Programming an Analog Computer for a Large Class of Trajectories
TR-1146
C
6
PROGRAMMING AN ANALOG COMPUTER
FOR A LARGE CLASS OF TRAJECTORIES
Albert I. Talkin
20 June 1963
HARRY 'DIAMOND
LABORATORIES
FORMERLY: DIAMONO ORDNANCE
ARMY
MATERIEL
WASHINGTO4
as.
FtUZe LAIORATORII[
COMMAND
D. C.
/
HARRY DIAMOND LABORATORIES
Robert W. McEvoy
B. M. Horton
LtCol, ord Corps
Technical Director
Commanding
MISSION
The mission of the Harry Diamond Laboratories is.
(1) To perform research and engineering on systems for detecting,
locating, and evaluating targets; for accomplishing safing, arming,
and munition control functions; and for providing initiation signals:
these systems include, but are not limited to, radio and non-radio
proximity fuzes, predictor-computer fuzes, electronic timers,
electrically-initiated fuzes, and related items.
(2) To perform research and engineering in fluid amplification
and fluid-actuated control systems.
(3) To perform research and engineering in instrumentation and
measurement in support of the above.
(4) To perform research and engineering in order to achieve
maximum immunity of systems to adverse influences, including countermeasures, nuclear radiation, battlefield conditions, and high-altitude
and space environments.
(5) To perform research and engineering on materials, components,
and subsystems in support of above.
(6) To conduct basic research in the. physical sciences in support
of the above.
(7) To provide consultative services to other Government agencies
when requested.
(8) To carry out special projects lying within installation
competence upon approval by the Director of Research and Development,
Army Materiel Command.
(9) To maintain a high degree of competence in the application
of the physical sciences to the solution of military problems.
report are not to be construed as ab
The findings in this
Department of the Army position.
official
UNITED S'TATES ARMY MATERIEL COMMAND
HARRY DIAMOND LABORATORIES
WASHINGTON 25, D.C.
TR-1146
DA-5N03-01-003
AMCMS Code 5010.11.71200
HDL Proj 31100
20 June 1963
PROGRAMMING AN ANALOG COMPUTER FOR
*ALARGE CLASS OF TRAJECTORIES
Albert I.
Talkin
FOR THE COMMANDER:
Approved by
Robert D. Hatcher
Chief, Laboratory 300
Qualified requesters may obtain copies of this report from ASTIA.
CONTENTS
ABSTRACT. ..................
..........
5
1.
INTRODUCTION. ........ ....... ............
5
1.1 Definition of the Problem...............5
1.2 Examples and Applications. ...............
6
2.
ANALYSIS ... ..........................
2.1 Partitioning the Problem .................
2.2 Closing the Switch .....................
3.
CONCLUSION
4.
REFERENCES .. .......................
..
..
..
..
..
..
..
..
..
..
7
7
..
..
.12
12
3
ABSTRACT
The negative gradient method is extended to the stable analog
computer programming of a class of time varying trajectory problems
defined by O(xt) = O f(x,ut) = O, u = i where X and u are n-dimensional vdctofs,
is an m-dimensional position vector function
(m < n), and f is An (n - m) dimensional velocity vector function.
This class of problems includes the amplitude-stabilized oscillator,
two-dimenSional contour tracing (ref 1), conic section generation
and three-dimensional trajectory plotting when at least the first
integral of the equations of motion are available.
4
An augmented velocity vector function is defined
f :f + d
dt
which provides n independent equations f/ = 0. For a fixed x,
these equations can be solved for u in the computer reset mode.
the compute mode u is connected to the integrator developing x.
resulting system is analyzed and shown to be easily stabilized.
1.
In
The
INTRODUCTION
1.1
Definition of the Problem
The problem will be defined in n-dimensional space. Vector/
matrix notation will be employed to reduce the labor of writing equations. The following definitions will be used:
x = [xl, x 2 , ... x n]
(1)
position vector
(2)
i-th position function 0i a i(xt); i =1...m, m < n
(t is the-rndependent variable time)
(3)
position function vector
(4)
velocity vector
(5)
j-th velocity function f
4
[$I'
E
u E [ul, u 2
2'
oUn]
.
= f (u,x,t); j = m + 1, m + 2,
(6)
velocity function vector
f _ [fm+'
(7)
position error vector
v
[vi, v 2 ,...
vn]
(8)
velocity error vector
a
[al, a2 ,o..
an]
(9)
gradient vectorV
fm+2;'
°
' fn
.oo
n
]
-n(10) gradient vector V
u
=[
a
u
(11) general matrix notation,
A
2
''
- (a..)
