(SANITIZED)UNCLASSIFIED SOVIET PAPERS ON AUTOMATIC CONTROL(SANITIZED)

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80T00246A022700330001-3
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 170
• Document Creation Date: January 4, 2017
• Document Release Date: December 13, 2012
• Sequence Number: 1
• Publication Date: August 5, 1963
• Content Type: MISC

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031/1
Two-positional Functional Frequency Device
for Automatic Regulation
I. A. MASLAROFF _ c) / 641e//
The complicated character of the technological processes has
developed in parallel with other research methods of ascertaining
ways of improving the qualities of the two-positional method
for regulation. The simplicity of the device and the low price
of the required elements have not detracted from its significance.
From all published literature on this subject the extensive work
of Campe Nemm1 is particularly noted. The author analyses
the existing methods of reducing the fluctuations of the unit to
be regulated: increasing the extent of current: the use of cut-off
two-positional regulation: and the introduction of inverse
connections on the first and second derivative, etc.
This paper gives some results of the methods undertaken to
improve the two-positional regulation by changing the frequency
of the influenced impulses. The methods are mainly directed
towards decreasing the fluctuations of the unit to be regulated.
The Essence of Two-positional Functional Frequency Regulation
The present survey refers to the monotonous varying
processes of a unit with a comparatively small changing rate
of regulation and the form of the equation to be used:
dA
Cdt =>Q
(1)
By using the method of full mathematical induction, we
determine that the value of the unit to be regulated after n
consecutive cycles (impulses and pauses) will be equal to:
A2n=AY(1-
n - E [tk+(n-a)tJ/T
-t;/T) Y e k=a
and after n + I serial impulses :
n
n - 1tk+[n-(a-1)ttll1T1
(3)
(4)
Eqns (3) and (4) show that by changing the duration of
pauses one -can effectively influence the unit to be regulated.
.In order to obtain the regulation we need the functional relation
t = 99(4A), at which the time of the pause will increase with the
decrease of the magnitude of the difference 4A. Such a depend-
ence may be realized simply by introducing the exponential
block in the scheme of the regulator (Figure 2).
The equation, characterizing the work of this scheme is:
kAA(1-e-t!T')=B
The time constant of the exponential block of the scheme must
be much smaller than the time constant of the object.
Then at JA = const. the time of the pause is equal to:
The principle of two-positional functional frequency regula-
tion consists in the addition to the object of previously fixed
identical portions of the utilized unit in the form of impulses.
The frequency of these impulses depends on the difference JA
between the given and actual value of the unit to be regulated.
Initially the influence of the net delay in the system is neglected
in the survey.
Figure 1 shows the change of the unit to be regulated.
During the time of impulses it is determined by: A = Az,
(1 - e-t/T)) and during the pauses, by: ' A = Ak e-t/T
(t = 0, A = Ak). These two expressions are the integrals of (1)
in the presence and absence of current. In such cases, at the end
of the impulses and pauses, the unit to be regulated will be
determined by:
A1=A,(1-e-t'/T)
A2=A1 e` IT =A,(1-e-to/T)e-t,IT
A3=A,(l -e-t`)+A2e-t`/T=Ay(1.-e-t,IT)e-tt+t,/T
t=Tlln k4A
kdA-B
(5)
Eqn (5) shows large values of the difference when the
percentage change in the pause time is insignificant. At an
established regime when there are small values of the difference
between the given and actual values of the unit to be regulated,
the time of the pause is determined only by the parameters of
the object (T > T1) where the delay due to the regulator is
slightly neglected in comparison with the common time of the
pause. In such a case the time of the pause is determined taking
into consideration that the consecutive fluctuations of the unit
to be regulated at a determined regime are also equal:
(2)
8A'= 8A" (6)
A'=A2+1 -A2n+2; 8A =A2n+3-A2n+2
A2n+3=AY(1.-e-t'lT)+A2n+2e-t;/T
A2n+2=A2n+1 e-t?+ ,/T
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031/2
to+1=Tin A2n+t -t/T
A2n+t-Ay(1-e ' )
(7)
By exerting an influence on the coefficient of amplification
and the internal limit of putting in motion B of the scheme it is
always possible to receive an equalization fo the maximal and
given values for the unit to be regulated. Then eqn (7) is modified
as:
-T 1_ Ag
Ag-Ay(1-e-t,IT)
(7 a)
From eqn (8) two fundamental parameters for the regulation
may be determined-the internal limit for setting in motion B
and the coefficient of the earlier amplification k. These para-
meters may be easily changed into parameters to be regulated
in large limits, depending on the requirements of the object
to be regulated.
Constructive Data of the Device for Functional Frequency
Regulation
The device uses a vacuum-tube scheme (Figure 4) consisting
of a measuring part 1, amplifier 2 and an integral group 3,
two channels for constant current amplifiers 4 and 4' and an
executive trigger 5. It differs from Figure 2 by the use of
a second channel for the constant current amplifier 4', which
is included in a circulating chain of the integrating group and
the base constant current amplifier 4. Its purpose is to accelerate
the process for establishing the regime. When there are many
large values of 4A the output voltage of 4' passes through the
logical scheme 'IF'-6 and sets in motion the executive trigger.
In this way the scheme works as an ordinary two-positional
regulator. Placed in a regime, close to the one established, the
output voltage of the second channel is not in position to set
in motion the executive trigger, and the device works like a
functional frequency regulator.
In parallel with the passing of each impulse from the trigger
exit 5 to the object 7 the signal for clearing the integrating chain
is simultaneously passed through an internal link.
The maximum value of the fluctuations of the unit to be
regulated is given by:
BA=AA= B =Ag-A2n+2=(Ay-Ag)(1-e-t?lT)e_tdT (8)
Eqn (8) shows that by decreasing the duration of the impulse
ti. the fluctuations of the to unit be regulated may be most effec-
tively reduced. The coefficient of amplification k may be deter-
mined at a previously chosen value B of the limit out of the
duration of the impulse.
Influence of the Net Delay on the Two-positional Functional
Frequency Method for Regulation
Usually, the effect of the delay which increases fluctuations
of the unit to be regulated is shown in the systems of the type
examined. In the following it is proved that the influence of the
net delay upon the value of fluctuations may be substantially
decreased using the functional frequency method for regulation.
Actually Figure 3 shows that the additional increase of fluctua-
tions 6AAt which follows from the delay of the system, is equal to:
8AAt=A2n+2 (1-e-AC/T)=Ag(1-a-At/T) (9)
With the usual two-positional regulation, the delay increases
the fluctuations of the unit to be regulated in the direction of its
decrease, as well as in the direction of its increase. These addi-
tional increases are of the same order.
It follows that with functional two-positional regulation the
fluctuation of the unit to be regulated increases in the direction
of its decrease and because of this the received additional
fluctuation is about twice lower.
The total value of fluctuations is:
8A?=8A+8AAt=(Ay-Ag)(1 -e-t-,lT)e-r,IT+Ag(l -e- ;t/T)
(10)
If it is accepted that 6A = 6AA,, then:
/T
AY (I y =1+( ) e-ti
A 9 (1 -e-At/T)
From eqn (11) some conclusions can be drawn for deter-
mining the parameters of the system to be regulated.
It is evident that at considerable values of the time of delay
At it is apt to accept z l,, > 4g, i.e. to use strong impulses.
However, at small values of At it is apt to accept Ag ~ A,,, i.e.
the impulses will be comparatively weaker.
Experimental Data
Initially the device was constructed and tested for regulating
the concentration of solutions. Conductive transformers linked
by a bridge scheme with temperature compensation were used
as a measuring device*.
The excutive trigger exerts influence on an electromagnetic
valve which adds a drop of concentrate to the solution at each
impulse. The results obtained at the time of regulation were
very good.
The device is used to regulate temperature, and for this
purpose the excutive trigger is replaced by a delay multivibrator.
The time of the impulse may be regulated at will by changing
the parameters of its device. Figure 5 shows the diagrams of
'temperature change of one and the same object, recorded with
the help of an electronic potentiometer. It is seen that the quality
of regulation with the functional frequency method is much
better than that of the ordinary two-positional method.
1. The two-positional functional frequency device for
regulation allows the possibility of decreasing the fluctuations
of the unit to be regulated, particularly those emerged out of
the delay in the system.
2. By the character of its work, the device approaches the
statistical regulators.
3. The devices for regulation can be realized by using
practical simple means.
4. The test results prove the expedience of using this method
for regulation in many cases.
* Eng. D. Detcheva took part in the computing of the construction
of the device.
031/2
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031/3
dt Time of the net delay
n Number of the impulses
C
Coefficient of the generalized capacity of the object to be regulated
T Time constant of the object to be regulated
A
The unit to be regulated
T, Time constant of the exponential block of the scheme
Ay
Fixed value of the unit to be regulated
B Internal limit for setting in motion the acting block of the s
cheme
Ag
Given value of the unit to be regulated
k Coefficient of amplification
AA Difference between the giver' and actual value of the unit to be
regulated
Q Generalized quantitative index of the process
(SA Variation of the unit to be regulated in the period of one impulse
or pause
t Time
i Time of the impulse
References
i CAMPE NEMM, A. A. Two-positional automatic regulation and
methods of improving its characteristics. Thermoenergical and
Chemicotechnological Devices and Regulators. 1961. Moscow-
Leningrad; Mashgiz
AA kC 3
kAA
9
A2n+1 A2n+3
MIT
KAA
2d A
-t
knA(1-eT,)
To the
object
Figure 3
Curve 1-Change of the regulated unit in close proximity to the source
of the impulses
Curve 2-Change of the regulated unit in the field of the sensitive element
Figure 5
(a) Change of temperature by using a contact thermometer for regulation
(b) Change of temperature by using functional frequency regulation
of the object
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A2n.2 ``. i up C
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502/1
Dynamic Planning for an Open-hearth Steel-making Plant
M. KOROBKO and Yu. SAMOILENKO
Introduction
The output of an open-hearth steel-making plant depends, in the
first place, on the output of individual furnaces. However, the
practice in works recently showed that the duration of the
melting operation could be reduced considerably as a result
of the use of oxygen. Under these conditions the output of the
plant is limited only by auxiliary equipment.
The operation of different furnaces of the plant is inter-
connected since they use the same machines and interchangeable
equipment, and also because they are served by the same
subsidiary works' departments and means of transport. All that
is needed in connection with the running of furnaces is therefore
conditionally described as `auxiliary'.
All auxiliaries are usually designed for a capacity margin
of 15-30 per cent, which when trying to force melting rate is
very often shown to be inadequate. In conventional programming
the interconnected operation of furnaces causes frequent
organizational delays in melting, which results in peak require-
ments exceeding the capacity of auxiliaries.
The capacity of some auxiliaries can be increased relatively
easily, but for the majority of them, for the increase required,
it would necessitate a complex reconstruction of the plant
involving the spending of large sums of money.
Since the.operation of furnaces depends on a large number
of factors, which change according to random laws, the
optimum operation of the whole plant can be obtained only by
continuous operational progamming, which may be called
`dynamic' programming.
The technical problem of dynamic programming for the ope-
ration of an open-hearth steel-making plant can be solved by
constructing a system with a computer capable, on the basis of
automatic processing of information concerning the progress of
melting and available capacities, of evaluating the volume of
work which could be done and, accordingly, of giving commands
.to furnacemen and automatic equipment, responsible for the
control of the melting operation, so that the maximum use could
be made of auxiliaries.
Theoretically, this problem may be considered as one of the
tasks of dynamic programming, the fundamentals of which are
explained in the works of L. S. Pontryagin, R. Bellman and
others. As is shown below, the presence of nodes in the optimum
phase trajectory is the main feature of the given problem. The
economic index, which represents the difference between the
value of increased production and the expenditure on automatic
control for a sufficiently long interval of time, is chosen as the
economic criterion for the quality of control. The term 'expendit-
ure on control' denotes the variable portion of operational
costs for the automated part of production, which is conditioned
by the necessary change of technology embodied in the process
of automatic control.
On the basis of analysis of the operation of works' furnaces
under actual operating conditions, logical differential equations
were constructed for the progress of the melting operation. The
choice of the optimum direction at the nodes is obtained by the
subsequent comparison of different variants of the automatic
control for the process. An assessment is-made of the increase
in output, when the number of comparable variants for
homogeneous node processes is increased.
Organizational Conditions for the Operation of Furnaces
The melting of steel in open-hearth furnaces is essentially
a cyclie process, which consists of successive technological
periods during which certain auxiliaries are engaged. On a
modern, open-hearth steel-making 'plant there are up to 12
furnaces. A typical general layout of equipment at a plant is
shown in Figure 1. Similar mechanisms move along past a number
of furnaces on the same rail track; thus, their relative disposition
is shown to be subordinated to ground connection. Mechanisms
used for different purposes are not subject to the interchanges
of position. The essential auxiliaries for the programming
of the operation of furnaces are the charging machines, casting
cranes, ladle cars, teeming cranes and casting bays.
The manceuvrability of machines along furnace runways
is unlimited, so that all the working machines and the inter-
changeable equipment may be used. In the teeming bay the
mobility of crane equipment is limited; therefore, the operations
in it are not always determined by the total number of the
mechanisms available. The effect of the possible idleness of
some teeming cranes caused by those currently in use should
be taken into account.
Every open-hearth plant has its own peculiarities; therefore,
the mathematical description should be made for the specific
plant. Thus, for example, on some plants the bunkers for the
charging of furnaces are not installed on the, furnace floor,
and during the entire period use is made of the casting
crane, which increases the engagement of the latter; for the
charging of the furnace on other plants two machines are used
for the same furnace, and so on.
So far as the control of the melting operation is concerned
its periods consist of controlled and uncontrolled periods; but
so far as the possibility of freeing the auxiliaries on one furnace
so that they could be transferred to another furnace is concerned,
the periods of the melting operation consist of intermittent and
continuous periods. In the first approximation it is considered
here that the durations of the latter periods of the melting
operation are independent of those of the former periods, since
in further considerations this condition is not of material
importance.
. In future, the existence of some relationship between the
durations of individual periods of melting may help to improve
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502 / 2
the programming. It is assumed also that in the course of each
period of the melting operation a certain amount of subsidiary
work is carried out.
The conditional graph for the melting operation, which
consists of the time periods used for the carrying out of such
work, with the furnace being served by all the auxiliaries, may
be termed the `condensed' graph.
By virtue of the effect of a large number of random factors,
the durations of periods of the condensed graph, strictly speak-
ing, represent random quantities and cannot be calculated
beforehand with the necessary degree of accuracy. Therefore,
the values for the assumed durations of the periods of the
condensed graph should be systematically corrected.
The volume of work (pi, actually carried out on the ith
furnace from the beginning of taking the readings to the instant
of time t, represents the basic coordinate, which determines the
progress of the melting operation in the furnace. The state of
a plant which consist of `n' furnaces is described by the values
of `n' coordinates cpl, 92, ..., cp,z, which may be considered as
the components of vector (p of the state of the melting operations
on the plant.
As an example, the engagement of the most essential
auxiliaries at different periods of time is shown diagrammatically
on the condensed graph for the melting operation (Figure 2).
The functions hi show how many units of i auxiliary are required
at different periods of the melting operation. These functions,
in the simplest case, take the values of only 0 or 1. For their
assignment it is necessary only to indicate the durations of
periods of engagement or the instants of time of their termina-
tion (with the known beginning of readings). In the first approx-
imation it is considered here that the durations of periods of
the condensed graph are assigned on the basis of mean statistical
data. The more precise data regarding the durations of the
periods may be obtained by using the results of the preceding
melting operations and by taking into account the current
information regarding the state of-the furnace and the quality
of the materials being used.
The technical and economic criterion of programming, which
makes possible the comparisons of different variants of the
progress of the melting operations in furnaces, should serve as
the basis for the choice of the optimum graph for the progress
of the melting operations.
This criterion which takes into account the expenditure on
automatic control is determined by the scalar product:
Q=[C{Jo cpdt - JQ f ((p)(w-cp)dt}] (1)
here c = c (cl, ..., cn), Ci is the output of the ith furnace;
f ((p) is the piecewise constant coefficient which depends on the
period of the melting operation and on individual characteristics
of furnaces; w = w (w1, ..., wn), wi are the rates of melting
operations according to the condensed graph which take into
account the provision for the general forcing of the progress of
melting for all the furnaces; and cp = cp (cpl, ..., cp? ), cpi are the
actual rates ? of melting operations which take into account the
delays caused by the separation of furnaces.
For the ease of planning of the progress of melting operations
it is possible to consider that the processes in furnaces are
intermittent in character, i.e., their rates may assume the values
of only 0 or 1. This procedure, which follows from the analysis
of phase trajectories, may be applied only in the programming of
the durations of periods and for the calculation of the mean
rates. But the actual process should proceed as uniformly as
possible in accordance with the mean rates found in programm-
ing. Otherwise, large fluctuations in the rate of the process would
lead to a strong increase in the cost of automatic control, which
would then no longer be taken into account by formula (1),
which is applicable only within a small range of rates. By
the interruption of operations, in future, if not specially stipu-
lated, only the corresponding increase in the duration required
for their completion will be understood.
In order to indicate the inadmissibility of delays during
continuous operations, multiply the right side of expression (1)
by the function
T n I
iE W`-Fi(cPi)dt (2)
o
=If where symbol J denotes the step function of the following form:
J(x)__''; $(O)=1 (3)
Fi ((pi) is the function which is equal to 1 for the continuous
operations and to 0 for intermittent operations.
Then, if any cpi during the continuous operation becomes
less than wi the integral will have a negative value and the entire
expression (2) will transform to zero. The criterion also trans-
forms to zero, thus indicating the inadmissibility of delay
during the progress of a continuous operation.
Finally, the technical and economic criterion has the form:
Q [J' dt- J of ((p)(w-0P)dt,CJJ { [wi--Fi((pi)]dt}
(4)
The Problem of Dynamic Programming
For automatic control it is essential to have information
regarding the volume of work (pi for each of the furnaces, and
also information regarding the expected functions hi for the
engagement of the auxiliaries. The finding of the latter can be
made by the prediction of the condensed graph on the basis of
the a posteriori distribution of the periods of duration. In this
way the last experience of the operation of the furnace is taken
into account. This problem may be solved by the known methods
of extrapolation of random sequences. Since the subsequent
programming does not change the condensed graph, then
its extrapolation may be considered as a problem independent
of the programming problem.
In the first approximation it is possible to be guided by
mathematical expectations for the duration periods of the
melting operation. In the second approximation it is necessary
to take into account the effect of the elapsed duration periods
on the subsequent duration periods. For this it is necessary to
have the information regarding the instants of time for the
beginning and the end of expired periods, which will be received
from the corresponding monitors. On the basis of this same
information, by the method of extrapolation, the current
values of (pi will also be calculated, as the initial conditions for
programming. In the first stage it is assumed that there is a
limitation to the calculation only of rate (pi, averaged out
according to the current periods.
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The continuous determination of the optimum controlling
actions in the process of automatic control is the essence of
dynamic programming. In the given case this is the determina-
tion of the maximum loading for the finding of the optimum
distribution of auxiliaries, in the process of control of the
work on an open-hearth furnace plant.
The qualitative analysis of the working of the furnace and
casting bays makes possible the construction of a system of
differential equations for the operation of furnaces. The right
side of each one of.the equations represents a function, which
depends on the volume of work completed at a given instant of
time and on the availability of the auxiliaries for the furnace.
