CONTROL THEORY AND NONLINEAR MECHANICS (A REPORT BASED ON SOVIET PAPERS 1955-56)

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• Publication Date: August 1, 1958
• Content Type: REPORT

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CENTRAL INTELLIGENCE AGENCY
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This material contains information affecting the National Defense of the United States within the meaning of the Espionage Laws, Title
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PROCESSING COP'
COUNTRY USSR
SUBJECT
DATE OF
INFO.
PLACE &
DATE ACQ.
Control Thwry and Nonlinear Mechanics
(i4 &Par 49,44-60 0,41- covif?er-
Phfittles /f ?i- so-6)
REPORT
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DATE DISTR.
NO. PAGES
REFERENCES
1 August 1958
1
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A report
"Control Theory and Nonlinear Mechanic"
Ofl
The report is based
on Soviet papers published on the subject during ly55 and 1956.
Distribution of Attachment:
OSI: Loan
S E-C-R E-T
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Eeport On 1955l956- Soviet Publications on Control
Theory and. Nonliziear Mechanis
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? Introduction
Part I. Wart On Control Theory
A. Survey of Publications On the Theory of Control Engineering
1. General Remarks
2. Mathematical Methods
3. Analogy Methods
Relationships Between Control Engineering and Nonlinear Mechanics
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Introduction
The following report refers to Soviet literature which appeared
in 1955 and 1956. Although not all publications were available it
is assumed that the most important works are covered. Some were
discussed in earlier reports and are thus not treated again here.
The report is divided as follows: The first part treats the works
on control engineering. Then the results are compiled. The subsequent
sections contain individual reports. A survey of the work is given by the
bibliography, in which the publications connected with the 2nd Soviet
Congress on Control Theory have been grouped together. The reports
in this group are arranged as In the-Congress report; they are cited
y volume number and chapter (for example Krug (1 .0). The other
reports are listed alphabetically and are in the numerical order given
in the 'references (for example, LEE. (16)), Section B of the first
part deals with the position of nonlinear control engineering within
Ulle framework of nonlinear mechanics and ?acts as an introduction to
the second part, which is devoted to nonlinear mechanics. It also
contains a compilation of results) a bibliography, and iudivi q-
rePorts$
;!ot,
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The reporters have taken the liberty of treating only very briefly
those partial problems which have already been treated in more detail
in earlier reports. For this reason, the second method of Lyapunov and
the Cypkin ffsypkin7 theory of relay controllers have not been developed
In detail. The present report, however, is in no place based directly
on any previous report, and can thus be read independently.
Part I
A. Surve Of Publications On The Theor of Control Engineerin
1. General Remarks
Although not exhaustive, the present report gives a reasonably
complete picture of Soviet publications in the field during 1955 and
1956.
The proceedings of the 2d All-Soviet Congress On The Theory Of
Automatic Control were published during the report period. The three-
volume work contains 63 papers with discussions. Ihdividual reports
on some of the papers are given (see Section D1). A discussion of all
the papers would have exceeded the scope of this work; nevertheless,
the table of contents of the three volumes, which is given in full,
should give an indication of what problems the Congress concerned
Itself with. It must be remembered that the Congress took place in
1953 and thus reflected the status of research work done up until 1952.
No essential changes in the research areas have been noted since,that
time, except that perhaps the areas of "statistical methods" and
"analog computers" have come more to the fore.
Since the Congress Report appeared in 1955, the number of works
to be considered in the present report increased considerably. Even
aside from the Congress Report itself, the number of works on the
subject is larger than in previous years. In part this is due to the
fact that Soviet publishers like to put out a whole series of articles
on a single theme, many of which differ very little from on nother.
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The works of Taypkin and Meyerov are an example. Increased Soviet
interest in control engineering also resulted in increased publication.
The most important journal on the subject Avtomatika I Telemekhanika,
appeared in 12 issues, rather than 6, during 1956; there are also other
journals which were not available to this reporter. The quality of
the publications has dropped somewhat, which is to be expected in view
of the increase in volume.
The five papers which were read by the Soviet scientists at the
International Conference on Control Engineering in Heidelberg are not,
strictly speaking, to be considered USSR publications, since the
Congress Report is to be published in Germany. They are nevertheless
included here because they are so closely connected with the Soviet
publications and also give a good indication of what subjects the
Soviets chose to discuss at the International Congress. It should be
mentioned here that the Soviets show a genuine desire to cooperate on
an international level in matters pertaining to control engineering;
the inclusion of foreign-language summaries in Avtomatika I Telemakhanika
Is evidence of this desire.
2. Mathematical Methods
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Most of the works discussed in the report were limited to a
presentation of the mathematical problem, A great deal of progress
in theory can hardly be expected in the short period of only two
years. Most of the works investigated known problems by means of
known methods; only in a few cases were there new points of departure
toward new developments. It must be remembered that many works are
intended only to provide practical aids for actual application. The
works are arranged according to linear and nonlinear problems, but it
must be remembered that a strict division into these categories is
not possible.
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a) Linear Methods
The use of the term "linear methods" assumes that the behavior
of a control system with respect to time can be described by a system
of linear differential equations with constant coefficients or by
means of a single differential equation equivalent to such a system and
of a correspondingly higher order. This assumption is correct for
most practical systems, at least in the first approximation. In
general, the differential equation itself is not used, but rather the
appropriate transmission function which is obtained as follows: let
a and xe be the output and input variables of the transmission system,
and the differential equation which connects them have the form
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Dnxa t. alDn-lxa 4.
, 41- anxa = boDmxe b1I)411 xe bmxe
where the operator D designates the derivative with respect to time,
D If this equation is subjected to the Laplace transformation,
dt
a relationship is obtained (in tlie case of vanishing initial conditions)
between the Laplace transform variables ia and ie in the form
xa : F(p) Xe
F(p) is a rational function of the Laplace transform variables p with
the denominator pr +. a1pn-1 ,
1- ? -1. a and the numerator
n
pm.+. b1Pin'.1+ ? ? 4. pm; it is called the transmission function
or the operator of the element of a closed loop (transmission element)
characterized by the differential equation. If a transmission system
consists of several transmission elements, then a transmission function
can be formed, according to fixed rules, from those of the individual
elements, and since an open and a closed control loop can always be
considered a transmission system, it is correct to speak of the trans-
mission function of the closed and of the open Control loop; the two
have a simple relationship. If the open loop has the transmission
function F(p), then that of the closed loop is
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A problem posed and solved by Kac (18) leads to a type of reversal
of this relationship. He considers the transmission function of the
controlled system as given and attempts to determine that of the
controller in such a way that the transmission function of the total
system has a given form. This problem can be solved by simple means;
it is also related in a certain way to the problem of the quality of
control.
