I, 2, 3, ...

311 313 314 31S 317 317

Appendix V ELEMENTARY STATEMENTS ON THE THEORY OF MARKOV PROCESSES RtLAT t;D LtTBR.ATURE

329

BIBLIOGRAP HY

332

.JJ9 SUBJF.C..'T INOt:X

341

0

INTRODUCTION

T h e purpose of any automatic system is the generation of suitable responses as a function of variations in the external conditions. T h e external effects can vary widely in character. For example, the operating condition of a servo system is determined by the motion of the control shaft, by variations in a load moment, in the voltage of a power input, in temperature and humidity of the environment, or in the effect of electromagnetic interferences. T h e reaction of an automatic system, i.e., the physical process that takes place inside the system, can be just as diversified. However, in dynamic studies most of the external factors are neglected, and only those that are significant are taken into account. We shall designate these basic external effects by the input signals Z,). Z(Z1 , 2, , Similarly, we confine ourselves to a study only of those reactions of the system that, to a significant degree, characterize its correct performance. These reactions are called the output signals X(X, , X , , ..., X,) of the given system. As a first approximation for a servo system one may assume that the input signals are given by the position of a shaft and the variations of the moment of the load, and the output signals by the position of the output shaft and the error signal. T he n a dynamic analysis of an arbitrary automatic system is reduced to the study of the transformation of the input signals Z to the output signals X. T h e properties of this transformation, generally speaking, determine the whole set of physical characteristics of the system. However, every real system can be represented by an idealized dynamic scheme that approximately describes (by means of a rule or an algorithm) the transformation of the signals. T h e study of these idealized schemes and the development of schemes with optimal characteristics is the basic objective of the dynamic theory of automatic systems which is known also as technical cybernetics. T h e characteristics of an idealized scheme should closely approxima’.,

1

2

Introduction

ate those of a real system; in developing the optimal scheme, it is necessary to choose characteristics that can be physically realized. If in the operation of the real system the value of the output signals at any instant depends only on the value of the input signals at the same instant, then the system can be made to correspond with the dynamic scheme of a lagless transformation (i.e., a transformation without memory). If the reaction of a real system to the combined action of several signals is equal to the sum of the reactions to each separate signal, and if the reaction of the system to a one-dimensional input signal is a one-dimensional output signal (i.e., assuming the superposition principle is valid in the system), then it can be made to correspond with the scheme of a linear transformation. T h e statement that a real system is linear or lagless always turns out to be valid only as an approximation which assumes certain operational conditions, and which imposes certain limitations on the input signals. A more or less complete specification of the response time of a system is considered, at the present, absolutely necessary for the design of almost any automatic system. At the same time, one usually finds that in studying the behavior of systems with random external influences the investigations are always limited to linear dynamic schemes. Further, it is assumed, and often correctly, that the inaccuracy of the dynamic idealization is compensated by the simplicity of the results of the analysis. However, in many important practical cases, the refusal to take nonlinearity into account leads to qualitatively false concepts about the operation of the real system, if such a system is already constructed, or to an underestimate of the possibilities for improving a system in the process of construction. I t should be stressed that, in the theory and practice of the construction of automatic systems, these “nonlinear effects” often do not require a study of a nonlinear transformation of any great complexity. It is completely permissible to limit oneself to rather simple schemes which are obtained by combining well-known linear and lagless nonlinear transformations. T h e form of these dynamic schemes frequently satisfies the structure of the real system and the properties of the elements of which it is composed. Beginning with the simplest schemes, we shall now give a short

Introduction

3

summary of the basic characteristics of transformations which are used in dynamic idealizations of automatic systems. For the sake of brevity, we shall limit our discussion to transformations with a single input and a single output. 1. Linear Lagless Transformations. T h e basic equation can be written in the form* x = hZ, (1.1)

where h is a constant coefficient or a given function of time. 2. Nonlinear Lagless Transformations. T h e relation between the input and output signals is given by the equation

x =f(Z),

(1 4

wheref(2) is a given function of 2 and, possibly, of time. Frequently, nonlinear transformations can be given in an implicit form, x XI, (1.3)

=w,

which corresponds to a scheme having feedback with respect to the output signal. 3. Linear Transformation with Lag. output signal X ( t ) at the given time t signal Z(t) at the same time, but also This transformation is written in the

x(t)=

1

I n this case, the value of the depends not only on the input on its value at different times. explicit form

m -m

h(t, T) z ( T ) dT,

(1.4)

where h(t, T ) is the impulse function (Green’sfunction) which has the physical meaning of the response (of the output signal) of the system at time t to a unit impulse 6 ( ~ at) the input. For physically realizable systems, we have, for t < T , h(t, T)

0.

(1.5)

* Here, and in the following part of the Introduction, a linear transformation signifies only a homogeneous transformation.

4

Introduction

Thus, the linear functional transformation (1.4) is completely determined by one function of two variables h(t, T ) . For the special case of a linear transformation with constant coefficients, we have the simpler expression

x(t)= jmh(f -

T)

Z(T)d T ,

-X

or, taking (1.5) into account, we obtain

x(t)=

It

h(t - T ) Z(T)d7 =

1 h ( ~z(t) m

- T ) dT.

0

--x

(1.7)

T h e linear transformation can also be given in the implicit form X(t) = J

03

-X

h,(t, T )

[z(.)-

Jm -m

k ( ~S), X ( S )ds] d r ,

(1.8)

which corresponds to the description of a system with feedback (that is, a closed system). Here, h,(t, T ) is the impulse function of the forward loop and k ( t , T ) is the feedback impulse function. T h e implicit equation (1.8) is essentially an equation for the output signals in terms of an integral equation with a kernel which depends on the values of the input signals at various instants. T h e equation for the linear transformation in terms of differential equations is more frequently encountered and usually follows from the analysis of transfer characteristics of the separate elements of the automatic system

where Q ( d / d t )and R ( d / d t )are polynomials in powers of the differential operator either with constant coefficients or with coefficients which are functions of time. We must specify the initial conditions for Eq. (1.9). Henceforth, we shall assume that they are all zero. Any equation can be reduced to this case by introducing nonzero initial conditions into the equivalent input signals. I n practice, the equations for the transformations in the forms (1.4) and (1.9) are not equivalent because the determination of an impulse function which corresponds to an equation with coefficients variable with respect to time is a rather difficult problem.

5

Introduction

However, for equations with constant coefficients, the implicit form of Eq. (1.9) can be made explicit quite easily and, moreover, it is often convenient to use it in this form. We shall now apply the Laplace transformation to Eq. (1.9). We obtain Q ( p )X = R ( p )Z or X =F@) Z , (1.10) where X and 2 are the Laplace transforms of the functions of time

X ( t ) and Z ( t ) , and where F(p) = R(p)/Q(p)is the transfer function

of the linear transformation. On the other hand, applying the Laplace transformation to (1.7), we obtain

X

=Z

and, consequently,

F(p) =

m

e-P'h(7) dT,

(1.1 1)

0

Irn h(~) e-pr

0

dT,

(1.12)

that is, the transfer function is the Laplace transform of the impulse function. Using transfer functions, it is easy to change the implicit equation for the linear transformation into an explicit one (the change is from a closed-loop to an equivalent open-loop circuit). Applying the Laplace transformation, e.g., Eq. (1.8), it is not difficult to find an explicit expression for the transfer function of the closed system (1.13)

whereF(p), F,(p) and K ( p )are the transfer functions which correspond to the closed system (with the explicit transformation), to the forward loop, and to the feedback loop, respectively. 4. Nonlinear Transformation with Lag. T h e fundamental form for this class of equations is the implicit differential equation (1.14)

T h e introduction of the differential operator as an argument of this function indicates that there is a functional dependence, not only

6

Introduction

between the signals X and 2, but also between the derivatives of arbitrarily high order. Because there is no general method for solving nonlinear differential equations of arbitrary form, it is impossible to write an explicit expression for the nonlinear transformation. It has already been pointed out that in the theory of automatic systems there are several very important subclasses of nonlinear transformations with lag. a. NONLINEAR TRANSFORMATIONS WHICHCAN BE REDUCEDTO LINEARTRANSFORMATIONS (NONLINEAR TRANSFORMATIONS WITHOUT FEEDBACK). Such transformations are the result of the sequential application of, first, a nonlinear transformation without lag, and, second, a linear transformation Y

=f(z),

or

x =1

h(t, T ) Y ( T )dT,

(1.15)

x =I

h(t, ~ ) f [ zd(r ~ . )]

(1.16)

m

-m

m

-m

One can associate with this subclass a more complicated transformation of the form (1.17)

which corresponds (Fig. 1) to the parallel application of several transformations of the form (1.16), and also to a transformation of the form

I

m

X

=

-m

[Im h,(~, ds]

h2(t,~ ) f

--m

FIGURE1

S)

Z(S)

dT.

(I,18)

7

Introduction

This corresponds to the sequential application of a linear transformation, a nonlinear transformation without lag, and, again, a linear transformation (Fig. 2).

FIGURE 2

Because of structure and comparative simplicity, the transformations described above which are similar to linear transformations are called reducible-to-linear transformations [65]. T h e subclass of nonlinear transformations which are reducibleto-linear transformations may be generalized by considering more complicated integral operators than those given in (I.17) and (I.18), for example, by substituting an operator of the following type [117, 1181 :

x

J

OD

=

(1.19)

F{z(T),~)~T,

-OD

where F is an arbitrary nonlinear function. However, in this case, we lose the simplicity of the relationship between the operator and its corresponding differential equation, and, as a result, we also complicate the relation of the operator to the physically realizable system by means of which the transformation is performed. Of more practical importance is the following subclass of nonlinear transformations with lag. b. NONLINEAR TRANSFORMATIONS WITH FEEDBACK. Transformations of this type are given by the implicit equation

1 W

X

=

-m

= ho(t,T)f

[Z(T)

-

Jm

k(7, S)

X ( S )ds]

dT,

-W

which corresponds to the block diagram shown in Fig. 3.

FIGURE 3

(1.20)

8

Introduction

Here h, ( t , T) is the impulse function for the linear part of the forward loop of the transformation, I(t, T) is the impulse function for the feedback loop, and f is a function which determines a lagless, nonlinear transformation for the forward loop. T h e study of the simplest types of nonlinear transformations indicated here will be the main subject of this book. Let us recall the methods by which signals undergoing transformations can be fed into a system. T h e signals can be: (a) random functions of time; (b) nonrandom functions of time. Physically real nonrandom signals, that is, signals which have exact values at each moment of time, are extraordinarily diversified. However, the study of the transformational characteristics of a system usually involves only a few types of functions; the most important among these are harmonic functions (or sums of harmonic functions) and functions the graphs of which have jump discontinuities. By using these, one takes advantage of the fact that in many cases real signals transformed by automatic systems can be approximated by certain typical functions and, further, that this provides a simpler solution of basic problems in the theory of transformations. It is convenient to choose harmonic functions because other functions belonging to a much broader class can be described by the Fourier transform in the form of a finite or infinite series of harmonic functions. A process Z ( t ) is expressed in terms of its Fourier transform (the complex spectrum)* Z ( j w ) in the following manner : 1 2l-r

~ ( t= ) -

J

-I

--a

Z ( j w ) ejcutdw,

(1.21)

that is, Z ( t ) is represented in the form of an integral sum of harmonic functions which have amplitudes i Z(jw) dw I and which have phases equal to arg Z(kw). For a given process Z ( t ) , the complex spectrum is Z(jw) =

jI Z ( f )e-jwt d t . --x

*

(1.22)

We are using the same notation for the transformed function as for the original function.

Introduction

9

If Z ( t ) is a periodic process, then its spectrum will be a line spectrum; that is, it can be represented by an even number of coefficients in its Fourier series. T h e spectral or frequency representation plays a very important role in the quantitative and, what is more important, the qualitative analysis of the transformed signals. Compared with the signals which are definite functions of time, signals which are random functions of time present an altogether different set of problems in the theoretical and practical study of dynamic automatic systems. By definition, a random function is a finite or infinite set of given functions whose variations are governed by probability relations. Thus, to study the rules for transforming a random signal is equivalent to studying the general probability characteristics of the transformation for a whole collection of definite signals which are given by a single mathematical description. Moreover, in many real cases it is not generally known hcw each of the signals enters this collection, whereas the general probability characteristics can provide an effective means for theoretical and experimental study. When the signal to be transformed is given in the form of a random function, exact values for the signal itself or the determination of the values which result from its transformation at any moment of time cannot be given. Instead, the problem of specifying the random signal reduces to the problem of specifying a system of functions which determine the probability of the limits within which the value of the random signal exists at one moment of time and for a sequence of moments of time. A complete description of the random function of time is given by the following infinite system of probability characteristics : (1) T h e function W,(z, , t,) is equal to the probability that Z ( t , ) < z1for t = t,; (2) T h e function W,(z, , t , ; z , , t,) is equal to the probability that Z ( t , ) < z1 for t = t , and Z ( t , ) < z , for t = t , , and so on. T h e function W,(z, , t , ; z , , t , ; ...; z , , t,) gives the probability that at the moments of time t = t , , t , , ..., t, all of the following conditions will be satisfied : Z(tl) < zl,Z(t,) < z, , ..., Z(t,) < z, . This function is called the n-dimensional distribution of the random process Z(t). If the functions W , ( n = 1, 2, ...) are differentiable with respect

10

Introduction

to each z , , then in addition to specifying the process by means of these functions it is convenient to make use of the system of ndimensional probability densities w o , ( z l , t,; z , , t,; ...; z , , t,), where w, =

anw

az, ax, ... az,

(1.23)

'

T h e quantity w, dz, dz, ... dz, is the probability of the joint event that for t = t , , z, < Z(t,) < z1 dz, , for t = 1, , 2, < Z(t,) < z, dz, ,

+ +

for

t

=

t, ,

z,

< Z(t,) < z, + dz,

Of particular significance in the theory of automatic systems is a special class of random functions-the class of stationary random functions (processes). A random process Z(t) is called stationary (in a limited sense of the word) if, for arbitrary t , , t, , ..., t, and T , we have the identity Wn(z, > t i ; zz > 1;, ...; zn tn) 9

Wn(z1 9

ti

+

7;~

2

t ,,

+

7;

...;zn

3

tn

+

T),

that is, if a translation along the time axis of arbitrary length T for every moment of observation t, (k = 1, 2, ..., n) does not change the character of the distribution of the process. T h e distribution of probabilities, generally speaking, gives a more complete characterization of the random functions. However, in many cases one can find the properties of transformations of random signals from their moment characteristics. T h e moment characteristics of nth order (or simply the moments of nth order) are defined as the mathematical expectations of the products of the values of the random function at the moments of time t, , t , , t, ..., t, nzn(t,

t

tz

9

...>tn)

= -

M { z ( t , ) z(t,)... Z(tn))

I' Jrn ... jz

--a

-m

--a

z,z,

... znZ4~,(z, , t,; z, , t,; ...; z , , t,) x dz, dz, ... dz, ,

(I 24)

that is, they are defined as values of products averaged over all possible

11

Introduction

realizations of the random function, taking into account the probability

of each realization.

Of special interest are the moments of first and second order, which are denoted by mz(t) and B,(t, , t,). T h e moment of first order, or the mathematical expectation of the random function Z ( t ) , is given by the equation (1.25)

and the moment of second order is given by W

l

!

t2)

=

M W , ) Z(t2))

=

j"r,J

io

-m

z1z2w2(z1,

tl;

z 2 ,t2)

dz, dz,

.

(1.26)

We introduce the concept of an unbiased function Z y t ) = Z ( t ) - m&).

(1.27)

T h e second moment of the unbiased random function is called the correlation function of the process RZ(t1

I

t2)

= M { [ Z ( ~ l l- mz(t,)l[Z(t2) - m,(tz>l>*

(1.28)

I t is obvious that RAtl

9

t2)

=

&(t,

9

t2)

- Ml)mz(t2).

(I .29)

For stationary processes we have m,(t) = m, = const,

(1.30)

t a ) = RZ(t1 - t 2 ) = RZb),

(1.31)

and RZ(t1

7

where T = t , - t , . T h e property (1.31), which leads to the proposition that the correlation function for the random process Z ( t ) depends only on the difference of the arguments t , and t,, generally speaking, is satisfied by a broader class of random processes than the one described above. This class is often called the class of random functions which are stationary in the general sense, or stationary in the sense of Khintchin.

12

Introduction

If the condition (1.31) is satisfied and if certain weak restrictions [65] are fulfilled, then we have the basic ergodic relations for the theory of stationary processes (1.32)

and (1.33)

that is, in order to obtain the mathematical expectation and the correlation function, one can pass from an average over the realizations of the random function to an average with respect to time for one of the realizations. An important property of stationary random processes is that they can be described by means of spectral representations. Let us examine the random function Z T ( t )defined by the conditions Z T ( t )E Z ( t )

0

zT(t)


for

-T

for

I t I > T.

(1.34)

We find the complex spectrum which corresponds to it,

j

a,

z T ( j w )=

z,(t)e - j w t

dt =

-a,

j

T

-T

zT(t) e-jwt

dt,

(1.35)

and which, likewise, represents a random function. T h e limit S,(W)= lim

T+m

1 2T

-M {I ZT(jw)12}

is called the spectral density of the process Z ( t ) .I t is not hard to show that the spectral density of a stationary process is related to its moment of second order by the equation

j

a,

s,(w) =

--P

e-jwr B,(T)dT,

(I .36)

that is, the spectral density is the Fourier transform of the moment of second order, and for an unbiased random function it is the Fourier transform of the correlation function.

Introduction

13

T h e description of a random function by its spectral density, of course, is not complete, and this distinguishes it from a nonrandom function. After this explanation of the basic types of dynamic characteristics for signals, we shall now give a general formulation of problems which arise in the study of automatic systems. T h e first of these is usually called the problem of analysis for a given system, while the second is called the problem of optimal synthesis for the system. 1. The Problem of Analysis. Given the characteristics of the input signals Z and given the dynamic representation of the system, or equivalently, the form of the transformation F which corresponds to this system, it is required to find the characteristics of the output signals X .

2. The Problem of Synthesis. T h e characteristics of the input signals Z are given. T h e desired characteristics of the output signals X also are given. I t is required to find the form of the transformation F out of a given class of transformations which will make F { Z } approximate X optimally (in some given sense). We shall now investigate the fundamental aspects of some specific problems and the principal means for solution. T h e analytical problem can be solved in two ways-either analytically or experimentally. If an apparatus for the system is already constructed, then a direct experimental study of the output signals of the system and their mathematical measurement is almost always the most convenient method. If the system exists only as a design, it is possible to make a theoretical or experimental study of the transform characteristics of its separate parts and their interrelations; on the basis of this study, a mathematical description of the transform characteristics of the system as a whole can be given. T h e effort to take into account all dynamic subtleties of the real system therefore is hardly necessary; besides, it will lead to such a complication of the problem that its analytical solution will become far too complex or even impossible while the experimental solution with the help of analog models will be much too labor consuming. And, more important, the search for the exact dynamic circuit usually makes it impossible to find the simple quantitative relations generally required by the engineer.

14

Introduction

Therefore, the analytical method of solving a given problem is acceptable only if it enables us to obtain sufficiently simple quantitative, and, more important, qualitative answers. This qualification requires that in the following exposition we pay attention to simple approximative methods regardless of the degree of accuracy and even when limitations make them valid only in a qualitative sense. Given certain random input signals, the analysis of output signals sometimes excludes the possibility of the experimental approach even in those cases when the random signal is given, not as a collection of all the realizations, but only as some bounded number of probability characteristics of the collection as a whole. At the same time, since we can generate random signals (cf., for example, Feldbaum [94]), in principle, we can also construct an arbitrary signal. This fact, together with the contemporary proficiency in making mathematical models with analog or digital computers, allows us to reduce the analytical problem to the problem of measuring statistically the experimental results of the output signals. A similar and practically identical method, widely known as the Monte-Carlo method, is always preferable to the purely analytical method when the dynamic circuit of the system and the characteristics of the signals are so complicated that simple approximative techniques of study are ineffective. As for the problem of synthesis, one would assume that the analytical methods would prevail; however, in practice they are generally replaced by the method of testing out several variants, a method which is based on intuition and on experimenting with the design and operation of similar systems. Obviously, this situation arises not only because the existing analytical methods of synthesis are very complicated, but also because in the majority of cases the necessity of passing from the optimal transformation which has been found (for the ideal dynamic circuit) to a real system is not taken into account. T h e real system usually includes the controlled variable and the power source, which is chosen on the basis of power and economic considerations, the characteristics of the power source and the controlled variable object itself often being of a pronounced nonlinear type. An analytical solution of the synthesis problem which does not account for these factors is useful only in the sense that it points the way to potentially achievable limits of operation associated with the characteristics of the real signals. T h e theory of synthesis of nonlinear

Introduction

15

transformations essentially broadens the possibilities for analytical study, both in the sense that it gives a more complete account of the real characteristics of an automatic system, and in the sense that it gives a more precise indication of the potential limits on improvements of dynamic properties. Unfortunately, the methods of nonlinear synthesis are still far from being completely perfected and, furthermore, they cannot always be reduced to practically acceptable results. Therefore, apart from introducing some of the clearer, modern methods of synthesis, the book is mainly concerned with the solution of analytical problems. Here, we keep in mind the fact that obtaining sufficiently simple results, by means of analysis, will facilitate the problem of finding the optimal choice of parameters in the system under study and will usually suggest a way to improve its structure; that is, in the last analysis, it will also simplify the solution for the general problem of synthesis.

chapter 1

NONLINEAR TRANSFORMATIONS WITHOUT FEEDBACK

1.l.Nonlinear Lagless Transformations

Nonlinear lagless transformations are the simplest form of nonlinear transformations. Their significance in the theory of nonlinear systems is determined by two factors: (1) For certain properties of signals the nonlinear lagless transformation is a satisfactory representation of the dynamic circuit for many real systems. (2) Nonlinear lagless transformations are often component elements of more complicated nonlinear transformations. In this section, we shall study only problems involving the analysis of output signals; moreover, we shall consider only transformations with a single input and a single output. Let f be a given function which defines the transformation of the input signal Z ( t ) into the output signal X ( t ) . T h e analytical problem is to determine the probability characteristics of X ( t ) knowing the probability characteristics of the function Z ( t ) . Since the transformation under consideration is a lagless transformation, the random variable X ( t J depends only on the random variable Z(t,), where ti is an arbitrary moment of time. Therefore, the problem of finding the characteristics of the random function x is the same as the problem of finding the characteristics of a nonlinear function of a random variable. T h e time t enters into consideration only as a parameter. Let us study the basic problem of finding the first two moments m, and B, of the signal X ( t ) . By definition [cf. (1.25) and (1.26)] the moment characteristics of X ( t ) are found from its probability density. However, it can be shown that, to find the mean of the function f ( Z ) , it is admissible to differentiate directly with respect to the probability distribution of its argument. 16

1.1. Nonlinear Lagless Transformations

17

Therefore, the moments of the first and second orders are given by the following equations : mJtl

M{x(t))=

--1c

and B,(t, T )

j

a

=

=

M { X ( t )x(t

1

ZI

=

-m

--co

+

fbl)wl(zl) dz, ,

(1.1)

T))

f ( z d f ( 4 wz(z1, 2 2 )

dz1 dz2

1

(1.2)

+

where 2, 3 Z ( t ) and 2, # Z(t T). We shall investigate a more exact process for computing these characteristics for several types of'signals Z ( t ) . Let the distribution of the signal Z ( t ) be normal. Then, we have (cf., for example, Pugachev [65])

and

where

For a stationary signal

Henceforth, we shall also be concerned with the case when only the unbiased component is a stationary function, while the mathematical

18

I . Nonlinear Transformations without Feedback

expectation m, is some function of time, that is, when the relation (1.4) is not satisfied. We see at once that, for a normally distributed input signal, the quantity m, is determined directly by the quantities m, and u, , mx =

m,(m,

>

az),

(1.6)

which may or may not depend on time [if Z ( t ) is stationary]. There is no explicit dependence on the variable t in (1.6). We shall now go on to a direct computation of the mathematical expectation m,(m, , u,) for several nonlinear transformations. A summary of the results of the computations for many typical nonlinearities often encountered in the study of automatic systems is given in Appendix I. Example 1 .

T h e stringent symmetrical bound :

For this transformation we have

which, after some simple manipulations, leads to

where we are using the notation

1.1. Nonlinear Lagless Transformations

19

and where

is the probability integral, for which tables are generally available. Figure 4 shows a graph of the dependence of m, on m , for various values of crl. From this graph one can draw definite qualitative

FIGURE 4

conclusions. In the first place, the presence of a random component tends to smooth out the nonlinearity of the resulting analytic characteristic. T h e range where, for practical purposes, there is linearity (where the deviation from linearity is about 5 % ) increases as the variance of the random component of the input signal increases. I n the second place, the value of the amplification factor on the linear portion of the characteristic m,(m,) decreases with increasing u1 (at cr, = 1 the amplification factor is equal to 0.6 of its value when there is no interference). These properties are shared by a large class of nonlinear elements. A decrease in the effective amplification factor am,/arn, in the zone of linearity (with increasing cr) is characteristic of all elements where the “differential” amplification factor aX/aZ decreases with increasing absolute value of the signal, because the random component causes an averaging for the characteristic over the whole range of values of the signal. This phenomenon, as will be shown in greater detail in Chapter 2, turns out to have a decisive effect in estimating the stability

20

1. Nonlinear Transformations without Feedback

and quality of nonlinear control systems which are operating in the presence of intense interference. Example

2. T h e ideal relay f(Z)

= -

:

1, -I,

z> 0, z< 0.

(1.10)

T h e effect of smoothing is still more pronounced for relays with sharper nonlinearities. T h e value of the mean component at the output of the ideal relay is given by the equation

(1.11)

This characteristic is practically linear in the sense indicated above when uzjmz < 1.4. T h e effect of linearizing the nonlinearity by the random component is of the same type as the well-known effect of oscillatory linearization by means of a periodic signal. I n this case too, the analysis of the behavior of the mean component when under the effect of either a random or a periodic oscillation is simplified; more specifically, it becomes possible to replace an essentially nonlinear transformation by an equivalent transformation which is, on the average, linear, and which has the transfer function

which is constant over a rather wide range of variations of the mean component of the signal. Example 3 . In Examples 1 and 2 only odd functions of Z were considered. Let us now look at some cases where this condition is not satisfied. In many control systems with devices that limit the output signal, it is not uncommon for the limiting equations to be nonsymmetric, that is, instead of (1.7),

1 . 1 . Nonlinear Lagless Transformations

we have j(Z) = -

z

1, 2,

-1(1

-1(1

-

a)

1,

< 2 < 1, 2

- a),

21

< -1(1

- a),(1.12)

where 0 < a < 1. Then the mean value of the output signal is given by the following equation :

Let the expectation of the input signal be equal to zero, that is, let the signal be random. Then

(1.14) It is not difficult to show that an expansion of m,il in a series in powers of a will begin with terms of order a, and only for a = 0 will *.r = 0.

I

(1.15)

Thus, the nonsymmetric property of the characteristic leads to a new qualitative concept : a nonlinear transformation makes it possible to detect the random component of the input signal. A mean component appears at the output signal even when there was no such component at the input. Example 4. We shall study the simplest example of a nonlinear transformation of two input signals especially for the case when the

I . Nonlinear Transformations without Feedback

22

output signal depends nonlinearly both on the input signal and its derivative x = f ( Z ,P Z ) . (1.16) T h e expectation can be found from the equation m, =

* --2

J

00 f(z9 -%

P.)

W,(.,

p z ) d. d ( p z ) ,

(1.17)

where w,(z, p z ) is the joint distribution function of the signal Z and of its derivative taken at the same instant. I t is a well-known fact (cf., for example, [49]) that for a normal signal these values are independent and that the quantity p Z also has a normal distribution %(Z,

P4

=Wl(4 Wl(P4

--

1

-(z

1

- m,)2

2/2.rra,

exp PZ

- pmA2 ( - ( P Z 2D;z--j

’

(1.18)

where

It should be emphasized that multivalued on-off-type nonlinearities with hysteresis encountered in the study of automatic systems are not of the type (1.16), although this description is often found in the literature. Let us now study an approximate solution of this problem, remembering that multivalued nonlinearities usually involve transformations with lag. Z A, Let X = f ( Z )for Z < - d, and for Z > A , ; when -dl the dependence of X on Z is multivalued : X = f i ( Z ) if the presence of Z in this interval was preceded by its presence in the interval Z > A , , and X = f,(Z) if Z was previously in the interval Z < - A , . Then, if Z ( t ) is a stationary random process, it can be assumed that the probability that Z has arrived from the right at the interval where f is multivalued is equal to the relative time of dwell in the interval Z > A , :

< <

P1 =

Jz$

SY2 w1(z) d z wi(z) d z

+ S“4

~ 1 ( zd)z

’

1.1. Nonlinear Lagless Transformations

23

Then, the probability of its having come in from the left is PZ = 1 - P 1 .

In accordance with this, we use the following equation to compute the expectation of X :

(1.19)

In particular, for an output signal of a relay with a symmetric hysteresis loop of width 24 we obtain m, =

A

[@ (-u-)

+

111,

-@

(A

uz

m=

11 1 - @ (

1 A+m,)-@(A-rnz)0 2

0 2

(1.20)

Next we compute a more complicated characteristic, the moment of second order. T o simplify the computation it is convenient to use an expansion of the two-dimensional probability density into a series of Chebyshev-Hermitian polynomials [49],

where H a ( ( ) is a polynomial given by the relation Hn(0

=

dn

(-1Y exp(+ C2/2)d5" exp(-P/2).

A well-known recursion formula for these polynomials is [48] Hn+,(5) = U f n ( C ) - nH,-1(5)*

and, moreover, it is obvious that Ho(0

=

1,

Hl(0

=

1.

24

1. Nonlinear Transformations without Feedback

T h e polynomials H,( () are orthogonal with weight exp( -- c2/2) on the line -m < 5 < 00:

or

- n!, = 0,

m = n, m # n,

that is, H,((z - m,)/u,) are orthogonal with weight wl(z) on the line -00 < z < 00. Substituting (1.21) into the general equation (1.2) we obtain (1.23)

where

1

"rn

(i

=

1,2)

(1.24)

or (1.25)

Thus, the problem of computing the moment of second order is reduced to the problem of computing a series of moments of first order while each of these moments depends explicitly on two parameters mzi and azi. This greatly simplifies the problem of tabulation. T h e series (1.23) is absolutely convergent. T h e coefficients a,, and a2, rapidly decrease (at least as fast as l/n!). Since

I . 1. Nonlinear Lagless Transformations

25

it follows that =

M{f(z,N,

a20 ==

M(f(41.

a10

Therefore, referring to (I.29), we can write the expression for the correlation function of the output signal in the form of a series in powers of the correlation function of the input signal : ~ z ( t T,)

=

3

pzn(T) a l n a z n

n=l

(I .26)

.

Here a,, and a2, depend identically on the arguments. For some types of nonlinearities these dependencies lead to the types of equations and graphs given in Appendix I. If the process Z ( t ) is stationary, that is, if mZ1= mz2 =

then a,,

=

and

7n2

5,1

=

5,2

=

5,,

aZn.= an and the series (1.26) can be written in the form

( I .27)

Thi s series has a clear physical meaning. Its first term corresponds to that component of the output signal which has a correlation function coinciding in form with the correlation function of the input signal. T h e remaining terms correspond to distortions related to the nonlinear elements. These distortions are usually not significant for two reasons : I n the first place, because the coefficients in the series diminish as l/n! and, second, because the quantities p , " ( ~ ) decrease since 1 p , ( ~ )I < 1 when T > 0. Significant distortions will take place only fot small T ; this corresponds to high-frequency distortions in the spectral density. Distortions lead to variations in the mean-square value of the random component of the output signal equal to

2 m

:u

=

an2

.

Tl=l

However, these variations usually are not significant.

(1.28)

26

1 . Nonlinear Transformations without Feedback

Let us construct, for example, the correlation function for a signal at the output of an ideal relay when m, = 0 and p , ( ~ ) = exp (- I 7 I). Figure 5 shows separately the first term of the series (1.27) and

4

FIGURE5

the exact value of R,(T),which is given by the equation 212 R,(T) = - a rcsin pZ(T). 7r

(1.29)

T h e derivation of this equation is carried out below. From Fig. 5 it can be seen that the basic form of the correlation function is given by the first term. T h e distortion is of greater importance when m, # 0 (Fig. 6), since the coefficient a2 increases as the ratio m,lu, increases.

51,o.

0

m;

I .o

FIGURE 6

uz

--c

2.0

1.1. Nonlinear Lagless Transformations

27

We shall now determine another important characteristic of the nonlinear transformation, namely, the cross-correlation function of the input and output signals; for this purpose we shall again use the expansion (1.21). T h e cross-correlation function R,,(t, T) is given by the relation %(t,

7) =

M { X ( t )z(t f

7))

- M { X ( t ) }M{Z(t $. 7)).

(1.30)

We compute the joint moment of second order

Writing the series (1.21) for the two-dimensional density w z ( z l , zz) we can express B,, by (1.32)

where a,, is given by Eq. (1.24) and b, =

d%n!

uz2

.

zH, --m

(

:.,""" ) exp[

- m 2 2 ) 2 ] dz.

2e2

We break down the right-hand side of (1.33) into two terms :

We introduce the new variable

5

= (z -

mzz)/uzzand obtain

(1.33)

28

1. Nonlinear Transformations without Feedback

Recalling that HO(0

=

Hl(1) = 1

1,

I

we can write the above expression in the form

+ mz, J"

io

Hn(1) ~ x P ( - P / ~d1] )

HO(O

-W

.

Then, from the orthogonal property of the polynomials H,(ZJ, we obtain b, = m z 2 , bl = 1, (1.34) and b, = 0 for n >, 2. (1.35) Hence, BZZ

= a1omz2

+

UllPZ

(1.36)

and, consequently, by the definition (1.30) (1.37)

because we know from (1.35) that a10

= m,,

*

Equation (1.37) expresses an important characteristic of nonlinear transformations for normal signals. T h e method of expanding the two-dimensional probability density into a series is very general and rather effective. However, its practical .application requires several lengthy calculations. In some cases other methods give results in a closed form such as in Eq. (1.29). We shall give two examples of the computation of the correlation function by applying the theorem of R. Price (cf. Appendix 11). This theorem, which will be used in its simplest form, asserts that for a normally distributed signal the following relation is true: (1.38)

1.1. Nonlinear Lagless Transformations

29

In many instances the computation of characteristics described by the derivatives of the resulting nonlinear functions is much easier. In particular, this is true when one is computing moments for nonlinearities at the output signal with piecewise-linear characteristics.

Exumple 5 .

Let us study, for example, the ideal relay:

f(Z)

= =

Then,

z > 0, z < 0.

1, -1,

f’(Z)

=

(1.39) (1.40)

216(Z).

We compute for this case

w Y(Z,)U22)1 It is assumed that

m,,

=

m,,

=

0.

.

o,, = u r P =

Making use of the properties of the delta functions, we immediately find that

( I .42) Consequently,

(1.43) We shall assume that for pz = 0, R, immediately gives Eq. ( I .29).

=

0. Then the integration of (1.43)

Example 6 . As another example, let us consider the determination of the correlation function for a nonlinear output signal with the characteristic -

f(2)=

,&

exp

(-

&)

dt.

(1.44)

This characteristic, as can be seen from Fig. 7, can approximate very satisfactorily many characteristics which arise in actual experiments. A variation in the values of the parameter b makes it possible to describe elements with different levels of “smooth” limiting. Let m, = 0. Then

30

1. Nonlinear Transformations without Feedback

1.0

0

2.0

3.0

4.0 - Z

FIGURE 7

since

Using new variables p1 and

al,

we obtain

M { f’(zl)f’(.z,))

where

(1.46)

The integral is equal to unity since the integrand is reduced to the form of a two-dimensional probability density. Consequently,

or

1.1. Nonlinear Lagless Transformations After integration this results in

R,

=

26%arcsin

(--f--) 1

n

+

P

31

( I .47)

(b2/%2)

The possibility of obtaining such a simple formula turns out to be the exception rather than the rule. Therefore, the basic method of computing the correlation function for an output signal with a normally distributed input signal is the one described above which uses a series expansion into Chebyshev-Hermitian polynomials.

Let us go on to study the transformations of signals for which the distribution function is other than normal. Let Z ( t ) be a harmonic function with a random phase. A stationary random function, namely, a harmonic function with a random phase and with a constant amplitude and frequency, is given by the equation Z ( t ) = a sin(wt

+ Y),

where Y is a random quantity uniformly distributed in the interval (0,277). Thus, Z ( t ) , on the one hand, can be considered a set of realizations each of which is a definite function a sin(wt -t#,J displaced relative to an arbitrary origin by a phase # k . On the other hand, the same process can be assumed to be a definite function of two parameters t and Y,where Y is a random variable. T h e latter assumption is more convenient for defining the moment characteristics of the signals X ( t ) = f [ Z ( t ) ]= f [ a sin(wt

+ Y ) ] = f*(Y).

We compute the expectation of the output signal as follows : m, =

J

ffi

-m

xw,(x) dx =

J

02

-.a

f*(4 W l ( 4 &.

(1.48)

T h e latter equation is true because of a general rule according to which the mean of a function can be taken with respect to the distribution of its argument. But the distribution of the phase Y is given in the form

=

0,

< 0,

I/J

> 277.

1. Nonlinear Transformations without Feedback

32

Hence, it follows that m, =

1 -J 2n

f*(+) d#

0

1

=

2n 0

f [ a sin(wt

1

,2n

2Tr

0

+ $11 dJ, = - J

f ( a sin v) dv, (1.49)

+ $. Analogously, B,(T) = Mif[a sin + Y ) ] f [ asin(wt + + Y)}= M { F ( Y ) }

where cp

=

wt

(wt

WT

and, consequently, 1 B,(T) = 277

J

f[a sin(wt 0

+ +)If[.

sin(wt

+ + +)] dJ, WT

It is not difficult to see that these results can be found by taking the average with respect to time over a period of the function for a determined harmonic signal;

m,

-J 257 w

=

znlw

f ( a sin w t ) dt,

(1.51)

0

One can show, in general, that the following equation is true:

M { f [ asin(wt w =

2s

In particular,

J

+ Y), a sin(w(t +

znlw 0

j [ a sin w t ,

+ Y),..., a sin w ( t + ..., I ~ )

TI),

+ + Y')]} a sin w ( t + dt. (1.53)

a sin(w(t

7,)

Tn)]

We note that m,, B,,(O), and BZ,(7r/2w) are the coefficients for the Fourier series of the periodic function x ( t ) : x ( t ) = f(a sin w t ) = m,

+ B,,(O) sin w t + B,, (2:)- coswt + ... .

(1.55)

Let Z ( t ) be the sum of two statistically independent processes, a harmonic signal with a random phase and a normally distributed signal : Z ( t ) = a sin(wt

+ Y )+ [ ( t ) .

(1.56)

1. I . Nonlinear Lagless Transformations

33

We compute the first two moments of the output signal

and

where w l ( ( ) and w ~ ( ( 5,~) , have the same form as in (1.3) with the change of notation of z for 6. Here, just as above, the mean with respect to the harmonic signal is replaced by the mean with respect to Y,which is equivalent. We note that the expressions (1.57) and (1.58) can also be found by another method. For this, one must average with respect to time over one period the values of the first and second moment of the process X ( t ) . This is obtained from the transformation of the nonstationary normal process Z ( t ) which has an unbiased component Zo(t) coinciding with the unbiased component C0(t) of the process ( ( t ) , and with an expectation m, equal to m , a sin ut :

+

(1.59)

and

if Z O ( t )and ( O ( t ) are stationary. T h e computations of these characteristics for concrete types of nonlinear dependencies f ( Z ) are rather complicated, although in each concrete case it is possible to carry them through. A much more obvious and general technique is the transition of f ( Z )to its integral representation in a complex region (cf. Appendix 11) and then making use of characteristic functions.

34

1. Nonlinear Transformations without Feedback

We shall describe this procedure in detail. We replacef(z) by its integral representation according to (A.7), (Appendix 11):

where F+(ju) =

1 f+(z) *

e-l"z dz,

-a,

f+(4= fb),

z 2 z <

f-(z)=

z z

0, 0,

= f(s),

F-(jic)

a

= --m

f-(z) e-'"* dz,

0, 0, > 0, < 0.

The contours for integration C- and C + in the u plane are shown in Fig. 8.

FIGURE 8

We write conditionally that

( I .62) Substituting this expression into (I.%),

we obtain

+ j [ t , -t a sin(wt + + $ ) l r ~ z ~ z ~ z (,t f, z ) UT

is the characteristic function for the process Z ( t ) . From this expression it follows that &(u1 , 1 4 =

, uz)4,(u1 , U?),

~ZS(U1

+ j u z f r ] } is the characteristic function = M{exp[ju,a sin(wt + Y) + ju,a sin(wt + + Y)]}

where 0 2 ~ ( u,Iu,) = M{exp[ju,t, the process f ( t ) and O2*(u1, u,)

is the characteristic function for the process a sin(wt

+ Y).

WT

for

35

1.1. Nonlinear Lagless Transformations

The separation of the characteristic function for the process Z ( t ) into two factors which represent the characteristic functions of its components is a manifestation of a general theorem on the characteristic function of the sum of statistically independent terms (cf., for example, Pugachev [65]). The characteristic function Bzt(ul , u z ) for the normal process has the form (cf. Appendix 11)

4,(ul

, ti2)

=

exp[j(mtlul

- $(q,ul 2

2

+ m& +

2

o,e,u,

2

+ 2PE~:1~~2"1~2)1.

(1.64)

~ ~ a Taylor u ~ u ~series, ] we change this expression Expanding exp[ - ~ ~ u E ~ u in into the following form:

(1.65) where

Cn(ui)= u," exp{jmEtui -

t $,ui2}

(i

=

1, 2).

The characteristic function B2.(ul , u,) also can be expressed in terms of well-known functions. In fact, B,&l

, u2)

J2n

= -

an

0

exp{ja[ul sin(wt

+ 4) + u2 sin(wt +

+ +)I} ci$ (1.66)

The phase angle po introduced into the intervening transformations does not depend on the variable of integration v. Therefore, the characteristic function Bta(ul , u 2 ) can be expressed in terms of a Bessel function of zero order l o . In the theory of Bessel functions, the following expansion is well known:

e Z a ( ~, ,t i , )

=

--

+

/ , ( a d ~ , ~uZ2 r

+ 2u,uz cos

WT)

(1.67)

36

1. Nonlinear Transformations without Feedback

Substituting the series (1.65) and (1.67) into (1.63), we find (1.68) where

or

n=O

where

It is not hard to prove that the coefficient aln(22n coincides with the time average of the derivatives of the coefficients ulnazn,which were introduced in Eq. (1.24), provided that me1 mZ2= me2

m,,

=

+ a sin w t , + a sin w ( t +

7).

Let us examine the physical meaning of the expansion (1.68). It can be represented in the form

Bz

&

B(O) +

B(1)

+ B(2) + B(3) z + B(4) z +

where

B(6) 2

'

(1.69)

BP) = h1.00h2,00 = m,,ni, , B'*)= h 1JOhZ .lO%%PE , 2

(1.70)

The term B:' is equal to the derivative of the mean values of the output signal T . The term BFI determines that part of the correlation at time t and at time t function of the output signal which has the same form as the correlation

+

I . 1. Nonlinear Lagless Transformations

37

function of the normal component of the input signal, while the term B:) determines that part which is the same as the correlation function for the harmonic component. BisBgives the distortion in the form of a correlation function for the normal component, and B:" does the same for the harmonic component. B:' determines the effect of the mutual superposition of both components. For odd nonlinear functionsf(z) the first three terms have the most important effect for the correlation function of the output signal. We now give some examples of how to compute the coefficients h,.,, .

Example 7.

Let

f ( Z ) = 1 sgn Z.

Using (1.61) we find that

( I .71) We assume, for the sake of simplicity, that the process f ( t ) is stationary and that mt = 0. Then

hl,,,

h.,

hZ.,r.

'I j

li"+"'

277

c-

u"-l/,(au) exp( -

4 o;u2) du

) - 4 o+c2) + J c- U ~ - ~ ] , ( U Uexp(

dzr'.

'

\

The integrands do not have singularities when u = 0 and, hence, both contours C- and C+ can be drawn together to make a single contour along the real axis. Consequently,

Hence, it is obvious that

2b*+k-l

h,k

= ___

\

30

~ " - ' ] ~ ( a exp(u)

$ o t u 2 )du,

- 0

since

=

0,

Jk(

-au)

= (-

+k n + k n

even odd

(1.72)

I)k Jk(uu).

We make use of an expansion of the Bessel function of the kth order in a power series [cf. Jahnke and Emde [98], p. 2241:

(1.73) Then ( - l)'(U/2)k+Z' i=O

exp( -

$ u$u2) du, (1.74)

38

1, Nonlinear Transformations without Feedback

but

Consequently,

2/5 2 -

Ij"+k+l

7

hrak

a€

>

( - I)i(n

i=U

+ k + 2i - 2 ) ! ! ( a / 2 0 E ) k + 2 1 i!(k + i ) !

.

(1.76)

If a / 2 o ~ I (i.e., if there is a small random component), the resulting expression will converge poorly. In this case it is convenient to use another expansion. Hankel's well-known formula [14, 721 is as follows:

(1.77) where ,F,(a, 8, -x) is the confluent hypergeometric function which has an asymptotic representation (for large x) in the following form ([98], p. 373):

. ( a

-

B I- 1)

+ 4 a + I)(a -- P2 !+x Z l)(a: - B + 2 ) + "'I { , ( I .78)

where

r is the gamma h,k

=

function. Making use of this representation we find

Ijn+*-1 2+ r ( k --

x [I

Example 8.

If n then

+

1)

r[(n

(1.79)

+

The smoothing function is

+ k is even, hnP

=

0 [because of the symmetry off(Z)]. If n

I

m

hnk

+ k)/2]

2i"fk-' 7

1

0

Jk(au)u"-' exp { -

4 u2(og* + b2)}du.

+k

is odd,

(1.80)

1.1. Nonlinear Lagless Transformations It is not difficult to prove that this holds also if we replace and if 1 equals 1 .