Each one of these functions is constructed in such a way that it
transforms to zero, if on a given furnace there is a lack of
auxiliaries, and transforms to w1, if the furnace at any given
distribution is served normally by auxiliaries.
Eqn (5)
Zi,k~! O,Zi,1+Zi 2+ ... +Zi nTo 1 (73)
(S2,>Q-1, i=1,...,N)
Every actual filter W (jw) = W1(jw) - W2 (jw) has a finite`
cut-off frequency w, (it is further considered that co* < w~,),
so that in accordance with (19), (32), (34) and (71)-(73) the
equations for the process of self-adjustment of the qth para-
meter may be put into the form
LdWL ~ (t
- 2 dvT1 J Gq(c),v,?r)di (74)
t T
Gq (w, v, T) = D ((o, v) [cos {(w - v) ,r + S. - 9v}
+ cos {(w + v) T + 9w + 9,,}] cos S2gT
where D (w, v), 0w and iv are defined by eqns (35), (36), (28)
and (29). The quantity Gq (co, v, t) is a sum of harmonic com-
ponents with frequencies Q equal to
Xq=~_- T-To 1?kq?7r-1
Gg(a))
N
+2E Di[g;(w,vq)+gi(w,v;q)+gq(w,vgq)]dw
i
where
(79)
Gq ((O)= 10T 00J) OT {i (w + 92q)} W(JW) W {J ((0+ S2q)}I
T -1 cos (9w - 9w+aq)
gi((0, v)= 2 T-1 IOT(j(O)OT(jV) W0(0) W001
(80)
[ Y? (co, v) cos (9w -
9v) - Vi, (w, v) sin (9w - 9A
(81)
1' (c),v)=-a, (w)IW
Vis(co,v)=bi(w)IW
(j(o)I-1+ai(v)IW(jv)I-
(jw)I-+bi(v)IW(jv)I-
(82)
Viq w+S
2i+S2q, viq =w+IS2i-QqI
(83)
[the quantities ai (co), bi (w) and 'bw being defined by eqns (28),
(29) and (36) and I the memory of filter W (jw)-see ? 3].
Considering the function (5) as a typical realization of a station-
ary random process {0(t)} and performing averaging according
to achievements, one can go from eqns (79)-(83) to equations in
the mean (as taken together) values Xq of the adjustable para-
meters. If here the interval T is taken large enough, then in the
right-hand sides of these equations one may replace the quantities
T_1 OT(jw) 0T(jv) I by characteristics like the mutual spectral
power densities12 of-the process {0 (t)} and certain random
processes obtained from {0 (t)} by simple transformations that
do not infringe the stationary condition.
76-1 cos(QT+9)di=cos9.8(Q) (77)
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This paper does not deal with the more detailed analysis of
the general case, but gives the results of the calculation for the
quasi-stationary mode of self-adjustment, i.e. the mode in which
w*>2QN (Oi>Qi-1, i=1,..., N)' (84)
with a test signal of white-noise type: f T-110T(jw)I2 lim T-110T(jw)I2= 0 for w < w* (85)
T-a ao GB for co > (t)*
Since eqns (30) and (31) are satisfied in quasi-stationary
modes, and furthermore t~., z~w+2n~ (co > w *), one may
neglect the terms Vas (w, v) sin (0 - 0) in (81), and so putting
OT(jw) - OT{j (w + 2QM)} and W(jw) = W{j(w + 2QM), the
following equations for the self-adjustment process are arrived at:
w~ If(D a ww
IW(jw)I2dw+?DeaxgJw~ IW(jw)I2dw
X ~
qko
+ z "~ i t ?i 6-Xi Jw. I W (jw)12 dw] (86)
i#q
k?=T? To t'k9=' 1 qc-Go
The following conclusions are evident from (86):
(1) In the mode of operation (84), (85) studied, minimization
of the quantity
J IW(jw)I2dw
w*
(87)
may be naturally considered the ideal result of the self-adjust-
ment process.
(2) The control signal for the qth self-adjusting network
contains derivatives of the quantity (87) being minimized, not
only w. r.,t. Xq but also w. r. t. all the other adjustable parameters
X,, so that one has not got a pure gradient system of extremal
control.,
(3) The equilibrium condition Xq = 0 (q = 1, ..., 'N) for the
system (86) is characterized for OZ = (i = 1, ..., N) by the
relations
a Cw (?D (N +1)I W (jw)I z do)= -I(o)I z do-) (88)
axi (i=1,...,N)
from which it can be seen that the more pronounced the extremal
nature of the dependence of quantity (87) on the parameters Xi,
and the less essentially attainable the minimum of this quantity,
the closer will this condition be to the ideal result of self-adjust-
ment.
(4) If quite large differences arise rapidly between the
frequency characteristics W1 (jw) and W2 (jw), the non-negative
term (87) on the right-hand side of eqn (86) will increase so
much that the operation of the self-adjusting network will be
reduced merely to increasing the parameter Xq (x q > 0), and
this may lead to the system's losing its required extremal
condition.
Finally it is noted that the equations given by Krasovskiy2
for quasi-stationary self-adjustment with a white-noise test
signal contain only terms analogous to the second term in the
right-hand side of equation (86).
The author expresses his gratitude to Ye. A. Barbashin and
I. N. Pechorina for their discussion of this paper.
KRASOVSKIY, A. A. Self-adjusting automatic control systems.
Automatic Control and Computer Engineering. 1961. No. 4.
Mashgiz
2 KRASOVSKw, A. A. The dynamics of continuous automatic control
systems with extremal self-adjustment of the correcting devices.
Automatic and Remote Control. 1960. London; Butterworths
KAZAKOV, I. YE. The dynamics of self-adjusting systems with
extremal continuous adjustment of the correcting networks in the
presence of random perturbations. Automat. Telemech. 21,
No. 11 (1960)
4 VARYGna, V. N. Some problems in the design of systems with
extremally self-adjusting correcting devices. Automat. Telemech.
22, No. 1 (1961)
a TAYLOR, W. K. An experimental control system with continuous
automatic optimization. Automatic and Remote Control. 1960.
London; Butterworths
6 MARGOLIS, M.., and LEONDES, K. T. On the theory of self-adjusting
control systems, the learning model method. Automatic and Remote
Control. 1960. London; Butterworths
ITSKHOKi, YA. S. Non-Linear Radio Engineering. 1955. Sovetskoye
Radio
8 KJIARKEVICH, A. A. Spectra and Analysis. 1953. Gostekhizdat
MALKIN, I. G. Some Problems in the Theory of Non-Linear
Oscillation. 1956. Gostekhizdat
10 Popov, YE. P. The Dynamics of Automatic Control Systems. 1954.
Gostekhizdat
CH'iEN HstiEH-SEN. Technical Cybernetics. 1956. Izd. Inostr. Lit.
12 CANING, G. H., and BETTIN, R. G. Random Processes in Automatic
Control Problems (Russian transi.). 1958. Izd. Inostr. Lit.
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Optimal Control of Systems with Distributed Parameters
A. G. BUTKOVSKIY
In many engineering applications the need arises for control of
systems with parameters that are distributed in space. A wide
class of industrial and non-industrial processes falls within this
category: production flow processes, heating of metal in metho-
dical or straight-through furnaces before rolling or during heat-
treatment, establishment of given temperature distributions in
`thick' ingots, growing of monocrystals, drying and calcining of
powdered materials, sintering, distillation, etc., right through to
the control of the weather.
The processes in such systems are normally described by
partial differential equations, integral equations, integro-
differential equations, etc.
The problem of obtaining the best operating conditions for
the installation (the highest productivity, minimum expenditure
of raw material and energy, etc.) under given additional con=
straints has required the development of an appropriate mathe-
matical apparatus capable of determining the optimal control
actions for the plant.
Pontryagin's maximum principle and Bellman's dynamic
programming method have been the most interesting results in
this direction for systems with lumped parameters.
A wide class of systems with distributed parameters is
described by a non-linear integral equation of the following
form :
Q(P)=fD [K[P,S,Q(S),U(S)]dS
H
ere the matrix
Q1(P)
Q(P)_ : = IIQ`(P)II
Q. (P)
(1)
(2)
describes the condition of the controlled system with distributed
parameters, while the matrix
U(P)=
U1(P)
11U`(P)II (3)
U' (P)
describes the control actions on the system. Here and in the
following, the index i will refer to a row number and j to a
column number in a matrix. The point P belongs to a certain
fixed m dimensional region D in Euclidean space.
The components of the single-column matrix
K 1(P, S, Q, U)
K (P, S, Q, U) =
K" (P, S, Q, U)
= II K` (P, S, Q, U) II (4)
belong to class L2 and have- continuous partial derivatives
w. r. t. the components of the matrix Q.
It will be assumed that the function U (P) is piecewise dis-
continuous, its values being chosen from a certain fixed permis-
sible set 0. Controls U (P) having this property will be called
permissible.
Further, from the set of conditions Q (P) and controls U (P),
related by integral eqn (1), let q functionals be determined,
having a continuous gradient (weak Gato differential).
I`=I` [Q(P)], i=0, 1, ..., l (5)
I`=I`[Q(P),U(P)]_1i (z), i=1+1,...,q (6)
? [S, Q (S), U (S)] dS
ID F
Z k fFk[S,Q(S),U(S)]dS
JF = [S, Q(S),U(S)]dS
D
(7)
The function (Di(z), i =1 q and Fi(S, Q, U), i = 0,1...,k,
are continuous and have continuous partial derivatives w. r. t.
the components of the matrices z and Q respectively. .
The optimal control problem is formulated in the following
manner.
It is required to find a permissible control U (P) such that
by virtue of equation (1)
I`=0, (8)
while the functional P assumes its smallest value. Here p is
a fixed index, 0 < p < q.
a~-
aF
ag
a0i
azj
aF
aQj.
j=0,1,...,k
2,...,n
(9)
(10)
grad I=11grad;I'll; i=1+1,...,q; j=1,2,...,n (11)
where grad 1P denotes the jth component of the vector grad Ii
w. r. t. the coordinate Qi.
The following theorems can be used as the basis of a solution
of the problem formulated above on the optimum control of a
'plant with distributed parameters.
Theorem. Let U = U (S) be a permissible control such that
by virtue of eqn (1) the conditions (8) are satisfied and the
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513/2
matrix function M (P, R) = IIM15 (P, R)II, 4j = 1, 2, ..., n,
satisfies the integral equation [linear in M (P, R)]
M (P, R) + aQ K [P, R, Q (R), U (R)]
J M (P, S)aQ K [S, R, Q (R), U (R)] dS
('D
= J a~K [P, S, Q (S), U (S)].M (S, R) dS (12)
D
1)
Then for this control, U (S), to be optimal there must exist
one-row numerical matrices
a=IIco,c1,...,clII and b=1Ici+i,...,X911 (13)
of which at least one is not null, and also cr, < 0, such that for
almost all fixed values of the argument S e D the function
n (S, U) = a' [grad I {Q (P)}, K {P, S, Q (S), U}
- J M (P, R) K {R, S, Q (S), U} dR]
D r ('
+baZ LJ F{P,Q(P),U(P)}dP]
LL D
CaQ F {P, Q (P), U (P)}, K {P, S, Q (S), U}
- J M (P, R) K {R, S, Q (S), U} dRJ
D ('
+ba (D 1JDF{P,Q(P), U(P)}dPl?F{S,Q(S), U}
(14)
of the variable U e .Q attains a maximum, i.e. for almost all
S e D the following relation holds:
7t (S, U)=H(S) (15)
H (S) = sup g (S, U) (16)
U E D
As an example of the application of this theorem, consider
the important practical problem of the heating of a massive
body in a furnace. Let the temperature distribution along the
x axis, 0 < x < L, at any instant t, 0 < t < T, be described by
the function Q = Q (x, t). Here the temperature U (t) of the
heating medium, which in this case is the controlling agent, is
a function constrained by the conditions
A1< U(t) 0, assuming that the magnit-
ude fo of the step cannot be measured. Reserving the freedom
to choose T, take the previous switching law given by eqn (4).
Clearly if K = Ko, eqn (3) after the step has passed will take the
form
E+aE+(b+KO)(e-82)=0 (6)
where e2 = bfo/(b + Ko).
Correspondingly, for K = - Ko one gets
E+as+(b-KO)(E-E1)=0 (7)
where el = bfo/(b - K0).
Assuming that Ko is large enough a qualitative plot in the
phase plane can be drawn for each of these equations without
difficulty. Equation (6) in the phase plane corresponds to a
family of spirals converging to a focus-type special point (e2, 0).
Equation (7) in the phase plane corresponds to a family of
integral curves of hyperbolic type, with a `saddle'-type special
point (el, 0) through which pass two integral straight lines whose
gradients are the roots of eqn (5).
Assuming now that the switching law is given by eqn (4),
the phase diagram shown in Figure 3 is obtained, provided only
that TA1 < - 1, where 21 is the negative root of eqn (5) is
assumed. If the latter inequality is not satisfied, an obviously
unsatisfactory result is arrived at, since the switching line (T)
given by the equation Ti + e = 0 will be cut by the integral
curves over the whole of its length with e < 0, while in our
case the straight line T is a sliding line everywhere except
over the segment EF, where E and F are the points of contact
with the integral curves corresponding to eqns (6) and (7). Thus
the switching line resulting from the relevant optimum criteria
will have been deliberately abandoned. If the representative
point M falls to the left of the line T, then 'it will slide along
this line as far as F, follow a curve of hyperbolic type as
far as the line e = 0, then a spiral as far as the right-hand part.
of line T, where it will again start to slide towards E. On arriving
at E it will approach the point (e2i 0) along a spiral if a > 0,
while if a < 0 it will start to move along a cycle consisting of the
segment GE of line T and the segment EHG of the spiral. Thus
any point in the plane arrives, eventually, either within a
sufficiently small region about the point (e2i 0) or at a limiting
cycle corresponding to some self-oscillatory mode. It should be
observed that the amplitude of the resulting self-oscillations is
of the same order as e2 = bfol(b + KO), and consequently can
be made as small as required by increasing Ko.
It should be noted that by increasing T the length of the
segment over which it cuts the integral curves is decreased, but
the speed of sliding along this line is also lessened since, as can
readily be seen, the sliding law is given by the relation e = eo exp
(- t/T). Thus proceeding from various quality criteria and
combining speculation with experiment a reasonable value for
the time constant T can be selected.
Toge ther with R. M. Yeydinov and I. N. Pechering the
author has been carrying out analogous investigations for a
third-order system. Here the main difficulty lies in the problem
of synthesizing a corresponding optima lsystem.
Connection with the Accumulated Disturbance Problem
Returning now to the problem formulated in the first
paragraph; as far as the approximation to the final section of
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515/3
the trajectory is concerned, our problem is directly related to
that of Bulgakov10 on the accumulation of disturbances in a
dynamic system.
Introducing the substitution z = x - ~ (t) into the system
of eqn (1), we transform it into the form
dt Z ,z, rl, i)+ r (c, y, O (t), t1(t), t) (8)
wh. re
Z(z,n, t) =J (z+'(?),n(t), t)-f(V/(t),r!(t), t)
r (c,) , / (t), r I (t), t) =.f (J (t), t1(; ,. t) - 0'(t) + u (c, y, t) = r (t)
System (8) is a system of equations for perturbed motion,
the function r (t) determines according to eqn (2) the approxima-
tion error of the programming or control functions, and the
deviation of the solution z (t) of system (8) from zero coincides
with the deviation of the solution x (t) of system (1) from the
given function t' (t).
If system (8) is linear, then for z (0) = 0, 0 < t < T < oo
we have z (t) = Ar (t), where A is a linear-bounded operator
transforming the function r (t) into the functions z (t). If II All is
the norm of the operator, then Ilz (t)II < hJA1l lIr (t)II is obtained.
The latter relation is also the most general expression of the
solution to the problem of disturbance accumulation. By taking
various norms for r (t) and z (t) and computing 11 All, the actual
inequalities that solve this problem", 12 are obtained.
Connection with the Theory of Approximations
If as an optimum criterion that of the minimum error r (t)
(in any dimension) is taken, then the problem of realizing the given
trajectory reduces to a problem in the theory of approximations.
This problem is most effectively solved in the case where r (t)
depends linearly on the programming parameters and functions,
and where we require a minimum of the mean-square approx-
imation error. In this case the elementary rules of the theory
of mean-square approximations are used for computing the
control. It should be observed that here two essentially different
cases are met. In the first case, by selecting the programming
parameters from a sufficiently large number of them the approx-
imation error can be made as small as required, i. e. realizing the
given motion as accurately as necessary. In the second case the
error of approximation cannot be made less than a certain value.
Here it is worth while to state the problem of simultaneously
choosing optimal values for the parameters and optimal pro-
gramming functions. The success of such a choice depends,
roughly speaking, on how well the given trajectory fits into
linear subspaces in the various dimensions11
Trajectory Realization and Stability Theory
If it is wished approximately to realize motion along a given
trajectory for the whole interval 0 < t < oo, certain difficulties
arise. It can readily be seen that such an approximate realization
is possible if the zero solution of the system
=Z (z, l, )
dz at
is stable in relation to continuously acting 'disturbances that
are limited relative to the dimension in which the approximation
error r (t) is evaluated. There exist13 stability criteria related to
continuously acting disturbances limited in modulus or in
mean value. Stability criteria can easily be deduced" for use
with continuously acting disturbances limited in their mean
square, which are of most interest in our problem. However, in
solving the problem it was required to find convenient evalu-
ations of continuously acting perturbations that were simulta-
neously evaluations of approximation errors.
Such evaluations14 were found and it turned out to be best
to make them in the dimension of space M with norm
ll r (1)112 = sup I r (t)I2 dt
where I r (t)l denotes the length of the vector r (t). Massera was
the first" to point out the important role of the space M in
stability theory.
Dwelling further on a question related to stability theory,
the operating mode ili (t) is called stable in relation to the
system z = X (x, t) if the zero solution of the system
z=X(z+~(t),t)-X(> (t), t)
is asymptotically stable. From the preceding argument it is clear
that only stable operating modes can claim to give a good
approximation. Unfortunately few criteria for operating mode
stability have so far been derived in relation to this system.
Clearly if the basic system is linear and asymptotically stable,
then any operating mode will be stable relative to it. The same
property is possessed by the systems considered by Krasov' kiy
in his paper" (theorem 3.1). These systems are determ n%,d by
the fact that for each of them a constant symmetrical aatrix A
can be defined having positive eigenvalues and sucti that the
symmetrized matrix
[B .J = [(A ax )ik + (A Bx ) ,J (W )~k - aX
has negative eigenvalues yi satisfying the inequality pi < - d,
where d > 0 at all points of the space - oo < xi < co, 0< t < oo.
The interesting result obtained by Letov17 is also noted,
concerning non-linear control systems. with parameters that
vary only slightly. He has proved for a large class of systems of
great importance in control engineering that the stability of a
given operating mode implies the stability of all sufficiently c1c'se
modes. In this case the closeness of th-- modes is assessed by the
magnitude of the modulus of the difference between the pro-
gramming functions.
Probably further results in this direction can be obtained
on the basis of both existing and new criteria for asymptotic
stability of linear systems with variable coefficients. It can easily
be verified that in the unidimensional case Krasovskiy's criterion
is a necessary and sufficient condition for the stability of any
mode. It would be interesting to know to what extent this
criterion is necessary for systems of a higher order.
