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The preparation of charts and tables for the optimization of automatic control systems with random inputs
The Preparation of Charts and Tables for the Optimization of Automatic Control Systems with Random Inputs J. F. COALES and P. J. LAWRENCE Introduction It is nearly 20 years since Wiener' first developed his theory of the optimization of linear filters and it is 12 years since the method was fully developed and published in The Theory ol Servomechanisms by James, Nichols and Phillips2. The method has, however, not been very much used and for a time was discredited in the eyes of many control engineers as not giving a useful result. There were probably two reasons for this: the first, that lengthy calculations are required and the second, that unless a constraint such as limited output power is applied to the system, a trivial answer is obtained. Newton's4 ,s development of the method for optimizing a control system with a power constraint has overcome the second difficulty but the need for complicated computation has remained a deterrent to its use by control engineers. Since the optimization procedure treats a control system as a linear filter, its advantages as a design method are most marked for a system, of which the output has a desired value which is a randomly varying function of time, and which has unwanted random fluctuations occurring at the input or at some other point within the loop. Such a system may be represented diagrammatically by Figure 1. In Figure 1 the
at the present time no generally applicable optimization procedures are available and it is, therefore, desirable that engineers should be able to apply the linear optimization procedure without having to make unduly laborious calculations. Consideration has, therefore, been given to the possibility of producing charts, which would enable a de.si~ner of a ~ingle control loop with some knowledge of the statistics of the Input and 'noise' to decide what compensating network and power source must be employed to meet a given performance specification with the greatest economy of power. It has been found that by the use of non-dimensional parameters very widely applicable results can be obtained and it appears that, because near the optimum the performance changes only slowly, a reasonable number of charts will cover all systems likely to be of interest to the control engineer. The preparation and use of these charts, which it is intended to publish later in book form, is described below. It is considered that the method of using the procedure will generally be as follows: (1) Either determine or estimate the power spectra of the input (or the desired output) and of the noise or unwanted fluctuations. (2) Choose the appropriate values of the spectral density parameters A , a, CI., (f, to get the best fit for a power spectrum of the form A2(W 2 + Cl. 2a 2 )(f3 + 1)2
F(!{ure I . Block diagralll of general single loop control system lI'ith randolll input and noise
unwanted fluctuations are shown as occurring at the input. ]f fluctuations occur at some other point in a linear system, they can, in some cases, be referred back to the input by operating on them by the appropriate transfer functions. . .. Up to the present time, it ha~ usually be~n only In pOSitIOn control problems associated With such th.Ings as radar and guided missiles, where high performance I~ the presence of unavoidable 'noise' is required, that the deSign procedure has been used with success. This is partly because in most process control problems the performance required in terms of accuracy has not warranted the use of such a sophisticated method but the drive for greater productivity in all sections of industr,Y results in a need for better control of all types of plant , ThiS means that more accurate design procedures must in the end be adopted to attain higher performance and it has bee~ s~own that where linear systems are employed a.n~ the St~tIStICS. of the input quantities are known, the statlstl~al deSign ,USing Newton's procedure does, in fac~, resul.t In the optimum design , Although it is almost certain t.hat In m~ny cases more economic designs can be developed USing non-lInear elements,
+ a2)(w2 + {J2a2)(r:J. + 1)2
for the expected spectral density of the input. . . . (3) Choose a value of N which makes band-lImIted white noise most nearly approximate to the expected power spectrum of the noise, (4) From a study of the specification estimate the mean and maximum output powers that are likely to be required and decide on the type of prime mover that will probably be used . Hence , estimate those factors of the transfer function of motor and load which depend on inertia and damping of the l?ad and ignore additional factors such as that due to the field time constant of an electric motor, which do not affect the output power, (5) From the values of A, a, CI., ~, Nand b calculate the non-dimensional parameters z; = NalA an~ p = bla. . (6) The specification must lay down either the maximum permissible mean squared error ~ or the maximum permissible mean power of the motor, a2 , or one of the values must be obtainable from the specification. Hence determine one of the non-dimensional parameters
E{ or
753
763
«((j2
=
(e2/mean-squared value of input signal)1/2}
,n ( = mean-square ;alue of input x ~) .
J. F. COALES AND P. J. LAWRENCE
(7) From the chart for the appropriate values of p, rx and fJ read off the value of E or fi, whichever is not previously determined, for the appropriate value of v and the other quantity. Hence the value of (!2 or a2 for the optimum system can be calculated. In general there will not be a chart or a curve on it for the required values of p, rx, fJ and v, and repeated linear interpolations will have to be employed to find the value of E or fi, whichever is required. (8) From the tables the values of the poles and zeros of the optimum open loop transfer function corresponding to the appropriate values of rx, fJ, p, v and fi are obtained. (9) Knowing these parameters, a check on the value of i! or a 2 , which will result from the use of this transfer function, can be obtained numerically, if required. (10) The design can now proceed on the basis of an actual prime mover. This means that the actual open loop transfer function must be made to conform to that determined for the optimum system by including in the compensating network the inverse of those factors other than K/p(p + b) which are introduced into the system either by the prime mover or amplifying stages. In some applications, the limitation will not be on mean output power but on some other quantity such as motor torque, in which case this constraint will have to be approximated by obtaining an equivalent value for the mean output power on the basis of a gaussian distribution of torque.
However, as a closed loop system with such an overall transfer function requires infinite loop-gain, this is clearly unpracticable. So in order that the optimum system shall be one that is useful 10r-~~~Tm-'--'''~-'-''Tn~
t
00_
a
Figure 2. Spectral density functions plotted against frequency
in practice, it is necessary to minimize the mean squared error subject to some constraint within the loop. The constraint chosen is the mean power at the input to the motor, since it is often found that the performance of a system is limited by the necessary limitation of power consumed by a power amplifier or motor. An alternative constraint, that of system bandwidth, has been suggested by Newton 5 • It can be shown 8 that this constraint can, in fact, be equivalent to the power constraint described here and that a system designed using the power constraint is a minimum bandwidth system.
Single-loop System We have considered a single-loop system as shown in Figure 1; the system is presumed to control a motor and load with transfer function K/p(p + b), i.e. with inertia and viscous
damping but no 'spring term'. For reasons that will be mentioned later, we apply a constraint to the mean power input to the motor and load. The choice of the form of spectral density function to be assumed for the input signal is difficult since, to our knowledge, very little has been published on the analysis of signals encountered in the various possible fields of application of this method. Accordingly, the chosen form must be as general as possible, but in order to keep the tabulation of the results of the calculations down to a reasonable size, the number of different parameters must be kept to a minimum. The form chosen is: A2(0)2 + rx 2a2)(fJ + 1)2 <1> (0) = -----,,--......,,--,,---------',,'--,c---,----'-----,: ss (0)2 + a2)( 0)2 + fJ 2a2)( rx + 1)2 By giving suitable values to rx and fJ a reasonable range of shapes of spectral densities can be fitted. The factor (fJ + 1)2/ (rx + 1)2 is included in order to allow for infinite values of rx or fJ. The graphs in Figure 2 show the result of plotting <1> against O)/a for various values of rx and /J. For the purpose of clarity, the function plotted has the factor (fJ2 + 1)/ (rx 2 + 1) instead of {(fJ + 1)2}/{(rx + 1)2}; in the subsequent calculations the latter factor has been used because this simplifies some of the algebra occurring. In most practical cases, it is found that if noise occurs in the input signal, the bandwidth of the noise is generally much greater than that of the signal, so the noise may be treated as 'white' as far as the design of the system is concerned. In the calculations described in this paper, we have not considered anything other than white noise in the input. It was mentioned above that a constraint is applied to the mean power; the reason for this is as follows: In the absence of noise it is obvious that the optimum system has a transfer function equal to unity, since this would give zero error.