-_nl
n
and I
1J
unity matrix
(6j)
ij
) i,j = 1....n
(12) diagonal gain matrix
K B (ki 8
(13) diagonal gain matrix
G a (gi 6 j) ij
= 1...
n
5
(14)
scalar or inner product of two n-dimensional vectors (p,q):
(pq).mp 1q 1 + p2 q 2 +... pnqn
(15)
positive definite matrix a a real symmetric matrix all of
whose characteristic roots are positive
Using the above definitions the problem can be stated as fol-
lows:
Given the set of independent equations
4(x,t) = 0
f(x,u,t)
=
(i
0
)
(1.2)
(1)
dx
U = j m dt-
(1.3)
program an analog computer to generate the unknowns x(t), u(t).
(1.1)
represents m equations, (1.2) represents n-m equations and (1.3) represents n equations for a total of 2n equations in 2n unknowns xl,," :
,
x , u , u2 ,... u .. The problem has been formulated in terms
ou
andnx rather than i and x because of the distinction between the
computer variable, i, which by definition is the total input to the
x integrator and the problem variable u (i.e. the * required as a
solution to the mathematical problem.). A full statement of the
problem also requires that n-m of the position coordinates be specified as initial conditions. Before continuing with the analysis of
such a system, it may be helpful to expand upon the terse problem
statement by considering some examples and applications in two- and
three-dimensional space.
1.2
Examples and Applications
In this section vector notation is momentarily abandoned
in favor of more conventional x,yz notation. Also u will be eliminated from the equations by the substitution x = U.
Consider the three-dimensional system (2):
Ol(x,y,z) a x
2 (xyz)
2
2
+ yl - C z2 = 0
+
+
+h.=.x=: 0
f3 (*,kjX,yz)M i2 + k2 + j-
(2)
s2 = 0
(c, a, P Y, h, s are arbitrary constants)
The sinmutaneous solution of (2) is a conic section traced at constant
speed s'.
In (2) the variable t does not appear explicitly in' or f.
Consider the two-dimensional system (3):
6
4l(x,y,t)
= x3+ y2 - r2 (t) = 0
f2 (i,,x,y,t) M 0
+
-
s2 (t) = 0
If r(t) is constant and s(t) is a linear ramp, (3) represents linear
frequency modulation without any amplitude modulation. If both r(t)
and s(t) are constant, (3) represents an oscillator with highly
If r(t) is constant
stable amplitude and frequency characteristics.
dO
and s(t) =-%- (3) performs trigonometric resolution (ref 2).
If
s(t) is a constant, (3) represents a complex modulation scheme obeying the law (AM) x (FM) = constant. The modulating intelligence r(t)
can be received by either an AM or FM receiver and is redundant.
Consider the system (4):
4l(x,y,t)
f
x1 + y2 - r2 (t) b 0
+
-
r2(t) s 2 (t)
(4)
=
0
If in (4) both r(t) and s(t) vary independently, the system will represent a simultaneous AM and FM waveform with independent messages'
r(t) and s(t).
2.
ANALYSIS
2.1
Partitioning the Problem
Consider first the situation in which the analog computer
is in the hold or reset mode. In this mode the correct stationary
values of x and u must be generated. Since + is not a function of u,
it is possible to generate x by programming the equations
+ 1(x)= 0, 02 (x) = o,...
m(X) = 0
(5)
with n-m of the coordinates of x specified as initial conditions.
Thisof course, results in a system of m independent equations with
m unknowns. Since these equations, in general, are nonlinear, the
gradient method* of programming is required to guarantee stability.
Turning now to the generation of u, the condition f = 0
provides only n-m equations to be solved for n unknowns.
To obtain
the m additional equations. an augmented velocityf unction vector (f')
must be defined.
* The gradient method is synonymous with least squares, steepest
descent, or transpose matrix method.
7
In vector notation
f'
dt
+ f
(6)
For the rigorous, interpretation of (6), the original definitions of
f and.4 must be expanded to n-dimensions, with.the first m components
of f.being identically zero, and the last n-m components of
bqing
identically zero. Written out:
4
[d4l
d42
dlm.
Sdt' dt'
dt
fm+l' fm+2'
°
(7)
f
If in the first m coordinates of fl the substitution of u for x is
made, then
f' =0
(8)
represents a system of n equations in n unknowns u1 ,
. o
The system (8) can be programmed by the gradient methoi and requires
that x developed from (5) be inserted as a parameter. The conditions
obtained in reset or hold then will be
0
(9)
0
f=
(9) represents the partitioned system.
2.2
Closing the Switch
Refer now to figure 1, which is a simplified schematic illustrating the connections for the i-th component of x and u. In
reset or hold, switch .Sis open and u and x assume their correct
stationary values. It is reasonable to suppose that if S were closed,
the resulting system would produce a very close approximation to the
desired trajectory, provided the system maintains dynamic stability.