The basic requirement of a technically feasible control loop is
that of stable behavior. The stability behavior can be described
conveniently with the aid of the transmission function. Stability
exists when all the poles of the transmission function are located
in the left half-plane of the complex variable p or, in other words,
when the denominator of the transmission function, which is also
called the characteristic polynomial of the corresponding differential
equation, has only zeros in the left hal)!,-plane. Various (mathematically
identical) stability criteria are known, which can be employed to
solve the stability problem in any individual case. Theoretically,
therefore, the problem is completely solved. The practical application
of the criteria, to be sure, especially in high-order systems, is
often rather difficult and laborious. For this reason processes are
suggested again and again, which are supposed to facilitate the comput-
ation, at least in special cases. Ostrovskiy (40) explains such methods
for systems of the orders four, five, and six. Kislov (20) treats
the stability problem for a control loop in which one transmission
element contains two variable parameters and gives the construction
of nomograms with which the stability ranges for the closed loop can
be read off. A satisfactory stability criterion, which is much easier
to apply than that of Hurwitz fflurvitg, is demonstrated by Dobronravov
(13).
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Some time ago, Ajzerman gytserme47 and his pupils introduced t.670_Hum
concept "structural stability." This concept can be defined as follows.
Consider the time constants and the amplification factors of an
arbitrary control loop to be arbitrarily variable. These parameters
obviously can assume only positive values. In general, the system
will be stable in the case of certain combinations of the parameter
values, and unstable in the case of other combinations. If, however,
even with an arbitrary choice of the variable parameter, it is not
possible to make the system stable in any way at all, then it is
designated "structurally unstable." For example, a system with the
characteristic function k(Tp - 1) is always unstable, no matter how
the positive values T and k are chosen, whereas the system with the
characteristic function kl(Tp - 1) k2 is structurally stable,
since the polynomial obtains a negative zero point in the case of
suitable values of ki and k2. This concept has proved to be of use
for all sorts of theoretical and practical investigations and indicates
the demand for criteria for structural stability or instability.
Such criteria can actually be established according to the ratio of
the total number of transmission elements to the numbers in which the
various types of transmission elements occur. Some of these criteria
are reported by Ajzerman 5Aserma7 (2) and Gantmacher (1, VI); they
involve propositions which can also be important for the practical
synthesis of control loops. Up until now only simplest cases have
been treated. (See Aytserman and Gantmacher, PMM 18, 103-122 /195g.
Further discussion of structural stability can also be found in the text
by Aytserman, mentioned in the introduction.)
As already mentioned, it is possible to assign the transmission
function of any complex control loop, if the transmission functions
of the individual elements and the manner in which they are connected
are known. The transmission function of a complex system depends on
the parameters of the individual elements in a manner which is so
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difficult to survey that generally valid statements are difficult w
arrive at. For this reason, the theory of multi-loop control systems
is still in its infancy, except for basic fundamentals. Meerov
gvieyerog has made some contributions to this theory. He is invest-
igating multi-loop circuits with several controlled variables and
assumes that the equations of motion have the form
,
k1Mi(p
aikxk 4. (Di(P)Mi(P)i. ki)xi = kiMi(p)fi (i
x is the controlled variable, f the external effect, D(p) and M(p)
the operators of the controlled system and the controller, k and k
amplification factors, and i refers to the values of the ith partial
system. The individual loops are thus coupled together in a definite
manner. Meerov (33, 34; 1, VIII) then treats the following question:
Under what assumptions does the total system remain stable, when the
Individual amplification factors ki are allowed to increase arbitrarily?
The necessary conditions refer to the location of the zeros of certain
polynomials which are determined by the coefficients of the equations
of motion. The order gi of the polynomials Di(p) also plays a role:
If individual values of gi are greater than two, an instability occurs
without fail when the coefficients ki increase; this instability can
be avoided by proper measures (introduction of differentiating effects
or use of stabilizing elements).
A second problem posed and solved by Meerov (35) concerns the
autonomy of the system under consideration: IS it (in cases of
permanent stability) possible to make the individual partial loops
of the system independent of one another by increasing the amplification
factors? It is demonstrated that such a possibility exists when gi
Is equal to or less than 2; otherwise, the system must be altered structurally
in an appropriate way, just as in the case of the stability problem.
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A favorite Soviet method of investigating stability is that of D-
separation, developed byNerjmark Lifeymarg; this method is also suitable
for judging the quality of control. The basic idea of D-separation
can be explained as follows: Let gp) be the characteristic polynomial
of a stable transmission system. Its zeros therefore all have negative
real parts. If the zeros are shifted in any way in the complex plane,
the stability of the transmission system will obviously disapper at
the very moment the moveable zero positions reach the imaginary axis.
If the polynomial H(p) (and thus its zeros) depends on certain para-
meters, then the "critical" parameter values, i.e.) those at which
the stability ceases to exist, are Obtained with the aid of the equation
H(jw) O. In this case, w is a real value. If this evatiOnlis
separated into its real and imaginary parts, and if w is eliminated,
a relationship between the parameters will be obtained, which can be
interpreted geometrically as a limitation of the stability range in
the space of the parameters and, correspondingly) is called the "limit
of the D-separation." A knowledge of this affords the possibility of
making statements on the stability behavior, especially on the dependence
of stability on the parameters. The simplest conditions exist when
the transmission function depends on only one single parameter, which
is linear. In this case, the limit of the D-separation in the plane
of this (complex) parameter can be established rather easily. It
forms a curve closely related to the locus of the complex frequency
response) as demonstrated by Meerov (21 III). Marjanovskij 5aryanovsk1e
(1, IX) used the method of D-separation in the investigation of a
transmission system which is made up of n similar series-connected
amplifiers with a transmission function of the order in and a feedback.