The Mathematical Problem The mathematical problem to be solved is, referring to Figure 1, to minimize the mean-squared error (s - y)2 subject to the constraint Z2 = a 2 • This can be done, using the general
method of Lagrangian multipliers; this was first applied to this type of problem by Newton 4 and the approach used here is due to Tsien 6 • It is well known that a system of overall transfer function H(p) which has an input consisting of a signal with spectral density <1>siw) with added noise of spectral density <1>nn(w) produces a mean squared error (i.e. signal output) given (if signal and noise are uncorrelated) by:
C2 = 217T L~£X){I-H(jO)j2<1>ssCO) + IHCjw)12<1>",,(W)} dw*
(I)
and that this is minimized by a transfer function given by 1 [<1>"';J
HCp) =lHp) 'F( -p) -;-
(2)
where 'F(jO)'F( -jw) = <1>siw) + <1>",,(w) 'F(p) is such that its poles and zeros lie in the left-hand (real part of p negative) half of the p plane and
[<1> ss
(5)]
IF(_p) -;-
* Depending on the definition of spectral density used, the factor preceding the integral may be Ij2 or 1/2rr.
754
764
CHARTS AND TABLES FOR THE OPTIMIZATION OF AUTOMATIC CONTROL SYSTEMS WITH RANDOM INPUTS
indicates that part of the partial fraction expansion of
We define non-dimensional parameters as follows: Na b a2, 1"A Y = A P = ~ ~ = ---y-
P\
and can now get expressions for A(jx) and B(jx):
'f'( -p)
A(jx) = a o
which has poles in the left-hand half of the p plane. This was originally shown by Wiener 7 • Applying the method of Lagrangian multipliers to the present problem, it is desired to minimize 2 + AZ 2 where }, is a Lagrangian multiplier. To obtain Z2, note that (referring to Figure 1 and using capital letters to indicate Laplace transforms)
+ aljx +
ao=
where
J
al =
and the quantity to be minimized, ~
t f+oo
2-
7T
{It -
+
Ho(j01)j2
+ ).z2 is given
(3)
by
+ IH o(j01)j2[
(4)
+ IHm (j01)12(
},[
'f'
rh
(-j01) = 'l-' s/(I})
+ +
1) (
[
rh + 'l-'n,,(w) +
Substituting the values of
I'f'0 w ) I -
}·
J(1)
1 = 'f'(p)
+ 2a Oa 2
1)2
+
a
(-y
(32)«(3
+
(_y2
+ (1..2)
I)(y
+
1)A
+
+
I)(y
(3)
A( -y)B( -y)
(7)
(3)A( -y)B( -y)
+ do + dly + d 2y2 + d 3y3 )
Cly
(5)]
A«(3
+ 1) a
+
Co
+
+
I)(y
(y
Cly
+ (3)
[
i.e. we now have an expression for Ho(p) in terms of only the non-dimensional parameters defined in equation 5. By substituting this in equations 3 and 1, expressions are obtained for the mean squared power and error. These are
+
and Hm(p) given
= A 2a 3
+ 1) a2
~f +OC \(Co + Cdx)jx(jx + P)\2dx (b o + bdx + b 2 )(jx)2
K2 . 27T
2" _ A2
(1
e - -- a 27T
+N
2) (
I
+
;.(1)2«(1)2 K
+ b2»)
A(jx)B(jx) (jx + l)(jx +
f +oo -
Of)
C o - Cdx )
(
x A(jx)B(jx)(jx
+ y2 • 2~
(8)
- 00
(A(jx)B(jx) -
Let O1/a = x and
J
+
+
\f'( -p)
and
= A«(3
+
«(3
l)(y + (3) A( -y)(3( -y) where Co, Cl' do ... d 3 can be obtained from a set of simultaneous linear equations generated by equating coefficients with those of equation 7. Hence
2
'F( '01)
+
1) 1)2 (y
Co (y
a
.
1)2
and therefore Ho(p)
The procedure outlined in The Mathematical Problem will now be applied to the problem we are considering. In the course of this, convenient non-dimensional parameters will be defined, Referring to equation 2 we first require 'f"(p) such that (jw)
A«(3 a«(1..
a
Non-dimensional Parameters
T
+
= A«(3
which will be given by equation 2. This will give Ho(p) as a function of A; by substituting this value of Ho(p) into equation 3. A is then determined by setting Z2 = a2 , the required mean power. A being known, Ho(p) and"? can now be calculated.
ur
+
«(1..
' /p2~2
Since A(y) and B(y) have zeros in the left-hand half-plane only, this expression can be expanded in partial fractions:
Comparing equation 4 with equation t it can be seen that the required Ho(p) is that of a filter for separating a signal with spectral density
=
+ 1)2 (_y2 + (1..2) + 1)2 (_y2 + 1)( _y2 +
A
bl
If the coefficients of powers of jx in 'f'(j01) are all positive, the condition on the poles and zeros of'f'(p) is satisfied, hence taking the coefficients defined above as positive is sufficient for this condition . The quantity {
- 00
I
=
b2 = ~
thus - 00
b
+ 2~ ---------------------1 y2(1 + (32)
Y(p) = Ho(p)X(p), hence Z(p) = {Ho(p)X(p)}/{H",(p)}
f +oo IHm(j01) H o(j01)1 2 {
((3
a2 = - (3 + 1
If the overall transfer function of the system is Ho(p), then
1
JC: If v! If o Y
Y(p) = Hm(p)Z(p)
2
+ bl(jx) + bz{jx)Z
B(jx) = b o
aijx)2
(6)
e
Z = 27T
(5)
+
1)
t+:\~(j~~1~ \dX)
Cl. Cl.
+
1
(1; :
+
(1..
n
JX. +
)
2
1 ~
(9)
The integrals in these expressions are entirely in terms of
(3) 755
765
Table I
{3=1
oc=1
p=1 G
Form of optimum open loop transrer function is - - - - - - - - -
(f!.a + /'
€
G
0-0001 0-0002 0·0005 0-0010 0-0020 0-0050 0-0100 0·0200 0-0500 0-1000 0·2000 0-5000 1·0000
0·064 0·080 0·109 0·136 0-171 0-229 0·285 0-352 0-457 0·546 0·639 0-756 0-831
2-72 X 10' 1·08 x 10" 31800 12700 5030 1490 596 239
0·284 0·287 0·296 0·306 0·321 0·353 0·387 0-434 0·515 0·589 0·669 0·773 0·841
·'I)(f!.a + ,,_)(f!.a +.,) {'l.