39 UE

by

__-

.\/of2

+ be

Next we shall investigate the probability distribution of the output signal for a nonlinear transformation without lag. We shall derive from first principles the relations for a one-dimensional distribution. T h e value of the output signal X ( t ) = x at time t is determined only by the value of the input signal Z ( t ) = z at the same instant : X =f ( Z ) . Z is 9 random quantity with probability density wlz(zl). Thus, the problem is to find the probability distribution Wl,(xl) for a function of a random quantity when the given probability density of its argument is wlz(zl). By definition, where P{X < xl} denotes the probability that X < x1 and where the integral is taken over all intervals of the z1axis where the condition f ( Z ) < x1 is satisfied. If the inverse transformation is single valued,

z= V ( W , then by changing the variable of integration in the integral (1.81) we have ( I .82)

and, consequently, (1.83)

If the inverse transformation breaks up into several branches,

zj = V i ( X )

(2 =

I , 2, ...),

(1.84)

on each of which it is single valued, then the region of integration in (1.81) breaks u p into a series of regions each of which satisfies the condition Z < rpi(x). I n this case, the integral (1.81) reduces to the form (1.85)

1. Nonlinear Transformations without Feedback

40

Differentiation with respect to x1 gives for the probability density (1 3 6 )

Let us study two simple examples. Example 9. distribution

Let X

I n this case

=f

( 2 ) = Z 3 and let 2 obey the normal

z = v ( X ) = .i?/z.

T h e transformation is single valued so long as we restrict ourselves to real numbers. Therefore,

Example 10. Let X 7u&)

= a

sin(wt 1

+ Z ) , where

=

5,

0

< z < 2r,

=

0.

z


z >2r.

T h e inverse transformation obviously is not single valued. In the interval (0, 2 ~ there ) are two branches : X vl(X) = arcsin - - w t , a

v z ( X )= r

-

Substituting into Eq. (1.86), we obtain

because

X

arcsin a

+ wt.

1.1. Nonlinear Lagless Transformations

41

Let us assume an n-dimensional probability density for a random process at the input, w,&1

, z2 *

...9

where zi is the value of the process Z(t) at the moment of time ti (i = I , 2, ..., n). Then one can find the n-dimensional distribution for the process X ( t ) which is related to Z(t) by the equation (1.87)

while T h e distribution function is obviously given by a multiple integral of nth order,

the integration is over the whole region where the inequalities f(2,)

< Xl

(i

=

1, 2, ..., n)

(1.89)

are valid. When the inverse transformation

z= T(X)

(1.90)

is single valued, the change of variables 2 3

=

dXt)

(1.91)

reduces the integral (1.88) to the form Wnz-(xl

7

~2

3

...>xn)

where (1.92)

is the Jacobian of the transformation.

1. Nonlinear Transformations without Feedback

42

T he differentiation of W,(x, , x 2 , ..., x,) sequentially with respect to each of its variables enables us to find the n-dimensional probability density of the output process : %(XI

!

x2

I

'..9

XTI)

= Iz~nz[P)(X1)r dxz),

'..)(P(xn>l.

(1.93)

Many typical nonlinearities lead to a transformation ( I .90)which is not single valued and which does not break down into a finite number of singlevalued branches because the graph of f ( Z ) contains portions which are parallel to the Z axis. Let f ( Z ) have the form shown in Fig. 9. In the range Z < zo the trans-

,-FIGURE 9

formation (1.90) is single valued and, hence, in the region X i< xo the probability density is given by the expression (1.93). In the regions where only one of the conditions xo < x i < co (i = I , 2, ..., n) is valid, it is clear that ZL'"(XI

, xz , ..., x,) = 0.

On the boundaries of the region the probability density is given by delta functions with coefficients determined by the probability of the output for the given boundary. Hence, one can write that W",(XI

, xz ,

...I

x,)

= J z ~ n a [ d x l ) l d x z )...) , dxn)lv

< xi <

--CO

1

( I .94) J

10

J TO

where

and where W ~ , ( Z ,, ~z S z ,..., z , k ) is the expression for the n-dimensional probability density of the input signal wnz(z1, zz, ..., z"), in which all the

1.2. Nonlinear Transformations with Lag

43

z i except zS1, za2, ..., msSkare replaced by their values in (1.91). The generalization for the case when there is also a lower boundary is not difficult. On the one-dimensional case the expression for w. has a very simple form:

where

XH

= f ( z H ) , XB = ~ ( z B ) , and

where

XH

< f ( z )< X B .

As a summary of this section, one can say that given the characteristic of the input signal it is always possible, in principle, to find some characteristic of the signal which has gone through to a lagless nonlinear transformation. I n most cases the computational difficulties can be overcome by using the standard techniques described above. However, it is often necessary to make a rough estimate of the fundamental characteristics of the output signal (the expectation and correlation of the function). I n that case, it is convenient to use the method of statistical linearization (cf. Section 1.5) and also the graphs of Appendices I1 and IV, which make it possible to find these characteristics for typical distributions of input signals for several typical nonlinearities. 1.2. Nonlinear Transformations with Lag

A nonlinear transformation with lag (without feedback) results from the successive application of a nonlinear transformation without lag and a linear transformation. T h e expectation of such a transformation in the simplest case is given by Eq. (1.15) or (1.16), written in the following form : (1.96)

or (1.97)

44

1. Nonlinear Transformations without Feedback

if it is assumed that the transformation is physically realizable and has constant parameters. T h e moments of the first and second order can be determined by substituting (1.96) or (1.97) into the general relations

(1.98)

(1.99)

(1.100)

Therefore, the computation of the moments of the output signal X ( t ) for the transformation as a whole includes the computation of the moments of the signal Y ( t ) at the output of the nonlinear transformation without lag and the insertion of these characteristics into the equations for the linear transformation. T h e latter equations are more easily understood if one passes from the moments of second order to the spectral densities by using the Fourier transform, S(W) = --m

eciwr B(T)d7,

1.2. Nonlinear Transformations with Lag

45

and from the impulse functions to the frequency characteristics,

In fact, from (1.100) and (1.101) it follows that but S,(w) =

m,

j

Cc

dT1 ejwrl h(7,) 0

=

( I . 102)

,y@(O),

"

j dT2

e-jwr2h(T2)

0

or S,(w) =

I @(h> l2 S,(w).

( 1.103)

T h e simple algebraic relations (1.102) and (1.103) are more convenient than the integrals (1.100) and (1.101). Moreover, the spectral density of the output signal is frequently of greater practical interest. Thus, the calculation for the transformation can be based on the moment characteristic B J T ) [or RY(7)], which one can find by the methods described in the previous section, and by introducing the spectral density expression

into Eq. (1.103). T h e mean-square value of the output signal is given by M{xz} = B,(O)

=

1 -J 257 1

=257

" -w

1 "

-w

S,(w)dw S,(w) I @ ( j w ) 1% dw.

(1.104)

For fractional rational functions S,(W)the integral (1.104) is tabulated (cf. Appendix 111); thus, the mean-square value M{x2}or the meansquare deviation (the variance) uz2= M{x2}- mz2 can be expressed quickly in terms of the parameters of the spectral density S,(W) and of the frequency characteristic @ ( j w ) .

1. Nonlinear Transformations without Feedback

46

Let us illustrate this computational technique by a very simple example. Example. Let f ( Z ) represent the characteristic of an ideal relay, i.e., f ( 2 ) = 1 sgn 2, and let 1 @(jw) = Tjw 1 '

+

T h e correlation function for the input signal has the form RAT) = u,2 exp(--e

I 7 I),

and mz = 0. T h e distribution of Z ( t ) is normal. We write the correlation function of the signal R,(T) in the form of a series

where 21"

a,2 = T

=

(2k - l)! (2k)!! (2k 1)'

+

0,

n=2k+1, n = 2k.

T h e corresponding spectral density is given by T

S,(w) =

&PSny(W),

( 1.105)

n=l

where

1

z

s,,,(w) =

e-iwTpzn(T) dT.

--m

In this case the computation of S,,(w) is very simple :

When p , ( ~ )is given by a more complicated expression, it is often convenient to use the recursion relation S,,(w)

=-

2.rr

1

m -a

S1(w - x) S,-,h) d x ,

which follows from the usual convolution formula.

(1.106)

1.2. Nonlinear Transformations with Lag

47

T h e spectral density of the output signal is given by Eq. (1.103). Substituting the expression for O ( j w ) into it, we finally obtain

1

4129

-~

n

T2w2

+1

g

(2k -- l)!! (2k)!!

1

w2

+ (2k + 1)W .

(1.107)

From Fig. 10 it is clear that the most significant effect of the signal

10

0.5

0 FIGURE 10

Y ( t ) on the spectral density SY(w) at the output of the lagless, nonlinear transformation comes from the first term, which is similar in form to S,(w). T h e frequency distortion becomes significant only for high frequencies. Therefore, if the linear portion has only lower frequencies, the error resulting from the use of the approximate formula aB

S,, = 1 cP( j w ) 12 -1;- S,(W)

( 1.108)

uzL

will be unimportant (see Fig. 11). T h e error in uz2 is of the order of 3 % when BT

=

1.

48

1. Nonlinear Transformations without Feedback

If f ( Z ) is not an odd function or if the mean value of the input signal is different from zero, it may be necessary to take the higher terms into account in an expansion of the type (1.33) for the moment

t

0

1.0

20

3.0

FIGURE 11

of second order or for the spectral density which corresponds to it. However, this calculation is not very difficult if one makes use of the dependencies a,(m,, u,) which are described in Appendix I for typical nonlinearities. We shall not give an example of the computation of S,(w) for the transformation described above when the mean component is different from zero. Let m, = oz. T h e n from the graphs in Appendix I we find that a, = 0.68, U, = 0.48, ~2

=

-0.34,

= 0.

Consequently, the mean value m y = a,, density of the central component is

=

0.68 and the spectral

From Fig. 12 it is clear that in this case it is perhaps not permissible

1.2. Nonlinear Transformations with Lag

49

to neglect the second term. Even after filtering (for BT error in the variance ax2is greater than 25% (Fig. 13).

=

l), the

f 05

0

1.0

2.0

3.0

4.0

~8

0

FIGURE12

1.0

2.0

3

FIGURE 13

Let us proceed to the analysis of a more complicated nonlinear transformation such as is given by Eq. (1.18). T h e input signal at first undergoes a linear transformation with lag, then, a nonlinear lagless transformation, and, again, a linear transformation with lag:

x(t)= y

i h , ( ~ )Y ( t OLi

- T ) dT,

=An

u(t)=

(1.109)

0

5

0

h , ( ~ z(t ) - T ) dT.

(1.110) (1.1 11)

We shall determine the two first moments of the output signal X ( t ) . I t follows from (1.98) and (1.99) that for this one must know the same moments for the signal Y ( t ) . However, to find the latter moments it is not sufficient to know only the first two moments of the signal U ( t ) .I n addition, one must also know the two-dimensional probability distribution. Therefore, one must first find the probability distribution for the signal at the output of the linear transformation with lag defined by Eq. (1.1 1 1). Generally speaking, this problem is very complicated. Only in the following two special, but very important, cases does it have an elementary solution.

50

1. Nonlinear Transformations without Feedback

( 1 ) The signal Z ( t ) is normally distributed. Then, one can show that an arbitrary linear transformation of Z ( t ) will result in a normally distributed signal (cf., for example, Pugachev [65], p. 415). A normal distribution is completely determined if the first two moments are given. Th en the problem for the normal input signal reduces to the simple problem of finding the moments. ( 2 ) The signal Z ( t ) is a harmonic signal with a random phase. I n this case the input signal U ( t ) will also be harmonic, that is, the form of the distribution is preserved, and the variation is changed only by a numerical parameter-the amplitude, au2

=

1

aiL.

(1.1 12)

T h e problem is more complicated when Z ( t ) is an arbitrary definite function of time t and when the random parameter rp (or several random parameters) has some well-known probability distribution. If it is possible to construct a definite solution U ( t , p’) for fixed, but arbitrary, values of the parameter p’, then the computation of the probability distribution for the signal U ( t , p’) reduces to computing the distribution for the function of the random parameter p’, which is a problem similar to that discussed in Section 1.1 for a harmonic signal with a random phase.* We shall now investigate a property of linear transformations which considerably simplifies many problems in the theory of automatic systems. Let us look at the basic formula for a linear transformation:

x(2)=

m

h ( ~z(t ) - T ) dT

0

* This method can be used in principle for solving a great variety of problems because an arbitrary random function Z(t) defined over a finite interval can be represented in the form of a series (a canonical expansion), (1.113) where V , is the mutually noncorrelated random variable, and where z,(t) is a fixed function of time. Computational techniques based on the use of canonical expansions and the theory for constructing these expansions are exhaustively described by Pugachev [65] and are therefore not developed here.

1.2. Nonlinear Transformations with Lag

51

We divide the interval of integration into subintervals, (0,

(71

9

~ n ) , ...>( T n

1

Tn+l),

...,

and then use the theorem of the mean.* The n we have m

oc

(1.114)

where dn

=

7*+1

h ( ~d )T ,

1,

=

dnZ(t - 7 , * ) ,

(1.115)

0.

(1.116)

Tn

and where 7,

<

Tn*

< Tn+l,

70

Let the subintervals of integration be chosen so that the random variables Z(t - T , * ) , and, consequently, the quantities 5, are statistically.independent. Thus, X ( t ) is represented as the sum of an infinite number of statistically independent random variables. From (1.1 16), in particular, one can deduce that if a process Z ( t ) is normally distributed, then X ( t ) is also normal because a sum of normally distributed random variables is normal. Moreover, the central limit theorem asserts that the distribution of the sum of a large number of random variables will approximate a normal distribution even if the distribution of each separate variable is not normal. T h e conditions for which these statements are true, roughly speaking, reduce to the requirement that all the summed quantities be approximately of equal proportions, that is, the variance of any one of the terms must not be significantly larger than the variance of the others and, in any case, it must be much less than the variance of the sum. These conditions, evidently, are satisfied if one can choose a sufficiently large number N of statistically independent subintervals A T , = T , + ~ - T , such that during the time from 0 to T , the value of the function does not diminish too much.

* It

is assumed that, within the subintervals, h ( t ) does not change signs.

52

1. Nonlinear Transformations without Feedback

Assuming as a first approximation that the concepts of statistical independence and absence of correlation are equivalent, one can state the above requirement as follows :

where /3 is the largest real part in absolute value of the poles @ ( p ) determining the attenuation h ( ~ )and , Tk is such that p , ( t ) M 0 for t > Tk . Since I /3 I determines the width of the passband of the system and l/Tk is the width of the spectrum of 2, that is, the region where S,(w) is significantly different from zero, the physical meaning of the condition (1.117) is as follows : In order to have a normalized output signal (that is, in order to get a normal distribution), the passband of the frequency characteristic I @(jw) I must be much smaller than the real frequency band for the spectral density S,(w) of the input signal. This statement has a purely qualitative character, but it is widely used (and will be used in this exposition) in computations for nonlinear automatic systems. However, it is worth noting that the statement about the “normalization” of the sum of a large number N of independent random variables

n=O

cannot always be interpreted in the strict sense of the central limit theorem, that is, in the sense that when N + 00 the distribution of X N comes arbitrarily close to a normal distribution. I n fact, the latter -is true only if several additional conditions are satisfied (the so-called conditions of Lindeberg), which, generally speaking, are not satisfied by processes defined by linear transformations without lag. Let us investigate this question in more detail. We introduce the notation

2 5,

Ai

N

SN

=

7l=O

9

mN

=

M { S N )=

mp n=iI

?

(1.1 18)

1.2. Nonlinear Transformations with Lag

53

Then Lindeberg's conditions ([90], p. 207) can be written in the form (1.119)

where

I tn I 5,

U n = C, , =

0,

~

-

mg. I mCn I

< ESN, >

ESN,

-

and where E is an arbitrary number greater than zero. T h e sign designates that sN + 30 as N + CO, and that the quotient of the two quantities linked by this sign approaches unity. A sufficient condition, obviously, would be the requirement of uniform boundedness for

15,l

( I . 120)

< A

as sN + CO. Usually h ( t ) can be represented in the form I

(1.121) k=O

where f l k are the roots of the characteristic equation for the system poles @(p); for a stable system Re f l k < 0. We shall now compute the coefficients d . in Eq. (1.1 14) assuming that

I < 1

, B&,

(AT,

=

T

-+

~

7" ~) .

Then,

=

h

(1.122)

AT,,.

( T ~ )

This expression could have been written immediately because of the slow variation of h ( ~ ) . We also suppose that the process Z ( t ) is stationary so that

D{Z(t -

T,*)}

= uZ2,

T h e computation of sN2 gives N

sN2 =

A r m 2h2(r,).

uz2 "=O

Obviously, the condition for the convergence to a normal distribution sN will not be satisfied, since for a stable transformation lim s N 2 m

N+x

1 r

uz2 AT

0

h2(r)dr

=

const

(AT = AT, = const). (1.123)

1. Nonlinear Transformations without Feedback

54

Hence, the effect of the normalization of a random signal in the operation of a narrow-band linear filter can be understood only in the sense that the distribution of the output signal will come closer to a normal distribution than the distribution of the input signal. The accuracy of the approximation to the normal distribution can be estimated by different methods. One of these is to estimate the higher moments of the output signal. It is well known ([37], p. 248), that an arbitrary one-dimensional probability density can be represented in the form of an orthogonal expansion, of which the first term coincides with the normal distribution and the remaining terms characterize its difference from this distribution *: (1.124) where $(z) = ( l / d % u z ) exp[-(z - m,)a/20,2], and where Ha(<) are the Chebyshev-Hermitian polynomials. The coefficients cn are not difficult to find if one makes use of the orthogonal property of the polynomials (1.22): (1 . I 25) Recalling the values of Hn for n = 0, 1, 2, 3 and 4 from Section 1.1, we obtain co = 1, Ca

where

=

1

c1 =

~3 , 3!

0,

CP

=

cz =

1

1

5 (Pa -

6112

-

I),

+ 3),

(1.126)

(1.127) is the normalized central moment of nth order. We choose the parameter u, so that pa = 1. Then cz = 0 and the expansion (1.124) is written in the form

(1.128) The coefficients ps and y = pa - 3 essentially determine the degree to which ml(z) differs from the normal probability density $(z) and are called the coefficients of asymmetry I p s ) and of excess ( y ) . The first really characterizes

* The

Gram-Charlier expansion (1.124) will converge at least in those cases when

x is bounded. Here, only the asymptotic properties of the series are of interest [37].

1.2. Nonlinear Transformations with Lag

55

the degree of nonsymmetry of the curve wl(z) [for a symmetrical curve all moments pn for odd n are zero]. The second, the coefficient of excess, characterizes the degree of smoothness of the curve wl(z). If the curve w,(z) has a higher or sharper apex than the normal probability density, then y > 0. We shall suppose that w,(z) is completely described by the first three terms of the series ( I . 128) and, consequently, is determined by four parameters: m, , uz , p 3 , y . Then the determination of a one-dimensional probability density for a signal at the output of a linear transformation reduces to the problem of finding these parameters. We introduce the concept of an n-dimensional correlation function %(Ti

, 7 2 , ..., 7%) = M { Z ( t ) Z ( t f

Here, we are assuming that m,

=

u,"p, =

Ti)

... z(t + Tn)}.

(1.129)

0. Then it follows from (1.127) that R,-,(O, 0,..., 0).

( I . 130)

Likewise, we introduce into our study the n-dimensional spectral density which is defined by a Fourier transform of nth degree of R, : Sn(w1

-

9

wz

, ..., ~

n

)

jm'.' jmR,,(T~, ..., -m

7 ),

-m

Obviously, the inverse relation RAT1

exp[-j(oJITl

+ ... -+

w,.,)]

dT,

... dTn . (1.131)

, ..., .,J

is also valid and, consequently,

The expression for the n-dimensional spectral density of the signal X ( t ) at the output of the linear transformation has the following form [42]:

When n = 1 in (l.l34), it is obvious that we obtain again Eq. (1.103). Therefore, we can find the variables pzn at the output of the linear transformation if the n-dimensional spectral density of the input signal is given (or if the n-dimensional correlation function is given). However, if the linear transformation is followed by a nonlinear transformation, as in (1.109)-(1.1 I I), then, after computing the moments pnm, it is not possible, as a rule, to find the correlation function of the signal Y ( t ) at the

56

1. Nonlinear Transformations without Feedback

output of the nonlinear transformation f(u), because for this we have to know the two-dimensional probability density w2(u, , u2). The direct computation of w2(u1, u2), even if one uses an expansion of the type (1.124), is extremely difficult (cf., for example, Tikhonov [87]). Therefore, one usually tries to use some approximative device. If the coefficients of asymmetry and excess, p3 and 7 , are small and, consequently, if the onedimensional distribution is close to normal, then one may assume that the two-dimensional distribution is also nearly Gaussian. If this is not true, then it is expedient to use an approximative technique for computing the correlation function, a technique which is based on the idea of statistical linearization (cf. Sections 1.5, 2.2 and 3.3) and for which it is not necessary to know the two-dimensional probability density.

1.3. The Problem of Synthesis. Optimal Conditions for Various Classes of Transformations

T h e problem of synthesis for transformations which gives the best approximation was formulated in the Introduction. We shall now describe a method which is based on the criterion of the minimum mean-square deviation (the minimum error). Assume that statistical characteristics determined by the random processes X and 2 are related to each other. We shall look for a transformation F such that u,2 =

M{[F{Z}- XI2} = min

(1.1 35)

among all possible transformations F which belong to the given class. I n the future, we shall limit our study only to those cases where X and 2 are stationary and are stationarily related. We shall denote by a line over the expression which we are averaging the operation of finding the expectation. Let the operator F { Z } be optimal in the given class, that is, let it satisfy (1.135). A variation of the mean-square deviation Su,2 which causes a small variation EG{Z}of the operator F { Z } is given by the equation 80;

==

2
+ E ~ ( G { Z,} ) ~

where G is an arbitrary operator which belongs to the same class as F and where E is some small number.

1.3. The Problem of Synthesis

57

I n order that F be optimal we must have 2G{Z)[ F { Z }- XI

= 0.

Here, as usual, we are neglecting the small higher-order terms. Thus, a necessary condition* for the operator F to be optimal is that

G{Z}[F{Z} - XI

= 0.

(1.136)

From (1.136) one can deduce several general properties which the optimal transformation must satisfy. Since G { Z } is arbitrary, (1.136) will be satisfied if we put G { Z } = F { Z } . T h e n we obtain --

~

(F{Z})Z = XF{Z}.

(1.137)

If a transformation of the form F { Z } = K where K is an arbitrary constant belongs to the given class, then by setting G{Z} = 1 in (1.136) we obtain -

-

F{Z} = X

=

m,

.

( I .I 38)

This means that the optimal transformation can be carried out with the exact reproduction of the expectations (the mean values). For the sake of simplicity, we write Y = F{Z}; then :u

By (1.137), Y X

= =

( Y - X)2

=

F - 2yx + xz.

Y 2 and, consequently, 0,2

-

= x2

_

- y2

but from (1.138) we have m, = my. Finally, we obtain the following relation which gives an equation between the mean-square error and the mean-square deviations of the input and output signals for the optimal transformation : u,2 = ux2 - uy2. (1.139) * One can prove its sufficiency (cf.. for example, the analogous calculations given by Laning and Batten [51]).

1. Nonlinear Transformations without Feedback

58

We now examine what form the optimal condition takes for various classes of transformations. 1. Linear Transformation (with Constant Parameters). I n order to make sure of the exact reproduction of the mean values, we use the nonhomogeneous transformation of the form F{Z}= h

+

ffi

0

h(7) z(t - T ) d7.

Since G(2) must belong to the same class, its most general form has the property G{Z}= f

+ j g(7) z(t m

T)

d7,

0

where g and g(7) are arbitrary (we require only that h(7) and g(7) satisfy the condition of physical realizability, i.e., that h(7) = g(7) = 0 when 7 < 0). Substituting the expressions for F ( 2 ) and G(2)into the optimal condition (1.136) gives f

[E + 2

h(7) d7 - 8 1 IJ

Since g and g(7) are arbitrary, the resulting condition can be satisfied only if

SQ d7247,) B z ( ~ l T ~ +) hZ= B,,(T,) -

0

>, 0).

( T ~

(1.140)

We note that, if the first of these conditions is satisfied, we have the equation

Bzz(71)-

a=

+ (2)’ 1 h(7) d7. 71

R+z(71)

But then the second condition can be rewritten in the following way

1.3. The Problem of Synthesis since

59

+

B z ( ~=) R z ( ~ ) (2)'.

We write the resulting expression for the optimal transformation in the form F {Z } = h =

+2/

8+

I

aD

ai

h ( ~dz )

0

+/

a 0

h ( ~ ) [ Z (t T ) - z]

dT

h ( ~Zo(t ) - T ) dT.

0

Thus, the optimal linear transformation is found to be a transformation which carries out the reproduction of the mean components; the impulse function of the transformation for the unbiased component is given by the integral equation (1.141), which is an equation of the Wiener-Hopf type. T h e methods for solving Wiener-Hopf equations are now well developed. They are discussed in detail by Laning and Battin [51], Morosanov [54], Pugachev [65], and Solodovnikov [80]. Hence, it is not necessary to outline them here. Finally, we shall look at the simplest case-the optimal approximation by a linear transformation with lag : F{Z}= h

+ h,Z

=

h + h,Z

+ h,ZO.

From the fundamental optimal condition ( 1.136) we obtain

h,

+ h,Z

-

=

8,

h,

xzo 02

=-

(1.142)

T h e coefficient h, is usually called the transmission coefficient with respect to the random (unbiased) component. T h e transformation of the mean components is defined by the coefficient (1.143)

Thus, the optimal approximation by a lagless linear transformation is given by the equation Y

=

home

+ h,Zo.

(1.144)

60

1. Nonlinear Transformations without Feedback

Of course, the transmission coefficient h, needs to be introduced only when for m, = 0 we have m, = 0. Otherwise it is easier to write directly Y = m2: h,Zo. (1.145)

+

Sometimes it is convenient to use an approximation which is still simpler, the homogeneous transformation F { Z } = h,Z.

(1.146)

Here, one does not get an exact reproduction of the mean components. T h e transmission coefficient h, is given by the following equation, which is easily deduced from the general condition (1.136) : (1.147)

I t is not hard to see that the coefficient h, is related to the coefficients h, and h, in the optimal nonhomogeneous transformation by the equation (1.148)

In other words, it is the weighted mean between them. I n conclusion, we note that to find the optimal values of the parameters in the linear lagless approximation it is not necessary to solve the general variational problem using the condition (1.136). I t is sufficient to require that the quantity u,2 be a minimum, not as a functional, but simply as a function of the unknown parameters h i . Minimization is guaranteed by the condition

&,2

- = 0.

(1.149)

ahi

T h e equivalence of Eq. (1.149) to the general equation can be checked by constructing the optimal approximation by the transformation (1.146). I n fact, (h,Z - .Y)' = h , 2 2 - 2 h , E - au 2 2 = 2(hcZ2- X Z ) , ah, ue2=

from which we obtain (1.147).

+ XZ,

1.3. The Problem of Synthesis

61

I n the general case we

2. T h e Nonlinear Lagless Transformation.

have

(1.150)

F{Z} =f(f(z),

where f is a nonlinear function. We shall look for an optimal transformation which satisfies condition (1.135) for a somewhat smaller class of nonlinear transformations, those which can be described by the equation (1.151) where k,, are constants and where O,(z) are orthonormal polynomials in the variable z with weight w l ( z ) and of degree n. Thus, for e(z) the following relations are valid :

We note that the first two terms of the system of orthogonal polynomials for an arbitrary probability density w1 can be found by using the elementary relations

-CX

wl(z) dz

=

.W

1,

J

--7)

zwl(z)dz = m, ,

--?c

(1.153)

( z - rn,)%,(z) dz =

02,

or

\

X

-x

1 . 1 . w,(z)dz

=

1,

1

X

1 . ( z - m,)w,(z)dz

= 0,

-'L

It follows from this that the first two orthogonal polynomials are =

z

-

m,

~

0,

(1.154)

62

1. Nonlinear Transformations without Feedback

T h e transformation G { Z} must also belong to this class (1.151), so that we have (1.I 55)

Condition (1.136) now takes the form

m=O

'n=U

From the orthogonality of the polynomials e,(z), we obtain

Since this condition must be satisfied for arbitrary k,, , the desired constants k,, are given by the following equations :

and, hence, from (1.154),

k,,

=

B,(Z)X

=

m,

.

(1.157)

Therefore, the optimal transformation without lag is given by the equation

2 u,D,%(Z) + m, , R

Y

= F{Z} =

(1.158)

n=l

where (1.159)

We shall now compute cr?. of 0, we find a;

=

( Y - m,)2

From (1.138) and the orthogonality =

( Y - mE)2 =

2 R

n=l

aX2~,,2.

(1.160)

1.3. The Problem of Synthesis

63

T h e error in the optimal nonlinear transformation is found by substituting (1.160) into the general equation (1.139) : (1.161)

T h e quantities D, can be found by using experimental data from a model which simulates the characteristics of the polynomials e,(z) and the signals X ( t ) and Z( t) , or by analytical means if the statistical characteristics of X ( t ) and Z ( t ) and the relations between them are given. If all the D, = 0 for n # 1, then the optimal transformation is linear. This is true, in particular, when X and Z are normally distributed and when the joint probability density is given by an equation of the form (1.3). We shall write w2(x,z ) in the form of a series in the orthogonal ChebyshevHermitian polynomials: w2(x, z) =

1

(1.162)

where

We also recall the fact that the system of polynomials H, , when normalized, is a system which satisfies the conditions (1.152) for the one-dimensional probability density

We now compute the coefficients

D,

1-

1

= - e,(Z)X = - e,(Z)(X - mz). 0%

0,

Taking ( I . 162) into account we can rewrite the expression for D. in the form

1. Nonlinear Transformations without Feedback

64 where

b, = But

5

=

5

ex~(5~/2)H,(5) d5.

HI(().Therefore, b,

=

0,

m # 1,

=

1,

m

that is,

=

1,

+

m,) At the same time, because the normally distributed polynomial O,(o,( is proportional to H,(5), al,, = 0 when n # 1, which also proves the proposition that the linear transformation is optimal in the class of lagless transformations if the input and output signals are normal.

3. The Nonlinear Transformation with Lag but without Feedback (the Transformation Which is Reducible t o a Linear One). Again, we shall

narrow the problem down somewhat by looking for a transformation which can be represented in the form

F{Z}=

2J R

.-m

m=O

0

htm(7)Bm[Z(t- T ) ] dT.

( I .163)

With this limitation G{Z}must also be representable in the form

0 and identically equal where h , , ( ~ )is an arbitrary function for T to zero for T < 0. Substitution into the optimal condition (1.136) in this case gives

1.3. The Problem of Synthesis

65

Since h,, is arbitrary, the following equation is valid :

2 j,“

m=o

h f , n ( 7 , ) ~ ~ m ) ( T l - 72)

dT2

=

c : ) ( ~ ~ (n ) = 0,1, ..., R ) ,

(1.165)

which is a generalization of conditions (1.140) and (1.156). We shall use the fact that Crrn)(7)= 0 ,

m # 0,

= 1,

m = 0.

CY’(7)= 0 , =

1,

12

# 0,

n

=

0.

From Eqs. (1.166) and (1.167) it follows that

and, hence, one can find the final expression for the mean-square error

which is a generalization of (1.161).

1. Nonlinear Transformations without Feedback

66

Thus, in order to solve the problem of synthesis for nonlinear transformations with lag, it is sufficient to calculate (or to determine and DZz(7)and to solve a system experimentally) the functions C~"")(T) of integral equations (1.167) of the Wiener-Hopf type; for example, by the method of undetermined coefficients [110]. = 0 when n # 1 and m # 1, and if Again, we note that if Cinm) DLt) E 0 when n # 1, then the optimal system will be linear, and the system (1.167) will reduce to (1.141). I t is not hard to show that this will be true, in particular, when the processes X and 2 are normal. T h e method of synthesis described above for nonlinear transformations which uses orthogonal polynomials leads to some very refined results, especially in the problem of constructing a transformation without lag, when it is possible to look for a decomposition of the system of equations in order to find the unknown coefficients of the optimal transformation. However, in applying this method one must also construct the system of orthogonal polynomials O n . For this purpose, one can, for example, use the following recursion relation [110] : (1.169)

Obviously, the procedure for constructing the 8, is rather complicated. Therefore, if we seek an optimal nonlinear transformation with lag in a case where the application of orthogonal polynomials proves fruitless,* then it may be expedient to apply the following method : T h e optimal transformation may be sought directly in the following form : (1.170)

Using the optimal condition (1.136) leads to the following system of Wiener-Hopf equations for determining the unknown impulse functions h,,, :

* There

is a simplification only when one is calculating the optimal error.

1.3. The Problem of Synthesis

67

where Bim"'(T, - T 2 ) B::'(T,)

= za(t- Tl)Zm(t- T..), =

B'"'(T,) = z"(t- T,)x(t).

(1.172)

I n any case, the solution of this system will be no more complicated than the solution of (1.165). Approximation by the criterion of the minimum mean-square error is not the only method. There are several well-known studies [3, 4, 651 on approximation in the sense of minimizing various probability functionals for the error. I n the theory of linear approximation, special significance is given to that criterion which determines the transformation F , which is that Y = F { Z } is to approximate X,by using the condition that the first two moments be equal : (1.173)

(1.174)

T h e first of these conditions was satisfied also in the synthesis using the criterion of the minimum mean-square error. T h e second condition leads to another determination for the unknown impulse function h(7) of the optimal transformation F. Replacing it by the requirement that the spectral densities be equal, (1.175) (1.176)

(1.177)

which expresses the amplitude-frequency characteristic of the desired optimal transformation in terms of the spectral densities S,(w) and S,(w) of the processes Z and X .

68

1. Nonlinear Transformations without Feedback

We expand the right-hand side of the expression (1.177) into factors :

Then, from the requirement that the system be physically realizable, we see that the zeroes and poles for the frequency characteristic must be located in the lower half of the complex plane in the variable w (cf. [19,51], etc.);we can, therefore,find an expression for the frequency characteristic of the optimal linear transformation :

(1.178) T h e parameters of the optimal, lagless transformation

Y

=F{Z} =

hornz

+ hlzo

can easily be shown to be determined by

(1.179)

x2.

if we require that 7 = 8, p% = An attempt to solve in an analogous manner the synthesis problem for nonlinear transformations of the type (1.142), that is, by starting with the requirement that the moments of order 1, 2, ..., R be equal, obviously will not lead to reasonable results. 1.4. The Application of Methods of Synthesis. Nonlinear Filters

T h e general synthesis method described in Section 1.3 can be applied in solving several classes of problems, of which the following are the most important : (1) Filtering, or, in the more general case, the optimal transformation of a signal which is mixed with interference; (2) Parallel compensation for a given system;

1.4. The Application of Methods of Synthesis

69

(3) T h e statistical determination of the characteristics of the transformation which the system realizes ; (4) Statistical linearization of nonlinear transformations. T h e most extensively studied problem is the one concerning optimal filtering. T h e filtering problem is the problem of separating two waveforms S ( t ) and N ( t ) , where S ( t ) is considered a distinguishable signal and where N ( t ) is considered to be interference, the effect of which on the output signal Y(t) must be reduced to a minimum. Often, the “pure” filter problem is associated with the problem of transforming the signal. Hence, in the general case, one can assume that the signal which is to be transformed is some definite function of the inputs S(t) and N(t) : Z ( t ) = UW),N(t)l, (1.180) and the signal X ( t ) , which must be optimally approximated by the signal which is to be transformed by the filter Y

=F{Z},

(1.181)

is related to the “useful” component of the input signal by a given equation, X ( t ) = H{S(t)}. (1.182) T h e cases of practical importance are as follows : (a) the signal and the noise are related additively, Z ( t ) = S(t)

+ N(t);

(1.183)

(b) the signal and the noise are related multiplicatively,

Z(t) = S(t)N(t).

(1.184)

We study the main method of constructing nonlinear filters when the signal S ( t )and the noise N ( t )are related additively and are statistically independent. By assuming that X ( t ) = S(t) we may consider the “pure” filter problem to be solved. We shall now give equations for the generalized moments BTm) and B g ) .

70

1. Nonlinear Transformations without Feedback

We use the following notation :

=

22

C,,T m S,iS,' N;-iN;-k

(1.185)

i=O k=O

and B g ) = z,"x,= ( S ,

+ N,)"S,

=

2

-__

C:S,iS, N;-l. (1.186)

i=O

T h e solution can be derived further by using Eq. (1.171), where coefficients represent directly the moments BLnm) and Bit) or using Eq. (1.165) and performing the preliminary construction of system of orthogonal polynomials e,(Z) for which one must also the moments Bknm). We shall next use both of these schemes.

the by the use

Example 1. T h e problem is to find the filter for a harmonic signal with a random phase in the presence of a stationary normal noise. T h e amplitude A of the signal and the variance of the interference unz(mn = 0) are given. Neither the frequency of the signal nor the spectral composition of the noise are known. Thus, filtering by linear stages with lag based on the use of spectral properties will be ineffective. We proceed to find the optimal nonlinear filter, limiting ourselves at the same time, for the sake of simplicity, to the class of polynomials of third order : Y

= f ( z )= a,z

+

a 3 ~ 3 .

(1.1 87)

T h e terms of even degree cancel because the distribution of the input signal is symmetric. T h e coefficients a , and a3 for the optimal filter

1.4. The Application of Methods of Synthesis

71

can be determined from the condition for the minimum mean-square error, which gives the equations

+ a,Z = Z X , a1z4 + a3ZB = z3x, a13

(1.188)

+

where Z = S N and X = S. We compute the moments z"and ZnX, which are the coefficients in these equations. Assuming that the signal and the noise are independent, we obtain

( 1 . 1 89)

Considering that N2m

NPmexp

= __

p m =-

sinem(wt

2r

0

(- -)N

Z

2Un2

dN

= (2m -

l)!!uim,

- l)!! + q ~ )dqI = A2m (2m2"m! '

(1.190) (1.191)

we find, finally,

+ 3A2un2+ 3un4,

3A4 8 5A6 2 6 z16 -

2 4 =

-

45 45 +A4un2+ 7A2un4+ 8

-

z x = -A2-2 ,

__ 3 Z3X = - A4 8

+ 23- A2un2.

1 5 ~ ~ 6 ,

(1.192)

72

1. Nonlinear Transformations without Feedback

It is obvious that the quantites a, and a3A2depend only on the ratios T h e calculations of these quantities for various values of ( U , / A ) ~ are shown in the accompanying table.

I

0.1

0.01

0

0.328

1.08

1.13

1

x

-0.114

-0.163

0

4

a1

a3A2

(%)2

(y)'

0

0.1 19 -6.30

0

x lo-*

-1.18

0.500

0.444

0.284

0.061

0.009

0

0.500

0.444

0.333

0.083

0.010

0

T h e table shows the values of the relative error for the filter (U,/A)~ as computed from the equation

and, also, the errors for the optimal, linear, lagless transformation Y

=

(1.194)

a,Z

In this case, a, =

1

1

+ ')(U,/A)B'

o ,:

A'L

-

1

+ 2(u,/A)..

(1.195)

T h e figures in the table show that the introduction of the nonlinearity renders a significant reduction in the error only when o,/A < 1. This example clearly distinguishes between the filter problem as a problem of finding the best approximation and the filter problem as a problem of increasing the ratio of the power of the signal to the noise (the signal-to-noise ratio). I t is obvious that for the second problem a linear lagless transformation serves no purpose since in this case the improvement of the approximation in the sense of the meansquare error turns out to be significant. Therefore, in problems

1.4. The Application of Methods of Synthesis

73

where special importance is attached to the form of the signal, the mean-square error criterion may not be acceptable.

Example 2. We suppose that the signal S can take only the values

+ a and - a with equal probability, W,(S)

=

i[S(S

+ S(S + a ) ] .

- Q)

T h e noise has a normal distribution with a mean-square deviation of u,. T h e signal and noise are mutually independent and, thus, additive. T o find the moments S?? and N2" we have Eq. (1.190) and

S2m

=

(1.196)

a2m.

Let us find the results of the calculations for the system of polynomials

e,(Z) ( n = 1, 3, 5) according to Eq. (1.169) for the case when a2 = 0.8 and un2 = 0.2 :

e,(q

=

z, e3(z)= 0.98423

O,(Z) = 0.612Z5 - 2.78Z3

T h e coefficients aD, all,

=

0.8,

=

-

1.692,

+ 2.3122.

en(Z)Sin this case are

a&

=

0.252,

aD, = 0.141, ... .

T h e equation for the optimal filter (among the class of fifth-order polynomials) has the form Y

=

1.55Z - 0.64Z3 + 0.087Z5.

Graphs for the optimal filters constructed for various values of the ratio a/., (where a2 un2 = 1) are shown in Fig. 14 (continuous lines). Graphs for the exact solution

+

Y

= a tanh

a2 ~

on2

(1.197)

are shown by dotted lines. This is the simple result for a particular problem [51, 1101. T h e dependence of the mean-square error on the ratio a/U, (in

1. Nonlinear Transformations without Feedback

74

its application to constructed filters of the fifth order) is shown in Fig. 15 (Curve 1). For the optimal linear filter the values of u, are related to the error u l c . T h e improvement when a/u, is large (i.e., when the noise level is low) turns out to be very significant.

Y

2

0.97 0.96 0.70

0.0 0.5 1.0

--c

1.5 2.0 2.5 2

FIGURE14

a /a,,

FIGUREI5

Comparing the form of the characteristics of the fifth order with the form of the exact solution (the optimal filter in the class of arbitrary functions) shows clearly that the approximation by a polynomial of the fifth order is not very good. However, one can see (Fig. 15, Curve 3) that the use of a filter constructed according to the exact solution (1.197) does not decrease the error significantly for small a/., . Even when a/u, = 2, we see that (uc/urc) = 0.45 for filters of the fifth order and that (ur/ulJ2 = 0.34 for a filter constructed according to (1.197). Only for large a/., does the difference become significant [for a/u, = 4,we obtain according to the exact solution (ur/ulJ2= 0.0041. We also note that for large ratios a/., in this problem, it is appropriate to apply, as a nonlinear filter, the simple relay*

Y where the coefficient

OL

= act, sgn 2,

(1.198)

is chosen in a corresponding manner.

* The choice of the relay characteristic becomes clear if one treats the resulting problem as a problem of potential interference-stability for the reception of a signal which takes the values f a [36j.

1.4. The Application of Methods of Synthesis

75

I t can be shown that for a filter with an arbitrary odd characteristic f ( 2 ) the variation of the error in filtration of the given input signal is equal to a :

=

J

OLi

--m

[a

-f(n

+

wl(n) dn,

where q ( n ) is the probability density of the noise. For the characteristic (1.198) with an optimal choice of CY we have (1.199)

where aopt=

U

2@ - . on

Computations (Fig. 15, Curve 2) show that, when a/un 3 1.5, filtering by the relay is always more effective (when a/un = 4, the quantity u,2/u: = 0.02) than filtering by a device which uses a characteristic in the form of a fifth-order polynomial. T h e problem of constructing the optimal nonlinear filter with lag with the restriction of using fifth-order polynomials is discussed byLubbock[45]. I t is assumed that the signal and the noise have the same correlation function ps = pn = e ~ @ (T~h )e .effect of introducing linear circuits turns out to be insignificant; it is less than passing from polynomials of the fifth order to the lagless transformation (1.197). Let us now consider a study of the problem of parallel compensation. Let the system carry out a given transformation

u, = F,{Z).

(1.200)

I t is necessary to construct a transformation

u, = F,{Z),

(1.201)

such that the summed signal

u = u, + u, = F,{Z) + F,{Z)

(1.202)

in the best possible way approximates some given S.*

* This problem can be considered as a special case of the general synthesis problem, if, in looking for the optimal F, one accepts the fact that the desired signal X will be given by the relation

x = s- F 0 [ Z ] .

76

1. Nonlinear Transformations without Feedback

Thus, the problem consists of designing an optimum parallel network Fl [Fig. 16 (a)] which compensates for the dynamic properties of the original system.

I t is obvious that the problem does not become more complicated mathematically if one introduces a more general type of compensation [Fig. 16 (b)], where the input signal of the parallel stage is not the input signal of the system F, , but instead is either some intermediate signal acting within the system, or the output of F,. If the system F, is given and if the statistical properties of the signal which is the input for the parallel link and, also, the properties of the output signal U , can be measured, then the general form of the transformation F,, is unimportant for the solution of the synthesis problem. T h e indicated technique can be used, in particular, to construct parallel correctional links and to introduce a disturbance effect (compounding) in systems with feedback. T h e following basic synthesis problem, and especially the problem of the statistical determination of the characteristics of a transformation which is carried out within a given system, is similar to the problem of compensation. I n this case, it is assumed that only the input and output signals of the system can be measured and that its internal structure is unknown (the system is considered to be a “black box”). T h e problem is to select a transformation F which will best replace the transformation of 2 to X which is performed by the system. T h e mathematical formulation of the problem, obviously, is exactly the same as the general formulation of the synthesis problem. We note that the result of the synthesis, that is, the transformation F which best replaces the system performance, depends greatly on the characteristics of the input signal 2. Therefore, it is appropriate to choose as our 2 that input signal which will be in use for the system we are studying under

1.5. Statistical Linearizntion

77

real working conditions. If its form is not known, it will be convenient to introduce as an “experimental” signal normal white noise, for which the polynomials 8, are the Chebyshev-Hermitian polynomials, and Cznrn(,)= 0 when n # m. Th en the system of equations (1.167) which determine the functions hln1(7)decomposes into a series of separate equations which coincide exactly with the classical WienerHopf equations. 1.5. Statistical Linearization

T h e problem of statistical linearization consists in finding the best description of a given nonlinear transformation in terms of a linear transformation. Its solution is important for studying complex systems where mechanisms which perform nonlinear transformations (which, in turn, can be linearized) are introduced as elements. Such systems are often important in relation to feedback systems, which are of fundamental importance in the theory of automatic control. T h e insertion of a nonlinear element into a linear transformation (the parameters of which naturally will depend on the statistical characteristics of the input signal) makes it possible in the study of closed nonlinear systems to use methods of linear analysis, which greatly simplify the problem. This section will only give a description of the method of constructing a statistically linearized transformation; the application to calculations for closed systems will be given in the following chapter. T h e problem of statistical linearization can be formulated mathematically in the following manner. Let F, be a given nonlinear transformation and let F be linear. We must select an F such that the signal Y = F { Z } approximates the signal X = F,{Z} in the best possible way. I t is obvious that this is a special case of the general synthesis problem of finding transformations which give the best approximation of some desired property. We limit ourselves to a detailed study of the case where the linearized transformation approximates a nonlinear, lagless transformation : I n the analysis of an open-loop nonlinear transformation with lag given by equations of the type (1.163), it is convenient to linearize

78

1. Nonlinear Transformations without Feedback

directly the elementary, lagless transformations which are part of it. I n more general cases, it is possible to apply directly Eq. (1.141) or (1.178), provided one can successfully determine theoretically or experimentally the characteristics of the input and output signals of the transformation which is being linearized. First, let us study a solution of the problem of statistical linearization which is based on the criterion of the minimum mean-square deviation. T h e solution is given by the conditions (1.140) and (1.141), which here take on the following form : (1.204)

and

Thus, in the general case the solution reduces to the computation of the moment characteristic R,(T) and the solution of the WienerHopf equation. If the process Z ( t ) is normal, the solution is quite simple. I n fact, in Section 1.1 it was shown that, for a lagless, nonlinear transformation of a normal process, the cross-correlation function R,,(T) of the input signal z ( t ) and the output signal x ( t ) is proportional to the autocorrelation function R,(T), (1.206)

where

fm

a,=-.