He shows that the consideration of the characteristic equation of the
total system which is of the order m n, can be traced back to a
system of n equations of the mth order.
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Gopp (16) presents an obvious modification of the method of D-
separation. He defines the limit of the DK-separation by means of
the equation H( iw)
0; 0PC is a real number (only negative ol<
are of importance), w is a real parameter. With the Do< -separation,
one can judge the degree of stability of a system which is defined
as follows: If the zero points 1)1, p pn of the characteristic
polynomial are all in the left semiplane, then the real part of the
zero point farthest to the right is designated the degree of stability.
Popovskij gopovskie (1) VIII) determines, with Doc -separation, the
degree of stability of an (idealized) system with two controlled
variables, such as occur in practice in the case of the control of
turbines and steam-boiler installations.
The latter works extend beyond the narrower scope of merely posing
the problem, since they take the quality of the control into account,
which is characteristic of the most recent developments in control
engineering. Whereas the prime consideration formerly was on stability
and the time response of the control action, the tendency now is to
concentrate more on the control action itself. Naturally, the external
interferences on the transmission or control system which play no part
In the stability problem must likewise be taken into account.
From a mathematical point of view, this leads to the solution of
a nonhomogeneous linear differential equation) which is simple enough
theoretically, but which is not always so simple from the practical
point of view. Moskvin (2, II) developed for this purpose an approxi-
mation method which makes use of matrix algebra. He substitutes for
the differential equation an equivalent system of the first order of
the form x = Ax f (A is a matrix with constant coefficients),
represents the solution in closed form with the aid of the matrix etA
and then derives an approximate expression. The method can, to a
great extent, be put into schematic form and is thus well suited for
treatment with computers. In the case of control problems, however,
there will be many instances when the solution, in the case of arbitrary
interference f, actually will not be carried out.
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It is obviously sufficient to know the reac ons o a
to certain pronounced interferences, which need be selected only so
that any arbitrary interference can be produced from them by means of
superimposition. The stationary reaction which occurs in the case of
a harmonic effect of a given frequency w, or the reaction on an abrupt
effect, is generally used. The synthesis of an arbitrary effect leads,
in the first case, to its Fourier representation, and in the second
case to the so-called Duhamel integral; thus it can be carried out
according to known methods. With a sinusoidal input variable xe =
A sin wt, the output variable (in the stationary state) has the form
xa A F(jw) sin wt Or A 12 F(Jw)eiw,
whereby F(p) is the transmission function. For the complex value F(jw)
the Soviets use the designation "amplitude-phase characteristic," or
"frequency characteristic." If the interference function is chosen
as the unit surge which is defined by z(t) = 0 (t 0),
then the transfer function CI) (t) is obtained as the reaction of the
system. In Soviet terminology this is called the "time characteristic."
If the symbol L is used for the Laplace transform, the relationship
between F(p) and 1)(0 can be represented by the formulas
L
and itl(t)
If the inverse of the Laplace transform is written as a complex
integral, a simple connection between frequency response and transfer
function (or between time characteristic and frequency characteristic)
Is obtained. Occasionally the impulse-transfer function k(t) thus
the reaction of the system to a unit surge, is used in place of the
function IS(t). It can be defined by the equations
L k(t)
= f(p) and k(t) L-1
and, naturally, has sense only when F(p) is a genuinely fractional
rational function.
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The connection between 0(t) and F(p) is used and investigated in
various works. Kagan (38a) investigates the conditions under which
the relationship
0(t) Re
car:7
F(j1.1) eiwtJL
tw
is differentiated for t. Block (2;3) shows how conclusions concerning
the transfer function 0(t) can be drawn from the behavior of the fre-
quency response F(jw). For example) he gives conditions for F(jw)
which guarantee a smooth response of 0 M. He further investigates
the extent to which the real portion of F(jw) alone) as compared with
the portion of F(jw) which stems from large w) participates in the
behavior of 0(0; he further obtains estimates for the time from the
beginning of the process to the first extreme of (t) and for similar
values. Oveseevic (42) derives other dissimilar terms. He considers
the transfer functions 01(t), 02(0 belonging to different transmission
functions Fl(p), F2(p) and estimates the expressions Di - 7 or
with the aid of the expression
(E)
hs2.(t)
F(jw) - F2(jw)
(>0
2
1
2
dw,
dt
Voronov (21II) gives a detailed survey of the various methods of
determining the transfer function graphically from the locus of the
frequency response) and vice versa, In most cases it involves graphical
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integrations whereby the integral is approximate by means or a sum.
Voronov also isketches methods of evaluating the parameters of the
transmission system on the basis of experimentally determined transfer
functions or frequency responses; these procedures are, to be sure,
limited to low orders.
The use of the locus of the frequency response for determining,
or at least for estimating, the transfer function is a recent develop-
ment. Originally, these loci were used only to obtain information on
the stability of the system with the aid of the Nyquist criterion.
This criterion is also the subject of the brief note of KUzovkov
LIC-utsovkoil (26), who investigates the degenerate Nyquist diagram for
transmission systems with one oscillating element (characteristic
polynomial T2p2 1), as well as the note of Demcenko (12), which
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deals with the construction of the locus of a rational transmission
function. More recently, the "logarithmic locus" is being used along
with the actual Nyquist diagram. The logarithmic locus is obtained
by a plotting of the logarithm of the value of F(jw), multiplied by a
suitable scale factor, as a function of w. This method of presentation
is now appearing in the Soviet literature, and has been explained in
detail in the textbooks of Aytserman and Popov. The note of Einovkov
(26) also considers the logarithmic locus and its degeneration for a
system with one oscillating element; in another work (27) he considers
the connections between the limits of the D-separation and the logarithmic
frequency characteristic, and gives algebraic-graphic criteria for
the stability in the case of variable parameters.
As remarked above, both the use of the D-separation and the
construction and evaluation of the transfer function are closely
associated with the question of the quality of the control. An exact
definition of the concepts "control quality" and "optimal control
action" is very difficult to give. One relatively simple definition
uses the "quadratic control surface:" the process x(t) is optimal
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when the integral (x(t)4dt becomes a minimum. This definition
Is frequently used, and can be evaluated mathematically, at least In
the case of low-order systems, It does not, to be sure, consider the
fact that any practical process is subject to certain "secondary
conditions" as a result of limitations on response and velocity, etc.
