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

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80T00246A023400480001-9
• Release Decision: RIPPUB
• Original Classification: U
• Document Page Count: 80
• Document Creation Date: December 27, 2016
• Document Release Date: February 13, 2013
• Sequence Number: 1
• Publication Date: October 16, 1963
• Content Type: MISC

STAT
Declassified in Part - Sanitized Copy Approved for Release
2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release
2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
031/1
Two-positional Functional Frequency Device
for Automatic Regulation
I. A. MASLAROFF
Introduction
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 Nemml 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
C?dt=EQ (1)
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 ZIA
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 --- A,
(1 ? et IT)) and during the pauses, by: A = Ak e?t/T
(t -= 0, A = AO. 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:
Al= Ay(1?e-t./T)
A 2 =A tin- =Ay _e?ti/T)e?ti/T
A3= Ay (1 ? e ti) +A2 e?ti/T=Ay(1?e?(0)e-rid-t1/r
(2)
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:
n ? E [tk+(n?a)ti]/T.
A2?=A3,(1 _e_ti/T) E e k=a
(3)
a =1
and after n 1 serial impulses:
1t1+ [n?(a-1)ti])
A2 n+1= Ay(1?e-tilT)
1+ E e k=a
(4)
a=1
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 ZIA. 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 :-
IulA(1?e-'17.1)=B
The time constant of the exponential block of the scheme must
be much smaller than the time constant of the object.
Then at ZIA = const. the time of the pause is equal to:
kdA
t=T1 In 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 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:
where
Since
031/1
aA!=sA" (6)
&4'=A2,1 ?A2,; &4"=A20+3?A21,+2
A20+3= Ay(' ?e-wT)+A2,7+2e- ti/T
A201-2 =A20+1 e-tn+IIT
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
031/2
the time of the pauses is equal to:
tn+i= Tin
A2n+1
A2 n+1? Ay(1?e-ti/T)
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
(7)
as:
A
t?,i= T ln g (7a)
Ay(1
The maximum value of the fluctuations of the unit to be
regulated is given by:
=LIA=--k-=Ag?A2?? =(Ay?A9)(1 ?e-t,/T)e-t,/T (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 Skit which follows from the delay of the system, is equal to:
"At= 4
2n+2(1 r=" Ag(1?e'lf/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:
Az A 3AA,=(Ay? Ag)(1 A0(1 ?e-`"/T)
(10)
If it is accepted that (SA = ?A,, then:
Ay (1? e'IT)
=1+ e-t,IT
Ag (1 ?e-AtIT)
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
A t it is apt to accept tiv> z1, i.e. to use strong impulses.
However, at small values of At it is apt to accept A,, A?, i.e.
the impulses will be comparatively weaker.
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 ZJA 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.
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.
Conclusions
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
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Nomenclature
Coefficient of the generalized capacity of the object to be regulated
The unit to be regulated
Fixed value of the unit to be regulated
Given value of the unit to be regulated
Difference between the given and actual value of the unit
regulated
Generalized quantitative index of the process
Variation of the unit to be regulated in the period of one impulse
or pause
Time
Time of the impulse
to be
Ag
A2n+1 42n43
? 42n A2n+2
AA
ttt t2. t,
Figure 1
KAA
/AA
0, and T = T (0, such that for all
trajectories {x (t)}n with initial values satisfying
{x (to)}o (n ? k)(H) t0 0 (11)
then (10) holds.
If the initial conditions were subjected to the following
restrictions
Ix (t0)}? e G? to > 0 (12)
then the above-mentioned stability, asymptotic stability and
equi-asymptotic stability are said to be stable, asymptotically
stable and equi-asymptotically stable under condition (12)
respectively.
In the sequel, the function V (xi., t) is called the Liapu-
nov function with respect to functions 0i, ..., Ok if
V (xi, xo, t) 0 as {x}o e3?.--(n ? k) (13)
and V is assumed to have continuous partial derivations.
103/1
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13 : CIA-RDP80T00246A023400480001-9
103/2
Definition 4. The function V (xi, ..., xn, t) is said to be
positive (negative) semi-definite with respect to (2) if
(a) (13) holds,
(b) V 0 [V Olin '5'? (n ? k)(H).
Definition 5. The function V (x1, ..., x,, t) is said to be positive
(negative) definite with respect to (2) if
(a) Definition 4 holds,
(b) there is a positive function Wi (yi, y,c) such that, in
(n ? k)(H),
V (xi, x?, t)_Wi[cp (x 1, ...,x), ..., (xi, ...,x)] (14)
{1/"(xi, xo, ? [01 (xi, ..., xo), cpic (x 1, ..., x?)]1
Definition 6. The function V is said to be uniformly small,
if for any given e> 0, there is 6 (t)> 0 such that the conditions
t> 0 and {x}? E. (n ? k) (6) imply V s.
Definition 7. The function V (x1, xn, t) is said to have
infinitely small upper bound with respect to (2) if there is a
continuous function W2 (y1, yk) such that
(a) W2 (OD 0) = 0,
(b) in. (n ? k) (H),
W2[01 (X13 ? ? ? 3 X n), ? ? ? 3 OK (X13 ? ? ? 3 X ...,Xn,t) (15)
Definition 8. The function V (xi, ..., xo, t) is said to have
the property A, if there are two positive continuous functions
(S) and W2 (S) such that
(a) W1(0)= Wa (0) = 0, Wi (00) = Wa(co)= + co (16)
(0 W2 (11011 x) V (x x?, t) W100110 (17)
Parallel to Definitons 1 and 3, one has the fundamental
theorems shown in the following section.
The Fundamental Theorems
(A) For the system (1), if if there is a Liapunov function
xn, t) such that
(a) V satisfies Definiton 5 and it is positive definite with re-
spectto (2),
(b) V satisfies Definition 7,
(c) the total derivative
dV a V
=--+ grad V ? X (18)
dt 1.1 at
is negative definite with respect to (2), then the system (1)
satisfies Definiton 3.
(B) If the system (1-) satisfies Definition 3, and the rank of
matrix
/41
0x1 ?' Oxo
D (4)1, 4)) _
D(x1,...,x?)
aOK a(bic
\ax,'??" axn /
is K, and the functions 0, (i = 1, ..., x) defined by (5) are
uniformly bounded in (n ? k) (H), then there is -a function V
(19)
which satisfies all conditions in (A). The proof of this theorem
is given in the Appendix. It is not difficult to prove the following
corollaries.
Corollary 1. If V satisfies Definitions 5 and 6, and d VIdt I w
is negative semi-definite with respect to (2), then the system (1)
satisfies Definition 1.
Corollary 2. If V satisfies Definition 8, and dV/dt I is
negative definite with respect to (2) then (9) holds for any
to > 0 and any x? in the space.
Let the set of position points at time of the motions which
take on the initial positions in G? be written as G(0 .
Corollary 3. If V satisfies (A), Corollary 1 or Corollary 2
in G(t) n (n ? k)(H), then the system (1) is stable, equi-
asymptotically stable, or asymptotically stable in the .whole
under condition (12) respectively.
Corollary 4. If the system (1) satisfies Definition 3 under
condition (12) and the rank of matrix (19) is K in the neigh-
bourhood ?F (n ? k) n G(t) and 02: (i = 1, k) are uniformly
bounded in G(t) n g?-? (n ? k)(H), then there is a function V
which satisfies the conditions in Corollary 3.
Example?Consider the system
I. = ay. ? cx (bx2 + a y2) sin
bx2 + ay2
1
57= ? bx ? ay (bx2 + a y2) sin bx2 + ay2
1
(20)
(a ? b ? c >0)
Obviously, if one takes 0 = bx2 + ay' then 0 = 111or
(k = 1, 2, ...) are the invariant sets of (20), they are closed
orbits. By means of the Liapunov functions V = (4) ? 11k7c)2
with respect to 0 ? 111(n, the following statements can be
proved:
(a) In the exterior of the ellipse 49 = 1/n, there is no closed
orbit;
(b) in the interior of the ellipse 4) = 1/7r, there are infinitely
many closed orbits;
(c) the closed orbit is asymptotically stable when K is even
and it is unstable when K is odd;
(d) the origin x y = 0 is a singular point of (20) and it is
stable. In any of its neighbourhood, there are infinitely many
closed orbits, and hence the origin is not asymptotically stable.
In the regulating or the dynamic systems, it is often necessary
to estimate the decaying time of perturbations for the standard
working state. In this paper the problem of estimating the
decaying time is considered. In the sequel it is assumed that
system (1) is equi-asymptotically stable with respect to (2), and
the following discussions are valid in certain attractive
regions of ,F (n ? k).
Let V be a Liapunov function of (1) which satisfies the
conditions of the fundamental theorem (A). In the general case,
there are two positive definite functions Wi (yr, ..., y,c) and
W2 (.})L, ? ?
103/2
such that
w2Di (X)3 ? ? ? 3 K (X)] V (Xi,
W1 brk 1 (X) ? ? ? 3 K (x)] (21)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
103/3
Besides, it is assumed that there are two functions f1 (s)
and f2 (s), such that in ?F (n ? k) (II) the inequalities
dv
A (v) f 2 (v)
1.1
hold. Furthermore, from (21) one has, in general,
1 . {x?} e {W2 Vol implies {x},, e {V Vol
' (22)
(23)
{x?} { V .8} implies {x}? e Wi el
where Vc, and s are given position numbers. Denote
f vo
Ti=_
(A)
v?
T2 =
f2(2) (24)
Then, the following theorem estimates the decaying time.
Theorem I. The decaying time T of the motion of the system
(1) from an initial point in the region
Wz [01 (x), ? ? K (X)] 17()
to a point in the region
Wi COI (x), ? ? ? K (x)] 8
satisfies the inequality
(25)
(26)
T T2 (27)
The decaying time T from an initial point in the region
W2[4)l(x),...,4)K(x)]-Vo
to a point in the region (26) satisfies
T
Let M2 (R) be the maximum value of W2 on the boundary
11 011. = R of (n ? k)(1?)
(28)
The above method is used to solve the following example.
Example?Consider an autonomous system
$=x? b2xy2
= y a2yx2 _ b2y2
and its unique closed orbit
= a2x2 b2y2 1 = 0
(a> b) (33)
(34)
If one selects the Liapunov function with respect to to be
_ (az x2 + b2y2_ 1)2
then it may be asserted that:
(a) the system is asymptotically stable with respect to 49 = 0;
(b) the decaying time T of the motion of the system from an
initial point in the region I 4) I ctio to a point in the region
I 4) I e satisfies T a2 log (1 -1-- e) 00/(1 -1- 00) s;
(c) the decaying time T of the motion from an initial point in the
region I 00 to a point in the region e satisfies
T L.' log (1 + 8) 00/(1 4))e.
(35)
On the Estimation of Decaying Time for Linear System with
Quasi-constant Coefficients
In the study of a practical dynamic system, one usually takes
the linear system with constant coefficients as its first approxima-
tion. In general, the frequency method may be applied to
estimate the time of transient process for the regulating system
with constant coefficients. However, this method is only applic-
able to the case of single output under specific initial conditions.
(29) In addition, the method is not rigorous. This paper gives the
formulae to estimate the decaying time in the general case, and
the method is rigorous.
A large amount of work9-12 is devoted to the estimation
of decaying time for the asymptotically stable system
= Psi x -1- ? ? ? PsNxN, S= 1, ..., (36)
(30) where the coefficients pij are constants. There results may be
summarized as the following. For any given positive definite
and let mi(y) be the minimum value on the boundary 11011. =71
of (n ? k)(y). Again denoting
ftni (y) fl 11 T2 = .1 ml (y) f2 (A)
\ SM2 (R)
? 3"1 2 (R) c12
T1=
the following theorem is obtained.
Theorem 2. The decaying time T of the motion of the system
(1) from an initial point in the region (n ? k)(R) to a point
in the region"- (n ? k) (y) satisfies (27), and the decaying time T
of the motion of the system (1) from an initial point in the region
11 (I) 11 R to a point in the region (n ? k)(y) satisfies (29),
where T1, T2 are defined by (30).
By taking
(v)= ? av f2 (v)= ? f3v (x> fl) (31)
one has
1
T1= ?log M2 (R)
a mi (r)
1 M2(R)
T2 = log
13 mi(r)
(32)
Particularly, when cki = x, i = 1, k < n one obtains the
formulae to estimate the decaying time for partial coordinates,
and when Oi = xi, i = 1, n, then one obtains the formulae
to estimate the decaying time for total coordinates (9), . (10)
and (11).
quadratic form
U=x'Ux
(37)
there is a positive definite quadratic form
V=x17x
(38)
such that
d V
? U
(39)
dt
(36)
If M1 and m1 are, respectively, the maximum and the minimum
eigenvalues of the matrix V, and M and m are the maximum and
the minimum eigenvalues of the matrix U, then the following
results are obtained.
'Theorem 3. The decaying time T of the motion of the system
(36) from an initial point in
103/3
E x s2 = R2
S = 1
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13 : CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
103/4
to a point in the region
L Xs r2
S=1
satisfies the inequalities
M R2 M1 MiR2
log 1 < T < log (40)
1V12 mi r2 ?
a.12 Mir2
In practice, it is of interest to select a suitable Liapunov
function V, such that for the given system (36) the range defined
by (40) is as accurate as it can be. It is very difficult to answer the
above question in the general case. But if the system (36) is
normal and the elementary divisors of the coefficient matrix P
are all simple, it may be proved that when
= E xs2
s=1
the equalities in (40) may be realized (i.e. the estimation is
accurate).
Let the normal transformation be
y = Cx (41)
where C is a matrix with real coefficients, and the system (36)
is reduced to the normal system
where
J=
? al
0
?ex
? /31 ?(01
col? fli
V = x'C'cx
? (DI
col? flu
(42)
(43)
(44)
may be taken as a Liapunov function of the
means of (42) the following results may be pro
Theorem 4. The 'decaying time T of the
system (36), from an initial point in the (n ?
ellipsoid V = Vo to a point in the ellipsoid V =
system (36). By
ved.
motion of the
1)-dimensional
e, satisfies
1 Vo 1
?2u log? 2
8 log -V--?
?
(45)
where u = max (ai,13;), v = min (ai, fl,). It is easy to select the
initial points such that the equalities in (45) hold (i.e. this
estimation is accurate).
In the following, the general formulae to estimate the
decaying time is given.
All roots of the characteristic equation D (2) = det (R ? 21)
= 0 are assumed to have negative real parts. Let 2, ..., Ai be
negative real roots, written as Ai = ? ai (i = 1, ..., 1), and let
NA, ? ? ?3 An be the remaining roots, written as 138+ cosi
(S = I. ....n ? 112 = k), and the order of the corresponding
elementary divisors be n1..... nh.
It is known that there is a non-singular linear transformation
y = Cx to reduce the system (36) to the normal form (42),
in which
J=
\O
?ai
0 ?1 0 ... 0
ai 1 ... 0
M1=\ \
N
NK
If one writes the mi x mi matrix
1 ? fli? coi 1 0 ... 0
coi? I3i 0 1 ... 0
(46)
0 /3, ?
0 0 ... co, ? fli/
(47)
. 1 1 1
frxi 2 ai 4 4 ???
2 cci ai 2 ai2 ? ? ?
1 [1 + 1 1
1
\1
\zi 4 ? ? ?
as acmui), this is constructed according to the following rule:
(a) when s = a, a(mi) is equal to (1/(x) (1 + a(mi) a), and let
(48)
,Oni) 1 ?
W11 =
(b) when
s> as(7,. = 1
2 ai s-107,1,
(c)
as(Ti) =
Thus the matrix is completely defined through the eigen-
values ? ai and the order of its elementary divisor. The maxi-
mum eigenvalve of the matrix a(mi) is assumed to be vi
1 when mi =1, vi=1
al
1
when mi = 2, vi= ? 1 [ iai ?+1 i4 4.. 1 (49)
oci 4 4 4
Following the method of construction of the matrix a(mi),
the 2 /4 x 2 ni matrix d2 74 may be constructed in the following
manner
103/4
(a)
(b)
ni) ni)
2i-1,2 j?"2i,2j-1=?, 1, j=1,?..,ni
A(2 ni) 4(2 ni) ??(ni) ? ?
21-1,2 j-1?"21-1,2 i-1=Gtij l,3? ,...,fl
(c) to replace ai by in the matrix PO.
For example ni = 2, one has
a(4) =
?, " (2) 0
v 12
0. (2)
"
l 0 12
,,(2) 0 al 0?(2)
)
"21 "22
\O ?(2)
"21 0
"22
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Obviously, the formula for the maximum eigenvalue of c/(2fli) is
the same as that of a(mi) in wich ai are replaced by j9.