I
i
2
,3
., J i
v=O 1·00 1·00 1·00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1·00
737 464 252 159 100 54·6 34-5 21-9 12·0 7-68 4-94 2-79 1-84
6·88 x 10" 2·73 x 10" 8·06 x 10' 3·21 x 10' 1·27 x 10-' 37800 15100 6050 1830 745 308 98·2 42-4
1·00 1·00 1-00 1·00 1·00 1·00 1·00 1·00 1-00 1·00 1·00 1·00 1·00
391 251 142 94·2 64·0 40·6 30·2 23·7 18·7 9·93 5·73 3·05 1-96
±391 i ±250 i ±141 i ±92'5 i ±61'5 i ±36'5 i ±24' 5 i ±15·8 i ±5'98 i 23·0 24-4 24·8 24-9
0·525 0·526 0·530 0·534 0·540 0·555 0·571 0·596 0·642 0·688 0·742 0-817 0·869
1·79 x 10" 7· 11 x 10" 2-10 x 10" 83400 33200 9820 3920 1570 473 193 79-4 25·2 10-9
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
413 261 143 91·6 58·9 33·5 22·3 15·3 9·83 7·42 5·89 4·70 2·36
±413 i ±261 i ±143 i ±91·5 i ±58-8 i ±33'2 i ±21'9 i ±14'6 i ±8'75 i ±5'91 i ± 3-84 i ±1 ·45 i 5·79
0·712 0·713 0·715 0·716 0-719 0-726 0·734 0·745 0-769 0·793 0·824 0·870 0·904
8-49 x 10' 3-37 x 10' 99400 39500 15700 4650 1850 743 223 90·6 37·2 11·7 5-02
1·00 1-00 1·00 1·00 1·00 1·00 1-00 1-00 1·00 1-00 1-00 1·00 1-00
468 295 161 102 64·9 36·0 23 ·3 15 ·3 9·03 6·29 4·56 3-21 2·6 1
±468 i ±295 i ±161 i ±102 i ±64'9 i ±35'9 i ±23 ·2 i ±15 ' 1 i ± 8'79 i ±5 '94 i ±4'06 i ±2'44 i ±1'57 i
72-1
29·5 12·2 3·89 1·68
" = 0-040
0-0001 0-0002 0-0005 0-0010 0·0020 0·0050 0·0100 0-0200 0·0500 0-1000 0-2000 0-5000 1-0000 ~'=
0-160 0·0001 0·0002 0·0005 0-0010 0-0020 0·0050 0·0100 0·0200 0·0500 0-1000 0·2000 0-5000 1-0000
}- = 0-360
0·0001 0-0082 0-0005 0·0010 0·0020 0-0050 0-0100 0·0200 0·0500 0-1000 0-2000 0·5000 1-0000
756
766
'X=2
{3=3
p
Table 2
= I
G
Form of optimum open-loop transfer function is
(~+
0) '
(~ + i'I)(~ + i'2)(~ + i'3)(~ + i") {t
l'
---
E
-
G
-
I
0
1' 1
" 2
I
;'3
it"
= 0,000 0,0001 0·0002 0·0005 0·0010 0·0020 0·0050 0,0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000
0·091 0·114 0·154 0·193 0·240 0·318 0·389 0·469 0·582 0·667 0,746 0·835 0·887
1·90 x 105 75500 22300 8890 3550 1060 428 175 54-4 22·9 9·79 3·25 1-43
2·00 2·00 2·01 2·02 2·02 2·04 2·06 2·10 2·16 2·22 2·29 2·39 2-46
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3·00 3·00 3,00 3·01 3,04 3' 16 3·51 2·50 2·02
0·323 0·329 0·343 0·359 0·383 0-429 0-477 0·536 0·626 0·698 0·767 0·846 0·894
6·45 x 106 2·57 x 106 7·58 x 105 3·02 x 105 1·21 x 105 36000 14500 5920 1840 773 330 109 47·9
2·03 2·03 2·04 2·04 2·05 2·07 2·09 2' 12 2'18 2·24 2·31 2·41 2·48
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3·00 3'00 3'00 3·01 3·03 3·11 3·76 2·66 2·07
337 219 127 86,0 60·5 40·6 31 ·9 26·3 13·5 7'53 ±0-46 i ±1'04i ±0'86 i
±336 i ±217 i ±124i ±82'7 i ±55'7 i ±33'1 i ±21.4 i ±1I '7 i 30·5 32·5 33 ·1 33·3 33·4
0·577 0·579 0·584 0·590 0·600 0·620 0·643 0·673 0'725 0·771 0·818 0·876 0·912
1·71 x 106 6·78 x 105 2·00 x 105 79700 31800 9480 3820 1550 479 200 84·8 27·9 12·2
2·10 2·10 2·11 2·11 2·12 2·13 2' 15 2·18 2·23 2·29 2·36 2-45 2' 52
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3'00 3·00 3·00 3·00 3·00 3·00 3·00 3'01 3·05 3-21 8·04 8-47
363 230 127 81 ·7 53·2 31 '0 21·2 15·0 10·2 8·00 6·53 2,97 2·2 1
±363 i ±230i ±127 i ±81'5 i ±52·8 i ±30·4i ±20'3 i ±13 '8 i ±8'27 i ±5'38 i ±2'83 i ±1 ' 18 i ±1·03 i
2·17 2·17 2' 17 2·18 2·19 2-20 2·22 2·24 2·29 2·34 2-40 2-49 2·55
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3'00 3·00 3·00 3·00 3·00 3·02 3·06 3·39 4·08
425 269 147 93'559·8 33 ·5 22·0 14·7 9·01 6'48 4·83 3·34 2·38
±425 i ±269 i ±147 i ±93-4i ±59·7 i ±33'4i ±21 '8 i ±14·5 i ±8'60i ±5·89 i ±4'02i ±2·25 i ±1'46 i
616 388 211
133 84·2 46·0 29·2 18·6 10·2 6,31 ±0'74i ±1'01 i ±0'81 i
v = 0·040 0·0001 0·0002 0·0005 0·0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000 v = 0·160 0·0001 0'0002 0·0005 0·0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000 v = 0·360 0·0001 0·0002 0·0005 0,0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000
0·760 0·761 0·763 0·765 0·769 0·778 0·787 0·801 0·827 0·851 0·877 0·912 0·936
8·23 x 105 3·27 x 105 96700 3850015300 4560 1830 740 227 94·1 39·6 12·9 5·61
757
767
J. F. COALES AND P. J. LAWRENCE
The tables of the parameters of the optimum filter show that as fl approaches zero, Y1 and Yz approach 1 and ,3
the non-dimensional parameters defined in equation 5, so two ae 2 K2(J2 further non-dimensional parameters equal to A2 and A 2a 3
e
could be defined. In fact, the parameters we have used are E -
Q;
J
~=3 cx=2 p= 1
QI
~ 0·8
2ae 2/J('l. + 1)2 A2(f/ + 1)( 'l.2 + /'J)
~tO'6
Ql
v =0·16
w
E 0.4 "U
These have the advantage of being quantities which have easily understood physical significances, E being the ratio of R.M.S. error to the R.M.S. value of the input signal, and fl being the ratio of the R.M.S. value of the input signal to the R.M.S. power at the motor input multiplied by the motor gain. Tables have been prepared by the authors, relating fl and E for given values of 'l., /'J, p and v. The tables were prepared using EDSAC 11, the computer at the Cambridge University Mathematical Laboratory. The value of ~ appropriate to a given value of fl is calculated by a process of repeated inverse linear interpolation between two initial estimates. The integrals in equations 8 and 9 are calculated by setting up and solving the set of linear equations which gives the integral as one of the unknowns (see Laning and Battin2). Having calculated ~, the transfer function Ho(p) is known, from the corresponding open loop transfer function =Hc(p)Hm(p) can be calculated as Co
v =0·36
eT
_ JA2a3(tJ + 1)('l.2 + /J) It 2K2(J2f/('l. + 1)2
HcCp)H",(p)
'·0
.~ 0·2 c;; E
o
Z
1-1-
Inverse of normalized power constraint
Figure 3. Relationship of normalized error E plotted against It, the inverse of normalized power constraint, for different values of normalized inverse noise/signal ratio
respectively and Y3 and Y4 are a complex pair with equal real and imaginary parts (i.e. a filter giving 1/21i2 damping factor). It can also be shown that the limiting value of is
e
vfJ fJ + 1
a --o