For this reason, Feldbaum (14; 2,V) uses another explanation: He
calls a control action optimal when it takes place in a minimum period
of time and, to be sure, with the secondary condition that a certain
expression, formed linearly with the aid of the values x, x, x(n)
?
remains below a fixed limit. As Feldmann geldbauaq shows, the evaluation
of this condition can be carried out most conveniently by following
the trajectories of the equations in the phase-space, and permits a
designation of the trajectories of an "optimal" process. It is
further shown that the optimal system, in this sense, can be obtained
only through nonlinear correction terms (2,V). The Feldbaum definition
of the control quality is used in a somewhat specialized form by Lerner
(2,V). He is interested primarily in fast-acting control systems and
gives a series of methods of arriving at the optimal process for actual
engineering applications (instrumentation). He, like Feldbaum, employs
the method of the phase-plane or phase-space and likewise considers
nonlinear correction terms.
The just-mentioned definitions for the quality of control are not
very well suited for practice. For this reason, more convenient means
have been sought. Among these belongs the already-mentioned degree
of stability which, as Meerov (36) explains, can be determined with
sufficient accuracy with the aid of the limit of the 1)(7separation.
Kalis (1,IX) uses the time in which the process decays to half its
value as a measure for the quality of an aperiodic process; in the
case of oscillatory processes, he uses the ratio of two successive
extreme values of the same sign, and shows how these values can be
determined graphically. Ocasionally, an attempt is made to determine
S"WIL'Ef'ilft"
LUOL
-
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Nor
P;411.
-1 ANIL
the control quality through the maximum value of
?
ii t
of the closed loop. Meerov (36) shows that this value is not always
suitable for an adequate designation of the control action. This
applies more or less for all those definitions of control quality
which do not take the total course of the control action into account.
The degree of stability) for example, depends only on one zero point
(or a pair of zero points) of the characteristic polynomial, whereas
the control action is influenced, naturally, by all zero points. The
methods which are based on the estimations of the transfer function
(Bloch (2;3); see above) all take the total fc-ontrof action into
account.
Generally) the control quality is investigated in connection with
the synthesis of control loops. Originally, the mathematical treatment
of control problems was essentially confined to the analysis of the
control loop) i.e.) limited to the setting up and discussion of the
equations of motion. Recently, the synthesis has gained in importance.
Even in the case of the design of the control system an attempt is
made to attain certain desired dynamic properties, one of which,
naturally, is stability. The mathematical task is to determine the
transmission function of a control loop in such a way that the control
action is stable and "optimal." Once the transmission function is
determined, the computed control loop must be realized physically.
Bloch (2,111) and Fateyey (21IV) describe a series of procedures for
the determination of the parameters of the system on the basis of
specifications for the control quality. General requirements, such
as maximal degree of stability, minimal control time, etc., are
discussed. One simple synthesis problem is treated by Kac (18a) (see
above), who attempts to determine the transmission function of the
controller on the basis of certain conditions. Sokolov (2,III) points
out that the synthesis should not be carried out solely from mathematical
points of view, but that the possibility of realizing computed trans-
- 14 -
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mission function in practice and the design characteristics or tne
components must be taken into account; he gives several examples.
Krasovskiy (21III) treats a special problem of synthesis: In a multi-
loop system, in which the transfer functions of all the elements
except one are known, the unknown function must be determined in such
a way that the total system will have predetermined properties. For
this purpose he uses methods for the approximate calculation of the
transfer function, when the transmission function is known. Lerner
(2,V) and Feldbaum (2,V) treat those cases where the limitations of
coordinates are taken into account in the synthesis of practical
control loops. Solodovnikav (21IV) points out that the synthesis of
correction elements for the improvement of the control quality is
mathematically equivalent to a problem from the Tchebycheff approxi-
mation theory; this involves the approximation of a given function
B(x)
f(x) in an interval a < x < b through a rational function TTA
with fixed degree of numerator and denominator is such a way that the
deviation
max
f(x)
t,-
B(x)
767)
?????11,
(a < x < b) becomes a minimum.
The definitions of the control quality mentioned up until now and
the corresponding problems of synthesis refer to an arbitrary
interference and make no assumptions regarding the external effect.
The tendency recently, however, has been not to give a general
explanation of the concept of optimal action, but to explain it in
regard to those interferences and effects which are considered typical
for the particular relay system. The concept "optimal" is, in such
cases, not defined absolutely, but in reference to the "environment"
of the relay system; it no longer depends solely on the left side of
the equations of motion, but also on the interference function which
appears on the right side. The problem of synthesis is altered like-
wise/ and now represents the attempt to compose the control loop in
the light of the typical interference functions.
ri 1"il ri
Oa=
KUlebakin (213)
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describes
e principle tor
the realization of loops which are optimal for certain interferences.
He explains as follows: The attempt is frequently made to compensate
rapidly the interferences effecting the system by measuring not only
the control deviation, but also the interference value directly.
This latter measurement is fed into the control loop through a suitable
transmission system, which causes an additional influence (so-called
"imposition of interference values") on the control action. KUlebakin
speaks of invariance when the influence takes place in such a way that
the control value becomes independent of the interference value.
Conditions can be established for this, and even carried out in
Instrumentation, obviously only for a certain class of interferences
(for example, those which are characterized by a differential equation).
Naturally, the principle can only be carried out approximately in
practice. Similar investigations were carried out by Ivachneko (2,3),
who also gives methods for computing the parameters.
The concept "typical interference" is an idealization. The
interferences which actually occur in practice do not obey any
devisable mathematical law, by which they can be predicted. They
cannot be considered known until they have actually occurred. The
concept "typical" interference can thus be interpreted only on the
basis of a statistical approach, wherein the interference is to be
looked upon as a value, the time behavior of which can be determined
statistically through a large number of observations; it is therefore
a probability function. Thus, even the coordinates of the control
system, especially the control variable and the control deviation
itself, now have the character of probability functions. Thus the
theory of linear communications systems must be expanded, if the
mathematical treatment of control Tnnpq 'which effect the probability
values is to be built up. At least it would be well to elaborate the
theory in the case of a number of problems of synthesis.