Consider
the Liapunov function for system (36) to be
V =x'C'ACx (50) satisfies
to a point in the sphere
where C is the normal transformation matrix, and
I a(m1)
A=
a(')
0
d2"
d
(2
'
It is not difficult to prove that V satisfies
d V
dt
2
2.1 V
(51)
(52)
where v is the maximum eigenvalue of the matrix A, and it can
be calculated by the aforesaid method. When mi = 1 or rni = 2
it can be calculated through (49). If the maximum and the
minimum eigenvalues of the symmetric matrix C'AC are
assumed to be M and m respectively, then the following theorem
is obtained.
Theorem 5. The decaying time T of the motion of the system
(36) from an initial point in the (N-1)-dimensional ellipsoid
?
V = Vo to a point in the (N-1)-dimensional ellipsoid V = e
satisfies
The decaying time T of the motion of the system (36) from
an initial point in the sphere
to a point in the sphere
s= 1
satisfies
2
xR2s =
2 2
= r2
MR2
T_vlog 2
mr
Moreover, the system
i=px+X(x,t)
(53)
is considered, where X is a vector function which contains the
non-linear terms and the unknown components. If one constructs
a Liapunov's function (50) of its principal linear system
(54)
and if one assumes X to satisfy the inequality
Igrad V ? XI < bx'C'Cx, (b < 2) (55)
then the following results are obtained.
Theorem 6. The decaying time T of the motion of the system
(53) from an initial point in the sphere
Exs2=R2
s=i
103 / 5
Exs2= r2
s = 1
v MR2
T < (56)
?2(1 ? b/2) log mr2
As an application of this theorem, an example of a forced
oscillation is considered.
Example?Consider the system
= Pu + sU (u, F (57)
where p, is assumed to have all its eigenvalues with negative real
parts, 8 is a small parameter, U is continuously differentiable
and (t) is the forcing term with period T.
Let the system (56) have a periodic solution
us= us? (t), us? (0=4 (t +7') (s = 1, ..., N) (58)
and let the linear transformation
y = Cu (59)
transform the system ? = pu into its normal form
.P= J./1 - (60)
By means of the transformation (59) the system (56) was reduced
to a system
3)=Jy+BY(y)+0(t) (61)
where 0 (t) = CF (t) has the same period as F (t).- Under this
transformation, the periodic solution (58) is reduced to
COS 14 (t)
a =1
(62)
Consider the perturbations x, = ys ? y (t) then x satisfies
Sc=Jx+sq(t)x+sX(x,t) (6)
where q (t) is a periodic matrix with period T, and it may be
evaluated through Y (t) ancLy?, (t). If one takes = x, then
= u ? u? (t) is the perturbation vector in u space.
By means of the above method the matrix A is constructed,
with its maximum eigenvalue v, and the maximum and
minimum eigenvalues of the matrix C' AC are M and m respect-
ively. The following results are obtained.
Theorem 7. The decaying time T of the motion of the system
(57) from an initial point in the R neighbourhood of the periodic
solution (58) to a point in the r neighbourhood of the periodic
solution (58) satisfies
T<
MR2
log (64)
mr
2[1 (b+8)E1
2
where the term X in (62) satisfies
!grad V ? Xi < bx'x (b < 2 = x'Ax) (65)
and C is the maximum eigenvalue of the matrix q (t) [q (t)]
when t a [0, T]. By the above-mentioned r neighbourhood of
?103/5
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
103 / 6
the periodic solution (58), is meant the set of points which
satisfies the inequality
E [us?us(t
o )]2 2
s= 1
On the Estimation of Decaying Time for Quasi reducible Linear
System
In the study of the dynamic systems, one may sometimes fail
to approximate it by a linear system with constant coefficients.
In this case one may take a reducible system as its approxima-
tions, and construct the corresponding Liapunov function and
estimate the decaying time.
Consider the non-linear system
Sc=p(t)x+X(x,t)
Let the linear approximation system
g=p(Ox
(66)
(67)
be a reducible system. Assuming the characteristic number to
be all positive, there is a Liapunov transformation
y=C(t)x (68)
which transforms the system (67) into its real normal form
= (69)
By means of the method mentioned in the previous section,
V = x'c'ACx (70)
is taken as a Liapunov's function for the system (67). Since (68)
is the Liapunov's transformation when t > to, the maximum
and minimum eigenvalues Mand m cannot equal zero. Obviously,
the maximum eigenvalue v of A can be calculated through the
characteristic numbers of (67) by the same method. Parallel
to Theorem 2 the following results may be obtained.
Theorem 8. The decaying time T of the motion of the system
(67) from an initial point in the sphere
to a point in the sphere
satisfies
to a point in the sphere
E xs2 = r2
s=1
satisfies
T<
MR2
(73)
? 2 (1 ? b 12) log mr2 (b 0 the region H > 11 45 II, > 77 is considered.
From the conditions mentioned in the theorem, the function V
takes on maximum M> 0 and minimum m >0 and the nega-
tive definite function d Vldt 11.1, with respect to (2), takes on
maximum ? cc 0, 0(k)= E p4-1
?=,
P=C(1? zv)2 (C C? P P 6v)
v v
N
Zy=ePvT , 5E
p=1p,+p?
(25)
The value of 0(0) is calculated separately from
C2 iT3 N cv Npv
0(0)-6 CP ?2c _ iT E ?2 E ,
v = Pv v=1 zv
(27)
The further procedure of calculation remains the same, ex-
cept that for 0 we substitute everywhere
c=c2iT3
(28)
and instead of coefficients Ai (i = 0, 1, n) we use everywhere
the coefficients Ai (i = 0, 1, 2, ..., N). Their relationship can be
seen from the arrangement of the denominator of the pulse-
transfer function G (z)
A (z)= Ao+ A iz- + Az-n
= (1 ? z- 1)(A-0+ A 1 + Apiz-N), N =n? 1 (29)
This arrahgement is made possible just beCause one pole of the
transfer function S(p) equals zero.
The last difference in comparison with case (a) lies in the
determination of the numerical values of 2 and K which are
used in the determination of the matrices pertaining to the
weighting function qct) = 1 (t ? T). They are calculated from
the formulas
= ??+c c I T2?o(o)? E ? ov
2 C_IT 6)
N C (1- - )
S = E it
p=1 Pv+PIL
NCy
K =-2-PC0C_iT2?C_IT ?(1?zr)
v=i Pv
v=1 Pv
(30)
Example
In order to illustrate the method of calculation described
generally in the preceding section, the calculation of a concrete
case is given below. The transfer function of the plant is
6p+4.5
S(P)=(p+ 2)(p+ 1)(P+")
All poles of this transfer function are different from zero, the
problem discussed is thus of the type of Case (a). The unit-step
function response of the system is
s(t)= .T-1 {S()} = C0 + C1 e+ C2 eP2t + C3eP3t
P1= ?2; P2= ?1; p3 = ?0.5; C0=4.5; CI =.2.5; C2= ?3;
The continuously acting member of the system has a pulse-
transfer function
G (z)=_B (z) 1.309 z- ?0-092'Z-2 + 0.248 Z73
(26)
A (Z)= 1 ?1110 Z-1 +0'355 Z-2-0.030 2-3
122/3
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
122/4
Let the calculation of the coefficients of polynomial D(z) for its
selected degree L = 1, 2, 3 be presented. By solving eqns (14)
and (13) we obtain
(51 = 49375;
P1 = -0.6875 ; p2= 3.5 ;
(53= -20
P3=16
In order to obtain the system of equations for the coefficients
bi pertaining to the weighting? funtion q (t) = 1 (t - T) it will
suffice, in accordance with relation (21), to add to each element
of matrices [Krs] and [R,t,] respectively the following quantities
A K = - 0.4548 and A R = - 0-0646
k
0(k)
= 0
= 18.813
1
10-899
2
6-347
3
3.743
4
2.229
5
1.337
6
0.805
obtained by the numerical solution of eqns (22) and (23). In this
way one obtains
Table 2 contains the calculation of the elements of the second
row (r = 2) of matrices K? and R,.0.
Table 2
k
21,k
2U(.
R2k
K2k
-3
2.406
-2
4.118
-1
.7.156
0
12.465
5.913
-0694
1
. 20.250
8.833
0-937
? 1.630
2
281035
9.771
2-567
3.260
3
33.344
9.045
4
36-382
8.720
5
'38.094
8.710
The first column in Table 2 has been compiled according to
eqns (15), the second and third have been calculated schemati-
cally according to Table /. The fourth column containing ele-
ments K2k has been obtained by means of relation (19).
The elements of the remaining two rows of matrices [K?]
and [Rro] are calculated in a similar way. As [K?] is a symmetri-
cal matrix, it is sufficient to calculate only its elements lying to
the left of the main diagonal and those on the diagonal itself.
The correctness of the calculation is checked by substituting into
relation (20) which is the means of checking almost all numerical
operations represented in Table 2 including the compilation of
the first column 'Pk.
By this method it has been possible to obtain a system of
linear equations for the sought after coefficients pertaining to
the weighting function q(t) = 1:
-6o
151
"2
:03
3
0.611
0-186
0.120
0.084
2
0.631
0.170
0.199
1
0.642
0.358
The pulse-transfer function of the continuously acting mem-
ber of the system G (z) = B (z)IA (z) and the polynomial D (z),
the coefficients of which have just been calculated, determine
completely the necessary transfer function (9) of the computer.
The respective curves of the controlled variable x following the
unit-step change of input signal w are represented in Figure 3
for She weighting function q (t) = 1, and in Figure 4 for the
function q (t) = 1 (t - T).
It can be seen from Figures 3 and 4 that, compared with the
minimum number of steps (L = 0), a considerable improvement
has been attained, especially in the case where in the minimaliza-
tion of the integral of squared error the errors have been con-
sidered as occurring only after the first sampling period.
Derivations and Proofs
First of all it will be proved that the above stated results hold
for the case where all the poles of transfer function S (p) are
different from. zero.
The sequence of the increments of the variable e2* (t)
Ae2 [i]= e (iT)- e2* (iT -T)
has, according to eqns (5) and (6), the z-transform of
(31)
L
Ae2 [i] =B (1) sE0 DA, (32)
where Ak = 0 for k < 0 and k > n; A e2[i] =0 for i > n +L.
Eqn (32) contains all the L + 1 coefficients of polynomial D (z);
however, only L of them can be selected, as it is necessary to
fulfil condition (8) that is D (1) = I. For the purpose of fulfilling
this condition let coefficient Do be detached
1.630 13.27
D1
0.380
-1)E2 (Z)A(z)
{Ae2 [i]} = (1 - z(Z) =
[1.697
1.630 3.260 2.890
D2
=
0.694
Bla)D
1.327 2-890 4.217
D3
0.704
From this z-transform it follows obiously
As the elements of matrices [K?] and [...Pro] are independent of
the chosen degree L of polynomial D(z) the mere reduction of
the respective matrices will suffice to meest the case of L = 1, 2.
By the solution of the above system of equations coefficients Di
are obtained for i 0, while the coefficient Do follows from
condition
(8)
Do =1- E D;
In this way the following results have been obtained
L
D0
D2
D3 ?
3
0.758
0.046
0.140
0.057
2
0.768
0.038
0.194
1
0.776
0.224
D = 1- E D,
s=1
and eliminated from eqn (32)
122/4
(33)
L
Ae2 [i] =B (1) LEI Ds(Ai_s-A1)? (34)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
122/5
Now, the time curve of the manipulated variable y (t) is dis-
solved into the sum of unit-step functions
n+L
y (t)= E Ae2 [i] 1 (t ? iT)
i=o
and the curve of the controlled varaible x (t) can then be repre-
sented by the superposition of the unit-step responses
n+L
X(t)= E Ae2 [i] s (t? iT) (35)
o
Then for error e, (t) it holds that
n+L
(t) =1? x(t)=1? E Ae2 [i] s (t? iT)
1=-0
where
n+L
= Ae2 [i] s (t ? iT)
i=o
s(t ? iT)=s(co)? s(t? iT)
t>iT,-s(t? iT)= ? En CseiT)
v=1
t n and
i < 0, it follows
122/5
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13 : CIA-RDP80T00246A023400480001-9
122 / 6
Krs= E E Ai) cri +s,
1=0 1=0
n 0
- E E a if
1=0 1=0
According to (45) the second term equals Rro, and let the first
be denoted
Rrs=E Ai ECTi+s,j(48)
= o = o
In this way relation (19) has been obtained
Krs=Rrs-Rr0
(49)
As the term Rro represents a special case of R? with s = 0,
it will suffice further to seek only the numerical solution of (48)
for R?. Let it be written in the following form
If we denote
'Rrs= E Ai E Aj(cri+s, j+r- ai+s,
1=0 j=0
rUi+s= E Aj(6i+s, j+r-6i+s,
j=0
we obtain relation (18)
(50)
Rrs= E AirUi s (51)
= o
All values of rU required for the calculation of the rth row
of matrices [1c0 and [Rm] can be obtained as the product of the
rectangular matrix
Eqn (52)
and of the column matrix
[A1]=[A0, A1, ..., A?]
U] = [a] [A1]
It will be proved that in the case of weighting function
q (t) = 1 matrix [a] has all its elements lying on the lines parallel
to the main diagonal of the same value. For the mth element of
the kth parallel above main diagonal it holds
n Cy
m,k+r+m- 0".?n,k+m=CM-PC0 E D -0(k+r)
v=i v
-[cm +Cov
E -0 (0] = 0 (k)- 0 (k + r)
v=i
" C
As this relation is independent of m, a 1 elements lying on
this parallel are equal, and they may be denoted by the same
symbol
(53)
r rk= ? (0- 61(k
Similarly it holds for the elements on the kth parallel below the
main diagonal
rrk=6k+m,r+m-6k+m,rn=C min (k, r)+ 0 (k)- 0 (Ir lc()
For the main diagonal k = 0.
Due to this property of matrix [a] it is possible to arrange
the numerical solution of matrix product (51) into a scheme
shown in Table I (a) which can be easily found by comparing
both methods of calculation.
It remains yet to prove the validity of formulas (21), (22),
and (23) by which the former results are to be corrected, if errors
are being considered only after the first sampling period. By
substituting into matrix (50) for ai; (46) the terms a (47) calcul-
ated for the weighting function q (t) = 1 (t - T), it can be seen
that only the first column has been altered. Obviously it holds that
r (71c=r k+(erk,r- Ck, r- k, 0 4- ak, 0) A0 (54)
When calculating the term in the parentheses, it is necessary to
differentiate two cases: k > 0 and k = 0. By substituting
relations (46) and (47), we obtain in the first case the relation
vn Cy ?
k>0,14?.- al"
k, r eric, 0+6k, 0= ,-.1) --D zv)=x
v=1 Uv
which is independent of k. Similarly for k = 0
a 0, r 60, r 60, 0+0, 0
n C n C
= -CoE - zo+ E cvz,j,+ co E v -fro)=K0
v=i v v=i v=1 Pv
With this denotation the relation (54) may be rewritten in the
form
k> 0,
k = 0,
k=rUk+KA0
r 6.0=rUO+K0A0
(55)
For the verification of formulas (21) and (22) it will suffice
to execute operations (51) and (49) with the relations (55), and
to denote lc, - k = 2.
The checking formula (20) can be be verified by substituting
relations (44) and (45) and by using the relation
that follows from eqn (46).
In Case (b), with the transfer function S (p) having one zero
pole, the continuously acting member of the system is astatic
[s (co) = co], and integrals (42) are not converging. It is possible
to by-pass this difficulty, if the curve of the controlled variable
is not represented as the superposition of unit-step responses,
but as the superpostion of responses to rectangular pulses.
Otherwise the procedure of derivation is the same as in Case (a).
Eqn (52):
[0-]=
a 0 ,r 60,0;
61,r 61,0;
62,r '2,0;
_an+L, r- 0;
60,1+r -60,1;
61, r 61, 1;
62,1+r Ta2,1;
an+L, 1+r-6n+L,I;
60,2+r 60, 2; ? ? ? 60, n+r
61,2+r 61, 2; ??? 61, n+r
62,2+r 2; ? ? ? 62, n+r
an+L, 2+r- 6n+L, 2; ? ? ? 17n+ +r-6n+L,n_
-61,n
2,n
(52)
122/6
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
References
1 RAGAZZINI, J. R. and FRANKLIN, G. F.
Systems. 1958. New York; McGraw-Hill
2 JURY, E. I. Sampled-data Control Systems.
Wiley
122/7
Sampled-data Control 3 Tou, J. T. Digital and Sampled-data Control Systems. 1959. New
York; McGraw-Hill
1958. New York; 4 STREJC, V. Ensuring reliability in complex automation by auto-
matic digital computers. Automatisace. V (1962) 5
Figure I
x(t)
lig
MEE=
=MI
w t)
(a)
Figure 2
15
? 1?0
0-5
sec
Figure 3
3
t
Figure 4
4
5
6
sec
(b)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/1
The Dynamic Properties of Rectification Stations
with Plate Columns
J. ZAVORKA
The control of rectification stations, as carried out at the
present time, is confined only to some control loops which are
designed without any thorough theoretical consideration. As far
as individual control diagrams are concerned, quite a number
of them have been designed; for instance, see Anizinovl. The
advantages and shortcomings of various connection schemes
have been published by the respective authors, however, and the
evaluation is mainly based on technical sense and experimental
results. Information on the general operational analysis of
rectification columns has been appearing only recently2, 3, 5, 7, 11.