Realization of Periodic Motions
Now let the right-hand side of.the system (1) and also the
function 0 (t) be periodic in t with period T. Assuming that the
(9)
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515/4
zero solution of system (9) is asymptotically stable to a first
approximation, we can again formulate the conditions for a
given motion to be realizable with the required accuracy. But in
this case these conditions can be set more simply, since here
the dimension in space M is given by
T
r(t)112= f o r(t)I2dt
urthermore it can be shown that even in the presence of an
F
approximation error different from zero there exists an asymp-
totically stable periodic motion lying within an e neighbourhood
of the given periodic motion.
It should be observed that the results obtained can be ex-
tended without difficulty to the case where the motion to be
realized is discontinuous, or more accurately has discontinuities
of the first sort14. In this case the programming functions will
appear as the sums of ordinary functions and linear combi-
nations of 6 functions.
The reduction or elimination of the effect of disturbance by
continuous tracking of it has found wide application in the
theory of automatic control, mainly in the theory of composite
control systems. This theoryuses the so-called invariance principle
developed by Academicians Luzin and Kulebakin, which has
served as the starting point for a large number of papers on
automatic control theory that have important applications.
Realization of Processes by means of Systems with Many-valued
Characteristics
Barbashin and Alimov22 have shown how to reduce systems
of differential equations with relay-type hysteresis, and in
general many-valued characteristics to a differential equation
in a normalized linear space. Thus in this case also all the
preceding results can be obtained by the same method as was
indicated for the programming of random processes.
Programme Control of Random Processes
Up to now attention has not been directed to the external
.influence or, more precisely, disturbance it (t). Normally 77 (t)
is a random function, and so the actual mode of operation will
be a random process. Naturally in this event the programmed
mode also is random. The extension of the preceding results to
the case of stochastic differential equations presents no difficult-
ies, provided the following points are borne in mind. A random
quantity, as is known, may be determined as a measurable
function defined in some choice space Q (or space of elementary
events). It is easy to see that the space o can be constructed in
such a way that it is the choice space for all random functions
11 (t), (t) and x (t) occurring in the equation
dt =f (x, t, ?1 (0) + u (t, ~ (t)) (10)
where ~ (t) is the distortion of the disturbance 91 (t) (see Figure 1).
If a norm is defined by any means in the linear space of
random quantities (as in the space of measurable functions
defined in the choice space _L), then differential eqn (10) is
transformed into a differential equation given in the ,linear
normalized space R, whose elements are random vectors. Here
one should take as initial vectors in the solution of Cauchy's
problem not only deterministic vectors but also any other random
vectors from R, while the derivative and integral of a random
function w. r. t. t should be understood as the derivative and
integral in Bochner's sense. In particular, if as the square of the
norm of a random vector the mathematical expectation of the
square of the length of the vector is taken, then the concept of
the derivative and integral of a random function coincides with
the generally accepted one.
It should be observed that the theory of differential equations
in a Banach space is well developed at the present day. By
making use of this theory, one can readily formulate conditions
for the existence, uniqueness and extensibility of solutions?s,
and consider questions of stabilityl9 or questions of the ex-
istence and research of periodic motions20. All this enables the
setting up of a completely analogous statement of the problem
of realizing random processes and to obtain results identical
to those presented above 21
It has been seen in this paper that the accuracy of approxima-
tion to the trajectory depends on the degree of stability of the
zero solution of the system (10). The better this stability is,
as judged by any of the existing quality criteria, the smaller
effect will approximation errors have on the deviation of the
trajectory from the given one. Thus the problem of improving
the response of programme control turns on the problem of
increasing the stability of motion. Here, in particular, the
theory of programme control again comes into. contact with the
theory of optimal control.
I BELLMAN, R. Notes on control processes, Pt I. On the minimum
of maximum deviation. Quart. appl. Math. 14 (1957)
2 PONTRYAGIN, L. S., BOLTNYANSKIY, V. G., GAMKRELIDZE, R. V.,
and MISHCHENKO, YE. F. The Mathematical Theory, of Optimal
Processes. Fizmatizdat (1961)
3 ROZONOER, L. I. Pontryagin's maximum principle in the theory
of optimal systems, Pt II. Automat. Telemech, 20, No. 11 (1959)
4 LETOV, A. M. The analytical design of controllers, Pt I. Automat.
Telemech. 21, No. 4 (1960)
5 LETOV, A. M. The analytical design of controllers, Pt II. Automat.
Telemech. 22, No. 4 (1961)
6 BARBASHIN, YE. A. On the approximate realization of motion
along a given trajectory. Automat. Telemech. 22, No. 6 (1961)
RoYTENBERG, YE. N. Some problems in the theory of dynamic
programming. Prikl. Matem. Mekh. 23, No. 4 (1959)
BARBASHIN, YA. A. On a problem in the theory of dynamic
programming. Prikl. Matem. Mekh. 24, No. 6 (1960)
9 YEMELYANOV, S. V., and FEDOTOVA, A. L The design of optimal
automatic control systems of the second order using limiting
values of the elements of the control circuit. Automat.Telemech.
21, No. 12 (1960)
10 BULGAKOV, B. V. On the accumulation of perturbations in linear
oscillatory systems with constant parameters. Dokl. Ak. Nauk
SSSR 51, No. 5 (1946)
11 BARBASHIN, YE. A. The evaluation of the mean-square deviation
from a given trajectory. Automat. Telemech. 21, No. 7 (1960)
12 BARBASHIN, YE. A. The evaluation of the maximum of the devia-
tion from a given trajectory. Automat. Telemech. 21, No. 10 (1960
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 Declassified in Part - Sanitized Copy Approved for Release 2012/12/13: CIA-RDP80T00246AO22700330001-3
13 GERMAIDZE, V. YE., and KRASOVSKIY, N. N. On stability in the
presence of continuously-acting perturbations. Pr1kl. Matem.
Mekh, 21, No. 6 (1957)
14 BARBASFIIN, YE. A. On the construction of periodic motions.
Prikl. Matem. Mekh. 15, No. 2 (1961)
'' MASSERA, J. L., and SCFIAEFER, J. J. Linear differential equations
and functional analysis, Pt I. Ann. Math. 57, No. 3 (1958)
16 KRASOVSKIY, N. N.. Stability with large initial perturbations.
Prikl. Matem. Mekh. 21, No. 3 (1957)
17 LETOV, A. M. The stability of non-linear controlled systems. 1955.
GITTL
18 KRASNOSELSKIY, M. A., and KREIN, S. G. Non-local existence
theorems and uniqueness theorems for systems of ordinary
differential equations. Dokl. Akad. Nauk SSSR 102, No. 1 (1955)
Figure 1
515/5
MASSERA, J. L. Contributions to stability theory. Ann. Math. 64,
No. 1 (1956)
MASSERA, J. L., and SCHAFFER, J. J. Linear differential equations
and functional analysis, Pt II. Equations with periodic coefficients.
Ann. Math. 69, No. 1 (1959)
BARBASHIN, YE. A. Programme control of systems with random
parameters. Prikl. Matem. Mekh. 25, No. 5 (1961)
BARBASHIN, YE. A., and ALIMOV, Yu. I. Contribution to the
theory of dynamic systems with non-single-valued and discon-
tinuous characteristics. Dokl. akad. Nauk SSSR 140, No. 1 (1961)
BARBASHIN, YE. A., and ALIMOV, Yu. I. Contribution to the
theory of relay-type differential equations. Izv. Vyssh. Ucheb.
Zav. Matemat., No. 1 (26), (1962)
1
L(P)
Figure 2
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517/1
Some Problems of the Dynamics of a Hydraulic
Throttle-Control Servo-motor with an Inertial Load
V. A. KHOKHLOV
Two questions are considered in this paper. The first concerns
the limiting conditions under which a hydraulic servo-motor
may still be treated as a linear system. This is investigated
without taking into account the compressibility of the liquid
in the hydraulic cylinder. The effect of this factor is taken into
account in the investigation of the second problem-that of the
limiting frequency of. oscillation of the servo-motor piston
at which cavitation of the liquid in the hydraulic cylinder does
not occur.
The following assumptions are made: there is no liquid
leakage from, or hydraulic loss in, the piping; the flow coefficient
in the control ports of the valve is constant; the working edges
of the sleeve and of the valve, in the mean position of the latter,
coincide; and the effective areas of the piston are the same on
both sides.
2
v~sin t - 4 sin 2 t +-(3 (3 sin 3 t + 5 sin t)
3 B3
+ 8 sin 3 t cost + ... (2)
where v is the dimensionless piston velocity.
dx
dt F y dx
v=
(dx _
)xx ?b
p* gpo dt
dt
(L,X) i s the no-load piston velocity corresponding to an amplit-
tude valve displacement of p *, -r is the dimensionless time
On the Limiting Conditions under which a Hydraulic Servo-motor
Working with an Inertial Load may be Considered as a Linear
System
Figure 1 shows an outline diagram of the hydraulic servo-
motor taken for analysis. The differential equation of motion
for the actuator neglecting the liquid compressibility, and
with only inertial loading,, has been derived ' by Katsl. In an
earlier paper2 the author has given the following general form
of differential equation for a servo-motor. under any kind. of
load:
dx _ g b
dt -p p /(Po-Ap'sgnp)='p (1)
where x is the displacement of the piston in the hydraulic
cylinder, measured from its central position; ,it is the liquid flow
coefficient in the valve ports; b is the length of the working slit
of the valve port; F is the effective area of the piston; po is the
pressure in the supply line; Ap is the pressure drop in the
capacities of the power hydraulic cylinder created by the external
load; p is the displacement of the valve; and sgn p is the sign
determining the direction in which the valve is displaced from
its central position.
Equation (1) is non-linear. It is of interest to determine the
limits of frequency and amplitude' of valve oscillation within
which the non-linear term Op sgn p may be neglected in eqn (1).
The solution of this problem is particularly interesting in the
case where an inertial load is displaced by the piston of the
hydraulic cylinder. Kats' has shown that for sinusoidal valve
motion the piston velocity may be expressed approximately in
the form of a series
t=(ot (4)
and B is the dimensionless parameter
6a)p*mib
F YPo
(5)
where m is the mass of the load applied to the piston.
In his paper he also gives two more approximate methods
for solving the forced periodic motion of the piston in a hydraulic
servo-motor, and :shows that all three methods give a satis-
factory approximation provided 0 < z
However with inertial loading the non-linearity' of the equa-
tion, of motion of the actuator ?[eqn (1)] is determined by a
term depending not on its output velocity but on the acce-
leration
m d2x
Op=F =T*_( Ft
According to the results of Kats' and making use of expres-
sions (3), (4) and (5), it is possible to obtain:
dv B 2 e e t 1
dt-cost- 4 cos T
[(1- 3 sin2z) +3 0 sin2z cost]
= 82 2 (1- 3 sin2z) +3 B (sin2z cost) (1- 3 sin2z)
+ 2 82 (sin 2z cost) J
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517 / 2
. Figures calculated from this equation for 0 = 0.5 and
0 = 0.1-are shown in graphical form in Figure 2.
These graphs show that with sinusoidal valve motion and for
0< 0.1 (6)
the piston accceleration for the main hydraulic cylinder follows
an approximately cosine law (error not more than 5 per cent).
Thus if condition (6) is satisfied the term Ap sgn p may be
neglected in eqn (1), and so one may treat the servo-motor
working on an unertial load as a linear integrating element.
In order for condition (6) to be satisfied, by virtue of (5) we
must have:
cop* 1 type
failure and the second a I - 0 type failure. Each such failure
transfers the basic state to an adjacent vertex of the many-
dimensional cube. The simultaneous failure of any two internal
elements transfers the basic state to a vertex two units removed
from the vertex selected for the given basic state; it is adjacent
to any vertex to which the basic state was transferred by the
failure of any one of these two elements.
In order to provide exact performance of the control
algorithm upon the failure of internal elements, each of the
states to which the basic state is transferred upon the failure of
any number of elements within the prescribed limits (that is,
inclusive to d) must compare in the right-hand side of the table
of states to the same state of outputs as the basic state. Therefore,
for each stable state of the table of transitions, for the case of
structural redundancy, there must correspond a particular
combination of states consisting of the basic state and all the
states to which it transfers upon failure of the internal elements.
All of these states are adjacent to one another, forming a certain
multiple of adjacent states. This multiple is called a set of basic
states.
Frist it is shown that the set of adjacent states, together with
the basic states, may be described by a symmetrical Boolean
function whose active numbers represent a natural series of
numbers from K - d to K.
Let there be any state fio corresponding to one of the basic
states and let this state be characterized by a row in the table
of states containing K1 zeros and K2 ones, where K1 + K2 = K.
Then, with d = 1, the collection of adjacent states Efil contains
all the states differing from the basic by the replacement of one
variable by its reciprocal. More precisely, they are K, while Kl
of them corresponds to a failure of the type 0 - 1 and K2 to
a failure of the type 1 0. It is easy to see that the sum of these
states may be characterized by the symmetrical function:
Y- fil = SK- 1 1x1, x2 -., XK1, XK, + i' XK1 + V
if the basic state is considered a symmmetrical function of those
variables with an active number equal to K, namely :
fio = SK(/ (Xl, X2,
The , sum of the basic and set of adjacent states is thus
characterized by the symmetrical Boolean function: .
A0+I Al. =SK-1, K(xle x2e ? ??, xK, XK+I, XK+2e ?? ?, XK,+K2)
If d = 2, the set of adjacent states consists of all states
differing from the basic by the replacement of one variable by
its reciprocal, the number of which, as was pointed out, is equal
to K = C'K, and two variables. The number of the latter is
obviously equal C2K, and since each of them differs from the
* All references made below to internal elements with an identical
base pertain to inputs and sensing elements.
basic by a change having a value of two variables, their total
Eft corresponds to the symmetrical function:
Y-fi2 = Sx- 2 (x 11 x21 ..., XKi, XKi + i 1 XKi + 2' ..., XKi +K2)
The Boolean function characterizing the basic state and the
entire set of adjacent states is thus a symmetrical function of the
type:
fio + E fi l +Y- fi2
= SK - 2, K-1, K (X, .X2, ..., XKi, XK1 + i)XK1 + 2, ..., XKI +K2~
It may be proved in an analogous manner that in the general
case, with the simultaneous failure of d internal elements, the
basic state and the set of adjacent states may be characterized
by a symmetrical Boolean function of the type:
SK -d, K-d+ 1, .... K (x1) x2, ..., XKI, XKI + i, XKi +2' ..., XK1 +K2)
Thus, the class of reliable structures of discrete devices is,
with respect to internal elements, a class described by symmetrical
Boolean functions of a special type, which facilitates their
realization since these functions have been most widely studied
and may be economically realized with the aid of different types
of threshold relay elements, including electromagnetic relay
elements with several windings'.
The basic state is designated as fi and the set of adjacent
states corresponding to it as Ni, assuming that fi + Ni = Fi.
The table of states of a discrete control device consists on
the left-hand side of all sets Fi combined with the corresponding
values of inputs. For each of these sets there corresponds on the
right-hand side of the table, as was pointed out above, a state
of outputs which provides for the performance of the cpntrol
algorithm. One more output is added for which is included in
the table of states a zero for each of the basic states and a one
for any of the states which are included in the sets of adjacent
states.
Since the latter corresponds to the failure of any one or to
the simultaneous failure of several internal elements, the appear-_
ance of a one at this output occurs only by means of a decrease
in the reliability of operation of the discrete device and may be
used to signal the presence of a failure.
For example, let there be a discrete device with three inputs
and one output (Figure 1) and an action, equal to one, must
appear at the latter in the subsequent sequence of change of the
states of the outputs :
000
1 0 0
110
111
011
Any subsequent change of inputs must lead to the appearance
of an action at an output to zero, while the further appearance
of an action at the output equal to one occurs only by the
repetition of the indicated sequence of change of the states of
the inputs. With any other sequence of change of the states of the
inputs, the action at the output must remain equal to zero.
The corresponding table of conversions is given in Table 1.
Here it may be seen that it is necessary to provide for four stable
states, which is possible with the aid of two internal elements.
When it is necessary that the aforementioned discrete device
performs exactly a preassigned control algorithm in the event
of the simultaneous failure of one of the internal elements, five
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525/3
000
100 I
110
010 1
011
111 I
101
001
(1)?
(1)?
2
4
(1)1
4
4
4
-
4
(2)?
4
-
3
-
-
-
-
4
-
1
(3)?
4
-
1
(4)?
(4)0
(4)?
(4)?
(4)?
(4)?
(4)?
internal elements are required, as seen in Table 5 of reference 3.
The following distribution for the basic states is chosen:
00000
10110
01011
11101
Then the table of states will have the form shown in Table 2.
In agreement with what was mentioned above, let us add the
output CO, in the column of which are written zeros in
all the rows of the table of states corresponding to fi and ones
in all the rows corresponding to Ni (Table 3). Then this output
will signal the presence of a failure of any one or several of the
internal elements.
A B C I
F I
X1
X2 X3 X4 X
5 I
Z
0 0 0
F.
0
0 0 0 0
0
0 0 0
F4
0
0 0 0 0
, 0
0 0 1
F1
1
1 1 0 1
0
0 0 1
F4
1
1 1 0 1
0
0 1 0
F1
1
1 1 0 1
0
0 1 0
F2
1
1 1 0 1
0
0 1 0
F4
1
1 1 0 1
0
0 1 1
F1
0
0 0. 0 0
1
0 1 1
F3
0
0 0 0 0
1
0 1 1
F4
1
1 1 0 1
0
1 0 0
F1
0
0 0 0 0
0
1 0 0
F2
1
1 1 0 1
0
1 0 0
F4
1
1 1 0 1
0
1 0 1
F1
1
1 1 0 1
0
1 0 1
F3
1
1 1 0 1
0
1 0 1
F4
1
1 1 0 1
0
1 1 0
F1
1
0 1 1 .0
0
1 1 0
F2
1
0 1 1 0
0
1 1 0
F3
1
1 1 0 1
0
1 1 0
F4
1
1 1 0 1
.
0
1 1 1
F1
1
1 1 0 1
0
1 .1 1
F3
0
1 0 1 1
0
1 1 1
F2
0
1 0 1 1
0
1 1 1
F4
1
1 1 0 1
0
00000
10110
01011
11101
1 0000
001 1 0
11011
01101
01000
11110
00011
10101
00100 F2
1001 0
01111
11001
00010
10100
01001
11111
00001
1011.1
01010.
11100
X1
X2
X3
X4
X5 I
C.
C1
C2
C.
C4 C5
0
0
0
0
0
0
0
0
0
0 0
1
0
0
0
0
1
1
0
0
.0 0.
0
1
0
0
0
1
0
1
0
0 0
0
0
1
0
0
1
0
0
1
0 0
0
0
0
1,
0
1
0
0
0
1 0
0
0
0
0
1
1
0
0
0
0 1
1
0
1
1
0
0
0
0
0
0 0
0
0
1
1
0
1
1
0
0
0 0
1
1
1
1
0
1
0
1
0
0 0
1
0
0
1
0
1
0
0
1
0 0
1
0
1
0
0
1
0
0
0
1 0
1
0
1
1
11
1
0
0
0
0 1
0
1
0
1
1
0
0
0
0
0 0
1
1
0
1
1
1
1
0
0
0 0
0
0
0
1
1
1
0
1
0
0 0
0
1
1
1
1
1
0
0
1
0 0
0
1
0
0
1
1
0
0
0
1 0
0
1
0
1
0
1
0
0
0
0 1
1
1
1
0
1.