,
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TO theory or transmission ana control systems wnich are subject
to probability effects has been built up within the last ten years.
It is based primarily on the investigations of N. Wiener and his
associates. In the USSR, the theory was introduced by Solodovnikov,
who, in keeping with the American works, carried out his own invest-
igations and even wrote a textbook. He caused a number of Soviet
"control mathematicians" to concern themselves with statistical methods.
The work here is generally limited to a consideration of stationary
probability functions. Such a function igt) can be characterized by
its autocorrelation function
Rm (Pt ) iim -r
?10110
m(t + t)m(t)dt
The functional value of Rm(5) for It: 0 is also designated as its
mean square value
1
(x(t))2dt
In place of the autocorrelation function, the so-called spectral
density Sm(w) can be given. There exist the relationships
+ (240
S(w
wt /
dw? S(w)
oe)
Let the transmission system be given (see above) by its transmission
function F(p) or its impulse-transfer function k(t). Let the input
variable be the probability function m(t), and the corresponding
output variable x(t). Between the spectral densities of these two
functions there then exists the relationship
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'rad
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Sx(w) =
F(jw)
2
+hug an analog of the known formula for the frequency response. For
the correlations
C>0
k(u)
Os!).
Rm( u v)k(v)dvdu
applies. The spectral density SE (w) of the deviation E (t) m(t)
x(t) is determined through the formula
2
S6 (w) = 1 - F(jw)
If, besides m(t), there is another interference value n(t), which is
independent of m(t), then we have the sum
On
I - F(jw)
??????..
2
Sm(w) [F(1w)
The mean square error can be computed from the equation
7
marova0
2
dw
Sm(w)
It plays a role in the definition of the control quality of a system
which is subject to probability effects. If )/4 (t) designates that
previously fixed time function which the system is supposed to simulate
(thus the command variable), and if the actual input signal (which
still contains secondary effects) is designated by Or(t), and the output
variable by x(t), then the system can be called optimal when the
expression
4-T
2
(/(t) x(t)) dt is a minimum
2T
rya
rillTri rt.11,C17H..
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This "condition of the minimal means square error" can be converted
mathematically and leads to an integral equation for the impulse-
transfer function k(t). It can also be put in the form
0
1116 (2,7" -u)k(u)du 0 ( t>, 0);
here Ro is the autocorrelation of the input signal and Rit49, is the
cross correlation of the probability functions /4c. and 0.
The criterion of the minimal mean square error is used by XPrakin
(25) to determine the optimal transmission function of a linear differ-
entiator under the assumption that the input variable is a probability
function of given spectral density superimposed by a white noise.
(White noise is a probability function, the values of which are
completely independent of one another; their autocorrelation is a
d-function, i.e., zero for all 1,?1"fr 00) KUrakin gives an
explicit expression for the transmission function. Solodovnikov and
Batkov (45) have a more general approach. They consider systems shich
are optimal not only in the case of an effect of a given form, but
which can adapt to a variable effect in such a way that the operating
state at any time is as close to the optimal as possible. Mathematically,
such a system, which is called "self-adjusting" by its authors, can be
characterized under somewhat general conditions by means of an integral
equation.
The above-mentioned connection between the spectral densities of
the input and output variables, which is completely analogous to the
formula of the frequency response in cases of continuous effects,
leads naturally to the use of the locus process even in the "statistical
theory," provided only the spectral densities are introduced. Idelson
(2,VII) uses the logarithmic frequency characteristic; his system is
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effected by an input variable, one consitituent of which is a white
noise, and the problem is to eliminate the noise as much as possible
by means of an appropriate filter without imparing the required
4
stability and reproduction quality. Cypkin gsypkig (11) treats a
pulse-control system (see below) which is influenced by a probability
tariablep Here, too, the introduction of the statistical values in
the analysis brings no new viewpoints. Kazakov (19) shows that even
nonlinear transmission systems can be made accessible to the statistical
method with the aid of a suitable linearization formula; the method
is treated briefly below.
The systems with lag time, which are described mathematically by
differential equations of finite differences, have a preferred position
among linear systems. Approximation methods are used for the most
part for the investigation of stability. Lag-time problems have
scarcely been treated during the report period. We might mention only
the work of Mjasnikav alyasnikog (1,IX), who investigates with graphic
methods a linear system of the third order (in addition to a nonlinear
system).
Occasionally) lay-time problems appear in other articles,
but without special emphasis.
b) Pulsed systems
Formally speaking, it would be justified to include the so-called
pulsed systems under the linear systems; a special treatment for such
systems, however, is to be recommended. These systems involve a dis-
continuous transmission and control. Their characteristic component
element) which transmits pulses in equidistant time
intervals. According to the design of the system, either the amplitude
or the width of the pulses is modified by the input variable. The
effect of this type of influence is that all the variables do not vary
continuously, but only at discrete points in time, and are thus defined
for discrete arguments. The proper facilities for a mathematical
F
20-
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treatment or sucn transmlssiuu byOto ms are, th 4-cAv14,
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equations, but rather equations of finite differences. Several years
ago, Cypkin (Tsypkini pointed out that the analysis of a pulsed control
system can be carried out in exactly the same manner as that of a
continuous system; the chief aid is the "discrete" Laplace transform.
An analog of the transmission function or the frequency response is
obtained, and a relationship of the same form as in the continuous
case is obtained between the transmission functions of the open and
of the closed pulsed control loop; a stability criterion can be
derived which is based on the course of the locus. One necessary
condition is that the so-called "linear" part of the control loop be
controlled, he., that the system be built of all the transmission
elements with the exception of the pulsing element. In the works of
Tsypkin which fall within the report period, this formalism is taken
for granted. He treats several individual problems. In his work
(6) he points out that his theory applies not only for square pulses
(for which it was devised), but also for pulses of a more general
form. In his notes (8) and (2,VI) he assumes that the pulse width
Is a nonlinear function of the input variable, and shows how the
corresponding nonlinear equations of finite differences can be looked
upon as recursion formulas and solved by a numerical method; the
method can be illustrated to a great extent graphically. Moreover,
the equations of finite differences are always nonlinear in the case
of modified pulse widths, even when the modification itself is not
linear. In the note (10) the transfer function of a pulsed relay
system is used to introduce the concept of the statistical error,
which can serve as a designation of the control action. A different
characterization is given in the note (11), where the discrete analog
of the control deviation and, in the case of the effect of probability
functions, of the mean square error is taken into account. This work
also considers questions of synthesis; the author is thinking primarily
of pulse corrections, i.e., the introduction of additional pulse trains
in order to improve the quality of the control.