In most of these papers the pressure and hold-up of the plate
have been considered as constant quantities. Due to this, the
validity of results is limited to cases with slow changes in the
input quantities; for instance, changes in feed composition or
changes occurring during the starting of the column. For
rapidly changing input variables, for instance pressure, the
results are erroneous. In view of these facts, an operational
analysis was worked out by Voetter and Houtappell3 where the
pressure and hold-up of the plate were considered as variables.
Starting from linearized equations the authors demonstrated
that, nevertheless, the results hold for a rather wide range of
input quantities. The same authors extended their study to
ternary mixtures, and used digital computers for the calculation
of dynamic properties. It has been found that the solutions of
these problems are exceedingly time consuming with regard to
the computer, and Rose and Williams5, 19, 11, attempted the
modelling of the system on an analogue computer. However,
these authors designed the model of the dynamics of the vapour
phase as single-capacity members connected in series which
does not correspond with reality. This deficiency has been
eliminated by the work of Rijnsdorp and Maarleveld9, who
succeeded in modelling a 32-plate column on an analogue com-
puter built from passive elements especially for this purpose.
The Bode frequency characteristics are the result of this work.
As an example, one of these characteristics is shown in Figure 1.
Obviously it cannot be evaluated, as the curve has no distinct
straight sections to permit the determination of the respective
intersects. Apart from this, it is not possible to agree with the
assumption made by the authors in the equations describing
the system, namely that the heat of evaporation is merely a
function of pressure and independent of the composition of
the mixture.
The aim of the present paper is to derive generally valid
relationships for the computation of transfer functions for the
individual input and output variables of the whole rectification
station, to create in this way the possibility of comparing and
assessing the advantages and shortcomings of various control
diagrams, and to obtain the data necessary for the synthesis of
control loops and for the complex automation of rectification
stations.
The task has been limited to rectification stations with plate
columns for the separation of binary mixtures.
The purpose of the work is to determine the transfer func-
tions of the system, which in turn determine the relationship
between the input variables (N: the flow rate of the feed; XN: the
composition of the feed; Pk: the pressure in the condenser;
: the flow rate of the heating steam) and the output variables
(A: the flow rate of the product; XA: the composition of the
product; B: the flow rate of the residue; XB: the composition
of the residue; Po: pressure at the first plate of the column) and
possibly between the concentration at some other plates.
The diagram of a rectification station with a plate column for
the continuous separation of binary mixtures is shown in
Figure 2.
For the investigation of dynamic properties let the rectifying
station be divided into three sections shown by the dash line in
the illustration. The first to be investigated is the independent
rectifying column, the second section consists of the bottom of
the column with the still, while the third section contains the
top of the column, the condenser, the cooler and the condensate
tank.
The rectifying column consists of plates that are to be
considered as separate units with regard to function and con-
struction. The diagram of a plate is shown in Figure 3. It can
be seen that the plate may be acted upon by the following nine
input variables:
The feed flow rate
Xis; The feed composition
The enthalpy of the feed
The flow rate of vapour from the plate below
The concentration of this vapour
Yn-1
The enthalpy of this vapour
L?+? The reflux from the plate above
.X.n+1 The composition of this reflux
H1 n+1 The enthalpy of this reflux
By these variables changes are produced in nine output
variables:
12711
The liquid hold-up of the plate
The vapour hold-up of the plate
The pressure on the plate
The flow rate of vapour streaming from the plate
The reflux from the plate
The enthalpy of vapour streaming from the plate
The enthalpy of the reflux from the plate
The composition of vapour
The composition of the reflux
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/2
The plate is described thus by a system of nine simultaneous
equations which are now derived.
First, the material balance of the plate is set up.
d1111, dM?, ?
dr+ dr
Ln + 1? Ln+ N (1)
" ?
By multiplying the individual terms by the corresponding con-
centrations the total material balance equation is transformed
into the material balance of the more volatile component:
d(Mi? X?) d
4. (M., . ? Y.)
dr dr
=L?,4.1X?_F 1? L,n? X,,+17?_i? Y,,_ V?Y?+ N ? X N (2)
In accordance with the material balance equation it is possible
to write the heat balance equation as follows:
d (M,, ? ? H,,?) +d(M?,?H?,?) v, .dP?
dr dr dr
II.,n- 1 N HN (3)
The last term on the left-hand side of the equation (which re-
presents the consideration given to the difference between the
enthalpy of the vapour phase and its internal energy for which
the equation holds) is neglected later with regard to the pressure
changes being of the order of millimetres of water gauge.
The vapour flow rate depends on the square root of the pres-
sure differential on two adjacent plates and on the density of the
vapour. In view of the fact that the difference in pressure on
two adjacent plates fluctuates within the range of 25-50 mm w.g.,
the influence of density may be neglected. The relationship be-
tween flow rate and pressure is then described by the equation
17.2 = ko (P.? P.+ 1) (4)
The following relationship should be further investigated,.
= M n(L
By the application of relation
Si =10
1 773 p ? y
one obtains
M 10 Si (10-5 ? pL
,'= 1. 773 p ? yi) + K (5)
Now consider the relationship between the concentration of the
more volatile component in the vapours and the concentration
of the more volatile component in the liquid during the state of
equilibrium of both phases at the boiling point temperature of
the binary mixture.
Y.= Yn(X (6)
The description of this relationship was attempted by a number
of equations (Wohl, Scatchard-Hammer, Van Laar, Margules,
symmetric4). However, they all contain constants that can be
determined only experimentally. Due to this, and also due to
their complexity, none of these equations has been accepted in
practice. The effect of the composition of the liquid upon the
composition of the vapours (established experimentally) is nor-
mally represented by the X-Y equilibrium diagram. This method
of representation has been accepted for the following sections
of this paper.
The remaining three equations are written in the form of
general relations:
M y,rs= M y, n(P
111,,,-= H1,,, (Pn, X,,)
H y, n= v, n(19 n,
(7)
(8)
(9)
The system of the above-stated nine simultaneous equations
describes one plate of the rectifying column. As interest here is
only in the non-steady states of pressure, composition of the
liquid phase and flow rate of the liquid phase, all other variables
will be eliminated. The transfer functions of pressure, composi-
tion of the liquid phase and flow rate for one plate are obtained
by the linearization of the equations or possibly by their trans-
formation into differential equations, followed by the LW trans-
formation and the arrangement of the equations. These transfer
functions are used for drawing the partial block diagrams of one
plate for the dynamic behaviour of the three variables. The block
diagrams are shown in Figure 4. The overall block diagram of
one plate is obtained by the interconnection of all three partial
diagrams. The complete block diagram of the whole rectifying
column is obtained by the interconnection of the block dia-
grams of the individual plates as shown in Figure 5. For the sake
of clarity the multiplication constants are not shown in Figure 5.
Now, it remains to conclude the block diagram of the column
by the connections of the condenser and of the still.
The block diagram of the bottom section of the column (the
first plate and still), and the block diagram of the top section of
the column (the highest plate, condenser, cooler of the conden-
sate, condensate tank and the piping) have been derived by a
similar method as used for the derivation of the block diagram
of the column proper. For the sake of brevity the respective
procedures are omitted, and only their results are given in
Figures 6 and 7.
The complete block 'diagrams of all sections of the rectifying
station have been obtained so far. The description may serve as
the source of some data for the modelling of the system. Owing
to the high complexity of the diagram, a large number of inte-
grating units will be required for the modelling and, therefore,
it should be possible to model only the simplest stations with a
small number of plates. For this reason the results of the preced-
ing chapters have been subjected to a further theoretical analysis.
The analysis follows the aim of simplifying the block diagram
of the column proper so that it is suited for modelling, or so that
it is possible to compute the transfer functions of the system.
First of all it was necessary to determine the zones within which
the values of individual design, physico-chemical and, operational
parameters can vary. Further the relations were to be stated that
were required for the numerical solution of various terms occur-
ring in the formulae for the time and multiplying constants. A
quantitative analysis of the time and multiplying constants was
made on the basis of these values and relations. The results
obtained were used for certain simplifications of the formulae.
Further, it appears that the dynamics of pressure and composi-
tion in the whole column are represented by block diagrams of
the same structure (Figure 8). The diagram is formed by single-
capacity members connected in series with feedbacks by-passing
two members that follow behind. The output signals of this
127/2
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/3
chain are formed by the algebraic sum of the signals of three ad-
jacent members and they link together the diagram of pressure
and the diagram of composition.
The general analysis of this block diagram was made; a
matrix calculation was used for deriving the matrices of the
transfer functions of this block diagram as the functions of the
number of the chain members (or of the member of the plates of
the column). A further analysis was used for establishing the
conditions at which the static value of the output signals of the
above chain is equal to zerci (the conditions are related to the
magnitude of the multiplying constants), and the conditions at
which it is possible also to neglect the dynamic value of the out-
put signals (the conditions are related to the number of plates).
It was proved by a further general procedure that the above-
stated conditions are fulfilled by each column. Assume for an
instant that, during the investigation of the dynamic properties
of the distilling column, there is no interest in the non-steady
states of pressure. Under this assumption, and owing to the
former conclusions, it is possible to interrupt in the block dia-
gram the connections of the pressure changes between the indi-
vidual plates. This can be done because any disturbance entering
any plate lying below or above the plate under investigation can
influence neither the flow rate, nor the pressure, but only the
pressure values at different points of the block diagram, or of the
column, and these values are of no interest for the time being.
Now consider composition in the same way?supposing that
one is not interested in the non-steady states of composition.
Similarly, as in the case of pressures, the connections between
individual plates may be interrupted. The block diagram is then
transformed into the form shown in Figure 9. The values cp,
and cpx are the sums of the input signals of the individual nodes
of the block diagrams of the dynamics of pressure and composi-
tion respectively. Now the non-steady states of pressure and
composition, that were formerly excluded from discussion, are
considered. The partial block diagrams of pressure and composi-
tion respectively are easily attached to the diagram in Figure 9
by introducing the signals Tv and cpx into the individual nodes
of the block diagrams of pressure and composition respectively.
The result is shown in Figure 10. The section of the block dia-
gram bordered by the dot-and-dash lines corresponds with one
plate of the rectifying column. By the solution of the system of
equations written for all three nodes of the block diagram of one
plate (naturally after the introduction of all multiplying con-
stants) the transfer functions of all output variables of the plate
are obtained. Finally, in the application of the transfer functions,
it is possible to re-draw the block diagram shown in Figure 6
into the final form according to Figure 11. This block diagram
holds for a general column with any arbitrary parameters with
regard to design, physico-chemical conditions and operation.
The block diagram shown in Figure 11 together with the per-
taining transfer functions and formulae for various constants.
and transfer functions, is the final product of the theoretical
part of the work. These results make possible the computation
of the transfer functions of a general rectifying station. During
the solution of concrete problems a number of possible simpli-
fications appeared that followed from the numerical evaluation
of individual constants and plate transfer functions. It is not
possible to prove the general validity of these simplifications.
However, it may be assumed that they will be identical in most
cases.
Further work16 contains the practical computation of several
transfer functions and step response curves of a concrete rectify-
ing station on the basis of the results obtained from a general
analysis. The necessary measurements were also made on this
station in operation. After a comparison, the results of the com-
putation were in very good agreement with the results of the
measurements.
Nomenclature
A
cn
HN
H,?
H?
H02
4+1
Mk
01 PI
Po
91
Q2
Qi bl
SiSI
Tssa
U01
UO2
V
V*
V00
X
XA
X Ao
X Ai
X_42
127/3
Flow rate of the product (mol /sec)
Flow rate of the residue (mol /sec)
Multiplying constants
Specific heat of heating wall (kcal /kg ?C)
Reflux ratio
Mass of the heating wall (kg)
Flow rate of the heating steam (kg/sec)
Enthalpy of the liquid (kcal/mol)
Enthalpy of the feed (kcal/mol)
Enthalpy of the vapour (kcal / mol)
Enthalpy of the heating steam (kcal/mol)
Enthalpy of the condensate from the still (kcal /mol)
Number of plates
Constants
Subscript of condenser
Reflux (mol /sec)
Reflux to the top (mol /sec)
Molar hold-up of the condenser (mol)
Liquid hold-up of the plate (mol)
Vapour hold-up of the plate (mol)
Feed flow rate (mol /sec)
Subscript of feed plate
Ordinal number of plate
Transfer function of the still
Pressure (atm)
Pressure in the heating system of the still (atm)
Pressure in the condenser (atm)
Heat flow to the heating wall (kcal /sec)
Heat flow from wall to substance (kcal /sec)
Elementary transfer function of the still
Latent heat (kcal/mol) .
Surface area of liquid hold-up (dm')
Heating wall area on steam side (m2)
Heating wall area on liquid side (m2)
Height of liquid level on plate
above the vapour nozzle of the bubble-cap (dm)
Mean temperature of heating wall (?C)
Mean tempetature of heating wall on the steam side (?C)
Temperature of heating wall on the side of the
heated substache (?C)
Free energy (kcal)
Free energy of the heating steam entering the still (kcal)
Free energy of condensate leaving the still (kcal)
Flow rate of vapour through column (mol/sec)
Volume (1)
Steam volume in the still heating system (1)
Concentration of the more volatile component in the liquid
(mol %)
Concentration of the more volatile component in the product
(mol %)
Concentration of the more volatile condensate component
after the condenser (mol %)
Concentration of the more volatile product component in the
cooler of condensate (mol %)
Concentration of the more volatile component in the reflux
(mol %)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/4
XB Concentration of the more volatile component in the residue
(mol %)
XN Concentration of the more volatile component in the liquid
on the feed plate (mol %)
? Concentration of the more volatile component in the vapour
(mol %)
Concentration of the more volatile component in the vapour
on the feed plate (mol %)
Heat transfer coefficient steam-heating wall (kcal/m2h?C)
Heat transfer coefficient heating wall-liquid (kcal/m2h?C)
Specific gravity of liquid (kg/1)
Specific gravity of vapour (kg/1)
Elementary transfer function of the flow rate .of the liquid
phase
? Molecular weight
(P) Elementary transfer function of concentration molar volume
(de/mol)
H (P) Elementary transfer function of pressure
Circumference of down-take pipe (dm)
T Time (sec)
Td Transport lag (sec)
? Time constant of the elementary transfer function of the flow
rate of the liquid phase (sec)
Tp Time constant of the elementary transfer function of pressure
(sec)
Derivative time constant of the pressure-concentration link
(sec)
Tif Time constant of the elementary transfer function of the still
(sec)
Time constant of the transfer function of the condensate (sec)
Derivative time constant of the transfer function of the con-
denser (sec)
Time constant of the elementary transfer function of the con-
centration (sec)
Txp Derivative time constant of the concentration?pressure link
(sec)
Time constant of the condensate tank (sec)
YN
Tpx
Tic
Tk3
?
TX
10
1.0
0.1
0?
-90?
-180?
-270?
References
1 ANIZINOV, J. V. Avtomatieeskoje regulirovanie proces rektifikacii.
1957. Moscow; Gostoptechizdat
? ARMSTRONG, W. D., and WILKINSON, W. L. Trans. Instn. chem.
Engrs, Lond. 35 (1957), 352
3 DAVIDSON, J. F. Trans. Instn. chem. Engrs, Lond. 34 (1956), 44
? HALA, E., PICK, J., FRIED, V., and Vittivi, 0. Rovnovaha kapalina-
para. 1955. Prague; NCSAV
5 HARNETT, R. T., ROSE, A., and WILLIAMS, T. J.
Chem. 48 (1956), 1008
6 JACKSON, F. R., and FIGFORD, R. L.
(1956), 1020
7 KIRSCHBAUM, E. Destilier- itnd Rektifiziertechnik. 1950
8 MARSHALL, W. R., and FIGFORD, R. L. The Applications of
Differential Equations to Chemical Engineering. 1947. University
Delaware
9 RIJNSDORP, J. E., and MAARLEVELD, A. Use of electrical analogues
in. the study of the dynamic behaviour and control of distillation
columns. J. Symp. Instrument Comp. Proc. Develop. Plant Design.