a1 - v (for v = 0 this expression reduces to 'l.) but this limit is not approached over the useful range of ft. Graphs such as Figure 3 could be used as design charts in order to simplify the interpretation of the tables. For example, it is easy to pick out on a graph the point beyond which an increase in power will not lead to any significant reduction of error. Ideally, the remainder of the tabulated information, i.e. the parameters of the optimum transfer function, could also be presented graphically. This has not been done, since the wide ranges of values recurring and the changes between real and complex values of the y's make interpretation of such graphs rather difficult. The method of interpolation between tabulated values of the parameters has been considered and the authors have attempted to ensure that linear interpolation in (1., fJ and It is sufficiently accurate. Linear interpolation in v1/2 also appears to be sufficient. Some checks of interpolated values against values calculated by the EDSAC have been undertaken; these have been satisfactory.
+ Cl e..a
cl!.. 1 a For the purpose of synthesizing Hc(p), the transfer function of the compensating network, it is convenient for the denominator of the above expression to be factorized into the form
The tables include the values of G, 0, Y1' Y2' Y3 and Y4 for each entry. In some cases the denominator is only a cubic in pia or one of the Y's may equal in which case the entry may appear to be incomplete. Tables 1 and 2 are typical tables, it is hoped to publish the complete set of tables elsewhere.
e,
The authors wish to thank the Director of the Cambridge Uni1"ersity Mathematical Laboratory for permission to use EDSAC Ilfor these calculations. They would also like to record their appreciation of preliminary work done on this subject by A. R. M. Notol1, J. F. Barretf and J. K. Bargh.
Information Plotting Some of the information in the tables can be conveniently presented graphically as plots of fl against E for fixed values of the other parameters. A typical plot is shown in Figure 3. Some features of the graphs are of interest. As ft approaches zero (i.e. as the power available increases), E approaches a limit dependent on)l. Interpreted physically, this implies that the following error which occurs is due almost entirely to the noise added to the signal to be followed and not to the power limitation. An explicit formula for the limiting value of E in terms of 1", 'l. and tJ is easily obtained but it is too complicated to be of interest. Conversely, as ft becomes large, E approaches unity. This is because as the power limitation becomes severe, the output vanishes so the error becomes equal to the input. It will also be noticed that variation with p is combined to the region of transition between the approaches to the limits mentioned above, and is comparatively small.
O~~~-L~~~~~-LLU~~-L~~~~. 0·0001 0'001 0·01 0·1
References BARRETT, l. F. Ph.D. Thesis, University of Cambridge, 1959 lAMES, H. N., NICHOLS, N. B. and PHILLIPS, R. S. The Theory of Servomechanisms. 1947. New York; McGraw-Hill 3 LANING, T. H. and BATTIN, R. H. Random Processes in Automatic Control. 1956. New York; McGraw-Hill , NEWTON, G. C Compensation of feedback control systems subject to saturation. J. Franklin Inst. 254 (1952) 281,391 5 NEWTO:--J, G. C. GOULD, L. A. and KAISER, l. F. Analytical Design of Linear Feedback Controls. 1957. New York; Wiley 6 TSIEN, H. S. £i1liineering Cybernetics. 1954. New York; McGrawHill 7 WIENER, N. Extrapolation, Interpolation and Smoothillli of Stationary Time-Series. 1949. Cambridge, Mass.; Technology Press 8 LAWRENCE, P. l. (to be published) 1
2
758
« 768
CHARTS AND TABLES FOR THE OPTIMIZATION OF AUTOMATIC CONTROL SYSTEMS WITH RANDOM INPUTS
Summary K
Although it is nearly twenty years since the Wiener-Kolmogorov theory of optimization of automatic control systems was first published, it has not been generally used because of the large amount of calculation required to solve a particular problem. With the use of non-dimensional parameters a small number of charts and tables can be prepared which enable the optimum design of a linear automatic control system of limited power output to be determined by a very simple procedure. Non-dimensional parameters relating to the performance of the system are defined. The relationships between these and the parameters of the input are calculated, using the Wiener-Kolmogorov theory. The optimum filter transfer function (closed loop) is calculated and hence the roots of the open loop transfer function are obtained. These relationships have been applied to a control system with
fixed parts having a transfer function of the form -C· -·--b with input
pp
.
A2(W 2
.
signal assumed to have a spectral density (
0
ur
+ )
+x2a 2)
.
+ a 2)( w· 2 + I'-alio ") wIth
white noise added. Charts are obtained which relate mean squared error to mean output power for different values of the input parameters. In designing a system the required mean output power is determined from these charts and then the required poles and zeros of the open loop transfer function can be read from tables of which samples are included. It is shown that this optimization procedure is quite general for linear systems with power constraint, provided the spectral density can be expressed approximately in the form given, which is believed to be the case for most commonly occurring random processes.
Sommaire se fixe certaines parties, ayant une fonction de transfert de la forme
Bien qu'il se so it ecoule pres de vingt ans depuis que la theorie de Wiener-Kolmogorov de l'optimisation des systemes de commande automatique a ete publiee pour la premiere fois, elle n'a pas ete utilisee de far;on generale en raison de la gran de quantite de calculs necessaires habitueIIement pour resoudre un probleme particulier. On a trouve qu'en utilisant des parametres dans dimensions il est possible de preparer un petit nombre de graphiques et de tableaux, qui permettent de determiner, par un pro cede simple, la conception optimale d'un systeme de commande automatique lineaire a puissance de sortie limitee. Dans le rapport on definit les parametres sans dimensions relatifs a la performance du systeme. On calcule les relations entre ces parametres et ceux de l'entree, pour le systeme optimise, en utilisant la theorie de Wiener-Kolmogorov. On calcule egalement la fonction de transfert du filtre optimal (chaine fermee), et on tire les racines de la fonction de transfert en chaine ouverte. Ces relations ont ete appliquees a un systeme de controle dont on
pep
K
+ b)'
et dont le signal d'entree est suppose avoir une densite
+ x 2a 2 ) + a2)(w2 + f3 2a2) a quoi s'ajoute le bruit blanc.