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Close4 aillea (A) uue pulbeu LAJLI Grol systems'is the use of digital
computers, with which Tsypkin (9;11) is more closely concerned. Such
an instrument converts the input variable, which is represented by a
series of numbers, into an output variable, likewise represented by a
series of numbers. The relationship between the input variable and
the output variable, that is to say the connection between the two
series of numbers, is expressed by a linear (or nonlinear) differential
equation with constant coefficients; such a differential equation
is formed in the same way as the equations of finite differences
which occur in the case of pulsed control systems. For this reason,
any digital computer can be assigned a "transmission function,"
obviously a transmission function for a discontinuous transmission,
and then a control loop, into which a digital computer is incorporated,
can be treated, theoretically, the same as a pulsed system, thus with
the discrete Laplace transform and the formulas set up with it.
Tsypkin (9) gives a report on the various possibilities of using
digital devices in control systems and the circuitsfor the individual
applications (element providing set-point adjustment, comparator or
final control element). He shows how the quality of control can be
improved with the aid of a digital device -- the instrument then works
as a pulse corrector -- and how the influence of a lag time can be
eliminated. Such a device is even useful when probability effects
occur (Tsypkin (11)).
Relay-Control Systems
The simplest nonlinear control systems are the relay control
systems, also called two-position action controllers or three-position
action controllers. Along with the linear part, they contain the
relay, the coordinates of which can assume only two or three values.
For this reason, the control process is discontinuous. A transmission
system with relays can be described mathematically by means of
differential equations with sectionwise constant coefficients. In
most of the practically important cases the differential equations
22
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themselves are linear. The nonlinearity of the process thus rests on
the fact that the description is given in various sectio.
%.?
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variable by means of equations with different constants. Other types
of nonlinearities also have the same property, for example, linear
systems in which an influence is exerted by dry friction or backlash.
Thus the same means can be employed to analyze such systems as are
used with the relay systems. In contrast to continuous controls, the
relay systems have, in many cases, the advantage of greater simplicity
and lower cost. They are especially suited for use where the processes
to be controlled are relatively insensitive to sudden changes of power
output. Thus the control of thermal processes represents a wide area
of application for the relay systems. Kampe-Nemm's monograph explains
the use of relay systems for temperature control. He investigates
(18 b) certain methods of improving temperature controls, for example,
the introduction of differentiating effects or the use of additional
relays which change the release times.
Several ways of analyzing a relay system are given. The most
obvious is the direct integration of the (linear) equations of motion
in the individual sections, within which the coefficients at all times
have the same value. The initial values of the unknown functions in
a section are, in such a case, equal to the output values from the
preceding section. This construction can be interpreted geometrically
in the phase plane or in the phase space; to be sure it gets out of
hand when there are a rather large number of degrees of freedom. The
method of point-transformation, the basic concept of which is supposed
to be explained by means of a system which can be discussed in the
phase plane, has been derived from the "integration in sections." Since
the process in a relay control system always has the character of an
undamped oscillation, the phase trajectories can, through the right
choice of coordinates, be made to move around the null point. For
this reason they will again and again strike a radial line through
the null point and thereby produce a topological mapping of the radial
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line, which will be unequivocal and continuc
i'411
50X1-HUM
systems. When this mapping is mastered, all the essential statements
regarding the course of the motion can be made. The same applies for
systems of a higher order; the mappings of planes or hyperplanes must
then be observed.
Alekseyev (1) investigates a temperature control with lag time
with the aid of such point transformations. The occurrence of the
lag time causes the motion to be described through differential
equations of finite differences. The exact analysis of the process
becomes very difficult; Questions of practical importances can be
answered, however, with one simple point transformation which must be
carried out with regard to the lag time in a multilayer phase plane.
Those points which overlap themselves in the mapping determine
periodic solutions of the equations of motion, and thus, simple or
complex natural oscillations of the physical system. In simple cases,
the formulas which define them can be set up explicitly; in complicated
cases, at least the dependence of the natural oscillations on the
parameter values can be investigated. Petrov and Rytkovskiy (43) treat
a control system with lag time in an analogous way, i.e., in a multi-
layer phase plane. They discuss the periodic solutions and show that,
under certain conditions, a so-called temporary "slippage state" can
occur, during which the relay does not release, but moves back and
forth between the null point and the upper or lower maximum deflection.
A study of such slippage states is of great importance for many
practical problems. Dolgolenko (1, VII) has given conditions for its
occurrence and estimates of its maximum duration.
Neymark (38) presents a very detailed theory of the relay systems.
u4^ chief aid is likewise the point transformation; he uses the
analytical presentation of the variables of the system with the aid
of the transfer function of the linear part. He studies primarily the
various types of natural oscillations and their stability. In order
to discuss stabilit
he forms the equations of the first approximation
a cr,
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and traces the problem back to a purely algebraic quest udwc.i.ty uv
the location of the zero positions of a certain polynomial. A shorter
work of Neymark (39) is devoted to relay systems with lag time.
Another method of treating relay systems has been developed by
Tsypkin. It makes use of the "characteristic of the relay system,"
which can easily be constructed, if the locus of the linear part and
the characteristic curve of the relay element are known. With the
aid of the relay characteristic, graphical procedures can be used to
obtain information on the existence of periodic solutions and their
properties, for example their stability behavior, and a study can be
made of forced oscillations which can occur in relay systems as a
result of nonlinearity. Tsypkin (5) gives a brief survey of his method,
which he explained in detail in the monograph. Korolev (24) is making
use of the characteristic to study a relay system with a retarded follow-
up. He will have to elaborate the theory of Tsypkin somewhat.