0
0
0
0
0 0
0
1
1
0
1
1
1
0
0
0 0
1
0
1
0
1
1
0
1
0
0 0
1
1
0
0
1
1
0
0
1
0 0
1
1
1
1
1
1
0
0
0
1 0
1
1
1
0
0
1
0
0
0
0 1
If one places the action from this output into a computer
and determines the number of times that actions equal to one
appear at this output during a certain time interval, the answers
from the computer may be used to predict an approximation
of reliable operation of the device.
The described principle of signalling and prediction has
significant advantages in the sense that neither the signalling nor
prediction requires the introduction of any additional internal
elements. Usually the performance of these functions relies upon
special units of the discrete device which require elements having,
in principle, a reliability as much as one order of magnitude
greater than the elements which make up the discrete device itself.
In the design examined above, comprising a structure of
signal outputs based on actuating devices already having internal
elements, and assuming that the connections between these
devices and the sensing signal and predicting devices have
100 per cent reliability, one would expect that the signalling of
failure would have absolute reliability in principle.
In fact, only two mutually exclusive events may occur:
(a) not one of the internal elements is faulty. Then the actions
equal to one appear at the corresponding operating outputs
and at the signal output the action is equal to zero; (b) failure
of one or several internal elements occurs within the limits of d.
Then an action equal to one appears both at the signal and
operating outputs.
It is noted that achieving reliable operation by means of the
introduction of structural redundancy according to the principles
previously presented by the author3 pertain to the internal
elements of the device as a whole, that is, both to the actuating
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525/4
and the reacting devices. Therefore, with respect to failures of the
actuating organs, the device retains its ability to perform
exactly the control algorithm upon the failure of either one or,
simultaneously, all of the actuating devices of a given internal
element for the conditions when these failures are all of a
single type.
The described principle of designing signal circuits makes it
possible to provide separately for signalling the number of fail-
ures greater than d, including those located between the limits of
d + 1 and d + J. Additional outputs must be added for this
purpose. This requires that ones be written in the specific rows
in the appropriate columns of the table of states; namely, for
signalling failures of elements within limits from d -{- 1 to d + 4
in the rows corresponding to failures in these limits, and for
signalling a large number of failures in the rows corresponding
to unused states.
It is obvious that the signalling of failures may be not only
general but also specific, or, for each of the internal elements
of the device separately. For this purpose one must have for
each of them an individual output, for which there must be
written in the columns of the table of states ones for all states
differing from the basic by the change in value of the correspond-
ing variable. For example, to signal the failure of element XI in
the above case, ones must be written for each first row of the
sets Ni for the corresponding output.
Table 3 gives the corresponding values of outputs for each
of the internal elements. The realization of such outputs pro-
vides, in the event of faulty elements in the device, for advance
notification as to which of the internal elements is malfunc-
tioning or, with prediction, an approximate indication, per-
mitting timely replacement or adjustment of the element for
proper action.
Obviously it is possible to provide not only for signalling of
failures of individual internal elements but for the separate
signalling of the nature of these failures as well. For example,
in Table 4, for the internal element XI examined above, are
shown the operating states corresponding to failures of the
type 0-* 1 [Table 4(a)] and failures of the type I 0
[Table 4(b)].
Table 4
As was pointed out above, the functions which realize the
basic states together with the sets of adjacent states are sym-
metrical with the operating numbers from K - d to K and for
their realization it is suitable to use so-called `threshold' elements.
When such elements are used it is advantageous to use the
structure of the discrete device having a form shown in Fig-
ure 2(b), where the [1, K] terminal network is based on thresh-
old elements according to the number of basic states. The [M,N]
terminal network has the same make-up as that shown in
Figure 2(a), while the output circuits for signalling and predic-
tion of failures are derived from the outputs of the threshold
elements by means of their series connection (providing an `and'
operation) and from circuits corresponding to the function A.
The latter may also be designed with the aid of threshold elements
having symmetrical functions with the operating number K.
In addition it is noted that, in the case examined above, it is
most rational from the viewpoint of the simplest physical
realization of the structure of a discrete device to choose the
operating levels of the symmetrical functions not from K - d
to K but from 0 to d, while simultaneously taking not the
variables but their inversions.
In conclusion one should note that the method considered
previously by the author3, as well as everything discussed in
this report, refer to the case in which the probability of failure
for all internal elements has a single value, the failures
are symmetrical (that is, the probability of failures of the
type 0 --> I is identical to that of type 1-+ 0), and, in addition,
failures of individual elements are mutually independent. Con-
ditions differing from these necessitate a somewhat different
approach to determining the minimum number of elements and
the distribution of the states. However, the principles of de-
signing signal circuit and of prediction remain the same, with
the exception that the functions characterizing the basic sets
and the sets of adjacent states may not prove symmetrical.
1 VON NEUMAN, S. Probabilistic logics and the synthesis of reliable
organisms from unreliable components. Automata Studies. 1956.
Princeton; Princeton University Press
a MOORE
E
F
and SHANNON
E
C
Reliable circuits usin
less
,
.
.
,
.
.
g
1 0
0
0
0
0
0
1
1
0
reliable relays. J. Franklin Inst. Vol. 262, No. 3 (1956) 191, 281
1 1
0
1
1
0
1
1
0
1
GAVRILOV, M. A. Structural redundancy and reliability of relay
(a) (b)
In conclusion some of the problems of realizing signalling
and prediction networks are considered. The circuit of each
output in the structure of a multi-cycle discrete device must
contain actuating devices of both internal and sensing relay
elements. The signal circuits must contain actuating devices of
only internal elements. Therefore the rational design of the
structure of a discrete device would be that shown in Figure 2,
namely, a structure in the form of a certain [1,K] terminal net-
work having at its outputs all the functions of fi. and Ni and
containing the actuating devices of only the internal elements,
and an [M, N] terminal network containing the actuating devices
of only the sensing elements.
circuits. Automatic and Remote Control. Vol. 2, p. 838. 1961.
London; Butterworths
4 ZAKROVSKIY, A. D. A method of synthesis of functionally stable
automata. Dok. AN SSSR Vol. 129, No. 4 (1959) 729
RAY-CHANDHURI, D. K. On the construction of minimally re-
dundant reliable system designs. B.S.T.J. Vol. 40, No. 2 (1961) 595
ARMSTRONG, D. B. A general method of applying error correction
to synchronous digital systems. B.S.T.J. Vol. 40, No. 2 (1961) 577
GAVRILOV, M. A. Basic terminology of automatic control. Auto-
matic and Remote Control. Vol. 2, p. 1052. 1961. London; Butter-
worths
GAVRILOV, M. A. The Structural Theory of Relay Devices, Part 3.
Contact less Relay Devices. 1961. Moscow; Publishing House of the
All Union Correspondence Power Engineering Institute
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527/1
On the Theory of Self-tuning Systems with a Search
of Gradient by the Method of Auxiliary Operator
I. E. KAZAKOV and L. G. EVLANOV
Structure and Equations of a Self-tuning System
In many cases important in practice, automatic control systems
may be represented in the form of a generalized system illus-
trated in Figure 1. The object of control is charaterized by an
operator of a given structure A (n), where 27 is a group of para-
meters for which a priori information is lacking. The system of-
control is described by an operator B(~) which depends on the
group of parameters ~i (i = 1, 2, ..., n) which may be tuned.
In actual systems, the aggregate of values of each parameter ~i
forms.a finite multitude Ei. The input signals of the system are
X(t), the useful random signal, and Z(t), U(t), random-
disturbances.
The equations of the automatic control system are as
follows :
Y=A(q)[V+U]
V=B( )e
e=X+Z-Y
L
(1)
In the particular case when the lower (upper) boundary of the
multitude ~i is attained within .~.i,
I0 = extremum I () (5)
For a complete description of the circuit for self-tuning it is
necessary to determine the method of computation of the com-
ponents of the gradient from the quality index for the tuned
parameters. In the given investigation a method is applied which,
in the following is termed the method of an auxiliary operator.
Its essence consists of the following.
If the information on operators B(~) and A(,q) is known
a priori, it is possible to construct a certain auxiliary operator
C (~, 77) whose application to the error of the tracking system
makes it possible to compute the components of the gradient
vector.
The derivative a I/a ~i is computed by the direct differenti-
ation of the expression (2) assuming that the operators N and
differentiations with respect to ~i are commutative.
In order to assure high quality functioning of the automatic
control system it is necessary to achieve tuning of parameters
of the operator B($) in the presence of variation of the char-
acteristics of the input useful signal X(t), of the characteristics
of disturbances Z(t), U(t), and also in the presence of variation
of parameters n of the operator of the object of control.
In order to construct a circuit for self-tuning, an index of
quality I of the automatic control system is introduced. The
index of quality I is a function, or in the general case it is a
functional of tuned parameters. Ordinarily the index of quality
I is computed on the basis of errors of the system:
I= Nf (E, c) (2)
where N is an operator or a functional, f (e, ~) is a function of
the error of the system depending upon the error e and the
tuned parameters .
In order to tune the parameters of the system use is made of
the broad possibilities offered by the method of steepest de-
scending slope or gradient, a discussion of which is considered
by Feldbauml. Applying this method for tuning parameters
one has:
~ = grad I (3)
where A is the scalar multiplier, and is a vector function of the
velocities of tuned parameters. In accordance with the gradient
method the self-tuning system assures the tuning of parameters
for the optimal value of index or quality I. In the general case
Ia inf I (~) or Io = sup 1( ) (4)
SiEdi ~iEdi
a7 _ a f (E' 0) ai afi (c, 5)
a5i - N ac a~i+ N aSi
(6)
The derivative a no ~i will be calculated by differentiating the
system of eqns (1). The derivative of the error s with respect
to $i is equal to
as aY
(7)
since the input. signals X(t), Z(t) do not depend upon ~i. The
derivatives of the output signal are computed:
A(rl)aa)e-+ A(q)B()ai (8)
Excluding from (7) and (8) a Yla ~i and transforming, one
obtains:
asz [1A(h)B(~)] 'A(q)aB
abi- aSi
Introducing the designation
(9)
Ci=[1+A(q)B( )] 'A(q)aB(~) : (lo)
ai> = -Ci(rl, )e
Vi
grad is C (q, ~) s (12)
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527/2
where C (77, ~) is an auxiliary operator-vector which is completely
determined by the operators A(n), B(~). Thus, the gradient of
the quality index for tuned parameters is determined by eqns (6)
and (11). .
The method of auxiliary operator requires an a priori
knowledge of information on the system, and this somewhat
restricts its generality. However, there exists in technology an
area of applicability of the method inasmuch as the predominant
majority of created automatic control systems can be described
.mathematically.
The advantages of the method are the absence of trial load
changes and the possibility of accelerating and simplifying the
process of computation of the gradient components. In self-
tuning systems with a search of gradient by the method of trial
load changes, a priori information on the object, other than the
knowledge of the band pass of the system, is not required. This
permits a correct selection of the frequency of the trial load
changes and constitutes the advantage of this method. However,
its basic shortcoming is the limited quick response imposed by
the finite band pass width of the system. In the considered
method the band pass of the mathematical model of the system
(operator C) may be artificially broadened by changing the
time scale of the solution. The possibility of simplifying the
process of computation is based on the substitution for a com-
plex operator C of an approximate and simpler expression.
The auxiliary operator r (?I, ~) depends upon the para-
meters of the object and the system of control. A typical case
is one of absence of a priori information on parameters 77. In-
formation on parameters of the object may be obtained on the
basis of application of a tracking system, certain aspects of
whose application were considered by Margolis and Leondes2,3
The structure of the operator of model A (C) is based on the
utilization of a priori information on the object. The aggregate
of,parameters C of the operator of the model is tuned for the
value 77. The circuit of the tracking model is constructed quite
analogously to the circuit for tuning. Introducing an index
of approximation J of parameters C into parameters 27,
J=LO (sl) (13)
where L is an operator for computing the index J, and 0 (e) is
a function of the error. The error is determined by the relation-
ship
E1=YM(t)-Y(t) (14)
Here YM(t) is an output signal of the model determined by the
expression
YM (t) = A (() V (15)
The change of the parameters of the model is carried out by
the method of steepest descending slope:
=,1 grad J
(16)
where 21 is a scalar multiplier, and I? is a vector function of the
velocities of the tuned parameters of the model.
In order to determine the components of the gradient one
applies the method of auxiliary operator:
aJ Lao(81) as1
a~i ac1 aii
(17)
Differentiating the relationship (14) with respect to C f one has:
as1 aYM a @A(~>
A(() V =
=
V
(18)
a(`
aft - ail
a~i
operator-vector G(C) with components
Gi (C) = aA (()
(19)
aSi
d J= L f" E E`) G (s) V}
(20)
gra
Equations (13), (14), (15), (16) and (20) describe the operation
of the tracking model. A useful output of the circuit of the model
is the aggregate of parameters of model C. For ideal operation
of the model C _- 77. An actual model assures the attainment of
parameters C close to values 27, and therefore, strictly speaking,
in the operator C it is necessary to replace parameters 77 by C.
The complete structural diagram of the self-tuning system in
accordance with eqns (1), (3), (6), (11), (14), (15), (16) and (20)
is presented in Figure 2. The schematic diagram was proposed
by Evlanov.
The structure of the self-tuning system contains three cir-
cuits: the basic circuit of the system, the circuit of the tracking
model, and the circuit of tuning of parameters. The circuit of
the tracking model assures the reception of information on the
parameters of the operator o' the object. In the following the
operation of the circuit of the t. -1.cking model is assumed to be
ideal, that is, C ?7. The circuit 1',r tuning the parameters as-
sures the tuning of parameters of th, control system in accord-
ance with the given optimal value of .he quality index of the
system.
Investigation of a Self-tuning System a Quasi-stationary Regime
A typical regime of operation of a self-tuning system is the
case of a change of parameters 77 of the operator A (77) of the
object and the characteristics of external random disturbances
X, Z, U which are slow compared with the duration of transi-
tional processes in the basic circuit of the system. In this case
it is permissible to consider the circuits of tuning parameters
and the tracking model on one hand, and the basic circuit on
the other hand, as being autonomous, since the tuned para-
meters ~ and parameters ' may be considered as constant during
the time of process control in the basic circuit. It is also assumed
that the tracking model carries out its functions in an ideal
manner. Under these conditions the process of self-tuning of
parameters ~ of operator B(~) is investigated in the vicinity of
extremum of the quality index I.
The presence of extremum in the quality index I of the
system with respect to all or several of the tuned parameters is
an important property of the self-tuning systems which permits
them to be tuned for an optimal regime. If the error of the
system e or another characteristics does not possess extremal
properties, then it is possible to construct an extremal quality
index by artificial means depending upon the direction of
aiming of the automat. This will be shown below by an example
of a typical tracking system. For the time being, however, it is
assumed that the quality index I possesses extremal properties.
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The random errors of the basic circuit can be expressed in
the form
(21)
where mE is the mathematical expectation, and s? is the centring
component of magnitude s. In the function of the error f (s, )
we shall also factor out the mathematical expectation
f (E,) = M f (s,) +f o (E,)
(22)
where M is the operation of mathematical expectation, f? (s, )
is the random centred component.
The quality index of control I introduced previously may
I*=NI*+Nf ?(s~) (23)
where the designation I* is introduced for the statistical,
quality index of control
I* = Mf (E, ) (24)
Computing the components of the gradient of the quality index
of control by parameters S i, one obtains :
ai W af? amE afo @8? afo
_
aSi-NaSi +N amE aSi +N~,_.a +N abi (25)
Representing the statistical quality index I* of control in the
vicinity of the investigated extremum by a quadratic form in
terms of deviations ui = ~i - $io of parameters ~i from the
optimal values $?, and considering that '
aI*
aSi
J 4i=sio V
values of i J*/a S i the. expressions :
aI* n 1 C all** 1
ai - J1 aSiaSiJO uj
(26)
Differentiating expressions (24) twice with respect to para-
meters . ~i, ~; and, utilizing a system of equations of the basic
circuit of control for optimal parameters ~iO of operator B(i;),
one computes the coefficients
a2I*KK
abi abj o
aI* a2f (80,4)
i a j-M as0 2 (C;oso) (Cioso)
ayy
S
+af (90, ~0)(CjoCio6o)+a2f (So, ~0)(Cioso)
ago zz aE a~ j
+a2f (80,S0)(C;0c0)+a2f (E0, o) (27)
aSaSj abia~j I
where Ci0 ($?in) are the auxiliary operators (10) for optimal
values of parameters ~i0.
Introduce the designations:
1
1 a21*
= ai;
~a~i ~; J
(28)
Taking into account also that
0
a `= - C1 M" a = - Cia? (29)
a i a i
the formula (25) is written for the components of the gradient of
the magnitude I in the form:
01 n Of 0 0
=N Y a.?u?-N a C?mE-N af Ca?+NafxY (30)
abt j=1 1.1 J amE t E~ t
Substituting the expression (30) into formula (3), one obtains
a system of equations of the circuits of tuning of parameters Si
in a scalar form:
n 0 0 f0
4i=2N E ai;u;-A.N am CimE-ANaa Cis?+2N a-t
j=1 E
(31)
From this one obtains a system of linear equations for the de-
termination of mathematical expectations of deviations mni of
tuned parameters from the optimal values:
n
inni-1N Y ai;u;=-i0
j=1
(32)
In order to determine random components of deviations of
tuned parameters ui? one obtains the following system of linear
equations:
n Of 0
u?-AN Y uai;)N CiorEo
j=1 amE
_AN 1f Ci0EO+~lN (33)
Caf O of
aE J 0 a i
An analysis of approximate linear equations (32) makes it
possible to evaluate the stability of the process and to determine
the systematic errors of self-tuning of parameters $i. In partic-
ular, if the basic circuit of control is stationary and possesses
astatism of the kth order, then for stationary random disturb-
ances Z and U, and for an additive component of the useful
signal X in the form of a polynomial of the kth order, the left-
hand carts of eqns (32) are stationary. In this widely encountered
case the stability of self-tuning of, the parameters is characterized
by properties of characteristic equation. In this case the in-
vestigation of stability is carried out by ordinary means. In the
general case the systematic components of the errors of para-
meters are computed by equations :
Y J t gij (t, z) 4jo (r) dx (34)
j=1 0
where gi; (t, z) are the weight functions of the system of eqns (32).
If ~i? = const., then the systematic values of errors of tuning
of parameters mni = 0. Dispersions of the errors of parameters
are determined on the basis of the system of egns (33) by
applying the theory of transformation of random functions4.
From the analysis of stability, duration of transitional pro-
cesses of tuning, and evaluation of the precision, one.chooses
the coefficient 2 and also other characteristics of the circuits of
tuning.
The final evaluation of mathematical expectation of the
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circuits of self-tuning is obtained by the formula :
n
mE=mEO+ me,
t=1
(35)
where mep is the mathematical expectation of the error of
control e for an optimal value of parameters.