London 11-13 (1959)
10 ROSE, A., JOHNSON, C. L., and
Chem. 48 (1956),.1173
11 ROSE, A., and WILLIAMS, T. J.
.2284
12 ROSENBROCK, H. H. Trans. Instn. chem. Engrs, Lond. 35 (1957) 347
13 VOETTER, H. Plant and Process Dynamic Characteristic. 1957.
London; Butterworths
14 YU-CHIN-CHU, BRENNECKE, R. J., GETTY, R. J. and RAJINDRA, P.
Destillation Equilibrium Data. 1950. New York; liii
15 ZAVORKA, J. Obecn3'T analytick3i rozbor dynamickY7ch vlastnosti
rektifikaCnich stanic s patrovrni kolonami pro deleni binarnich
smesi. OTIA tSAV 68 (September 1960)
16 ZAVORKA, J. Wpoe et nekterYch pfenosu kolony 31 (provoz 03)
ye Stalinov3'Tch zavodech, e srovnani s vysledky meteni. OTIA
CSAV, 86 (September 1961)
Industr. Engng.
Industr. Engng. Chem. 48
WILLIAMS, T. J. Industr. Engng.
Industr. Engng. Chem. 47 (1955),
100
40
lo-3
10
4
1 1 TT, 11 1 11hTt I 1 11 1 1 11
Figure 1
127/4
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Condenser
Figure 2
1,n.1 Pn Vn
M v,n Yn
Fiv,n
V
Figure 3
Zn n.2
A (P)
Figure 4
A (p)
irovri(p) okr-;0144
(P
TI (p)
Pi 441iff
M
(p) (p)
(P)
Figure 5
127/5
127/5
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/6
?
Figure 6
Vn-1 4
1
1
1
/9 4/ I\
/ I \ // I\
/ \ / 1
/ i ,
.
\ /
// Yn-2 \Yn // Yn-1
(11)\
\ Yn?1
N / \ /
/ Yn-1 1 \ / Yn I \ //
\...
// 1 \ / I \ ,/
\ I /' \
1/
7in
gn.1 I ?
'4,11
Vna2
Yn?2 Yn.1
,? I \
// ,n?21 \
I \
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
127/7
'ex,n+1
cPpp
I
Pn,n (P)
Tx,n
x5T,,,n(p)
'-,,__----
Figure 9
XN
A(p)
2n42
-4---Yr216
(
P r
cfx,n+1
E-
-
Figure 10
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
. 190/1
Some Recent Results in the Computer Control of Energy Systems
T. Vi&MOS, S. BENEDIKT and M. UZSOKI
The results of automation in the Hungarian energy system
were reported in ealier papersi, 2 .The most important equip-
ment realized is an automatic economic load dispatcher, based
on new principles, calculating the effect of the network losses
(differing from the usual solutions) from the actual network
configuration. The special analogue computer of Figures / and 2
calculates the matrix B using the well-known formulag, 4
P, = P B P, according to the scheme of Figure 3. The poten-
tiometers kg in section G of the figure have the values
=
-JT).
Ui
characterizing the generators, where QiIPi is the active and
reactive power rate, Ui the generator terminal voltage, while
k=
Ii
111
E
r=1
in section L characterizes the proportions of the loads on the
total loss, where In is the load current at the ith node; /i, is the
power station current at the ith node; and N is the direct
current model of the actual network.
By a more detailed analysis it may be proved5 that if the
voltages corresponding to the generator powers Pig (without
the network losses and formed similarly to the previous analogue
computer solutions) are the driving voltages of the scheme's
input, the voltages iBik Pk, being proportional to the network
incremental losses, are obtained at the output, which provide
with a suitable feedback the optimum load distribution consider-
ing the network losses. The papers quoted prove that accuracy
of the method (considering the neglections) exceeds the practi-
cally reasonable limits.
The analogue single-purpose machine for such a solution,
installed in the system control, takes into account, as compared
with the former solutions, the active-reactive power proportions
of the generator bus-bars, as well as variations in the load
proportions.
With the part simulating the actual network (matrix N), one
obtains an adaptive system, resetting the economic load distribu-
tion by switching the elements of network H (that is, the actual
lines, by hand, or automatically with remote control).
The experiences with the automatic load dispatcher have
shown that the methods developed up to now for evaluating
the economy (the incremental heat rate curves, plotted on
statistical bases) are not satisfactory. In connection with this,
the following problems have arisen:
(a) Continuous evaluation of the economy characteristics
(efficiency, increment costs).
(b) Determination of the estimation periods for the data
processing and economic load distribution, permitting filtering
of the measurement uncertainties and other short cycle, transitory
disturbances, but giving information about the effect of the
system variations (e. g. fluctuations in connection with the
frequency and power control).
(c) Measurement accuracy corresponding to better calcu-
lating and data processing possibilities and improvement of
the sensing elements.
(d) Calculation of the transient phenomena effect (e.g. un-
load, increase in load) in the automatic load distribution system.
(e) Problems of availability, probability of breakdown,
objective judgement of the operation during partial disturb-
ances, or unfavourable service conditions for an automatic
dispatcher.
(f) The complex logical decision problems of the automatic
energy system dispatcher control for searching the most
favourable network cennction manipulations.
Among the above problems (a) is generally solved, and a
great number of power system data processers are operating.
It is worth mentioning, that as regards development, these
problems are not resolved. The endeavour for a practically
pprfect service safety, the complications of practice in connection
with the electromechanical output equipment, the reasonable
combination of the analogue and digital elements, and the
development of a more reliable and cheaper annunciator system,
rendering the whole apparatus less expensive, justify numerous
new solutions.
Problem (b) must be regarded as the most open one. The
digital instruments have generally a class accuracy of 0.1 per
cent, while the digital computing technique is practically of
absolute accuracy. At the same time the power system measuring
and control instruments are of class 1-2 per cent, but in practice
instruments and sensing devices for a higher accuracy can be
reproduced, but these are accuracy limits under service con-
ditions. Determination of the most important quantities, such
as fluid and solid material flows, heat content, ash content, etc.,
leads to the greatest number of uncertainties, and here the
measurement accuracy is 2-5 per cent. The error is increasal
with the data calculated from such uncertainly measured values,
e.g. with the quotient formation necessary for the efficiency. This
is the pivotal question and basic contradiction of the whole
energetical optimization. We want to attain prospective efficiency
improvements of 0.5-1 per cent, the sum of which may be in one
country in an integrated average many millions, perhaps many
tens of millions, of dollars per year, based on a measurement
uncertainty of 1-5 per cent.
Up to now attention has been concentrated on the prob-
lem that has not yet been completely solved, i.e. continuous
190/1
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
190/2
and more accurate control of the coal heat content value, this
being all the more justified in Hungary as the fuel quality varies
considerably, and most of the power stations are thermal ones
working with coal. Based on some former results 6-8, 11, a definite
improvement has been attained in the field of coal analysis by
radioisotopes', 1". We have succeeded in deriving a method
permitting the continuous control of the heat content of the
coal, at least within the accuracy of the laboratory calometric
method, which is unacceptable from the statistical sampling
point of view. The main experiences were as follows:
(a) In the case of thorough sample preparation a correlation
of better than 0-9 may be reached with laboratory calorimetric
control, which covers the uncertainty range of the laboratory
method measurement accuracy.
(b) With mechanical sample preparation, the measurement
accuracy is much influenced.
(c) In the case of considerably varying coal composition, a
multi-ray method of different discrete energies can be advised,
combined with a suitable, simple computer.
Generally it may be concluded that the next most important
step in process automatization will not so much concern the
automatic system itself, but rather the development of the
quality analysing and quantity measuring devices and sensing
elements.
In designing the process optimization a very important and
insufficiently considered viewpoint is the determination of the
estimation period of the characteristic to be optimalized.
This view is especially clear when optimizing the efficiency,
as efficiency samples taken for too short a time may lead, for
example, to an efficiency value exceeding one after a former
storage period, while sampling periods which are too long eli-
minate the possibilities of estimation of system variations that
may be important for optimization. Determination of the ideal
sampling period is complicated by the fact that in a boiler the
transit and storage time constants of the single energy quantities
are extremely different, changing even during the operation.
The ideal accurate evaluation of the efficiency could be accepted
only for a complete start-operation-stop period. The criterion
of duration regarding the estimation period xis that the deviation
between the efficiency calculated from the efficiency taken for
the total operation time and from that taken for the partial
times should not exceed the error e, caused by the measurement
inaccuracy, i.e.
xou, dt
1 n 1 n
= n Erli?8= E
n
ti-1-
xoutdt
+8
n=
xi,, dt
x,,,dt
where xi,, = power quantity supplied during the measurement;
)(out = power quantity taken out during the measurement;
tb = initial time of measurement; t = final time of measure-
ment; 17 = process efficiency; ni = efficiency of the ith partial
time, and n = number of samples taken during the whole
process.
The estimation period must be chosen as the shortest one
meeting this accuracy criterion.
There may be several practical solutions among which the
most simple is the working with a time x fixed by experience
on the basis of the above criterion. With the boilers used in
Hungary there is an interval of 10-15 min, taking values into
consideration only if deviation between the output and input
energy levels is less than 3-5 per. cent from the beginning to the
end of the estimation. The greater variations are, in any case,
to be processed separately. The other system adjusts adaptively
the evaluation interval on the basis of the auto and cross correla-
tions of the output and input energy characteristic. The numeri-
cal results show also, in an apparently entirely identical mode
of operation and circumstances, efficiency changes of 2-6 per
cent. This is partly due to the considerable variations in the fuel
quality. A test made in Czechoslovakian shows variations of
-I- 10 per cent for coal quality fluctuations within a very short
time. The experiences in Hungary gave similar, or even worse,
results, and coal quality fluctuates sometimes by minutes. The
effect of the system power and frequency control on the change
of efficiency is also most interesting, the load fluctuations
having relatively rapid frequencies resulted in an efficiency
deterioration of 2-3 per cent in some cases, against the same
level steady state operation. The experience in Czechoslovakia
justifies the introduction of a corrective control working on the
basis of quick coal analysis, while that in Hungary demonstrates
the necessity of sensing the effect of the relatively faster changes
upon the efficiency.
From the foregoing it follows that the former view of the
static load distribution is not satisfactory for calculating the
economic load distribution, and the costs of the necessary
alterations (heating, unload, switchover, etc.) must be con-
sidered.
The problem is clarified by the following example. A power
station is operating with four identical boilers, each being
loaded to 90 per cent. If the demand increases so that loading
of the boilers is to be raised to 100 per cent, the alternative may
be considered, i.e. starting a fifth boiler of similar capacity, as
a consequence of which the single boilers may operate with
80 per cent load, generally the optimum efficiency level. In this
case the expenses of the transients (start, possible later stop,
loss of life due to manifold start and stop) must be compared
with the savings of the more economical steady-state operation
for the expected interval. These circumstances are taken into
account already, though in a more simple way, in the present
load distribution practice.
The former static load distribution methods are to be gen-
eralized to an optimum energetical programming, taking into
consideration also the presumable changes. These methods
start, as a rule, from Lagrange's method of constrained extrema
and are calculated on the basis of the equal incremental costs.
The generalized task is the typical case of the multi-step decision
problem. On the basis of the power demand given, the system
must be programmed in an optimal way, considering afterwards
the transition to the power demand expected for subsequent
periods and the optimal mode of operation on the new levels.
Considering the calculating difficulties and practical demands,
the programme was realized for two steps, consequently,
besides the system performance level given, the search for the
optimum is realized for the next two levels. Consideration of
the second change provides information about the first alteration
being justified. (In our example the heating up of the new boiler
is made reasonable by the time elapsing until the next change
and by the direction of the next variation.)
190/2
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
190/3
The optimum energetical programming must calculate the
availability of the system and its units also, and therefore the
probability factors must be considered not only when estimating
the power demand, but also when calculating the available
power system capacities and network interconnections. In the
period to be planned, the system service conditions are charac-
terized by the prospective capacity distributions of the individual
units (power stations, machine units, etc.), that is, the probabili-
ties of the available capacities as a function of time, and further,
the probability cost values relative to these. These cost values
are probability variables not only in the sense that they belong
to probable power values, but they are also' in themselves only
probable values, e. g. the efficiency of the condensation machines
is considerably dependent on the cooling water temperature,
and consequently on the probable factors of the weather. The
third important characteristic, from the optimization point of
view, as mentioned earlier, is the excess cost of the transient
states, this corresponding not to the expenses integral taken
along the static time diagram of the given capacities, but gene-
rally exceeding it.
Consequently, for the predictive characterization of the
availability, the following are needed. (1) Probability distribution
of the capacities depending on the time and direction of tran-
sients, (2) the probability cost distributions belonging to these
values, and (3) the time integrals of the expense distributions
along time tables to be considered.
That is, the availability A is a set:
A = {pi =fi (P, t); 1)2= f (K , , ; p3 = f 3 (P , K , K dt)}
where pi, p2, 1,3 are the probabilities discussed above.
Accordingly, the availability A at instant t is the set of the
possible power capacity values P, where to each value P belongs
a value K (the costs of the service in steady state conditions) and
to each curve Pi = F (t) belongs an integral cost curve S K dt.
The task of optimization is as follows. The lines of constraint
of the possible power capacities Pi = P (t) are given, that is,
the boundary surfaces of a solution space of dimension n, the
lower and upper power limits belonging to the individual
units and changing in time. Given the probable system power
E (t) = P1 (1). The trajectory of the vector P of n dimensions
is to be determined (the vector characterizing the power output
condition of the system units), the vector being, under the above
conditions,
.11
E Ki [P opt ( t)] dt
t=i to
that is, providing the minimum of the cost integral taken along
the trajectory. The line integral taken along the trajectory in
the coordinate space P forms no conservative space, as the line
integral is not independent of the path and the integrals taken
along the closed curves (the cost of returning to the same power
distribution) is not zero.
The probability influence of the availability and of the costs
have been derived according to the following considerations:
(a) One determines for all equipments the operation time
permitted on an experimental base, that may be considered?if
there is no special fault indication?as a time of practically
perfect safety. During this time the service costs of operating
the apparatus in steady-state conditions correspond to the
value calculated in general till now.
(b) At the beginning of operation (primary disorders) and
over the service time permitted, the probability of outage is
greater. Here a penalty tariff is stated, depending on the time
and calculating from the former outfall statistics and from the
probable economic consequences of the outfall.
(c) Similar penalty tariff is stipulated in case of some error
signals (fault indications).
(d) For all important units the transient costs (the expenses
of the transient conditions) obtained by experience or calculation
are stored, adding to this in some cases the penalty tariff cal-
culated from the disturbance danger relative to the transition.
The above data can be elaborated by the individual power
station data processers with a relatively small storage and time
requisition to data necessary for the load distribution. These
are the curves corresponding to the classical increment cost
curves, corrected by the penalty tariffs considering the avail-
ability, the possible time functions of the transients and the
integral cost curves of the transient conditions. For power
stations a relatively slow processing of about 10-20,000 data
is needed and the communication of about 300-400 data with
the central load dispatcher, as a result of the above calculation.
The latter must be dispatched only in case of and to the degree
of change. The knowledge of these 300-400 data per power
station accomplishes the two-step optimalizing programme
mentioned earlier.
In this manner, with the aid of suitable power station data
processers, by the otherwise available telemetering channels and
by a central, medium size computer, energetical automatic
optimization may be realized, which takes into account the
economic consequences of the power system transient conditions
and of its availability, and also the changes in production costs
and efficiencies during operation.
The optimum system control referring to the whole power
system does not make superfluous the optimization of the indivi-
dual control circuits, which may be considered partly to be
autonomous. Reference is made here, for example, to the control
of coal pulverizers, which may be controlled directly by a
continuous analyser of ash, assuring the given fuel quantity as a
primary condition. As against the non-interacting control
systems suggested recently by many authors, installing fixed
matrix connections into the control circuits considered previ-
ously autonomous, we think to be rather practicable such semi-
autonomous adaptive circuits, as the rigid functional connec-
tions give suitable results only under perfectly steady-state
conditions (e.g. time constants), this condition being chiefly
realized with boilers.
In the course of the dispatcher control automization, the
question arose to what extent the dispatcher work may be
mechanized in addition to chart preparation and beyond the
tasks of the continuous economic load distribution. This idea
is supported by the fact that the switching, manipulating and
failure suppression activity of the dispatcher control is motivated
by subjective factors; extremely hazardous decisions must be
made in a short time, and the presence of mind, momentary
mood and luck of the dispatcher influence considerably his
activity in this field. Mechanization of the task is complicated
by the fact that the methods of judgement of the situations were
partly subjective ones, based on the experiences and intuitive
improvization capabilities of the dispatcher, as there is no
possibility for accurate analysis in the case of rapid decisions.
190/3
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
190/4
Accordingly, mechanization of the dispatcher control permits-
the application in practice of cybernetics in a narrow sense, and
the adoption of recognition and heuristic search.
In the course of elaborating the problem the method of
approximating the tasks step by step has been chosen, selecting
a single logical task of the dispatcher control. It is seen, taking
into account the present machine capacities, that the question
arises as to how this task can be elaborated and, after solving
this, the kind of further tasks that remain for the dispatcher to
solve. By this one can remove from the total dispatcher's activity
the parts having not been exactly formulated up to now and
examine their weight in the total work and how to handle them.