0 2

Substituting (1.206) into condition (1.205), we see that it is satisfied when h ( ~= ) a16(T). Hence, when Z ( t ) is normal, the optimal approximation is a lagless transformation, (1.207) Y = hornz hJO,

+

I . 5 . Statistical Linearization

79

where

I t is not difficult to see that the coefficients for the optima1 transformation depend only on the parameters m, and uZ . We shall compute the approximate value of the correlation function for the output signal X ( t ) of the nonlinear transformation f o ( Z ) from its approximation in (1.207) : R,(T)

M

Ry(7) = hz2R2(T)= a12p2(T).

(1.208)

By comparing this expression with the exact expression for R,(T), which was found in Section 1.1,

we see that R,(T) coincides with the first term of the exact series expansion. When p ( T ) > 0, it is always true that R&) > RW(4

(1.209)

that is, the statistical linearization by the criterion of the minimum mean-square error gives a lower value for the correlation function of the output signal. T h e inaccuracy (at least when m, = 0) is not very great, since, as shown in Sections 1.1 and 1.2, only the first term is of great importance in the determination of R,(T) and, consequently, in the determination of the spectral density s,(~), which as a result of the statistical linearization turns out to coincide in form with S,(W). As shown by Barret and Lampard [loo], the proportionality of R,(T) and R,(T) holds for a whole class of signals whose two-dimensional probability densities w z ( z l , zz)can be uniquely expanded in a Fourier series of the form m

WAZ,

3

~ z = > w ~ ( z ~ ) w ~ ( z AanOn(z1Yn(ZA* n=O

(1.210)

where O,(z) are polynomials of weight w l ( z ) which are orthogonal in the interval where z is defined.

80

1. Nonlinear Transformations without Feedback

Naturally, for these signals the optimal property of a lagless linear approximation will be valid in the general class of linear approximations. Together with a normal signal in the given class, we have a harmonic signal with a random phase, 2

=

a sin(wt

+ Y).

Without showing that its two-dimensional probability density can be expanded into the series (1.210), we shall go on to compute R,(T) and R,(T) directly. From (1.53) we have fo (a sin t ) sin t dt,

R,

a2 2

= - COS

(1.21 1)

T.

Hence, the best linear approximation is given by the equation Y

=

=

2?T1

my

+ h,Z,

(1.212)

where m y = fo(z)

1

Dl

0

fo(a sin t ) dt,

T h e right-hand side of (1.212) is the sum of the first two terms of the Fourier series of the function fo(a sin t ) ; this means that, in the theory of nonlinear transformation of determined harmonic signals, the method of statistical linearization is equivalent to harmonic linearization. We note that this method of harmonic linearization can be considered the best approximation in the sense of the minimum mean-square deviation, where the mean is taken with respect to time over a certain period. This fact is a special case of the so-called minimal property of coefficients in a Fourier series. Statistical linearization by the method described above can be

1.5. Statistical Linearization

81

difficult or simple according to the nature of the problem. Thus, in several cases it is convenient to use an approximation by a simpler homogeneous linear transformation : Y

(1.213)

= h,Z.

From (1.148) the transmission coefficient is equal to (1.214) which is unique for both the mean and the random component. I n problems where a nonlinear transformation has several input signals, that is, where

it is usually reasonable to look for a linear approximation in the form

Y

=

2 hiZi .

(1.216)

i=O

We shall assume that the signals Zi ( i = 1, 2, ..., n ) are random and are mutually uncorrelated, that mZi = 0, and that the signal 2, does not have a random component. Then

(1.217)

As an illustration, we take a nonlinear transformation of a sum of statistically independent signals, one normal and the other harmonic with a random phase : Z

=

6 + a sin(wt

+ Y ) = ml + to+ a sin(wt + Y).

Approximating the nonlinear transformation f ( Z ) by a linear one Y

=

horn(

+ h,a sin(wt

Y )+ h,tO,

1

(1.218)

82

1. Nonlinear Transformations without Feedback

from Eq. (1.217) we find that

I t is not hard to see that the coefficients ho , h, and h, are simply related to the coefficients of the exact equation (1.68) for B,(T) : h

-lo!!

O -

h,

m5'

2hOl

=, U

h,

=

h,, ,

B,(T) computed from (1.218) corresponds to the first three terms of the series (1.69). Graphs showing the dependencies of the coefficients h, and h, on the parameters uE and a (when m E = 0) for several typical nonlinearities are in Appendix IV. As another example, we shall look at the nonlinear transformation of a normal signal and of its derivative :

x = f(Z, P Z ) .

(1.220)

We look for its approximation in the form Y

= h,Z

+ h,pZ

= h,Z,

+ h,Z,,

(1.221)

where 2, = p Z . Here, we shall assume, for the sake of simplicity, that m, = 0. Then

fkl * ~ , ) ~ l W l ( ~ l > W , ( ~dz, , ) dzz h,

where

1

= -5

02

j j "

--I,

"

-n

9

(1.222) f(z1

?

~ Z ) ~ Z W l ( ~ l ) W Z ( ~d Zz ,)

dz,

7

1.5. Statistical Linearization

83

Again, just as in Section 1.1, we note that these equations are not applicable to multivalued, nonlinear dependencies with branch characteristics. T h e approximate expression for the transfer function in terms of the random conponent for a nonlinearity of this type, which is derived under the same assumptions as the expression (1.19) for the expectation, has the form u,2hl

(z

=

-

m,)j(z)wl(z)d z

-ffi

z

-

mz)[tl1f1(4

+J

+

m

(z A2

-

m , ) j ( z ) w l ( ~dz)

cL2fi(41 W l W

(1.223)

dz,

where

Here, we are using the same notation as we did for the branches of the multivalued function in (1.19). For a relay with a symmetric hysteresis loop with width 24, we obtain

Next we develop the method of statistical linearization based on approximation with respect to the criterion that the first two moments be equal. T h e requirement of equality for the expectation and the correlation function actually guarantees an exact description of the nonlinear transformation within the bounds of correlation theory. T h e solution is given by Eqs. (1.173), (1.144) and (1.178), where

84

1. Nonlinear Transformations without Feedback

T h e problem of finding the optimal linear transformation reduces to the problem of computing the mean value and the spectral density of the signal at the output of the nonlinear transformation without lag and, then, of expanding into factors for the ratio S,(w)/S,(w). T h e first part of this problem was discussed in detail in Sections 1.1 and 1.2. T h e second part, also, is not very difficult; it is well known from many other problems in the theory of automatic systems.

Example 1. Let f o ( Z ) = Z3, and let Z ( t ) be a stationary random process with a spectral density of

We compute the correlation function for the output signal R,(7)

= a?p,(~)

+

U32p,3(T) = a12 e-'lr'

+ a32

e-3eiri.

T h e corresponding spectral density is equal to

Hence, we find that

Separating out the factors with roots lying below the half-plane, we obtain, finally, @ ( j w )= k T l j w T2jw where k(u,) and

+1

+1 '

G12 + Qa32is the

_____.

= (1/us)

-.

' 1'

z=

dI2

+ 3a32

2/302(3012

+ a;2)

statistical transfer function,

and

T,

=

1/30

1.5. Statistical Linearization

85

are time constants for the optimal linear approximation. I t is often convenient to use a weaker criterion for approximation, namely, that there be equality for the mean values and variances of the output signals of the approximate linear lagless transformation Y

=

hp,

+ hl*Zo

and of the given nonlinear transformation

x =f(Z). From the requirements uU2= ax2

m, = m , ,

we immediately obtain equations for the coefficients h, and h,* : (1.227) h l * ( m 2 , u,)

0 ,

=-

.

(1.228)

=2

We compute the correlation function for the signal Y at the output of the approximate transformation as follows : RU(7) = (h1*)2Rz(7) = O X ~ P ~ ( ~ ) .

Let Z ( t ) have a normal distribution. Then,

n=1

and

n=l

Hence, we have the inequality

86

1. Nonlinear Transformations without Feedback

where p , ( ~ )> 0; this means that a linear approximation constructed by this criterion gives an estimate for the correlation function above the correlation function of the actual output signal, whereas the linear approximation by the criterion of the minimum mean-square deviation gives an estimate which is below [cf. Eq. (1.209)]. In the first case we obtained a better approximation for large p , ( ~ ) (that is, for small T ) , and in the second case it was better for small p , ( ~ ) [that is, for large TI. I n general, the first case is more important because the resulting values of p,(.) are usually more valid for small T ; this has been borne out by experimental data for input disturbances (cf., for example, Pugachev [65]). However, the computation of the transfer function with respect to the random component* by applying the minimum, mean-square deviation criterion is simpler because h, is given by a linear operation onf(z) which makes it easier to tabulate. At the same time, if the coefficient h,* has already been computed (the dependency h,*(m, , uz) for typical nonlinearities is given by Pugachev [65]), then one can use the averaged coefficient hi

+ hi* 2

.

I .o

0.5

0

0.5

1.0

1.5

2.0

r

FIGURE17

* The coefficient h,

is, obviously, the same no matter which criterion is used.

1.5. Statistical Linearization

87

T o illustrate the effectiveness of this technique, graphs for the correlation function of a signal at the output of a relay for which R,(T) = -exp(I T I) and m, = 0 are shown in Fig. 17 for the following cases : (1) the (2) the (3) the (4) the tion.

exact solution, Eq. (1.29); approximation by the minimum, mean-square deviation ; approximation by equating variances; and approximation with the averaged coefficient for lineariza-

chapter 2

NONLINEAR TRANSFORMATIONS WITH FEEDBACK STAT10NARY STATES

2.1. A Short Description of the Basic Methods of Investigation

T h e fundamental characteristic of a nonlinear transformation with feedback is the fact that it is impossible to find in explicit form an expression for the dependence between the input and output signals. Hence, the techniques of computation described in Chapter 1 cannot be applied directly in order to find the statistical characteristics of the transformed signal. We now give a short summary of the current methods for dealing with nonlinear transformations which do not require an explicit expression for the depeqdency between the input and output signals. * (1) The method of direct linearization. T h e nonlinear functions at the input of a transformation with feedback can be replaced by linear functions if one considers only the first two terms of the Taylor series. Wherever this operation is feasible (when the nonlinearities are analytical and when the signals at the input are small), the problem loses its special nature and becomes a problem of linear transformations of random functions. We shall not discuss the method of direct linearization in detail because it is assumed that, wherever such a method is applicable, it has already been incorporated in the process of transforming from the real system to its dynamic counterpart. ( 2 ) Methods based on the application of canonical expansions of

* We do not pretend that this classification is complete; our objective is only to bring out the basic methods for solving this problem, and the degree to which they are developed in this book. 88

2.1. Short Description of Basic Methods of Investigation

89

random signals. Here, a random process is represented, over a finite interval of time, by the sum of definite functions of time with coefficients which are mutually independent random variables (cf. Section 1.2). I n principle, this kind of representation reduces the problem to the task of integrating nonlinear differential equations which contain only definite functions of time. (3) Methods based on representing the output signals by Markov processes (either one-dimensional or multidimensional) and, subsequently, on using Kolmogorov dtfferential equations to compute the probability distribution of these signals (cf. Appendix V ) . T h e complexity of this procedure in general, limits the scope of its application to analytical problems of (a) transformations which are defined by differential equations of the first order, or, in some cases, of the second order, and (b) of transformations which lead to these by way of introducing auxiliary transformations, such as harmonic linearization. T h e possibility of using Markov processes for exact solutions, even though feasible only for a limited number of problems, has attracted the attention of many researchers. This book gives a brief description of these methods, illustrated by examples (Section 2.6 and, in part, Sections 4.2 and 4.3). (4) The method of dealing with transformations which are piecewise linear functions, based on the sequential lacing together (alignment) of solutions for each region of the phase space where the transformation is linear. This method is applicable in analyzing vibrational states where there are small random disturbances from some source (Section 3.4 and, also, Section 4.5). ( 5 ) The method of successive approximations. This method derives from physical representations of the process of establishing an operating condition in a system with feedback in terms of an iterated process of a circulating external disturbance around the closed loop. Here, the integral equation (1.20), which implicitly expresses the transformation with feedback, can be solved by the method (2.1). .x

Xi(t)= J

--x

[Z(T) - jxk ( ~ , X i - l ( ~d) ~ ]

h,(t, ~ ) f

S)

X

dT,

(2.1)

where X o ( t ) = Z ( t ) , that is, the value of the signal X ( t ) is assumed to lag each time it is taken from the previous iteration of the cycle [43, 691.

90

2. Nonlinear Transformations-Stationary

States

Formally, of course, one can think of the method (2.1) as an ordinary mathematical method for sequential approximations which does not have to be associated with any kind of physical interpretation. I t is obvious that the application of this method changes the problem from that of a closed to that of an open system. A variation of the method of successive approximations is described in detail by Pugachev [65, p. 5241.

( 6 ) Approximation methods based on the assumption that the character of the distribution of the signal a t the input of the nonlinear, lagless transformation is unknown. In this case one tries to find several of the numerical parameters which are undefined in the equation for the distribution. T h e implicit relations in these parameters (they are usually transcendental equations) can be solved, for example, by graphical techniques. Remembering that in the process of filtering the approximation for the distribution is taken to be normal (cf. Section 1.2), one can usually assume that here the distribution is also normal. A normal distribution is completely determined by the values of the mean m, and the mean-square deviation u, and so, too, is the form of the correlation function. As an additional technique, it is imperative to use, in particular, the method of statistical linearization of a lagless nonlinear transformation. Hence, we can dssume that it is necessary to preserve only that term which is proportional to the correlation function of the signal X ( t ) at the input in the expression for the correlation function of the signal at the output of this transformation. This assumption considerably simplifies the problem and involves only the parameters m, and u I . Using the idea of expanding in terms of some small parameter broadens the applicability of the method and helps in studying the effect of small distortions in the form of the correlation function or of a deviation in the distribution from the normal. Because of its generality and comparative simplicity, this method of statistical linearization is of the greatest interest for practical computation. I t is convenient to divide the problems of nonlinear transformations with feedback into two parts : T h e first part (Chapter 2) is concerned with stationary states, that is, states where the signal which is acting outside the feedback loop is a stationary function of time; the second part (Chapter 3) considers nonstationary states.

2.2. Application of Statistical Linearization

91

T h e type of state which is realized in a given system (or transformation) is determined, not by its structure, but by the characteristics of the input signals and the parameters of the system. I n the study of real systems it is usually necessary to analyze both stationary and nonstationary states. I t is of very great practical importance to determine the conditions under which the transition from one state to another occurs when the parameters of the signal and the system are changing. These conditions frequently determine the so-called interferencestability of a system, that is, the potential loss of stability because of random interference. Of special importance is that situation where both stationary and nonstationary states can be studied on the general basis of statistical and harmonic linearization. T h e development of these approximative methods takes u p the larger part of this and the following chapters. 2.2. The Application of Statistical Linearization t o the Analysis of Nonlinear Transformations with Normally Distributed Stationary Signals

Consider the nonlinear transformation with feedback of the signal Z ( t ) to the signal X ( t ) , which is given by the system of differential equations

where Q(p), R( p ) and S ( p ) are linear differential operators (i.e., polynomials with constant coefficients in the differential operator p = d/dt. T h e system of equations (2.2) corresponds to the block diagram shown in Fig. 18.

FIGURE 18

92

2. Nonlinear Transformations-Stationary

States

Let the input signal Z ( t ) be a stationary random process with a normal distribution. Physically, Z ( t ) can represent any signal or any combination of a signal and noise. We now introduce the basic proposition that the distribution of the signal X ( t ) at the input of the nonlinear element be normal. It is equivalent to the proposition that the output signal U ( t ) of the linear transformation be normal (Fig. 18). T h e latter proposition refers to the effect of normalization of the signal which passes through a linear transformation (cf. Chapter 1.2). This effect does not take place under all circumstances, nor is it ever complete. However, in most practical cases the assumption that X ( t ) [or U ( t ) ]is normal leads to sufficiently reasonable results. (For a more detailed statement of the conditions under which the method of statistical linearization is applicable, see Section 2.4.) Starting with this assumption, we construct the optimal linear lagless approximation to the nonlinear function f ( X ) , which we shall first suppose to be odd and single valued :

Y

= horn,

+ h,XO.

(2.3)

As was shown in Section 1.5, the coefficients ho and h, are functions only of the parameters m, and ox :

T h e further development of the method does not depend on the concrete form of the functions (2.4) and, therefore, it does not depend on the accepted criterion for approximation. Let us separate from (2.2) equations for the mean and random components of the signals :

2.2. Application of Statistical Linearization

93

These equations can be solved formally for m, and XO : m, = @,,(P, m, , O

xo = @ d p , m,

I

, h ,

(2.7)

ux)Z0,

(2.8)

Here we have adopted the following notation : @, is the transfer function of the system with respect to the mean component, and is the transfer function with respect to the random component. Equations (2.7) and (2.8) are related because the transfer constants h, and h, depend both on m, and on u, . Therefore, they must be solved simultaneously. First of all, it is important to note that Eq. (2.5) has the solution* m, = @,,(O, m, , a,)m, = const

or

(2.10)

We have at our disposal the family of curves m v ( m x , a,) = a , (m,, 0), which have already been constructed for typical nonlinearities (cf. Appendix I) corresponding to various values of u,. We now draw on the same diagram the line [Fig. 19(a)] (2.1 1)

At the points of intersection we can immediately find the dependence m,

=

(2.12)

mx(4

Then, it is not difficult to calculate the dependence of the transfer constant h,* with respect to its random component directly on the variable ux : h*(%) = h,[m&Jz), %I. (2.13) From (2.8) and (2.9), the basic equations for the linear transformations, we obtain the following expression for the spectral density of the process Xo(t) : Sz(w) = I @l(ju9 m,

* From the

condition that

Z

I

0,)

I2~~b),

is stationary it follows that m,

=

const.

(2.14)

94

2. Nonlinear Transformations-Stationary

States

where S,(w) is the spectral density of the external disturbance Z o ( t ) . We shall compute the mean-square deviation u, in terms of S,(w)

T h e integral at the right is tabulated in Appendix 111. Its value can be expressed directly in terms of the values of the coefficients in S,(W),that is, in terms of the parameters of the linear operators which determine the form of Sz(w), and the transfer constant with respect to the random component.

Thus, Eq. (2.15) can be considered as an implicit expression which relates ux to the parameters of the system and of the external disturbance from which, for example, u, can be found by graphical means. Equation (2.12) then gives immediately the value of the mean component. Of course, the proposed procedure for finding m, and u, from the system of implicit equations (2.10) and (2.15) is not the only one possible, but it is quite convenient for computation. Its only drawback is that the dependence hl*(uz) has to be computed numerically. However, this drawback can be eliminated if a second, slightly different step is used in the computation. Equation (2.15) can also be considered a functional dependence of u, on the transfer constant h, : = oz(h,).

We construct this dependence in the plane (u,, h,) on the same diagram which shows the tabulated dependence h,(a, , m,) [Appendix I], constructed for various values of the parameter m, [Fig. 19(b)]. Along the points of intersection of the curve u,(h,) with the various curves of the family h,(m, , a,), we find a dependence of the form m3c = %(ax).

Its graph in the u,m, plane, together with the graph for the function

2.2. Application of Statistical Linearization

95

(2.12) gives the coordinates of the point of intersection mz0 ,

uzo,

which obviously is the desired point representing the steady state [Fig. 19(c)]. This whole procedure is illustrated graphically in Fig. 19. T h e visual representation and the elimination of intermediate analytical calculations are advantageous.

FIGURE19

A significant simplification in the method can be obtained if the random component of the external disturbance, which acts at the input of the nonlinearity Z?(t), is a high-frequency one, that is, if for all the existing frequencies in S,(W)the following condition is satisfied: (2.16)

96

2. Nonlinear Transformations-Stationary

States

In this case, we have the simple equation (2.17)

and rn, is given by Eq. (2.12) as before. We now assume that f ( X ) is not an odd function of X . (It has been described in detail in Section 1.1.) I t is obvious from physical considerations that in this case the nonlinear element corresponding to f ( X ) will detect the random signal, and, therefore, even in the absence of a DC component in the external influence, there can be a signal in the circuit. This fact does not allow the formal application of the concept of the transfer constant with respect to the mean component, which was introduced previously. I n fact, if m, is replaced according to the equation rn, '= hOmxin (2.19), one immediately arrives at the physically meaningless result that m, = 0 when m, = 0. T h e representation of m, in the form hOms entails the implicit assumption that, when m = 0, the expectation of the output signal is also zero; but, for an arbitrary (even) functionf(z), this cannot be true. Therefore, in making calculations for systems with this kind of nonlinear element, one must reject the concept of a transfer function Q0 with respect to the mean component of the signal. However, this does not lead us to impossible complications in computation. I n fact, we can use the first equation of the system (2.5). Since m, = const, it follows from (2.5) that

or (2.19)

Hence, one can directly apply the grapho-analytical technique developed above for systems with odd nonlinearities. Let us now study the case when the nonlinear dependence has the form y = f(X, P X ) . (2.20)

2.2. Application of Statistical Linearization

97

In Section 1.5 it was shown that similar functions can be effectively approximated by linear functions :

Y

= m,

+ hlXo + h,pXo,

(2.21)

where the coefficients h, , h, and m y are determined by Eqs. (1.17) and (1.22), with 2 replaced by X ; the coefficients h, , h, and m y are thus functions of the parameters m, , ux and cr,, . I n order to find these we have the following three implicit equations : (2.22)

where

From these equations, the graphical determination of the parameters m, , crx and u , ) ~is inordinately difficult. A more convenient solution is given by an iterative method according to the equations

(2.26)

98

2. Nonlinear Transformations-Stationary

States

T h e suitability of this method of computation is obvious from physical considerations. We note that a similar iterative method can be used for solving the problem of single-valued nonlinearities, if for some reason the graphical representation is not convenient. Because of the importance of these methods in practical computations, several detailed examples are given. Example 1. Let us study as a simple system the block diagram of Fig. 20. T h e corresponding differential equation is 1

Y f k ( T p + l)X=Z.

(2.27)

U

FIGURE 20

Let Y

=

X

+ ax3,and Z ( t ) be white noise with density d , that is, R,(7) = d8(T).

We replace the nonlinear function by its linear approximation in accordance with the criterion of the minimum mean-square deviation Y

=

where h, = 1

h,X,

+

3UOrZ.

Then, for the signal X one can write the linear equation

where

99

2.2. Application of Statistical Linearization We now determine

ux

:

We can find this integral from the table in Appendix I11 (although in the present case it can be found by elementary methods). For this integral

+

Hence, and

h ( p ) = P B, &(PI = 1,

a. = 1,

6,

=

a,

=

B,

1.

6 1 I 1 ----o=2aoa1 28

(2.28)

We substitute into this equation the expression for h, in terms of and solve the resulting quadratic equation for uT2:

uX2

Thus, in this simple case it is possible to find an explicit expression for the variance in terms of the parameters of the system and the characteristics of the signal.

Example 2. T h e system is the same as in Example 1, but the nonlinearity has the characteristic of an ideal relay : f ( X ) = I sgn X .

T h e input signal 2 has a constant mean component m, and a random component which is white noise with the intensity d. For the mean components, we have the equation my

=

m,

1 -7;"r.

(2.30)

According to the graph of Appendix I, we construct the family of

100

2 . Nonlinear Transformations-Stationary

States

1.5

I .o

0.5

0

FIGURE 21

I

2

3 Y

2.2. Application of Statistical Linearization curves (l/l)m&mz/l) for several values of u,/l diagram we draw the line

= ul.

101

In the same

'

mr - mz,- 1 m, k 1 for the case when m,, = 1.5 and k = 1 [Fig. 21(a)]. By plotting the points of intersection, we find the graph for the function ml(ul) [Fig. 21(b)]. Analogously, we construct the graph [Fig. 21(c)] for the family of curves hl( l/ul) for various values of m, and draw onto it a section of the parabola (2.28) : my -

"2

I

1

Here, we assign the values 0.5, 1.0, 2.0 and 3.0 to y = dk/2TI2. T h e values of ml(ul) for various values of y are shown in Fig. 21(b). T h e points of intersection give the solution, that is, the quantities m 1 and u,, for the respective values of y . T h e results are given in the form of graphs of m l ( y ) and ul(y) in Fig. 21(d). Example 3. Consider the problem of transforming random noise at the input of a low-power servo system. T h e block diagram for a servo system is shown in Fig. 22, and the corresponding ideal transfer function block diagram is shown in Fig. 23. Reduction

Tacho-

I

gene rotor

,

J

FIGURE 22

"I.C.

FIGURE 23

I

102

2. Nonlinear Transformations-Stationary States

T h e input signal Uin consists of two parts, the component min and the random noise Upn . We write the equations which relate the mean values (expectations) and the random components of the output signal* lJout and of the signal at the input of the limiter to corresponding quantities for the input signal Uin. On the basis of the block diagram and with the aid of the equation for statistical linearization, y

=

+hlh,

ho(m,,

, u?Jxo,

we can find the following equations : mout =

K",,(k)%; (2.31)

We investigate two cases. (a) Let mi, = const. Then m, given by the implicit equation

= 0.

T h e mean-square deviation is

* The voltage of the potentiometer for the feedback may be taken as an output signal.

2.2. Application of Statistical Linearization

103

T h e spectral density for the interference has the form

T h e integral at the right in (2.32) can be computed either graphically for various h, or directly from the equations for I, in Appendix 111. We carry out the analytical calculations neglecting the time constant of the amplifier T , . I n this case the polynomials h,(p) and g,(p) in the equation for I, will have the form h n ( ~= ) (P

+ 0) 11 + ~ P ( T , P+ 1) [I +

1

(Tmp

+ 1111 .

+

gn(P) = - ~ @ J ~ [ - T ; , P1]& ~ ,

and, hence,

b,

= 0,

b,

=

29T$~:,, ,

ba

=

-2eu?,,,

b,

=

0.

Substituting the value of the coefficients b, into the general equation for I4 , we obtain az'L =

+-

a3Tm2 a, ala2a3- aoa32- aiza,,

eU;,

(2.33)

.

T h e curves u,.l = u1 = ul(hl) are constructed in Fig. 24 for the following values of the parameters of the system :

k,

=

150,

km = 13.4 rev/sec . volt, T

.

1 z=90 '

k,

=

Tm = 0.1 sec,

= 0.013 volt . sec/rev

7 2 voltslrev,

k,

=

0.5,

T f = 0.05 sec,

1 = 12 volts

and for various mean-square values of the input noise for 0

=

20 sec-'

104

2. Nonlinear Transformations-Stationary States

10

5

0

FIGURE24

T h e curve for the transfer constant h,(u,) taken from Appendix I is drawn on the same diagram. T h e points of intersection correspond to the required values of ux = u,l and h, . T h e function axil = u,(oin) for B = 20 sec-' is shown in Fig. 25. T h e function axil = ul(0) when 01, = 3 volts is constructed in an analogous fashion, and is shown in Fig. 26. T h e values for the transfer constant h , , which were found, are used to construct the functions uo"t(oin) for O = 20 sec-' (Fig. 25) and uout(B)for 0131 = 3 volts (Fig. 26). T h e computation is based on the equation 2

Oout --

u:,,

%a2 - %a3 ala2a3- a,a,2 - a12a4'

which was derived in the same way as (2.33).

(2.34)

2.2. Application of Statistical Linearization

FIGURE 25

FIGURE 26

105

I06

2. Nonlinear Transformations-Stationary

States

These graphs show clearly that particularly the output in the region of nonlinearity leads to a sharp increase in the variance of the signals uout and X . Here, the output in the region of nonlinearity can be ( Gi n > 2 volts for B = 20 sec-l) looked at both as an increase of and as a variation in the spectral density component of the interference (0 > 5 sec-' for ui,, = 3 volts). (b) Let the signal be a linear function of time : m,,

= At.

Then, for the steady state, (2.35)

Along the points of intersection of the family of curves m,(m, , u,) with -the horizontal line which has the ordinate X/k,,ik,, one can find the function mx = m z ( 4

Substituting into the expression for the transfer constant in terms of the random component, we obtain -ml),

Q,(

(2.36)

01

where m,

=

mX I '

0

' = -1

O X

'

which makes it possible to find directly the dependence of h, on O x . A construction has been made for X = 1 volt sec-l and with the same parameters of the system as in case (a). I t turns out that for practical purposes, the form of the resulting explicit function h,*(u,) is identical with that of the function h,(o,) when m, = 0 (Fig. 24). T h e latter fact can also be derived from general considerations. I n fact, it is not difficult to see from (2.36) that when X = 1 volt sec-', the quantity m, is very small, and that the expansion of h,(m, , 0), in degrees of m , will not contain terms of the first degree. Hence, the graphs of case (a) [Figs. 25 and 261 for ooutand u, remain valid in this case. We now evaluate the variation in the settling error in the signal,

2.3. Computation of Frequency Distortions

107

the so-called noise effect; in other words, we evaluate the variation in the quality of the servo system. I n the absence of interference, the quality of the system is given by. the equation

e

=

i p (1

1 + -), P

(2.37)

and, in the presence of interference, by the equation (2.38)

If uin = 3 volts and 0 = 20 sec-l, we see from Fig. 25 that U, = 9.6 M 10. Moreover, from the graph in Fig. 27, we can find the point

0.1

0

0.2

c ml

FIGURE 27

of intersection of the curve (l/Z)mv(m1, 10) with the horizontal which has the ordinate h/k,,,k,i = 0.0078 and gives rn, = 0.1 and h, = 0.08. Therefore, 1 0.04 _ Be = 0.7. e 1 0.04(i/o.o8)

+

+

Here, the presence of interference reduces the quality of the system by a factor of 13. 2.3. Computation of Frequency Distortions Introduced by Nonlinear Elements

T h e method for the statistical linearization of a lagless, nonlinear transformation f ( X ) investigated above for nonlinear closed-loop

108

2. Nonlinear Transformations-Stationary

States

systems consisted in finding the optimally close linear, lagless transformation on the assumption that the random component of the signal X was normally distributed. We shall describe in greater detail a method of calculating the moment characteristics, which is based only on the assumption that X is normally distributed, and which makes it possible, by means of a nonlinear transformation, to take into account the distortions in the form of the correlation function. Let us take another look at a system with one nonlinear element which is described by Eq. (2.2), where Z ( t ) is assumed to be a stationary, random process. This equation can be written in the simple form x+ u=z,, (2.39) where

Z

N(p)

We write the equation which links the expectations m,, , m, and mu to the processes Z,(t), X ( t ) and U(t), m,, = m,

+ mu;

(2.40)

the equation for the random component is

z,o

=

xo + uo.

(2.41)

Equation (2.40) is equivalent to Eq. (2.9), N(O)m, = m,

+ K(O)m,(m,,

uz),

which was obtained by the method of statistical linearization.

(2.42)

2.3. Computation of Frequency Distortions

109

From Eq. (2.41) for the random component, we have dependence of the correlation function R,,(T) for the process Zl0 on the functions R,,(T)and R,(T) and on the cross-correlation functions R,,(T) and R,,(T) for the processes Uo and X o :

Ri(7) = Rd7)

+ R J T ) + R A T ) + RUd7).

(2.43)

Applying the Fourier transform to this identity, we obtain

Thus, we find the following well-known equations from the theory of linear transformations :

(2.45)

and, also, the condition

This follows from a property of the cross-correlation function of the input and output signals of a nonlinear element, which was discussed in Section 1.1 on the assumption that the input signal is distributed normally. We also remember that, in the method of statistical linearization, the coefficient a , is proportional to the transfer constant with respect to the random component h, which was found by the criterion of the least, mean-square deviation : (2.47)

110

2. Nonlinear Transformations-Stationary

or

I N ( j w ) 12Ss,(w)= [l

States

+ 2 4 Re K ( j w ) ] S , ( w )

+ I KCjw)

(2.48)

I”S,(w).

But SJw) is related to SJw) by the following equations (cf. Section 1.2) : ac

S,(w) =

j

e-jwT & ( T )

d7,

-W

(2.49)

or (2.50)

where

I

W

S,(w) =

--a

e-jwr p , ” ( ~ ) d7 =

,s:

e-jwT[-

1 ux2 27r

la

e3”’ S,(w) dw]” d7.

--iL

Therefore, Eq. (2.48) can be thought of as a complex integral equation relative to the spectral density SJw) of the signal at the input of the nonlinear element. For its solution, one can use the well-known technique of successive approximations. T h e method is based on the fact that in the expansion (2.50) only the first term is significant, while all the following terms contribute, at most, only small corrections. T h e limitation to the first term in (2.50) corresponds to a solution based on the method of statistical linearization. In fact,

and Eq. (2.48) reduces to the relation (2.14), (2.52)

in the method of statistical linearization.

2.3. Computation of Frequency Distortions

111

or

where

Taking as first approximations the quantities mi1)and a;’),obtained graphically (cf. Section 2.2) by the method of statistical linearization, we find

(2.55)

We can find the last approximations in the following way : SLk+l)(w)= I W ) ( j w ) 12Ssz(w) -

[uq+1)]2

1

1 @ y ) ( j w )( 2

=-

\“

277.

m l-2

u,2(m(zk),af))Sy,

(2.56)

SLk+”(w) dw,

-%

(k = 1, 2, ...),

112

2. Nonlinear Transformations-Stationary

States

where

I t is difficult to give a rigorous proof of the convergence of this series, although it is physically clear that it does converge and, in fact, rather quickly. Ineach of the approximations it is very difficult to compute the functions W k ) ( w )One . can use the recursion relation (1.106) W

(2.57)

where

1

S:"'(w) = -s;)(w). [u;']~

Nevertheless, the computation is still very laborious. One way of avoiding complicated calculations is to confine oneself to the construction of a second approximation* (the definition of the quantities m;') and oLz)) in the following simplified manner : T h e expression for Sil)(w) which was found by the method of statistical linearization is approximated by the simple relation (2.58)

or

T h e corresponding expressions for the correlation coefficients are ~ ( ~ ' ( =7 e-elrI )

* Or, more precisely,

and

p : ) ( ~ ) = e-elrl

cos BT.

to an improvement over the first approximation.

2.3. Computation of Frequency Distortions

113

I n the first case we have S P ' ( w ) = w2

2v0

+ (,e)z

(v =

1, 2, 3 , ...),

(2.60)

and, in the second,

For v > 3, this way of finding Sj') cannot conveniently be used, both because of the inaccuracy of the resulting approximation and because of the rate of decrease in the coefficients up2. T h e correction in the variance is given by the equation

T h e exact value of the mean component can be found graphically from the abscissa of the points of intersection of the line

7%"(O)/W)l

-

[l/~(O)lm7

with the family of curves m 2 / ( m xu, x ) , which corresponds to the value ux =

up.

I n conclusion, let us note that there is little justification in attempting a construction for a greater number of approximations, since this method is quite rough, based as it is on the assumption that the input of the nonlinear element is normally distributed. Moreover, in this technique an increase in the corrections has a bearing on the deviation of the distribution from the normal. I t is

114

2. Nonlinear Transformations-Stationary States

physically obvious, for example, that these corrections are more important when the frequency band of the spectrum of the signal at the input of the linear part of the system is narrower than the output band. Hence, as shown in Section 1.2, the effect of normalization is diminished.

Example 1. We shall study the relay system which is described by the equations 1 Y +,(TP 1)X = 2, (2.63) Y = lsgn X,

+

where Z has a random component in the form of white noise with intensity d. T h e system was analyzed by the method of statistical linearization in Example 2 of Section 2.2. We shall use the results to construct a second approximation. Let us consider the case

where

By Eq. (2.60) we find

We compute the correction to the variance by the approximate equation (2.62), which here has the form

since

2.4. Restrictions Imposed for Normal Input Signal

115

From the table of integrals in Appendix 111, we find that

Just as before, we set m,/l = 1.5 and k = 1 ; then we find from the graph of Fig. 21d, the quantities (l/l)mkl) and (l/l)u~’)for different values of the parameter y = d/2T12. For example, let y = 1. Th en (l/l)mil) m (l/l)@ m 0.8, and h,(mi’), u:’)) = 0.6. T h e functions u2(m, , u,) and u3(mx, u,) are given in Appendix I. In this case a2 = 0.34,

u3 = 0.

We find immediately - [ u : ) ] ~ = 0.01512.

T h e exact value of

ui2) differs

little from its initial value :

u z ) = 0.791.

Similarly, there is little change in the quantity m, . For y = 0.5, the equation mkl)/ubl) = 1.27 is even less satisfactory, although the correction to a, does not exceed 3 yo. 2.4. Restrictions Imposed by the Requirement That the Input Signal of the Nonlinear System be Normal

As shown above, the condition that the input of a nonlinear element have a normally distributed signal was the only limitation which prevented the method of statistical linearization from “turning” into a precise method which would, at least in principle, be as desired. It is of obvious interest to find the practical importance of this restriction. We shall study once more a system with only one, nonlinear, zero dead time element. We shall first consider the important case for which distribution of the external influence Z,(t), imposed at the input of the nonlinear element, can be assumed to be normal. I n this case, the distribution of the output signal of the nonlinear element will obviously not be normal. T h e distribution at the output

116

2. Nonlinear Transformations-Stationary States

of the linear portion will be given essentially by the frequency characteristics of the linear part of the system and by the frequency characteristics of the signal at the input of the nonlinear element (or, roughly speaking, those characteristics which the external disturbance imposes on the input). Usually, the linear part of an automatic control system can be thought of as a filter for low frequencies with a passband over the w w C p ,where wen is some boundary frequency (the range 0 cutoff frequency) above which the amplitude-frequency characteristic begins to fall off sharply.* T h e relations between the passband for the linear part of the system and the band of effective frequencies in the spectrum of the input signal can take on several forms :

< <

1. A Wide-Band Signal. Assume that its highest frequency is far above wcp . I n this case, there will be a normalization of the output signal U ( t ) of the linear part of the system, and, therefore, the signal X ( t ) at the input of the nonlinear element will have normal distribution. We can apply the method of statistical linearization directly to a nonlinear system of this type.

2. A Narrow-Band Signal.

Here, there are three possible cases :

(a) T h e spectral density of the signal differs significantly from zero only for lower frequencies. With respect to such a signal the system as a whole can then be considered to be lagless.+ Provided we find the static characteristics of the closed loop (cf., for example, Voronov [15]), the problem can be reduced to that described in Chapter 1, of finding the probability characteristics of a signal at the output of a lagless, nonlinear transformation. T h e distribution of the signal X ( t ) in this case turns out to differ from a normal distribution. (b) T h e frequency band of the input signal lies above the passband of the linear part of the system. With respect to such a signal, the system turns out to be open. T h e methods developed in Chapter 1 can again be applied to find the transformation in this system.

* A rigorous definition of the passband is given, for example, by Levin [49]; however, here we are interested only in the qualitative term. t This applies to static systems. For an astatic system of the first order, the problem reduces to the study of a nonlinear system of the first order.

2.4. Restrictions Imposed for Normal Input Signal

117

We note that in cases (a) and (b) it is not necessary to postulate that the external disturbance Z ( t ) be normal. (c) T h e frequency band of the external signal ( w 0 - d < w < w0 + A ) lies within the passband of the linear part of the system. Let us investigate case (c) in detail. From the assumption that the spectral density of the random signal has a narrow frequency band, we deduce that it car? be represented in the form (cf., for example, Levin [49]) Z ( t ) = az(t)sin[wot

+ Yz(t)1,

(2.64)

where a,(t) and Y s ( t ) are the amplitude and phase of the random process, which can be represented in the form of slowly changing functions with respect to time, or in the form Z ( t ) = B , ( t ) sin w0t + C , ( t ) cos wot,

(2.65)

where B,(t) and C,(t) are also slowly changing functions of time. If the process is normal and if the spectrum not only has a narrow band, but also is symmetric relative to the central frequency w o , then B,(t) and C,(t) are mutually uncorrelated normal processes with the same spectral densities (correlation functions). One can show that SBb)

=

SCb) =

- wo),

(2.66)

that is, that the spectral density of the “amplitudes” B, and C, is concentrated in a narrow band near the point w = 0, and, thus, we see that B, and C, are slowly varying functions of time. Consider the nonlinear system

O(p)Y

+ R(p)X = S(p)%, y =f(X),

(2.67)

where f ( X ) is an odd function of X ; we shall try to find a solution for this system in the form X

=

B,(t) sin wot

+ C,(t) cos wot,

(2.W

where B,(t) and Cz(t)are slowly varying functions. This is permissible

118

2. Nonlinear Transformations-Stationary

States

if the linear part of the system satisfies additional restrictions of the form Q[i(wo + w)l = K[j(w,

R [ j ( w o.tw)l

+ w)] > K[jn(w,+ w ) ] ,

(2.69)

where w

Ikl
=f ( x ) m q

+ CZ2)[ B , sin w,t

(.\/BZ2

i- C,

cos wet],

(2.70)

where

is the coefficient of harmonic linearization. T h e substitution of the expressions (2.65), (2.68) and (2.69) into Eq. (2.67) gives

+

[Q(p)q(2/Bz2 Cs")

+ R ( p ) ] [ B ,sin + C , cos wot

=

wot]

S ( p ) [ B ,sin w,t

+ C, cos wet].

(2.71)

We now make use of the combination theorem for the differential operator and assert that for a linear operator D ( p ) of the form

2.4. Restrictions Imposed for Normal Input Signal

119

we have the following equations* : D ( ~ > [ 4 sin t) ~

0 2 = 1

sin wOtD1(~)[4t)l

+ cos WD2@)[4t)l9

(2.72)

D(p)[.(t) cos wotl = cos ~ , t ~ l ( P ) [ ~ ( t-) lsin w,to2(P)[wl,

where

N P ) = Re J q P +jwo), D2(P> = Im D(P +jwo).

T h e application of the combination theorem makes it possible to reduce Eq. (2.71) to the form F ( t ) sin wot

+ G ( t )cos w,t

(2.73)

= 0,

where F ( t ) and G(t) are given by slowly varying functions of time of the following form :

From the assumption that C, and B, are slowly varying functions it follows that Eq. (2.73) is satisfied only if F=0,

(2.75)

G=0.

T h e resulting system of equations can be simplified considerably. Actually, the variation with respect to time of the desired processes B,(t) and CJt) is determined by frequencies which are of the order d w,. Therefore, all the operators of the form D(p j w , ) can be linearized close to the frequency w,, according to the rule

<

+

D(P

+jw,)

E D(jwo)

+ ~D'o'w,),

(2.76)

* The combination theorem is not difficult to deduce from Leibnitz's classical theorem for the differentiation of the product of two functions. One can also show that Eq. (2.72) still holds if D ( p ) contains two polynomials in the variable p .

120

2. Nonlinear Transformations-Stationary

States

and, consequently, (2.77)

I Re D(jwo) I > w I Re D’(jwo)I, I Im D ( j w ) I > w 1 Im D’(jwo)1,

(2.78)

then Eqs. (2.75) determine the lagless, nonlinear transformation of the functions B, and C, into B, and C, . T h e conditions (2.78) are always fulfilled if the operators Q ( p ) and R ( p ) do not contain factors of the type p or of the type p 2 + 25w0p wo2where 5 41, and if the bandwidth of frequencies A is sufficiently narrow. As an example, consider the very simple system

+

f(X)

+ k1 (0+ 1)X

=

2.

In this case,

J u P ) = 1 (0+ 11, el@)= Sl(P) = 1,

RAP)

=

QdP)

=

1

Two

S4P)

1

= 0.

In order that the filtering condition be satisfied, it is necessary that the quantity Two be of the order of unity. If the bandwidth of the wo , then it is obvious that, input signal has the property that A with respect to the processes B, and C,, the operator R,(p) can be thought of as lagless. We note that if conditions (2.78) are satisfied, there is no need to find Eqs. (2.75). I n this case it is simpler to write the equivalent equations for the amplitude and the phase of the process X ( t ) in the form

<

(2.79) (2.80)

where

2.4. Restrictions Imposed for Normal Input Signal

121

T h e implicit equation (2.79) can be solved for a, and the probability characteristics of a and Y can be found by the rules developed in Section 1.1. If the condition (2.78) is not satisfied, then one can by approximation linearize directly the linear functions B,q ( d B z z + Czz) and C,q d(Z?l.z CZ2),which usually are smoother than the initial nonlinearities. I n some cases the processes B,(t) and C Z ( t )may be rather wide banded in a system defined by (2.75). In particular, this may occur in many problems of radio physics where the linear part consists of high-Q resonant networks. There, the conditions for the applicability of the method of statistical linearization in the system (2.75) are satisfied. I n conclusion, it can be said that statistical linearization together with the well-known methods of harmonic linearization gives effective solutions for a wide variety of dynamic problems for nonlinear systems which are under the influence of stationary, normal disturbances. Far more complicated is “the critical” case in which the frequency band of the external disturbance coincides with, or is perhaps somewhat narrower than, the frequency band of the linear system. T h e methods described above will, in general, give doubtful results-although for some of the simpler systems there is fairly good agreement between the experimental results and the estimated variance. ~~

+

T h e problem is considerably more complicated when the distribution of the external disturbance Z ( t ) is not normal. Below, we shall describe a modified method of statistical linearization [28] which contains a solution to this problem and also takes into consideration these differences. T h e basic idea of statistical linearization in the analysis of closed systems, as has been shown above, is to look for the distribution of the signals in a predetermined functional form, at the input and output of the nonlinear element. T h e undetermined parameters can be found easily from implicit finite equations. If the distribution of the external disturbance is not normal, then there is no basis for seeking a solution in the form of a normally distributed signal. However, it was shown in Section 1.2 that any differential form of distribution zcl(x) can be represented in the form of a series (1.128). If we restrict ourselves to the first three terms of this series and if these describe w l ( x ) adequately, the distribution can be found from four fundamental parameters: the mean value rn, , the mean-square error u z , the coefficient of asymmetry u s = , and the coefficient of excess yz .