The magnitudes mE, are determined by the expressions:
me, = C1p (o) [cimn,]
where C;0 (~0) are the auxiliary operators for optimal values of
parameters ~,? and the magnitudes bi are equal to
b =
oB(~)
36
)
m (
is computed by the formula:
De=Dep+2 Y kEpe,+ Y ke,E) (37)
i=1 i,J=1
where DE0 is the dispersion for optimal values of parameters ~;0,
KEOE,, Ke:e, are the coefficients of correlation of random com-
ponents of the error of control ei?, and the magnitudes ei? are
e? = -,U? [C,0 (w meo]
. (38)
Linear Tracking System with One Tuned Parameter
The application of the method to a linear tracking system,
with one tuned parameter, is now described. In tracking systems,
as a rule, the index of quality of control is assumed to be the
second initial moment of error e. This magnitude does not
possess extremal properties with respect to parameters ~ corre-
sponding to the change of input random actions X, Z, U.
Now consider an example of a tracking system having the
following characteristics: A(77) = D, B() X = at, U = 0,
mz 0, SZ DZ
and values of parameters given by 17l = 10, a = 0, 1, D0 = 10-4,
# = 100. The second initial moment of error e in a stabilized
regime is equal to:
a2 D/3.
ae= _i_2_72 +
(39)
This relationship has no extremum with respect to parameter sl
In the theory of optimal filtration the magnitude e* _
- Z = X - Y is considered as an error. The second initial
moment of this magnitude possesses extremal properties. Thus,
under the conditions of the preceding example the magnitude
aE is equal to:
a2 D
* z111 1
ae ~i17i~~1111+/3
. (40)
the information that the spectrum of the frequencies of the
disturbance Z, as a rule, is substantially broader than the spec-
trum of the useful signal X. Then the function Zl will possess
characteristics which are close to the characteristics of the
function Z.
Measuring the magnitudes e and Z1 it is possible to formulate
artificially a quality index having an extremal characteristic with
respect to coefficient of amplification ~1 of the correcting circuit
B(e). For this the function of the error is assumed to have the
form :
J (6, b)=E2+0 (~1)Z1
(41)
The function V (~1) may be chosen in a specific case, for instance,
from the condition of proximity of the extrema of functions
M [e- Z]2 and M [e2 + V(t1) Z2) with respect to parameter
for statistically prescribed input signal.
As an illustration of the method of prescribing, a function
WO let us consider the case of good filtration when it is possible
to neglect the component X in function Z1. Let us determine
V(~1) = v~l, where v is a constant coefficient computed from
the condition of proximity of the values of parameters ~10 for
extremal values of the functions a = Mee + v~1DZ and aE _
M(e-Z)2.
In Figure 3 there are presented graphs of functions a and ae
corresponding to the minimal value and computed for the
preceding example. For v = 0.1 the minima of the functions,
(curves with an index 1) coincide closely, and the optimal value
of parameter X10 = 3.0. The change in a sufficiently broad range
of probability characteristics of disturbance Z, useful signal X,
and parameter q leads to a distortion of the form of the curves
a and ae. However, their minima coincide, but are not reached
for other values of parameter X10 as shown in Figure 3. In
Figure 3 the index 2 denotes curves for D2 = 10-3 and the pre-
vious values of other parameters.
In Figure 4 there is shown a schematic diagram of a linear
tracking system with tuning of the amplification coefficient S1
for 2i1(Sl) = val. The function Z1 is separated with the aid of a
band pass filter or a filter of high frequencies. Then the signal is
supplied to a square wave generator and a circuit with amplifica-
tion coefficient v~1, and then to a low frequency filter. Now
consider the quasi-stationary regime of self-tuning of para-
meters. Eqn (31) of tuning of parameter S1 stated with respect to
deviation U1 assumes the form:
[(TD+1)D-7..a1] ul = -22m,,0 [C10(0)+C10 (D)] co
- D~0 + 21 vm0Z1?
(42)
a1=M{[C10(D)e0]2+[ec (Ci0(D)so]}>0
(43)
From these one obtains the following equation for the de-
termination of mathematical expectation mu 1:
:
(TD2+D-2a1) m?,= -D~10
(44)
This function has an extremum with respect to parameter ~,.
It is possible to measure directly the magnitude e* in tracking
systems using a priori information on the statistical properties
of the input useful signal and the disturbances. In practice it is
possible to measure the error e and the signal Zl = Z1 (Z, X)
related to Z. For instance, the function Z1 may be obtained by
filtering with special filters the input signal X + Z and utilizing
For 1 < 0 the stable process of tuning is assured. When one
determines the centred random component ui , one obtains the
equations :
[TD2+D-gal] u?= -22mE0 [Cio (0)+C1o (D)] eo
+2.tvm,,Z?
(45)
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527/5
The magnitude m2, may be set equal to zero by proper selection
of the corresponding filter. Taking this into account and also
utilizing expressions for ED in terms of X? + Z?, one obtains
from eqn (45)
u0 =d1(D)(X?+Z?)
(46)
0
(D)
22 2m.0 [C1.0 (0) + C10 (D)]
(47)
1
-
(TD +D-2a1)[1+A(D)B0(D)]
In this case, for computing the dispersion of parameter u1 in a
stabilized regime, one obtains:
Du, =f I(D1(iw)I2[S,, (w)+S.(w)]dco (48)
where S. and Sz are the spectral densities of random functions
X and Z. For ~I? = const._ the magnitude mu, = 0 in the sta-
bilized regime. In this case the systematic error of a following
system with self-tuning in a stabilized regime of operation is
equal to m,, = mE0, that is, equal to systematic error for an
optimal value of parameter X10. The random component of the
error of following is equal to:
C1 + 1 +A (D) B0 (D) 1(D)] 1 + A (D) B0 (D)(`Y? + Z?) (49)
where the magnitude b1 according to formula (36) is given by
bl_ aB0 (~0)
ab 10 , .
(50)
=f 00
1 ~2
b1A (ico)
D
1+
(
)]
FELD
Mo
MAR
BAUM, A.
scow; GI
GOLIS, M.
A.
FM
an
Computers in Automatic Control Systems. 1959.
L
d LEONDES, C. T. A parameter tracking servo
-w
lo
1+ A(iw)B(iw)
1
1+A(i(o)B(i(
o) ,j
for
control s
yste
ms. Trans. Inst. Radio Engrs, N. Y. AC-4, N 2
(19
59)
[S. (o)) + S. ((o)] dco (51)
MAR
GOLIS, M.
an
d LEONDES, C. T. On the theory of adaptive
The calculations carried out for a tracking, system (Figure 4)
.con
Re
trol syste
mote Cont
ms;
rol.
the learning model approach. Automatic and
196]. London; Butterworths
having the values of the preceding example for 7~ = 105, T= 1.0,
PUGA
CHYOV, V
. S.
Theory of random functions and its application
and the optimal value of parameter S10 = 3.0, show a sufficiently
to p
roblems
of a
utomatic control. 1960. Moscow; GIFML
X+Z
good effectiveness of tuning. Thus, the mathematical expectation
of tuned parameter S1 is equal to mCi = X10, and the dispersion
of the error of tuning computed by formula (48) is given by
DC1 = D?1 = 4 x 10-7. From these calculations it follows that
the maximum relative error of tuning the parameter ~1, is equal
to 6.3 x 10-2. per cent. As regards the error of tracking by the
following system, the mathematical expectation of this error in
tuning coincides with the value of this magnitude in an optimal
system mE = mE0 = 0.33 x 10-2.
The dispersion of the error of tracking in a 'self-tuning system
computed by formula (51) coincides with a precision to three
significant figures with a value of dispersion of the error of
tracking in the optimal system DE . DEO = 2.31 x 10-5. Thus,
in the considered example the self-tuning system with the utiliza-
tion of the method of auxiliary operator assures an effective
tuning for the minimum of the second initial moment of error
in the presence of random disturbances.
The considered scheme of a self-tuning system may be
effectively utilized both for the direct control of objects and the
synthesis of automatic control systems during their design. The
advantages of the system of self-tuning utilizing the method of
auxiliary operator are: relative simplicity of achieving tuning
circuits, effectiveness of operation in the presence of disturb-
ances, and the possibility of obtaining high values of quick
response.
Figurg I
527/5
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1 I I 1 '-w
2 3 4 5 6 7 8 9 10
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One Self-adjusting Control Systems Without Test
Disturbance Signals
E.P. POPOV, G.M. LOSKUTOV and R.M. YUSUPOV
In this paper, the term `self-adjusting control system' means a
system which performs the following three operations:
(1) Measures by means of automatic search or computes
from the results of measurements the dynamic characteristics
of the system, and possibly the characteristics of the disturbances
as well.
(2) On the basis of this or that criterion defines the controller
setting, parameters or structure needed for calibration (or opti-
mization).
(3) Realizes the resultant controller structure, parameter or
setting values.
Many studies of the theory and practice of self-adjusting
control systems for stationary controlled plants have so far
appeared in the world literature. There have also been con-
tributions on self-adjusting of quasi-stationary systems. But there
is almost a complete lack of contributions dealing more or less
specifically with problems of synthesis and analysis of self-adjust-
ing control systems for essentially non-stationary controlled
plants. Moreover, as far as the authors are aware, even in the
case of stationary and quasi-stationary systems, the process of
self-adjustment is frequently effected solely on the basis of an
analysis of the dynamic characteristics of the system, without
taking into account the unmeasured external disturbances acting,
upon the controlled plant. At the same time it is obvious that
external disturbance, besides the dynamic characteristics of the
system, determines the quality of the process of control.
Another drawback of many of the self-adjusting systems in
existence and proposed in the literature is the need to use
special test signals to check the dynamic characteristics of the
system.
This paper proposes, and attempts to validate, one of the
possible principles for the creation of a self-adjusting control
system for a particular class of non-stationary controlled plants.
The main advantage of the principle in question is the
opportunity it provides to take account of both internal
(system parameters) and external (harmful and controlling
disturbances) conditions of operation of the system. In contrast
to the self-adjusting systems known, a system created in accord-
ance with the principle proposed will make it possible to obtain
automatically the fullest possible information about the process
under control without the use of test signals.
For the operation of a self-adjusting control system created
on the basis of the principle proposed, a mathematical model
of a reference (calculated) control system must be constructed.
A `reference system' is understood to be a system the controller
of which is designed in accordance with the requirements on
the quality of the control process, with the assumption that the
mode of variation in time of the system's parameters as well as
the disturbance effects is known.
The structure of the mathematical approximation of the real
process is selected to match that of the mathematical model of
the reference process. The self-adjusting system operates in such
a way as to ensure continuous identity between the mathematical
approximation of the real process and the model of the reference
system. In this connection, the problem is posed of making the
mathematical approximation of the real process as close as
possible to the model of the reference process.
Without loss of generality, the case of control of only one
variable is considered, which is denoted by x, and the correspond-
ing reference differential equation is written in the form
x(n)+n~1aE(t)xEi)= >J bn(t)fE) (1)
i=o i=o
The real process is approximated by a linear differential equation
of the same structure:
n-1 in
x(n)+ E ai(t)x(`)= Y bm(t).fEi) (2)
i=o i=o
t = to, x(`) (to) = -Eo (i = 0, 1, ..., n -1)
The operation of the proposed self-adjusting control system
will be examined in accordance with the sequence of the process
of self-adjustment, indicated at the beginning of the definition.
General Case of Determination of the Dynamic Characteristics
of a System
In order to create an engineering method of determining the
dynamic characteristics of non-stationary systems in the construc-
tion of a self-adjusting control system, this paper proposes the
use of the methods of stationary systems. For this purpose, the
non-stationary system (1) is replaced by an equivalent system
with piecewise-constant coefficients. (The methods of stationary
systems are used on the intervals of constancy of the coefficients.).
The transfer from a system with variable coefficients to one with
piecewise-constant coefficients is effected on the basis of a
theorem which can be formulated with the assistance of a
number of the propositions of the theory of ordinary differential
equations. In accordance with this theorem, the solution of a
differential equation of form (1) with piecewise-continuous
coefficients (a finite number of discontinuities of the first kind
is assumed) can be obtained with any degree of accuracy in a
preset finite interval (to, To) by breaking down the latter into
a finite number of sub-intervals (tK, tK+I) and replacement of
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the variable coefficients within each sub-interval by constants,
equal to any values of the corresponding coefficients inside or
on the boundaries of the sub-intervals under consideration.
In the general case, it is expedient to effect the breakdown
process by the method of multiple iteration of solutions on a
high-speed computer.
Let the differential equation with variable coefficients (1)
be approximated by an equation with piecewise-constant
coefficients.
Then, for t e (tK, tK+1), one may write
n-1 m
x(n)+ aE x(`) bE (i)
E iK E M f
i=0 I i=0
(3)
the approximation error is absent,,and the connection of the
coefficients of eqns (4) and (5) is expressed by the equalities:
aiK=aiK(i=O, 1,...,n-1)
biK=CK biK(i=0,1, ..., m)
(7)
Equation (5) is used (henceforward, to simplify the notation,
the dashes over the coefficients and the variable x are dropped)
for definition of the coefficients aiK and biK. It is assumed that
measurements x, x', ..., x(n) are performed at the points
tK = T1, r2, ..., Ts = tK+1 - At.
The values of fE, fE, ..., f. ("n) are known. Then, for the
definition of (n + m + 1) desired coefficients in each interval
(tK, tK+1) one obtains the following system of S algebraic
equations, which will be written in abbreviated form thus :
n-1 m
E x(`)(rJ?)aiK - Y ff`)(iJ?)biK x(n)(zJ?) (J=1, 2, ..., S)
- -
i=o i=o (8)
It is not always expedient to solve directly system (8) for
S = m + n + 1, since, on account of the existence of measuring
instrument errors and random high-frequency control process
oscillations, the. accuracy of definition of the coefficients will
be very low. Moreover, for the same reasons, system (8) may
be altogether incompatible.
To eliminate the case of incompatibility and to increase the
accuracy of definition of the searched coefficients the method of
least squares is employed', 2. In so doing, the problem of
approximation is also solved. When utilizing this method,
it is expedient to take S > m + n + 1.
Using the method of least squares, the coefficients aiK, biK
are defined, minimizing according to these coefficients the
In accordance with differential equation (3), the real process
is approximated by the equation
n-1 M
a,?,, ~ x(i)= E bjxf(i) (4)
i=0 i=0
As the dynamic characteristics of the system at the first
stage of operation of the self-adjusting system on each interval
(tK, tK+1), the coefficients aiK (i = 0, 1, . ., n - 1), biK (i = 0,
1, ..., m) are defined.
The simplest way to define these coefficients lies in defining
the values of x and f and their corresponding derivatives at the
points tK = Z1, Z2, ..., r8 = tK+1 - At-
By substituting 'these values into eqn (4), one obtains for
each interval (tK, tK+1) a system of S algebraic dissimilar
equations for defining the searched coefficients.
In practice it is not always possible to measure the disturbing
effect f and its derivatives. Therefore, in the general case, the
above-mentioned method of defining the coefficients aiK and biK
cannot be directly employed.
This difficulty may be avoided in the following way. The real
process is approximated, not by differential eqn (4), but by a
differential equation of the form
n-1 m
x(n) + y aiK x(`) L biK f (1) (5)
In eqn (5) the disturbing effect and its corresponding derivat-
ives are taken to equal the reference values. This avoids
the need to measure the real disturbance f, and makes it
possible to use the above-mentioned means of defining the
coefficients of the differential equation approximating the real
control process. The non-agreement of the real disturbances
with the reference ones are taken into account through the
coefficients aiK and biK. Therefore dashes are placed over them.
In the general case x(i) * x(i) (i = 0, 1, ..., n) i.e., there
is an approximation error. In view of this, in the transfer from
eqn (4) to eqn (5), it is necessary to evaluate the maximum
possible value of this approximation error, using for this
purpose the assumed values of the limits of variation of
disturbance f.
If for some class of controlled plants it can be assumed that
in the process of operation only scale of the disturbance changes,
i.e., the equality
f (t) = CK fE (t), t e (tK, tK+ 1) (6)
where CK is the random scale of disturbance, is satisfied, then
L= E p(Tj)L;
j=1
where
L-n-1 m
x(`) .r a (i) X b +x(n) 2
J ( J) ~K - fE ( J) ,K ( )
i=0 i=0
is the disagreement, and p (r;) are weight coefficients which
define the value of each measurement and, accordingly, of
each of equation of system (8).
The necessary condition of the minimum of function L is the
equality to zero of its first-order partial derivatives according to
aiK and biK. Having computed the partial derivatives and
equated them to zero, one obtains an already compatible
system of m + n + 1 linear algebraic equations for the defini-
tion of m + n + 1 coefficients :
s
aL = p(ij)LjaLj-0(i=0,1,...,n-1)
aaiK j=1 aaiK
s
ab - l p(ij)LjabJ=0(i=0,l,...,m)
1K j=1 ,K
(9)
Solving system (9) by known methods, one obtains the
values of aiK and biK.
In certain cases the process of control at intervals may be
approximated by a differential equation of the form
n~-1
x(n)+ Y_ aiK x(`)= coEK (t) (10)
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This coarser approximation will make it possible to reduce
computing time considerably by a reduction of the quantity of
searched coefficients; in the given case only the coefficients aiK
are desired.
In the given approximation the deviations of the values of
real coefficients biK and real disturbances f will be taken into
account in the system via the values of the coefficients aiK.
System (11) will be the initial algebraic system for definition of
the coefficients:
Y x(`)(r)aiK-(PEK(Tj)-x(n)(v) (j=1,2,...,S) (11)
For definition of the searched coefficients aiK by the method
of least squares, one minimizes the function
(PEK (t) _ Y biK fEi) (t)
S
L1= P(Tj)L; (12)
j=1
n-1
Lj= E x(`) (Tj) aiK+
i=o
n)(T) -(PEK(r )
Using the necessary condition of the existence of a minimum
of function (12) for the definition of n, coefficients aiK (i = 0,
1, ..., n - 1), one obtains a system of n algebraic equations:
aL1 s a-j _
aaiK j~1P(Tj)L1Oaig 0 (i=0, 1;...,n-1) (13)
All the above discussion and the operations were performed
on the assumption that the values of the control variable
and the necessary quantity of derivatives at the moments
of time of interest are available. In practice, however, one is
usually limited to second-order derivatives.
In a number of cases real high-order systems may be
approximated by second-order differential equations, preserving
the description of their main dynamic properties. But even in
the case of more complex high-order systems it is possible to
suggest a number of algorithms for defining the searched
coefficients, given the existence of a limited quantity of derivat-
ives, some of which are as follows:
(a) Derivatives of higher orders of the control variable can
be calculated with the assistance of a digital computer on the
basis of the Lagrange and Newton interpolation formulae or
according to the formulae of quadratic interpolation (method
of least squares).
(b) If one integrates each term of eqns (5) and (10) n - q
times, where q is the order of the senior derivative of the control
variable, which one can measure in a system with the requisite
accuracy, then, taking the limits of integration tK, t; (j = 1,
2, ..., S), one obtains the integral forms of eqns (8) and (11)
respectively. If reference values are given to the magnitudes
x (n-1) (tK) x (n-2) (tK) x (n-q+l) (tK) in these equations,
then for /defining the coefficients aiK (i = 0, 1, ..., n - 1) and
biK (i = 0, 1, ..., m) it is sufficient to measure the derivatives
to the qth order.