In any case, as more tasks -are mechanized and separated from
the dispatcher's control, the more time and possibility remain
for accomplishing the part demanding the most complicated
intellectual activity.
As a first task, estimation of the possible circuit diagram was
examined from the overloading point of view. Similar calcula-
tions (load flow programmes in the network) have been made
regularly for more than a decade on digital machines, but this
was the first digital computer application in the power systems.
At the same time the method S complying with computer re-
quirements are not fully practicable for automatic control
purposes, due to their other demands. Here the analogy of
differences between the measuring instruments and the sensing
elements of automatic control must be referred to. In sensing
systems, however, detecting identical quantities using identical
physical principles as measuring instruments the difference in
their field of application, demands different approaches. For
automatic control we have confined ourselves to a load flow
computing method of an accuracy of 5-8 per cent, but being
most rapid, providing the results for a 40 node network on a
medium size machine in less than 1 sec. The storage capacity
demanded is about 1,000 words over the programme. Otherwise
the method was a generalization of the well-known method of
current distribution factors and imaginary loads, reducing the
evaluation of a network of n nodes to the solution of a complex,
linear equation system of about i unknowns, if the calculated
network differs from a basic configuration with i lines.
The next step was the determination of the optimum connec-
tion configuration of the connection manipulations (main-
tenance, disturbance) tested from a safety point of view. When
elaborating the programme, the theory of games has been
adopted, interpreting the dispatcher's work as a two-step game
of two persons, a game against nature. The pure strategies of
one of the players, i.e. the dispatcher, are the available connec-
tion manipulations, while those of the other player, that is,
the nature, are the disturbances imaginable in the system. The
game is two-stepped, first a favourable main network connection
diagram is selected by the dispatcher, not knowing yet what kind
of disturbances may arise during the validity of this circuit
diagram. After this the nature 'moves', a possible disturbance
ensues, and as a last step, a changeover strategy is chosen by the
dispatcher which reduces the power limitation produced by the
disturbance to the minimum. To adopt the minimax principle,
a suitable pay-off criterion had to be found by which the elements
of the game matrix may be filled and the optimal strategy may
be evaluated. This criterion is established on the basis of the
damage caused by the possible power outage and the weight
functions formed by the outage probability. On the basis of
several considerations, the outage probability p is not directly
applied for weighting, but this is done, however, with the
relation
1
k=f(pi)=1_ in pi
so the criterion of the optimal game is:
min max III; IC-" 1
1-1n pj
where Wi; is the power outage caused by the ith dispatcher's
strategy and the jth disturbance possibility (kWh), Ic; is the
specific damage due to the above disturbance (S/kWh), and
p; is the probability of the jth disturbance.
The machine time for analysing a complete situation in the
case of a medium size machine and of a starting position deviating
not from the normal one but at most with the state of the four
lines is about 4-5 min for a 40 node network, its storage demand
being without programme about 800-1,000 words.
The availability of the network, and that of the power
stations, may be considered along similar lines making use of
the suggestions mentioned earlier, thus extending further the
possibility of the objective evaluation of the network configura-
tion. The programme evaluating the manipulations may include
the data referring also to the stability. As examination of the
stability conditions of a single situation demands considerable
time even by a computer, the application of the pre-calcu-
lated, stored stability data, as well as the continuous proce-
ssing of the data of the stability reserve indicators, are referred
to here.
Control of the dispatcher by computers would not make
superfluous the application of less complicated network auto-
matics, such as protections, overswitch and backswitch auto-
matics, etc.
It must be emphasized that in the field of the present sum-
marizing report on the authors' developments and ideas, these
are up to now mainly theoretical achievements calculated for a
mathematical model, prepared for simulation on a digital com-
puter. Their expediency and adaptability must be decided, how-
eyer, by practice, for many technological and other realization
difficulties must be overcome.
References
Uzsom, M. and VAMOS, T. Some questions regarding control of
power systems. Automatic and Remote Control. 1961. London;
Butterworths
2 VAMOS, T., Uzsom, M. and BOROVSZKY, L. Novilj, nyeposz-
redsztvennyj, masinniij szposzob ekonomicsnova raszpregyeljenyija
nagruzki mezsdu elektrosztancijami i nyeszkoljko voproszov
szvjazanntich sz optimizaciej enyergoszisztyem. Symposium, of
Automation of Large Energetical Units. 1961; Prague.
3 KRON, G. Tensorial analysis of integrated transmission systems.
The six basic reference frames 1. Trans. Amer. Inst. elect. Engrv,
70 Pt 11. (1951) 1239
4 KIRCHMAYER, L. K. Economic Operation of Power Systems. 1958.
New York; Wiley
5 UZSOKI, M. Uj, &pi modszer a gazdasagos teherelosztas szami-
tasara. Colloquium of Automatic Control. 1962. Budapest
6 NAUMOV, A. A. 0 primenyenyii obratnovo rasszejannovo ?iz-
lucsenyija dija avtomaticseszkovo kontrolja szosztava szlozsntich
szred. Avtornaticseszkoje upravlenyije, pp. 152-159. 1959. Moscow;
lzdatyelsztvo Acad. Nauk SSSR
190/4
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
7 PIVOVAROV, L. L. 0 primenyenyii javlenyija.pogloscenyija y--iz-
lucsevyija dlja avtomaticsdszkovo kontrolja szosztava mnogo-
komponentniich szred.
8 DIJKSTRA, H. and STESWERDA, B. S. Apparatus for continuous
determination of .the ash content of coal. Int. Coal Prep.`Conl,
1958; Liege
9 BISZTRAY-BALKU, S., Dr. LtvAt, A., ,KAKAS, J., NAGY, M. and
VARGA, K. Szenek filtdertekonek meghatarozasa radiologiai mod-
szerrel.' Energia ?Atomtechnika, 6 (1960) 472
190/5
1? 'BISZTRAY-BALKU, S., KAKAS; J., NAGY, M., VARGA, K. and
LtvAi, A. Die Bestimmung des HeiZwerts von Kohlen durch
radioaktive Strahlung. Isotopentechnik, Nr. 5-6(1960-61)
" BELUGOU, P and CONJEANUD, P. The determination of the ash
content of coals by means of x-rays. 1st Int. Coal Prep, Conf, 1950.
Patis
12 BLER, J. Trebovanyija k regulirovanyiju energeticseszkiehblokov
sz tocski zrenyija upravlenyija energeticseszkoj szisztyemii. Sympo-
sium of Automation of Large Energetical Units. 1961. Prague
Figure 1
,
Figure 2
1 Al
Figitre 3
190/5
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
195/1
The Problems, Operation and Calculation of a New Component
to be Applied in Certain Control Circuits
o. BENEDIKT
Introduction
The object of the paper is to describe the physical operation of
a new component for the stabilization of oscillatory processes
arising in certain control circuits, as well as to give account of a
new practical method for calculating the parameters of this
component. The component inspires a lively scientific interest,
not merely because it can stabilize most effectively an otherwise
entirely unstable control circuit in certain cases, but also from
a theoretical respect, since there is no need to connect it to the
external circuit of the machine. Moreover, without increasing
the size of the machine to be stabilized, an effect is realized which
up to now could be attained only by a relatively large set con-
sisting of auxiliary devices, with an increase in machine size.
The circuits to be stabilized by the component in question
are control circuits, in which the newly developed electrical
amplifier `autodyne' is applied to maintain the load current at a
constant value (e.g. for the automatic charging of accumulator
batteries, for automatic welding, for supplying motors in
series, etc.).
In his paper 'The New Electrical Amplifier', presented at
the 1st IFAC Congress the author gave a general report on the
theoretical bases, the main application field and control circuit
connections of the autodyne, mentioning the autodyne for the
above purpose only in short. Csaki, Fekete and Borka, however,
referred to the experimental test of another kind of autodyne,
namely an autodyne maintaining the output voltage constant.
The following shows in detail the characteristics of the
transient phenomena in the autodyne maintaining the load
current, as the task and problems of the new stabilizing com-
ponent of the control circuit of this machine may be understood
only in this relation.
Comparison of the Stability Criteria of the Autodynes Controlling
Voltage and Current
The operation of all autodynes working as amplifiers is based,
independently of their concrete connection, upon the physical
phenomenon that the spatial fundamental harmonic res of the
main flux of a converter (Figure I) may theoretically take up
any spatial position (in a different state of equilibrium) with
a suitable arrangement of the split poles and a synchronous
speed no of the rotor. At the same time, this flux is produced by
a magnetizing excitation, the direction of which is set auto-
matically to the flux direction. (Regarding the problems dealt
with below, this magnetizing excitation is of no practical
importance and therefore is not shown in the figures.) In con-
sequence, with the appearance of a small positive, or negative
excitation of + rl A W' in the control winding Wy, the control
torque A M produced by this excitation and the main flux,
and also the small rotor lag or lead caused by the main flux,
the same time, the internal phase voltage , rea, (Tbeiinresg. At
U
change considerably the spatial position of the _ flux
equilibrium with the terminal voltage vector Enna, can be
displaced between the limits of 16 = 0 and # = 1800, while the
output voltage U is varying continuously between the limits
Uniax. If the output voltage U of the amplifier realized, or
another control circuit parameter depending on U, is fed back
negatively, then the parameter may be maintained automatically
at a constant value. For example in Figure 1 an autodyne is
shown stabilizing the output voltage U to the value of the con-
trol voltage Uy.
In the publications of the USSR Academy of Sciences
Technical Section, Energetics and Automatics, No. 2., 1962, the
author examined the transient phenomenon taking place in the
autodyne controlling voltage (Figure 1), using a different
simplifying supposition and neglecting the relatively small rotor
resistances.
The characteristic equation of the control circuit, using the
operator calculus, yields
A' + pB' +p2C' +p3D'=0
As a stability criterion, the following relation is obtained:
(1)
> WC4 (C C2n0+ xo) Wy? C4 ' C6
0X
a ? ry? Ci. C2 (2) 2, +a?ry?Ct ? C2
The quantities A', B', C', D', C1, C2, C4 and C6 are constants
depending on the machine dimensions.
The physical meaning of these two formulas may be illustrated
briefly as follows. Suppose the synchronous speed no of the rotor
is decreased to a value n hardly deviating from no (Figure 2), as
a consequence of which the vectors & rex and El ?a rotate by
a small angle ZI ,63 anticlockwise. Meanwhile U is increased by
U, and a control current A 1y arises, producing an excitation
ZI A W' downwards. The resulting accelerating torque Z1 M is
greatest when the vectorsand Ara, reach their dotted
the central
upper limit position. At position of the two vectors
ck
shown by the full line ZI # = 0. Evidently, ZI A W' causes the
vectors to oscillate freely around their central position, and
such oscillations would appear if the values B' and D' in eqn (1)
were equal to zero.
Nevertheless, in addition to the voltage A U, the control
winding is effected also by the voltage induced by the increment
of the direct axis component ZI of the flux AILes (resulting
from the rotation of &res), which lags the incrementrirA 01' by 90?.
The additional control current being formed evidently establishes
195/1
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
195/2
an excitation upwards when i res
has returned to its central
position, thus the excitation has a damping effect. This effect is
represented in eqn (1) by the damping term p B'. The unit of
the stabilizing flux LI C' corresponds to the left-hand side of
eqn (2) (the right-hand side is, for the time being, zero).
The voltages induced in the control winding by the fluxes
proportional to the current LI I must now be considered.
The excitation d A W' produced by 4 4? being proportional
to it and arising?in the rotor winding, is short-circuited by the
a. c. network, since according to the converter theory, all
excitation arising in addition to the magnetizing excitation is
cancelled out almost completely as a consequence of the com-
pensating current LI 4'. Nevertheless, because of the leakage
reactance x4, of the phase winding, the excitation of the current
is smaller by few per cent than LI A W', and consequently
a small flux difference LI Ox appears. As this, compared with the
flux LI is of opposite direction, it induces a current in the
control winding, which reduces the damping effect of the current
induced by the flux LI 4j.'. At the right-hand side of eqn (2) the
second term, proportional to x4? corresponds to the flux 4 Ox.
In addition to this, the leakage flux LI C. must also be con-
sidered. This is caused by the current LI 4, which passes through
the control winding. To this corresponds the first term, propor-
tional to the leakage factor A5, at the right-hand side of eqn (2),
which proves that the stability degree is now even lower. The
effect of fluxes LI 4 and LI OA is represented in eqn (1) by the
term D' p3.
The right-hand side of eqn (2) is proportional to the expres-
sion Wvla ? rv; where W is the number of turns of the control
winding, a is the number of parallel branches of the control
winding, and rv is the resistance of this circuit. This expression
is obviously proportional to the cross section of one turn of the
control winding. The greater this is the greater is the increment
of the current LI 4 corresponding to the angle LI e8, as well as
the value of LI A W', and, evidently, the steady-state control
accuracy, also. On the other hand, the value CI Ow + LI OA is
increasing together with LI 4. However, owing to the fact that
at a given value of LI /3, LI 4?,' remains constant, it may be con-
cluded physically?as shown mathematically by eqn (2)?that
as the cross section increases, the stability is reduced. If the sum
LI ckx ? LI 01 were equal to LI 4),', then obviously no voltage
would be induced in the control winding and free oscillations
would again arise, while the two sides of eqn (2) would be equal.
In practice this never occurs, as a satisfactory control accur-
acy may be realized by small values of Wv/a ? rv, at which the
stability limit is very great.
The case is quite different with an autodyne used for control-
ling the load current to a constant value, e.g. in spite of the
variation in the internal voltage EA of an accumulator (Figure 3).
To demonstrate this question theoretically in a more simple way,
compare Figure 3 with Figure 1.
At first sight the difference is great. Actually, to the winding
W' (working in this case instead of WO the loading current I is
fed back, not the voltage U. Further, in this winding, not two
voltages (U and Uv), but two excitations are compared, that is,
the excitation I W' with the excitation i, Wv of the continuously
controllable regulating current iv. Consequently, instead of the
law U = U5 the law I = i, W,/W' is valid here.
However, examining the problem of the transient phenomena,
an important analogy of principle may be observed between the
two connections at once, as in this question the magnitude of the
current iv is practically of no importance and it may be made
equal to zero. In this case, however, the connection of Figure 3
does not differ in any respect from that of Figure I, as EA may be
regarded as the given control voltage, while the control winding
is connected to EA and to the voltage U. In consequence the
factors illustrated in Figure 2, affecting the stability, may be
distinguished also in the autodyne shown in Figure 3 and if
= 4 4 + LI OA, free oscillations also arise here. Moreover,
it may be seen that in this case the presence of winding. Wv may
not practically cause any deviation either, as the fluxes mentioned
pass also through this winding and so to the latter, and if the
fluxes balance each other mutually, no voltage is induced.
Accordingly under the same conditions as have produced eqn (2),
a stability criterion corresponding theoretically to eqn (2) must
also be obtained. From this, however, follows the interesting
fact mentioned below.
While, in the case of Figure 2, the control winding forms a
shunt winding, and consequently the cross section of its turns is
very small; with an autodyne maintaining its load current at a
constant value, the cross section is very large, because the W' is
series connected. This means, however, that the right-hand side
of eqn (2) is, in this case, incomparably greater, i.e., there is an
actual danger of oscillations arising. This has in fact occurred
in practice at an early stage in the development of the autodynes.
It is to be considered that (compared with Figure 2) in the
case of Figure 3 the resistance of the rotor may not be neglected
with respect to the actually small resistance of winding W'.
As the current LI I must now overcome the resistance of winding
W', in addition to series-connected resistance R, the effect of
Wvla ? rv will be somewhat smaller. It is clear, however, that if
this tefrn is replaced by W'/L' R, being physically analogous,
the latter will still be incomparably greater, than W5/? rv in
the case of the autodyne controlling its output voltage to a
constant value. So it is proved that the autodyne shown in
Figure 3 can perform its task only if provision is made for its
stability by some supplementary means.
Problems Concerning the Development of a Suitable Stabilizing
Device and the Way Leading to the Solution
The auxiliary devices for stabilizing circuits, in which the
loading current is to be maintained at a constant value, are
theoretically known.. This is obtained as follows (Figure 4).
Assume the autodyne operates just at the limit of lability,
as a consequence of which sinusoidal currents LI I are super-
posed on the current I. These would induce sinusoidal voltages
in the transformer T, the primary coil of which is series connected
with the load circuit. If the capacity of this voltage is increased
by the amplifier A and the stabilizing winding W, is joined to
windings WI, and W' of Figure 3, with a suitable connection
there arises in W' an excitation leading in time with regard to the
excitation LI I W' and proportional to it. In this way effect of
fluxes LI Ox + LI 4 could be theoretically reduced by well-known
means.
Nevertheless, this arrangement has several great disadvant-
ages. The additional winding increases the machine dimensions.