122

2. Nonlinear Transformations-Stationary

States

Then the best linear approximation (without lag) for the nonlinear function f ( X ) can be written in the form

Y

=

m,

+ h,X',

(2.8 I )

where the coefficients for the approximation are

mu = M { f ( - V ) ,

h

=

1

7 M{f(X)X"J,

which in this case are functions of the parameters m, , uz , pLSZand y z (they are linearly dependent on the latter two). It is not difficult to see that the expressions for the coefficients h, and h, can be written in the following form:

mZh0= a,

P3 +-

d3!

a3

Y +

d41 ' (2.82)

where a n ( m z ,u,) are the coefficients introduced in Section 1 . 1 [cf. (l.24)]. The functions a,(m,, u,) were already used earlier. For typical nonlinearities these are described in Appendix I. After linearizing the nonlinear equation

(2.83) (2.84) Equation (2.84) enables us to write the following equations which are derived from the equations given in Section 1.2 for a linear transformation with dead time: 1

"'x

(2.85)

2.5. Synthesis of Linear Compensation Networks

123

Equations (2.85) together with (2.83) constitute a system of nonlinear dependencies which determine the unknown parameters m,, uz , p S z , pdz (or y,), and which can be solved most conveniently by the method of successive approximations. For the initial approximation we assume a solution according to the usual method of statistical linearization, that is, we let p 3 , = y z = 0. The most complicated and laborious step of this procedure is the calculation of the moments from Eq. (2.85). If @,(p) is the least bit complicated, this computation becomes extremely involved. Therefore, the method is applicable in practice only over a limited range.*

2.5. The Synthesis of Linear Compensation Networks in Closed-Loop Systems with Nonlinearities

Let the input signal Z ( t ) be represented as the sum of two stationary random signals : Z(t) =

W )+ N ( t ) ,

(2.86)

where S ( t ) is the signal reproduced as accurately as possible, N ( t ) is the interference, and m, = m,9 = m,n = 0. T h e problem of constructing a system which is optimal in the sense that there is minimum mean-square error in the reproduction of the signal was studied in Sections 1.3 and 1.4. There, it was shown that, for a normal distribution of Z ( t ) , the optimal system would be linear. A number of widely read books [65, 80, 51,] have investigated, in detail, the synthesis problem for optimal linear systems, and have proposed quite simple and effective methods for its solution. However, the practical application of the theory of linear synthesis to the construction of automatic systems is complicated by the fact that the actual systems often have pronounced nonlinear properties. I n some cases, these difficulties can be avoided quite easily. Let us study the system shown in Fig. 29(a). We assume that the characteristic of the nonlinear element f ( X ) and the transfer function of the linear dynamic part K,(p) are given. T h e problem is to make the optimal choice of the linear compensation

* Obviously, it makes no sens3 to try to use this method for improving the statistical linearization if Z ( t ) is normally distributed. In fact in this case Eqs. (2.85)automatically give ps = y = 0 since the transformation (2.84) is linear.

124

2. Nonlinear Transformations-Stationary

States

FIGURE 29

circuit K,(p) which can be included in the forward loop following the part of the system which is given. We assume that the distribution of the signal and noise is normal (just as in all the problems that we shall study further on). Then the optimal transformation which the system as a whole must carry out is a linear transformation with a transfer function F(p) which is given in terms of the spectral characteristics of the signal and of the noise. Once F ( p ) has been determined, it is easy to find the spectral density of the signal X at the input of the nonlinear element from the equation Sx(w) = I K,( j w ) [ l - F(jw)] 12&(w).

(2.87)

T h e spectral density S,,(w) of the signal at the output of the nonlinear transformation f ( X ) can be computed from the equations given in Sections 1.1 and I .2 T h e frequency characteristic, and, therefore, the transfer function of the desired compensating network, is not given by the equation (2.88)

T h e nonlinear system constructed in this way will obviously be equivalent to an optimal linear system from the point of view of the level of the mean-square error. Of course, the above reasoning assumes implicitly that in the constructed nonlinear system the distribution of the signal at the input of the nonlinearityf(X) is normal.

2.5. Synthesis of Linear Compensation Networks

125

In the practical computation of the spectral density S Z 2 ( w )at the output of the nonlinear element it is expedient to restrict oneself to the first term of the series (1.26) which gives an expansion of the correlation function of the output signal in powers of the correlation function of the input signal, that is, to restrict oneself to the term which is obtained by statistical linearization according to the criterion of the minimum mean-square deviation. T he n (2.89) S,,(w) = hl.(a,Kqw), where

T h e method described above can be applied practically unchanged; for the system under discussion, namely, the block diagram which is shown in Fig. 29(b), it is applicable if, as before, we are looking for the transfer function K,(p) of the compensating network which can be placed in the forward path after the nonlinear element. T h e solution is more complicated if the compensating network can be placed only at the input of the given nonlinear element. Going back to the block diagram of Fig. 29(a), we shall now assume that the operator K,(p) is given and that we are looking for the operator K,(p). I n this case, we first find the optimal transfer function F(p) for the system as a whole, and then compute the spectral density of the signal XI at the input of the network we are looking for, K,(p), SZl(w) =

I 1 - F(jw) I2S,(w),

as well as the spectral density of the signal

nonlinearity f ( X ) , which is given,

(2.90)

X , at the output of the (2.91)

T h e problem will be the same as the previous one if we can compute the spectral density S,(W) at the input of the nonlinearity in terms of S,,(W). Naturally no difficulties will arise when the inverse transformation f - l ( X , ) is single valued, for example, when X , = f ( X ) = X3, and, therefore, X = f - l ( X , ) = Xi'3. However, automatic systems often have nonlinear characteristics with portions

126

2. Nonlinear Transformations-Stationary

States

that are parallel to the axis of the abscissa where an inversion operation cannot be carried out. T o a certain extent, the approximation method of statistical linearization provides a way of avoiding this dilemma. In fact, we replace the nonlinear characteristic f ( X ) by a linear one with a transfer constant h,(u,). Then uz [and, therefore, the value of the coefficient h,(u,)] can be found from the implicit equation

or

where the constant A can be calculated once the transfer function F(p) of the system has been defined. We rewrite Eq. (2.92) in the form

(2.93) Nonlinear elements which have the property that the graph of the dependence of h, on l/u, is limited, can be represented by a curve which smoothly increases from the point zero where l/u, = 0, that is, where uJ = 00. I n particular, for a typieal element which is linear over a limited range, we have* (Appendix I) h,(o,)

=

1

LxI

2@ -

(2.94)

T h e graph of the dependence of h, on l/u, is shown in Fig. 30. A graphical solution of Eq. (2.93) leads to the determination of the points of intersection of the line A/l, Z/u, with the curve hl(Z/uz). It is obvious that this solution is impossible if

A11 > dTF, that is, if the level of the upper limit 1 is too low.

* For simplicity,

we assume in Eq. (2.94) that d = 1.

(2.95)

2.5. Synthesis of Linear Compensation Networks

127

h

0.96

0.!

2 /

.7-

U X

I

T

X

FIGURE 30

Besides, even if the point of intersection can be found, such a solution often will be unsatisfactory if the value of the transfer constant is too small. I n fact, the determination of the transfer function K,*(p) = h,k,(p) for the linear element, which is equivalent to the series connection of the desired network K,(p) and the given nonlinearity f ( X ) , follows from the solution of the problem of optimizing the system as a whole : (2.96) Therefore, a decrease in the coefficient h, makes it necessary to increase the static transfer constant for the circuit K,(p). However, in practice, there is an upper limit to this gain coefficient. Hence, the inequality (2.95) which determines the maximum limiting level for the nonlinear element, and which, in turn, is subject to optimization for a given construction, must be made considerably more restrictive. T h e lack of a solution to this problem in the presence of severe restrictions can also be explained by a simple physical argument. If a nonlinearity has some characteristic which is bounded, then the maximum power which can be realized (the variance) from the output

128

2. Nonlinear Transformations-Stationary

States

signal will be bounded. In exactly the same way, the power of the signal at the output of the system is bounded because the linear element K,(p) is fixed. Therefore, if the power of the signal exceeds the level of this bound, then it cannot be reproduced exactly, even in the absence of interference. I t should not be inferred from the preceding discussion that it is often impossible to find a practical and convenient way to determine the compensating network when the system has a bounded nonlinearity. Actually, it is possible to find the compensating networks. However, in this particular case it is necessary to reduce somewhat the restriction on the system as a whole in a qualitative sense, and to take into account in the construction of the optimal transfer function the presence of a bound in the given part of the system. We shall formulate the mathematical problem of optimization taking these bounds into consideration. Consider the block diagram of Fig. 31. Assume that the operator

FIGURE3 1

K,(p) is given, and that the operator K,(p) must be found, so that U$

while

= M { ( Y - S ) 2 }= min,

(2.97) (2.98)

u:, = A2 = const.

T h e interference N , is present at the input of the system and the interference N , is present at the input of the given operator K,(p). On the basis of the block diagram (Fig. 31), we write Y

=

HK,(S

X,

=

H(S

+ N1) + Kz(1

-

+ N,) - HKZN,,

HKz)NZ,

(2.99)

where we are using the notation (2.100)

2.5. Synthesis of Linear Compensation Networks

129

Henceforth, we shall be looking for the operator H ( p ) which is linearly inserted into the expressions (2.99) for Y and X , . After finding H , it is not difficult to find the operator K,* from the equation (2.101) From (2.99) we can write in an explicit form the function for the error variance, E

=

Y

-

s = (HK,

-

+ HK2N,.

1)(S - K 2 N 2 )

(2.102)

For the signal X , on the operator H , where, in order to simplify the calculations, we assume that all the signals S , N , and N , are statistically independent, we can write

(2.103)

u:, =

& j",I H ( j w )

Iz[S,(w)

+ I K,(jw) I2Sn2+ Snl(w)]d w .

(2.104)

It is well known that the problem of minimizing uC2on condition retain some fixed value is equivalent to the problem of minimizthat CJ:, ing the functional 7'

j"

j w1 )1 2 2x -* {[I ~ ( j w ) ~ ~ (-

= uc $- h U2Z 2 = -

+ A I W j w ) l213[s,(w)+ I K&) 12ss,2(w)l + I H ( j w ) 12"1 K 2 ( j w )l2 + AI&dw)} dw,

where h is some constant (Lagrangian multiplier). By making several transformations we find I H(jw)K2(jw)- 1 =

l2

+ h I H ( j w ) l2

[H(jw)K,(jw) - l ] [ H ( - j w ) K , ( - j w ) A I H ( j w ) l2

=

-

I H ( j w ) I2[l K2(jw) l 2

- H( -jw)Kp( -jw)

+ 1.

11

+ A1

-

H(jw)K2(jw)

130

2. Nonlinear Transformations-Stationary

States

We introduce the notation

where K,(jw), and C ( j w ) have zeroes and poles only in the upper half of the complex plane of the variable w . Finally, we obtain

If the operator H ( p ) is optimal (if T is minimized), then the first variation in T for an arbitrarily small change in H ( p ) must be equal to zero. Substituting into (2.106) in place of H ( p ) the expression

and separating out the small terms of first order, we find the first differential :

Since we are looking for an operator which corresponds to a stable ) be in the left half-plane system, all the poles of H ( p ) and ~ ( pmust in the variable p and, hence, all the poles of H(jw) and q ( j w ) must be in the upper half-plane in the variable w .

2.5. Synthesis of Linear Compensation Networks

131

We divide the integral at the right in (2.107) into two parts so that each of the integrands is a complex conjugate :

1 ” + 2v(-jw)[H(jw) 1 K,(jw)C(jw)1s - K,(-jw)B(w)] dw. rr 1-“

(2.108)

T h e variation in 6T will be zero if the poles of the fractional rational function v( -jw)[HO’w) I K,Cjw)C(,jw) 1’ - Kd-iw)B(w)l are all in the same half-plane,* which is equivalent to the restriction that the function Q ( j w ) = H ( j w ) I K ~ ( j w ) C o ’ w l2) - K,(-jw)B(w)

have no poles in the upper half-plane in the variable We express Q ( j w ) in the form

=

(2.109)

w.

KJ.(-jw)C( -jw)[KA(jw)C(iw)H() - L+cjw) - L-uw)],

where L+(jw) has poles only in the upper half-plane and L-(jw) has poles only in the lower half-plane, Then, in order that Q u w ) have the indicated properties and, consequently, that the variation in 6T be zero, it is sufficient to set the following condition : K l ( j w ) C ( j w ) H ( j w )- L+o’w) = 0.

(2.110)

Therefore, the optimal operator is given by the relation (2.111)

and, hence, K,*(p) can be found from Eq. (2.101).

* Here, it can be assumed that as 1 w 1 -+ w the given function will converge to zero at least as rapidly as I w and, consequently, the integral along the semicircle of infinite radius in the upper half-plane will be equal to zero.

132

2. Nonlinear Transformations-Stationary

States

T h e parameters of the operator H ( p ) and, therefore, of K,*(p) depend on the value of the quantity h which must be determined by inserting H ( p ) in the condition (2.98). From the table of integrals in Appendix I11 we obtain for the latter condition an implicit equation of the form (2.112)

oZ2(h) = A2,

which for a fixed A can always be solved graphically. Operator H ( p ) is completely determined in the same way. This makes it possible to find the minimum error variance for a given level of the bound A. It is obvious that for some A we will have h = 0 and that the error variance will reach its lowest level which is the solution of the problem in the absence of any bounds. However, this value of A may be infinitely large. Let us apply these results to the solution of the original problem of synthesizing compensating networks in systems which contain nonlinearities with bounds. A straightforward application is evidently impossible because in the scheme described above a bound is imposed on the variance while in the initial problem the variable X 2 ( t )is bounded. We make use of the following approximation technique, which is based on the idea of statistical linearization. We construct the operator K,*(p) for . an arbitrary value of A 2 of the bound of the variance We shall assume that the effect of the operator K,*(p) on the signal X , ( t ) is approximately equivalent (in the sense of the method of statistical linearization) to the sequential application of the unknown operator for the compensating network K,(p) and of the given nonlinear transformation f ( X ) [Fig. 321. Then

4,

YGHTpqqlKl*(P) = h,(%)Kl(P).

(2.113)

FIGURE32

If the technical conditions impose a maximum value on the static gain coefficient for the compensating network K,(O) which is equal

2.5. Synthesis of Linear Compensation Networks

133

to k,,,, then the transfer constant of the nonlinear element may be written in the form (2.1 14) which relates h,(u,) to the chosen level for A. On the other hand, the following equation will hold: uzz = u , ~ ~ ~ ( u= , )A’ or A h,(U,) = - . (2.115) (JX

T h e simultaneous solution of Eqs. (2.1 14) and (2.115) makes it possible to find the maximum value of A on the condition that Ki(0)

< hmax

(2.116)

where the nonlinearity is bounded. Identically, one can immediately find the parameters for the operator K,(p) and also the minimum value of the mean-square error u , . This method leads to rather cumbersome calculations because one must find Kl*(0) as a function of the parameter A , and solve Eqs. (2.114) and (2.1 15) graphically. However, one can often use a simpler way. I n particular, for a typical element which is linear over a limited region (cf. Fig. 30) it is sufficient to choose a quotient somewhat smaller than the limiting value = 0.8. For example, the value A / l = 0.7 practically guarantees that Kl(0) will satisfy condition (2.1 16) if, of course, the quantity k,,,, has a reasonable value. At the same time a variation in the quantity A in the range from 0.71 to 0.81 can hardly have a significant effect on the quality of the system, that is, on the level of the error variance. But if a definite A is chosen, the solution of Eq. (2.112) g‘ives a value of h which, in turn, completely determines the operator K,(p) which we are looking for.

a

Example 1. Consider the synthesis problem of finding the compensating network K,(p) for the block diagram of Fig. 32, where 1

&(PI

=

p”,

f(W = 1,

=x, - -1,

x > 1, (XI
134

2. Nonlinear Transformations-Stationary States

There is no interference N , at the input of the system. T h e spectral densities of the signal S and of the interference N , are equal, respectively, to 2or S,(W)= 7 ,a : S , ~ ( W= ) d. wz + or2 If the parameters of the signal and of the interference have the values ci = 4 3 6 ' 13, d = 4/36/13, a,2 =

4,

then, as shown by Pelegren [55, p. 811, the optimal transfer function for the system as a whole has the form 0.45

F ( p ) = (P

+ 2 ) ( p + 3) .

We now compute the variance of the signal X, :

Computation based on the formulas of Appendix 111 gives and, hence,

a:.

=

A' A

%

0.097,

%

0.31.

Applying the method of statistical linearization, we find or

u,h,(u,) = 0.31

1

2@(-)

O X

=--*

0.31 1 1

ax

It is possible to find h,(u,) [Fig. 301 if 1 > 0.31

d q = 0.39.

For example, let 1 = 0.62. Then, h, operator K,(p) is given by the equation

=

0.96, and the desired

2.5. Synthesis of Linear Compensation Networks

135

Let us now look at the following example, which deals with the case in which the presence of a bound, generally speaking, makes it impossible to realize the optimal transfer function which is found by disregarding the effect of this bound.

+

Example 2. Let K,(p)[l/(T, l)] be a nonlinearity of the same form as in the previous example, where there is no internal interference N , and where the spectral densities of the signal S and of the external interference N l are given by 2a

S,b) = -&-5uA

STIkJJ)= d .

As we shall see in this case, it is impossible, in general, to realize the transfer function, which is found regardless of the effect of the bound. We find the solution by determining the optimal operator

which takes into consideration the bound imposed on the variance of the signal X , at the output of the nonlinearity ur2 =

A.

T h e optimal operator H ( p ) is given by Eq. (2.111). We shall find expressions for the functions K A ( p ) ,C ( p ) , L+(p). which appear in this equation. I n the present example

and, hence, K,(jw) =

Moreover,

+ d/1-tx +1

T dijw Tjw

136

2. Nonlinear Transformations-Stationary

States

and, consequently, C(jw) =

dc-i-j w + B JW

+

OL

where we use the notation

We can find in a like manner the expression L ( w ) = K2(-jw)+) Kn( -jw)C( - j w ) ’

where B(w) = S,(w). After substituting for the functions which appear in the expression for L(w), we find 2cia,2

L(w) = d/do’W

+ ci)(-jw + ,B)(-T d i j w + d-+j

’

Expanding L(w) into common fractions, we can separate out the termL+(w), which has poles in the upper half-plane of w : L+(W) =

2aa,2

d d ( a + B ) ( ~ di+ T diTij j w

1

+

‘

Moreover, putting all these expressions into Eq. (2.11 l), we find, finally, H(P) =

Tp

2or0,2

+1

4.. + B)(aT dh+ d/r+) ( p + B)(T d/hp + d m )’

and, hence,

Further calculations are made for specific values of the parameters : 2aiZ= 7,

ci =

3,

d

=

Graphs of the functions uz,(h) and

1,

T

uC2(X) are

=

1,

p

= 4.

shown in Fig. 33.

2.5. Synthesis of Linear Compensation Networks

137

FIGURE 33

Let the boundary level for the nonlinearity 1 be equal to 1.5. Then, following the simplified method developed above we obtain A

= ox2 = 0.71 =

1.05.

From the graph in Fig. 33, we ascertain the corresponding value of h to be equal to 0.27 and the mean-square error uE M 1.65. T h e graph in Fig. 30 for All = 0.7 shows that h,(u,) is equal to 0.65. Using these data, we obtain, finally,

We,also note that in this problem A = co when h = 0; therefore, it is impossible to construct the optimal circuit unless the bound restriction is considered in advance, regardless of how weak this restriction.

2. Nonlinear Transformations- Stationary States

138

2.6. Application of the Theory of Markov Processes in the Study of Some Nonlinear Systems

We shall describe one class of nonlinear problems for which, in principle, it is possible to obtain exact solutions. Suppose that the equations for the nonlinear system can be written in the form dX, dt

-= f i ( X l , ..., X , , i1, ..., 5,)

(i = 1, 2, ..., n),

(2.117)

where fi are nonlinear functions and ti are random functions of time, while the quantities ci(tl) and ci(t2),where t , and t , are arbitrary moments of time, are statistically independent. Without loss of generality we assume that M{ci} = 0. We assume that the cross-correlation functions for the processes and are given by the expressions

ci

ci

T h e processes for the variation in X i ( t ) are Markov processes.* A precise definition of a Markov process is as follows. “Let t - A,
+

+

+

* An elementary introduction to the theory of Markov processes is given in Appendix V.

139

2.6. Application of the Theory of Markov Processes

then we obtain information only about the values of the functions li in the interval of time ( t - d,, t ) . However, from our original assumption of the character of ti,such information does not change the probable values of li in the following interval ( t , t d), and, hence, does not change the probability distribution of the variation in Xiover this interval of time. If we make the additional assumption that X i ( t )varies continuously, then one can use Kolmogorov's equation* (Appendix V) to find the differential function for the probability distribution w ( X , , ..., X ,, t ) . This equation can be written in the form

+

aw

-- =

-

at

" a

-[ A & ,

8%

...,x, , t ) w ]

where the coefficients A and B are defined in terms of the transition d) in the following probabilities p ( x l , ..., x,,, , t ; y 1 , ..., yq L, t manner :

+

1

A i = lim -M{Yi A-o A

-

1 24

Bij = lim -M{( Yi- X i ) ( Y j- X , ) } A+O

=

lim -J (yi - xi)(yj - x , ) p ( x l , ...,x, , t ; A - o 24 -= ~ 1 ...,yn , > t 1

-

+A )

4

1

9

...*dyn. (2.120)

We shall show how to find the coefficients Ai and Bij for this type of Markov process in a special case which is frequently encountered, that is, when f i ( X , , ..., xn;5 1 , ..., 5,)

* Equation

= fa"X,

, ..., X,)

+

(2.1 19) is often called the Fokker-Planck equation.

ti

.

(2.121)

140

2. Nonlinear Transformations-Stationary

States

For small A , we have

Xi

= Afp

M{Yi - Xi}

= Afro

Yi

-

+ Jt

t+A

tidt,

and, hence, and, consequently,

A i = f:. From (2.118) we also obtain

and

/:,d 1

t+A

B '' ..bij - lim A+O 24

8(t,

t

-

t s ) dt, dt,

bij 2 '

=-

Therefore, the basic equation for this case, Eq. (2.119), takes the following form :

For the probability distributions in the steady state, we obtain (2.123)

T h e integration of Eq. (2.123), and, particularly, of Eq. (2.122), is a very complicated mathematical problem; in general, it is not solved. However, for several special cases which are of definite practical interest, the integration can be carried out. Consider, for example, the equation of the first order dX

dt = f " X )

+5

(2.124)

and let us try to find the steady-state probability distribution for the process X ( t ) .

W(X)

2.6. Application of the Theory of Markov Processes

141

Equation (2.123) takes the form

or

d - [f"X)W] dx

f"x)w

b d2w

(2.125)

=--

2 dx2

b dw

- - - = c,

(2.126)

2 dx

where C = const. We assume that when 1 x I -+ co w ( x ) and dw!dx decrease at the same rate at which the left-hand side of Eq. (2.126) approaches zero. Then C = 0 and, consequently,

and, hence, (2.127)

where the constant C , is determined by the normalization condition

$-,

oc

In the case of a stable linear system,fO(X) for w ( x ) takes the form W(X)

=

(2.128)

W ( X )dx = 1. =

- k X , and the equation

1 3

C, exp - - x2 ,

(2.129)

that is, the probability distribution is normal (Gaussian). Since the distribution of C i , generally speaking, is not normal, one can see in this result a manifestation of the general rule that a signal will be normalized by a linear filter. We shall proceed to study a concrete example of a very simple nonlinear closed system by using Kolmogorov's equations. The block diagram for the system is given in Fig. 20. The differential equation which corresponds to this system is -T2 + + X1+ f ( X ) = Z , k

(2.130)

142

2. Nonlinear Transformations-Stationary

States

where we are assuming that R,(T) = d 8 ( ~ ) . Equation (2.130) can be reduced to the standard form (2.117). I n this case

T h e steady-state distribution is given by Eq. (2.127) :

1

w(x) = C , exp -

where

L x a

b T [2k

+

1 11 , f(x) dx

(2.131)

k2

b =-d.

Let f ( x )

= x

+ ax3. Then

T2

(2.132)

We shall compute the constant C, from the normalization condition :

For small nonlinear distortions in the region of the significant values of X, the integrand can be transformed with the help of the approximate equation ka

exp)--xx" 1 2bT

!

w 1

ka

--x4.

2b T

The n we can find an approximate value of the constant C, from

where bT -Y = 2(k

+ 1) .

2.6. Application of the Theory of Markov Processes

143

With the aid of all these assumptions in computing the meansquare value of oZ, we find ' , a

=y

(1 - 67'

kci F ).

(2.133)

One can show that with an accuracy up to terms of the order of a3, this result is identical with the result obtained for an analogous system by the method of statistical linearization (Section 2.2, Example 1). It is of special interest for several problems in the theory of automatic systems to find the probability characteristics at the time when the system reaches some definite boundary value. If the variation of the coordinates of the system is represented by a Markov process, then one can use the differential equations developed by L. S. Pontryagin for the solution of this problem (cf. Appendix V). For a first-order differential equation which will determine the mathematical expectation m ( x ) at the time when the point x will first arrive at one of the boundaries of the interval x1 x x2, we have

< <

d2m dm b -+ f " x ) dx2 dx

+ 1 = 0,

(2.134)

with the obvious boundary conditions m(x,) = 0 .

m(xl) = 0 ,

T h e solution of Eq. (2.134) is easy to find by squaring. We specify P

=

dm x

Then for the variable p we have a linear equation of the first order. Its solution is well known :

where ~ ( x = ) J f o ( x )d x . Hence,

144

2. Nonlinear Transformations-Stationary States

Making use of the boundary conditions, we find the final equation

where

Let us consider the application of this method for solving a problem which is of particular practical interest. The operation of a radar tracking system is very roughly described by the block diagram of Fig. 34.

FIGURE 34 The form of the nonlinear dependence -f (.X ) which remesents the characteristic of the sensing element is such that for some definite value of the input signal I X I > I, the output signal will be zero and there will be “target loss.” It is of interest to find the expected value of the length of the time interval during which normal tracking is possible, that is, the amount of time before 66 target loss.” Let the control signal be a linear function of time, zo = at, and let the random interference be noise in the amplification channel, which one can assume to be white noise with relatively little effectiveness; then the interval of time before “target loss’ is much greater than the time it takes to establish the dynamic process from the initial conditions at the time of target acquisition. Thus, one can assume that the quantity at the input of the nonlinear element (the error) is composed of a constant component xo = aT and a random component which represents the presence of interference. Therefore, the problem of finding the mean time of operation before “target loss” reduces to the problem of finding the expectation of the time it will take coordinate X (which has an initial value equal to a T ) to reach the boundary of the interval - 1 X 1. Writing the equation for the system in the form (2.1 I7), we obtain I

< <

f“X)

= -

1

-ff(X), T

5

1

=T

z.

2.6. Application of the Theory of Markov Processes

145

Let R,(T) = & ( T ) . Then b = ( 1 / T 2 ) d .The boundaries of the interval in this case are such that x, = -1, x2 = 1. In order to find the relation m ( x ) = m(aT ), one can apply Eq. (2.135) directly.

Chapter 4, in the study of extremal systems, gives a number of examples of the application of the theory of Markov processes.

Chapter 3

0

NONLINEAR TRAN SFO RMATIO N S WITH FEEDBACK NONSTATIONARY STATES

3.1. The Transformation of a Slowly Changing Signal in the Presence of a High-Frequency Random Interference

Nonlinear transformations of a slowly changing signal side by side with the simultaneous transformation of a high-frequency random interference are of interest in many technical applications. Let us look again at Eq. (2.2) for a transformation with feedback which includes a single nonlinearity with zero dead time. Let the input signal Z ( t ) be the sum* of a high-frequency unbiased, normal, stationary process Z o ( t ) and of some given, slowly changing function of time mz(t), which can be considered the mathematical expectation of the nonstationary process Z ( t ) . Separating out of (2.2) the equations for the expectation and the random component, we obtain

and

+

Q(p)Y0 R ( p ) X 0 = s(p)zo, Y" = f ( X ) - 112,.

We shall look for a solution in the form X

= mx

+ Xu,

* T h e method developed here can be generalized easily for the case when 2 is the result of an arbitrary transformation without feedback of two signals, one of which is a high-frequency, random signal and the other some definite, slowly changing signal.

146

3.1. The Transformation of a Slowly Changing Signal

147

where m, is a slowly changing function of time and X o is an unbiased, normal, random process which, in general, is nonstationary. We shall assume, however, that its correlation function R,(t, T ) is slowly changing with t , that is, aR, -
aR,

(3.3)

aT

T h e spectral density S,(W, t ) , which is given by the equation S,(w, t ) =

/

m --9

R,(t,

7 ) e-iwr

d7,

(3.4)

also is assumed to be a slowly changing function of time, just as is the mean-square deviation o,(t). We apply the method of statistical linearization to the nonlinear function f ( X ) , Y = f ( X ) m my

+ hlXo,

(3.5)

where the coefficient hl

= h l h ! 9 %)

also is a slowly changing function of time. Equation (3.2) for the random components takes the form

T h e resulting equation, which is a linear differential equation with coefficients which are functions of time, can be solved by the method of “freezing,” that is, one can look for the solution as if h, were a constant parameter. Th en (2.8) can be written in the form where

(3.7)

is the transfer function for the random component. We find the mean-square deviation ( T ~from the usual equation of a linear transformation :

148

3. Nonlinear Transformations-Nonstationary

States

From a table of integrals, we obtain the algebraic relation UZ2

=

w%! , m,)l.

From the graphical solution, we find the explicit relation

By using this function, we can write the equation for the slowly varying components in the form

Equation (3.10) is a nonlinear equation which contains only definite functions of time and which defines the transformation for the slowly changing component of the input signal. It differs from the original equation (2.2) only in that, instead of the nonlinear functionf(X), we have now the “reduced” nonlinearity F(m,) [according to the terminology used in [64] this is a composite function]. If the frequency characteristic of the linear transformation

satisfies the condition of complete filtering of (2.16), then the equation for the slowly changing component can be written in a much simpler form. I n fact, the random component of the signal X ( t ) then will be stationary, while the mean-square value will be given by Eq. (2.17), that is, it will not depend on m, and, therefore, W

Z

)

=

m,(mz

I

u,).

(3.11)

The characteristic of the ‘Lreduced’’linearity turns out to be identical with the functional dependence between the mathematical expectation of the signals X and Y , whereas the function uz , in the definition of (2.17), serves as parameter. Let us analyze Eq. (3.10). For this purpose one can use any of the convenient approximation techniques for nonlinear definite functions.

3.1. The Transformation of a Slowly Changing Signal

149

First of all, we see that, as the result of the linearization of the nonlinear transformation of the random component, the nonlinear functions m,(m,) and, therefore, F(m,) always have a sufficiently “smooth” character. For a sufficiently high value of the random component one can write the approximation (3.12)

and m,, is given by the implicit equation

where m,, is the constant component in m,(t). T h e coefficient k obviously depends both on the properties of the system and on the properties of the random component of the input signal (i.e., on the properties of the interference). T h e nonlinear equation (3.10) reduces to the linear form (3.14)

S(P)

= kQ(p)

+ H(p) ’

Am, = m, - m,,

.

Equation (3.14) enables us to compute easily the form of the signal d m , ( t ) for a given d m , ( t ) , and also to solve another problem of practical importance, which can be called the problem of detecting a slow loss of the stability of the original system (2.2) in the presence of a high-frequency random influence. I n fact, for stationary random input signals the solution to Eq. (2.2) is not always a stationary random process. T h e trivial solution ( m , = const) of Eq. (2.7) may turn out to be unstable. If the loss of stability varies slowly with respect to the random external influence, then the method described above can be applied. T h e problem will then be to find the properties of the roots of the characteristic polynomial

150

3. Nonlinear Transformations-Nonstationary

States

which can be solved by any of the well-known techniques, for example, by using frequency criteria. T h e effect of the random components on the transformed properties which most often appear in automatic systems for nonlinearities with bounds is essentially to decrease the real amplification coefficient. On the basis of this fact, one can easily give a qualitative description of the effect of random components (interference) on the stability of closed nonlinear systems. Consider, for example, a static system. Let the amplitude phase characteristic of an open loop have the form shown in Fig. 35(a),

that is, assume a system with an “absolute” stability with respect to the static amplification coefficient. Th en the effect of the interference will lead only to a decrease in the static accuracy of the system. For an astatic system with the same conditions, we find that the interference increases the error with respect to the velocity of the system. If there is no “absolute” stability [Fig. 35(b)], then the initial interference has more serious consequences, insofar as it can bring on a loss of stability in a system which without interference is stable (in its linear approximation). Therefore, servo mechanisms which are to operate under conditions of intense interference are designed to provide “absolute” stability (cf., for example, Besekerskiy et al. [lo]). If the nonlinearity is not an odd function, then there will be the so-called detection effect, that is, there will be a false signal and, consequently, a reduction in the static accuracy. These ideas are illustrated in greater detail in the examples at the end of the chapter.

3.1. The Transformation of a Slowly Changing Signal

151

For a comparatively high signal-to-noise ratio, the approximation (3.12) may be inadequate. In that case, one must apply methods from the nonlinear theory for an investigation of the system given by (3.10). We describe in greater detail the application of the method of harmonic linearization, assuming, for the sake of simplicity, that the characteristic for the nonlinear element f ( X ) is an odd function (mZo= 0) and that the frequency characteristic of the linear part of the system satisfies the restriction (2.16). Then the characteristic of the “reduced” nonlinearity F(mx) = m,(m,)

will also be odd. We shall look for a solution to Eq. (3.10) in the form (3.16)

mx = a sin p,.

We obtain the approximate expression for F(m,) :

where 1

h,(a, u,) = -

=a

or h,(a, 0,) =

/

2n

1/

=a

0

2x

0

mv(u sin v) sin 9: dp,,

sin p, dp,

j

m

-m

(3.18)

)

1 ( x - a sin p,>” f(x) -exp( dx.

d2T

202

(3.19)

I t is not difficult to see that the coefficient h, is the same as the transfer constant for the harmonic signal (1.219) with statistical linearization of the nonlinear function f ( X ) when it is replaced by the optimal linear combination of a normal random signal and a harmonic signal with a random phase. A table for the coefficient h,(a, a,) for various typical nonlinearities is given in Appendix IV. Equation (3.10) can be rewritten in the form (3.20)

152

3. Nonlinear Transformations-Nonstationary States

If one is looking for spontaneous oscillations,* then mz = 0, and mot, where wo is the desired frequency of the oscillation. T o determine the unknown parameters a and wo,we have the conditions =

(3.21) (3.22)

It is obvious that the value of the amplitude of the self-oscillations

and actually the very possibility of realization depend on the form of the coefficient h,(a, uz),and, consequently, on the parameters of the random component of the external disturbance. T h e frequency w o , for the state which is self-oscillatory “in the mean,” is given by Eq. (3.22) and turns out to be independent of the parameters of the external disturbance (in the approximate method considered here); it is the same as the self-oscillation frequency calculated for the case in which there is no external disturbance. This frequency (from the conditions of the applied method) must be considerably lower than the bound of the range of significant frequencies for the external disturbance. We note, also, the order in the design for the system when there is an external harmonic disturbance m, = a, sin wot with a frequency wo which is low by comparison with the frequencies of S,(w). Instead of (3.21) and (3.22), we have two conditions for defining the amplitude a and the phase 4,1 : (3.23) a = I @O’Wo) I a , , 4 = arg@O’Wo), (3.24) where S(P) @(’I = Q(p)Ma, c,) R(p) ’

+

It is obvious that the amplitude and the phase of the forced oscillations

depend on the random component. Analogously, one can apply the approximation methods to study processes of establishing periodic states, insofar as they are frequently based on the hypothesis that the process of establishing the state is slow.

* Of course, such a state can be called self-oscillatory only in a conditional sense. It would be more precise to speak of a state which is self-oscillatory “in the mean.”

3.1. The Transformation of a Slowly Changing Signal

153

When f ( X ) is not an odd function or when it does not satisfy the condition of complete filtering (2.16), the characteristic F(m,), generally speaking, also is not odd. Therefore, one must look for a solution of m,(t) in the form of the sum of a constant and a periodic term [64], ml(t)

=

m,,

+ a sin 9,

while for F(m,) we can use the approximate expression

Fl(ml0 , a) =

&

rF(mlo

+ a sin rp) sin 9’ drp.

T o find the parameters m X o ,a and wo for the case where there are self-oscillations, we use the following equations : Q(O)Fo(m,o

?

4 + R(O)m,o = 0, (3.26)

T h e third condition gives the frequency w o , and the first two give implicit expressions for the constant term and the amplitude of the periodic term. I t is clear that “constancy” and “periodicity” must be interpreted here as constancy and periodicity “in the mean.” We also note that if the condition of complete filtering is not satisfied, the function F(mJ and, consequently, the functions Fo(mz0, a ) and F,(m,, , a ) are determined by the characteristics of the system as a whole and cannot be tabulated in advance for each nonlinear element. However, in practice, the function F(m,) can always be approximated by one of the typical nonlinearities. Considering the approximate nature of the method itself, it makes no sense to pay special attention to the accuracy of this approximation.

3. Nonlinear Transformations-Nonstationary States

154

Example 1. We return to the study of the servo system that was analyzed in Example 3, Section 2.2. Let m,(t) be an arbitrary, slowly varying function of time. Generally speaking, the variance of the signal at the input of the nonlinear element ax2also is a slowly varying function of time. T h e relation between uz and m, is given by Eq. (2.32) where K,,(jw) depends on the transfer constant hdmx 0x1. I n principle, one can find an implicit expression for a,(m,) from 9

(2.32).

We substitute it into the equation for the slowly varying component:

Because the system is astatic, m, = 0 when m, = const. We study the stability in a microscopic region of a solution which is restricted only by the slowly varying forms of the loss of stability. This study is very simple. I n fact, [for odd functionsf(x)] the transfer constant with respect to the random component is equal to

-

-

1 ax3 d 2 n

j

a

-m

[

(x - rn,)f(x) exp -

1 x'2

1( dx

- mX (JX

)'I

dx

+ O(mxX)= hl(ax)+ O(mX2). (3.28)

Thus, in the analysis of stability when small terms beyond the firstorder terms are neglected, the transfer constant related to the random component does not depend on m, and, therefore, the quantity uz determined by (2.32) does not depend on m, .

3.1. The Transformation of a Slowly Changing Signal

155

On the other hand, the transfer constant related to the slowly varying component ho(m, , u,) is equal to hob, -

> ax!

1

f ( x ) exp[-

(

- mx UX

)'I

dx

Hence, the characteristic equation for the system in the presence of interference can be written in the form:

Disregarding the small constant T, , we rewrite the condition for stability (the Routh-Hurwitz criterion) as follows : (3.30)

For a servo system with the parameters of Example 3, Section 2.2, we obtain h, > 0.05. It is not hard to calculate the frequency w0 at which loss of stability can occur. From the characteristic equation, it is obvious that (3.31)

and, hence, wo rn 14 sec-'. Therefore, strictly speaking, the loss of stability under the influence of interference can be considered as real only if the spectral density of the interference is significantly larger than zero for frequencies higher than wo = 14 sec-l.

Example 2. We shall study the state of a system with a nonlinear element which is an ideal relay. T h e block diagram of the system is

156

3. Nonlinear Transformations-Nonstationary States

shown in Fig. 36. I n the absence of an external disturbance there will be self-oscillations. T o find out what they are, we apply the method of harmonic linearization.

FIGURE36

Here, the system of equations has the form

k P(TP

Let X

= a sin mot;

+

Y

+X

= 0,.

Y

=

sgn X .

(3.32)

then y

=d

4X,

where ql(a) = 4 / r a is the harmonic coefficient of linearization for the ideal relay. T h e amplitude a and the frequency w 0 of the self-oscillations are given by the conditions ql(a) = -

1

Re[jwo(Tiwo

+ 1)7,

0 = Im[jwo(Tjwo + 1 ) 7 ,

(3.33)

and, hence, we obtain 1

wo=T'

2kT a=-

7r

Self-oscillations will occur for arbitrary values of the amplification coefficient k. Now impose upon the system a high-frequency external disturbance Z ( t ) [Fig. 361, while for all significant frequencies in S,(W)the condition for complete filtering is satisfied in the linear part of the system. Then a, = a,.

3.1. The Transformation of a Slowly Changing Signal

157

T o determine the amplitude of the harmonic component in the signal x we have the equation (3.21), 2T 2 h,(u, uz) = 7 wo. = kT

.

(3.34)

T h e function h,(a, u,) for the ideal relay is derived in Appendix IV : h,(u, a,)

where

1

= -&(a) U

a=-

[ a dn

=-

a

2a 1 - -

+ -1

,

(3.35)

U

dZ

0,

T h e abscissa of the point of intersection of the curve B,(a) with the line 2 2 d? ah&, u,) = - a = -ur(Y = La (3.36) k 1‘ kT gives the desired value of the parameter a, and, therefore, of the amplitudes a (Fig. 37) for arbitrary values of the combined parameter

FIGURE 37

L. T h e parameter L increases with an increase i n t h e interference level and when L > 214; or when U, > kTjd2.-, the harmonic component vanishes. There is a “break” in the self-oscillations. T h e phenomenon of this “break” in the self-oscillations with random external disturbances has in its very character much in common with the phenomenon of locking under the effect of harmonic disturbances, although, obviously, it has a slightly different form.

158

3. Nonlinear Transformations-Nonstationary

States

3.2. Passage of a Slowly Varying Random Signal through a System in a State with Periodic Oscillations

Let the nonlinear system operate in a state with periodic oscillations. These periodic oscillations can be produced either by an external periodic disturbance (forced oscillations) or by the internal properties of the system (self-oscillations). We shall study a method of finding the dynamic characteristics of such a system when it is under the effect of a signal which is a slowly changing, stationary, normal, random function of time Zo(t). “Slowness” in this connection means that the band of significant frequencies in the spectrum of Z o ( t ) lies considerably below the frequencies of the undisturbed periodic state. For disturbances in the self-oscillatory state, this defines the low-frequency characteristics of the spectrum Z o ( t ) with respect to the passband of the open loop. We shall restrict our investigation to the case in which the system contains only one nonlinear element and is in a state of forced oscillations because of a harmonic signal* : Q(P)Y

+ R(PW = S(p)Z, y

where Z

=

= f(X),

m, + Zo,

m, = a, sin w,t.

(3.37)

(3.38)

We look for a solution of the system (3.37) in the form of a sum of a high-frequency harmonic component XIand a slowly varying, normal component X, :

x = x,+ x,.

If it is assumed that the linear part of the system (3.37) satisfies the well-known conditions for the applicability of the method of harmonic linearization, then XI = a sin(w,t +),

+

* Computations for

a self-oscillatory system are not significantly different [57].

3.2. Passage of a Slowly Varying Random Signal

159

and when we expand Y in a Fourier series, we can limit ourselves to only the first terms : y

where*

= Fob,

.u,) + Q l ( 4X0)Xl

j

1 zn ~ , ( ax,) , = - j ( a sin v ql(a, x,) =

2~ 1

n

J

2n

j ( a sin ‘p

9

(3.39)

+ x,)dv, + x,)sin v dv.

Here, the amplitude a is, in general, a random, slowly varying function of time. Taking the mean with respect to time and assuming that the slowly varying functions are constant over one period, we can separate from the system (3.37) an equation for the periodic components, Q(PIY1

+ W)X, = S ( P h ,

Y,

= q1(a, Xo)X19

(3.40)

and for the slowly varying components,

+

Q ( ~ ) y o W)Xo

=

S(P)ZO,

Yo = Fo(a, Xo)*

(3.41)

T o solve the system (3.40), we use the method of “frozen coefficients.” Then the amplitude a and the phase y5 of the process X , , which are functions of X o ( t ) and, therefore, are slowly changing, random functions of time, are given by the equations (3.42)

where

We shall consider the first equation in (3.42) as an equation which gives the dependence of the amplitude a on the slowly varying component X , : F(a, Xn) = 0. (3.43) *Equations and graphs for the coefficients F,, and q1 are given by Popov and Pal’tov

[a].

160

3 . Nonlinear Transformations-Nonstationary

States

This function can be constructed in an explicit form, for example, by graphical means. Thus, we find a =

a(Xo).

(3.44)

Then, the slowly varying component can be expressed only in terms of the slowly varying component of the input signal, and the system (3.41) reduces to the form

where FO*(XO) = Fo[a(Xo),Xol.

If the condition for complete filtering is satisfied,

the system turns out to be open with respect to the high-frequency component and (3.46)

that is, the amplitude does not depend on X o . Therefore, FO*(XO) = FOWO).

(3.47)

If the condition for filtering is not satisfied, that is, if w 0 lies inside the limits of the passband for the open system, then the equation for the slowly changing component can usually be greatly simplified by neglecting terms of higher order. For a very low-frequency Zo, the system can be regarded as lagless (with respect to ZO); that is, X , will be given by the implicit equation

+

Q(0)Fo*(Xo) R(0)Xo = S(0)Zo.