(c) Practically all existing controlled plants and control
systems can be described by a set of differential equations, each
of which characterizes one degree of freedom of movement and
therefore has an order no higher than second.
(d) Sometimes, to reduce the order of the derivatives required
for measurement, one may also take advantage of a number of
coarse assumptions in relation to the terms of eqns (5) and (10),
which contain derivatives of high orders.
For example, in these equations the values of the derivatives
X(n), x (n-1), x (n-q+l) can be assumed equal to the reference values.
(e) The coefficients of approximating eqns (5) and (10) can
be defined without any recourse to algebraic systems (8) and (11),
if one uses the following methods.
Let the composition of the control system include an analogue
simulator, on which is set up a differential equation of form (5)
or (10). In this simulator there is a controlling device, which
provides an opportunity to effect variation of coefficients aiK
and biK in a certain way.
The control system memorizes the curve of the real process
in the interval (tK, tK+1 - At), and selection of the coefficients
aiK and biK is performed on the simulator in such a way as to
bring together in a certain sense the real process and the solution
of the equation set up on the simulator.
When the quantitative value of the proximity evaluation
reaches the predetermined value, the magnitudes of coefficients
aiK and biK, are fixed and extracted for subsequent employment
in the self-adjusting control system. Obviously the simulator
operation time scale must be many times less than the real time
scale of the system. Only under this condition can the requisite
high speed of self-adjustment be achieved. Practically any time
scale may be realized with the assistance of analogue computing
techniques.
Automatic Synthesis of Controller Parameters
For the operation of the majority of self-adjusting systems,
the system operation quality criterion is. set in advance. For
systems constructed on the basis of the proposed principle, it
is generally expedient to use as the criterion the expression
n-1 m
E 2 E 2
M= (aiK -aiK) + Y (biK - bix)
i=o i=o
This criterion generalizes both the methods of approximation
of the real control process expounded above.
To simplify subsequent operations, the following notations
are introduced.
(14)
b?K=a,,K; b1K=an+1,K,...,bmK=am+n,K
Expression (14) can then be rewritten in the form
M= (aiK - a E iK)2
i=o
_ Jn+m for (5)
no n -1 for (10)
(15)
On each interval (tK,.tK+1) the adjustable parameters are so
selected as to bring expression (15) to the minimum. The ideal,
i.e., most favourable, case would be one when M would reach
zero as the result of selection of the adjustable parameters. This
is not always possible, however. In the first place, not all the
coefficients aiK (i = 0, 1, ..., n?) are controllable. Second, in
multi-loop non-autonomous systems even the values of the
controllable coefficients cannot all be tuned up to the reference
values simultaneously, since the relationship of the coefficients
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ai to the adjustable parameters, although usually linear, is
nevertheless arbitrary in relation to the quantity of adjustable
parameters, the sign and the coefficients with which these
parameters enter into expressions for ai.
The second difficulty may be avoided by means of successful
selection of the reference system or by complete disconnection of
the loops (channels) of control of the main variables, i.e., by
satisfying the conditions of autonomy.
It is assumed that all the coefficients ai (i = 0, 1, ..., n?) are
controllable (in practice the values of uncontrollable coefficients
may be reckoned to be reference values). Then, for the coeffici-
ents ai one may write
ai= ai (Kl, K2, ..., Ks,; T1, T2, ..., T,; 11, 12, ..., 1r)
(i = 0, 1, ..., no)
where K1, K2, ..., K, are the gains of the controlled plant;
Tl, T2i ..., Tq are the time constants of the controlled plant and
the controller, and 11, l2, ..., 1, are the gains of the controller
(adjustable parameters).
Since the coefficients ai usually depend on the adjustable
parameters linearly, one may write
r
ai Y_ ?ijlj+Vi (i=0,1,...,no) (16)
j=1
1tij=,Uj(K1,K2i...,Kp; T1,T2,...,Tq);
vi = Vi (K1, ..., Kp; T1, T2, ..., Tq)
Using the necessary condition for the existence of a minimum
of function M, one obtains the following algebraic system for
determination of the setting values 11, 12, ..., 1,
no
E aaiK (ll, l2
Y_ , ..., lr)
1aiK(11,12, lr)
..., -aix] alp =0
i=o
(j =1, 2, ..., S) (17)
It is assumed that when the system is in operation, the
adjustable parameter values only change in accordance with
their computed values, i.e., at any moment of time one knows
the magnitudes of ll, 12, ...,1,. Then, for the interval (tK, tK_1)
until the moment of correction of the adjustable parameters in
accordance with expression (16), one can write:
Block-circuit with a Self-adjusting System using a Digital Computer
The duration of the intervals of constancy of the coefficients
of reference eqn (3), when 'a digital computer is used in the
control system, must satisfy correlation
tK+1-tK=T1+T2+T3+At
(20)
where Tl = Ar (S - 1) is the time required to carry out measure-
ments; T2 = N/n? is the time required for the computations;
T3 is the time of actuator generation; 0 < At < tK+1 - tx;
Ar = rj+1 - rj is the period of measurements (j = 1, 2, ..., S);
no is the computer speed of action, and N is the number of
operations required to define coefficients ljK (j = 1, 2, ..., r).
It is obvious that to ensure better operation of the self-
adjusting system, it is necessary to reduce as much as possible
the magnitude T = Tl + T2 + T3.
Now the opportunities for reducing .the time T3 are dealt
with. This question is directly linked with the choice of the
actuator. Electromechanical servosystems with a considerable
time constant are usually employed as actuators at the present
time. But it turns out that it is possible to suggest a number of
purely circuit variants of the change of the transfer functions or
of gains of the correcting devices (regulators) of the system.
These inertia-less actuators are termed `static'. It is particularly
advantageous to produce static actuators with the aid of non-
linear resistors (varistors), valves with variable gains (varimu),
electronic multipliers, etc.
Consider, for example, one of the variants of a static
actuator based on an electronic multiplier. Let the made of
control have the form
r
Y= Y, 1.xW
and let the jth adjustable parameter have the value lj? at moment
t?j= l (start of operation of the system). While the system
operates in accordance with the signals of the computer, the
value lj is constantly being corrected.
Thus, at the end of the interval (tK, tK+1) one has
ljK=.ljo+dljK
r
Y= Y loxcj)+ E dljKxu) (21)
j=1 j=1
Obviously each addend in the right-hand side of expression
(21) can be instrumented with the aid of the circuit in Figure 1,
where EM is the electronic multiplier, and AD the adder.
The following are self-adjusting system computer operating
algorithms: when the real process is approximated by
differential eqns (5), the algebraic systems (9), (18), and (19);
when the real process is approximated by differential eqns (5),
the algebraic systems (13), (18), and (19).
It is obvious that in the general case it is more convenient
to solve the problem of self-adjustment according to the proposed
principle with the aid of a high-speed digital computer. It can
be specialized for solving systems of algebraic equations.
Figure 2 shows the block diagram of a self-adjusting system
with a digital computer.
r
aiK- Y_ 1tijK 1j, K-1+ViK
j=1
From system (18) one may determine the magnitudes of
MijK and viK (i = 0, 1, ..., n?; j = 1, 2, ..., r) since the values
of aiK (i = 0, 1, ..., n?) and 1j, K_1 (j = 1, 2, ..., r) are known.
Taking into account eqn (16), after substitution of the
values of MijK and viK the algebraic system (17) for defining
14K, 12K, ..., l,K takes the form
no
E
MijK 1jK+ViK -aiK 1tijK=0
i=0 j=1
(j=1,2,..,r)
(18)
(19)
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In the preceding sections the proposed principle for creating
a self-adjusting control system for non-stationary objects was
expounded in general form. In practice, one may naturally
encounter cases when the given principle can be used in more
simplified variants. Several such opportunities are considered.
(1) Obviously, the entire theory expounded above can be
applied fully to stationary and quasi-stationary systems, which
are particular instances of non-stationary systems. In this case
the durations of the intervals of constancy of the coefficients
(1K, tx+1) equal, for stationary systems
K=0, tK+1-tK=tl-to=To-to (22)
for quasi-stationary systems
tK+ 1 - tK ~!Otp (23)
where Otp is the control time (duration of the transient process).
As can be seen from relations (22) and (23), in stationary and
quasi-stationary systems one is less rigidly confined to the time
of analysis of the real process and synthesis of controller para-
meters. It is therefore possible to define coefficients aix and bix
more accurately and to use criteria which reduce the self-
adjustment process speed, but make it possible to increase
the accuracy of operation of the system. Among such criteria
one may cite, in particular, the integral criteria for the evaluation
of the quality of a transient process3.
For stationary and quasi-stationary systems the problem of
self-adjustment in accordance with the principle proposed above
may be solved as a problem of the change in position of the
roots of the transfer function of a closed system, i.e., the self-
adjustment problem may be solved in accordance with the
requirements of the root-locus method, which is extensively
employed in automatic control theory. A feature of the use of the
proposition of the root-locus method in accordance with the
principle under consideration is that the zeros and poles defined
by the coefficients aix and biK are fictions since they not only
depend on the parameters of the controlled plant and controller,
but also depend on real disturbances as well.
(2) In practice, one may encounter cases when a controller
is required to ensure only the stability of a system in the course
of operation. As is known, the stability of linear stationary
systems is determined by the coefficients of the characteristic
equation. This proposition is also valid for certain quasi-
stationary systems (method of frozen coefficients).
Therefore to solve the problem posed (the provision of
stability), the control system must define the actual values of
the coefficients of the left-hand side of the differential equation
of the system and must set on the controller such gains
factors as will satisfy the conditions of stability, for example the
conditions of the Hurwitzian algebraic criterion. On the assump-
tion that disturbance.f is constant in the interval (tx, tx+1) the
coefficients of the characteristic equation of the system on this
interval are determined in the following way.
The differential equation of the system for t e (tx, tx+1) is
written in the form
x(n)+ E aiK x(i)=F
K
i=0
where FK is in the general case the unknown right-hand side,
constant for t e (1K, tx+1). The algebraic system for determining
the described coefficients will then be written thus :
n-1
x(n)(rj)+ E x(`)(rj)aix=FK (j = 1, 2, ..., S) (24)
i=O
Since Fx is unknown, but is constant in the interval (tx, tx+1) it
is eliminated with the assistance of one of the equations of
system (24). For this purpose one uses the equation
n~-1
x(n)(til)+ Lr xW (x1) ai
=FK(1 to > 0 is determined, as is well known, by the
history x (to + 0) (- h < 0 < 0) of this motion. The initial
function x (to + 0) (- h < 0 < 0) will therefore be called the
initial disturbances (with t = to). It is also convenient to con-
sider, as quantities describing the state of system (1) at instants
t > to, and determining its future motion when r > t, sections
of the trajectories x (t + 0) (- It G 0 < 0). It is therefore
suitable to form the control signal $ (t) at each instant t on the
basis of information on the whole of the realized trajectory
x (t + 0) with - h < 0 < 0. In other words, analytic con-
struction of the regulator24 means finding ~ in the form of a some
functional ~ (t) = $ [t, x (t + 0)], determined on the curves
x (t + 0) = {xi (t + 0), - h < 0 < 0, i = 1, ..., n}. In future
it will be assumed that the argument 0 varies within the limits
- h < 0 < 0. The continuous functions x (0) or x (t + 0) of
the argument 0 are assumed to be elements of a certain space X
with a matrix
IIx (0)II =max (xi (0)+ ... +xn (0))
Also used is the notation
IIx(o)II =(x1 (0) + ... +xn (0)) ,
Ilx(t)II =(xl (t) + ... +x? (t))f
Three problems are considered:
Problem 1. Find a control signal _ ~? t, x (0) such that the
motion x = 0 in a closed system (1) (that is, with ~ (t) = ~? (t, x
(t + 0)) is asymptotically stable29 with respect to the disturbances
x? (to + 0) (t? > 0) from a region
IIx?(0)II 0 and x? (to + 0) out of (2) there
holds a minimum
J [to, x?, ~?] = min J [to, x?, ]
(3)
J [to, x?, ~] =J co [t, x (t; to, x?, ), ~ (t)] dt (4)
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529/2
where w is a given non-negative function, x (t, to, x", ~) is the
trajectory of (1) with initial conditions to and x? (to + 8) and
a selected law of control ~ (t) = ~ [t, x, (t + 0)]. The control
signal ~ can be constrained by a supplementary restriction
E - (for instance, f ~ I < 1).
Problem 2. Find a control signal _ ~? t, x (0) assuring
a minimum of
lo
+ 0 [x (T, to, x?, )] (6)
and T < oo is a given instant of time, while lx? (to + 0)ff < Go.
Problem 3. Find a control signal _ ~? [t, x (0)] assuring
minimum of
J~ [to, x?, ~?] = min J., [to, x?, ] (7)
where lfxo (to + 0)ff < G? and
J~ [to, x?, c] = lim TJTt when T-* oo
0
(8)
In Problems 2 and 3, as in 1, it is assumed that the initial
conditions x? and trajectories x (t, to, x?, ~0) do not go beyond
certain previously fixed regions.
The sufficient conditions of optimality of the control signal ~?
will be formulated for Problems 1 and 2.
Theorem 1. Let it be possible to indicate functionals
v [t, x (0)] and ~? [t, x (0)], defined and satisfying in some
region fix (0)11 < G the following conditions:
(1) The functional v is positive definite with respect to
11X(0)11-
(2) The functional v admits an upper limit with respect
to ffx (e)lf.
(3) The following inequality is satisfied:
in f [v [t, x (0)] when if x (0) li = G,
f f x (0) = G] >- sup [v [t, x (0)] when Ii x (6)11 < G?]
(4) Along trajectories of (1)29 the derivative (dv/dt), of the
functional v satisfies the condition
()o + w [t, x (t), ?]=min[() + [t,x (t), = 0 (9)
4E
in the region f f x (t + 0)f1 < G, and is negative definite with
respect to if x (t)f I in this region.
Then ~? [t, x (t + 0)] is the optimal control signal for
Problem 1, and the following equality is valid:
v[to,x0(to+0)]=J[to,x?(to+0),~?] (10)
Note. Properties (1) and (2) generalize in a natural way the
corresponding properties of Liapunov's functions27 that is (1)
means that there exists a function w (r) > 0 with r 0, such
that v [t, ,x (0)] > w (ix (0)11) with fix (0)11 = JJx (0)11, and (2)
means that there exists a function W (r) satisfying the conditions
W (0) = 0, v [t, x (0)] < W (fix (0)11). If in Problem 1 the region
Go encompasses any possible large initial disturbances xo
(the problem of optimal stabilization as a whole), the region G
must coincide with the whole of the space X, and (1) is replaced
by the condition
limv[t,x(0)]=0o when 11x(0)11-->oo, lix(0)=IIx(0)11 (il)
uniformly with respect to t.
The demonstration of Theorem I is made by reasoning
typical for the theory of stability of motion29, but taking into
account the principles of dynamic programming4.
The sufficient criterion of optimality for Problem 2 is for-
muiated as follows:
Theorem 2. Let there exist for every if x? (to + 0) < G? and
to e [0, T) an admissible control signal (t), that is, a control
signal for which the trajectory x (t, to, x0, ~) may be prolonged
in some finite region G until the instant t = T, and therefore the
integral (6) is finite. If one can find in the region G functionals v
[t, x (0)] and 1=? [t, x (0)] satisfying conditions (9), and
v [T, x (0)] = / [x (0)] (12)
then ~? is the optimal control signal for Problem 2, and the
following equality is valid:
v[to,x?(to+0)]=JT[t0,x?(to+0),~0] (13)
The solution of Problem 3 can be obtained by passage to the
limit from the solution of the problem when T-* oo.
Note. If the load (t) is random or the system is subject to
random disturbance, Problems 1 to 3 are modified as follows:
integrals (4), (6) and (8) are replaced by their mathematical
expectations (the conditional mathematical expectations for
the appropriate initial conditions to, x?, 72?), and in Problem 1
the requirement of stability is replaced by the requirement of
stochastic stability30. In this case seek the control signal ~? in
the form of a functional ~? [t, x (t + 0), (t + r)], where
- h < 0 < 0 and - h* < z < 0, while h* = 0 is the value
of the maximal after-action for the probability process 77 (t)
(if n (t) is a Markov process, then h* = 0). The criteria of
optimality given above preserve their form, with the modifica-
tion that v must here also be a functional v [t, x (0), (r)],
and the derivative (dv/dt)4 is replaced by its average value30
(dM{v}/dt)i;.
Conditions (9) reduce to partial derivative equations of
a special kind. The solution of these equations in the general
case is cumbersome; it is possible, however, to indicate a number
of cases when an explicit form can be found for the optimal
control signal, or when a numerical procedure for its deter-
mination can be indicated.
The results of applying the proposed criteria to systems
described by equations of actual form will be illustrated.
Let the transient process be described by the linear differen-
tial equations
dx. n
-
dt Y aii(t)xi(t)+ Y cii(t)xi(t-h)+bi~+ai11(t) (14)
i=1 j=1
where ail, ci;, ai and bi are known functions of time or constants.
First assume that 77 (t) = 0, and then consider Problem 1 for
JT [to, x?, ~?] = min JT [to, x?, S] (0< to < T) (5)
~E_
T
w [t, to, x?, ), (t)] dt
JT [to, x?, ] = f
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f tao J= [~ x?(t)+.1 2(t) dt,
o i=1
J>0-const (15)
any initial disturbances x? (to + 0) are admissible.
Here the functional v from Theorem 1 must be chosen in
the form
v [t, x (0)] = Y [d i j (t) xi (0) x j (0)
i,j=1
+2xi(0)J ? /jij(t,0)xj(0)dO
+ J O j ? Yij (t, 0, i) xi (0) xj (r) dOdi]
h
(16)
which generalizes in a natural way the Liapunov function
widely used in stability theory, as a quadratic form. If for every
initial condition x?, to there exists an admissible control signal
$ (t), that is, a control signal (t) for which integral (15) converges
uniformly with respect to to, then there exists a functional v (16)
satisfying the conditions of Theorem 1. From this it is directly
concluded that in this case there exists an optimal control
signal S? having the form
? [t, x(t+9)]
(17)
="[?i(t)xi(t)+J hvi(t,9)xi(t+9)d9I
The optimal regulator 4? in system (14) with condition of
minimum (15) is seen to be the regulator B, which applies to the
input of the controlled plant A at every instant t a quantity
~? (17), worked out on the basis of a measurement of the
error x at the given instant of time t and at previous instants
t - h < r < t, while the results of measurement of the previous
errors x (x) = x (t + 0) must be processed in the integrators
fvi (t, 0) xj (t + 0) d& The control signal S? depends linearly
onx(t+z0)(-h E Iki
k=1 i=1
is the full amount of coordinates of the system (including the
additional ones),
(2)
rm : m Pk;
Ak (S) Rki (S) Fi (S) + I Ckij F* (z) Cki J (S)
1=1 i=1 j=1
akj (s) = Wki (s) , akk (S) = Wkk (S) - I.; aK'j (s) = bK'i j (s)
and are numbered in accordance with (2).
System (3) formally contains N equations with 2 Nunknowns.
xj (s) and xj* (z). As in ref. 17, the terms containing transforms
of the coordinates will be transferred to the right-hand side.