Further, through the application of auxiliary devices, the service
safety is reduced. It must also be taken into account that the
dimensions of the transformer T are considerably increased,
because its primary coil must be dimensioned for the total
195/2
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: ClA-RDP80T00246A023400480001-9
load current I and saturation of the iron core of the transformer
by the load current I must be avoided. The other stabilizing
devices of the classical control technique to be adopted here
have similar disadvantages.
Accordingly, it has become essential to seek a novel device
for additional stabilization, permitting elimination of the
disadvantages mentioned above.
Actually, it has been proved that the physical processes
corresponding to Figure 4 may be realized without the applica-
tion of a transformer or amplifier, while the required additional
winding may be placed in the machine in a way which does not
reduce the useful winding area.
This problem is to be solved step by step as follows. (1) In
order to spare the primary coil and flux of the transformer T,
instead of this flux another existing flux, already in the autodyne
and being proportional to the current zI I, is applied to produce
a current in the winding to be placed in the autodyne, playing
the role of the secondary coil. This current must lag behind zI I.
(2) To amplify the effect of this current, a generated voltage
proportional to it is established in the autodyne. (3) To eliminate
also winding Wc, this generated voltage is established in the
winding W' itself, that is, between the main brushes.
Meanwhile, two difficult problems arise. First, (3) obviously
necessitates that the winding sought should operate in the direct
axis of the autodyne. But then it is inductively coupled with the
winding -Yr, which eliminates the effect wanted, i.e. only a single
suitable additional generated voltage should affect this winding.
On the other hand, the following problem arises. If the new
winding is placed in the direct axis, then the fluxes LI 0, 4 OA,
being proportional to current LI I?, will pass through it, and,also
the flux LI0,'.
It is already known, that free oscillations arise, when the
sum of these fluxes is zero, in which case no current is induced
in the winding, and therefore the desired effect does not arise at
the occurrence of the free oscillations. From this it follows that
the tested winding should fulfil the following, apparently con-
tradictory conditions: on the one hand, the magnetic effect of
the current arising in it should fall into the direct axis of the
machine, but, on the other hand, the direct axis flux LI 0.1'
should not be enclosed by the winding, consequently, the flux
enclosed by it and being proportional to the current I must
not exercise any effect in the direction of the direct axis.
This problem may be solved by a special shape of the tested
winding and also by the winding having a particular physical
function.
Physical Operation and Method of Calculation of the
Stabilizing Winding
In view of the fact that the autodynes as Figures 1 and 3 have
a practically analogous behaviour regarding the transient
phenomena, the following consideration should be valid also
in the case shown in Figure 1. Therefore, instead of LI land W',
the physically similar symbols 4 4 and Wy shall be applied.
The stabilizing winding, as shown in Figure 5, has the shape
of a figure eight and is placed, according to Figure 6, to the pole
shoes of the half-pole I and II belonging to the pole pitch.
Thus the condition that they should not be inductively coupled
with the winding Wy, is fulfilled. The condition, that the flux,
proportional to the current LI 4 and enclosed by the winding,
195/3
does not exert any effect in the direct axis of the machine, may
be fulfilled on the basis of the following consideration.
As is known, the compensation current /1 /1', corresponding
to the excitation 4 4 W5, is proportional to the current LI 4.
In the airgaps below the half-poles the induction of the flux
produced by A corresponds evidently, within a pole pitch 25,
to the ordinates of curve 1 2 3-4 5 6 7 8 9 10 in Figure 7.
If, everywhere, constant inductions of the flux of the same
magnitude are represented with the aid of line 1-11-12-13-5-
6-14-15-16-10, it becomes apparent that the area 12-3-4-13-12
is equal to the difference of areas 2-11-17-2 and 17-3-12-17.
As a result of this, the part of the area 3-4-13-12-3 of the flux
proportional to LI 4, as shown in Figure 5, enters the half-pole
through one half of the stabilizing winding and leaves on the
side of its other half, that is, it is twofold inductively coupled
with this coil. Obviously, the situation is just the same in the
other half-poles. If the total flux being established is denoted by
LI 08 and the ordinates of curves 2-4, 7-9 by LI B (x), then,
adopting the above symbols
x (I + a)
?
A 08=K 21f (x)dx C,2 .m; (3)
X =
4
= (1+
4
? MI
and / AB (x)dx = C5
2
x=.32-(1 -a)
4
(4).
where 1 is the active length, C5 LI 4' is the flux produced by
ZI /1', and K is the factor considering the saturation. (The cause
of C5 being constant in spite of the saturation is explained in the
paper mentioned previously.)
It follows from eqn (3) and (4) that
Tr ? OC
1 - COS --
4 C8L1F,
- (5)
A08
7r-oc 2
sin
4
On the other hand
= K, ? yWy (6)
where, as is known, K, is a constant depending on x The flux
LI 08 induces a current of
AI8= 1 c1408 (7)
r 8 d t
in the stabilizing winding, where r8 is the resistance of the winding.
For the sake of simplicity, the inducing effect of the stray
magnetic field of the winding is neglected here. As shown by
theory and practice, this is permissible, because the frequency of
the free oscillations is insignificant.
Thus, up to now the secondary coil of the transformer T
has been replaced by a winding corresponding to Figure 5,
while the transformer primary coil and its iron core become
195/3
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
195/4
superfluous. The production of the generated voltage mentioned
in (3) between the main brushes of the rotor, will be attempted
with the aid of current A /8.
At first sight this seems to be impossible. Namely, the
excitation A 0, produced by current A /8 is obviously
0,=4/8 (8)
that is, of the same magnitude but of opposite direction in the
two half-parts of the figure-of-eight winding. Thus, the excita-
tion is divided on one pole pitch according to the line 15-1-2-3-
4-5-6-7-8-9-10-11-12-13-14-17 of Figure 8. As the induction
LI Bg established by A 0, is distributed practically in a similar
way, and as a result of this the total flux arising on one pole
pitch yields
=r,?
zIckg=fx 409c1x=0
x=0
(9)
it may be concluded that the flux produced by the current 4 /8
of the winding cannot produce the generated voltage required
in the d. c. winding of the rotor.
The problem can be solved if it is considered that the excita-
tion zI 0, must have a positive fundamental harmonic in the
case of the distribution in Figure 8, because the positive areas
4-5-6-7 and 8-9-10-11 are closer to the central line than the
negative areas 1-2-3-4 and 11-12-13-14. The harmonic analysis
proves that the amplitude value of the fundamental harmonic
is
8 n?a
0,, = eg 7c.\ /2(1 cos) (10)
2
Accordingly, this excitation has an inducing effect on the phase
winding of the rotor and consequently is practically eliminated
by a compensating current A /?, because 4 i is proportional
to the excitation, A 09? having produced it, i. e.
21/91 = K24 0,, (11)
where K2 is another constant depending on the value of xd,. The
induction produced by the excitation of sinusoidal distribution
of this current A 4, is evidently distributed in the same way as
the induction produced by the excitation of the current A Jr',
but in the opposite direction. Consequently, the integral of this
induction taken within the section -r? establishes the flux formed
by the current A 4,, which ?produces the generated voltage
wanted between the main brushes, the latter operating opposite
to the voltage induced by AOx and A OA in W,. Considering the
fact that this voltage is proportional to 4 4,, as well as eqn (6),
(5), (7), (8), (10) and (11), the generated voltage may be made
equal to
K8 ' zlIy(p)
where K8 is constant. On the other hand, the voltage produced
by the fluxes A Ox and AI OA is obtained as p D ? A Iy(p), where D
is constant. Performing the substitution
K'8 =K8 (C1 ? no ? C2 + .X0)
0 ? co
pit
where co is the angular frequency of the rotor, 0 is the moment
of inertia of the rotor, and p, is the number of pairs of poles, the
characteristic equaton is, after all
(12)
A' - F p/3' + p2 ? + p3 ? (EY ? 0 (13)
r8
The stability criterion:
1 >2 5 1 2 0 +
a ? r
W [ C4.(Cn)
C
CiC2 C4C61 Kf8'
Ci C2 r8
where K"8 is a constant, in which W, does not figure. It is recog-
nized that the stabilizing winding is actually in possession of the
effect demanded, as for instance the term comprising p3,
reducing the stability and, in an analogous way, the right-hand
side of eqn (14) may be decreased most effectively, if the cross
section of the winding, i.e. 1/r8, is suitably inEreased. With
extremely high values of Wy the first term of the right-hand side
is increased and there is no place in the machine for giving a
cross section so large to the stabilizing winding that would
suffice for a sensible decrease of the first term. Therefore, in the
cases illustrated in Figure I, that is, in the autodyne controlling
the voltage to a constant value, this winding has not been
applied. Nevertheless, in the cases of Figure 3, where the value W'
taking the place of Wy, is small, calculation shows that the right-
hand side of eqn (14) will be zero with such small cross sections,
which (with the stabilizing winding set on the pole shoes) has
practically no effect upon the machine dimensions.
The autodyne of serial production, provided with such a
winding and maintaining the load current at a constant value,
would operate without the above- mentioned winding far within
the unstable range and prove itself entirely stable in practice.
(14)
195/4
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
195/5
Figure I.
Figure 2.
195!5
lp
wp
wc
Figure 3.
Figure 4.
Figure 5.
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
195/6
15
pH 5
4
Tp
-2-
Figure 6.
13
14
Tp
Figure 7.
9
AOg'i
118
I *
2
11
12
--aeg
14
13
Figure 8.
195/6 .
17
?
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
283/1
Optimalization of Non-linear Random Control Processes
R. KULIKOWSKI* p
Introduction
In the theory of optimum control systems it is usually as-
sumed that the plant differential or operator equations are
completely known to the controller. In such cases, through the
application of known optimalization techniques, the optimum
control signal can be derived by an analogue or digital computer
and applied to the plant during any time interval. However,
there are known systems such as chemical plants and aircraft
whose differential equations are not known completely to the
controller because of environmental changes, ageing, etc. In
many systems of this kind the best that can be accomplished is to
construct a multistage optimalizing process which converges to
the optimum control. All the necessary information for the
construction of such a process can be obtained by observing
outputs of the plant at every stage for known inputs. Applying
this approach to non-linear, zero-memory plants, the gradient
of the performance measure can be determined and known
iteration methods, based on the gradient concept (such as
steepest descent, non-linear programming, contracting iterations,
Newton method, etc.) can be appliedl.
If it is desired to extend these methods for the case of non-
linear and random plants having memory (i.e., possessing
inertial elements), with the object of obtaining a stable opti-
malizing process, one should first define and determine ex-
perimentally the generalized gradient of the performance measure
and then construct the convergent iteration process. It will be
shown that these problems can be solved successfully, at least in
the case of certain classes of non-linear, inertial, random plants,
by using some concepts of non-linear and probabilistic functional
analysis. However, since the writer realizes that one of the main
purposes of a short technical paper is to present the arguments
and results in a form which is understandable for the majority
of engineers, an attempt has been made to avoid abstract
formulations. The more delicate, formal questions are therefore
explained in Remarks I, II, III which can be omitted during the
first reading.
Assumptions
(1) It will be assumed that the controller generates signals
x (t) which may be subject to certain constraints, such as
volume or energy constraints, i. e.
SoIX (t)IP dt -,L=const. where p= 1,2
or amplitude constraints, i.e.
max Ix (01 M = const. etc.
(I)
(2)
* This research was partially supported by the National Science
Foundation under Grant NSFG-14514.
The controller can observe the output y (t) of the plant for
every x (t) applied to the input by the feedback loop (see
Figure 1).
(2) We shall also assume that the form of the output-input
relation of the controlled system can be described with sufficient
accuracy by a non-linear, twice differentiable, integral operator.
This operator for example, may be of the polynomial type:
y = A (x)= A0 (t)
m
+E
i = 1
fT
0
ki (t ;
fT 0
dTi
(3)
where the kernels k, and the function Ao (t) are generally un-
known to the controller.
The differential dA (x, h) of the operator A (x), which is an
extension of the usual concept of the differential of a function,
can be defined as
. 1
dA (x, h)= hrn? {A [x (t)+ yh (t)] ? A Ex (t)]}
Y
dy
?d A [x (t) + yh (t)] y=0 (4)
where h (t) is an arbitrary function subject to the same constraints
as x (t). We assume also that it is possible to determine the
approximate value of (4) experimentally by observing the
outputs of the plant for x (t) and .x (t) yh (t) and computing3
1- {A Ex (t)+ yh (t)] ? A Ex (t)]} dA (x, h) (5)
where y is a sufficiently small number.
(3) It is assumed that a performance measure F (x) is given
IT
F (x)= I G [x, y, yd] dt (6)
Jo
where G [x, y, yd] is a known, twice differentiable function of the
arguments x, y.
As an example, consider a chemical plant (for instance, a
reactor, distillation column, etc.) described by the positive
operator A (x) [which is non-negative for any x (t)]. The amount
of steam, fuel or electrical energy delivered to the plant within the
time interval [0, T] will be equal to S I x (t) I dt, where p 1
or 2. The output product obtained in time T from the plant will
be A (x) dt. Then as the cost of running the plant in the time
T, we can take the performance expression
283/1
In (X) = (t)IP dt .1 A (x) dt
(7)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
283/2
where 21, 22 = positive coefficients which express the cost in the
accepted currency.
As the next example consideration can be given to an
autopilot-controller, which minimizes the integral error between
the desired yd (t) and the actual y = A (x) path angle of an
aircraft:
F (x) = w (t)
1Yd (t)? A (x)IP dt (8)
+ T
where w (t) --=? the given weighting function and x (t) is subject
to the amplitude or integral constraints (2) or (1).
In the case where we want to minimize the final deflection
E (T) = yd (T)? y (T) and its derivatives s(1) (t)i t = T,
F (x)= E
i= o
the weighting function
d'
de
t = T
i.e., when
(9)
W (0= E 1)i 6(i) (T ? t) and p1
i= o
should be substituted into (8). The problem becomes more
complicated when one wants to minimize the time T subject to
the constraints s(i) (T) = 0, and (1) or (2).
(4) In the general case A (x) may be a random operator, i.e.,
for the same x (t) it may be the case that y (t) = A (x) is a
random function. Therefore, in the performance measures (7),
(8), (9) the expected values will be assumed, i.e. E IA (x)}
instead of A (x).
In many cases yet (t) is not known a priori and the transient
term Ao (t) caused by the non-zero initial conditions of A (x) is
not known as well. Therefore in the case of (8), (9) it will be
assumed that the function yd (t) ? Ao (t) can be predicted so
that it will be known, at least approximately, in the interval
[0, Ti and A (x) will not depend on the initial conditions. In the
case of (7) the output y (t) is the sum of the processes due to
x (t) acting within [0, T] and x (t) acting in the past, and this
last term contributes to the output.
Now the goal can be formulated, and it is necessary to find a
control signal x (t) which will minimize the performance
measure F (x). In order to solve this problem one has to de-
termine the conditions of optimality and construct the opti-
malizing process which will converge to the best x (t).
Remark I. Speaking more precisely, one wants to minimize
the twice-weakly differentiable functional F (x), determined on
the open or closed sphere of LP [0, T] space
11X11= {foix(01,dt}"P,R,
The norm 114 in the space Loo should be defined as the
so-called 'essential maximum' or
114= inf { sup IX (t)1} , mes E= 0
E
which is, roughly, equivalent to (2). The functional (9) should
be regarded as the so-called Schwartz distribution or generalized
function. The c5(i) (t) functions can then be defined as the limits.
of weakly converging linear functionals.
The concept of a random operator is based on the notion of
the so-called generalized random variable2. Usually, in control
theory, random phenomena are described by random numbers
or stochastic processes, which are, roughly, random numbers for
any fixed time moments. It is known that the random numbers
can be defined in the axiomatic way as the mapping of the space
of events into the space of real numbers. It is possible to extend
the notion? of random numbers to the generalized random
variable, which is a Borel measurable mapping of the space of
events into some topological or metric space (in our case only,
a sphere of LP [0, T] space). More precisely2, let (Q, S) be a
measurable space and X a non-empty metric space with the
a-algebra Z of all Borel subsets of the space X. Then the mapping
V of the space D into X is called a generalized random variable
if the inverse image under the mapping V of each Borel set B
belongs to the a-algebra S, or in symbols, if {[co: V(co) a B]
: B e Z} c S.
The random operator, which can be denoted by A (co, x),
a) a Q, can be defined as the operator which for every fixed x
is a generalized random variable.
The expected value of A (co, x) can be defined as the Bochner
integral over the space Q:
E {A (x)} A =f A (co, x) dit (co)
where y is the probability measure, i.e. a non-negative, count-
able, additive, real set function with the property pc (Q) = 1. It
is assumed that E {A} exists and the expectation sign will be
treated as a linear operator acting from the random variable
space into the output signal space Y.