(3.48)

I n the general case, Eq. (3.45) is an equation for a system with one nonlinear element which has an effectively real (“reduced”) characteristic given by the function Fo*(Xo).Let us look for an ap-

3.2. Passage of a Slowly Varying Random Signal

161

proximate method of solution for these equations. T h e simplest method is by straightforward linearization with a Taylor expansion of the function F,*(X,). Th en it will not be necessary to find the implicity function (3.43). I n fact,

But for an odd, piecewise differentiable functionf(X) one can show that aF,/aa = 0 when X, = 0. Hence,

while in the expression for the derivative obviously a = a , where a, is the amplitude of the periodic signal in the absence of the slowly varying component. T h e system (3.45) reduces to the linear equation where

x,=

@0(p)Z0,

(3.50)

which enables us to compute the mathematical expectation and variance of X , in the usual way. If direct linearization is not admissible (if the amplitude of the high-frequency component a, is comparatively small), then obviously statistical linearization is to be applied. T h e method for solving Eqs. (3.45) is no different from that outlined in Section 2.2 except that here the transfer constants must be constructed not from the real characteristic of the nonlinear element, but from the “reduced” characteristic F,*(X,); in other words, the transfer constant is given by the equation

(3.51) where wl(x) =

- ( ). uxo4%exp - 2“:,

162

3. Nonlinear Transformations-Nonstationary

States

If the filtering condition is satisfied, then (3.52)

or

that is, the transfer constant is the same as the one derived in Section 1.5 for the transfer constant h, for the random component using statistical linearization of a nonlinear transformation of the sum of a random, normal signal and of a harmonic signal with a random phase [cf. (1.219)]. We note that the direct application of the method of statistical linearization to Eq. (3.37) would have been very difficult because the transfer constants would then be rapidly changing functions with respect to time. In problems of this type, it is often of interest to find only the characteristics of the slowly changing component X , . If it is required to give an estimate of the fluctuations in the amplitude of the periodic component, then one can use Eq. (3.44) which can be written roughly in the form

from whence

One can also compute directly act from the nonlinear equation (3.44) if one assumes that X,has a normal distribution, an assumption which was already made in applying the method of statistical linearization. Then

J

-x

3.2. Passage of a Slowly Varying Random Signal

163

We shall give a summary of the material developed in the previous two sections. Essentially, the techniques described depend on the successive analysis of the processes in the two components operating in the system, the low-frequency and the high-frequency component. T h e basic premises of the method do not depend on whether these components are random or definite functions of time. T h e solution is required in the form of the sum of a slowly changing and a high-frequency component. From the equations for the system, we find, by averaging with respect to time (harmonic linearization) or with respect to the set of possible values (statistical linearization), the equations for the slowly varying components. T h e dependence of the coefficients of these components on the high-frequency components of the system hinges only upon the unspecified character of the parameter (the amplitude or the mean-square deviation at the input of the nonlinear element). This parameter, or more precisely, its dependence on the slowly varying component, is found either by statistical or harmonic linearization if one assumes that the slowly varying coefficients in the equations can be considered as “frozen.” T h e substitution of this dependence into the equation for the lowfrequency component makes it possible to analyze independently the transformation process of this component. For this, the resulting nonlinear functions are replaced by the smooth “reduced” functions. I n most cases it is possible to linearize them directly and to think of the equations for the low-frequency component as if they were linear. I t will then be comparatively easy to determine the transforming properties of the system with respect to this component. If direct linearization is not possible (either because of the properties of the external disturbance or because of the internal properties of the system), then any of the methods in the theory of nonlinear functions must be applied, and one may use once again harmonic or statistical linearization to compute the nature of the resulting states. T h e presence of a high-frequency component in the external disturbance can cause significant qualitative changes both in the properties of the system with respect to the transformation of the external slowly varying signal, and in the internal dynamic properties, such as the stability of the equilibrium state and the possibility of attaining a self-oscillatory state. It is obvious that these properties

164

3. Nonlinear Transformations-Nonstationary

States

are considered realizable (or unrealizable) “in the mean,” that is, averaged over the long period of the high-frequency component. T h e method of separating the frequencies agrees with the physical nature of the processes in nonlinear systems. Its application is always productive in the analysis of both periodic and random processes. It is especially useful in the composite problems described above, where one of the components is periodic while the other is a random function of time.

Example I . Consider a servo system with a sensitive element which has relay characteristics. Disregarding the time constant of the amplifier and the hysteresis of the relay, one can write the equation for the system in the form (3.56)

+

where Z = a, sin w,t Zo, and I is the time constant of the motor. T h e frequency of the harmonic interference w , is much higher than the significant frequencies in the spectral density S,(w) of the signal Zo. T h e amplitude a of the interference at the input of the relay is given by the equation

(3.57) where the transfer constant ql(a, X,) is equal to [64]

T h e phase relations are usually of no interest. We take the parameters to have the following values : W,

=2n.

50 sec-l,

T

= 0.1

sec,

k

=

50.

I t is not difficult to see that in this case the linear part of the system completely filters the interference and a = a , . For the useful signal (which is slowly changing), we have the equation

(3.58)

3.2. Passage of

a

Slowly Varying Random Signal

165

where Fo(a, X , )

2

.x o= 2- arcsin XO ,

= - arcsin

n

If the quotient

o,o/az is

a

n

( a > I X , I).

az

(3.59)

small enough, one can linearize directly : Fo(a, X,)

2 =a,

-X,

.

(3.60)

T h e variance of the signal X o (the error in the output signal) is found from the usual equation for a linear transformation (3.61)

If S,(W) = [2O/(w2

+ 02)]

oz2, we

obtain (cf. Appendix 111)

where a,

=

T,

a, =

1

+ BT,

a2 = k,

bo

= T2,

b,

+ 8,

a3 = Bk, ,

= -1.

Finally, we have (3.62)

When T < l/O, an increase in the amplitude of the interference increases the variance of the error, but when T > l/O, there will be a decrease. Naturally, this statement is valid only if the condition that u+,/az is small is satisfied. Suppose now that this condition is not satisfied; then one must use statistical linearization.

166

3 . Nonlinear Transformations-Nonstationary

States

In this case, the mean-square error o,, is given by an equation of the type (3.61) or (3.62), but the parameter k, turns out to depend on uI itself : ko = Ah,(%

1

UE0).

It is shown in Appendix IV that

where LY = a,/2/2 us0 and the function CO(a)are given by graphs. A graph of the function a, h, = aC0(a) = Co*(a)

d2

is shown in Fig. 38.

FIGURE38

Moreover, it follows that

3.3. Transformation of Sum of Random and Harmonic Signals

167

or (Y

=

.(Co*).

T h e abscissa of the point of intersection of the functions a(C,*) and Co*(a)gives the desired value of a (Fig. 38). T h e construction is made for the values B = 5 sec-', T = 0.1 sec, K = 50, and u, = 4 2 , and various values of a,. Figure 39 shows the final graph of the function u,u(a,/u,). I t is

FIGURE 39

interesting to note that for small a,/., an increase in the amplitude of the interference brings an increase in the error of the output signal. Computation by the use of Eq. (3.62), that is, by direct linearization, gives a very good approximation when a,/u, 3 1. 3.3. Transformation of the Sum of Wide-Band, Normal, Random Signals and Harmonic Signals in a Nonlinear System with Feedback (Method of Statistical Linearization)

I n the preceding sections we solved the problem of the transformation of the sum Z ( t ) of a harmonic signal Z , ( t ) and a normal, random signal Z z ( t )on the basis of certain assumptions regarding the frequency characteristics of the random component. Let us denote the spectral density of the stationary, random component &(t) by S,(w), and the frequency of the periodic component by woe

168

3 . Nonlinear Transformations-Nonstationary States

I n the problems considered above it was assumed that the spectral density S,(w) was significantly different from zero, at least in some w u p . T h e following variations were frequency range w1 analyzed :

< <

>

(1) T h e high-frequency random signal (wl w0). T h e solution was sought in the form of a sum of a harmonic function and of a nonstationary, normal process with a mean-square value uz periodically varying with respect to time; (2) T h e low-frequency random signal ( w 2 w,,). T h e solution had the form of a sum of a stationary, normal process and of a harmonic function with random amplitude; (3) T h e case where the spectrum of the signal is concentrated in a narrow band near the frequency wo (wl w0 w 2 , w1 - w 2 = 24, where d / w o 1) is analyzed in a manner which, in principle, is no different from the technique described in Section 2.4 for the case of a narrow-band random disturbance of a stationary state.

<

< <

<

T h e external disturbance is written in the form Z = B , sin

w,t

+ C, cos wot,

and a solution is sought in the form X

=

B , sin w,t

+ C, cos w,t,

where B,(t), Cz(t),B,(t) and C,(t) are slowly varying functions of time. If the original periodic motion was forced oscillations, then mB, # 0, and we will find in general that mRz # 0 and mcz # 0. For an undisturbed periodic motion which is a spontaneous oscillation m , = mcz = 0, but mB2# 0. A solution by such a means is possible if the linear part of the system satisfies the usual requirements for applying harmonic linearization, that is, if it either filters low frequencies and, thus, effectively screens out the higher harmonics or if it has a resonance characteristic with a maximum near wo . As has been shown in Section 2.4, the problem of finding the solution depends in most cases on the development of a lagless, nonlinear relation between the envelopes and phases of the signals X and 2, which is found in the usual manner-by harmonic linearization. If

3.3. Transformation of Sum of Random and Harmonic Signals

169

this is not possible [if the conditions (2.78) are not satisfied], then it is practical to apply direct linearization near the point where there is periodic motion. T h e computational variations which we have covered do not, however, exhaust the whole range of problems which appear in practice. I n many cases the spectral density of the input signal does not satisfy any one of the above conditions. T h e solution of the general problem is complicated because of inherent difficulties. In fact, assuming that the signal X ( t ) at the input of the nonlinear, lagless transformation (Fig. 40) is the sum of

FIGURE 40

harmonic and of normal random signals, X ( t ) = a sin(+

+ 4) + [ ( t ) .

(3.63)

We find the correlation function of the output signal Y ( t ) ,

which turns out to be a periodic function of time t ; in other words, the random component of the signal Y ( t )is seen to be nonstationary. T h e result of its linear transformation U ( t )will also be nonstationary. Hence, the assumption that the random component of the signal X(t) = Z(t) = U(t),

is stationary, in general, cannot be sustained. However, one can construct an approximate method of investigation which is satisfactory for most analyses of nonstationary states in automatic systems which are subjected to the influence of wide-band random signals.

170

3. Nonlinear Transformations-Nonstationary States

T he physical basis for the method is as follows. We assume, for example, that the input signal has the form (3.63). I t can be thought of as a set of realizations, each of which is the sum of a sinusoidal segment and some realization of the component [ ( t ) . T h e choice of the initial phase in each sinusoidal segment is, by its very nature, arbitrary. Therefore, in constructing the best linear approximation in the sense of the minimum of the square of the deviation averaged over all possible realizations, one can assume that X ( t )is the sum of a harmonic signal with a random phase uniformly distributed in the interval (0, 27~)and of a normally distributed signal [ ( t ) . T h e construction of such an approximation was carried out in Section 1.5. T h e signal at the output of the nonlinear, lagless transformation can be represented in the form

where r i i , , h, and h, are constant coefficients. Here, we do not consider the distorted form of the spectral density of the normal component or of the higher harmonics in the superposition of both components [cf. the exact equation (1.69)]. This omission is admissible if the linear part of the system has an effective filtering influence on the transmitted signal Y ( t ) .I n the approximation given by Eq. (3.65) the random component of the signal Y ( t ) turns out to be stationary, and this justifies the assumption (3.63). T h e coefficients in the linearization 6,, h, and h, are functions of the unknown parameters, namely, the amplitude a of the harmonic component, the mean value mt and the variance at2 of the component [(t):

In Sections 3.1 and 3.2, we made use of the coefficients h, and h, to solve problems concerned with low- and high-frequency random

3.3. Transformation of Sum of Random and Harmonic Signals

171

signals. Graphs for the functions h,(a, 0, o f ) and h,(a, 0, up) which are needed to solve the problem when me = 0, are given in Appendix IV. ( T he graphs are computed for several typical nonlinearities.) T h e equation for the nonlinear transformation where

+

Q(P)[f(X)l R(P)X = S(P)z, Z

=

mt

(3.67)

+ a, sin wot + lo,

after substitution of (3.65) breaks down into three equations : (a) For the constant signal, Q(0)fiy

+ R(O)mt

=

S(O)m,;

(3.68)

(b) For the harmonic signal, a sin(wot

+ 4) = @Pl(p)[aZ sin wet],

(3.69)

where

(c) For the random signal, where

5

=

(3.70)

@,(P)L

From Eq. (3.69), we find expressions for the amplitude and phase of the harmonic component relative to the harmonic component of the input signal : a = I W w o ) I az

II= arg @,(jw0).

9

(3.71) (3.72)

Equation (3.70) enables us to find the variance of the random component :

172

3. Nonlinear Transformations-Nonstationary

States

or 05

I[h,(a, mc

,Of)]

Equations (3.71), (3.73) and (3.68) are a system of implicit equations which connect the three unknown parameters, a, mEand uC . I n the general case, the most effective way of solving this system is through the method of successive approximations. T h e sequential operations are given by the recursion relations

T h e choice of the zero approximation, in general, is determined by the relative frequency characteristics of the signal and of the operator Q ( p ) / R ( p ) for the linear transformation which approximates the nonlinearity. However, usually it is sufficient to assume that the scheme is for an open-loop system where

(3.75)

If we have a closed loop with a large static amplification coefficient, it is more convenient to assume a zero approximation, my)=O.

(3.76)

We consider in more detail a very important special situation, namely, that in which the constant component m f of the signal X ( t ) ,in general, is not present. Then, there can be two propositions : (a) mE = 0 and +$,(a, 0, uc) = 0 (the nonlinearity is odd); ( b ) mE= 0, but the system is astatic relative to the signal 2, that is, Q(p)/R(p)+ 03 when p + 0, and S ( p ) / R ( p ) remains finite (the nonlinearity again is assumed to be odd).

3.3. Transformation of Sum of Random and Harmonic Signals

173

T h e system of equations (3.77)

is comparatively easy to solve graphically by a method which is analogous to that already described in Section 2.2. T h e computations can be separated into three steps. T h e first step is to construct a graph of Eq. (3.77) for the amplitude a with respect to the transfer constant h,. In the same plane (defined by a and h,), we construct the graphs of h,(a) from the data given in Appendix IV for various values of the parameter uc. At the points of intersection, we find values of a which correspond to each (I,, and can then construct a graph of the function a(a,). In the second step, we find the points of intersection of the function (3.78) for the mean-square deviation (I,with respect to the transfer constant h, with the graphs of the functions h,(a,), which have been constructed for various values of a. T h e values of (
T o find the parameters a and

+ f(t).

wo from

Eq. (3.37), we postulate

Ww0) hl(a, 0 , at) = -Re -

and

Q(iwo) ’

(3.79)

(3.80)

174

3. Nonlinear Transformations-Nonstationary

States

Condition (3.80) determines the frequency of the oscillations which, obviously, are no different from self-oscillations in an undisturbed system. From condition (3.79), it is easy to find by graphical means the function a ( u e ) , which corresponds to the equilibrium state we are looking for. T h e second function for the computation of the forced oscillations can be found also by graphical means from Eq. (3.78). T h e intersection of the graphs of these functions gives the desired parameters for the equilibrium state. Together with this computational scheme for the analysis of the dynamics of systems which are affected simultaneously by harmonic and by normal random signals, one can apply a simpler scheme which uses only one coefficient in the statistical linearization, i.e., it uses the same coefficient for both the random and the harmonic components. This coefficient can be found if one constructs the linear approximation in the form Y = f ( X ) e h,X

=

h,[a sin(w,t

+ v) + t ( t ) ]

and uses the minimum, mean-square deviation to find the coefficient hc * It is not difficult to show [55] that, when m1 = 0, the coefficient h, can be expressed in terms of the coefficients h, and h, introduced earlier : h, =

h,

+( 2 m 4 h 2 + .

1

(2a,”/a2)

However, the use of this device is admissible for the analysis of selfoscillations with random disturbances Z ( t ) only when S,(wo) = 0, that is, in the “nonresonant” case. If this condition is not satisfied, the computation of the variance u, from an equation of the type (3.73) leads to a divergent integral. I n conclusion, we note that these methods of statistical linearization will give good results only when the distribution of the random component of the signal at the input of the nonlinear element is nearly normal. For this method of computing nonstationary states, we have the same restrictions as for the method of statistical linearization in the stationary case, which reduce essentially to the requirement that the random disturbance have a sufficiently wide bandwidth (cf. Section

3.3. Transformation of Sum of Random and Harmonic Signals

175

2.4). If this condition is not satisfied, one has to use the computational schemes described at the beginning of this section. These schemes yield results which give better agreement with the reality andwhich take the nonstationary nature of the problem directly into consideration. * Let us give some examples to illustrate the application of the method of statistical linearization, paying special attention to some of the more common qualitative properties. Example 1. We shall consider once again a servo system with a sensitive relay element as described in a simplified way in the example of Section 3.2; but now we shall assume that the harmonic component of the input signal has a frequency wo , which lies within the region of significant frequencies for the spectrum of the random component. T h e basic equations for finding (1) the amplitude of the harmonic component of the error (of the signal at the input of the relay) and (2) the variance at2 of the random component of the error have the form a2 _ az2

w:(T'w: [&(a,

ue) -

+

1) TWO']~ wo2 '

+

(3.81)

T h e expressions are analogous to Eqs. (3.83) and (3.88). T h e families of functions h, of the variable a for various values of uc and the families of functions h, of the variable at for various values of a are constructed from the graphs given in Appendix I V and are shown in Figs. 41 and 42. Moreover, Fig. 41 shows a as a function of h, computed from (3.81) and Fig. 42 shows at as a function of h, computed from (3.82). T h e computations were carried out for the following values of the parameters : T = 0.1 sec; K = 50; w,, = 10 sec-l; 0 = 5 sec-'; u, = 4 2 ; a, = 1 ; 4 2 ; 4 3 .

* Obviously, these schemes do not exhaust all possibilities. For example, consider the problem of a random disturbance of a periodic state where the spectral density of this disturbance is concentrated in a narrow band which does not include the initial periodic motion.

176

3. Nonlinear Transformations-Nonstationary States h4

1

I

I

0

I

2 FIGURE 41

FIGURE 42

3

0

3.3. Transformation of Sum of Random and Harmonic Signals

177

T h e construction of the points of intersection of the functions

u(ac)and .[(a) can be seen in Fig.

43.

FIGURE 43

Finally, we find uf =

1.08

for

a = 0.75,

uE =

1.09

for a,

a = 1.0,

uf =

1.10

for a, =

a

=

0.5,

a, = 1,

=

d2, -

v3.

T h e dependence of the error variance on variations in the amplitude of the harmonic component turns out to be a very weak one. Analogously, the amplitude of the error does not depend strongly on the variance of the interference. Therefore, one can propose the following simplified means of computation : determine the amplitude of the error a by the method of harmonic balance (letting 6 = 0), and then find the error variance, using the value of the coefficient h,(a, ae), which corresponds to a ; or, conversely, find the approximate value of uf by the method of statistical linearization (letting a, = 0), and then use the value of h,(a, uc) which corresponds to the value we found for ut to find a.

178

3. Nonlinear Transformations-Nonstationary States

Example 2. T h e behavior of a self-oscillatory system described by the equations k Y+X=Z, Y = sgn X, (3.83) P(TP

+

was studied earlier in Section 3.1, under the assumption that the external disturbance 2 had a high frequency. We shall now analyze the dynamics of a system which has a random disturbance with a spectral density of the form 2e S,(W) = a,2 ___ w2

+ 8”

e = - 1=

for

T

“O

Here, the frequency of the self-oscillations lies within the band

of significant values for S,(w). We shall look for a solution of Eq. (3.83)

in the form

x = a sin w0t

+ I,

that is, we assume that there are periodic oscillations in the system. T h e amplitude a and the variance oE2are given by the conditions h,(a, at)

=

-1

=

2 kT

-= d

1 “ 2TaZ2w2 2rr I jw(Tjw kh2 1’ --lo

+ +

(3.84)

O ’

1

& I - - - -

2 do

do - h2

U*2.

(3.85)

T h e parameter do is inversely proportional to the amplitude of the self-oscillations a,, in the undisturbed system do

=

4

To

Applying the usual graphical technique, the successive steps of which are shown in Figs. 44,45 and 46, we find values* for the quantities a and u E .

* From Fig. 46, it is obvious that each pair of values for do and uz correspond to two solutions. The usual considerations for the method of harmonic balance show that stable, periodic oscillations will correspond to only one of these solutions. Hence, the problem demands some special investigation.

3.3. Transformation of Sum of Random and Harmonic Signals h,=dof I .o

0.8 06 0.5 0.4

I

2

3

a

I

I 3

I Ze

FIGURE 44

I

0.5

V

0

1

2

'I

FIGURE 45

179

180

3. Nonlinear Transformations-Nonstationary

States

FIGURE 46

I t is important to realize that the solution can be found only for certain relations between the parameter do and the variance uz2of the random disturbance; therefore, there will be periodic waves in the system only for certain relations between the amplitude of the undisturbed, self-oscillatory state (a, = 4/7~d,)and the magnitude of the disturbance. Figure 47 shows a graph of the curve (the continuous line) which

‘333

0.5

0

0

I

2

3

00

FIGURE 41

separates the region in the plane (a,, u,) where there are periodic oscillations (region I) and the region where they do not exist (region 11), (where the self-oscillations are broken off). For the sake of comparison, we show in the same diagram a graph of the boundary which was found in Section 3.1 for high-frequency disturbances (broken line).

3.3. Transformation of Sum of Random and Harmonic Signals

181

A break in the self-oscillations will take place for small variances in the random disturbance if the frequency of the self-oscillations lies within the band of significant frequencies in its spectrum. Example 3. T h e block diagram of Fig. 48 shows a typical servomechanism the load of which has oscillatory dynamic properties.

I

--

I

FIGURE48

T h e nonlinearity in the amplifier and the presence of a random interference have been taken into account. T h e spectral density of the interference is given by the expression

We assume the following numerical values of the parameters for the load and the interference spectrum :

First, we shall study the dynamics of the system without any interference. For small amplification coefficients in the control section, the system is stable ; but with increasing amplification self-oscillations can appear. T h e characteristic equation of the system has the form (3.86)

We can write the condition for stability as (3:87)

182

3. Nonlinear Transformations-Nonstationary

States

For the values of the parameter of the load we have assumed that

k

- < 0.6. Wl

Self-oscillations which arise if this condition is violated can be found by the method of harmonic balance. We obtain the following expression for the frequency of the self-oscillations w0 and the amplitude no of the signal at the input of the nonlinearity : wo = w 1 , WlW12

where

q(ao) = 2[ k kt

w1 = 0.6 - ,

(3.88)

k

is the harmonic transfer constant for the bounded nonlinear element. T h e graphical solution of the resulting equation (Fig. 49) gives in explicit form the amplitude of the self-oscillations for variations in the coefficient k / w , .

9 8

6 4

2

0 2 4 6 8 looo -G

0

I

2

3

k

WL

FIGURE 49

Under the influence of random interference the effective amplification coefficient decreases, and, therefore, one can expect periodic oscillations in the system to take place only for very high values* of k / w l . This assumption is confirmed by the calculations carried out

* The random interference for this scheme has a stabilizing effect in contrast to the effect it had for the servo system studied in Example 3 of Section 2.2 and in Example 1 of Section 3.1.

3.3. Transformation of Sum of Random and Harmonic Signals

183

by Sawaragi et al. [ 1151 by graphical means, which, in principle, are no different from the ones described in this section. T h e desired values of the amplitude a of the harmonic component and the variance up2 of the random component are found by the graphical solution of the implicit equations of the type (3.78) and (3.79) :

(3.89)

and

(3.90)

or for the particular values of the parameters we have assumed,

h,(a, uC)

w1

= 0.6 -

k We shall restrict ourselves to studying only those final results which are of definite qualitative interest. as functions of the variable Figure 50 shows graphs of a2/12,and uE2/12

FIGURE 50

WL

184

3. Nonlinear Transformations-Nonstationary

States

k / w l , computed for the values of the parameters given above. For the sake of comparison, as a function of k / w l is drawn in the same diagram; the latter characterizes the state of the system without interference. Periodic oscillations can appear only for larger values of the amplification coefficient ( k / w , = 1.52), while the amplitude of the oscillations in transition across the boundary value increases not from zero, but from some sufficiently large value. At this point, the magnitude of the random component decreases sharply, and there is a redistribution of the energy between the two types of oscillations. I t has been observed experimentally [115] that the character of the appearance of the oscillations depends on the direction of the variation in the parameter k / w , . There are hysteresis phenomena (protractions), which to a certain extent are analogous to the well-known effect that takes place when a harmonic impulse force acts on a nonlinear system. T h e complete determination of the moment when periodic oscillations appear as the result of a slow increase in the parameter k / w l cannot be carried out, in general, within the framework of this theory. In fact, if there were interference acting on the system, then for the first value of the coefficient K I W I , there would be no conditions for the appearance of periodic oscillations. Only beyond the boundary value K/wl = 1.52 given by Eqs. (3.27) and (3.28) do these conditions take effect. However, in principle, there are conditions for the preservation of a state with purely random oscillations; its parameters are given by the implicit equation

where hl(uc) is the coefficient for the statistical linearization with respect to the normal random signal. A solution satisfying this equation exists for k / w l > 1.52. T h e variance of the signal at the input of the nonlinearity increases with increasing k / w , ; but the effective amplification coefficient (kjw I ) /zl(uE),although it increases at first to some extent, remains below the limiting value 0.6 for arbitrarily large k / w l . I t is physically clear that such a state is not

3.4. Random Disturbances of Periodic States

185

stable and must transform into a state with periodic oscillations, although a rigorous study of this problem is difficult.* 3.4. Random Disturbances of Periodic States in Relay Systems (Exact Solution by the Method of Alignment)

Among the analytical problems for the dynamic properties of essentially nonlinear closed systems, there are some separate, comparatively small classes for which solutions can be found by exact methods. One of the most interesting problems of this type is the problem of small random disturbances of periodic states in systems with nonlinearities of the relay type. We shall limit ourselves to an investigation of a system with one nonlinear element : (3.91)

where K ( p ) = Q ( p ) / R ( p ) We . assume that the polynomial R ( p ) is representable in the form R(P) = P W P ) ,

where all the roots hp(p = 1, 2, ..., n ) of the polynomial R,(p) are different and the real parts are negative. We can then write an expansion of K ( p ) into simple fractions : (3.92)

It is not difficult to see that by introducing the new variables

w,= -1 Y , P

w,= Y P - A,

(3.93) ( p = 1 , 2 ,...) n),

* The explanation of the phenomenon of protraction given by Sawaragi et al. [ I 151, where the present example can be found, is questionable.

186

3. Nonlinear Transformations-Nonstationary

States

one can write the system (3.91) in the following form :

pw, = Y , ( p x = z+ Y ,

-

hp)W,= Y

( p = 1, 2,

..., n),

(3.94)

Y =f(X),

This form of the nonlinear equations, namely, the canonical form

[50],considerably simplifies the study of piecewise linear transforma-

tions and, particularly, of relay systems. Let the random component of the input signal Zo(t) be equal to zero. T h e equilibrium state of the system (3.91), which we will call undisturbed, can in many cases have a periodic character. We shall now study three types of periodic states : (a) self-oscillatory states in which m, = 0; (b) states with forced oscillations of the form m, = a, sin w o t ; and (c) states with a linearly increasing signal, i.e., m, = It. We shall analyze the disturbances of these periodic states by stationary random influence.

OF SELF-OSCILLATORY STATES.We shall a. RANDOMDISTURBANCES give a short summary of the well-known method of computing oscillations in a relay system [50]. Let the relay characteristic have the form shown in Fig. 51. T h e signal Y at the output of the relay can have only two values, fo and -fo . We denote by index 1 the values of the variables which correspond to the interval T ~ where , Y = fo, and by index 2 the values of the variables which correspond to the interval T~ , where Y =: -fo .

FIGURE51

3.4. Random Disturbances of Periodic States Integration of the system (3.94) gives (for k

+

w o k . = (-l)'-'fOt

wp.k

CO.k

1, 2),

t

(3.95)

f + Cp,keApPt

=

=

187

( p = 1, 2,

AP

... n). 3

T h e constants C p , k ,where ( p = 0, 1, 2, ..., n) correspond to the values of W, at t = 0 (we shall read off the time from the beginning of each half-period). We use the conditions of continuity and periodicity of the processes in the system : wp.l(o) = w P . 2 ( 7 2 ) ,

wp,l(71) =

(P = 0, 1,

Wp,,(O)

*..>.I.

(3.96)

Substituting the expressions (3.95) into these conditions, we find Co.1 = C0,Z

c

-foT2

+

=foQ

- - fo_ - c,,,~e A,

o =C Cp,leAppTt- fP.2 AP

+ cod

9

C0,l 1

~

+y fo- ,

(3.97)

A,

o + fA,

( p = 1, 2, ..., n ) .

Combining the switch-over conditions

p=1

(3.98)

with' Eq. (3.97), we obtain a complete system of equations to find the unknown quantities T ~T , ~ CPel , and C p , 2 where , p = 0, 1, ...,n. From the first two equations of (3.97) we see that T1

=72 =7

and co.2 - Co.1 =foT.

188

3. Nonlinear Transformations-Nonstationary

States

T h e second pair in (3.97) gives

(3.99)

cp,l= -Co,2 = C ,

(p =

1, 2 , ..., n ) .

T h e sum of Eqs. (3.98) gives

and, hence,

x - for co,l= 2

Bo

(3.100)

*

Subtracting the second equation of (3.98) from the first and substitut, obtain an equation ing the values found for the constants C p , k we which determines the period : POT

B P

-2-+zxtanhI p=1

APT

P

=

A

--

fo

(3.101)

T h e half-period T does not depend on the displacement of the characteristic of the relay x. T h e latter effects only the quantities Co.l and C o , 2 .We also note that the value found for 7 must have the correct direction in the switch-over X1(O)

>0

or p=l

We now analyze the disturbed state. We denote by T ~ where , = 1, 2, ..., n ) the successive intervals of time (the “half-periods”) during each of which the output signal of the relay Y is constant (taking the values of either + f a or -fa). T o be specific, we assume that the disturbance Zo(t) is introduced into the system at the initial moment t o , which is within the interval 71, where Y takes the value

(K

189

3.4. Random Disturbances of Periodic States

T~ , that is, at the moment of switch-over, is denoted by Zko . We also introduce the index k to denote processes in the variation of the variables over the period T,,. . Then

f a . T h e value of Zo(t) at the end of the interval

+ (-l)k-lfot,

= C0.k

W0.k

+

W P m=k Cp,keApT

f

(p = 1,2, ..., n ) .

( - 1 ) k L AP

We take the beginning of each interval as the initial moment of time t . T h e continuity conditions ( p = 0,

IVp,k+l(o)

=

wp,k(Tk)

1 , 2, ...)n)

make it possible to derive the recursion relations

+

cO,k

C p , ke A p T k

=

(-l)"-'fOTk

+ (-

I)k

cO,k+l

9

f2 = Cp . k + l + (-I)'.+'

f.

(p =

A,

AP

1,2, ...,n ) . (3.102)

We introduce variables which characterize the deviation from an undisturbed state : A7k =

where

T

(-I)'((.

- Tk),

ACp.,

= cp,k - cz.k

I

is given by the equation of the periods (3.101), and z

2fo ___ 1 -

A, 1

+

(-1)k-1

eapT

=

cp ( - l ) -

C:,k = Co,l, = C0,*,

( p = 1,2, ..., n ) ,

k k

= 2m -

1,

= 2m.

We assume further that the disturbance Zk0 is small in comparison with the amplitudes of the values of the input signal X ( t ) in the undisturbed state. Because of the stability of the indicated state, the deviations AT^ and AC,,, , where p = 0, 1, ..., n, are in general, small by comparison with the corresponding quantities. Using Eq. (3.97) we can linearize the continuity conditions (3.102) : AC0,k-1 +f"4

ACO,,

=

AC,,,

= eApTACp,k-l

- - 1

,

+ C,e'pThpATk-1.

(3.103)

190

3. Nonlinear Transformations-Nonstationary States

We see that Eqs. (3.103) are difference equations of the first order relative to AC,,, , where p = 0, 1, ..., n ; it is easy to find expressions for the dependence of these quantities on the fluctuations in the halfperiods A T , : i=l

(3.104)

We now use the switch-over conditions Vk(0) = ( - l ) k - l A

or AVk(0)

+x

-

2,' - 1,

(3.105) (3.106)

-Z;-1

Moreover, n

n

= cppAWp,k(O)

Avk(o)

p=o

=

c p p A c p , k . p=0

(3.107)

T h e substitution of (3.104) into Eq. (3.107) gives an infinite system of linear equations for the determination of AT^ :

2 AT^ + 2 5

pofo

i=l

A,

B,C,

p=O

e2p7(k-iJ

Ari

=

-Zok-1

i=l

(k = 2, 3, ...). (3.108)

Substracting the (k - 1)th equation from each kth equation, we can put the system in the form

$

i=l

where

ak-i+l

=

-(z:

- Z,"-l),

(3.109)

3.4. Random Disturbances of Periodic States

191

I t is obvious that the coefficients ai rapidly decrease as the index i increases. T h e physical meaning is very simple. Quantity a, represents the difference between the values at the moment iT and at the moment ( i - 1). of the reaction of the linear part of the system to the impulse of the disturbance, though not for zero initial conditions-which would give the impulse function which corresponds to the operator K(p),-but for conditions which will guarantee the continuity and the periodicity of the process in an undisturbed system. T h e solution of the system (3.109) is not difficult because the matrix of the coefficients ai is triangular and because the values of Ark can be successively expressed in terms of all the 2: where i k. For an arbitrary k, )

<

(3.1 1 1 )

where the coefficients A, are given by the recursion relation (3.112)

while it is obvious that 1

A1=--,

A,=-

01

Equation (3.112) can be proved by direct substitution of (3.111) into Eq. (3.109) and equating the coefficients of each Zko in the resulting identity. T h e solution (3.1 1 l), in general, completely determines the disturbed state. Substituting (3.1 11) into (3.104) and changing the order of summation, we obtain

i=1

i=l

(3.1 13)

192

3. Nonlinear Transformations-Nonstationary

States

T h e fluctuation of the signal V ( t ) is given by the equation

vk(t)= p0 A C , . ~+ CpPeApt dc,,, , n

p=1

or, taking (3.106) and (3.107) into account, n

dVk(t)

-2E-1

-

Cpp(1 - d p t ) dcp,, . p=l

T h e first term characterizes the immediate effect of the disturbance for the kth switch-over, while the last term characterizes the influence of the fluctuation in the previous half-periods on the disturbance; this effect becomes steadily weaker because of the damping on the system. Finally, we write

2z;-ioj(t), A- 1

dvk(t) =

(3.1 14)

i=1

where

Let Zo(t)be a stationary random process with mathematical expectation equal to zero and a correlation function given by

Rz(4

=

%w4.

I t is obvious that the mathematical expectation of the fluctuation of a period and of the signal V ( t )will also be zero. T h e variance for a period is given by the equation (3.115)

(3.116)

3.4. Random Disturbances of Periodic States

193

If the random disturbance has a wide band and if p,[(i

-

j j ~< ] 1

for i # j ,

then one can find the simpler equations : (3.1 17)

and

2 D?(t).

k-1

(3.1 18)

= uz2

&(t)

i=l

I n an analogous manner, it is possible to find the correlation coefficient between half-periods :

2 2 ~ , ~ , p , [ (-i j + m k

M { A T ~A T , )

= uzz

m

i = l j=1

-

k)T].

(3.1 19)

T h e equation for the equilibrium value of u, is given by (3.115) [or by (3.1 17)], if we replace the upper limit of the sum by co. T h e fluctation of the signal V ( t ) has a nonstationary character. However, the change in d V ( t ) over the period T is completely determined by the fluctuations in the initial values dC,,, , where p = 0, 1, ..., n, which are elements of a random sequence, each of which has a stable character for sufficiently large k. In this sense, one can speak of the equilibrium fluctuations P ( t ) as fluctuations over a period sufficiently far removed from the initial moment of the impact of the random disturbance Zo(t). T h e expression for the variance of the equilibrium fluctuations will have the following form :

For small correlations, we have (3.121)

Computation by the use of these equations is not extremely difficult in spite of inherent complexity, because the coefficients A j and D j

194

3. Nonlinear Transformations-Nonstationary States

rapidly decrease with an increase in number, which is the direct consequence of the steady state. We shall evaluate the quantity called the “phase shift,” that is, the net increase in the phase AOk in the disturbed state by comparison with what it would be in the undisturbed state. By virtue of our assumed notations, we can write

We shall start from the basic system of equations (3.109). Consider the equations for the values of k between 1 a n d j :

-zl0, u1 4~~+ a24~~ = + ZIO, a1 471 =

-2,O

. .

a,

ATj

+

a2

d7j-1

.

+ .“ +

.

. .

U j 471

=

.

-zp + z,”_l

We multiply by -1 both sides of each of the equations which corresponds to an even number k : a1 471 = -zio, -al 4~~- U ~ A T= , Z,O . . . . . . . . . . . . . . . al( - I)’-’ dTi

+ a,( -1

)’-I

LIT^-^ -+ ... + a3(-1)’-’ LIT^

=

-

Z 1”,

(-I)’(.Z:-Z~-~) 0

Summing the left- and right-hand sides of all these equations, we find

We change the order of summation in both sums

or

3.4. Random Disturbances of Periodic States

195

Now we introduce the quantities

Then Eq. (3.122) can be written in the form

z j

a,Sj-i+l = 2

(-1)j-l

i=l

2 zk0(-l)' + z?(-I) j-1

k=l

or j

a,-,+lS,

=2

i=l

z j-1

Zk0(-l)*+j-' - z?.

(3.123)

k=l

Since this equation is valid for j = 1, 2, ... (with the assumption that 2 : = 0 for j < l), it is possible to consider the infinite system of linear equations as determining the quantities S, . The solution of the system (3.123) obviously can be written in the form (3.124)

where the coefficients Bk are given by direct substitution of (3.124) into the equation of the system and by equating the coefficients of each z k o . Substituting (3.124) into (3.123), we obtain

and, hence,

2 i-1

aj-i+lBi-k+l = - 1, = 2(-1)k+i-11

k=j, k = 1,2, ...,j - 1 ,

which is equivalent to the 'equations alBl

= 1,

2 i=l

aj-i+lBi = 2(-1)'

(j

= 2,

3, ...).

196

3. Nonlinear Transformations-Nonstationary

States

These equations can be put in the form of a recursion formula to find the coefficients B, : 2(-1)J

2 aj-i+,Bi] , j-1

-

(3.125)

i=l

while

T h e presence of the term 2/a, attests to the fact that the coefficients B j do not decrease with increasing j. Therefore, the variance of the “phase shift” mik is not bounded for increasing j, that is, for increasing time,

or for the uncorrelated variables Zi0 and Z,”(i # j) (3.127)

This physical phenomenon is well known, and for many simple systems (especially in radio physics) several studies have been carried out by various techniques (cf., for example, Bershteyn [9] and Rytov

[731)-

An example of the computation of the variance of the half-periods is given at the end of the section. OSCILLATIONS. b. FORCED Let m, = a, sin (mot + p)). Consider the periodic state ( Z o = 0), whose period is the same as the period 27 = 2 ~ r / mof~ the external influence, while the signal X(2) has a zero phase* so that p) represents the relative phase of m,(t) and X ( t ) . Again, making use of the canonical form of writing the equations for the problem (3.94), we immediately find from the continuity and

* Usually, the phase is included in the signal X ( t ) in its position relative to the external impulse. However, the reference system proposed here makes it possible to simplify the resulting equations to some extent without making any significant changes.

3.4. Random Disturbances of Periodic States

197

periodicity conditions (3.96) and, also, from the switch-over conditions, Xl(0)

=

VdO)

+m

+ ZB, (11

m

= BOCO,,

p=1

h, + Gl) + a2 sin ?J

fo

=x+4 (3.128)

equations for the constants Cp,l, C p S zwhere , p = 0, 1, ..., n, which agree with the equations found above (3.99) and (3.100), and an equation from which we can find the relative phase : (3.129)

T h e sign of the phase angle is given by the condition which specifies the direction of the switch-over :

We shall consider further the calculation when a stationary random disturbance Zo(t) is added to the periodic disturbance. We use the same notation as in Example a and assume again that the random components are small. Then for a fluctuation in the initial values AC,,, , where p = 0, 1, ..., n, we obtain the same equations as (3.104) in Example a, since the form of the continuity condition is preserved. T h e switch-over condition (3.106) in this case must be written in the following form : Vk(0)

= (-l)k-lA

+X -

zk-1,

198

3. Nonlinear Transformations-Nonstationary States

where

Carrying out the linearization by taking into account condition (3.128) for an undisturbed state, we obtain

2p, dc,,,

9COS q~

= -

z k- 1

AT^( - l ) i + k

- zo k-1

(3.131)

i=l

p=o

Substitution into (3.131) of the quantities AC,,, as described by Eqs. (3.104) gives a system of linear equations which determine the fluctuations in the half-periods AT^ :

Subtracting the (k obtain

-

2)th equation from each kth equation, we

z k

ak-ifl

= -(zk

0

i=l

where

a, = Pojo -

a n

cos v

+

- z”,z),

(3.132)

n

A,B,C,e A p 7 , p=1

(i >, 3). p=1

T h e solution of the resulting infinite system of linear equations with a triangular matrix has the form

hi= $Ai-j+lZ,O j-1

= ZloAi

+ Z,OAi-l + ... + AIZio.

(3.133)

3.4. Random Disturbances of Periodic States

199

We find the values of the undetermined coefficients Ai by substituting (3.133) into the system (3.132). In the resulting identity,

we equate the coefficients of each 2:. As a preliminary step we change the order of summation :

Then, it is obvious that

or alAl

a,Al

+ a,A2 + a1A3

=

-1,

= 1,

(3.1 34)

i=l

Finally, we find the following recursion relation : (3.135)

while

Starting with k = 3, the coefficient A, rapidly decreases. We also find an expression for the magnitude of the "phase incidence" A Q k .

200

3. Nonlinear Transformations-Nonstationary

States

We take the sequential sum of the equations (3.132), and multiply each even equation by - 1, to obtain

i=l

a=l

k=l

After changing the order of summation we find

= Z,"( - 1 ) j

+ z;-l(

(3.136)

-1 ) j - l .

We introduce the quantities k s k =

CdTi(-l)i+k. i=l

Then the system of equations (3.136) takes the form (3.137) T h e resulting system is the same as the system of equations of (3.109) considered above. Its solution can be written in the form (3.138) where and

20 1

3.4. Random Disturbances of Periodic States

T h e coefficients B, decrease rapidly because of the decrease in the a, . Therefore, the variance of S , is bounded, which is the opposite of what was found for the self-oscillatory system. T h e expressions for the correlation function and for the variance of the fluctuations in the half-periods are obviously the same as the expressions (3.115), (3.1 17) and (3.1 19) which were derived in Example a of this section. A computational example will be given later. c. THEOUTPUTOF THE SUMOF A SIGNALVARYINGLINEARLY TIME A N D OF A STATIONARY RANDOM SIGNAL.I n this case

WITH

m,(t)

where

(3.139)

It.

=

1 = const.

Consider first the undisturbed equilibrium state ( Z o = 0). T h e variation in X ( t ) is periodic, although the mean value of X ( t ) over a period is different from zero. T h e output signal V ( t ) of the linear part of the system must have a term which linearly varies with time [otherwise there would be no compensation for the input signal

mz(0l.

From Eqs (3.95) which give the results for integration in the original system, it is obvious that this term can appear only in the , in view of the above discussion, will not component W O , kwhich, have a periodic character. From the continuity conditions, WO.k(rk)

it follows that (-l)B-lfOrk

We assume that 7k

=

+

= 71,

-r2,

WO,k+l(o)~ CO.k

=

CO.k+l.

(3.140) (3.141)

k Odd, k even.

T h e process of variation in the coordinate of W , , where p

2, ..., n, is periodic, and, thus, we have

=

1,

(3.142)

202

3. Nonlinear Transformations-Nonstationary

States

Moreover, from the continuity conditions, WP.k(Tk)

= WP,k+l(o);

this gives

Cp.2 = Cp.2,

2f0

1

- eAprl

= -Xp 1 - e a p ( 7 i + T z )

( p = 1 , 2 ,..., m,

m=O,l,

...).

(3.143)

T h e switch-over conditions can be written in the form

(3.144)

Subtracting the second equation from the first and taking into account (3.141), (3.142) and (3.143), we obtain, after some fairly simple manipulations, the equation for the periods

and, hence, considering (3.142), we obtain (3.146)

3.4. Random Disturbances of Periodic States

203

Analogously,

+

1

C0.2m+2

- Co.2m = - (71 BO

72).

(3.147)

These relations together with (3.141) make it possible to write a relation between the intervals T~ and T~ : (3.148)

T h e quantities T~ and T~ are then found by the simultaneous solution of Eqs. (3.145) and (3.148). From (3.148) one can also find a condition for the tracking stability, Ill

(3.149)


which, by the way, is obvious from physical considerations. We shall go on to estimate the effect of random disturbances .To. T h e switch-over condition can be written in the form (3.150)

We carry out the linearization taking into account the equations for an undisturbed state (3.144) :

-z",_,.

i=O

p=o

(3.151)

We determine the quantities AC,,, , where ( p = 0, 1 , 2, ..., n). T h e continuity conditions (3.102) obviously remain in effect in this case. Using the corresponding relations for the undisturbed state and carrying out the linearization, we obtain "0.k

= dcO,k-l

+fO

ACp,k= eAPTk*dCp,k-l

d7k-l

1

+ Czkh

(3.152) d7k-1(-1)k-19

eaPTk4

204

3. Nonlinear Transformations-Nonstationary States

where

Tk*

=7 1 ,

k odd,

Tk*

=72,

k even.

From this follows that

zc;,~ k- 1

AC,,,

=

i=l

exp(A,

zTs*)

k-1

A, d T i .

(3.153)

s=i

Substituting these equations into (3.151), we obtain an infinite system of linear equations for the determination of AT^ :

Subtracting the (k - 2)th equation from the kth equation, we find

=

-2:

+ ZE-,

(k

=

1 , 2 , ...).

We break down the system (3.155) into two equations : T h e first corresponds to odd values of the index k and the second corresponds to even values of K.

3.4. Random Disturbances of Periodic States

205

I t is not difficult to see that these systems can be written in the following form :

=

-Z!r

+ Zir-,

where we introduce the following notation :

P=l

(I =

1, 2, ...),

(3.157)

206

3. Nonlinear Transformations-Nonstationary

States

We shall look for a solution of Eqs. (3.156) and (3.157) in the form

T h e coefficients A, and B, are found in the usual manner. Substitution of the expressions (3.158) into the system (3.156) gives

(3.159)

After changing the order of summation, we have

i=l

it1

a+2 +)

i=l

m-($,

):

Comparing the coefficients of 2 : with the same index, we find

2

i+l m=- 2

azr-zrn+lAzm-i

i+2

*T

+ m=-

2

i+l 2

b2r-zmB2rn-i+l

i

'Y - -1, 1, - 0,

i = 2r - 1 , i = 2r - 3, i#2r-ll,2r-3,

which is equivalent to the following conditions : alAl = 1 , (3.160)

3.4. Random Disturbances of Periodic States

207

T h e resulting conditions can be thought of as recursion relations which enable us to compute the coefficients A, and B, with comparative ease for arbitrary k. Beginning with k = 4, these coefficients rapidly decrease. Here, we write out the first four coefficients :

c)]

1 [b4 - -- b, (1 - azb2 + b A, = -

a81

a1

a1b1

T h e coefficients B , can be found from the expressions for A, by direct interchange of the quantity a for b and of b for a. T h e equations for the variance of the fluctuations of the “halfperiods” AT,< obtained from (3.158) for uncorrelated 2; and Zio ( i # j ) are the following : 2m-I

= u:

ZA:,

k

= 2m -

i=l

2 B:, zm

= u,2

1,

(3.1 62)

k

= 2m.

i=l

Example I . Consider a relay system, with an operator in the linear part of the form

Let us assume that the relay is ideal.