A special form of relay-control systems are the so-called vibration
controllers. They are based on the following phenomenon: If natural
oscillations of very high frequency and small amplitude occur in a
relay system, the system behaves toward a slowly varying input variable
In a manner very similar to that of a linear system. We therefore
find the expression "linearization of vibration," Pospelov (1, VII)
portrays a series of technical realizations of this principle and
various applications. Bernshteyn (1, VII) emphasized the use of the
principle in the control of electrical machines; especially in the
voltage regulation of small generators.
Finally, let us mention the works of V. V. Petrov (1, VII) and
Fufayev (117). They are concerned with transmission systems or control
loops containing two transmission elements with relay-like characteristic,
which considerably complicates the picture. Nevertheless, the point-
transformation method or the method of trajectories in the phase space
can be used successfully to obtain general information on the possible
Sion,
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s
I:
periodic solut ons an L.e r s a
all (46) employs the point-
transformation method in the treatment of a control loop without
relays, but with dry friction.
Other Nonlinear Systems
Nonlinear transmission systems are those in which the relationship
between input variable and output variable is expressed by means of
a nonlinear function or a nonlinear differential equation. No general,
universally applicable methods, such as can be given for linear systems,
can be given here, however. Nonlinear problems require special methods
from case to case; thus the survey of works on nonlinear control
engineering presents a picture which is anything but uniform. Some
of the work can be grouped, but it must be considered within the
wider scope of nonlinear mechanics.
As mentioned already, some nonlinear systems, such as systems with
dry friction, can be described, just like the relay systems, by
differential equations with coefficients which are constant within
steps only. The motions within them have the character of damped
vibrations, and the position of rest is one possible position of
equilibrium. Tal' (46) investigates the influence of dry friction on
the operation of a centrifugal tachometer and gives conditions for
the stability of the position of rest, which naturally depend on the
constants of the friction and the friction of rest. He uses the
point-transformation method.
Maslennikov (32) likewise treats a system with sectionwise constant
coefficients; he poses the problem in an entirely different way,
kx 0(x),
however. The equations of motion have the form lc HL ay
where
A(/ \
tik)() "
1 evle, ? 1
-
The problem is to select the function
0(x), i.e., the law according to which the control takes place, in
such a way that the process occurs in a proscribed manner. It is thus
a matter of the influence of a nonlinearity on the control action.
The author gives formulas for the solution, but does
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practical applications. A work by Ostrovskiy (41) belongs in the same
category. It de Aatich is described by an
equation in the form.Oraa I Cenr17T- 26 -
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The nonlinear function F is to be chosen (under certain secondary
conditions) in such a way that the control action is optimal. It is
?
shown that F must be equal to kx; the constant k has two different
values.
If the characteristic curves of the nonlinear transmission elements
can not be composed at the outset from linear segments: it is often
possible to use such a representation at least as an approximation.
Kagan (21 II) studies and compares certain methods of replacing a non-
linear characteristic curve with straight-line sections and shows
that the transfer process can be constructed with the methods applicable
In the linear Case, for example, with the frequency method (Voronov
(2,II), see above). A method developed by Bashkirov (21II) is based
on the same idea, thus on the replacement of the curve by a polygon
series) but is to be carried out essentially in a graphical way. This
permits the approximate construction of the transfer process in a
system which is described by one or several differential equations of
the type
? ? ?
x ax 4- bx = F(x) x, t),
and thus takes interference functions into account. The method is
simple and can be applied with quite good accuracy to current conditions,
Naumov (37) suggests another approximation-construction for the
transfer process. it is based on the rewriting of the nonlinear
differential equation into an integral equation of the Volterra type
and the replacing of the integral by a suitable approximate expression.
The integral equation then goes over into a finite system of algebraic
equations and thus becomes accessible to treatment. Tsypkin (8) touches
on similar recursively solvable nonlinear equations in posing certain
problems in the theory of pulsed systems.)
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The treatment of the transfer processes is, in most cases, quite
laborious, even when limited to approxin
therefore, this treatment has been circumvented, and mere general
statements on the connections between the time behavior of the system
and its parameters, especially the nonlinearities, have been considered
sufficient. The tendency has been to linearize immediately the stability
behavior, in order to study the influence which the nonlinearity
exerts on the stability behavior. Krinekiy (1,V) replaces the non-
linear right side F(x) of an equation having a linear left side by a
linear function hx with suitable h, and investigates the resultant
linear system for control quality, using the degree of stability (see
above). Krug (1,V) works with the "description function," i.e., he
uses the "harmonic linearization" or the "harmonic balance." He then
investigates the linearized system with the D-separation and obtains
information on natural oscillations and their stability. Kislav (20)
shows how the linearized system can be discussed with the aid of
certain nomograms; he is interested primarily in the stability ranges
of the parameters (see above).
Tsypkin (6) gives a brief account on the calculation of the des-
cription function. Kbrolev (24) and Kagan (2,II) give examples to
show that the method of harmonic linearization by no means always
leads to useful results and thus must be used with a certain caution.
Levitan (31) attempts to attain linearization by first Emplacing the
nonlinear transmission term with a lag-time term and then using a
differential equation of a higher order in place of the differential
equation of finite difarences. The method might perhaps lead to
useful results only in special cases. Myasnikav (1) Ix) considers a
nonlinear system with lag time and linearizes it with the aid of the
description function.
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The second method of Lyapunov, used so frequently in former years,
has been applied only in very few cases during the report period. The
most important contribution, the monograph of Letov (28), actually
does not belong in the report period, since it represents a compilation
of earlier works of the author. Letov's presentation is centered
around several modifications of a problem, stemming probably from
Lur'ye, which can be expressed in general form as follows: Let a
control loop be described by the equations
Xi
aik xk biy, y f(s), s =
p ry;
o<
f(s) is the characteristic of the servomotor and s the command signal
for the motor; r designates the feedback. The parameter p 0( of the
controller is supposed to be fixed in such a way that the system is
stable with any input variables and especially with any form of the
nonlinear function f(s)0 insofar as the latter satisfies only the
(obvious) conditions f(0) : 0, sf(s) > 0 for s
0. In the
solution of' this problem the equation system is first converted into
a "cononical ford' by means of a linear transformation (for which there
are several possibilities); then, in addition to the cononical equations,
an appropriate Lyapunov function is constructed in various ways,
according to whether the linear portion of the control loop is itself
stable or not. The analysis of the conditions which the Lyapunov
function has to satisfy leads to algebraic equations for the parameters.