Conditions of Optimality
When x (t) is optimum, any variation yh (t) of x (t) should
not decrease F (x). For example, taking G [x, y, yd] = Ax2 (t)
? g [y, yd] one can express this condition in the form:
dF (x, h)= d?yF [x
=2AS x(t)h(t)dt?fT d
?dg[y,y ?dA[x+yh]dt
0 dy dy
= 22 x (t) h (t) dt ? 1 g' [y, yd] dA (x, h)dt =0 (10)
.10 .
assum ng that the second differential d2/dy2 F [x yh]ly= 0 is
positive for all h (t).
It is more convenient to formulate this condition in a form
which does not depend upon the arbitrary function h (t). If an
example is taken of the operator
A (x)= f ki(t? k2(x ? 1) x(T 1) ri] dt (11)
which has the following differential
1?
dA (x, h)=n f ki(t?)[I k2(t_ti)x(rodt, 1 IT
k 2 (-c ? h 1) r
283/2
(12)
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
one can substitute (12) into (10) and by interchanging the
integration order there is obtained
dF (x, h)= h (t)dt {2 Ax (t)? dA* (x, g')} =0
where
I Eqn ( 13)
Then it can be observed that dF (x, h) = 0 for every h (t) if
f (x)= 2 Ax (t)? dA* (x, g') = 0 (14)
The operator f (x) will be called the gradient of F (x),
[ f (x) = grad F (x)] because it can be regarded as a generaliza-
tion of the notion of gradient as commonly thought of in
analytic geometry.
When the gradient f (x) of F (x) in the neighbourhood of a
certain x (t) = xo (t) is known, it is possible to express the
decrease of F (x) along the trajectory x0 (t) + cc [x (t) x, (t)]
(where cc is changing from cc = 0 up to a = 1) by the mean
value
f(xo, x)= f. daf {xo (t) + a [x (t)? x (t)]}
.
of the gradient along this trajectory. Indeed, one obtains see
Remark II) the following inequality:
IF (x 0) ? F (x)1
re/ jf T 1.1/P
.4fT If(X0,x)Ig dt Ixo (t)?x (t)1P dt (15)
o
which becomes an equality when the two arguments (x0, x)
and x0 (t) ? x (t) are adjuncts, i.e.
Ixo x (t)IP = const.lAxo, x)1g , p-1 + q- 1 =1
Then from (15) it follows that the best change or variation of the
control signal should be subordinate to the mean gradient of
performance measure.
Remark II. It is assumed that the weak differential dF (x, h)
of F (x) is a linear functional with respect to h and therefore it
can be written in the scalar product form dF (x, h) = [ f (x), h],
where h, x e LP [0, T], f (x) e L [0, T].
To prove inequality (15) let us observe that for every number
e [0, 1] we have
?da F [x0+ a (x ? x0)]= dF [xo + a (x ? x0), x ? xo]
= { f [xo + (x ? x0)], x ? xol
Integrating this relation we obtain:
F (x)? F (x0) =f {da f Ex + (x ? x0)], (xo ? x)}
o
Applying the Holder inequality, the 'maximum principle' is
expressed by formula (15).
The necessary condition for a minimum of F (x) can be
283/3
written in the form grad F (x) = 0, where 1101I = 0,4 and for the
sufficient condition the following formula is obtained.
, d2F (x, h,h) (111111)11h11
where y (z) is a non-negative function having the property
lim y (z) = oo .
In the case of conditional minimums it must be assumed that
the functionals are strongly differentiable or, what is equivalent,
that the weak differentials are continuous with respect to x4.
When, for instance, it is required to minimize a certain F1 (x)
subject to the condition F2 (x) = c = const., then, for the
necessary condition, the following equation is obtained'.
grad F1 (x)= A grad F2 (x)
where A is a number and, in addition, at point x one has
igrad F2 WI >0.
In the case when the time T should be minimized subject to
the constraints s() (T) = 0, i = 0, 1, n, in a closed sphere of
D [0, T] space 114 R, the problem can be solved in two
independent steps:
(1) Fix T and solve the conditional optimalization problem
in an open sphere of LP [0, T], by minimizing the functional
F(x) = E (T), where Ai = constant multipliers
i=o
determined by the constraints: e(i) (T) = 0.
(2) Assuming that the norm of the solution of (1) depends
monotonically on T, the minimum T which satisfies the condition
iixii
R is found.
Optimalizing Processes
When A (x) is unknown one cannot solve equation (14) and
find the best x (t) in the first interval [0, T]. But it is sometimes
possible to construct an optimalizing process x? (t), n = 0, 1,
2, ... in the consecutive intervals [nT, (n + 1) T], which con-
verges to the best control signal. Consider, for example, the
problem of minimizing (8) which is equivalent to the solution of
the equation ye (t) ? A (x) = 0, or the equivalent equation
x = x +K [ye (t)? A (x)] = T (x) (16)
where K is a number. This equation can be solved by the iteration
xn +1(0 = T [x], n=0, 1, ...
(17)
where x0 (t) is an arbitrary function, provided the process
converges, i.e. the integral distance between x?.? 1 and xn is
smaller than the distance between xi,, and x_1
}1/P
ff
kn+i ? Xn(t)1P dt
0
r 7'
=1 j 01T (X0? 7' (Xn_i)IP dt}lIP
- m> 0.
The consideration of normalized Krasovskii evaluation and
normalized transfer functions F (p) does not affect the generality
of the assumptions.
It has been shown', 4, 5 that the Markov stability criterion
enables a solution of the inverse stability problem of linear
systems to be obtained. The generalized notion of determinant
indices of stability margin has also been introduced', 4, 5; the?
indices will be denoted by SMI (Stability Margin Indices).
(1)
(2)
287/1 .
The determination of the values of the coefficients of the
characteristic equation corresponding to arbitrary values of the
SMI is obtained according to the developed method" by inter-
mediate determination of the Markov parameters. To omit the
intermediate stage (the determination and calculation of Markov
parameters), which is specially convenient in the case of syn-
thesis of linear systems based on the qualitative Krasovskii's
integral criterion, a new method has been developed for establish-
ing characteristic equations, corresponding to any prescribed
conditions concerning the SM/6, 7. It presents a new and
independent solution of the inverse stability problem.
2. Expansion of the Coefficients of the Characteristic Equation
in Terms of SMI
The new solution . of the inverse stability problem consists
in expansion of the coefficients of the characteristic equation
in terms of determinant indices of stability margin and, in par-
ticular cases, in terms of the Hurwitz or Markov determinants
or Routh parameters. As an example of the expansion of the
coefficients of the normalized characteristic equation in terms
of Hurwitz determinants, mention should be made of Table 2,
Reference 1. To generalize the results obtained there to the case
of any degree 'n' write the characteristic equation An (p) = 0 in
the following form7:
An(p)= pn+ a 1,. pn- a2, on- 2
ak,npn-k + an, n
(3)
By considering the sequence of Routh's matrices correspond-
ing to successive values of the degree n and the equivalent
sequence of Hurwitz matrices, it can be shown that the coefficients
ak (k = 1, 2, ..., n) of the characteristic polynomial (3) can
be expressed in a unique form in terms of Hurwitz determinants7.
In particular, the following expansions of the coefficients ak,n
are obtained in terms of Hurwitz determinants A
287/1
A
;
A2 n-2 A, _
E
Ai+ 2
A21-
1
(4)
(5)
a n=?+
Al 1=1 Ai
A3 At nV Ai
Ai+1
--2
Ai+ 2
a3 =-A-- L A
n 1-11 LIO
A2k-1+ A 2k
A
1-11-1-1
k-1 A
-21-1
a2k-1,2k= A
2k-2A Ll2k-
E
i=1 A2i-2
A2i
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
287/2
A2k _LA2k+1(AO k-1 2i A2
a2k, 2k+1 = A m A A E A
L12k-1 1-12k i=1 ?-'21-1 2i+1
an,? An _
Table 1 has been prepared on the basis of eqns (4)-(7).
In the general case the expansion of the coefficients in terms
of Hurwitz determinants is expressed by the following algorithm7:
ak, n= n-i+ AA Lik - 2, n- 2 (8)
_2n _1
where a18 1 in the case of i = 0 and 0 < s < n
a6,2,-7.- 1 in the case of? a = 0, ? 1, ? 2, ...
(4,8 0 in the case of i < 0 or I> s
The expressions for the expansion of the coefficients ak,?
can be most easily obtained by means of the recurrence equations
(6)
(7)
Ak_ 3 A, A
Ak(p)= pAk -1 + 1 Pik - 2 1.(
1-')
(9)
where Ak (p) is a polynomial of degree k and Ai 1 for
1 = 0, ? 1, ? 2, ... The recurrence equation (9) holds for
1 < k < n (Reference 7). In particular, in the case of k = 1, 2
one obtains A0 (p)= A-1(p)E..- 1.
Analogous expressions may be derived for the remaining
forms of the SM/7.
Equations (4)-(9) enable the inverse stability problem of
linear systems to be easily solved. The selection of appropriate
values of the SMI should be done on the basis of a suitable
qualitative criterion of transient responses.
3. The Transformed Krasovskii Integral Criterion
As an estimation of quality of transient responses assume
the Krasovskii integral criterion J(m). It has been shown', 5, 8
that as a result of a suitable transformation the integral square
estimation J(0) can be expressed in a simple manner in terms
of the indices of stability margin. The transformed evaluation
./n(3) takes, when Hurwitz determinants are used, the form
-0) 1 a2 k - 1 ao A2A2
J 2k= 2 + + + +
a2k a1 1A3 A3A5
A2k-3 A2k- 1
1 a2k- 1 AO k 112(i- 1) 212(i- 1)
+A + E
2 a2, i=2 "21-3 A21-1
7(0)
'2k+1
(10)
1 a2k AIA, 4343 A5A5
=
+ +
A042 A2A4 A4A, Ll
+ ... + A,
+
2 a2k+1 2k-2 A2k
A
1 ( k a2 k + E L12. i- 1 A2 1-1
L' a2k+1 i=1 A21-2 A21
In order to obtain a complete transformation of the evaluation
4,(?) make use of the relations (5)-(7) and express the ratios
an ? 1/an fo the coefficients of the characteristic equation
An(p) in terms of Hurwitz determinants. It is found that
a2 k- 1 = A21-1. A2i-1
a2k 1=1 A21-2 2i
a2k ,k 2i A2i
=
a2k+1 A1 1=1 A21-1 2i+1
(12)
Observe that the evaluations (10) and (11) have different
forms in case of even (n = 2 k) and odd (n = 2 k + 1) degrees n
of the characteristic equation. Substituting in (10) the expression
(12) and in (11) the expression (13) and performing an appro-
priate change of summation indices one obtains a single general
expression
j(0)
where A, E--_- 1 is arbitrarily assumed.
The expression (14) is the transformed Krasovskii evaluation
expressed exclusively in terms of indices of stability margin*.
It holds for both even and odd degrees; that is, for any degree n
of the characteristic equation.
The above transformation of the Krasovskii evaluation may
be considered as a transition from one set of independent
variables to another. The independent variables of the first set
are the coefficients of transfer function; those of the other, the
SMI. Further investigations show that this transformation is of
essential importance chiefly because the SMI furnish much more
necessary information on the control system than the transfer
function coefficients. It is also of importance that the new
expressions of the integral square estimation take a much simpler
analytical form, which is essential for the synthesis of control
systems.
To generalize the results obtained to systems of the non-zero
class (m 0; n> in> 0) consider some of the relations
between the Krasovskii determinants and the SMI.
(14)
4. Expansion of the Krasovskii Determinants in Terms of SM/
In the general case the normalized t integral square estima-
tion takes the form
17" A(?7)
.7,1")= E B?? A -1 bmi
_
a = 0 Lin
where
xT)a,=bm2_,, ?2 b?,?+ibm-a- + 2 b.-.+2b.-.-2
+ + 2 ( ?1r -"b?,bm- 2.
(15)
(16)
for a = 0, 1, 2, ..., m and b. 1; bk 0 (k m).
The expression of the normalized evaluation (15) in terms of
SMI requires, above all, the expansion of the Krasovskii
determinants A(n),,,,_?/A? in terms of SM/7, 8. The elements of
these determinants are exclusively the coefficients of the charac-
teristic equation, therefore the unique expansion A(n),?_,,/A?
in terms of SMI may be done on the basis of eqns (4)-(9). As an
example a few of the relations obtained are quoted:
* Other forms of the transformed Krasovskii evaluation may be
found in Reference 7.
t The normalized evaluation .in(m) corresponds to the normalized
transfer function (f) p, for which one has .20 = an = b. 1.
287/2
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Table I. Expansion of the coefficients of the characteristic equation in terms of indices of stability margin of the group H (the general case)
Coefficients of the characteristic equation
?
a,
a,.
a2
a,
a4
.
a,
.
a,
1
1
A1
a, = 67)
- .
2
1
A1.
a1=--3
A2
a , = 6,1
3
1
A,
a, = A0
A2 A0A3
A,
as= -3,7
a2 - Ai + AA 2
4
1
A,
a,== A.
A2 A0A3 A?A?
A, A, A,A,
A,
=
a ?
4 Aa
a2 = Ai. + Ai A2 ? A2 A3
a3 =zi2 + Ao z12 A 3
5
1
A,
al =-a-0-
A2 A,A,
A3
a, =
(A,A, A2A5
A4
a, = A3
A, A0 A2 A2
A,
a, = A4
a2 - A1 + Ai A2
Al A4 A2 A5A,
+ +
A2 A3 A3 A4
+ Ao ,a,,,6,3 H._ A3A,
? +
A4 Ai Ai 6'3 )
6
1
A,
al = ,3
A2 AO A3
Ao
a - ?,,,
3 - "2
A1 / AiA4 AA A3As)
A4
(24 - A3
A5 ( AO A2 A2 )
A5
a =
5 A4
A A3A3
65
a6= A6
5
a2 - A). + AlA2
Ai Ai A2 A, A3 A6
+ _ +
A A +A A
A2 A3 -3 -4 -4 -5 ,
+ ?
+ AAA A A A A i
-3 -1 -4 -5 /
4_ +
A, \ Ai Ai AS I
A5 ( Al A2A2 A0A3A3 \
(AlAi +
+
A A A2 A2 A4
0 )
+ + +
A, \ 6.2 6/16,4 Al A2A4 i
Notes: (1) In the case of normalized characteristic equation one should substitute a == 1.
n
(2) n is the degree of the characteristic equation.
(3) A0 -= 1 (assumed arbitrarily).
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
287/4
vn Ai-I. At-I
An ? Ai_ 2 Ai
A(nri') An _ 2
A? An
A(n)A
m - 2 n ? 3
An An_
ALT) 3 An- 4 An ? 3 An- 3
An An _ 2 An_ 2 An _1
The determination of the values of the coefficients of the
(17) optimum polynomial /3 (p)opt reduces to that of the extremum
(minimum) value of a function of many independent variables.
To do this one must equal to zero the partial derivatives of the
(18) evaluation .77,(m) with respect to the coefficients of the poly-
nomial 11(p). One has
aj(m) a:/;,m) ai(:n) Om)
(19) ?0; =u; ... ? ? 0; ... ; ?0 (22)
b 0 Obi a b k abm- I
Solving successively for each m the system of m equations
(20) ofn(m)/abk 0 (k = 0, 1, 2, ..., m ? 1) the values of the
coefficients of the polynomial B (p)opt will be obtained, i.e., the
optimum in the sense of SMI. Thus, for instance, in the case of
m = 3 one has:
The expansions of the remaining expressions of the form
A
An
can be represented in a similar manner. Consider the sequence
{.64,71
An
of these expressions:
At(nn) Ann) AO) AO)
An An An " ; An
(21)
It can be shown that the structure of the equation obtained for
the expansion of each particular expression
A(n)
m ,
An
in terms of the SM/ is independent of the degrees a and m of
the transfer function polynomials and depends only the ordinal
number cc in the sequence
t An
This property is very useful for the generalization of the con-
sidered problem for the case of m> 0.
5. The Optimum Integral Estimation in(m) in the Sense of SMI
The value of the evaluation (15) depends on the distribution
of poles and zeros of the transfer function (2). Assume that in
the general case the distribution of the zeros is independent of
that of the poles. Then, the coefficients ./3(m),c, are also inde-
pendent of the coefficients of the characteristic equation and
cannot, in general, be expressed in terms of SMI. For any
assigned distribution of transfer function poles there exists only
one distribution of zeros of the polynomial B (p) in the numer-
ator of the transfer function, which, for the given assumptions,
corresponds to the minimum value of the evaluation J,(m).
Such a distribution of zeros will be called, in what follows,
optimum in relation to the SMI. The determination of the cor-
responding optimum polynomial B (p) = B (P)opt will be
called the optimization of the integral square estimation j(m)
in the sense of SM/7.