The resultiqg computer differential equation will now be derived.
In accordance with the gradient method,
let
v = -grad
a
-
(0,0) = -Vx(,4)
gradu (f'",f')
u
= -V u (f',f')
(10)
(11)
From figure 1 we have
8
x = OU +1W
(12)
u = Ga
-(13)
(axis the potentiometer setting, figure 1, and is the time scale
factor)
Differentiating (12) with respect to time and substituting (13)
xW = 0Ga + K4
(14)
The system stability may be investigated by linearizing .the system
about some arbitrary operating point .and examining the effect of
small perturbations 6x and 6u.
Operating on equations (12) and
(14) with the variational operator 6
d(6x)
dt
(15)
+ K6v
+ K d(6v)
_u
d(6x)
+
dt-
(16)
dt
It is now necessary to express 6v and 8a in terms. of 6x.
variations of both sides of (10)
6v"
n
2 m a
j=l
+n
Taking
6x.
k=l
2 2
k x
)
6
xj
(17)
Since variations are taken starting from an assumed equilibrium
state 0 = 0 the second term on the right in (17) vanishes. The
matrix
A M(a
2
iim
-
is positive definite since it can be factored into the product of
a matrix by its transpose; (17) then becomes
(18)
6v = -A 6x
Note that since 0 is not a function of u, only the variation with
respect to x had to be considered in (17). To .fi'd 6a, take variations of both sides of (11), noting now that both 6u and 6x will
contribute to 6a.
6ai =
n
/ - nf fk
( -2
.. i
'\
) 6Uj
k -l@U
j=l
+
u-
n(2 zu.
J=1
6
axj k)
6xj
(19)
k=l
9
cI
A
0
0
.
,
w
0
U0
I
bb
C,)
i0
314.
(to
-aa
00
-4
w
Ill.
0
In (19) terms involving second partial derivatives are absent be= 0 and f' = 0
cause of the assumption that the initial conditions
are satisfied.
Rewriting (19)
(20)
6a = -B 6u -H 6x
where
(b2
jna
a /
k=l
i
Y2
k
H = (h..j)
f
__
The matrix B is positive definite but the matrix H is not so restricted.
Substituting (18) and (20) in (16)
d2
dt
(6x) + OaGB~u + OGH6x + K T
d2(6x)
da
+ cB6u +
GH6x + K
d
(21)
(A6x) = 0
dA
d
6x + KA -. (q x) = 0
t
dt
dtW
(22)
From (15) and (18)
d
O~5u
=
KA~x +
(6x)
(23)
Substituting (23) in (22)
ds
I d2 (6x) + (KA + GB)
T7dt
d
(
dA)
x) + (GBKA + caGH + K-
dt
6x = 0
(24)
(24) is the characteristic matrix differential equation of the sysNote that (I) the unity matrix is positive definite and
tem.
(KA + GB) is positive definite, since it is the sum of two positive
definite matrices KA and GB. Now it car be-shown by an extension
of the argument of Bellma1 (ref 3) that (24) will be stable if'the
matrix sum
dA
(5
GBKA + OGH + K d(25)
can be expressed as the sum of a positive definite matrix and a skew
symmetric matrix. The matrix product GBKA is positive definite and
dA
and the matrix K T is symmetric, Let H be expressed as the sum of
a symmetric part H and a skew symmetric part S
11
H =r+ S
(26)
Let the notation A > B for two symmetric matrices denote the fact
that A - B is positive definite. Then the condition for stability
can be written
GBKA + COH + K L
dA > 0
(27)
dt
or
GBKA > - OGH -
K
(28)
dt
It is now evident that if the term aGH is causing instability, the
scale factor a must be decreased ioe., the trajectory is run slower
than the real time case (a = 1).
If the term K dA is causing in-
stability K can be decreased. If only the full set of velocity
functions is given (f1, f2,oo, f ), then A m 0 and the condition
for stability becomes H,> 0.
Tfis may be impossible to satisfy,
in which case at least one function 4K must be found by integration
from the system f = 0.
Since at lea~t 4.is available, A is reinstated and system stability is obtained ai'before.
3.
CONCLUSION
It has been shown that the powerful negative gradient technique
can be extended to successfully program trajectory problems as defined in the introduction. Computer stabilization may require time
scaling.(decreasing a) or reducing the gain K.
ACKNOWLEDGMENT
The problem generalization and analysis in this report was inspired by the work of A. Hausner, who intuitively used this method
to program a special problem proposed by the author,
4.
REFERENCES
(1) "Contour Tracing with an Analog Computer, " H. K. Skramstad;
AIEE Winter General Meeting Feb 1-61 1959, N. Y., paper no, 59-461,
(unpublished).