A2(W 2
spectrale (w 2
Les relations entre les parametres sans dimensions peuvent etre tirees des graphiques qui lient I'erreur quadratique moyenne a la limitation de la puissance moyenne de sortie. Dans le projet d'un systeme la puissance moyenne de sortie desiree est determinee d'apres ces graphiques et ensuite on lit sur des tableaux, dont on donne quelques exemples, les poles et les zeros recherches de la fonction de transfert en chaine ouverte. On montre que ce procede d'optimisation est tout a fait general pour les systemes lineaires a puissance limitee pourvu que la densite spectra le puisse etre exprimee approximativement dans la forme donnee, ce que I'on croit etre le cas pour les processus aleatoires les plus communement rencontn!s.
Zusammenfassung Obzwar seit der ersten Veroffentlichung der Optimierungstheorie flir Regelungsanordnungen von Wiener-Kolmogorov fast 20 lahre vergangen sind, so ist diese Theorie wegen des zur Losung einer bestimmten Aufgabe anfallenden Rechenaufwandes bisher nicht allgemein angewandt worden. Bei Verwendung normierter Parameter findet man jedoch, daLl einige Diagramme und Tabellen aufgestellt werden konnen, welche die optimale Bemessung einer linearen Regelungsanordnung mit begrenztem Leistungsausgang in sehr einfacher Weise ermoglichen. Im Beitrag werden die das Verhalten der Anordnung beschreibenden normierten Parameter definiert. lhre Verkniipfung mit den Parametern der Eingangsgr6f3e flir eine optimierte Anordnung wird noch der Theorie von Wiener-Kolmogorov bestimmt. .. Weiter werden die optimale Filter-Ubergangs~~nktion (geschlossener Kreis) berechnet und daraus die Wurzeln der Ubergangsfunktion des offenen Kreises ermittelt. Diese Beziehungen werden auf eine ruhende Regelungsanordnung mit einem Frequenzgang der Form
pep
K
+ b) angewandt, wobei eine EingangsgrOf.le mit einer spektralen
Dichte
+ x 2a 2 ) + a2)(w 2 + f3 2a2)
A2(W 2
(w 2
mit zusatzlichen weil.len Rauschen angenommen ist. Die Beziehungen zwischen den normierten Parametern lassen sich in Diagrammen darstellen, welche das mittlere Fehlerquadrat der optimierten Anordnung mit der mittleren Leistungsbegrenzung der AusgangsgroLle in Beziehung setzen. Beim Entwurf einer Anordnung werden die mittlere Ausgangsleistung aus den Diagrammen bestimmt und darauf die benotigten Pole und Nullstellen des Frequenzganges des offenen Kreises aus den Tabellen abgelesen. Es wird gezeigt, daf3 dieses Optimierungsverfahren ziemlich allgemein flir lineare Systeme mit Randbedingungen flir die Leistung anwendbar ist, wenn die spektrale Dichte etwa mit der angegebenen Form iibereinstimmt.
DISCUSSION E.
BLANDHOL
(Norway)
The attempt to bring statistical methods down to the practical stage is a very important step in this field of automatic control. It is quite true that statistical methods have not yet been much applied to process control problems. I think that another reason for this, not mentioned in the paper. is that processes are usually very slow, and thus extremely long averaging times are needed to obtain a sufficient accuracy in the statistical estimates. Very often, however, there exists an upper practical limit to the possible averaging time, and this time may not give the desired accuracy . There also exist practical difficulties due to drift in the measuring equipment, possible non-statiomirity of the signals, etc.
Newton's use of a power constraint in the statistical sense means that the system is allowed to operate in its non-linear range only a small fraction of the time, thus forcing the system to be approximately linear and amenable to mathematical treatment, but this is not the best solution. A limiter characteristic should be incorporated in the power source dynamics, and the system should be allowed to operate at maximum available power as much as it likes. To my knowledge, no sufficiently general solution exists to this important non-linear problem. Further efforts should, in my opinion, be directed more towards the evaluation of charts and tables for an exact or approximate solution to the non-linear case than to the restricted class of systems considered by the authors. although their tables will of course be very useful in a number of cases.
759
769
J. F. COALES AND P. J. LAWRENCE
The power spectrum (Il" (w) chosen by the authors has a specific form depending on four parameters. I would like to ask the authors if they have considered the possibility of using the first few terms of some series expansion for the power spectrum, for instance terms representing low-pass filtered or band-pass filtered white noise, or if they have found their expression to be the simplest one and yet sufficiently versatile. When fitting an estimated power spectrum to the form chosen by the authors four parameters have to be determined . The estimate will always have a statistical uncertainty due to the finite ensemble or time average used. It would be interesting to know how accurately each of the parameters can be determined for a given uncertainty in the power spectrum estimate. Work along these lines has just been started at the Automatic Control Laboratory in Trondheim, Norway. In particular, we will try to determine how the parameters of the optimum system function are influenced by the statistical uncertainty in the estimates of the various signals and disturbances involved. Finally, I would like to ask the authors if they use any specific graphical or numerical method for the practical determination of the four parameters in the power spectrum.
A. N.
POKROYSKI
it is difficult to obtain sufficiently long records to give sufficient accuracy in the statistical estimates is a very real one. However, the results of the work described in the paper show that the parameters of the optimum system change only slowly with the input parameters and therefore high accuracy in the estimates is not likely to be required. In fact, the results lead me to conclude that an optimum system based on guessed spectral densities is likely to be better than one designed for step inputs . I entirely agree with Dr. Blandhol that a limiter characteristic should be incorporated in the power source dynamics in order to give the best solution and, in fact, this work was started in order to get a comparison between our relay system with predicted change-over! and the optimum linear system. The result of this comparison was that the best relay system with predicted change-over will give the same mean square error as the optimum linear system for one-third of the maximum output torque, and it has the added advantage that it is to some extent self-adjusting, since a change in the statistical power of the input does not require a change in the parameters of the system, whereas the parameters of the optimum linear system are different for different input powers. Unfortunately, it is not easy to compare the relay system with the optimum linear system on the basis of output powers, since this depends on the form of the input signal, because the relay system will hunt in periods when the input signal is constant for any appreciable time. The power spectrum
(U.S.S.R.)