208

3. Nonlinear Transformations-Nonstationary

States

T h e half-period of the self-oscillations is given by the expression

We carry out the computation for the case when T , = 1, T , = 2. Then h1 -- -1, A, = -0.5, Po = -1, P1 = -1, /I2 = -2. From the equation for the periods we find that T = 4.6 sec. We compute the coefficients ak from the equation (3.1 10) : )

a, =

-0.656f0;

a2 =

-o.326f0;

a3 = -0.0328f0.

T h e coefficients ak for k > 3 are too small. Moreover, we find from (3.1 12) the quantity Akfo: Alfo

=

1.53,

A2fo = -2.29,

A3f0

=

-0.414,

A4f0

=

A5f0

= 0.153.

1.06,

We assume that the correlation function for the random component is equal to

R,(O)

=

e-a(e).

Then (3.1 15) makes it possible to find the mean-square value of the fluctuation of the half-period : U

0, =

0.84"

when

01

= 0.01,

=

1.6-0 2

when

01

= 0.1.

0,

fo

fo

If there is no correlation, we find from (3.1 18) that u7 =

U

3.6"

fo

T h e wider the band of the interference, the more significant will be the choice of the values of the parameters for the self-oscillations.

3.4. Random Disturbances of Periodic States

209

Example 2. Let us analyze disturbances of a state with forced oscillations in a simple system where

T h e periodic signal has the amplitude a, = f,,/2. Then, A, = - 1, = 1, and the phase angle is given by the equation = - 1,

so

+ tanh 2

sin

-

g~ =

0.

When T = 0.4 sec, we obtain sin y = 0.8. From the condition for the correct direction of the switch-over, it follows that C O S T = 0.6. We compute the coefficients which are necessary to evaluate the disturbed state :

From Eqs. (3.132) we can also calculate a, = a2 =

a3

=

a4 =

= - 0.198f o , a, = - 0.133f o , a , = - 0.089 fo .

2.55 fo, + 1.88 fo , -0.44 f,, , - 0.295 fo ,

as

-

Because of (3.135), f o A , = 0.393, foAz = 0.290, fOA3 = -0.246, fOA4 = -0.277,

fOA5 foAs fOA7

= = ==

-0.226, -0.233, -0.035.

T h e results of the computation of the mean-square value of the fluctuations in the half-period in the presence of interference with a correlation function of the form R,(e) = ~

~ 2 e - ~ ' " l

are shown in the accompanying table.

2 10

3 . Nonlinear Transformations-Nonstationary a

LI ~~~

States

~

0.01 0.05 0.1

0.147 0.25 0.368

0.5 1 .O

2.0

0.6 0.76 0.75

For large values of a , when one can neglect the correlation between the values of the interference at successives switch-over moments, we have

It can be seen from the table that the effect of the fluctuations depends significantly on the width of the spectrum.

chapter 4

E X T R E M A L SYSTEMS

4.1. Basic Principles of the Operation of Extremal Systems

Consider a dynamic system the operation of which is characterized by some unique index. T h e index could be the efficiency coefficient in a system which is an energy device, or it could be the performance per unit of time in a system which consists of a combination of machines carrying out some technological process; or it could be any quality index, for example, the mean-square error, if the system were an automatic control system. Henceforth, we shall call this index the quality index Y of the given system. T h e value of the quality index depends on the choice of the parameters of the system, which are called the tuning adjustments X ( X , , ! **’, X,). We assume that the tuning adjustments can be varied inside some closed region Q which is defined by technical conditions. T h e quality index Y as a function of X has an extremum in the region Q (either in its interior or on the boundary) which corresponds to the optimal operating conditions of the dynamic system. A system* in which there is an automatic search for, and the establishment of, the quality index on a level which is near the extremum in the given region of tuning adjustments is called extremal. Thus, an extremal system is a system which dynamically optimizes itself; the characteristic for optimization is defined by

x,

and supplementary control devices vary the tuning of Xc so as to make Y take on its extremal value. If the exact form of the characteristic f ( X ) is known in advance, then the tuning values Xext can be calculated to correspond with the

* The

~~

~

definition of an “extrernal system” given here is not the generally accepted one (cf., for example, Feldbaurn [94]).

21 1

212

4. Extremal Systems

desired extremum of Y , and, thus, the extremal system can be constructed as an ordinary system for automatic control which is guided by the error signals A X = x - Xext. Its special nature as an extremal system is manifested only in the terminology and not in its dynamic structure. Of special interest, is the case where the form of the characteristic of the controlled system is unknown, and where the values of the quality index Y for the operating system, as measured directly, are the single source of information. Of course, a necessary condition for the realization of an extremal system is the inclusion in the algorithm for its performance of an operator which defines the form of the characteristic, that is, which defines the values of the index Y for various values of the tuning adjustments X in the given region Q. We shall confine ourselves to studing objects with characteristics which have only one extremum (to be specific, let us say, a maximum) which is inside the region Q. T o find whether a given value of Y is extremal, it is sufficient to study the values of Y in the neighborhood of the tuning adjustments, that is, to “examine” the local neighborhood of the point. Thus, one must perform two operations in such an extremal system. (1) One must show the relation between the initial and the extremal value of the quality index, that is, one must find out in what direction and (in some cases) by how much one must vary the tuning to achieve the extremal state (a test or search operation). (2) One must analyze the values found for the variation in the tuning adjustments (the working operation).

T o explain the search operation, we shall discuss the simple case in which there is only one tuning adjustment (the one-dimensional problem). I n the initial position let X = X,, . We increase the tuning adjustment : XI = xo c. If as a result of this change,

+

4.1. Basic Principles of Operation

213

this means that the increase has been in the direction of the maximum, that is, in the right direction. If we find, on the other hand, that Y(X,) < Y(X,), then we must change the direction of the adjustment in the variable X . T h e result Y(X,) = Y(X,) means that the adjustment lies near the optimal value.* I t is obvious that this operation leads to an approximate determination of the differential characteristic for the object near the given point or to the determination of the “slope” S of the characteristic at the given point

Such a method of determining the slope is not unique. If the characteristic is such that continuous measurement of the extremal index is compatible with the indicated method, one can feed into the input of the object a test signal in the form of a periodic signal and measure the first harmonic of the signal at the output of the object. Let the test signal have the form x,, = c sin w o t .

+

Then, Y = f(X) = f(X, x n ) = f(X, test signal has a small amplitude, then Y

=f ( X , )

+f’(Xo)csin

+ c sin wet).

OJot

When the

+ ...,

and, consequently, the amplitude of the first harmonic appears to be approximately proportional to the unknown slope f’(X). If the characteristicf’(X) can be written in the form of a polynomial of the second order, Y

= f ( X ) = a,

+ a,X

1 .a J 2 ,

then S

=

a,

+ 2a,X,

=

w

+ c sin w o t ) sin w,t dt,

2 r’wof(Xo nc.

0

that is, the slope is exactly equal to the harmonic transfer constant for the first harmonic.

* For the

sake of simplicity, we assume here that there is no inflection point.

214

4. Extremal Systems

T h e mechanism which enables us to find this constant can be constructed either directly in the form of a computer which carries out the integration over a period by Eq. (4.2), or in the form of a combination of a selective filter tuned to the frequency w 0 and a demodulator. It is obvious that the slope can also be found by another similar operation, namely, by using a periodic signal of an arbitrary form. I t is particularly convenient to use a signal which is a rectified wave :

that is, this operation is the same as that described above for comparing the values of the extremal index with two values of the parameter X nearby; however, here the measurement is continuous only over each half-period of the test signal. We shall not state a property of extremal systems which appears only in search operations, that is, operations which determine the slope S. Since at the extremum S = 0, the operation of the extremal system is that of an ordinary control system trying to reduce the error, i.e., the current value of S to zero. Therefore, extremal systems of this type (with an object characteristic which is a priori unknown) are often called automatic search systems. I n principle, the operation of automatic search systems can be divided into three classes :

(1) systems with a time separation between testing and operation (discrete systems); (2) systems where testing and operation are separated by a frequency band; (3) systems in which testing and operation are simultaneous.

4.1. Basic Principles of Operation

215

T h e first of these classes is, in principle, the simplest. T h e process is broken down into a series of successive steps. During the first part of each step the slope of the characteristic is found (or, perhaps, only its sign) by measuring the increment in the value of the quality index and corresponding increases in the tuning adjustments, that is, by feeding test signals into the tuning adjustments. I n the second part of each step the adjustment is changed by an amount which depends on the value found for the slope. Usually, one makes a distinction between systems with constant correctional steps and systems with proportional corrections. I n the first case, thevalue of the adjustment inXafter the kth step is correlated with the value of the adjustment made after the (k - 1)th step by the following equation : X , = XkP1-t- h sgn SkPl, (4.4) where h is the magnitude of the step. In the second case

Xk

= Xk-1

+

w,-1,

(4.5)

where h, is the coefficient of proportionality. I n general, the correction can be made to take into account all the previously found values of the slope of the characteristic which have to be remembered. If one knows these values, it is possible to extrapolate and to take greater steps without having to fear that one “jumps” past the maximum value. I n multidimensional problems with a large number of control parameters, the algorithm for the system is obviously considerably more complicated. We shall discuss, by way of illustration, one of these algorithms which is based on the application of the so-called gradient method. I t is desired to find an extremum for a function of n variables (i.e., with n tuning parameters) : y

=

frx, , x,, ..., X,J

=f ( X ) .

At the initial point X = X,,we vary the values of each of the variables and determine the partial derivatives

216

4. Extremal Systems

Then, a step is made at once for each of the variables AX,

= hS,

X i ,where

.

I n this way, a translation is made in the direction of the gradient of the function f( X). T h e system will continue in motion as long as 1 gradf(X, , ..., X,) l 2

2 S: n

=

i=l

does not vanish, that is, just until the system arrives at the extremum. There are other possible algorithms for the organization of the operating motion (cf., for example, Gel’fand and Tsetlin [16] and Feldbaum [94]). I t is important to mention that in principle they are all based on the measurement and use of the quantities Si. Now let us consider extremal systems in which the output and testing operations are separated by a frequency band. T h e basic scheme for this case is shown in Fig. 52. A test signal

ii

4 1

Correction device

I

t

1

S FIGURE 52

generator ( T S G) produces periodic oscillations in the turning parameters. T h e value of the index for the extremal of Y varies continuously. First we must find the amplitude of the harmonic oscillations in Y which correspond in frequency to the frequency of the test signal; as was shown above, this enables us to find the slopes of the controlled object characteristic. T h e value of S is used as the input of the correction device, which continuously carries out the operational variation in the tuning.

4.1. Basic Principles of Operation

217

I n order to prevent a distortion in the results of the measurement of S , this variation must be carried out at a frequency considerably lower than the frequency of the test signal. In the search for several of the tuning adjustments one must introduce test signals for each adjustment; obviously to obtain undistorted values for the quantities Si one must separate each of the test signals from the others by a frequency band and at the same time separate all of them from the operating frequency. T h e simplest methods of forming operating signals are the following. I n a system with proportional action, dX

2 = h,S. dt

I n a system with relay action, d Xv

dt

- h, sgn S .

We note that, in fact, a measurement of the amplitude of Y and, consequently, a measurement of S cannot be made without some lag and distortion. This is related both to an operation of the type ( j ( x ) sin

w,,t

dt,

and also to the operation carried out by the demodulator. However, the distortions which are caused by the higher harmonics are smoothed out in the operating signal, and, thus, continuous operation is possible. We also note that the limitation in the choice of the type of search operation restricts the range of applicability of extremal systems of this type to objects for which continuous measurement of the quality index can be made. We shall now consider systems in which testing and operation are simultaneous. In this case, we have two subclasses : (a) systems of the step type; and (b) continuous systems. Systems of the step type have the following operation algorithm. T h e first step, that is, the first variation in the tuning for X,is made in an arbitrary (but known) direction by a fixed amount : XI = x,+ c .

4. Extremal Systems

218

T h e next step is made in accordance with the results obtained by comparing the measured values of the quality index, Yo = Y ( X o )and Y , = Y ( X o c). For example, if we are looking for a maximum and c > 0, then

+

x, - x,= h,(Y, - Yo) or

x, x,= F(X, - Yo), -

where F is an arbitrary odd function. T h e next operation is carried out in accordance with the equation

or, in the more general case (for extrapolation systems), AX,

= F(d Y,-l

,A Y k - 2 , ..., d Y l ) ,

(4.7)

where

Y, =f(X,). T h e essence of such a process is the necessity of introducing the first arbitrary step (the necessity of self-starting). We shall not describe the operation of a continuous system. T h e first direction in the variation of the tuning, is again arbitrary. For example, let the tuning vary in accordance with the equation dX dt

-=

C,

where X(t

= 0) =

Xo,

C

=

const.

We assume that it is possible to measure continuously the quality index Y ( t ) which corresponds to the current value of X ( t ) . T h e data can be used in different ways to decide whether the choice of direction is correct, that is, whether it is toward the extremum, or whether it is necessary to make the variation in the other direction (to make a reverse step). Obviously, it is necessary to check continuously whether the chosen direction is the right one because a change in direction must be made immediately after the extremal point has been passed.

4.1. Basic Principles of Operation

219

We shall give a short summary of one of the methods of using the information (for the sake of simplicity, we shall assume a maximum is desired). T h e current value of Y ( t )is compared with the maximum of all the former values. If the difference E between them turns out to be negative and greater in absolute value than some fixed quantity xo, then a reverse step is made. If, in spite of this reverse, the measured value of the extremal index decreases as before, that is, E < -x0 (for example, because of the dead time of the object or of the measuring devices), then obviously there cannot have been a second switch-over on passing the level -xo with increasing E. It is not hard to prove that the steady state of such a system is self-oscillatory relative to the maximum point, which has an amplitude determined by the magnitude of xo and the possible lag in the object T . Figure 53 shows the curve for the process of establishing equilibrium and the steady-state limiting cycle in the phase plane in the variables Y*, X , where Y* is the measured value of the quality index

I

FIGURE 53

220

4. Extremal Systems

which is delayed relative to Y by the time T. When xu+O, T + O and the cycle shrinks to a point. An algorithm for the operation of the system can be summarized as follows. T h e motion begins in an arbitrary direction; the criterion for its being incorrect is whether or not the quantity E passes the level x0 in a definite direction (if a maximum is sought, then the conditions are E = -x0 < 0, d Y / d t < 0 ; if a minimum is sought, the conditions are E = xo > 0, d Y / d t > 0). T h e device which carries out these operations must, in principle, consist of two blocks : the first gives the value E (for the extremal index) and the second is a nonlinear indeterminate device with the characteristic shown in Fig. 54.

c

KO

FIGURE 54

T h e first block must have a memory storage element, the contents of which (the extremal value of Y for the previous period) must change with a reverse. T h e indeterminate device can be realized in the form of a relay in series with a ratchet mechanism (for details see Morosanov [53]) or through various electronic means. This algorithm is typical for many extremal systems with continuous action. T h e control mechanisms differ only in the form of the extremal index E. T hi s index can also serve as the derivative with respect to time of the quality index Y ( t ) or as the adjustment over a fixed interval of time E = Y(t T) - Y ( t ) or as the corresponding integral estimate E = J ' + ~y ( t >dt. 1 All the above descriptions of the various principles for constructing extremal systems involve the assumption that in operation the quality index is related to the tuning value by the equation

+

4.1, Basic Principles of Operation

22 1

where f is a nonlinear, lagless transformation which has an extremum at X inside some given region Q of variations in the tuning adjustments. However, the characteristics of real objects are considerably more complicated. T h e complication is caused by two basic factors. (1) T h e dependence of the measurable values of the extremal index Y* on the tuning adjustments is not lagless, so that the ideal characteristic (4.9) corresponds only to a state which is being established some time after the change in the tuning adjustments.

(2) T h e value of the extremal index depends not only on the tuning control X , but also on many other uncontrolled factors which must be considered as interference in the extremal system. T h e very measurements of the index Y are made by nonideal measuring instruments, the errors of which also must be considered to be interference. Actually one must divide into two types the uncontrolled disturbances which act on the object. (a) Disturbances which are relatively small. These disturbances are often called the drift characteristics. An extremal system is usually constructed so that it can shift for a slowly changing position of the extremal. In a sense, this change in the position of the extremal can be called the effective signal. (b) High-frequency disturbances in the quality index; this is, in fact, real interference and must generally be filtered out before one can evaluate the results of the measuring. T h e presence of these nonideal properties in the quality index as a measurable function of the tuning control greatly complicates the problem of extremal systems and creates many important dynamic problems related to the requirements of fast reactions and accuracy. T h e physical meaning is quite clear. Because of the dead time of the object and the interference in the search operation (the operation of measuring the slope characteristic), some time interval is required for the reaction. This interval must be large enough to decrease the dynamic error and the error due to incomplete filtering. On the other hand, because of the drift, the characteristic may change during the interval and the value of S which was “accurate” may turn out to be considerably different from its current value.

222

4. Extremal Systems

Moreover, even if there is no drift in the characteristic and if the optimal tuning values are constant (although unknown in advance,) the loss, over a long period of time, for each measurement made to improve the accuracy lengthens rather than shortens the over-all time which it takes the system to arrive at the extremum (even though each operating step is made in the right direction). Therefore, the problem is to make a compromise in choosing the time for each measurement by the criterion of the minimum over-all time which is to be spent in arriving at the extremum. Another problem is to select the magnitude of the test signal. I t is physically clear that the greater the power of this test signal, the easier it will be to distinguish it from the background interference. On the other hand, an increase in the level of the test signal means decreased accuracy in maintaining the extremal in the steady state. An important part of this chapter is devoted to explaining a method of studying dynamic properties of several types of extremal systems, with special attention given to the analysis of interference effects. I n many cases the results of this study can be used to solve problems described above of choosing the optimal control system. 4.2. Extremal Systems with a Time Separation between Testing and Operation; Systems with Proportional Action

We shall consider two basic types of systems with a time separation between testing and operation, namely, systems in which the operating step has a constant magnitude (i.e., systems with relay action), and systems with proportional action. Here we make the following assumptions : (a) Each measurement of the quality index requires a fixed interval of time which is determined by the properties of the index itself, or by the inertia of the system (in a number of cases, it will be assumed that the measuring can be done continuously and that the controlled object does not have any significant inertial effects); (b) T h e random errors in the measurement of the quality index Y are caused by interference which is added to the test signal (2,) or to the output signal (Z,),

y* = f ( X

+ 4)+ 2,.

(4.10)

4.2. Systems with Proportional Action

223

We also assume that 2, includes a slow drift (displacement) in the position of the extremum. An error in the measurement of Y generates an error in the measurement of the slope. We shall analyze the character of these errors for the two operation algorithms used to determine the slope as described in Section 4.1. (1) Let the measuring of Y take place only at discrete moments of 7 , which are separated by the fixed interval T , and time tk and tk let the extremal index be given by the equation

+

Assuming that the magnitude of the test step c is small and disregarding a change of the interference Z , with time T , we find from (4.10) and (4.11) the approximate equation

where zl,k

= Z1(tk),

AZ2.k

= ZZ(tk

+

52.k 7)-

"2,k

=-9

C

Z,(t,).

(2) Let the measuring of Y take place continuously; we then compute S * ( X ) by taking the average over the period 7 of the product of the output signal and the test signal. T h e test signal has the form of a rectified wave c sgn[sh ( 2 7 ' r / T ) t ] . Then,

This equation can be rewritten in the form (4.12), with the exception that

4. Extremal Systems

224

If the interference dZ2.,(t)is white noise, that is, if

&do)

=

dka(e),

then the error variance for the measurement is given by (4.15) If the interference has the correlation function

Rzk(e)=

u:ke-aleI

we obtain (4.16) The Dresence of a constant comDonent in the interference Z , has no effect on the results of the measurements. Consider the case in which the mean value of the interference increases linearly with time:

m&)

(4.17)

= a,t.

It is not hard to see that (4.18) that is, the mean component in the error of the measurement increases with an increase in the time interval T . However, because of (4.15) and (4.16), the opposite is true for the variance.

Let us study the dynamics of a system with proportional action, restricting ourselves to the case of a one-dimensional search. T h e basic equation for the operating motion has the form x k

- Xk-1 = hk-]S*(Xk-l),

or xk

=

xJ:-l

f

hk-l[S(Xk-l

+

‘1.k-1)

f

“2.k-11‘

(4.19)

+

We introduce the new variable ck = X , Z l , k , which immediately characterizes the magnitude of the deviation of the characteristic of the object from the current position of the extremum; its drift is defined by the interference Z l , k . T h e specific equation for the error ek has the form ck

= ck-1

f

hk-ls(ck-l)

+

5,-1,

(4.20)

4.2. Systems with Proportional Action where Ck-1

= L,J:-l

51.k-1

= Z1.k

+ h5,,,-, - Z1.k-1

225

,

'

First, we shall analyze the simple case in which S ( X ) = -1X (the search for the maximum of a parabolic function) and hkp1 = h = const. Equation (4.20) becomes linear with constant coefficients = acI;-l

where 01 = 1 - hl. Its solution has the form

c

+ L1,

1;

E,k

=

ak-"<"-,

(4.21)

+ a%O.

(4.22)

"-1

Obviously, the system is stable when 1 01 1 < 1. T h e expectation of the deviations ek from the optimum is given by the equation (4.23) where mt = M{<,} = const, and m, = M{e,} can be assumed to be zero when the initial deviation is uniformly distributed. We also compute the mean-square deviation I)

+ a"",

,

(4.24)

is the reciprocal moment between different values in the random sequence t k , which is assumed to be stationary. On the basis of the assumption that the initial deviation is uniformly distributed in the interval -L, L , where L is the range of the possible tuning positions, we find that

v,="3. LZ

(4.25)

226

4. Extremal Systems

If the values of simple equation

ck are assumed to be uncorrelated, we have the (4.26)

Passing to the limit as k m

= limm,: =

+ CO,

_ -mC

mT

(4.27)

hl'

1-a

k+m

we find

If the quantities tk are distributed normally, then cL are distributed normally so that the quantities we have found completely determine the distribution of the deviations from the extremum. For sufficiently small coefficients h the distribution of the deviations in the steady state will be nearly normal regardless of the distribution of Ck . (Cf. the analogous discussion in Section 1.2.)

h.

Equations (4.27) enable us to find the optimal values of the coefficient

We can determine, for example, the optimal value of h from the condition of the minimum mean-square value of the established error when mC = 0. Because of (4.21) and (4.27), we find V=

U

lh(2 - Ih)

-

o:,

+ h2u;,

Ih(2 - Ih) '

(4.28)

where u;, = M{(,2}, and u : ~= M{Cz2},while the interference and c2 is assumed to be independent. From (4.28) we find (4.29)

where fl = In many extremal systems the first established state is the most important. Equation (4.26) makes it possible to find the number of steps n needed to reach the value of the mean-square deviation which is equal to V , : n

1 V,( 1 - a2) - i 2 log a log VO(l - a 2 ) - UC2 .

= ~-

(4.30)

4.2. Systems with Proportional Action

227

Besides the deviations in the operation, there are test oscillations which are necessary to find S ( X ) .We designate by c2 the mean-square error caused by the test oscillator. Then, if we take ezdm as the admissible mean-square deviation after n steps, the number of steps necessary to reach this level in the error is given by the equation (4.31)

T h e resulting equations make it possible to find either the magnitude of the coefficient h which minimizes V , for a fixed number n, or the number of steps n for a , admissible value of the error E & . It should be noted that the problem of the extremal search for a fixed parabolic characteristic can be studied under considerably more general assumptions than those made above since we had a linear equation for the motion. I n particular, Chang [lo51 studied the problem for the optimal synthesis of a system with extrapolation from the condition of the minimum mean square of the quality index Y in the steady state reaching the extremal point. Moreover, it is obvious that fixing the characteristic for the object, in general, is in contradiction with the very principle of the operation of extremal systems, that is, systems which carry on a search for the extremum of previously K unknown control parameters. Therefore, in determining the dynamic properties, one must take into consideration the set of all possible characteristicsf(X). This set is limited by an assumption made in advance that, in the range where the search is carried out, f ( X ) can have only one extremum specifically (let us say a maximum) for a number of values of X ; further, we assume this extremum to be at the coordinate origin. Thus, we impose on the slope characteristic the restriction sgn X . sgn S ( X ) < 0.

(4.32)

It is practically always possible to impose more rigid conditions of the type b

< I xS ( X ) I < a,

(4.33)

where 0 < b < a < 00, that is, S ( X ) would be an arbitrary function lying between two straight lines.

228

4. Extremal Systems

There are two basically different approaches to determine the dynamic properties of an extremal system. T h e first is called Bayes’ approach. I t is assumed that the distribution of the random function S ( X ) which satisfies the conditions (4.32) and (4.33) is known a priori; the optimum dynamic estimate is computed by averaging over the whole set of realizations of S ( X ) . I n its simplest form, Bayes’ approach is to represent S ( X ) in the form of some function of X with random coefficients, for example, S(X)=

-zx,

(4.34)

where 1 is a random parameter with a known distribution. T h e results found above for a parabolic characteristic (and also the results of Chang [lOS]) can be easily generalized in the same way. I t is obvious that the generalization can be carried out simply by taking the average of the found equations over the set of realizations 1. However, the notion of having an apriori knowledge of the distribution of I is usually purely hypothetical and, consequently, Bayes’ approach is not fully effective. We also note that the expression of the slope in the form (4.34), that is, the specification of the parabolic dependence of the quality index on the tuning parameters, is admissible only for small deviations from the extremum, so that, in general, it is suitable only to study the operation of an extremal system in the steady state. T h e second approach to evaluate the properties of the system is the so-called minimax approach. I n this case the estimate is made with respect to the worst function S ( X ) [in the sense of some given dynamic criterion] compatible with the restrictions. We shall describe in greater detail the method of applying the minimax approach to the problem under consideration. Because of the initial equation (4.21), and assuming the statistical independence of the quantities 5, = (l,k + h k [ 2 , k ,we obtain (where h, = h = const)

4.2. Systems with Proportional Action

229

Using the restrictions (4.32) and (4.33), we thus find that

while it is assumed that h < 1 / b (this condition guarantees the stability of the search process for any of the possible characteristics). Performing the iteration from k = n to k = 0, we find from (4.35) that (4.36)

where a0 = 1 - hb. Comparing this with Eq. (4.26) we note that the upper limit for V , is the same as it was for V , when there was a linearly alternating slope if we let* 1 -= b. Hence, it follows that, for all characteristics satisfying the restrictions (4.32) and (4.33), the number of steps required to reach the extremum within a given error range €id,,, satisfies the inequality n<-

1

(rhl

-

?)(I

- a0Z) - 0 5 3

(4.37)

Here, just as in (4.30), it is assumed that m: = 0. Since the initial discrepancy is unknown, we obtain V o = 4L2 by the sequential application of the minimax principle. T h e optimal choice of the coefficient h in the indicated minimax sense can be made either by minimizing the right side of the inequality (4.36) with n fixed or by minimizing the right side of the inequality (4.37) with a fixed value of the error €id,.,, . In the absence of drift = 0), when the steady state is fixed, one can solve the more complicated problem of an optimal choice (in the minimax sense) of the sequence of values of the amplification coefficient h, at each of the various steps. -

* In

the derivation of (4.26) it was assumed that m,

=

0.

4. Extremal Systems

230

We shall prove that the following rule for variation in the coefficients h, is an optimal condition : h -

- 052

+ b”bV0 o(K + I ) ’

(4.38)

where at2 3 a & . First, we shall give an estimate of the current value of the meansquare deviation from the extremum if h, is chosen in accordance with (4.38) :

< v ~max{(l - ~

(1

- hk-1b)2,

-

hk-1a121

We demand that all k, beginning with k h,

=

+ hr-loc . 2

2

(4.39)

1, satisfy the condition

<-a +2 b ’

(4.40)

This corresponds to the following restriction for the initial deviation : V <

20 52

‘b(a - b )

(4.41)

T h e inequality (4.39) can then be written in the form VI,

k

< VI,-l(l

-

hk-lb)2

+ hi-1q2.

(4.42)

Substituting expression (4.38) into this equation and iterating from n to k = 0, we obtain

=

(4.43)

Under this control rule the steady-state error (due to the operation) obviously will be zero.

4.2. Systems with Proportional Action

23 1

Now, let I S(E)/EI = b, and at, = ut; then the inequality (4.42) becomes an equality : V k = Vk--l(1 - hk-,b)'

+ h2,-la:.

We minimize V , with respect to h,-l for fixed VkPl . T h e corresponding minimum value of V , is given by the equation (4.44)

where

From (4.44) we obtain for the minimum value of

v 7 l

Consequently, for any rule for the variation in h, , which is different from (4.38), (4.45)

that is, the definition in (4.38) is optimal in the minimax sense described above. It is obvious that for some particular variation in the slope which satisfies the restrictions (4.33), the principle of (4.38) may not be optimal, so that, for the optimum a more rapid convergence to the extremum must be found than is given by the estimate (4.23). We now assume that the value of the slope S(E,-~)used to form each Kth step is the average of the results of some measured number n, steps, while the errors of each of these measurements are independent and the variance does not exceed uZ2. T h e error variance in S ( E , _ ~will ) decrease in proportion with increasing n, , although the time spent in making the measurements will increase. T h e total time after p searching steps is proportional to n = E[=l n, , if the time spent for each working step itself is negligible.

4. Extremal Systems

232

I t is not hard to show [lo21 by using the iteration device already described that, in this case, a variation in h,. according to the rule (4.46)

will be optimal, while (4.47)

and the restrictions h,

26 ZFli n ,
2uc2

Q =

qz-73 1,

(4.48)

are imposed on the quantities h,. . T h u s , the introduction of an intermediate averaging process does not in principle improve the estimate of the speed at which the extremum is approached. At the same time, the averaging method is effective if considerable time is spent on the operating step. Nevertheless, we note that for several types of characteristics and error distributions this deduction is not valid. I n particular, Feldbaum [91] has shown that, even if the time spent on the operating step is negligible, in the presence of a piecewise linear characteristic and a normally distributed error, the averaging method can improve the action speed. Next, we shall describe a method of solving minimax problems based on the application of statistical linearization. We return to Eq. (4.20). Its nonlinear function S ( E )can be replaced by a linear one

S(€) * --l&

nearest to it in the sense of the minimum mean-square deviation. Because of (1.2 I4), we have

4.2. Systems with Proportional Action where

W(E)

is the one-dimensional probability of

4, =

s

c2

1

F.

S ( c ) / c 1 w(c) dc

J €%(€) d€

233

It is obvious that (4.49)

and, consequently, with the restrictions of (4.33), l,.t

2b

(4.50)

does not depend on the form of w ( E ) . After statistical linearization, Eq. (4.20) takes the form, analogous to (4.21), = actck-1

<

+

<,,-I

9

(4.51)

where act = 1 - l,th 1 - bh = ( y o . T h e quantity V , given by the equation

is a monotonic increasing function a c t . With these restrictions on S(c), the largest possible value of the error is reached when act = a. , that is, it corresponds to the searching error for the characteristic with the slope S ( e ) = - b e , which is in agreement with the above argument. Statistical linearization can be used to find approximate solutions of problems with more general postulates regarding the form of the characteristic of the object. Letf(c) be a differentiable function which has a unique extremum at s = 0, while in a given range of tuning values it satisfies the condition

(4.53) that is, it lies between two parabolas of the second order. Moreover, we assume that f(c) is symmetric relative to the axis at c = 0 (the slope S ( c ) = f'(s) is an odd function of c ) , and that the mean value of the interference nz: is zero. Because of the restrictions imposed on the steady state, m, - 0. Thus,

234

4. Extremal Systems

where, assuming an approximately normal signal at the input of the nonlinear element, we have

(4.54) We shall estimate the limits of variation in the coefficient k(o) for an arbitrary symmetric characteristic which satisfies the restrictions (4.53). Integrating by parts, we obtain

Therefore, the value of k(u) is determined by two terms,

k(0)

=

k’ + k”,

(4.55) (4.56)

where

Hence, it is clear that

kmm

=

-mini k’ I

+ max k”,

kmin =

-maxi k’ I

+ min k”.

(4.57)

It is also obvious that mini k’ 1 and min k” reach the curvef(c) = - ( b / 2 ) r Z , which gives the upper limit of the region where the possible characteristics can be located, whereas maxi k’ 1 and max k” reach the curve f(r) = - (a/2)e2, which is the lower limit of this region. Hence, the minimum value of k ( o ) is reached (Fig. 55) on Curve 1 and the maximum value is reached on Curve 2, which are composed of boundary sections of the region.

FIGURE 55

4.2. Systems with Proportional Action

235

Curve I does not have a single local extremum and, therefore, strictly speaking, the resulting estimate for the minimum value of K(o) is too low and gives a somewhat greater significance to the steady-state mean-square deviation than is actually possible for the class of characteristics under consideration (cf. also Section 4.3). Computation of the limits of k ( o ) gives

+ A) - bA,

kmax =

a(l

kmin =

b(l -t A)

-

(4.58)

ad,

where

Because of the monotonic dependence of V on value of V is reached when a<,t,niax =

<

I - kminh

=

I

-

h[b(l

, the maximum possible

a<.,

+ A) - ad].

(4.59)

Since A I , the resulting estimate differs little from what was found under the restrictions (4.33). For still more general restrictions of the form -

a, - a-

<

€2

2

f(6)

a, >, b,, ,

< a

2b

one can show that kmax =

+

~ ( l A)

-

-

bA

b, -

by, €2

> 0,

ao + + ___ 40,

(4.60)

60

U2

(4.61)

where

A corresponding estimate for

Vmax

follows from (4.61).

I n the above discussion on how to estimate the neighborhood of the extremum, we have used the value of the mean-square deviation of the control parameter as the value which corresponds to the extremum. However, a more natural, and practically a more important, estimating method is to measure the loss in quality, in the sense of the deviation of the current value of the quality index Y from the extremal value (taking its drift into account).

236

4. Extremal Systems

W e shall give an approximate estimate of the maximum possible quality loss in the steady state when a search is made of the set of characteristics which satisfy the restriction (4.53). T h e extremal value of the quality index is zero, and, therefore, the quality loss O v is given by the equation

where W ( E ) is the probability density Because of (4.53),

< J-:, ;

in the steady state. a

5

dy

E

€'LW(E) dr = -

2

a

V <2

Vmaa

.

(4.62)

W e find the quantity Vmax in its turn with respect to act,max, which is given by Eq. (4.59).We note that the estimate of AYobtained by this means is not the exact upper limit dy for the given set of characteristics, but only an estimate of such an upper bound. W e shall describe another way of estimating the action speed of a system based on the theory of Markov processes (a brief introduction to this theory will be found in Appendix V). We shall assume that there is no drift and that the errors Ck are mutually independent and have probability densities zot(z). W e shall find v(x, n) which is the probability that ~ ( t . ) c " , , where en, is a tuning adjustment at an admissible distance from the extremum, while e ( t ) > e m when t < t, , and € ( t o )= x > 0, that is, the probability of the first output in the c,,~ neighborhood of the extremum c = 0 after n steps, if the first operating position is located on the right branch of the characteristic. I n accordance with Eqs. (A. 19) and (A.20) for determining v(x, n) we have the following recursion relations:

<

v(x, n)

=

1"

v(x, n

-

I)p(x, y ) dy

(n

> I),

(4.63)

'rn

while v(x, I ) =

p ( x , y ) Q.

Here, p ( x , y ) is the probability of passing in one step from the point x in the interval ( y , y d y ) . In correspondence with the basic equation (4.20) we have

+

(4.64) where we assume that h ,

=

h

=

const.

4.2. Systems with Proportional Action

237

We introduce the generating function

c; si

g(x, s) =

S"Z'("X,

n).

Then, we obtain from (4.63)

We substitute into this relation the expression for the transition probabilities (4.64) and introduce a new integration variable,

z

=

y

-

x

-

hS(x).

T h e equation for g(x, s) is written in the form

1

* X

J

-g(x, s) =

g[z

+ x + h S ( x ) , s]zug(z) dz + j-s(z) w& z) dz. --a

r,-z-hS(Z)

We introduce an assumption regarding the low probability that the transition from the starting point F = x to the extremum will occur in one step. Then we obtain finally

Integration of Eq. (4.65) is difficult for any form of the extremal characteristic However, for the interesting case of the piecewise linear characteristic

f(c).

f(<) it can be performed exactly. For 0 < c < co we have S ( c ) form

=

-hi

1

'

-b, and the solution can be sought in the

(4.66)

g(x, s) = erf's).

In fact, the equation

where B

=

bh, is satisfied if t ( s ) is chosen from the condition

(4.67) or where F ( t )

,s

=

X

F(s, t) e%&)

=

In s

+ In ~ ( t-) pt

=

0,

dz is the characteristic error function l.

4. Extremal Systems

238

Without going on to the solution of Eq. (4.67) with the goal of finding an explicit expression for t(s), we restrict ourselves to finding only the first two moments. It is not difficult to show that

where u = Ins. From Eq. (4.67) it follows that 1

dt

-

(z),=o = (aF/at),,, d2t

-

(l)lllo = -

-

(a”F/at2),-o (“/at):=,

-

-B

=

1

+ ( y ’ / ~ ) ~ ’. . ~

+ (V’/V) 1

- ((V’/T) - (V’Y

-B

u-o

But when o = 0, Eq. (4.67) has a root at t = 0, because ~ ( 0= ) 1. Hence, it is clear that

V”t(41 L

o = V’(O),

p’”t(o)l

L o = T”(0).

On the other hand, .oc

~’(0) =

J

z w c ( z ) dz = -m

0,

~“(0)=

J

X

-m

z2wwl(z) dz

= ocz.

Therefore, we have finally

M - - x1 ;

“-B

D , = - x02.

B3

(4.68)

Equation (4.68) shows, in particular, that it is not always possible to estimate the dynamic properties of extremal systems only on the basis of the expected time of search M , , first introduced by Feldbaum [91], because in that case the value of M , does not depend on the magnitude of the interference and does not characterize the interference stability of the system. T o find a probability estimate of the number of steps necessary to reach the extremum, one can use an approximate method which is suitable when the probable number of steps is rather large. We restrict ourselves to finding the mean time m(x) required for the transition from the point s = x to the point E , ~ .From Eq. (4.63) it is easy to see that -x

m(x) =

-r

m[z

+ x + hS(.r)]wr(x)d z + T ,

(4.69)

where T is the time required or one step. We expand the expression m[z x hS(x)] into a series

+ +

m[z

dm + x + hS(x)] = m ( x ) + [ z + hS(x)]+ h [ z + hS(x)lad2m - + .. dx dxp

4.2. Systems with Proportional Action

239

and, using only the first three terms of the expansion, we substitute them into (4.69). The result can be written in the following form:

d2m hS(x) dm + [hS(x)l2)+- - + 1 = 0. dxa T dx

I

- {1

2T

05

For the parabolic characteristic f(x) uc2

-

+ b2h2x2d2m 2T

=

-(b/2)x2 we obtain

bhx dm +1=0. T dx

dx2

(4.70)

(4.71)

The boundary conditions for Eq. (4.71) are physically obvious.

(4.72) After integration we find

where 8 = hb. For very intensive interference and a comparatively small useful component of the operating step, ac2

> 82x2,

Eq. (4.71) can be written in the form aC2 d2m 2T dx2

8x dm +1=0. T dx

(4.74)

An analogous result can be obtained if we start from the differential equation which corresponds to the difference equation (4.19). Integration of (4.74) under the conditions of (4.72) and neglecting the small quantity em gives

(4.75) where m

exp(f2) dt

I ( a ) = --

4,

0

exp(y2) d y .

(4.76)

1

For the range of small values of a which interest us a graph of I(a) is shown in Fig. 56. We note that, in practice, the only case of interest is that in which there is intensive interference. In fact, if u and 8x are of the same order and x is sufficiently large (i.e., 8 is small), it is not difficult by means of successive approximations to construct a solution of the form

m(x)

-- =

T

1

-

8

In

x

+ E*

(2n - I)!! 2 (F)P-', 2n m

a<

2"

(4.77)

240

4. Extremal Systems

which, when ut = 0, becomes an approximate solution for the transition period from x to c,,, , which in turn was computed from the initial equation (4.19) for 5 = 0. T h e correction terms of the series are barely larger than I / x , and, therefore, they are not significant.

FIGURE 56

Just as in the analysis of an object with piecewise linear characteristics, we have here a case where m, in practice does not depend on the magnitude of the interference.

Example I . Consider the operation of a device which automatically searches the extremum by the gradient method proposed by Krug [41], and let us limit the analysis to a search along one coordinate. T h e generator of the test signals produces a signal in the form of a rectified wave c sgn[sin 2wot],

which arrives at the input of the object with the characteristic f ( x ) during the time interval T = n / w o (the amplitude of the operating signal remains unchanged). At the output of the object the slope is changed by integration of the output signal as a function of the rectified wave of the same frequency. T h e computation of the errors in the measurement caused by interference can be done in this case by using Eqs. (4.15) and (4.16). T h e output signal at the measuring device is equal to C T [ ~ ’ ( X ) (1, and is used to determine the magnitude and the sign of the operation step AX, which is taken during the following interval by continuation T - T = n / w o , according to the equation,

+

Ax = kCT[Y(X)f (1, which obviously agrees with the general equation (4.19) where hk-l = KCT and 2, = 0. Therefore, the results developed above can be used to find the time of search for the described system.

4.2, Systems with Proportional Action

24 I

I t is interesting to compare the calculated results with experimental data produced by a model of the system (cf. the accompanying table, where xo = 40).

All of the previous considerations have been concerned with the analysis of a one-dimensional search; however, several interesting and rather simple results can be found by investigating the effect of random errors on a search process involving several optimal tuning adjustments. Let the search for the extremum be carried out by varying the parameters X , , X , , ..., X,,, . T h e extremum is displaced by the effect of a random drift which has the components Zl', Z2',..., Z,,' on the X , , X , , ..., X, axes. We assume that near the extremum (which is a minimum), the characteristic for the object can be approximated by a positive definite form of the second order with respect to the errors ei = Xi Zj', that is,

+

(4.78) i=l k = l

In matrix form this equation can be written Y

=F(8) =

p w = &(X + Z')'B(X + Z ' ) ,

(4.79)

where 8 , X , 2' are column matrices composed of the quantities e i , X i ,Zi', and B is a square ( m x m ) matrix composed of the coefficients bi, , while the index t denotes the transition to the transposed matrix. We shall describe the search process by the gradient method. T h e value of the quality index is measured at the moment ti . T h e measurement takes place with an error of Z " ( t j ) :

+

Y*(t,) = Y ( t j ) Z"(tj).

242

4. Extremal Systems

Further, let there be an increase cj in the parameter X i and a measurement of the corresponding value of the quality index at the ( l/m ) T. T h e slope, with respect to the parameter Xj , moment t j can then be approximated in the following manner :

+

or rn

sj ‘v z b j k ( X k

+

zk’)

+

[j”

=

sj +

(4.80)

[j“,

k=l

where we are using the notation

Here we neglect both quantities of the order cj2, as well as the magnitude of the drift during the measuring time. Similarly, successive measurements of the slope are made for all parameters, after which, the operating step is made simultaneously with respect to all of the parameters. T h e increase in the parameter Xj at the nth step is given by the equation Xj,,l = Xj,n-l

-

hSj*

( j = 1,2, ..., WZ)

(4.8 1)

or, in accordance with (4.80),

k=l

Changing to the variables e i , n we obtain (4.82)

4.2. Systems with Proportional Action

243

where

Equation (4.82) can be conveniently written in matrix form :

a,, = ( E - hB)CFn-, + z - 1 ,

(4.83)

where E is the unit matrix and Zn-l is a column matrix composed of the elements < j . n - l . T h e linear difference equation (4.83) can be written in the form of two equations : the first one for the mean components (4.84) E, = (E - hB)Bn-, Z n 4 ;

+

and the second for the unbiased components

+ z:-,

ano= (E - hB)&-,

.

(4.85)

+ ( E - hB)"g0,

(4.86)

T h e solutions have a form analogous to (4.22) CFn =

2( E

- hB)n-"Zv-l

"-1

c n

8,O

=

"-1

(E - hB)7'-'z~-l+ (E

-

hB)b,O .

(4.87)

Assuming that the mean components of the drift Zj' and of the interference Z" change with constant velocity, we obtain -

-

2, = Z = const,

and, hence, it follows that

8"=

c n

( E - hB)"-"Z

"-1

+ ( E + hB)"ao,

or, if we use the formula for the sum of a geometric progression,

8"= [ E - ( E - hB)]-"E =

1

- ( E - hB)"]Z

B-'[E - ( E - hB)"]Z

+ (E

-

+ ( E - hB)"do

hB)"g0.

(4.88)

244

4. Extremal Systems

Here, the index - 1 denotes the inverse of the matrix. We note that Eq. (4.88) is completely analogous to Eq. (4.23) derived for the one-dimensional case if we replace the coefficient b by the matrix B and the coefficient CY = 1 - hb by the matrix A

E

-

hB.

A condition for stability in this case would be the requirement that all the roots of the equation IAE-AI = O

(4.89)

lie inside the unit circle. T o estimate the dynamic properties of the system, we define the loss in the quality at the nth step AT,

=

M { Y , - Yext}= M{Y,}

=

&M{BntB8,}.

(4.90)

T h e quantity A T , can be broken down into two terms : =

;(An'

+ A,"),

(4.91)

where A,'

M{(&?ptO)'B&?,"}, -_ A," = B,'B&?, =

We shall compute the first term using Eq. (4.87) and assuming that all the variables ti,, where ( j = 1,2, ..., m ; n = 1, 2, ...), are mutually uncorrelated, while (4.92)

We also take into account the fact that the matrices A and B are commutative AB

=

( E - hB)B = B

-

hB2 = B(E - hB) = BA ,

and symmetric B

= Bt,

A

=

E - hB

=

( E - hB)t = A t .

4.2. Systems with Proportional Action

245

Then, we find that

or, because of the conditions (4.92),

(4.93) where cii,? are diagonal elements of the matrix C, = BAZT

We introduce the matrix

Then (4.93) can be written in the form m

(4.94) T h e matrices D, can be computed comparatively easily if, once again, we use the sum of a geometric progression :

T h e second term in (4.91) can be computed directly by using Eq. (4.88). Without writing a sufficiently complex general expression for d Y , , we limit ourselves to a more detailed analysis of the steady state (n +a),since, in general, the initial approximation (4.78) is valid only near the extremum.