The resultant system will be solved relative to the arbitrary
coordinate xj (s). This gives:
Xi (S) ,Od(s)) (4)
-
aii (s), ..., a1N(s)I
aN1 (s), ..., aNN (s)I
(5)
is the common determinant of a purely continuous system.
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which column of the common determinant 4 is subject to
substitution, while the lower index indicates substitution by.
coefficients for particular variables. Thus, 41 x*j means that the
kth column of the common determinant d is to be replaced
by coefficients at the jth discrete coordinate.
System (10) can be solved, relative to the coordinates of
interest, by ordinary, algebraic methods. Sampled-data systems
with various types of link will now be considered.
The first of the determinants entering into (7) will be denoted
by 4A2 (s), and the remainder by J!,,; (s). Bearing in mind the
notation adopted:
x; (s) _ (S) xk (Z) Cod (s) (8)
Subjecting.(8) to a z transform and cancelling out like terms, the
following relationn is obtained:
) 1=1 xk (z) (A;*k)*()
xT (
Z) Ci +(A. J*)*(z)] _ (4J)* (Z
e(k j) (9)
Sampled-data Systems with Continuous Compounding Links
An automatic control system with one pulse element, which
can be described by a system of three linear equations with
constant coefficients, is studied. The block diagram of the system
is given in Figure 2, which also shows the transfer functions
of both the main loop and the additional links.
The initial system of equations is:
cp(s)+0-W,,W A(s)+0=(s)A(s)
(11)
Thus the initial system (3) can immediately be raised to a
- W. (s) W"' (s) cp (s) + v (s) - W?a (s) it (s) + 0
full system by equations of type (9). The full system of equations
of a multiloop sampled-data system has the form:
= W?a (s) A (s) + W"'E (s) tli (s)
N N
Y_ a j (s) xj (s) + E a ij (s) xi (z) = A, (s)
j=.1 J=1
N N
aNj (s) xj (s) + Y aNj (s) x j (z) = AN (s)
j=1
j=1 [1+(*z)]xz)
+i ~4drt')* (z) x; (z)= (Ai)* (z);
. . . . . . . .
(10)
T
N ee')*(z)x~ (Z)+L1+' 44)(z)]xN(z)(e)*(z);
(j, N)
When writing the determinants forming part of (10), the
following symbolization is accepted. The upper index shows
- W" (s) cp (s) + 0 + ? (s) - W(s) v* (z)
= W?2 (s) ). (s) + 6P?, (s) ,i (s)
In accordance with ' the method expounded above, this
system is made into a full one by the deficient equation:
2
[1+(J)(z)]v*(z)=()
* (z) (12)
Henceforw ard, only programme and servosystems will be
considered; hence, in (11), A (s) = 0.
From (11) and (12) one can easily find an expression for the
controlled coordinate in which one is interested.
cp (s) = K2 (s) f (s)
+ K3 (S) {[(K5 +K2K6) 01* (Z)} (13)
1-K* (z)-K,K6 (z)
~N`
a11(s),...,alj-1(S);Al(s)-.L alj(s)xj(z);alj+1(s),...,a1N(S)
j=1
N
aNl (S), ..., aNj - 1 (s); AN (S) - S aNj (s) xj (z); aNj+ 1 (S), ..., aNN (S)
j=1
N
- E xk (Z)
a11(s),...,a1j+1(s);A,(s);a1j+1(s),...,a1N(s)
aN1 (S), ..., aNj+ l (s); AN (S); aNj+ i (S), ..., aNN (S)
all (s),..., a l j -1 (s);a l k (s); a l j+ l (s), ... , a NN (S)
aN 1 (S), .... aNj - l (s); aNk (s); aNj + 1 (S), .... aNN (S)
5312
(7)
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K2 (s) = W(s) W Y, (s) K3 (s) = Wuv (s) W" (s) .
1- W~? (s) W?~ (s)' 1- Wv? (s) W, (s)'
K s (s) = WvE (s) + W?N, (s) W?P (s)
K6 (s) = W,,, (s) i1'n (s) + Wv? (s) W,, (s)
K, (s) = Wv? (s) W?v (s)
Conditions of Absolute Invariance. The condition of absolute
invariance for servo and programme systems is:
tp (s) = K2 (s) 0 (s)
K3 (s)
+ {[(K5 + K2K6) J]* (z)} = 0 (s)
or 1-K; (z)-K3K6 (z)
-e(s)=K2' (s)0 (s) (14a)
+ K3 (s) {[(Ks +K2K6) ~]* (z)} =0
1-K; (z)-K3K6 (z) (14b)
where E (s) _ V (s) - cp (s) is the system error of the system;
K'2 (s) = K2 (S) - 1.
The basic differences between the conditions of invariance
for continuous and sampled-data systems is emphasized. While
in continuous systems the conditions of absolute invariance do
not depend on the form of V, and are determined only by the
parameters of the components of the system, in the sampled-data
system under consideration, these conditions (14) essentially
depend on the form of the input signal v.
It can be shown that the condition of absolute invariance
physically signifies the equality to zero of the sum of the indivi-
dual components of the coordinate e produced both as a result
of the direct effect V upon the system, and also on account of the
effect via the additional (compounding) links.
Invariance Conditions for Discrete Moments of Time. The
invariance conditions (14) were obtained from the requirement
of the equality to zero of coordinate e at any moments of time.
One may pose a less rigid requirement-the equality to zero of e
at the sampling instants, i.e.,
e [nT] =0 (15)
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By equating the right-hand side of (16) to zero, the following
invariance conditions are obtained for discrete moments of time:
1-K*(z)+[K3(Ks+K6)]*(z)=0 (17)
K2
The conditions of absolute invariance for a similar con-
tinuous system (i.e., a system having the same structure) can be
given in the form:
1-K7 (s)+K3 (s) [K5 (s)+K6 (s)] =0 (18)
K2 (s)
If (18) is subjected to a z transform, eqn (17) is obtained,
i.e., the introduction of a pulse element into an absolutely invariant
continuous system does not impair the conditions of invariance
for discrete moments of time for the so-called `fictitious coordinate'
E (s) = I (s) 8(s)
K2 (s)
As shown by Krementulo10 from the equality to zero of
E. [nT], there still does not follow the equality to zero of e [nT].
The additional conditions will be given, under which e [nT] = 0,
and does not depend on the form of V. (14b) is subjected to a
z transform, and then 1 - K* (z), found from (17), is substituted:
K3 (z)
{K3 (K +K6)}*
K' J (z)+K3K6 (z)
2
{K50* (z)+[(KZ + 1) K60]* (z)}
Sampled-data Systems with Discrete Compounding Links
A brief examination will be made of the properties of a
typical sampled-data servo-system, the block diagram of
which is given in Figure 3. The expression of the system
error s is:
[(1(5?1(6 + K6) K2U] (z)=(Ks+K6+K6 (z) K' (z)
K2 K2
(20)
Condition (20) is satisfied if [(K5 + K6)/K2'] + K6 contains
proportional components or components with a pure time lag.
From (20) and (17) can be found the transfer functions of
continuous compounding links.
The conditions under which (15) is satisfied are called
`conditions of invariance for discrete moments of time'. If (14)
is subjected to a z transform, then the problem is solved at first
sight. However, it is easy to show that the invariance conditions
for discrete moments of time as well, will depend upon V.
An attempt is made to obtain the conditions, independent
of o. Both parts of (14b) are multiplied by
K. (s)+K2 (s) K6 (s)
K' (s)
and then subjected to a z transform.
-(Kl E)(z)=(l0)*(z)+(K3ll*(Z) (10)*(Z)
2 K/2 1-K7(z)-K3K6(Z)
(16)
is obtained, where l (s) = K5 (s) + K2 (s) K6 (s).
E*(Z)-1-W,,,~(Z)i1'*,(Z)(Z) (21)
1 + W E (z) Wv? (z)
The condition of invariance at discrete moments of time is:
W*
Al (Z) _
W,*, (Z)
(19)
(22)
In the general case, W E (z) and W,u (z) are the ratio of
polynomials according to the positive powers of z, the power
of the numerator being less than that of the denominator.
Since Wt*,o (z) must be inverse to WW, (z), then it cannot
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be physically realized (advancing components are required
for this).
It is important to note that the introduction of the link
W,,, (z) and the satisfaction of the invariance condition (22)
do not alter the characteristic equation of the system:
Ka (z)P*(z)+Ki (z)Q*(z)=0;
(!~ W(Z); P* (z)= * /
1 (Z)vg Q* (Z)-Ww~y (Z)
and therefore do not influence the stability of the system.
Examples were given by Kuntsevich12 to show that even in
those cases when W,*, (z), obtained from condition (22) cannot
be realized, provided it is selected in a particular way, it is
possible to increase considerably the accuracy of a sampling
servosystem.
When for any reasons it is inconvenient or impossible to
introduce the compounding link W,*y, (z), one may introduce
into the system additional links, equivalent to the direct com-
pounding link W,*, (z). Eqn (21) can be brought to the form:
1.
E (z) _ 1 + W E (z) W* (z) ~* (z)
W?# WW? (z) [e* (z) + (P* (z)] (24)
1 + WV E (Z) WW' (z)
It is not difficult to see that (24) is met by the scheme
shown in Figure 3 (b).
If (22) is satisfied, then the condition of absolute invariance
has the form:
The signal of the compounding link v, (s) equals:
vl (S) W ? (s) + WUE (s) [ WE, 0* (z) - G* (z)] (28)
This signal can be realized with the aid of the scheme shown
in Figure 4 (b). In a similar continuous system, the compounding
link with respect to 7p, chosen from the conditions of absolute
invariance, equals:
W W o(s) W"I (s)+ W. (s)[WW'(s)-1] (29)
It can be seen that for both the sampled-data and the con-
tinuous system the compounding link has one and the same
structure and consists of identical components. The difference
lies in the fact that in an absolutely invariant sampled-data
system some of the components are connected up via additional
pulse elements operating synchronously and in phase with the
main one. What has already been said also holds in the case
when real pulse elements are used.
(23)
Systems without Compounding Links
Today a large number of extremal sampled-data systems of
various types are known, which have been studied by many
scientists. But certain specific features of these systems remain
unexplained. Of the known extremal sampled-data systems an
analysis will be made on the basis of full and precise equations
of dynamics of only one system which, as was shown in (29),
provides the best tracking quality with continuous drift of the
extremum, and whose properties are at the same time closest
to those of a hypothetical system measuring the position of the
extremum point without any errors.
As in most works, the controlled plant with extremal
characteristics will be considered to be one which consists of a
linear inertial component and an inertia-less component with
extremal characteristics.
The equation of the non-linear component, taking into
account the action of two kinds of disturbances (or two com-
ponents of one and the same disturbance), which displace the
extremum point, will be written in the form:
ui (/s) _ W,, /(s)
W * (z) WWU (z)
The latter equality can be satisfied only in some particular
cases, and, as shown by Krementulo", requires the inclusion
of advancing components if V [0] = 0.
Sampled-data Systems With Pulse-continuous Compounding Links
In this section a servosystem will be used as an example to
show that when pulse-continuous links are used it is in principle
possible to achieve absolute invariance in a combined sampled-
data system.
Assume that the block diagram is predetermined, i. e.,
W, (s), Wq,,, (s) and WE, (s) are known. A compounding link
with respect to the input signal 0 W,,y, (s) is introduced to
improve the dynamic properties. The transfer function of this
link has to be determined.
The expression for the system erroris:
E(s)_[Wu,'(S) W"(s)-110 (s)
W?E(s)W" (s) [O*
+ (z)+WN,PW,?WEpp*(z)] (26)
1+W?EW"w p(z)
Having equated E (s) to zero the condition of invariance of
the system is obtained from which the transfer function of the
compounding link can be determined:
W?, (s) K? (s) + 0 (s)) [WW, (z) - (z)] (27)
(25)
9= -a3 (x+~)Z+~ (30)
wherecp is the index of the extremum, and V, 2 are disturbances
of an arbitrary kind, inaccessible for direct measurement by
virtue of the conditions of the problem. Let the remaining
equations of the extremal system (see Figure 5) in the absence
of the components shown in Figure 5 by the dotted line, be:*.
X (s) = WXM (s) M (s)
(31)
M=u+m
(31 a)
* Since the system under review is non-linear, then strictly speak-
ing, neither the ordinary nor the discrete Laplace transform is applic-
able to it. Therefore the final results will be obtained with the aid of
a set of non-linear difference equations. To simplify things, the
Laplace transform will only be used in application to the linear
components.
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in (s) = Wo (s) m* (z) (32)
where z
m* (z) = aM
l+z
? (s) = J4' (s) u * (z)
Yn=dcon -1 (-1)n
u * (z) = W?y (z) YA(z)
(32 a)
(33)
(34)
(35)
system parameters and the speed of variation of disturbances
cpn, 2,2 the stability of the system is impaired, whereas analysis
of the linearized equation obtained from (41), disregarding the
non-linear terms (as done by Chang25, Van-Neis26 and Ivakh-
nenko27) does not permit one to detect this phenomenon.
Therefore the feasibility of constructing an adaptive system, the
error of which would be invariant in relation to V., 2,,, acquires
particular interest, since it involves not only the improvement of
the quality of the system, but also the increasing of its stability
margin.
Here (31) is the equation of the linear part of the plant,
(32) the equation of the modulation circuit, (34) the equation
of a controller with synchronous detector, (35) the equation
of the correcting elements, (33) the equation of the servo-
motor and x, ,u, cp, u and y the controlled coordinates.
Henceforward it is taken that the dynamic properties of the
plant and the slope a3 of the extremal characteristic are constant
,or quasi-constant.
The error of the system is denoted as :
e=p'+A
and also the notations are introduced
(36)
x*(Z)=h'*(z)+n1'*(Z)
(37)
?'*(z)=WxMW? (z)u*(z)
(37a)
m'* (z) = Wo W?u (z) m* (z)
(37b)
On the basis of (37b) and (32, 32a), the modulating effect
m',n, scaled to the input of the non-linear element, can be
represented in the form
m?= am COS nn =aM (-1)" (38)
where am is determined from the particular solution of the
difference equation
aM(-1)"=aM WXM Wo(E)(-1)" (39)
which is obtained following the replacement of (32) by the
difference equation corresponding to it.
Solving jointly (30), (36), (37) and (38) gives
yn=-2aMa3(en+en -1)+42n-1(-1)"-a3(en-en-1)(-1)n
(40)
From (40) it can be seen that the signal on the output of the
component (34), apart from the useful component proportional
to the error contains further additional terms, one of which
d2,n_1 (- 1)n reflects the influence of the disturbance 2,n, and
the third term shows that the measurement of the position of the
system relative to the extremum point is not ideal.
Further replacing (35) and (33) by their corresponding
difference equations, and solving then jointly with (40) and
(37 a), the equation of the dynamics of the system is obtained
in the form of a non-linear difference equation with time-
varying coefficients
[2aMa3W(E)(E+1.)+E]en -c3W(E)[en+1-en)cosrcn]
= 0n+ 1 - W (E) [d An cos 7rn]
where W (E) = WXM W,," (E) W?y (E) (41)
As was shown by Kuntsevichss, 31the non-linear eqn (41)
has the peculiarity that at a particular correlation between the
Invariance of Extremal Control Systems with Indirect Com-
pounding Links
Since, by virtue of the conditions of the problem, the
possibility of direct measurement of the signals V and A is
excluded, the possibility will be considered of using indirect
compounding links with respect to 1p and A similar to those
considered above.
Consideration will first be given to the possibility of attaining
invariance of system error at discrete moments of time, relative
to 1p,,, *.
From (41), (36), (42), and (42a) and also from Figure 5, it
(z)=e*(z)-?'*(z) (42)
or Ji* (z) = e* (z) - WXM ?* (z) (42a)
For the construction of the correcting link with respect to
y, in accordance with (42 a), the variable a',, can be obtained
with the aid of a model of the linear part of the controlled plant
(see Figure 5*). A signal proportional to em (or, more strictly,
containing en) can be obtained on the output of an additional
synchronous detector (see the part of Figure 5 outlined by
broken line), the equation of which is:
Y. = con (1)n
(43)
Solving (30), (36) and (43) jointly, gives
Yn=-2aMa3en-93 (en+aL1)(-1)"+A?(-1)"
(44)
For filtration of the parasitic quasi-periodic terms of signal
(44) on the output of the detector in the network in Figure 5,
a low-frequency filter is provided.
Taking this into account, the signal on the output of the
additional control loop is written in the form
W. D(E)Wn
D (E) = 2 aMa3 WW (E) WK (E)
(45)
Omitting the intermediate operations, the equation of the
dynamics of the system in Figure 5, with an additional control
loop, is obtained, on the basis of the equations cited above and
also eqn (45), in the form
[2 aM o3 W (E) (E + 1) +E] en - a3 W (E) [(en+ 1- en) cos 7,17]
=[1-2aMa3WxMW?"(E)WW(E)WK(E)]Jn+1
- W (E) 42n cos 1zn (46)
By equating to zero the operator comultiplier for y, in the
right-hand side of (46), an expression is obtained of the impulse
* It is noted that in contrast to ordinary servosystems, in which
the input signal may also contain a noise which has to be suppressed
as effectively as possible, the task of an extremal system in all cases is
complete performance of signal V.
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transfer function WK (E), which ensures the invariance of the
system from 'n at discrete moments of time
WK (E) = 1' 1 (47)
2 aMa3 WW (E) W (E)
From (46) it can be seen that the satisfaction of the con-
ditions of invariance (47), and the presence of the filter in the
compounding-link network (as distinct from the filter in the
main network of the controller), do not alter the form and
coefficients of the left-hand side of the equation of the dynamics
of the system, i. e., do not directly influence the stability of the
system. .
When the required transfer function W;* (z) is physically
unrealizable, then, as for ordinary servosystems, a considerable
improvement of accuracy (increasing of the degree of astatism)
can be achieved by appropriate selection of the transfer function
WK* (z). An example is given in the Appendix of the method
of selection of the coefficients of the transfer function WK* (z).
- In deriving the conditions of invariance (47), the quasi-
periodic non-linear terms in (44) were disregarded in order to
simplify the investigation. As follows from the example in the
Appendix (see also Figure 6), the influence of these terms is in
fact small. *
A brief examination will now be made of the possibility
of minimization (or complete elimination) of the system error
due to 2. From the equation of the system dynamics (46) and
(40), it follows that for the predetermined structure the possibi-
lity of constructing a correcting link with respect to A (t) in a
similar way as with respect to V, without constructing an analogue
of the non-linear component, is excluded. By virtue of this, with
the scheme structure adopted, only methods of minimizing the
influenceof A (t) can be considered. One such method, based on
the selection of the corresponding function W,,*, (z) was con-
sidered by Chang25, Van-Neis26 and Ivakhnenko27. The results
obtained by Tou24 may also be used here.
Appendix
Example-In Figure 5 let
.
WxMF(s)= a1 '
als+1
to which there corresponds
11'x, W*?n (Z) = a1a2 (1-dl)z
.
d
kz - 1)h -
1)
* B1 z)
WnY (Z) BZ (z)
W?u (S) = a2
S
where Bi (z) and B2 (z) are polynomials from z, d1= e-T i 1 .