(2) R. M. Howe and E. G. Gilbert, "Trigonometric Resolution in
Analog Computers by Means of Multiplies Elements," IRE Transactions,
Vol EC 6, June 1957.
(3) R. Bellman, "Introduction to Matrix Analysis," McGraw-Hill,
1960, p 246.
12
DISTRIBUTION
NOTE:
Corrections to the following list would be appreciated
and should, -along x4.h the report number, be addressed
to Department of the Army, Harry Diamond Laboratories,
Washington 25, D. C., Attn: Technical Reports Unit.
Commanding General
U.S. Army Materiel Command
Washington 25, D. C.
Attn:
AMCRD-RS-PE-E
Commanding General
U.S. Army Missile Command
Redstone Arsenal, Alabama
Attn: Redstone Scientific Information Center
Chief, Documents Section
Commanding Officer
U.S. Army Munitions Command
Frankford Arsenal
Philadelphia 37, Pennsylvania
Attn:
Library, 0270
Commanding General
Aberdeen Proving Ground, Maryland
Attn:
'STEAP-TL, Tech Library (Bldg 313)
Commanding Officer
U.S. Army Munitions Command
Picatinny Arsenal
Dover, New Jersey
Attn:
Library, SMUPA-VA6
Commanding Officer
U.S. Army Research Office (Durham)
Box CM, Duke Station
Durham, North Carolina
Attn:
Dr., F. J. Murray
Commanding General
White Sands Missile Range, New Mexico
Attn: :STEWS-AMTED-SA
Commanding Officer
U.S. Army Foreign Science and Technology Center
Washington 25, D. C.
Attn:
AMXST-SD-TD
13
DISTRIBUTION (Cont'd)
Commander
U.S. Naval Ordnance Laboratory
Corona, California
Attn: Dr. H. K. Skramstad
Commander
Naval Research Laboratory
Washington 25, D. C.
Attn: Tech Library
Commander
U.S. Naval Air Missile Test Center
Point Mugu, California
Commander
U.S. Naval Proving Ground
Dahlgren, Virginia
Commander
Rome Air Development Center
uriffiss Air F'orce Base, New York
Air Force Cambridge Research Labs
Bedford, Mass.
Attn: R. Roberts
Commander
Armed Services Technical Information Agency
Arlington Hall Station
Arlington 12, Virginia
Attn:
TIPDR (10 copies)
Commander
Air Proving Ground Center
Eglin Air Force Base, Florida
Attn: Technical Library
Scientific and Technical Information Facility
P.O. Box 5700
Bethesda, Maryland
Attn:
NASA Representative (S-AK/DL)-A366
National Aeronautics & Space Administration
George C. Marshall Space Flight Center
Huntsville, Alabama
Attn: M-G&C-NS
Dr. D. Greenberg
David Taylor Model Basin
Washington 25J, D. C.
14
DISTRIBUTION (Cont'd)
National Aeronautics & Space Administration
Ames Research Center
Moffett Field, California
National Aeronautics & Space Administration
Lewis Research Center
21000 Brookpark Road
Cleveland 35, Ohio
National Bureau of Standards
Washington 25, D. C.
Attn:
Library
University of Florida
1104 Seagle Bldg
Gainesville, Florida
Attn: R. C. Johnson, Jr
-Internal
Horton, B.M./McEvoy, R. W., Lt. Col.
Apstein, Mo/Gerwin, H. L./Guarino, P. A./Kalmus, H. P.
Spates, J. E./Schwenk, C. C.
Hardin, C. D., Lab 100
Sommer, H., Lab 200
Hatcher, R. D., Lab 300
Hoff, R. S., Lab 400
Nilson, H., Lab 500
Flyer, I. N., Lab 600
Campagna, J. H., Div 700
DeMasi, R., Div 800
Landis, P. E., Lab 900
Seaton, J. W., 260
Kohler, H., Lab 200
Kirshner, J. M., 310
Talkin, A. I., 310 (5 copies)
Cuneo, J., 310
Ravilious, F., 310
Burton, W. C., 310
Hausner, A., 330
Butler, R., 330
Kurkjian, B. M., 330
Technical Reports Unit, 800
Technical Information Office, 010 (10 copies)
HWD Library (5 copies)
(Two pages of abstract cards follow.)
15
-
r-i~~~Is
----""
II"Et
I
-
.
I
I
l
, .I
"
-
1
I
ml I
-q-0
1
-0
1-
4-1
0
*-j
U
..
C
II
0eq
II l
""U.
Il
ll>
..-
1006
~.4
',...*r
810
4
*dm.
A
*
~0
-O
4
-
U ,.,(:, a.
j ~
I-
I"
.-
'~[fit
- I#s
,a' It,
c,,ai