A An -2 A,,_1 An_ 3 An_b3 = 1 (23)
b _ , b
b? = AAA_n-4' An-3 2 n ?2 An -4 n-2
Table 2 contains expressions for the coefficients of optimum
polynomials B. (p)opt, obtained as a result of solution of the set
of eqns (22) for a few successive values of the degree in of the
polynomial 11(p).
The integral evaluation in(m), that satisfies the set of con-
ditions (22), will be called optimum in the sense of stability
margin and denoted by in(n,')opt. Optimum evaluations in the
sense of SMI, have a number of valuable properties. Some of
then-i will be considered below. Of particular importance is the
fact that for full analytical description of the evaluation J(m)0pt
the SM/ are required only.
6. The Two Equivalent Forms of the Integral Evaluation 4,(m)
In the general case the integral evaluation in(rn) does not
satisfy the optimum conditions (22), and therefore it cannot be
expressed in terms of the SM/ only. This follows directly from
the assumption, that the coefficients of the polynomial .11 (p) are
independent of the coefficients of the characteristic equation
A (p). In this connection try to separate in the integral evalua-
tion in(m) a component depending exclusively on the SMI from
another component in which the influence of the polynomial
11(p) is taken into account. The introduction of the SMI and
the notion of optimum conditions in the sense of SMI enables
two new equivalent forms of the integral evaluation ..17,(m) to
be established, that is:
j(m)j(0)+M(m)
and
J(m)=J(:n.)+Am(,-)
(24)
(25)
A detailed analysis of expressions (24) and (25) will be shown
later. Now one is satisfied with the statement that for the
determination of the first components, that is ./n(?) and J7i(m)0pt,
only SMI are needed. To find the remaining components, that
is Mn(m) and AMn(m), the knowledge of the polynomial B (p)
is also needed. In particular, the component Mn(m) expresses
the increase of the evaluation J12(m) due to the fact that the
polynomial increase B(p) = (p) ? Lim has been taken into
consideration, and the component AM(m) is the increase due
to the introduction of the polynomial AB (p) = B (p) ? 11 (P)opt
in the numerator of the transfer function F (p). For further
investigation form (25) will be of particular use.
287/4
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Table 2.
287 / 5
Coefficients bi (i 0, 1, 2, ..., in) of the polynomials B(p) = B(p)0t, satisfying the optimum conditions in the sense of the indices of
stability margin(qi?
Coefficients of the polynomial B(p) = b0 pm b1pm-1 bni_,p bm= B(p)0
in
b,
b2
b,
b,
?b5
1
qn_2
2
qn_ 3
qn-2
1
?
?
3
qn-4
qn-3
, qn-4

1
q -I-
n-2
qn-3
q_5
q_4
qn-5
.4_
'
qn-5
' qn-2
, qn-4
1
_L.,
q_3 , ,
,
Nn- 4
q --r- n-2
qn-3
5
qn-5
qn-5
qn-6
+
qn- 6
qn-2 +
6/12-6
qn-3
? qn-5
q n-5
qn-2
qn-4
qn-6
1
qn-1 +
qn-3
qn -4
qn-5
qn-3 + ,
N n- 3
+ ,
' N n-4
qn-2 + ,, ?
Vn-3

q_5
Notes: (1) n is the degree of the characteristic equation A (p) = 0
(2) m is the degree of the polynomial B(p); 0 m < n
(3) B(p) is the polynomial in the numerator of the normalized
A (p)
t transfer function P(p) ? B (p)
7. The Primary Spectrum of the Integral Evaluation in")
Under the term of primary spectrum of the integral evalua-
tion in(m) one will understand the expression
Ru=1??(r1,r2,r3,???,r) (26)
The elements of the spectrum Kr, are r1, r2, rn. They are
related to the SMI by the formulae
9i-2 Ai- l? Ai-1 I
Ti? = ? (i =1,2, ..., n) (27)
qi-1 S7
where qi are the Routh parameters, Ai the Hurwitz determinants,
and Si* the Markov determinants (Ai = a1i ? Si*).
Example:
Ao A A
A2A2 An_i - _I
? r = =
406.2' 3 A/A3 _ '2 ? An
Knowing the values of the elements r1, r2, rn the values of
the corresponding indices of stability margin can easily be
determined:
Routh parameters q,
=rir2,???,rk= fl ri (k= 1,2, ..., n) (28)
9k-1 i= 1
Hurwitz determinants Ak
k
1. k k-1 k- 2
?= rir2 r3 , ? ? ? , rk= II?
r:i- i - for k= 1, 2, .., n (29)
Ak a = 1
Markov determinants Si*; (Si* 1)
11 k - 2 rk _ ra k+ for k= 2, 3, ..., n
= rz r3 , ? ? ? , ?
Sk a= 2
The primary spectrum R? determines uniquely the first
components of the forms (24) and (25) of the integral evaluation
L(m). In particular, by virtue of eqns (14) and (27), one can
write at once
r? (31)
?
It can also be shown7 that when the optimum conditions (22)
are satisfied, the expression of the evaluation .in(m)opt. takes
the following exceptionally simple form
1 n-m
(32)
where 0 < m < n.
From (32) it follows that evaluation .?(m)op/ depends only on
the first n ? in elements of the spectrum Rn and is invariant in
relation to the remaining ones. Thus, the elements r1, r2,
r?_?, will be called weight (influence) elements and the remaining
ones, that is r?_(,,,), independent or free ones*.
Observe that although the independent elements of the spectrum
Rn show no influence on the value of the evaluation iii"opt,
(30)
* in the case of normalized transfer function the condition
fl
III r 1 should be satisfied.
i-1
287/5
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
287 /6
they influence the character of the transient response. This is a
separate problem and is not dealt with in this paper.
On the basis of eqns (27)-(30) the stability of a control
system can easily be analysed. From this analysis it follows
that if a control system is stable, all the elements of the spectrum
are positive. If, in addition, the system is physically real,
these elements are bounded. The spectrum Rn, of which all the
elements are different from zero and positive will be called
'essentially positive'.
Another interesting property of the spectrum Rn is now
shown. It is known that the stability margin of the system is
greater for greater values of SMI3.4, Hurwitz determinants,
for instance. This means that the stability margin is greater for
smaller values of elements of the spectrum R.
On the basis of the above results and considerations the
following cardinal properties of the spectrum Rn can be for-
mulated:
Property I: In order that a linear control system with the
characteristic equation A (p) _ pn = 0 is
stable and physically real it is necessary and sufficient that the
primary spectrum Rn= Rn (r1, r2, ..,rn) of the evaluation fn(m)
of this system is essentially positive and bounded, that is
0 < ri < oo for i = 1, 2, ..., n.
Property II: The primary spectrum it, characterizes the
transient performance in a linear control system of the order n
and class m, because the sum of its weight elements ri (1 = 1,
2. ..., n ? m) determines the value of the evaluation in("opt
satisfying the optimum conditions in the sense of stability
margin (SMI), that is
) 1 "v7
r
for m = 0, 1, 2, ..., n ? 1.
8. The Secondary Spectrum of the Integral Evaluation in(m)
The components M(m) and AM(m) in expressions (24) and
(25) for the evaluations j(m) depend in the general case on the
spectrum Rn and the polynomial n(p) = b0 pm ? b, pm-1 . . .
bm-ip bm or the equivalent polynomial C(p)
where
C(p)=c?,pm +cm_ipm-i+ +cip+co-B(p);(co=b,n=1)
(33)
The task now is to find a set of m parameters such that their
structure contains as much information as possible on the
transient performance in a control system and would enable
the determination, in a unique form, of the values of the coeffi-
cients c? = bm_a, and the easy computation of the component
Mn(m) or 1M(m) of in(m). To this aim consider the partial
derivatives
a./Witn)
w.=
ac,^ ac
a- -i
One has, in the case of odd i:
E rm+11
L 2 J A(n)
wi = 1 + E . ?
i=1 An
E pn+11
L 2 J
\(i+/+1)
W2i+1= E
i=1 A?
and
for i = 1,2, ..., m (34)
(35)
where 1 < < E [m-1/2]. One finds, for even i:
A(n) L 2 J
Erml
A?,("1 +0
w21= (-1)I E ( 1)(1"?
i =1 C2 i
for 1 < / < k = E [m12].
The expressions (35) constitute a set of equations for the odd
coefficients ea, of the polynomial (33) and the expressions (36)
are a set of equations for even coefficients cH of that polynomial.
The principal determinants of these systems will be denoted by
W?,(1) and Wm(2) respectively, where
(36)
w(2)=
or
Ow, awl
aci ac3
Ow, aw3
0c1 003
314'2 a w2
ac2 ac4
a w4 a w4
ac2 ac4 ?
.=
W11 W13
W31 W33 W35 ???
W51 W53 W55 ???
W22 W24 W26 ???
W42 W44 W46 ???
W62 W64 W66 ???
w(1)=a (W1, W39???/ W2k-13???) 1
a(c1,c3,???,c2k-1,???) qn-2qn-3, ? ? ?, qn?k
k= 2 E [1 +
2
w(2) =0 (w2, W45???, W2k, ???) = 1
?
qn-2qn? ...,q_1
1=2E [2 +
2
(37)
(38)
(39)
From eqns (37)-(39) it fol ows that the determinants Wm(1)
and Wm(2) are the Jacobians of the transformation. The elements
TV"; of these Jacobians are Krasovskii determinants
A (n)
with appropriate signs and
_Om a2M,(,m
w )
(1, j=1,2,3, m) (40)
0c1 = Oci0ci
Assume that the system is stable and its spectrum R? is in-
variable (constant); then, assume also that the Jacobians Wm11)
and Wn,(2) have, in agreement with (39), constant values different
from zero and positive. From the analysis it followsu that in
this case all the necessary and sufficient conditions are satisfied
for the transformation considered to be homeomorphic. It
follows that the representation of the set of parameters wi (i =1,
2, ..., m) in an m-dimensional space L(m) on the set of para-
meters ci (i =1, 2, ..., m) in an m-dimensional space D("') is
one-to-one, and that the homeomorphic representation of a
space region is a space region and the representation of an arc
287/6
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
is an arc. The set of the parameters wi (i = 1, 2, ..., m) will be
called the secondary spectrum of the integral evaluation J0(m)
and denoted by
Wm = Wm (WI, W2 t ? ? ? Wm) (41)
If the values of the elements iv, of the spectrum wm, are known,
it is easy to calculate all the coefficients cc, of the polynomial
(33). To do this it suffices to solve in relation to cc, the matrix
equations*:
II kv,;z1)11 = 11 v,V)II and C;n2)II = V,2)II (42)
The spectrum wni is called positive, zero or negative if all
its elements wi (i = 1, 2, ..., m) are, respectively, positive or
zero or negative. A spectrum Wm may also be of a mixed type.
In particular, from the solution of eqns (42) it follows that if
the spectrum Wm is zero, the set of eqns (22) is satisfied. This
impotant feature of the spectrum win concerning the optimum
evaluation j(m) in the sense of SMI can be expressed in the
form of the following.
Property of the spectrum wm: In order that the Krasovskii
integral evaluation J(m) .should satisfy the optimum conditions
in the sense of stability margin (SMI) it is necessary and suffi-
cient, that its secondary spectrum wn, is zero; that is, Nit 0
= 1, 2, ..., m).
Now pass to another form of the spectrum Wm connected
with the increment AM(m) of the evaluation ./n(m). For this
purpose the coefficients of the polynomial C(p) should first be
represented in the form
Ci = Ci opt + hi (i=1, 2, ? in) (43)
where ci opt satisfy the optimum conditions in the sense of SMI
and expand Mn(m) in Taylor's series for functions of more than
one independent variable
Mr) (C1 opt + /11, C2 opt + h25 ? ? ?1 Cm opt ? hm)
(n) dM(m) d2M(nm)
= Mn opt + 1 !n + 21 ? ? ? ? + (k-1)! k (44)
In the general case the derivatives dvM?(m) and the rest Rk
of the expansion (44) are
a moo a c aMm) y (45)
112+--ivim) h2 + ... ? ,.,C ' h,,,
(
"1 u(-2. ? m
and
dkM(m)
Rk?
k!
(46)
where the derivatives &M0(m) for v < k should be determined at
the point Q0= pt Q opt (Ci opt, C2 opt, ? ? ? Cm opt) and the rest Rk at
an intermediate point (C10pt + f9h1, c20 pt ? h2, ..., cmopt +
Ohm), where ,0 < < 1. One obtains
amon)in
dM n
(?m)= E h?= E w?h? (47)
i=1 aci
(m 2 rn m
Rk= R2 = am(m) ? E h)
? E w0
??+ E wi?hih ?
2 i ?I 2 " =1, j= 2
* The solution of eqn (42) are given in Table.
where
a2m(m)
= aciac
The expansion of the function Mn(m) in Taylor's series, taking
into account eqns (47) and (48), will now be written
287/7
or
Onin) =IV/Mt + dM;Im) +R2
AM,m) = Mr) ? M,M2pt = R2
In agreement with the optimum theorem of the evaluation
fn(m) the point Q opt (ctopt, c2 opt, ..., cm opt) corresponds to a zero
spectrum Wm. In other words the derivative (47) is at this point
equal to zero, i.e.
(49)
dM;r)(Qopt)= 0 (50)
It can easily be shown that the second partial derivatives wi;
do not depend on the choice of the intermediate point. There-
fore the expression of the component 1M(m) of the evaluation
J(m) takes the following very simple form
1
in
2 i=1 " i= 1, j=2
(i 1, S-2 is defined by eqn (4). If r = 1, S-2 th is t if co is
h=1 , h=1
u, *, ?,:lf and is 7 if w is T or I ; in (13), zi; will be sub-
stituted by La (xp) and therefore by zfl or .Tp.
II. Between the expressions, a relation of equivalence = may
take place, satisfying the following conditions if
di=
cld,=12
Cosi,=0g,
Evidently, u, satisfy condition II.
then
304/1
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
Declassified in Part - Sanitized Copy Approved for Release 2013/02/13: CIA-RDP80T00246A023400480001-9
304/2
II. The application of Quine's method is based on the
formulae
Therefore
x u =1 ,
I X = X
z0z1 Zr L.) fozt Zr L.) t1 L.) L.) ts
=(z0 Li 20) zi Zr L.) ti L.) ... L.) ts
= 1 zi u ti u u
=z1 Zr Uts ts
(14)
(15)
In order to apply this method, the terms z?, 20 Zr
must be brought to be neighbours and the variables must be
arranged in a definite order. Therefore, it will be assumed that.
III. Q and 0 are commutative; that is to say if 7r is a per-
mutation of the indexes 1, ..., r, then
2,? (1)0 OZR 09= Z10 ... OZ,.
cozi, = zico . . . coz,.
This property allows the expression to take the form
(16)
(zoOziO 04)co(20(9z10 Oz,.)cotico ... cots
The commutativity is valid for all the operations given as
examples: Li, T,I, +, . Yet, in order to make the sim-
plification, it is not necessary that all the steps in eqn (15)
could be made. It is sufficient that
IV. The following equality be true
(zoOziO Oz,.) co (200z10 Oz,.)cotico ... cots
(z10 Oz,.)0.)tico ... cots
(17)
It is important to emphasize that for all the pairs of opera-
tions co, 0, from eqns (6), (7) and (9), eqn (14) remains true.
That is so much more remarkable, since the various steps made
in eqn (15), such as the associativity of L.), the distributivity of
with regard to L.), etc. are not valid for some of these pairs
(I?X) of co, 0 operations.
III. This first stage of simplification is valid for:
(a) the dipoles H with contacts, as well in the normally
disjunctive form (co = U, 0 = .) as in the normally conjunctive
form (co = = u);
(b) the diode circuits, in the same cases;
(c) the triode circuits, of the two following forms
co= u, O=T (form III)
co=T, O=T (form VIII)
(d) the transistor circuits of the eight forms 1?VIII;
(e) the transistor circuits of the form IX, X;
(f) the cryotron circuits of the following forms
co =1, 0= 1 (form VII)
= T, O==T (form VIII)
IV. In the classical case, the following simplification is made
xyz U XyZ U xyz U Xyf
xyz U XZ U xyz L.) xyz U xyz L)xyf
=(x U x)yz u (y y)xz L.)(z U f)xy
= yz XZ u xy (18)
by virtue of the idempotence law
Z=Z -(19)
To indulge in this type of computation, it is necessary to
assume that
V. The following equality is true
dowslowdico ?? ? cod,
?docodico ??? cod,
This property is valid for the operations L.), T, I, but it
is not valid for + and 1Z- .
V. It is known that in the classical case, there can be the
following type of simplification
xy L.) j7z U xz=xyU yz U xyz L.) xyz
=(xy L.) xy.z) U (yz U xyz)
= xy U yz
(21)
The problems arising from this type of simplication con-
stitute the originality of Quine's method.
A start is made with an expression such as eqn (13) where the
za have been replaced by La (xp) as in the expressions provided
by eqn (4).
An expression of the form