246

4. Extremal Systems

From (4.94)when n D = lim D, n-tm

1

=

B(E - A2)-'

h 2

= B(-h2B2

+ 2hB)-'

-l

= -(E - -B)

2h

it follows (assuming a stable system) that

+ CO,

(4.96)

For sufficiently small h, we can use the approxlmate expression D

a1 ( E + h2 B) ,

N

that is, dii

Passing to the limit as n A'

= lim An' = n-Mo

N

1

2h (1

(4.97)

+ zh bii) .

so in (4.94),we obtain

--f

c m

i=l

2:

1 " diiuii _v - (1 '2 i=l

+ h bii) oii ,

(4.98)

since, because of the stability of the system, lim cii,%= 0.

n-tw

From (4.88)we find an expression for the mean component of the steady-state error : -

6 Z - hB - 1 z .

Hence, A"

= lim

n--M

-

-

A," = 6 t B g

1 -

= - z t B - l B B - 1 ~=

h2

(4.99)

&h2

zts-1~.

(4.100)

Finally, we can write

(4.101)

where b ~ 'are the elements of the matrix B-l.

4.2. Systems with Proportional Action

247

We note that an analogous problem has been solved by Feldbaun

[93],which takes up in greater detail the realization of the search

by the gradient method, although the final result is not given in a closed form. Example 2. We shall now give an example of the calculation of the quality loss in a two-dimensional search, where the characteristic for the object can be written in the form Y

=

&(q2

+ 28€,C2+

c22).

(4.102)

The matrix B has the form

and, hence, B-1

1 1-8'11

= ___

and

-B

1

The matrix D can be written in the form

1 D = 2h

h

- - p1

1 -p3

l - -

2

-l

2 h 2

From this we find immediately

Restricting ourselves to the lowest terms of the first power of h, we find

which agrees with the result found directly from the approximation equation

(4.97).

Finally, we see

From Eq. (4.103) it follows directly that an increase in 8, that is, a deterioration in the circular symmetry of the plane (4.102), will increase the quality loss. This phenomenon, which has already been noted by Feldbaum [93], can be explained by the choice of a single amplification coefficient h with respect

4. Extremal Systems

248

to both the control parameters, that is, by the basic characteristics of the gradient method. With the kelp of (4.103), one can make the optimal choice of h in order to minimize AY. It is obvious that if there is no drift (5, = 0), then hopt = 0, since u;, =

0’1

5. I

+ h”o’5”, ,

__ q q+hc?”. =

We find the value of hop, for the more important case when -

u;,, =

0,

5;.

=

0.

After some fairly simple manipulations, and neglecting the higher powers of h, we find that

(4.104)

4.3. Discrete Extremal Systems with Constant Steps

T h e basic equation for the operation of a system with constant steps mentioned in Section 4.2 can be written in the form x k = xk-1

f h Sgn[S(Xk-l

+

‘1.k-1)

+

62.k-11

(k = 1, 2, ...), (4.105)

where h is the magnitude of the step, S ( X ) = f ’ ( x ) is the slope of the characteristic for the object, Z , is the drift in the position of the extremum, and c2 is the random error in the measurement of the slope. Just as in Section 4.2, we change to the variable ck = x k Zl,k, which characterizes the deviation of the control parameter from the displaced (because of the drift) optimal value :

+

cl,k-l Z l , k - Z l , k - l . Henceforth, we shall assume that c2.1iare mutually uncorrelated sequences for independent random

where

,

=

variables with constant mean values mt, , ms, and constant variances sequence of values of Ek forms a Markov chain and, this chain is discrete since ck can take only the values 1, f 2, ... . c j = c,, + j h w h e r e j = 0,

u:~, u f 2 . T h e when 5, = 0,

4.3. Discrete Extremal Systems with Constant Steps

249

First, we shall consider an approximate method of estimating the dynamic properties of the system which is applicable when there is very heavy interference We shall find the mean and the mean-square expressions for

c2.

f[S(fk-l),

62.k-11

= Sgn[S(Ek-l)

+ 62.k-11

when the ekpl are fixed :

where (4.108) Thus, the equation for the operation can be rewritten in the form Ek

= Ek-I f

51.k-1

4-h [ 2 p k - 1

-

1

7k-119

(4.109)

where the variance of the random variables vk does not exceed unity. In fact, ogk-, = 1 - [2p,-, - 11’ < 1. T h e closer the variance is to unity, the more intensive is the random interference. Intensive random interference also causes, in general, a linearization of the nonlinear function f ( S , 5,) [cf. Section 1.11, that is,

Wf/%-1}= 2 P k - 1

-

1

= X [ S ( C k - , ) + mc,],

(4.110)

where the coefficient X depends on the distribution of the random If the probability density of the random component component , because of (4.108), is W ; ~ ( Z ) then,

c2.

where a ,

=

-[S(r,)

+ m;?], and, consequently, X

=

2

@k

(r) = 2W<,(O). ak=O

250

4. Extremal Systems

In satisfying the approximate equation (4.1 lo), the basic equation (4.109) takes the form

+

+

where tr,= tl,k h(.rk Amc,),and h, = hX. But Eq. (4.112) is analogous to (4.20). Thus, all the results of Section 4.2 can be adopted for a system with constant recovery steps, provided, of course, that the condition of very intensive interference c2 is satisfied. I t is of interest to study another means of analyzing systems with constant recovery steps which will give a sufficiently complete picture of the processes. T h e method is based on the theory of discrete Markov chains (cf. Appendix V), which is useful in the analysis of this problem when* t1 = 0. As a result of the Kth step, let the coordinate E have the value c j . Its further variation because of the interference will be random. As a result of the next step the coordinate can take either one of the two values, ej+, or ej-,. T h e probability of moving to the right, from the point cj to the point E ~ + is ~ ,equal to pj . T h e probability of its moving to the left q j = 1 - pi . If the range of possible values of 6 is bounded, then the transition conditions must necessarily change at the end points of this range ( j , < j < j J . At the lower limit we can assume that the probability of making one step to the right i s p j L , and the probability of staying in the same place is q j , . Analogously, at the upper limit the probability of passing to the left is equal to qj, , and the probability of remaining in the same place is pj, . Because of the assumptions we have made, we see that the indicated transition probabilities do not depend on the previous history of the motion of the system (in particular, on the number of steps), but only on the value at the given moment. We shall develop a method of computing the steady state. We adopt the usual notation of pii for the probability of going from one state i to another state j in one step.

* I n general, this method can be considered for the case when the drift has a constant velocity ( m c= const), assuming that m c and h are whole multiples of some number h, which is taken to have a discrete level.

4.3. Discrete Extremal Systems with Constant Steps

251

For the system under consideration,

p . . = p*. , = q i ,

=

0,

j=i+l, . . J=Z-l, j#i+l,i-l,

(4.113)

with the exception of the boundary points where we have the conditions

p3.1 . 1.

-

p.

~- 9il ,

=

0,

j J

.

=it + 1 ,

p1.1 , 1. --p . 3 u ,

.

j#jl,jl+l,

-

qj,,

=

0,

j =j. j

= j.,-

1,

(4.114)

j # j $ , , j u- 1 .

T h e probability uj that the error E will have the value ei in the steady state can be found by using the obvious recursion relation

Writing this relation for all j in the indicated range and adding the boundary conditions (4.1 16)

as well as the condition for normalization Czj, ui = 1, we obtain a system of linear algebraic equations which determine u j . Solving this directly, or using the general formula for Markov chains, we arrive at the result that

We shall apply this general formula to several concrete examples. Example 1. function

Let the characteristic of the object be the piecewise linear f(€) =

--bl

E

I.

(4.118)

4. Extremal Systems

252

We introduce the discrete variable 1, which determines the amount of the initial deviation from the extremum: =

€1

€0

+ lh < 0 < (I + 1)h +

=

€0

E1,l.

(4.

Here, we shall restrict ourselves to the case of intensive interference, so that the approximate equation (4.1 10) is satisfied, that is, P k

=

I

2

+ ms,].

+ 2h

(4.120)

Then, because of (4.1 18), (4.1 19) and (4.120), we obtain

Pi

=

4 [1

+

=

3 [l

+

+ b)l, +is,

-

jl

b)],

1

< j < 1,

+ 1 <j <j,,

(4.121)

Consider, first, the case when mg. = 0. Then, by applying the usual formula for the sum of a geometric progression and using (4.117), we find

-

where

I-, -_ 2

(4.122) mi-l--l

[I -

,=-

1 1

-

t

(,l--il+1

+ ai"-')]-l,

hb

+ hb

+

We shall now assume that the numbers 1 - j L 1 and ju- 1 are large, that is, that the distance from the extremum to the boundaries of the possible variation in the tuning is large. We can deduce the simple expressions

(4.123)

It is not difficult to prove that

M{j}= 1

+ 4.

The variance of the distribution also can be easily calculated: u,2

=

M !(j- 1 -

-1'1 1 2

1

2a

=i+---(1 - a)'

(4.124)

4.3. Discrete Extremal Systems with Constant Steps

253

Now, let mc, = const # 0,that is, let the interference Z , , which distorts the correct measurement of the quality index, have a mean component which is linearly dependent on the time. By disregarding the boundaries of the basic range (i.e., by letting j , - 0 0 , j . + CO, we find from Eq. (4.1 17) that -+

i-1-1

where

[=+GI’ 1

1

(4.125)

-I

T h e distribution (4.125) clearly is asymmetric. T h e deviation of the mathematical expectation from the extremal tuning value E = 0 is given by the equation (4.126) and, if

K

~

< 1, .

~

(4.127) An approximate expression for the variance also can be written in a simple form: (4.128) T o estimate the degree of the approximation, one can compare it with the exact equation

found for mCt = 0, that is, when

Example 2.

K,

=

K,

=

K.

Let the characteristic for the object have the form f(6)

=

-4b~’.

(4.129)

4. Extremal Systems

254

Let mC,= 0, and let 5, be uniformly distributed:

0,

Then, because of (4.1 1 I),

>a.

I Z I

(4.130) =

0,

S ( € j )

<

- a,

or for the given characteristic of the object, P j

U

= 1,

€ j

< --,

0,

€ j

>-

=

b

a

b

In order to make the computations simple, we shall assume that co = 0 (this assumption does not have any special significance since we are trying to find the steady-state distribution). By definition, we have c j = E~ j h = jh.

+

Thus, the expression for the transition probability can, finally, be written in a very simple form:

P,

=

1,

=

&(I

i<-B,

- 7i ) '

lil < B ,

(4.131)

= 0, i > B, where B = a/bh is assumed to be a whole number. It is not difficult to see from the general formula (4.1 17) that the steady-state probability is given by the binomial distribution. 241

=

c$;i(&)2@,

-B
The expectation, obviously, is zero and the variance is

(4.132)

B (4.133) 2' T h e derived expressions make it possible to solve several problems with respect to the optimal choice of parameters in an extremal system whose fundamental purpose it is to remain at the extremum in the steady state. oi2

= -

4.3. Discrete Extremal Systems with Constant Steps

255

By way of an estimate of the dynamic properties of a system, one can choose the magnitude of its quality loss =

I

Yext -

WmJ I.

(4.134)

A more detailed investigation of the process of measuring the slope, for example, by the method of (4.13), permits us to write

(4.135) where c is the magnitude of the test step and where TIT is the ratio of the time spent on measuring the total time of a single step. Calculations from Eq. (4.134) or (4.135) are not particularly difficult if the probabilities of the possible values of E are known. Thus, one can find from (4.125) and (4.134) the following value for the quality loss for a piecewise linear characteritic: (4.136)

Here it was assumed for the.computation that

c,,

=

-(I

f 4)h.

For a parabolic characteristic, the quantity A T , computed from Eq. (4.135) is expressed in terms of the mean-square deviation from the extremum as

-

LlY When

=

b 2

[m,2

+

0 , '

+f C Z ]

.

(4.137)

is uniformly distributed, we can find by using (4.138) that

(4.138)

For the optimal choice of the parameters it is, of course, necessary to take into consideration the dependence of the error characteristic l2 on the magnitude of the test step and on the time taken for the measurement T. T o illustrate one of the qualitatively uniform characteristics, we shall use a simpler function to estimate the deviation from the extreme.

4. Extremal Systems

256

We shall choose the optimal values of the measuring time T and the magnitude of the test step c from the condition of minimizing the following function : L

=

I m, I

+ 2(. + ,JJ.

(4.139)

We shall carry out the calculations for the concrete cases when the characteristic of the object is piecewise linear and the interference Z , ( t ) is normal white noise with an expectation which linearly increases with time. T h e computation of the slope is made in accordance with the technique of (4.13). Using (4.27) and (4.28), we can express the estimating function in terms of the parameters K, , K~ :

where K, =

(h(mC,

+ b),

K~

=

h(mtl

-

b).

Because of (4.15), (4.18) and (4.16),

Cumbersome, but nonetheless fairly simple, calculations give the following expression for the optimal period for measurement: (4.140) that is, the optimal period depends on the deviation of the test step in the quantity Y from the rate of change of the mean component of the interference. * We note that the choice of the period does not depend on the noise level d (in this instance d is sufficiently large). Having picked T in an optimal manner, one can also compute the optimal value for the test oscillations. As a result of computations and a graphical construction, we find Copt %

2

21 s

h2da, b3 ’

--

(4.141)

It is important to note that the optimal amplitude of the test oscillations increases very slightly with an increase in the noise level. * A graphical construction will show that the value of L depends very little on the magnitude of the coefficient of proportionality k ; to find Lminone can choose k in the range between 1 and 2.

4.3. Discrete Extremal Systems with Constant Steps

257

T h e results derived above are excessively cumbersome and demand lengthy computations for each specific type of characteristic function for the object. We shall describe an approximate method which makes it possible to find an expression in a closed form for the probability distribution for an arbitrary characteristic. Consider the basic difference equation (4.1 15) for the quantities u j , that is, the difference equation for the probabilities that in the steady state the value of the deviation from the extremum will be e j . We rewrite it in the form U(Ej)

=

U(.j

-

+

h)P(€, - h )

U(Cj

+ h)q(e, + h ) ,

(4.142)

replacing the indices for the points by coordinates. We recall that a

P(4

w&)

=

d=

* 2 + 3 [S(Ej) 1

h

+ m1J.

Thinking of p ( c j ) and u(cj) as continuous functions of the argument e j = x , we expand p ( x f h) and u(x & h ) in Taylor series :

p(. f h)

= p(x)

u ( x f h) = u ( x )

dp h2 d2p f h - + -dx 2 dx2

kh -+ du

dx

+ ...

h2 d”u - - A ... 2 dx2

(4.143)

Substituting these expansions into Eq. (4.142) and preserving only terms of order h2 we obtain h2 d2u(x) +__ = 0. 2 dx2

d - {[1 - 2p(x)]u(x)} dx

(4.144)

Integrating (4.144), we find 2 -du(x) +,[I dx

- 2P(.)]u(x) =

c.

T h e function X ( x ) = 2 p ( x ) - 1 is such that when x and when x > 0, X ( x ) < 0 .

(4.145)

< 0, X ( x ) > 0,

258

4. Extremal Systems

Andronov et al. [2] have shown that, under these conditions, we have U ( X ) 4 0 ,

du -40, dx

(4.146)

when x f co so fast that the left side of Eq. (4.145) also converges to zero and, consequently, C = 0. Thus, the function u(x), which in a sense is a probability density, satisfies a linear equation of the first order : --f

+ 2h [l -

dx

- 2p(x)]u(x) = 0 .

(4.147)

Hence, from the condition for normalization,

1

m

u ( x ) d x = 1,

--m

(4.148)

we find the final solution

If we use the approximate expression

Eq. (4.149) takes an especially simple form :

After determining ~ ( x we ) can compute the initial probability u j in the following way : u(x) dx.

uj =

(4.152)

(i-l-l)h

T h e accuracy of the resulting approximation will be clear from the following examples.

4.3. Discrete Extremal Systems with Constant Steps Example 3 .

259

For the piecewise linear characteristic f(x)

=

-bl x I,

we find (when mc, = 0)

(4.153) We shall find the probability u, when j

1

+ I:

which agrees fairly well with the exact solution (4.123) when h2b' is, when there is intensive interference, because

< 1, that

1 - hb ___ - a.

e-PAb

1

+ hb

T h e variance has the same degree of accuracy.

Example 4 .

Calculations for the parabolic characteristic

f(x)

b

= - 2

X2

give

(4.154) that is, we get a normal distribution with a variance equal to 02

=

h -

(4.155)

2hb '

which agrees well with the exact result of (4.132) since the binomial distribution at the transition limit becomes normal, and the exact value of the variance h2(B/2)= (ha/2b), completely agrees with the approximate value, because for a uniform distribution h = I / a .

T h e resulting approximate expression for the probability density in the steady state (4.151) can be used to find an estimate in the case in which the characteristic of the object is not exactly fixed. We shall find, for example, an estimate of the mean-square deviation of the tuning from the extremal value with the assumption that the form of the characteristic f ( r ) is arbitrary to the extent that it satisfies the condition ax2 bx2 -
(a>b>O).

(4.156)

4. Extremal Systems

260

Because of (4.151), we can write for an arbitrary fixed function f ( x ) , when mc,= 0, (4.157)

where A = 2h/X. We add to the function f ( x ) the small variation Sf(x). T h e variance computed for the altered characteristicf(x) Sf(x) is equal to

+

u2

+ Sa2 =

OCI

J-m

+ A S_",x 2 e A f ( l )Sf(x) dx , dx + A S_",e A f ( z )S f ( x ) dx

x 2 e A f ( x )dx

S-Z

eAf(s)

(4.15s)

from which we find the amount of the variation in the variance:

Therefore, an arbitrary variation Sf(x) > 0 somewhere in the interval x2 < u2 causes a decrease in the variance from what it was for the initial characteristic. Analogously, an arbitrary variation Sf(x) < 0 in the interval x2 > u2 also causes a decrease in the variance. Thus, if for the initial curvef ( x ) we choose the function which coincides with the lower boundary of the region of the possible values when x2 < u2 and the function which coincides with the upper boundary of the region when x2 > u2, that is, f(x)

a

= --2

x2

< uz,

-

x2

> 02,

2 ' b - -x2, 2

(4.160)

then arbitrary variations S f which satisfy the given restrictions can cause a decrease only in u2 .

4.3. Discrete Extremal Systems with Constant Steps

261

Consequently, the curve (4.160) is in a sense extremal for the given variational problem, and the variance u2 computed for it is the maximum possible with these restrictions. We note only that Eq. (4.160) gives the desired extremal in terms of the unknown parameter u2, which one must find in the usual way, requiring that 02

=

sum

x2 exp[ -(Ab/2)x2] dx

4-

xs exp[ -(Aa/2)x2] dx

s," exp[-(Ab/2)x2] dx + j,"exp[-(Aa/2)x]

dx

- max.

(4.161)

I n this manner, the reasoning given above makes it possible to reduce the general variational problem to finding the maximum of a function of one variable. Analogously, the exact lower bound of all the possible variances is given by the minimum of the quantity u z , determined by the expression 02

s,"

=

+ si +

exp[ - ( A a / 2 ) x 2 ] dx xE exp[ - ( A b / 2 ) x 2 ] dx ______ . exp[ - ( A a / 2 ) x s ] dx j,"exp[ -(Ab/2)x2] dx

(4.162)

Thus, we find urnin ==

Sorx2 exp(- x2/2uo2)dx + s:x2[exp(-

J-texp(-x2/2ub2)dx

- Ju5

xs/20,") - exp(- x2/2oO2)J dx

[exp(-x2/2ub2) - exp( -x2/2an2)] dx

1 ; x2 exp(- x2/2ab2)dx J.0 x2 [exp(- x2//2ub2) exp(dx < j,"exp( - ~ ~ / 2 a ,dx~ )- j,"[exp(-x2/2a,2) - exp(- x ~ / ~ u , ~dx) ] -

2 - urnax

where u,

=

-

X ~ / ~ U , , ~ ) ]

(4.163)

I

l / d A a , and

(J,,= l/z/Ab, while ub 3 uo . Hence, since

262

4. Extremal Systems

we find simple estimates for the upper limit of the largest values and for the lower limit of the smallest values :

aka,

< s:

x2 exp( -x2/2ub2) d x

somexp(-x2/2u,2) dx

I-

= ub2?

=

CT,,~,/~

(4.165)

ua

These estimates could be made precise, but it is hardly necessary since the initial restrictions (4.156) are also made with considerable leeway. T h e physical meaning of these estimates is immediately clear if one recalls that the quantities u, and u,, are equal to the mean-square deviations from the extremum when the range (4.156) of possible functions f ( x ) is within the upper and lower boundaries. We must stress that the curve (4.160) which we have taken to be extremal is not a curve with one extremum and, therefore, in general, is not in the class of curves under consideration. T h e extremal which satisfies the "single-hump" requirement will, evidently, be a curve of the following type (Fig. 57) : f(x)

=

4x212,

= -312, =

-bx2/2,


x2

FIGURE57

2 c2/b,

(4.166)

4.3. Discrete Extremal Systems with Constant Steps

263

where c is an unknown constant which is defined by the condition

.=

J c v b x2 exp(-Abx2/2) dx +

x2 dx exp( -Ac2/2)

+

-

exp( -Abx2/2) dx

+

ryi 0

x2 exp( -Aax2/2) dx

cvb

+

';1

exp( -Aax2/2) dx

= max.

(4.167)

T h e condition for the extremum lJ1'- I,l;

=0

(4.168)

can be written in the form (4.169)

Performing the corresponding manipulations, we find

4. Extremal Systems

264

Equating these expressions gives the following transcendental equation :

4:

x exp(-s2/2)

f,

i

- 1 - - G(x)

Y3 +p-1

-

Y --y-I

x3 - 0,

3

(4.170)

where

(4.17 1 ) For the case when y = 1, that is, the case when the specified region is subtended by a curve, we have the obvious result x =

1,

OhaX

=

(Tb’.

Solving Eq. (4.170) graphically, we construct the dependence of x on the parameter y , that is, on the degree of separation between the two boundary parabolas. This dependence is shown graphically in Fig. 58(a). From it, we have computed the curve u2max(y)/ab2,shown in Fig. 58(b).

x 1.8 1.6

( a ) 14 12 10

::

13 (b)

10

1.0

2.0

3.0

FIGURE 58

4.0 y

4.3. Discrete Extremal Systems with Constant Steps From a consideration of the latter curve it follows that when 2

amax -- -

0.ly

Ub‘L

+ 0.9,

265 y

< 4,

(4.172)

which makes the above estimate considerably more precise 2

omax

-< Y .

(4.173)

These estimates are analogous to those found in Section 4.2 and can be used to find the optimal (in the steady state) parameters of the extremal system. I n conclusion: we shall describe a way to estimate the steady-state process which is also based on the theory of Markov processes. T h e steady-state process can be treated as a random process of transition from some level = E~ (the initial tuning adjustment) to the level E == e l (the optimal tuning adjustment). T h e number of steps n, and, consequently, the amount of time spent in making this transition, is a random variable. Estimates of this quantity (the expectation and the mean-square value), as has already been shown in Section 4.2, can serve as a measure of the action speed of the extrema1 system. We shall denote by ~ ( jn ), the probability that ~ ( t , )3 e l , while for E(t) < E, . to < t < t , , Then, as shown in Appendix V [cf. (A.31)], we have the following equation : (4.174) V ( j , n t I ) = plv(j + I , n) [ I - p&(j - I , n).

+

We shall impose initial boundary conditions on this difference equation. Let the search begin at a point lying at the left of the extremum ( j l j < 1). We introduce the obvious conditions for the initial position,

<

(4.175) and, also, the “absorption” condition (for ending the process) when j = 1: v(l, n) = 0 (n 3 1). (4.176)

4. Extremal Systems

266

At the other boundary, we impose conditions which express the impossibility of going beyond the limit cj, :

T o solve this problem we apply the method of generating functions (the discrete form of the Laplace transformation). We introduce the generating function for the probability v(j, n) of the number of steps taken before the extremum is first passed : (4.178) Multiplying Eq. (4.174) by

sn+l

and summing over n, we obtain (4.179)

n=o

n=U

If we take into account the initial conditions, this equation will have the form

while

T h e solution of Eq. (4.180) for the case pi = po = const (that is, for a search along one branch of a piecewise linear characteristic when there is sufficiently intensive interference) has a very simple form : g(s)

= u1-(L-j)(s),

(4.181)

where u l ( s ) is larger in absolute value than the root of the equation

We can find an expression for the expected number of steps prior

4.3. Discrete Extremal Systems with Constant Steps

267

to the first arrival at the extremum by direct differentiation of Eq. (4.18 l), Mi=idS) - 1-j (4.182) *=I 2p, - 1 ' and the variance of this quantity is

Moreover, by expanding into elementary fractions, one can find the inverse transform of g(s), that is, directly,

Here, z = 1 - j , and the binomial coefficient is taken to be zero if ( n - z)/2 is not a whole number in the interval (0, n). This last equation, of course, contains the former results of (4.182) and (4.183); however, in practice, it is necessary only to find the mean value and the variance. T h e quantity M j / z serves as an estimate of the action speed of the system and the quantity D j can be used to find the accuracy of this estimate. If there are large errors in the measurement of the slope, p , will differ little from $. In this case it is comparatively easy to find the expected number of steps. Thus, if p , = s/8, then M i= 42, and D j = 602. However, for large z the accuracy of the estimate increases. Set, for example, z = 60; then M i= 240, and the mean-square deviation is d D J r= 60. We shall study further the case when the probability of transition in one step depends on the coordinate of the operating point. We limit ourselves to finding the expectation M i .T h e difference equation Mj

==

PjMj+,

+ ( 1 - Pj)Mj-I i-I

(4.185)

can be derived by differentiating Eq. (4.180) with respect to s or from direct considerations. Its solution has the form

4. Extremal Systems

268

Computations on the basis of this equation are complicated. I t is obvious that there can be performed when the number z = 1 - j is quite small and also when the errors in measurement are small, 1 . T h e opposite limiting case is conveniently that is, lipj - 1 handled by using an approximate equation which is based on the assumption that the length of the operating step is very small in comparison with the dimensions of the region in which the search is conducted. We write the difference equation (4.185) in the form

<

m(x) = p ( x ) m ( x

+ h ) + [I

-

p(x)lm(.r - h)

+ T,

(4.187)

where m(x) = T M ( x )is the expected time required for the transition from the point ei = x to the point , and T is the length of the interval between steps. Expanding m ( x 3 h) into a Taylor series and using only the first three terms of the series, we obtain h

dm

7 [ 2 p ( x ) - 11 -dx

+ 27he

d2m dx2

+ 1 = 0.

(4.188)

77

This equation must be integrated with the following boundary conditions : m(x) = 0 for x = , and

T h e latter condition is found by transforming (4.177). T h e solution of Eq. (4.188) under these conditions has the form

X

[$+ f 1: exp(a 1' [ 2 p ( t )

- 11 d t ) dy] d y .

(4.189)

f'l

'I

Example 5 . Let the characteristic for the object be piecewise linear; then = po = const (when x < c l ) and

p(x)

m(x) =

T ( c I - x) (2po - 1)h

-

exp

1 2 (-') (p. q) (ecX" - XEj

-

-

e - X f l ) , (4.190)

4.4. Testing and Operation Separated

269

wherex = (2/h)(2p0 - I ) = (2/h)Ab. In the limiting case when ejl + -- 00 we find the simple result (4.182). For the parabolic characteristic f ( x ) = - (bx*/2), when c l l + -a and e l = 0, one can write the following equation: (4.191) where ~ ( a)

2

J: exp t* J

d;

~

a

exp -yz dy.

(4.192)

t

The graph Z(a) a < 2 was given earlier in Fig. 56. When a 2 2, one can use the asymptotic equation Z(a)

=

1

6

(1

+ In a).

(4.193)

I n summarizing the study of steady-state processes based on the theory of Markov chains, we note that it is rather complicated and that it can be used to a large extent only for certain types of characteristics. I n most cases it is, therefore, convenient to use in estimating steady-state processes the computational schemes developed in Section 4.2, which can be applied within the bounds of the validity of the approximate equation (4.1 12). 4.4. Extremal Systems in Which Testing and Operation Are Separated by a Frequency Band

T h e principle for the operation of extremal systems in which testing and operation are separated by a frequency band was described in Section 4.1. In the following discussion, we shall consider only systems of the proportional type with a one-dimensional search. T h e very nature of the problem demands that we consider the nonideal dependence of the measured value of the quality index Y* on the variation in the tuning, that is, of the presence of dead time and of the influence of the uncontrolled external disturbances. We assume that this dependence can be represented in the form of an open nonlinear transformation (Fig. 59). T h e tuning signal first is transformed in the linear dynamic input portion with the transfer function K,(p), then undergoes the nonlinear lagless transformation

270

4. Extremal Systems

f ( X * ) and, finally, the linear input transformation K,(p), where K,(O) = K,(O) = 1. T h e disturbances are composed of two normal random signals : One of them, 2, , is additive to the tuning signal, and the other, 2, , is additive to the measured value of the quality index.

FIGURE 59

Each of these signals can be decomposed into a slowly changing component (the drift characteristic) Zi,and the high-frequency , where i = 1, 2. T h e component Z,, defines the component Zi, uncontrolled slipping in the coordinate which corresponds to the extremal tuning adjustment. T h e speed of its variation gives the necessary action speed of the system with respect to the basic (the operating) signal. We shall think of the simple component as undesirable interference.* T h e operation of producing the value of the slope at the point of the characteristic which corresponds to the current value of the slowly changing tuning adjustment, as has already been indicated, is reduced to the operation of demodulating the signal Y * , that is, to finding the amplitude of the harmonic with the frequency w,, , which is the frequency of the periodic test signal. We shall consider a circuit for demodulation whichis realized by a synchronous detector. T h e signal Y* is multiplied by the periodic signal which has the same frequency as the test signal and is displaced relative to it by the phase angle (to compensate for the phase shift that the test signal undergoes in passing through the object). T h e p) passes through the filter resulting product U = Y* sin(w,t with the transfer function R ( p ) which has a cutoff frequency lower than w, :

+

I RO’w,)I

< 1.

(4.194)

* Such a distinction is not absolute: one can imagine a system in which the required extremal tuning adjustment is fixed, although it may be unknown in advance. For similar systems the component Z , , is interference.

4.4. Testing and Operation Separated

27 1

Thus, the system as a whole can be described by the following equations : (1) the equations for the object,

Y*

=

K,(P)Y

+

2 . 29

X* = K,(p)X+ c sin w,t

y

= f(X*),

+ 2,;

(4.195)

(2) the equations for the demodulator, U =

W = (3) the equation for the cofitrol element,

x = hP“ w .

(4.197)

T o simplify the study, the disturbance 2, and the test signal are fed into the nonlinear transformation. We assume that the amplitude of the test signal and the power of the high-frequency interference Z,, are relatively small and we replace the expression for Y , by the first terms of its Taylor expansion, Y

= f(c) + f ” ( c ) [ c

sin O J , ~

+ Z,,] -+ if’’(c)[c sin wot + Z11]2+ ... ,

(4.198)

+

where E = K,(p) X Z,, is the error of tracking beyond the extremum which is displaced because of Z,, . In the high-frequency components Z,, , Z,, one needs to consider only that part of the spectrum which is near the frequency of the test oscillations w , , since the higher frequencies can, in practice, be suppressed without distorting the dynamic properties of the system as a whole. We represent the components Z,, , Z,, in the form Z,,

=

A , sin w,t

Z,,

=

A, sin w,t

+ B, cos mot, + B , cos wet,

(4.199)

where A , , A , , B , and B , are slowly varying (with respect to sin mot) random functions and mutually uncorrelated (if, for the sake of sim-

4. Extremal Systems

272

plicity, it is assumed that the spectra of Z,, , Z,, are symmetric relative to w o ) [49]. T he n we retain in the expression for the signal Y , only those components which have spectra near the fundamental frequency, wg

:

Y

=f ( c ) [ ( A , =

+ c ) sin mot + B, cos wet]

A , sin w0t t B , cos w o t .

(4.200)

Moreover, referring to Eqs. (2.72), we can write y*

=

2,

where K&)

=

Re K,(P

+ sin WOt[K,l(P)A, - K,,(P)B,I

+ cos WOt[KZ,(P)A,+ K,l(P)B,I,

(4.201 )

+iwo),

(4.202)

&dp)

=

Im K,(P + i 4 .

After the demodulation and the averaging in the filter and the control mechanism, the signal is transformed to the form

We now write the equation for the closed system relative to the tracking error c :

T h e resulting equation is nonlinear. Moreover, if A , , B , # 0, its coefficients are random functions of time. In the general case, its analysis is very complex. We shall study the dynamics of the system on condition that there is no high-frequency interference Z,, . I n this case, the problem reduces to investigating a closed nonlinear transformation with one lagless nonlinearity f’(~).

4.4. Testing and Operation Separated

273

We apply statistical linearization. Let f ’ ( ~ )be an odd function of and m,, = const, then m, = 0 and f ( ~ )can be approximately represented in the form = h(u,)r, (4.205)

E

where

If we impose the restriction

I b l E I e If’(€)

1

eI

Qlf

!

(4.206)

on the characteristic of the object, then it is obvious that in searching for the maximum I b l l
while h ( c ) < 0. Equation (4.204) can be written in the form

where

T o find the variance uc2we use the expression

where S,(W)is the spectral density of the random component in Z,, and SZ(w) f S A , ( w )= SB2(w) is the spectral density of the “amplitude” of the high-frequency interference Z,, . When there exists an expression for the spectrum of the inter-

274

4. Extremal Systems

ference itself SZ2(w),the spectral density S 2 ( w )is given by the equation (see C491)

: 7

S Z ( w )= -

,.=

cos

WT

dT

.a

J

SZAw)COS(W

-

w 0 )T

dT.

(4.210)

JO

I n computing the integral (4.209), it is convenient to use simplified with due consideration expressions for R(jw),K ( j w ) ,K21(jw),K22(jw), of the fact that the spectral densities Sl(w), S,(w) have values significantly different from zero only when w w,, . I n particular, because of (2.77) the operators K2,(p),K22(p)can be represented by ratios of the form (aop a)/(bop bl), preserving terms of only the first power of p in both the numerator and the denominator. From (4.209), and using the tables of Appendix 111, we find an explicit dependence of uC2on h(o,). We propose the obvious assumption that in the range of possible variations of h(o,), given by the inequality (4.207), this dependence is monotonic. T h e maximum possible value of a: is reached when h(u) = b or h(a) = a. We shall give an elementary example. Let us assume that in the low-frequency range under consideration,

<

+

+

Kdp) =

1 +

1

9

K d p ) = 1,

W) =1

and Then, where T

,

ChOh(0)

Integration gives u,2

2

= -

= a,2

eT,

+ T(e +TJ +

e~ + q e + T , ) .

022

..h 1 ".

8

Let 0 >, T ; this means that the process Zlo defining the displacement of the extremum has a frequency band which is no wider than the passband for the dynamic element at the input of the object. T he n the error variance increases monotonically with decreasing -h(a), and the worst possible choice will be h(a) = --b, .

4.4. Testing and Operation Separated

275

T hi s method shows a way of solving the minimax problem, that is, the choice of the parameters and the structure of the filter hoR(p), which guarantees minimum error variance uC2for the worst form of the curvefl(c) in the range of possible forms given by the inequality (4.207). T h e problem can be solved by the usual methods of synthesis of linear circuits. Obviously, the results of the synthesis determine only the low-frequency part of the characteristic for the filter. T h e behavior of the high-frequency portion is given by the suppression restriction (4.194). We now consider the simple problem in which the effect of the high-frequency input interference is taken into account. We assume that Zl0 = const, 2, = const, K,(p) = K,(p) = R ( p ) = 1, and afl(c) = -b1c. Then, Eq. (4.204) can be rewritten in the form de

nt

where or

+ b(Ai +

dc

- 4-&)€ dt

C) =

(4.211)

0,

= 0,

while m p = bc, R,(T) = b2R,(7). I n this manner, the problem becomes one of finding a linear equation of the first order with a parameter which is a slowly varying random function of time. Using the method described by Tikhonov [84],* one can find the following expressions for the expectation and error variance c :

u,z((t) = co2exp'-2 I

[m p

-

2

RE(7)d ~ ( t]

expl -2 [m,- 2 1

t-t0

-c02

0

-

t o ) -4

R,(T)d ~ ( ]t - t o ) -2

1

t-to

0

l-to 0

7RE(7)d r /

TR,(T)dT!1 .

(4.213)

* Cf., also, a discussion on this topic in the journal Automotion ond Telemechonics 21, No. 7 (5960). ~

~~

4. Extremal Systems

216

T h e values of m, and ue as t on the signs of the quantities h=nt-

co depend in an obvious manner

--f

j-;

R,(7) d7, (4.2 14)

p = m, - 2

-00

J H , ( T ) d7. 0

Here, it is assumemd that .(to) = q,. Insofar as T = R,(T)d~ > 0 and p following possibilitils : l.X>O, 11. h > 0, 111. h > 0 , IV. h = 0 , V.h<0,

=

X - T , there can be the

p>o, p = 0,

p /L

< 0, < 0,

(4.215)

p
I n accordance with these possibilities, the following results can be obtained : I. m,(co) = 0, oC2(co)= 0 for arbitrary initial conditions. 11. m,(co) = 0 (for arbitrary initial conditions) and

111. m , ( m )

=

0 (for arbitrary initial conditions) but u:(t), when

q,# 0, increases with time without bound, or, as is more often said,

the system is unstable with respect to the variance. IV. m , ( m ) = c0 exp[- J," TR,(T)d T ] ; the system is unstable with respect to the variance. V. T h e system is unstable relative to both the variance and the mean value.

Hence, it is clear that the boundedness of the variance in the steady state is guaranteed only by the condition that p >, 0, or bc

.a

-

262 J RA(7)d~

3 0.

(4.216)

4.5. Simultaneous Testing and Operation

277

Thus, in choosing the amplitude of the test signal one must impose the condition that c

where T~

=

> 2b7,0A2,

(4.217)

1 -It 7J R,(T) d7 gA

0

is the time of correlation of the process A,(t). A study using an analog computer confirms this conclusion. T h e breakdown in the stability with respect to the variance takes place aperiodically and, as is evident from (4.213), for small deviations c from the boundary value as defined by (4.217), at a very slow rate. I n spite of the simplicity of the initial statement of the problem, we were able to obtain a definite physical result, namely, that the parametric effect of the interference which is additive to the tuning signal restricts the choice of the amplitude of the test oscillations, and, consequently, restricts the possibility of reducing the loss in the search. Evidently, this qualitative result will hold also for more complicated dynamic objects. 4.5. An Automatic Extremal System with Simultaneous Testing and Operation

Since the extremal system must carry out a continuous search, the steady state of its operation must be an oscillation near the extremum. In a system with simultaneous testing and operating steps, these oscillations are created by the intrinsic properties of the system. I n the introductory systern,the operation of the system was described under the assumption that the dependence of the measured value of the characteristic of the extremum Y * on the tuning of X was of an ideal nature. Here, we shall study the same problem, taking into account the dead-time character of this dependence and the influence of uncontrolled random effects. T h e schematic structure for the system can be represented by the diagram of Fig. 60. T h e corresponding equations are y*

=

K,(P)Y,

y

=f(X*),

x*

=

K,(p)X

+ z,,

(4.218)

278

4. Extremal Systems

In the first line are the equations for the extremal dependence (in the same schematization as in Section 4.4)and in the second line are the equations for the control device. T h e extremal disturbances 2, and 2, are fed, respectively, into the input of the static characteristic and the input of the nonlinear control system with the indeterminate characteristic F described in Section 4.I .

1

I

E

U

FIGURE 60

T h e operators K,(p), K,(p) [ p being the differential operator] characterize the dynamic properties of the objects with respect to input and the output. I t is assumed that, near the extremum, the static characteristic of the object can be approximated by a quadratic parabola, f(X*)

=

b(X*)Z.

T h e determination of the steady-state parameters for the described system with 2, = 2, = 0 was developed by Morosanov [53] by the method of harmonic balance, and later, by Dolgolenko [21] by the method of alignment. In both of these studies, the technique was to establish a dependence of the basic parameters of the steady state, ) the oscillations namely, the frequency W , (or the half-period T ~ of and the hunting amplitude A,, on the parameters of the scheme, in particular, on the operating level of the relay K~ :

Where these functions are sufficiently smooth and within small ranges of variation of K , they can easily be approximated by linear functions.

4.5. Simultaneous Testing and Operation

279

We consider first the case in which the interference acts only at the output of the object and, thus, can be reduced to a linear transformation at the input of the control system. T h e effect of the interference 2, on the state of the system can be only a variation of the actual level of operation on the relay in the control system, that is, the relay does not operate under the conditions

but only under the conditions

We assume a sufficiently low level of interference uZz< A.

(4.220)

T h e state can then change with small fluctuations in the switch-over moments and the amplitude moments, i.e., in the limit there will be a tightening of the cycle into a fairly small ring. A rigorous study of the dynamics of such an arrangement leads to the search for steady-state equations with finite differences such as were developed in Section 3.4. Hence, one can now consider two more elementary cases, which we shall describe below.

( I ) We assume that the rate of change of the interference Z,, and of the corresponding fluctuation of the level K is rather slow. More precisely, the values of K will coincide when the switch-over takes place during one period. With the method given in Section 3.2 it can be shown that the basic quantities which are characteristic for the periodic component of the motion will vary according to the equations w = w~(K), T = T,(K), A = Ao(~), (4.221) where w , 7, A are slowly changing random functions of time (more precisely, random sequences).

280

4. Extremal Systems

From (4.220) we find that

AA = A - A o

= -

(

T h e mean-square values of the fluctuations d w , 47,d A are related to the mean-square value of the interference az2by the elementary equations

(4.222)

T h e values of the partial derivatives in (4.222) can be found either graphically or analytically. Consider, for example, the simple case when

K , ( p ) = 1,

K,(p)

=

1

1)

We shall compute the undisturbed state 2, of the given half-period, let

F(0 Then,

+1 =

0. During the course

= 4 0 .

x = A, - j o t , 1 Y b

-

1

=

A , - 2A0f0t +fo2t2,

- 6 = Boe-t/T- 2fo(A0

b

(4.223)

+ lyo)+ 2f;t,

4.5. Simultaneous Testing and Operation

28 1

where A, and B , are constants defined by the condition that the self-oscillations are symmetric and the switch-over conditions are

x(0) = -X(To), 530) =

[(To)

=

KO

.

It is not difficult to see that

+ f270 + 2f:". I t is obvious that the quantity A, is the hunting amplitude of the tuning of X . Now, the equation for finding the half-period 7, can be written in the form (4.224)

I t follows from the resulting equations that

-_-

1 (1 - e--o/r)2 bf,2 1 - 2 ( ~ , / T ) e - - ~e~- 2~s /~~/ T - -fo7, 1 870 277 a7, -- ifK 2 dK 8K 70' aK 87,

a~

awe=

(4.225)

(2) T h e problem can be solved in a more elementary way with some other assumptions. We shall consider only those systems for which (a) the time for the transfer processes in the linear parts of the system is shorter than the half-period 7 , , that is, hi70

>1

B

(4.226)

where Xi are the roots of the denominators K , ( p ) , K,(p), and for which (b) the successive values of the interference at the switch-over moments are independent random variables. It is physically clear that with these assumptions the disturbance of each half-cycle can be considered independent. T h e values of the quantities T,, A , w for the successive half-cycles transform into a sequence of independent

282

4. Extremal Systems

random variables, each of which is related to the random operating level according to Eqs. (4.221). Thus, in this case one can use Eqs. (4.222) to find the mean-square deviations uT, uA, 0,; however, to find the partial derivatives for equations of the type (4.225), it is desirable first to make a simplification by taking into account the initial assumption (4.226).In particular, for a system in which (4.226) holds, we have the simple equations

Consider the more complicated problem when the interference is acting at the input of the object (2, # 0, 2, = 0). T h e approximate study of this problem, again, can be initiated by using two schematic forms which correspond to the ones described above for the action of the interference only at the output of the object. (1) We assume that Z , ( t ) can be approximated by a piecewise linear process (a delta function of time) with angles of inclination that vary from period to period in a random manner. I n other words, the drift velocity for the characteristic is a random variable with a constant value over each period. T h e variation in the drift velocity is thus assumed to be slow. T o find the fluctuation in the parameters of the self-oscillations one can, therefore, use the results obtained for a system with a constant drift velocity. I n this case, the symmetry of the oscillations is obviously lost but the computations could be based on the method of curve fitting, for instance. T h e equation for the periods of the simplest system with an operator of the form (4.223) is

-

where

f o e[coth 71+ coth --__ (fo+ 4f0T

T~

L

2T

=

dz 1

dt ’

L)T1

(fo - L)2T

Tp = 7 1

f--o +L , fo - L

] + 1 = 0,

(4.227)

4.5. Simultaneous Testing and Operation

283

and T ~ 7,, are the lengths of the time intervals corresponding to the motion of the control element in one direction. Assuming that L

= Lo

+ AL,

where d L is a small random variable which has independent values for each period, one can find the equation (4.228)

Here, 710 is the solution of the equation F ( L o , T ~ =~ 0.) T h e partial derivatives have the following form :

For the second half-period we have = a,

.fo ---.

-

Lo

' f o -Lo

We shall also find an expression for the mean-square value of the fluctuation of the largest deviation A of the coordinate x(t) at the input of the object. From the solution of the fixed problem, we find that

284

4. Extremal Systems

A relation between the fluctuation A and OA -- w 1 0 or1

T~

is given by the equation (4.230)

*Lo),

with the specification

(2) Consider the system where the condition (4.226) is satisfied. I n this case there is a signal at the output of the object : Y

=

b[Xo + 2J2,

(4.231)

where X o = K,(p) X , if we neglect terms of the second order, Y

We denote

+ 2bZ12,.

= bXiZ

(4.232)

c1 = 2bX0Z,.

(4.233)

T h e values of the interference at the input of the control device (at the moment of switch-over r ) , depend on in the following manner (with an accuracy within terms of the first order) :

c1

where h, is the impulse function corresponding to the operator pK2(p). Because of the assumptions (4.226), ZZ(7) %%

fh2(e)r,(7 0

- 0 ) do,

(4.235)

that is, in computing 2, it is possible to use only the values of X , for the half-periods under consideration. T h e mean-square value of uz2can be found by the equation

x

- ~ o ( e z ) f z , ( e , - 0,)

de14

1

where u~lp,,(~)is the correlation function for Z,(t).

(4.236)

4.5. Simultaneous Testing and Operation

285

For a system with operators of the form (4.223) and when , the computation can be carried out fairly easily in an explicit form expressing C J , , / U , ~ in terms of the parameters of the system and the exponent a . T h e equation is rather complicated but when a T > 1, we obtain

P z , - ,-a10

( J z p GZ

(Jzp.fo

+

.