In recent years methods have been developed for the synthesis of optimum impulse systems and continuously-acting systems with random disturbances. When the random disturbances are stationary comparatively simple solutions are obtained which find ever greater application in engineering practice . However, in order to bring the problem to a practical conclusion, it is usually necessary (except in a few trivial cases) to make very laborious calculations. This particularly applies to the calculation of impulse systems and it hinders utilization of the available results in engineering problems. Moreover, as the calculations become more complicated, the probability of error increases. It is therefore advisable to tabulate the most laborious part of the calculation and solution of integral equation used for the synthesis of continuously acting systems and to solve systems of linear, algebraic equations for the synthesis of impulse systems. Bearing in mind the known formulation of the problem of Zadeh and Ragazzini for continuously-acting systems, or that of V. P. Perov and K. 10hnson for impulse systems, we should apparently limit ourselves to examination of useful 'regular' signals in the form of polynomials of not higher than second or third power and steadystate random interference with correlation functions of the form R(T) =
(J2
e-o;(T) {cos {Jr
+ X ~ sin fJH}
Allowing for all possible combinations of parameters of the correlation function and memory time of the system, this will cover a considerable number of problems encountered in engineering practice. The author of this communication has carried out considerable preparatory work for setting up the necessary tables. It is proposed to make the calculations on universal digital computers.
References COALES, 1. F. and NOToN, A . R. M. An On-Off Servo Mechanism with Predicted Change-over. Proc. I.E.E. Pt. B (1956) 449-460 2 BURT, E. G . C. Self-optimising Servo-systems with Random Inputs. Cambridge Seminar on Non-linear Control Problems. 1954 3 BURT, E. G. C. Self-optimising Systems. 1956. Heidelberg, Fachtagung Regelungstechnik
1
1. F. COALES, in reply. Dr. Blandhol's point that statistical methods have not yet been much applied to process control problems because
760
770
at the present time no generally applicable optimization procedures are available and it is, therefore, desirable that engineers should be able to apply the linear optimization procedure without having to make unduly laborious calculations. Consideration has, therefore, been given to the possibility of producing charts, which would enable a de.si~ner of a ~ingle control loop with some knowledge of the statistics of the Input and 'noise' to decide what compensating network and power source must be employed to meet a given performance specification with the greatest economy of power. It has been found that by the use of non-dimensional parameters very widely applicable results can be obtained and it appears that, because near the optimum the performance changes only slowly, a reasonable number of charts will cover all systems likely to be of interest to the control engineer. The preparation and use of these charts, which it is intended to publish later in book form, is described below. It is considered that the method of using the procedure will generally be as follows: (1) Either determine or estimate the power spectra of the input (or the desired output) and of the noise or unwanted fluctuations. (2) Choose the appropriate values of the spectral density parameters A , a, CI., (f, to get the best fit for a power spectrum of the form A2(W 2 + Cl. 2a 2 )(f3 + 1)2
F(!{ure I . Block diagralll of general single loop control system lI'ith randolll input and noise
unwanted fluctuations are shown as occurring at the input. ]f fluctuations occur at some other point in a linear system, they can, in some cases, be referred back to the input by operating on them by the appropriate transfer functions. . .. Up to the present time, it ha~ usually be~n only In pOSitIOn control problems associated With such th.Ings as radar and guided missiles, where high performance I~ the presence of unavoidable 'noise' is required, that the deSign procedure has been used with success. This is partly because in most process control problems the performance required in terms of accuracy has not warranted the use of such a sophisticated method but the drive for greater productivity in all sections of industr,Y results in a need for better control of all types of plant , ThiS means that more accurate design procedures must in the end be adopted to attain higher performance and it has bee~ s~own that where linear systems are employed a.n~ the St~tIStICS. of the input quantities are known, the statlstl~al deSign ,USing Newton's procedure does, in fac~, resul.t In the optimum design , Although it is almost certain t.hat In m~ny cases more economic designs can be developed USing non-lInear elements,
+ a2)(w2 + {J2a2)(r:J. + 1)2
for the expected spectral density of the input. . . . (3) Choose a value of N which makes band-lImIted white noise most nearly approximate to the expected power spectrum of the noise, (4) From a study of the specification estimate the mean and maximum output powers that are likely to be required and decide on the type of prime mover that will probably be used . Hence , estimate those factors of the transfer function of motor and load which depend on inertia and damping of the l?ad and ignore additional factors such as that due to the field time constant of an electric motor, which do not affect the output power, (5) From the values of A, a, CI., ~, Nand b calculate the non-dimensional parameters z; = NalA an~ p = bla. . (6) The specification must lay down either the maximum permissible mean squared error ~ or the maximum permissible mean power of the motor, a2 , or one of the values must be obtainable from the specification. Hence determine one of the non-dimensional parameters
E{ or
753
763
«((j2
=
(e2/mean-squared value of input signal)1/2}
,n ( = mean-square ;alue of input x ~) .
J. F. COALES AND P. J. LAWRENCE
(7) From the chart for the appropriate values of p, rx and fJ read off the value of E or fi, whichever is not previously determined, for the appropriate value of v and the other quantity. Hence the value of (!2 or a2 for the optimum system can be calculated. In general there will not be a chart or a curve on it for the required values of p, rx, fJ and v, and repeated linear interpolations will have to be employed to find the value of E or fi, whichever is required. (8) From the tables the values of the poles and zeros of the optimum open loop transfer function corresponding to the appropriate values of rx, fJ, p, v and fi are obtained. (9) Knowing these parameters, a check on the value of i! or a 2 , which will result from the use of this transfer function, can be obtained numerically, if required. (10) The design can now proceed on the basis of an actual prime mover. This means that the actual open loop transfer function must be made to conform to that determined for the optimum system by including in the compensating network the inverse of those factors other than K/p(p + b) which are introduced into the system either by the prime mover or amplifying stages. In some applications, the limitation will not be on mean output power but on some other quantity such as motor torque, in which case this constraint will have to be approximated by obtaining an equivalent value for the mean output power on the basis of a gaussian distribution of torque.
However, as a closed loop system with such an overall transfer function requires infinite loop-gain, this is clearly unpracticable. So in order that the optimum system shall be one that is useful 10r-~~~Tm-'--'''~-'-''Tn~
t
00_
a
Figure 2. Spectral density functions plotted against frequency
in practice, it is necessary to minimize the mean squared error subject to some constraint within the loop. The constraint chosen is the mean power at the input to the motor, since it is often found that the performance of a system is limited by the necessary limitation of power consumed by a power amplifier or motor. An alternative constraint, that of system bandwidth, has been suggested by Newton 5 • It can be shown 8 that this constraint can, in fact, be equivalent to the power constraint described here and that a system designed using the power constraint is a minimum bandwidth system.