For general explanations, Letov uses a control loop, the linear portion
of which has the form
x 4- aix ax a3y = 0,
without going into its technical significance. (The corresponding
equation system has also been used as an example by other Soviet
authc
interpreted as an automatic pilot
for aircraft.) Numerical examples ge_pot glv
Ft fanar
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A slight elaboration of Lurgye's problem is given in a brief
account (29) by Letov, in which the command signal s for the motor has
an additional feedback term
s pix, 4- pnxt - ry NY
A study is made of the influence of the variable N; the method used
is the same. In the work (30) (and in the last chapter of 28) Letov
concerns himself for the first time with nonstationary control loops.
He considers the coefficients alk in the above equations as functions
of time. As far as the rest is concerned, the treatment of the problem
is the same. Formally, the method is the same; naturally, the Lyapunov
theory must be drawn upon for nonautonomous systems. The analysis of
the results becomes correspondingly more difficult. It is not yet
possible to tell whether this elaboration of the presentation of the
problem will be of importance in practice, especially since control
loops with tit time-dependent parameters have been treated only very
little up until now.
In addition to the problem of Lurlye, the Lyapunov method of
presenting problems of control engineering is also used in the case
of the Aytserman problem, which has been the basis of several articles
during recent years. Generally, it is a case of a system with several
nonlinearities which is described in the form
a X. b fifN-
ik K x
l" ." ' xn)
For the functions f linear computations
= 1
C x f (x
ik k -
? 0 0
,x)
k : 1
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likas
liu
' ?
?
r 7 ,
should be given in sucn a way -un t the system is stable in the case of
any input variables) if only the nonlinear functions satisfy these
computations. In general, the problem can
second method of Lyapunov, but leads to somewhat laborious computations.
RazumicOin ffiatsmikhig (4)4) investigates a special case, where the
functions fi agree for all i and depend on a linear connection of the
xk, and, through several deft manipulations, arrives to the point
where the solution can be obtained by means of a quadratic equation.
Certain attempts to use the second method for an investigation of
the quality of the control action (Letov (28)) Plishkin, Avtomatika
Telemekhanika, 16, 19-26 (1955)) will only be mentioned here, since
they have been reported previously.
If only the symmetrical natural oscillations of a relay system)
i.e., the symmetrical solutions of the corresponding differential
equations, are to be computed, a method developed earlier by Luriye,
which leads to transcendental equations for the length of the periods
and the initial values of the periodic solution can be used; the
stability problem can be treated by going over to a "contiguous"
solution and discussing the resultant system of equations of the first
approximation. Trojckij groykie (43) extends this method to relay
systems with two servomotors. He is consistent in using matrix algebra
to make the quite cumbersome mathematical treatment reasonably clear.
Formally, the results correspond to those already obtained by Luriye
and others in the case of the servomotor. Whether or not the practical
utilization of these results can be recommended in regard to a saving
of time seems questionable.
pizAknv treats a nonlinear problem of a special form (19). He
investigates a nonlinear transmission system
Y(t) f(X(t))
under the assumption that the input variable X(t) is a probability
function. In order to arrive at a linearization, he considers the
31
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variable X to be divided into two terms of a sum, X : mx -1?X.0, the
first of which represents the mean value (the mathematical expectation)
of X, whereas the second is a probability function with the mean value
0. The equation for the approximate linearization is
Y(t) f(X(t)) kornx kiXo
The author gives various ways of calculating the constants ko, kl,
which, naturally, assume that the nonlinear function is known. This
"statistical linearization" has all sorts of advantages and could
facilitate the treatment of nonlinear transmission systems within the
framework of "statistical" control theory.
The Analog Method
For SOMB time the USSR has been interested in the investigation
of control engineering problems with the aid of analog computers,
preferably electronic computers. The amount of literature on analog
computers has increased in proportion with this interest. It is well
known that electrical circuits, the time behavior of which is described
directly by the given differential equation, can be formed with satisfactory
accuracy for any linear differential equation with constant coefficients.
This is possible, up to a certain point, even for differential equations
with variable coefficients and for nonlinear equations. The most
important component for such an analog system is the amplifier, a
diode with an impedance. Depending on the switching and the type of
impedance, it can be used as amplifier, integrator, or dLfferentiator,
i.e., the input voltage, which can be considered the input variable,
is amplified) integrated) or differentiated. In the treatment of a
concrete problem, the differential equation must first be simulated
through the required components in a suitable circuit, and then the
individual amplification factors must be chosen so that the model
circuit reproduces the correct parameter values. In addition, of
course, tae accuracy of the reproduction, the errors, etc., must also
be known.
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The work of Kbgan (22) gives a detailed view of the basic principles
in the case of the solution of differential ecitAtionkr-Vith the aid of
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analog computers and the mathematical and instrument skills necessary
for such a method. Talancev tfalantseil (47) compares various possi-
bilities for the switching of diodes and impedances and suggests
modern circuits which, according to his data, afford considerable
advantages. They have been used in the EMU-5 which was set up in 1955
at the Institute of Automatics and Telemechanics in Moscow. Gurov and
his associates (17) give a report on this installation. Kogan (3,II)
reports on earlier computers of the institute (EMU4 to EMU-4). These
instruments were built individually. G. M. Petrov (3,II) gives a
survey of the analog computers which have been built in the USSR and
on the experiences obtained with them.
The treatment of nonlinear differential equations with analog
computers is of special interest, particularly for the purposes of
control engineering. Kogan (21;23) gives a series of circuits for the
realization of nonlinear relationships. He differentiates between the
"typical" nonlinearities and the rest. Systems with typical nonlinearities
are those which are represented by differential equations with section-
wise constant coefficients, thus relay systems, systems with dry friction
or backlash, etc. They can be simulated rather easily, for example
through diode pairs in push-pull. Either approximations through polgyon
series or special functional transformations, as described by Vitenberg
(3,III), must be used in the case of nonlinearities with constant
characteristics. Even components which simulate lag times are used
(Gurov (3,III)).
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