(4.237)

After uzphas been calculated from (4.236), the computation of the mean-square values of the fluctuation of the parameters of the oscillatory state can be done by Eqs. (4.222) where the interference was reduced to a linear input in the control device. It is obvious that this method of computation can be easily extended to the case in which drift characteristic with constant velocity is superposed on the disturbance. We proceed to a more rigorous method of study. Since the calculations necessary for the analysis of more complex systems are much more cumbersome, we restrict ourselves to a solution of the case in which the dynamic properties of the object can be described by the simplest operators (4.223). T h e equations for the system then can be written in the following form :

We denote the duration of the successive intervals during which the sign of 7 is preserved by r k , while it is assumed that in the first interval 7 = fo . I n the absence of disturbances (2, 2 2, = 0), it is not difficult to see that in the kth interval

X I ; = c,* +fot(-l),.-l, 1

Y,.

=

(C,*)Z

+ 2C,*f0t( -

1 5,. = Bk*e-'IT b

-

I),.-1

+f ; v ,

+ 2f02(t - 7') + 2f0(-l)'-'C,*

(4.239)

(k = 1, 2, ...),

where the origin for the time coordinate is taken to be the beginning

286

4. Extremal Systems

of the interval. T h e constants C,*, B,* and T,* (the duration of the interval in the undisturbed state) are determined by the conditions Sk(Tk*)

=

(,(r,*) =

Xk+l(O),

(4.240)

&+l(0)= Kg

(k = 1 , 2, ...).

As a result, we find

c,* = Q f ~ T k * ( - l ) ~ , B,*

=

B,

=

* K +f,2

(T+

+ T)

(4.241)

(k

=

1,2, ...),

and rk* = T , , where ro is given by the equation for the periods given above (4.224). We also assume that the disturbances Z , , 2, are imposed on the system at the beginning of the first interval. We make the same assumptions as in Section 3.4 regarding the order of magnitude of these disturbances. Thus, it is not difficult to find the following equations which characterize the variation in the coordinates of the system in the disturbed state :

x

j:;

{ Z l , k [ C k+j,v(-l)~-l]}eu/T dv.

(4.243)

4.5. Simultaneous Testing and Operation

287

T h e conditions for finding the constants C , , B, and the duration of the interval T , are =

Xk(7,)

Xk+d0)9 -

= 6k1-1(0) =

'$k(Tk)

(k = 1, 23 ...).

Z2.k

(4.244)

Here, by Z2,, we mean the value of the interference Z,(t) at the moment which corresponds to the end of the kth interval. From conditions (4.244) we find c k +foT/~(-l)'-~

Bke-'klT =

B,,,

+ 2f,,'(~, -

-

2f,"T

T)

=

ck + l

+ 2f0(-l),-lC, + &,,(

$. 2f0(-1)~-'Ck+1

=

51

(KO

--I),-'

(4.245)

- Z2.k)

(k = 1, 2, ...), where

We go on to linearize these equations, taking into account the triviality of the deviations of the quantities B, , C , , T , from the corresponding parameters in the undisturbed state. Using the notation ACk = C k - C,* , AB,

=

( B , - BO)(-l)'-',

AT^

=

( T ~

-T~)(-I)~-'

(4.246)

(k = 1 , 2, ...),

we find, after some fairly simple transformations, the equations which relate the fluctuations of the parameters A C , , A B , , 47, to the external disturbances : ACktl -AB,

e-'olT

-

[2fO2-

= ABk+l

=

B

AC, + f a & , , e - ' ~ / ~A ]T , - 2f0 AC,

+ 2fo ACk+l = 51 z z , k ( - l ) k - l (k = 1 , 2, ...).

-zl,k

(4.247)

4. Extremal Systems

288 Considering that

we eliminate these variables from the second equation of the system

(4.247).

Thus, the difference equation for the fluctuation of AC, in the amplitude of the tuning values X has the following form :

where

51.k

=

-

21.k 2f,,[l - (R0/27",,2)e - ' ~ / ~' ]

T h e fact that /3 < 1 proves that the initial self-oscillatory state is stable. T h e solution of the difference equation (4.248) of the first order is well known : (4.249) i=l

For the steady state we obtain the solution in the form of a series AC, = lim AC,; . k

(4.250)

T h e effect of the previous disturbances subsides as /? < 1. We write an expression for the variance uA2of the quantity dC, under the

4.5. Simultaneous Testing und Operation

289

assumption that

(4.251)

and M { [ , ,

=

0 for all i and k :

If we assume that there is very rapid damping of the transfer processes, we obtain (for Z , ( t ) = 0) (4.253)

which agrees with the approximate results which follow from Eqs. (4.222) and (4.225). Now, let Z , ( t ) vary slowly so one can assume that ZZtk= const during the course of those periods k for which Pk 1, while Z , ( t ) = 0. Then,

<

from which, substituting our value for the parameter

P,

we find

From Eq. (4.244) and the equation for the periods (4.224), we find (4.256)

Eliminating the exponential terms in the expressions (4.222), (4.225) and (4.255) by using the above equation, it is not difficult to see that

290

4. Extremal Systems

we have in both cases the same equation for the variance of the amplitude : D

O*f

U = 22 -

4b2 B,[( 1I T ) + ( I/T,)]

I

---

-

(Ho2/2T7,)- fi'

'

(4.257)

This also proves the validity of the approximate methods described above for an object with dynamic characteristics given by the operators

(4.223).

I t is reasonable to assume that this statement is also valid for more complicated objects. T h e simplest system considered was investigated experimentally with an analog computer. A shot-noise generator was used as a source for the random influence. T h e coefficient of noise correlation was closely approximated by the exponential function

T h e computation of the random noise at the input of the control device was made by using Eq. (4.225') for b = 0.03 sec-' and xo = 3.6 volts (see Table I). TABLE I

(volts)

(see)

(see-I)

(volts)

10

4.0

1.1

10

1.o

2.1

0.68 8.3 I .02 2.54 8.3

Experimental Computed

0.068 0.026

0.063

0.194 0.154 0.120

0.233

For a random disturbance superimposed on the control signal, the computation was carried out by using Eq. (4.237) for T = 4 sec, b = 0.03 sec-' and xo = 3.6 volts (see Table 11).

29 1

4.5. Simultaneous Testing and Operation TABLE I1 fo

WO

0 2I

(volts)

(sec-')

(volts)

10

1.1

1.41 8.3 I .4l 8.3 1.41

15

24

I .8 3.0

u,/u,,

(volts-' sec-l)

Experimental

Computed

0.024

0.027

0.013

0.05I

0.030

0.018

0.063

0.03 1

A comparison of the calculations with the experimental results shows an acceptable agreement for small values of the random influence. T h e results of the theoretical computations become more precise , is, as the conthe more w o is decreased (with increasing T ~ ) that formance with condition (4.226) is improved. We note that for all the parameters studied above, the quotient r O / Texceeds unity by very little or is even less than unity. T h e acceptable accuracy of the resulting estimates shows that the approximate method is not too critical to the damping conditions

(4.226).

Appendix 1 0 F U N C T I O N S m,(mN, a,), AND a,(m,, a,) FOR SEVERAL TYPICAL NONLINEARITIES a2(mXD

ht(mX,

ON),

This appendix introduces the equations and graphs necessary for computing the coefficients a,(rn,, u,) = ( l / d d ) M ( f ( X ) H , ((X- rn,)/o,)} used in the study of nonlinear transformations of the form Y = f ( X ) [Figs. 61-66], where X ( t ) is a random signal with a normal or a nearly normal distribution. T h e coefficient a, is the expectation of the output signal Y( t) , %(ma!, u,) = adma! ,uz),

and the coefficient a, is simply related to the transfer constant by its

FIGURE 61

*; FIGURE 62

FIGURE 63

Yl

-A

FIGURE 64

FIGURE 65 292

FIGURE 66

293

Functions for Typical Nonlinearities

random component h, in the case of statistical linearization according to the criterion of the minimum mean-square deviation :

Since graphical techniques of computation by the method of statistical linearization use the coefficients mJm,, 0,) and h,(m, , uZ), they can be graphically constructed as a rule. It is not hard to prove that there is a very simple recursion relation for the coefficients a,, of the form

and, therefore, we shall give only the equations for n following examples. 1. The Ideal Relay Y

in the

= 1 sgn X

1 =

<3

m12

. ,(

-

1)

01

Graphs of the functions (a,/l)(mz/a,) for n = 0, 1, 2, 3 are given in Fig. 67.* For direct computations, more convenient graphs of

-

~~

*

~~~~

Here, just as in the main text, the function @(x) stands for the probability integral,

Appendix Z

294

a,(m,) = m,(m,) in the form of a family of curves constructed for various values of the parameter uz and graphs of h,( l/uz) for different values of m, are given in the text, Section 2.2 [Figs. 21(a) and 21(b)].

10

0.9 06 07 06 05 04 03 02 01

0 -0 I -02 -03 -04

04

08

I2

1.6 2.0

FIGURE 67

2.4

2.8

Functions for Typical Nonlinearities

where inl

=

171, -

295

"'=o' UX

9

Dependence of the functions mull and h,A/l on the variables m , and

u1 is graphically shown in Figs. 68(a) and 68(b).

"Y L

I .o

0.0 0.6

0.4 0.2 0

0.0

0.6 04

0.2

0

1.0

2.0

3.0 (b)

FIGURE 68

4.0

5.0

6.0 1/17

Appendix I

296

1.0

0.0 0.6 0.4

0.2 I .o

0

2.o

3.0 m,

(4

yl, A 10

0.8 0.6

04

0.2 ~~

0

I0

2.0

3.0

4.0

5.0

6.0

Functions for Typical Nonlinearities

297

3. An Element with a Bounded Zone of Linearity (Fig. 62)

exP( a,

= o&

a3 =

-

=

[ (

lo, @

1 [(I 2 dG

+ m1 01

) + Q, (

+ m,) exp(- (

- *I Ul

+ 01m1

j]

1

r1

where mX rn, = A

3

o1=-

(JX

A '

Dependence of the functions (l/l) m y and ( A l l ) h, on the variables m, and 0 , is given graphically in Figs. 69(a) and 69(b). 4. An Element with a Dead Zone (Fig. 63)

I t is obvious that a signal at the output of such a nonlinearity can be represented in the form of different signals at the output of a linear element with an amplification factor k = tan OL and of an element bounded by ( / ! A ) = k. Thus, in this case we have

where, on the right, one must substitute the corresponding functions for an element with a bounded zone of nonlinearity.

Appendix I

298

5. An Element with a Bounded Zone of Linearity and a Dead Zone (Fig. 64)

a, = uxhl

-@

=1 -v

(

--lml

)]

1

Graphs of ( 1 / l ) m y , ( A z / l )h, for v 70(b) .

=

0.5 are given in Figs. 70(a) and

Functions for Typical Nonlinearities

1.0

08 0.6 04

0.2

0

1.8 I.6

I .4

1.2 1.0 I.8

0.6 0.4

0.2 0

1.0

2.0

3.0 (b)

FIGURE 70

40

5.0

6.0

299

Appendix 1

300

6. An Element with a Characteristic of the Form Y a, = my =

~ N U , ~

[

a, = u,hl = 4NuZ2

Graphs of m,/Nuxa and h1/2Nuxare given in Fig. 71.

0.5

0

1.0

1.5

-

2.0 m,

U l

FIGURE71

7. An Element with the Characteristic Y

a2 =

3

5,

~ / Z N U ~ ~

= Nx’

a3 = d/i5Nux3.

Ur

Graphs of m,/Nux3 and hl/Nux2are given in Fig. 72.

= Nx’sgn

x

Functions .for Typical Nonlinearities

301

t

0.5

0

1.5

1.0

-

2.0 m, Q,

FIGURE 12

8. A Relay with a Hysteresis Loop (Fig. 65) m , = 21 [A@( -

1 +m, 01

)-

P2@

(

- m1 (J1

)]

(Jr

p1

=

1 -/+

=

1 - @((I

;- @((I

+

- m1)/(JA ml)/ol) - @((I - ~ 1 ) / ( ~ 1' )

where mz m, = A ,

(JX

(Jl=d'

Graphs of (l/l)rn, and ( A l l ) h, are given in Figs. 73(a) and 73(b).

Appendix I

302

I .o

0

0.2 (0

3.0 rn,

1

0.5

0.4

0.3

0.2 0.1

0

1.0

2.0

3.0

(b)

FIGURE 73

4.0

5.0

1 6

Functions for Typical Nonlinearities

303

9. An Element with a Bounded Zone of Linearity with Nonsymmetrical Bounds (Fig. 66)

1

- exp( - 2 (

a2 =

-[exp(- 1 ( I - - + m l 2

dv

01

,3

- exP(-

1

2(

1fa-m, 01

1+a-m1 01

j2)1

+ (1 + a - m,) exp where m l = - m3. , A

*X

=l

=

2i

It is not difficult to show that the equations for a , , a2 and a3 are equivalent to the equations for a nonlinearity with symmetrical bounds if we assume mz A

m,=--a,

U - -

U X

' - A

This follows from simple geometrical considerations.

Appendix I .

REPRESENTATION OF A LAGLESS N O N L I N E A R T R A N S F O R M A T I O N IN T H E FORM OF AN INTEGRAL T R A N S F O R M A T I O N IN A COMPLEX REGION. T H E T H E O R E M O F R. PRICE

Consider the nonlinear transformation

where for the simplicity of this discussion we assume that f ( z ) is an odd function. We represent f ( z ) as the sum of two functions,

such that

f+(.)

=f(z),

z

-

0, 0,

z

=f(z),

z

f-(z)=

z

> 0, < 0, > 0, < 0.

Because f ( z ) is odd, Consider the Laplace transform of these functions : F+(p) =

-I

.x --x

1

f+(z)e-Dzdz,

U

F-(p)

=

f-(z)e-IJZ dz,

-cc

304

(A.4)

The Theorem of R . Price

305

while it follows from (A.3) that F-(P)

=

-F+(-P).

(‘4.5)

T h e transforms F + ( p )and F J p ) have different regions of convergence in the complex plane p , but each of them has a singularity at the point p = 0. T h e inverse equation gives

where the contours P , and P- are coincident on the imaginary axis except in the neighborhood of the point p = 0, where it may be necessary to approach the singularity from the right (for P + ) or from the left (for P-).* Thus, the nonlinear transformation (A.l) can be changed to an equivalent integral transformation :

where j u = p , and the contours of integration C, and C- (Fig. 8) both are on the real axis in the u plane with the exception of the neighborhood of u = 0, where one can approach either from below (for C+) or from above (for C-). T h e representation (A.7) may turn out to be useful in computing the moment characteristics of the output signal for a nonlinear transformation. We shall write a general equation to calculate the reciprocal moments of the second order for the signals Then,

Xl

=

fi(G)*

x,= fZ(Z2). (‘4.8 1

* It is assumed that the abscissa of

convergence for f ( z ) is at zero.

306

Appendix II

Substitution of the identity (A.7) into (A.8) gives

Fl-(jul)~zl(j~~,)e,(u, , u 2 )du,

where ~ z ( u l ul z ) = Wexp[j(zlul

4,

+ ZZUZ)~}

(A.lO)

is the characteristic function of the two-dimensional probability density w2(z1, z,). One can conditionally write Eq. (A.9) in the form

where the contours C and C’ (in the complex planes u1 and u,, respectively) are the same as the contours C, and C-. I n the special case when

fi(z)Z f 2 ( Z ) 3 f ( z ) , we obtain

2 1 = z(t), 2,

B,,.T* = B,(t,

=

z(t +

T),

T),

and from (A.11) we find an expression for the second-order moment of the signal at the output of the nonlinear transformation f ( Z ) : (A.12)

Iffl(Zl)

= f(Z1),fz(Z,) = Z , , then F,,

=

I/(ju),and

307

The Theorem of R . Price

and this makes it possible to find the reciprocal moment of second order between the input and output signals of the nonlinear transformation. * We shall deduce one important corollary from Eq. (A.9) which is valid when w2(z1, z2) is the normal probability density. T h e characteristic function which corresponds to w2(z1 , z2) has the form

which can be deduced directly by substituting an expression for

w2(z1, z2) into (A.lO) (the necessary computations have been carried

out, for example, by Levin [49]). We shall compute the kth derivative of B2(u, , u2) with respsct to

Pz

: ake, a p z2 -

=

(- ~ ) " ~ ~ l ~ z , ~ , ~ z ) *kU:Pl) ) P ( ~ l (.3,.,)"(i.l)"(iz~2)~~z(z41

1

UP).

(A.15)

From (A.11) and (A.15), it follows that

We substitute into (A.16) the original expression (A.lO) for the characteristic function and change the order of integration. Then,

but

* Equations (A.12) and A.13), as well as (A.1 I ) , have a conditional character. T o compute them, one must use an expanded equation of the type (A.9).

308

Appendix I I

By the same token, one can find a simple relation between the derivaZ respect to pz and the derivatives of the tives of the moment B Z I I with nonlinear functionsf,, f , : (A.18) This equation was first derived by Price [114] (in a more general form for moments of arbitrary order). H e also proved that it is valid only for normally distributed variables 2, , 2,. T h e effectiveness of applying this equation to find the correlation function of a signal at the output of a nonlinear transformation was demonstrated in the examples of Section 1.1. With the help of (A.18) it is elementary to , prove the property (1.34) of the mutual correlation function RZIZ2 where X , == f(Z,), X, = 2,. I n fact, in this case (A.18) can be written in the form (for K = 1) aBz1Xz

-= U , ~ U , ~ M { ~ ' (= Z ,C) }= const, aPi

and, consequently,

but, when pz

=

0,

Consequently, Rx,x2= BXlr2- m,,mx2 = ~ p , ,

(A.19)

that is, the mutual correlation function is proportional to the selfcorrelation function of the input signal.

Appendix 111 COMPUTATION O F T H E INTEGRALS I, Computation of the Integrals I,

g,,(p)

= bopzn-'

+ blPZn--l + ... + b,,-, .

All the roots h,,(p) are assumed to lie in the left half-plane

where A n - , are the Hurwitz determinants for the polynomial h,(p) : A2

=U

~ U Z-~

0

~

3

,

A3 = U ~ U , U , - ~ 0 ~ 3 '~

A,

=

1

~

~

4

,

+ ~ u ~ u ~ +u U~, Uu, U,~ U , + w,&Ja4

-u,~u,'

-UOU~'U~

-~

1

~

~ ~4

'1

~

2

~

9

iV15 = (aou4a5- u ~ -uaiLu5 ~ $~ u,u~u,)~,, ('2u5

+

a,a4)b1

+0 (Qoa1a5 a5 U

+

aO(ala,

- aOa,)b2

- uou32 - Ul%Z4

+

aO(a~a2

+ a,a2a3)h4.

- a0a3)b3

T h e computation of integrals of higher order can be conveniently carried out graphically, even though the integrands depend on an unknown parameter (for example, h, in the equations of Section 2.2). 309

5

Appendix

Iv

T H E COEFFICIENTS O F STATISTICAL LINEARIZATION hi(a, U) A N D h l ( ~U) , FOR TYPICAL NONLI NEARlTlES

I n the analysis of the nonlinear transformation of the sum of normal and harmonic signals (Sections 3.1-3.3), we use the coefficierlts of statistical linearization hl(a, m,u) and h,(a, m, u); graphs and equations for these coefficients are given below for several typical nonlinearities with the assumption that m = 0. T h e computations are made both by direct use of the original equations, h,(a, 0, 0 ) = h,(&

1

=7702

0)

,, I dp,

‘ L

-m

xf(x

1 + a sin y ) exp(- -)

X2

u

1/2=

dx,

and by using an integral expression in a complex region. In the latter case [for odd functionsf(x)], we apply the equations hl(a, u) =

h,(a, u) =

3 1 F+(ju)Jl(au)exp(-i =a c,

I= Ic+F+(ju)JO(uu)exp(-$

where F+(ju) =

a

e - i u z f ( x ) dx,

0

and J o , J1 are Bessel functions. 310

uzu2)du,

OW)du,

Coeficients of Statistical Linearization

31 1

These equations are equivalent to the original ones but are often more convenient for finding asymptotic representations as was shown in particular in the example of Section 1.1. Unfortunately, in most cases of practical interest the functions h,(a, u) and h,(a, u) cannot be expressed in closed form in terms of certain special elementary or tabulated functions. Therefore, we give below expressions for h, and h, in the formof series. T h e coefficients of the series are expressed in terms of the confluent (degenerate) hypergeometric function, which in this case appears to be a function of only one parameter a = ajq2u. Let us remember the basic data for the hypergeometric function lFl(%8 9 4 . This function is given by the series

which converges for arbitrary values of a , /3, z. Usually, ,F,(a, /3, z ) is tabulated for z > 0. I n the formulas given below, the values of ,F1for z < 0, which are not difficult to find, are given by using the equation

B,

1F1(a,

-z) = e-Z,F,(B - a , B, z).

T o compute this function for very large absolute values of 2 not listed in most tables, one can use the asymptotic formula

For all the functions studied below, we give both the basic and asymptotic descriptions and the graphical representation. 1. The Ideal Relay*

. . _ .-

* Here,

-

and in the remainder of the discussion, we consider the same typical nonlinearities as in Appendix I (Figs. 61-66).

Appendix I V

312

-

4 7l

1.2 1.0

0.8 0.6

0.4 0.2

1.0

0

2.0

3.0

4.0 a

FIGURE 74

where

2

So(a)= -(Y F (', 2, -a2),

6

C,(a)

For small

a

=

$

l 2

1,

-a2).

he following representations are appropria e :

And, for large a, we have the asymptotic expressions,

1 + [ (2(Y)2

1 C,((Y) = 7ra

~

7 5 1 +-2 0 4 2 (2a)S + ...I . 9 1

+

Graphs of the functions Bo(a),and C,(a) are shown in Fig. 74.

Coefficients of Statistical Linearization

313

2. A Relay with a Dead Zone

where

For small a

=

4 4 2 ; we have the convenient forms

2

4

&(a) = -

z/F k=o k=o

+

(-1)"+"[2(k n)]! (27~)!(k!)~((Kn ) ! ( k +T

+

(-1)k+n[2(k + n)]! (2n)!(k!)Y(K n)!

+

Bk+l

6)

'

(3

"

For large a there are the asymptotic formulas

a'""C,(a)

+ 3)"2n + 5 ) + -1 , + (2n - 1)(2n + 61)2(2n (20!)~ (2n + + 3)Z e + (2n + 21)2(2n (244 ( 2 n ) ! [ r ( g- n)]Z pn+l

1)i

+ 3)2(2n + 5)' + ...I . + (2n + 1)2(2n6(2a)6

Graphs of the functions B,(a) and Co(a),as was already indicated, are shown in Fig. 74. Graphs of the functions B,(a) and C,(a) for n = 1, 2, 3 in the domain from a = 0 to a = 1 are shown in Figs. 75(a) and 75(b).

314

Appendix I V

0.6

0.4

0.2

0

1.0 0.8

0.6

0.4

0.2

o

0.2

0.4

0.6

0.8

-

1.0

a

(b)

FIGURE 15

For the domain between 01 = 1 and 01 = co,Figs. 76(a) and 76(b), show the functions aZnB,(or)and 01"+~C,(a)for n = 1, 2, 3 with the function 1/ a . 3. An Element with a Bounded Zone of Linearity

Coeficients of Statistical Linearization

315

where the functions B,(a) and C,(a) are described above in Appendices

I and 11.

FIGURE 76

4. An Element with a Bounded Zone of Linearity and Dead Zone

316

Appendix I V

30 28

2.6 24 2.2 2.0 I .8 I.6

0.0 0.2 0.4 0.6 0.0 1.0 1.2

yzu

(a 1

(b)

FIGURE 77a

a=a

AND

b

317

Coefficients of Statistical Linearization

where Y = d,/d, and the functions B,(a) and Cn(a)are the same as those used in the description of a relay with a dead zone. 5. An Element with the Characteristic f(X)

h,(a, u )

=

= NX'sgn x

2 d2 Nu -

d.rr

d2

2

hz(0, U ) = NU -[ I

6

= N u - [4 1 77

+2

-

(9)4 + 32 (?)2 6 ...I ,

+-+----,+...I, 3 1 (20r)~

-

-

1

a

< 1,

a 3 1.

16 (2a)

Graphs of the functions h, and h, are shown in Figs. 77(a) and ,77(b). 6. Elements with the Characteristic f(X) n = 1,2,3,

...

In particular, for n

=

3 we have h,(a, u) =

h,(a, u) = $(a2

+ 407, + 202).

= NX'"+'

where

k@pendix k‘

ELEMENTARY STATEMENTS ON T H E T H E O R Y O F M A R K O V PROCESSES

A set of random variables X which depend on a single parameter and which can take on only discrete values (..., t , , tk+l , ...) is a random sequence, i.e., a random sequence is a random process with a discrete argument (discrete in time). Just as a random process, a random sequence can be completely described by a denumerable set of compatible functions with probability distributions W,(xl , t , , ..., x, , t,). T h e simplest form of a random sequence is a sequence of independent random variables. T o characterize it entirely, it is sufficient to give a one-dimensional distribution function Wl(x,, t l ) , such as

T h e next class, in the sense of the degree of its complexity, is the so-called Markov chain. A Markov chain is a sequence in which the probability characteristics of the variable X ( t i ) depend only on the probability characteristics of the variable X(ti - l), that is, only on the values of the process at the previous instant and not on the whole “past” of the sequence. Thus, for a complete description of a Markov chain, it is sufficient to give the initial conditions, that is, the probability distribution W(x, to) of the variable X at some moment t = t o , and the probability of passing from the state X = x to the state X y in one step (from t , to ti+l),that is, the conditional probability

<

or its derivative with respect to y ,

318

Theory of Markov Processes

319

If the function p ( x , t i , y , td+l)does not depend on the transition moments (the number of steps) P(X,

ti

9

Y , t i + l ) = P(.,

then the chain is called homogeneous with respect to time. First, consider the case in which the variable X can take only discrete values x k (either a finite or a denumerable number of values). T h e initial distribution (for t = to) is given not as function of one variable x , as in the general case, but as a column matrix of numbers uk

:

.... u k ( t O ) ,

U(l0) = { u l ( t O ) y

where

Uk(t0)

**.>,

is the probability that x(t0)= X k

.

T h e probability of passing from one state to another in one step (from ti to ti+l) is given by the square matrix Pi of numbers P j k , i , characterized by the probability that X(ti+,) = x k under the condition that X(t i ) = x j : Pl1.i

P12.i

"'

PZ1.i

P22.i

''. P 2 k . i ..'

P1k.i

Pk1.i

Pk2,i "'Pkk.i

"'

p 1. = ...........................

1,

...........................

'**

For the homogeneous chain, which, henceforth, will be the only one with which we are concerned, Pi

E

P.

T h e absolute distribution at the moment t summation over all possible transitions

=

ti+l is given by the (A.20)

(A.21)

320

Appendix V

that is, the column matrix of absolute probabilities that X(&+,) = xk is equal to the product of the transposed matrix of transition probabilities and the column matrix of absolute probabilities that X ( t i ) = xi . I t is obvious that the absolute distribution at the moment t, > ti+l can be found by sequential application of the transition operators : U(tn)= (P"n-'U(ti)

=

(P"-'))"U(t,).

(A.22)

We denote the probability that X(ti+,) = x k , provided that X(t,) = x i , by p$) and the matrix composed of the magnitudes pj;) by P ( , ) . From Eq. (A.22), it is obvious that

We write an equation which relates the transition probabilities of the moments t = t , to the moments t = t,+,, and t = t,+,+, : p(7n+n)= p m i - n

=

pmpn

= p(mp)[n)

(A.23)

or, in scalar form, p;y+n) =

Cp;ypt;'.

(A.24)

Y

This equation is called the Kolmogorov-Chapman equation and is fundamental for the theory of Markov chains. For the nonhomogeneous chain it can be written in the form (A.25)

T h e transition probabilities depend on the number of the initial points at which each transition takes place. I n most physical problems it is of special interest to determine the equilibrium (limiting) probabilities, that is, to find U = lim U(tn)as n+m. T h e limiting probabilities U = { u l , u 2 , ..., u v ,...} can be found from the homogeneous matrix equation

u = PtU,

(A.26)

Theory of Markov Processes

32 1

which is obtained from (A.21) by the substitutions U(ti+J = U ( t i ) = U . This equation is equivalent to the system of linear equations Ilk

=

CPjkUi

(K

=

1 , 2, ...).

(A.27)

j

T h e probabilities u j obviously must also satisfy the normalization condition (A.28) c u j = 1. T h e characteristic equation A(h)

=

I AE

-

PI

= 0,

where E is the unit matrix, always has a root at h = 1 ; when pi > 0, this is a simple root [71], that is, the chain is regular. Therefore, the system (A.26) has a nonzero solution which, by taking into account the condition (A.28), can be written in the form (A.29)

where A,,(h) are the principal minors of the characteristic determinant

44.

Another problem which often comes u p in practice is to find the probability distribution of the number of steps N (of moments t N )for which the variable X will first become equal to or greater than x I if its initial value is xj < x 1 when t = t o . We denote by v ( j , n) the probability that X(t,) 2 x I , while X ( t ) x I for to t t, . It is obvious that

<

< <

since v ( j , 1) is the probability of going through the “boundary” x I in one step. T h e transition from the value xi to the value x l in n steps can be carried out only in the following manner : After one step, X appears equal to x, < x i , but after n - 1 steps it appears that there has been a transition from x, to x 2 x l .

Appendix V

322

Hence, it follows from the theorem of probability combinations that v(j,

=

Z p j v v ( v ,12 - 1).

(A.31)

”=1

Equations (A.30) and (A.31) are a system of recursion equations which uniquely define the probabilities v ( j , n). Let us go on to a more detailed study of the general case. T h e absolute probability distribution of the variable X at the moment ti+l , that is, the probability that X y at the moment ti+l , is related to the absolute distribution of X at the moment t , by the following integral equation :

<

for the probability densities we have (A.33)

<

< +

that is, the probability w(y , ti+l) d y that y X(ti+,) y d y is the integral sum of the transition probabilities p ( x , y ) dy of all possible values of X ( t , ) , where these probabilities are weighted in accordance with the corresponding absolute probabilities. It is obvious that Eq. (A.20) above is a special case of Eq. (A.33). T h e latter can be considered a recursion relation which makes it possible (at least, in principle) to find the absolute probabilities in terms of any number of steps. T h e Kolmogorov-Chapman equation for a homogeneous chain takes the form p ( m + n ) ( x ,y ) =

/ p y x , z)p‘n)(z,y ) dz,

(A.34)

where P ( ~ ) ( xy, ) d y is the transition probability from X = x to y X y d y in k steps. I n the general case of a nonhomogeneous chain with a continuous representation of a random variable, we have

< < +

(A.35)

323

Theory of Markov Processes

By analogy with Eq. (A.26), one can write an equation for the steadystate (limiting) probability densities (considered as a homogeneous chain) : W(4

while

=

J P(% r ) w W 4,

J-w(+)dx =

(A.36)

I,

(A.37)

T h e integral equation (A.36) is the homogeneous Fredholm equation of second order. Given several supplementary conditions, it has a solution which can be found by the method of successive approximations or by introducing a discrete scheme. We shall go on to the solution of the problem of the time distribution of the first crossing. We denote by v ( x , n ) the probability that X(t,) 2 x I while X ( t ) < x l for to t < t , . T h e probability of crossing beyond the boundary x = x , in one step obviously is equal to

<

(A.38)

By a process of reasoning analogous to the considerations for the discrete case, we find v(x, ).

=

J

=i

-m

P(X,Y)4Y,

-

1) dY

(.

> 1).

(A. 39)

Equations (A.36) and (A.37) uniquely determine the probabilities v ( x , n ) for arbitrary n. Methods for studying these equations for the case when P(X, Y ) = P ( X - Y

)

are explained by Bartlett [8]. T h e generalization of the theory of Markov chains for the case when the parameter t changes continuously is the theory of Markov processes (processes without aftereffects). A complete probability characterization of a Markov process is given by the function P ( x , t,; y , t z ) , which is equal to the probability

324

Appendix V

<

that at the moment t , the variable X(t,) y with the condition that at the moment t , the condition X(t,) = x was satisfied. Additional knowledge of the values of X ( t ) when t < t, will not change the character of the probability distribution of X ( t ) for t > t , . For the function P(x, t,; y , t,) the fundamental KolmogorovChapman equation can be written in the form P(X, t,;y, t z ) =

j

ffi

-m

(A.40)

P(S, t ; y , t 2 ) d,P(X, t,; z, t ) ,

where t , < t < t , . If P(x, t,; y , t,) is differentiable as a function of y , then one can introduce the differential distribution function p ( x , t,; y , t,) :

and instead of (A.40) write the equation

P(xl ti; y , b) =

1

--P

(A.41)

p(z, t ; y , tn)p(x,t,; z, t ) dz.

At the moment t = to let the differential distribution function be, w ( x , to). T he n for an arbitrary t > to one can find w(y, t ) by an equation analogous to (A.33), w(y, t )

=

J^

m

-m

p(.,

(A.42)

t o ; Y ; t)w(x, t o ) dx.

We assume that X ( t ) has a continuously varying character, that is, that X ( t ) varies without jumps of finite length. More precisely, we assume that the moments of Kth order (where k = 1, 2, 3, ...), of the absolute value of the increment Y - x possible during the time from t to t d, approach zero as d -+ 0 :

+

m,(.r, t , A ) =

j

m --m

Iy

-x

IWX,

t;y ; t

+ A ) dy

--f

0

as d -+0, while mk/mk-, + 0 as d 3 0. With these assumptions from (A.42), one can arrive at the partial differential equation given by Kolmogorov [35], (A.43)

Theory of Markov Processes

325

Here,

.m

I

are quantities which characterize the mean velocity of the variation in X ( t ) during the infinitely small interval of time d and the differential variance of the process. For the steady-state distribution w ( x ) = liml- w ( x , t ) , Eq. (A.43) takes the form -A(x)w

d + d,x - [B(x)w] = c,

(A.45)

where C = const. Here, it is assumed that the process is homogeneous with respect to time and that the coefficients A and B depend only on y . If it is assumed that w and w' approach zero as I y I + CO, then c must equal zero. Hence, (A.46)

T h e function w naturally must satisfy the normalization condition m

-m

w ( x ) d x = 1.

We note that Eq. (A.43) can be extended to the multidimensional case in which X ( t ) is a set of n processes X , ( t ) , X,(t), ..., X n ( t ) .This equation is written in the form

.(A.47)

326

Appendix V

where

1

B . - lim'j - A+O 24

J"

--oo

(yl -

I n conclusion, we shall use the Markov process to consider the amount of time required for the first transition beyond a certain level. T h e probability of passing beyond the boundary level x = x l in the time t A is related to the transition probability in the time t by an equation analogous to (A.39) :

+

4x9 t

+4 = r

21

J

-a

p(x, t;y, t

+ d ) v ( y , t ) dy.

(A.48)

We assume that the quantity I xi - x I is sufficiently large compared with the possible amount I Y - x I of the variation in X ( t ) during the short interval A. Thus, we can replace the upper limit in equation (A.48) by co. Expanding a(y, t) in a Taylor series, we find

and substituting this result into (A.48), we have

327

Theory of M a r k o v Processes Passing to the limit as A equation,

+ 0,

we obtain the desired differential

av -- A - av _ + B - . a+ at ax ax2

(A.49)

We write the initial and the boundary conditions for the function

v ( x , t). From the condition of the choice of origin, v(x, 0) = 0 .

We also assume that the probability of passing beyond the limit in an arbitrarily large number of steps (the probability of “absorption”) for ~ ( xco) , is equal to one. I t is obvious to assume that, when x --+ x l , v ( x , t ) + 1,

and that as x - - + - co, v ( x , t ) --t 0

for any arbitrary fixed t > 0. Consider the case of a process homogeneous with respect to time where the coefficients A and B do not depend on time. We shall find the equation for the expected time required to first pass beyond the point x = x 1 : m(x) =

I, m

av tzdt.

From Eq. (A.49), it follows that

/

* azv dt t at

0

=

A(x)

2 / ax

O0

av t - dt at

+ B ( x )I ax2 a2

t

av at

- dt.

Integrating by parts the expression on the left part gives

I

m

t -a dZ vt = t z I t - av o at2

t=m

- / o mz davt = - [ ~ ( x , t ) ] I t-m = - l a t-0

Thus, the desired differential equation is

+

d2m dm B(x) 7 A ( x ) dx dx

+ 1 = 0.

(A.50)

328

Appendix V

T h e boundary conditions obviously can be written in the form, m ( x l ) = 0, and m ( x ) + 00 as x 4 m. Equation (A.49) can be generalized for the multidimensional case. Let X ( t , , x l , ..., x,) be a point in n-dimensional space distributed inside a region G of this space with a bound A . We call co(xl, ..., x , , t ) the probability that the point X ( t ) will pass outside the indicated region through a part of the boundary y. T h e process X ( t ) is a Markov process with transition probabilities p ( x l , ..., x,; t , ;y1 , ..., y n , t z ) . Thus, we have the following partial differential equation for the function v : (A.51)

with the initial conditions v(x,

..., .r, , t o ) = 0

for all X inside the region, and with the boundary conditions v ( x l , ..., x,, t ) = 1 for all points on y whereas co(xl, ..., x , , t ) = 0 for all the points on the rest of the boundary A .

RELATED LlTE RAT URE

In the preface we have already mentioned the books which discuss the general development and the application of the theory of random processes to problems of automation. These are the works of V. S. Pugachev [65], J. Kh. Laning and R. G. Battin [51], and V. V. Solodovnikov [79, 801. One must also refer to the books by M. Pelegren [55], 0. J. M. Smith [78], and B. R. Levin [49]. T h e latter book contains an adequate and concise explanation of the general theory; it also discusses several nonlinear problems of special interest in automation and radio engineering. Together with these general works, it is appropriate to point out other publications that were used in writing this book and, also, to mention those which will help the reader to increase his knowledge, at least in respect to those problems which are extensively discussed. List for Chapter I

T h e list questions to be investigated dealt with the analysis of nonlinear transformations without feedback; there is a vast amount of literature on this subject. As basic sources, we used material from books by the following authors : V. S. Pugachev [65], V. I. Bunimovich [ l l ] , B. R. Levin [49], I. N. Amiantov and V. I. Tikhonov [I], P. I. Kuznetsov, R. L. Stratonovich, and V. I. Tikhonov [42, 431, Barret and Lampard [lOO], Baum [ l o l l , Galey [108], Gold and Young [109], Siegert [116], Middleton [ l l 11, and Price [114]. For the very condensed development of the general questions of synthesis and the theory of nonlinear filters we used a book by V. S. Pugachev [65] and also some papers by Lubbock [45, 1101. A more detailed presentation of the modern theory of synthesis can be obtained from the same book by V. S. Pugachev (Chapters 16-18) and also from a book by L.A. Vaynshteyn and V. D. Zubakov [12], which contains a systematization of the problems of synthesis related to the problem of separating signals from random interference. Problems of statistical linearization are described in accordance with the concepts of I. Ye. Kazakov [28-311 and Booton [103, 1041. 329

330

Related Literature

One ought to mention also the publications by Sawaragi, Sugai, and Sunakhara [75, 1151, the article by the present writer [56] on statistical linearization in the presence of a sum of harmonic and normal signals, and the work by K. A. Pupkov [66]. List for Chapter 2

A compilation of the most important ideas on the statistical theory of the nonlinear system with feedback is given essentially in the text of Section 2.1; a discussion of the method of statistical linearization

is presented in Section 2.2, and its modification for the case when the distribution of the input signal is somewhat different from normal (the final part of Section 2.4) is generally given in accordance with the work by I. Ye. Kazakov [28, 291. T h e concept of a more precise method for the investigation of nonlinear distortions in the form of the correlation function has been borrowed from the works of G. I. Pyatnitskiy [67]. I n describing the synthesis problem of correcting links in a system with given nonlinearities several general remarks were used from a book by M. Pelegren [55] and from studies by Newton [112] and Westcott [89]. One should also mention an article by Ye. P. Merkulova [52], which gives a formal scheme for synthesis in the presence of nonstationary signals. T h e application of the theory of Markov processes to the solution of nonlinear problems has been the subject of a large amount of literature. In particular, we refer to the works by I. L. Bershteyn [9], V. I. Tikhonov [85, 861, R. L. Stratonovich [81-831, P. S. Landa [46], P. S. Landa and S. P. Strelkov [47], Chuang and Kazda [106], and Barret [7]. This important topic was developed mainly in connection with questions of radio engineering and, to a certain extent, it is discussed in this book. T h e exposition of Section 2.5 is based almost entirely on the classical work of A. A. Andronov, L. S. Pontryagin, and A. A. Vitt [2]. For appendix V, which attempts to give a short summary of the elements of the theory of Markov chains and processes, the reference material is based on the works of A. N. Kolmogorov [35], V. I. Romanovskiy [71], Barret [8], and Feller [90].

Related Literature

33 1

List for Chapter 3

T h e study of nonstationary states was derived essentially from the works of Ye. P. Popov [63], Sawaragi, Sugai, and Sunakhara [75, 1151, of the author [56-581, and of the author in collaboration with V. Ya. Katkovnik [34]. T h e investigation of the problem of random disturbances in selfoscillatory states with piecewise linear systems can be traced back to articles by S. M. Rytov [74] and R. Kho Sadekov [76]. One must emphasize that most of the references cited above on the application of the theory of Markov processes discuss nonstationary states with equations which are first transformed by means of a slowly changing amplitude (cf. Section 3.3). Several other ideas can be found in the paper of S. M. Rytov [73], where the technique of using a small parameter is applied to a number of problems. List for Chapter 4

T h e fundamental works on the theory and practice of extremal systems are by V. V. Kazakevich [25-271 and by C. S. Draper and I. T. Lee [23]. A study of the general principles of the construction of extremal systems is given by Tsyan’ Syue-Sen’ [97], A. A. Feldbaum [94], and A. G. Ivakhnenko [24]. A development of a statistical theory for extremal systems is begun in the works of A. A. Feldbaum [91-931 and Serdenzhekti [77]. Of the most recent works, one ought to mention the articles by A. A. Krasovskiy [38-40], I. Ye. Kazakov [32], I. S. Morosanov [54], V. N. Varygin [13], S. A. Doganovskiy [20], V. M . Kuntsevich [MI, T. I. Tovstukha [88], Chang [105], and also A. A. Feldbaum’s study on the theory of dual systems [95] for the very special case when they are extremal systems. T h e basic reference material in Chapter 4 (Sections 4 . 2 4 5 ) is the work of the author [59-611 and the publications of V. S. Baranova, [6], Ye. P. Gilbo [18], and V. Ya. Katkovnik [34], all written in collaboration with the author. Important results were obtained from the mathematical studies of Dvoretskiy [lo71 and Block [lo21 concerning problems of statistical approximation.

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AUTHOR INDEX Amiantov, I. N., 329 Andronov, A. A., 330

Khintchin, A. Ya., 11, 138 Kolmogorov, A. N., 139, 141, 324, 330 Krasovskiy, A. A., 331 Krug, Ye. K., 240 Kiintsevich, V. M., 331 Kuznetsov, P. I., 329

Baranova, V. S., 331 Barret, J. F., 79, 329, 330 Bartlett, M. S., 323 Battin, R. G., viii, 57, 59, 329 Baum, R. F., 329 Bernshteyn, I. I,., 196, 330 Besekerskiy, V. A., 150 Block, H. D., 331 Booton, R. C., 329 Bunimovich, V. I., 329

Lampard, D. G., 79, 329 Landa, P. S., 330 Laning, J. Kh., viii, 57, 59, 329 Lee, I. T., 331 Leibnitz, 119 Levin, B. R., 116, 117, 307, 329 Lindeberg, 52, 53 Lubbock, J., 75, 329 Lurie, A. I., viii, ix

Chang, S. S. L., 227, 228, 331 Chuang, K., 330

Merkulova, Y. P., 330 Middleton, D., 329 Morosanov, I. S., 59, 220, 278, 331

Doganovskiy, S. A., 331 Dolgolenko, Yu. V., 278 Draper, C. S., 331 Dvoretskiy, A., 331

Newton, G. C., 330

Emde, F., 37 Feldbaum, A. A., ix, 14, 21 1, 216, 232, 238, 247, 331 Feller, V., 330 Galey, J., 329 Gel’fand, A. O., 216 Gilbo, Ye. P., 331 Gold, B., 329

Pal’tov, I. P., v, 159 Pelegren, M., 134, 329, 330 Pontryagin, L. S., 143, 330 Popov, Ye. P., v, viii, ix, 159, 331 Price, R., 28, 304, 305, 308, 329 Pugachev, V. S., v, viii, 17, 35, 50, 59, 86, 90,329 Pupkov, K. A., 330 Pyatnitskiy, G. I., 330 Romanovskiy, V. I., 330 Rytov, S. M., 196, 331

Ivakhnenko, A. G., 331 Jahnke, E., 37 Katkovnik, V. Ya., 331 Kazakevich, V. V., 331 Kazakov, I. Ye., 329, 330, 331 Kazda, L. F., 330

Sadekov, R. Kho., 331 Sawaragi, Y., 183, 185, 330, 331 Serdenzhekti, 331 Siegert, A. J. F., 329 Smith, 0. J. M., 329 Sobolev, 0. K., ix 339

340

Author Index

Solodovnikov, V. V., viii, 59, 329 Stratonovich, R. L., 329, 330 Strelkov, S. P., 330 Sugai, N., 330, 331 Sunakhara, Y., 330, 331 Syue-Sen, T., 331

Varygin, V. N., 331 Vaynshteyn, L. A., 329 Vitt, A. A., 330 Voronov, A. A., 116

Tikhonov, V. I., 56, 275, 329, 330 Tovstukha, T. I., 331 Tsetlin, M. L., 216

Young, C. O., 329

Westcott, J. G., 330

Zubakov, V. D., 329

SUBJECT INDEX Analysis, 13

Random functions, 8 Relay ideal, 293, 311 with dead zone, 294

Canonical expansions, 88 Ergodic, 12 Frequency distortions, 107 Harmonic balance, 178 High-frequency random interference, 146 High signal-to-noise ratio, 151 Hysteresis loop, 301 Linear compensation networks, 123 Linearization direct, 88 statistical, 77, 167, 233, 310 Markov chains, 318 processes, 89, 138, 318 Method of alignment, 185 Moment characteristics, 10

Signal narrow-band, I16 slowly changing, 146 wide-band, 116 Simultaneous testing and operation, 277 Stationary, 1 1 Successive approximations, 89 Synthesis, 13 Systems ,discrete extremal, 248 extremal, 21 1 radar tracking, 144 relay, 185 Transformations lagless, 3 linear, 58 nonlinear lagless, 16, 17 with lag, 43 reducible-to-linear, 7

Nonstationary states, 150 Oscillations forced, 196 periodic, 158 spontaneous, 152

White noise, 224 Wiener-Hopf equations, 59 Zone bounded, 297, 314 dead, 297, 313

Proportional action, 222

341