Single-loop System We have considered a single-loop system as shown in Figure 1; the system is presumed to control a motor and load with transfer function K/p(p + b), i.e. with inertia and viscous
damping but no 'spring term'. For reasons that will be mentioned later, we apply a constraint to the mean power input to the motor and load. The choice of the form of spectral density function to be assumed for the input signal is difficult since, to our knowledge, very little has been published on the analysis of signals encountered in the various possible fields of application of this method. Accordingly, the chosen form must be as general as possible, but in order to keep the tabulation of the results of the calculations down to a reasonable size, the number of different parameters must be kept to a minimum. The form chosen is: A2(0)2 + rx 2a2)(fJ + 1)2 <1> (0) = -----,,--......,,--,,---------',,'--,c---,----'-----,: ss (0)2 + a2)( 0)2 + fJ 2a2)( rx + 1)2 By giving suitable values to rx and fJ a reasonable range of shapes of spectral densities can be fitted. The factor (fJ + 1)2/ (rx + 1)2 is included in order to allow for infinite values of rx or fJ. The graphs in Figure 2 show the result of plotting <1> against O)/a for various values of rx and /J. For the purpose of clarity, the function plotted has the factor (fJ2 + 1)/ (rx 2 + 1) instead of {(fJ + 1)2}/{(rx + 1)2}; in the subsequent calculations the latter factor has been used because this simplifies some of the algebra occurring. In most practical cases, it is found that if noise occurs in the input signal, the bandwidth of the noise is generally much greater than that of the signal, so the noise may be treated as 'white' as far as the design of the system is concerned. In the calculations described in this paper, we have not considered anything other than white noise in the input. It was mentioned above that a constraint is applied to the mean power; the reason for this is as follows: In the absence of noise it is obvious that the optimum system has a transfer function equal to unity, since this would give zero error.
The Mathematical Problem The mathematical problem to be solved is, referring to Figure 1, to minimize the mean-squared error (s - y)2 subject to the constraint Z2 = a 2 • This can be done, using the general
method of Lagrangian multipliers; this was first applied to this type of problem by Newton 4 and the approach used here is due to Tsien 6 • It is well known that a system of overall transfer function H(p) which has an input consisting of a signal with spectral density <1>siw) with added noise of spectral density <1>nn(w) produces a mean squared error (i.e. signal output) given (if signal and noise are uncorrelated) by:
C2 = 217T L~£X){I-H(jO)j2<1>ssCO) + IHCjw)12<1>",,(W)} dw*
(I)
and that this is minimized by a transfer function given by 1 [<1>"';J
HCp) =lHp) 'F( -p) -;-
(2)
where 'F(jO)'F( -jw) = <1>siw) + <1>",,(w) 'F(p) is such that its poles and zeros lie in the left-hand (real part of p negative) half of the p plane and
[<1> ss
(5)]
IF(_p) -;-
* Depending on the definition of spectral density used, the factor preceding the integral may be Ij2 or 1/2rr.
754
764
CHARTS AND TABLES FOR THE OPTIMIZATION OF AUTOMATIC CONTROL SYSTEMS WITH RANDOM INPUTS
indicates that part of the partial fraction expansion of
We define non-dimensional parameters as follows: Na b a2, 1"A Y = A P = ~ ~ = ---y-
P\
and can now get expressions for A(jx) and B(jx):
'f'( -p)
A(jx) = a o
which has poles in the left-hand half of the p plane. This was originally shown by Wiener 7 • Applying the method of Lagrangian multipliers to the present problem, it is desired to minimize 2 + AZ 2 where }, is a Lagrangian multiplier. To obtain Z2, note that (referring to Figure 1 and using capital letters to indicate Laplace transforms)
+ aljx +
ao=
where
J
al =
and the quantity to be minimized, ~
t f+oo
2-
7T
{It -
+
Ho(j01)j2
+ ).z2 is given
(3)
by
+ IH o(j01)j2[
(4)
+ IHm (j01)12(
},[
'f'
rh
(-j01) = 'l-' s/(I})
+ +
1) (
[
rh + 'l-'n,,(w) +
Substituting the values of
I'f'0 w ) I -
}·
J(1)
1 = 'f'(p)
+ 2a Oa 2
1)2
+
a
(-y
(32)«(3
+
(_y2
+ (1..2)
I)(y
+
1)A
+
+
I)(y
(3)
A( -y)B( -y)
(7)
(3)A( -y)B( -y)
+ do + dly + d 2y2 + d 3y3 )
Cly
(5)]
A«(3
+ 1) a
+
Co
+
+
I)(y
(y
Cly
+ (3)
[
i.e. we now have an expression for Ho(p) in terms of only the non-dimensional parameters defined in equation 5. By substituting this in equations 3 and 1, expressions are obtained for the mean squared power and error. These are
+
and Hm(p) given
= A 2a 3
+ 1) a2
~f +OC \(Co + Cdx)jx(jx + P)\2dx (b o + bdx + b 2 )(jx)2
K2 . 27T
2" _ A2
(1
e - -- a 27T
+N
2) (
I
+
;.(1)2«(1)2 K
+ b2»)
A(jx)B(jx) (jx + l)(jx +
f +oo -
Of)
C o - Cdx )
(
x A(jx)B(jx)(jx
+ y2 • 2~
(8)
- 00
(A(jx)B(jx) -
Let O1/a = x and
J
+
+
\f'( -p)
and
= A«(3
+
«(3
l)(y + (3) A( -y)(3( -y) where Co, Cl' do ... d 3 can be obtained from a set of simultaneous linear equations generated by equating coefficients with those of equation 7. Hence
2
'F( '01)
+
1) 1)2 (y
Co (y
a
.
1)2
and therefore Ho(p)
The procedure outlined in The Mathematical Problem will now be applied to the problem we are considering. In the course of this, convenient non-dimensional parameters will be defined, Referring to equation 2 we first require 'f"(p) such that (jw)
A«(3 a«(1..
a
Non-dimensional Parameters
T
+
= A«(3
which will be given by equation 2. This will give Ho(p) as a function of A; by substituting this value of Ho(p) into equation 3. A is then determined by setting Z2 = a2 , the required mean power. A being known, Ho(p) and"? can now be calculated.
ur
+
«(1..
' /p2~2
Since A(y) and B(y) have zeros in the left-hand half-plane only, this expression can be expanded in partial fractions:
Comparing equation 4 with equation t it can be seen that the required Ho(p) is that of a filter for separating a signal with spectral density
=
+ 1)2 (_y2 + (1..2) + 1)2 (_y2 + 1)( _y2 +
A
bl
If the coefficients of powers of jx in 'f'(j01) are all positive, the condition on the poles and zeros of'f'(p) is satisfied, hence taking the coefficients defined above as positive is sufficient for this condition . The quantity {
- 00
I
=
b2 = ~
thus - 00
b
+ 2~ ---------------------1 y2(1 + (32)
Y(p) = Ho(p)X(p), hence Z(p) = {Ho(p)X(p)}/{H",(p)}
f +oo IHm(j01) H o(j01)1 2 {
((3
a2 = - (3 + 1
If the overall transfer function of the system is Ho(p), then
1
JC: If v! If o Y
Y(p) = Hm(p)Z(p)
2
+ bl(jx) + bz{jx)Z
B(jx) = b o
aijx)2
(6)
e
Z = 27T
(5)
+
1)
t+:\~(j~~1~ \dX)
Cl. Cl.