It will be taken that
N, (s) _
to which there corresponds
ww(z)=I-d2; (d2=e-TIt2)
z-d2
* The system in Figure 5 was checked experimentally on an elec-
tronic analogue by A. A. Tunik, and the check confirmed the effective-
ness of the introduction of indirect correction31
It is not difficult to see that in the given case the impulse
transfer function WIC (z), as determined from (47), which is
required for attainment of the conditions of invariance, is
physically unrealizable, and only the approximate satisfaction
of the conditions of invariance can be spoken of; by virtue of this,
WIi (z) will be sought in the form of the series
WK (z) - > C` (:ii:Z -11t
(48)
Denoting the left-hand side of equation (46) by L (E)e,, in order
to abbreviate the notation, one can write it for 4 2n = 0 for the
given example, bearing in mind (48), in the form:
L(E)en=EB2(E){-2aMala2a3(1-dl)(1-d2)
X [C1d0n+C2420n-1+ ... +CKAKVf-K+1]
+430n+420n[(1-dl)+(1-d2)]+A0n(1-dl)(1-d2)}
(49)
Provided 1
Cl =2 aMOC1a2a3
(50)
the error from the first difference 7N,, is eliminated, since, when
this is satisfied, the equation of the system adopts the form
L (E) en
EB2(E){-2aMala2a3(1-dl)(1-d2)[C2A2`Yn_l+ ...
+CKAK> ',-K+l]+A 3t /1n+(2-d1-d2)A2'n
(51)
Further taking
(2-d1-d2)
(52)
2 aMala2a3 (1-d1) (1-d2)
and bearing in mind that
4 awn-A~'Nn-1=4i+14' -1
(51) can be rewritten in the form . WY
L (E) en
=EB2(E){-2aMala2a3(l.-d1)(1-d2)[C3A3li _2+ ???
+CKAK0n-K+1] +43tIin-C2A3t//n-1 (53)
from which it will be seen that, irrespective of the coefficients
W y (z) the error is eliminated from the second difference ~,,.
Since further increasing of the degree of astatism on account
of the correcting link is impossible in the given example, Ci = 0
will be taken for i > 3.
. For quantitive evaluation of the quasi-periodic terms in (46),
which have not been taken into account, in Figure 6 the transient
in an extremal system is plotted, taking into account these
terms for y1,, = j9n, d A,, = 0 for eqn (46).
For the transfer function of the components cited in the
example under consideration and for W , z = 1, the precise
equation of the dynamics of the system has the form:
Aoen+3+Alen+2+A2en+1+A3en
=aE(1-d1)[e2 n+2-e2 n+1+d2(e2 n+1-e2n)](-1)n
+al (1-dl)(1-d2)[en+l+en+2aM](54)
where
Ao=]; A1=2aMaz(1-dl)=(1+dl+d2);
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A2=2aMaz(1-dl)(1-d2)+dl+d2+d1d2;
A3=-dld2-2aMaz(1-d1)d2; 01E=ala2a3
Here, for comparison, the transient processes in an extremal
system without correcting link with respect to V. have been
plotted, in which WxM (s) and W,,,, (s) are the same as given
above, and low-frequency filter with transfer function
1-e-sT 1
S r3S+1
is included into the main extremal-control network z transform
of We (s) is
where d3 = eTiTS.
W" (z)-1-d3
z-d3
Bearing this remark in mind, for the given case, the equation
of the dynamics (41) of the system adopts the form
A0'en+3+A1'e?+2+A2en+i+A3en
-ar(1I1''-d1)[en+2-en+1+d3(en+1-en)](-1)n
=43'Yn+I+[(i-d1)+(1-d3)] 1 2 II,n+I
+(i-d1)(1-d3)dWn+I
(55)
A0' =1; A1' =2aMa(1-d,)-(1+d1+d3);
A2=2aMaE(I.-dl)(1-d3)+d, +d3+d1d3;
A3=-2aMaz(1-dl)d3-dld3i a?=a1a2a3
As can be seen from the curves in Figure 6, an increase in 9
(the rate of drift of the extremum) leads to the loss of the
stability of the system (55). Thus the introduction of compound-
ing links with respect to V. not only improves the quality of
the system, but also preserves its stability, thus extending the
sphere of application of extremal systems to the case of high
extremum drift rates.
ScHIPANOV, G. V. Theory and method of design of automatic
controllers. Automat. Telemech., Moscow 1 (1939)
KULEBAKIN, V. S. The theory of invariance of regulating and
control systems. Automatic and Remote Control. p. 106. 1961.
London; Butterworths
PETROV, B. N. The invariance principle and the conditions for its
application during the calculation in the design of linear and
non-linear systems. Automatic and Remote Control. p. 117. 1961.
London; Butterworths
IVAKHENKO, O. G. Automatika (1961)
KosTYUK, O. M. Automatika 1 (1961)
BELYA, K. K. The invariance of the controlled magnitude of an
automatic device from certain of its parameters. Izv. Akad. Nauk
SSSR, Otdel Tekhn. Nauk, Energ. Automat. 6 (1961)
Invariance theory and its application in automatic devices. Trud.
Soveshch. Sostoyavshegosya v g. Kiev, 16-20 sent., 1958 (Proc. of
a meeting held in Kiev, Sept. 16-20, 1958), Moscow, 1959
TsYPKIN, YA. Z. Automatika 1 (1958)
Tou, J. Digital compensation. for control and simulation. Proc.
Inst. Radio Engrs, N.Y. Vol. 45, No. 9 (1957)
KREMENTULO, Yu. V. Automatika 1 (1962)
KREMENTULO, Yu. V. Automatika 2 (1960)
KUNTSEVICH, V. M. Automatika 1 (1962)
GRISHCHENKO, L. Z., and BOLDYREVA, D. F. The invariance of
automatic sampled-data control systems. Automatika 2 (1962)
STREITZ, V., and RuzHICHKA, I. The theory of autonomy and
invariance of multiparameter control systems with digital con-
trollers. Izv. Akad. Nauk SSSR, Otdel Tekhn. Nauk. Energ.
Automat. 5 (1961)
FEDOROV, S. M. Delay in the synthesis of servosystems with
digital computers. Izv. Akad. Nauk SSSR, Otdel Tekhn. Nauk
Energ. Automat. 4 (1961)
TsYPKIN, YA. Z. Teoriya Impulsnykh Sistem (Theory of sampled-
data systems) 1958. Moscow; Fizmatgiz
BURSHTEIN, I. M. Solving equations of multiloop sampled-data
systems. Automat. Telemech., Moscow 12 (1961)
RAGAZZINI, J. R., and FRANKLIN, G. F. Sampled-data Control
Systems. 1958. New York; McGraw-Hill
JURY, E. J. Sampled-data Control Systems. lliJ. New York;/
John Wiley. Jill. London; Chapman and Hall
LENDARis, G. G. and JURY, E. J. Input-output Relationships for
Multisampled-loop Systems Applications and Industry. Jan. 1960
Tou, J. A simplified technique for determination of output trans-
forms of multiloop multisampler variable-rate discrete-data sys-
tems. Proc. Inst. Radio Engrs, N. Y. 49, 3 Jill
Tou, J. Digital and Sampled-data Control Systems. 1959. New
York; McGraw-Hill
SALZER, G. M. Signal-flow reduction in sampled-data systems.
Wescon Conventional Record, Inst. Radio Engrs, N. Y. Pt IV
(1957)
Tou, J. Statistical design of linear discrete-data control systems
via the modified z-transform method. J. Franklin Inst. 271, 4
(1961)
CHANG, S. S. L. Optimization of the adaptive function by the
z-transform method. A.I.E.C. Conf. Pap. NCP 59-1296 (see also
Synthesis of Optimum Control Systems. Ch. 10, 11. 1961. New
York; McGraw-Hill)
VAN-NEis, R. I. Automatika 1, 2 (1961)
IvAKHNENKO, A. G. Comparison of cybernetic extremal sampled-
data systems characterized by extremum search strategy. Auto-
matika 3 (1961)
FELDBAUM, A. A. Vychislitelnye Ustroistva v Avtomatika (Com-
puters and Automation) 1959. Moscow; Jill
KUNTSEVICH, V. M. A study of sampled-data extremal systems
with extremum drift. Automat. Telemech., Moscow 7 (1962)
KUNTSEVICH, V. M. Invariance of sampled-data extremal systems
without disturbance links. Automatika 3 (1962)
TuNIx, A. A. Automatika 6 (1962)
bkil bkil
bki2 bk12
bkilki
Figure 1. Block diagram
of combined control system.
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Figure 2. Block diagram of combined control system
I: plant
(a)
Figure 5. Block diagram of difference-type sampled-data extremal
system with indirect compounding link
I: plant; 1: multiplying unit; 2: memory element
(b)
Figure 3. Block diagram of servosystems: (a) with direct link with
respect to assignment; (b) with indirect link with respect to assignment
(a)
(b)
I I I(e,. 10)
5 ; ..410_ 15
Figure 6. Transients of extremal system for 1n = fln, 42m = 0
I: in system (54) with compounding link for satisfaction of con-
dition (50); (a1a2 = 0.4; a3 = 1; dl = 0.4; d2 = 0.8; fl = 3.5);
II: in system (55) (ditto, but dl - d2 = 0.4; = 2);
Figure 4. Block diagram Structural scheme of combined servosystem 1.II: in system (55) (ditto, but for 3 = 3.5)
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Optimization and Invariance in Control Systems
with Constant and Variable Structure
B. N. PETROV, G. M. ULANOV and S.V. EMELYANOV
Optimization of Automatic Control Systems and K (D) Image
Theory
The object of the general theory of optimization of automatic
control systems with respect to accuracy is the optimal synthesis
of control systems operating under conditions of continuously-
acting disturbances.,
In the deterministic set-up of the problem1-3' ', 8 the optimal-
ity criterion is the achievement of the highest degree of accuracy
of the automatic control system, as measured by the error e,
which is equal to the difference between the desired g (t) and
the realized x (t) value of the state of the system s ~ g (t) - x (t).
In the case of static synthesis the optimal system found from
the probability characteristics, of the controlling signal and the
interference, has a transfer function (Dopt, and possesses the
greatest accuracy only in the mean.
The main results relating to the construction of optimal
systems in the case of the deterministic set-up, have been
obtained by the theory of invariance, on the basis of which
there can be effected the construction of automatic control
systems with an error e, equal to zero or extremely small in the
presence of disturbances, the measurement or use of.which for
the purposes of control is feasible. The conditions of the theory
of invariance of automatic control systems, in the case when
disturbance links do not nullify the numerator of the transfer
function (and thus the corresponding transfer function), and
when f (t) is specified, are expressed with the aid of the K (D)
image introduced by Kulebakin
K(D)?f (t)=0, K(D)*0, f (t)*O... (1)
K (D) and f (t) are linked by the conditions of the. operator
K (D) image of the functions'. In this case for a stable system
its transfer function must either be the conform K (D) image
or have this operator K (D) image as co-multiplier.
In the statistical set-up, with regard to determination of the
transfer function of a control. system in the case when it has an
infinite memory, according to the mean-square error minimum
criterion, one of the main results was obtained by Wiener. Ob-
viously, in one case it is possible to establish precisely the corre-
spondence of optimal systems in the case of the statistical and
deterministic set-up of the problem. When the dispersion f (t)
tends to zero, Wiener's optimal system and the optimal system
as determined by the conditions of invariance coincide and
should, strictly speaking, lead to the same results. The generality
of systems obtained in this case according to Wiener, and of
invariant systems, in particular systems meeting the condition
of Kulebakin's K (D) image, are demonstrated. Taking the
interval of observation of f (t) to be infinite, and thus -being
concerned only with the forced output of the system, the
transfer function of a Wiener optimal system is characterized
by the magnitude of the MS error e2 (ref. 6):
e2= J {Sn((O)-/(DoPt(j()/2Sf((o)}dww...
(2)
Sf (w) is the spectral density of f (t), S? (w) the spectral density
of the desired output signal. In the reviewed problems of
control for stabilization S,, (w) is conformally equal to zero,
since, with complete filtration of external disturbance f (t), the
desired output of the system must be conformally equal to zero.
The conditions of zeroth error e2,,,;n = 0 lead to the following
requirement in respect of the optimal transfer function of an
automatic control system:
.
Z2=0
I I0P,(J(0)I2
S?((o)=0 (3)
S f (c)) = 0 (4)
The latter can be satisfied for (D (p) ? f (t) = 0, which is a
sufficient condition.
In the case indicated, when
(P) _ A, (p) -=o
0 (p)
where Ol (p) is the numerator of the transfer function, and 0 (p)
is the characteristic polynomial of the automatic control system,
expression (4) can be found for (a) A (p) = 0 or (b) K (p)er oo,
where K (p) is the coefficient of transfer of the automatic control
system (the characteristic equation of the control system is
A (p) = K (p) + 1 = 0).
The above-mentioned conditions correspond to the known
conditions of invariance, the realization of which in physical
systems is determined specially.
Without individually examining the above-mentioned
possibilities (for (D (p) = 0), the case of the non-zero operator
(D (p) 0 0 will be considered.
If (Dopt * 0 and Sf * 0 the satisfaction of condition (4)
is possible when
Oopt(p)'.f (t)=0 (5)
This requirement corresponds to the condition of invariance
optimal according to Wiener in respect of disturbance f (t),
and coincides with the K (D) imager. An analogous method is
used to establish the community of invariant systems and
systems optimal according to Wiener, in the case of other
control problems. Thus the K (D) image can serve as a tool
for automatic control systems optimatization theory.
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As an example, consideration is given to the forced motion
of an automatic control system under the influence of an external
disturbance, which, is described by, the equation
O(p) x(t)=(p2+(oK)sinwKt
The transfer function of system ( (p) = p2 + wK2/A (p),
by virtue of condition (5) corresponds to an optimal system,
since it contains the K (D) image of the action f (t) as a comul=
tiplier (p2 + a)k is the K(D) image off (t) = sin wkt).
Then, according to condition (4), the function 1(D (jw)12 and
Sf ((o) will respectively have the form of Figure 1.
The product of the function II (jw)12 Sf (w) equals zero,
since I( (j(0)12 > 0 when to 0 (K, 1 (1) (jw)12 = 02 when co = wK
Sf((0)=S 1w-(OKI
j0 uW 0OJK
1 Sfunct w = wK
Generalization of K (D) Image Theory for the Case of Statistically.
Given Disturbances f (t)
The K (D) image theory expounded in the works of Kuleba-
kin, was developed for the case of a disturbance f (t), preset as
a determined function of time t. To the class of. functions
particular, those which permit approximation of f (t), as
accurate as one likes, by 'integrals of linear differential equa-
tions, homogeneous and having constant coefficients. Shannon
has shown that a very broad class of functions, with the
exception of hyper-transcendental functions and ,` functions,
may also be approximated by the solutions of homogeneous
differential equations with constant oefficients.
The need to develop statistical methods in the theory of
invariance and in particular in the case of K (D) images is
explained by the following. The theory of invariance up to e
depends essentially upon the form of f (t). The absolute invari-
ance of automatic regulation and'control systems in 'the case
when the transfer function of the systems, asl1 function from
f (t) equals zero, is generally speaking real for any f (t), con-
strained with respect to the modulus, in particular in relation
to those about which information is missing.
In the case of the K (D) image the effect of absolute invariance
may only be observed for a completely defined function f(t),
knowledge of which, as a determined function of t, must be
available with a probability of 1. Thus. essential for the theory
of invariance is knowledge about f (t), which is nesessary in
different cases with a probability from 0 to 1, particularly when
investigating invariance with accuracy up to e. In the case when
f (t) is given in a probabilistic sense, the effect of invariance-
particularly from the viewpoint of the K (D) image theory-was
not examined, and the theory of invariance itself is not developed
at the present time. An attempt is made below to apply the.
theory of statistical optimization to the determination of the
statistical probabilistic conditions of automatic control systems
invariance, and generalize the theory of K(D) images for this case.
Henceforward, as previously, we are examining the effect of in-
variance, the class of statistical actions f (t) and control systems
relating only to stationary systems and stationary actions f (t).
Approximate Conditions of Optimalization Using the K (D) Image
in the Case when Dispersion is Present
In the well-known works of Kolmogorov10 and others it is
shown that any stationary random process may be represented
as the limit of a sequence of processes with a discrete spectrum.
The general expression of a' stationary random process f (t)
in this case may be as follows:
Q
//t a sin(/w t+ (6)
~ K l K cPK) \ )
where a1, a2, a3, ..., aK, ..., an are uncorrelated random magni-
tudes with mean value zero, i.e.,
Mai=o, i=1,2,...,n
Mp.Ma.=0 i 0 j
where M is the sign of the mathematiccl expectation.
It is also known?? 10 that for each stationary process f (t) it
is possible to indicate. a number e as small as desired and as
large as convenient an observation time range thereof T, for
which there exist such pairwise uncorrelated random magnitudes
a1, a2, ..., an that the completeness of approximation to the
n
series E aK sin (wKt + 01f), determined by the mean-square
K-1
difference, will be such that
n
Mlx(t)- E aKsin(wKt+(pK)IZ 0, fN < 0, where fv and fN are the projections
of the vectors f+ and f- on to the normal to the hyperplane S,
directed from G- to G+). Then, when Z Q) hits U there arises
the so-called sliding mode and the solution of system (13) does
not depend on ai, bi, bi*., gi(t).. In fact in this case, as shown by
Filippov13, in the domain U- 'there exists a solution E(t) of
system (13), and the vector d s / dt = f O (E; g (t)), where f O _
(fo fo) lies in the hyperplane S and is determined by the
values of the vector functions f+ and f .
From the condition that f ? (E, g(t)) e S there follows the
linear relationship of the components of the vector!
n
cjf,9=0 (14)
j=l
where f ? is the jth component of the vector f ? whence
{{ -l n-1 {{
j 0 E cif"
c,n j=l
Hence the solution of system (13) for E(t) e U coincides with
the solution of the system of similar homogeneous differential
equations
dt=JO(E)
(16)
1 n-1
{{~
Jj =ej+1(.1=1,2,...,n-1),fnO=C cjej+1
n j=1
cj are constants.
Obviously the solution of system (16) does not depend on
a1, bi, bi*, gi (t). Use will be made of this property of the solution
of the system of non-homogeneous differential equations with
a discontinuous right-hand side for the construction of a com-
bined tracking system with variable structure.
E _ (E 1, ..., En)
Conditions of Invariance in Combined Tracking Systems with
Variable Structure
In the domain, ? G, of an n dimensional space el, ..., en let
the motion of a dynamic system be described, by a system of
non-homogeneous differential equations with a discontinuous
right-hand side
dt .f (E14 (t))
e=(E1e...l80,9=(gl,...,gm), =(J1e?..ef)
Jn = - aiei+ ~i (E, 9 (t) gi (t)
i=1 i=1
(13)
532/4
t In the case cj ej I gi (t) = 0
=1 I
+V(E~g(t) = bi for ,Elcjejgi(t)-+0
= bZ for (
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532 / 5
Let the structure, selected in a definite way, of the open-loop
cycle of a combined tracking system [Figure. 3(b)] change
stepwise on some hyperplane S = E c; e; = 0 in such a way
j=1
that the movement of this servosystem is described by a system
of non-homogeneous differential equations with a discontinuous
right-hand side (13), where ~i (s, g(t)) = F [4)i (s, g(t))]
(Di (?, 9 (t)) _
K1 for ci?j g1(t)> 0
1-1
(i =1, 2, ...,.n)
K* for cj?~~ gi(t)