+
1
(1; :
+
(1..
n
JX. +
)
2
1 ~
(9)
The integrals in these expressions are entirely in terms of
(3) 755
765
Table I
{3=1
oc=1
p=1 G
Form of optimum open loop transrer function is - - - - - - - - -
(f!.a + /'
€
G
0-0001 0-0002 0·0005 0-0010 0-0020 0-0050 0-0100 0·0200 0-0500 0-1000 0·2000 0-5000 1·0000
0·064 0·080 0·109 0·136 0-171 0-229 0·285 0-352 0-457 0·546 0·639 0-756 0-831
2-72 X 10' 1·08 x 10" 31800 12700 5030 1490 596 239
0·284 0·287 0·296 0·306 0·321 0·353 0·387 0-434 0·515 0·589 0·669 0·773 0·841
·'I)(f!.a + ,,_)(f!.a +.,) {'l.
I
i
2
,3
., J i
v=O 1·00 1·00 1·00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1-00 1·00
737 464 252 159 100 54·6 34-5 21-9 12·0 7-68 4-94 2-79 1-84
6·88 x 10" 2·73 x 10" 8·06 x 10' 3·21 x 10' 1·27 x 10-' 37800 15100 6050 1830 745 308 98·2 42-4
1·00 1·00 1-00 1·00 1·00 1·00 1·00 1·00 1-00 1·00 1·00 1·00 1·00
391 251 142 94·2 64·0 40·6 30·2 23·7 18·7 9·93 5·73 3·05 1-96
±391 i ±250 i ±141 i ±92'5 i ±61'5 i ±36'5 i ±24' 5 i ±15·8 i ±5'98 i 23·0 24-4 24·8 24-9
0·525 0·526 0·530 0·534 0·540 0·555 0·571 0·596 0·642 0·688 0·742 0-817 0·869
1·79 x 10" 7· 11 x 10" 2-10 x 10" 83400 33200 9820 3920 1570 473 193 79-4 25·2 10-9
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
413 261 143 91·6 58·9 33·5 22·3 15·3 9·83 7·42 5·89 4·70 2·36
±413 i ±261 i ±143 i ±91·5 i ±58-8 i ±33'2 i ±21'9 i ±14'6 i ±8'75 i ±5'91 i ± 3-84 i ±1 ·45 i 5·79
0·712 0·713 0·715 0·716 0-719 0-726 0·734 0·745 0-769 0·793 0·824 0·870 0·904
8-49 x 10' 3-37 x 10' 99400 39500 15700 4650 1850 743 223 90·6 37·2 11·7 5-02
1·00 1-00 1·00 1·00 1·00 1·00 1-00 1-00 1·00 1-00 1-00 1·00 1-00
468 295 161 102 64·9 36·0 23 ·3 15 ·3 9·03 6·29 4·56 3-21 2·6 1
±468 i ±295 i ±161 i ±102 i ±64'9 i ±35'9 i ±23 ·2 i ±15 ' 1 i ± 8'79 i ±5 '94 i ±4'06 i ±2'44 i ±1'57 i
72-1
29·5 12·2 3·89 1·68
" = 0-040
0-0001 0-0002 0-0005 0-0010 0·0020 0·0050 0·0100 0-0200 0·0500 0-1000 0-2000 0-5000 1-0000 ~'=
0-160 0·0001 0·0002 0·0005 0-0010 0-0020 0·0050 0·0100 0·0200 0·0500 0-1000 0·2000 0-5000 1-0000
}- = 0-360
0·0001 0-0082 0-0005 0·0010 0·0020 0-0050 0-0100 0·0200 0·0500 0-1000 0-2000 0·5000 1-0000
756
766
'X=2
{3=3
p
Table 2
= I
G
Form of optimum open-loop transfer function is
(~+
0) '
(~ + i'I)(~ + i'2)(~ + i'3)(~ + i") {t
l'
---
E
-
G
-
I
0
1' 1
" 2
I
;'3
it"
= 0,000 0,0001 0·0002 0·0005 0·0010 0·0020 0·0050 0,0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000
0·091 0·114 0·154 0·193 0·240 0·318 0·389 0·469 0·582 0·667 0,746 0·835 0·887
1·90 x 105 75500 22300 8890 3550 1060 428 175 54-4 22·9 9·79 3·25 1-43
2·00 2·00 2·01 2·02 2·02 2·04 2·06 2·10 2·16 2·22 2·29 2·39 2-46
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3·00 3·00 3,00 3·01 3,04 3' 16 3·51 2·50 2·02
0·323 0·329 0·343 0·359 0·383 0-429 0-477 0·536 0·626 0·698 0·767 0·846 0·894
6·45 x 106 2·57 x 106 7·58 x 105 3·02 x 105 1·21 x 105 36000 14500 5920 1840 773 330 109 47·9
2·03 2·03 2·04 2·04 2·05 2·07 2·09 2' 12 2'18 2·24 2·31 2·41 2·48
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3·00 3'00 3'00 3·01 3·03 3·11 3·76 2·66 2·07
337 219 127 86,0 60·5 40·6 31 ·9 26·3 13·5 7'53 ±0-46 i ±1'04i ±0'86 i
±336 i ±217 i ±124i ±82'7 i ±55'7 i ±33'1 i ±21.4 i ±1I '7 i 30·5 32·5 33 ·1 33·3 33·4
0·577 0·579 0·584 0·590 0·600 0·620 0·643 0·673 0'725 0·771 0·818 0·876 0·912
1·71 x 106 6·78 x 105 2·00 x 105 79700 31800 9480 3820 1550 479 200 84·8 27·9 12·2
2·10 2·10 2·11 2·11 2·12 2·13 2' 15 2·18 2·23 2·29 2·36 2-45 2' 52
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3'00 3·00 3·00 3·00 3·00 3·00 3·00 3'01 3·05 3-21 8·04 8-47
363 230 127 81 ·7 53·2 31 '0 21·2 15·0 10·2 8·00 6·53 2,97 2·2 1
±363 i ±230i ±127 i ±81'5 i ±52·8 i ±30·4i ±20'3 i ±13 '8 i ±8'27 i ±5'38 i ±2'83 i ±1 ' 18 i ±1·03 i
2·17 2·17 2' 17 2·18 2·19 2-20 2·22 2·24 2·29 2·34 2-40 2-49 2·55
1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00 1·00
3·00 3·00 3·00 3·00 3'00 3·00 3·00 3·00 3·00 3·02 3·06 3·39 4·08
425 269 147 93'559·8 33 ·5 22·0 14·7 9·01 6'48 4·83 3·34 2·38
±425 i ±269 i ±147 i ±93-4i ±59·7 i ±33'4i ±21 '8 i ±14·5 i ±8'60i ±5·89 i ±4'02i ±2·25 i ±1'46 i
616 388 211
133 84·2 46·0 29·2 18·6 10·2 6,31 ±0'74i ±1'01 i ±0'81 i
v = 0·040 0·0001 0·0002 0·0005 0·0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000 v = 0·160 0·0001 0'0002 0·0005 0·0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000 v = 0·360 0·0001 0·0002 0·0005 0,0010 0·0020 0·0050 0·0100 0·0200 0·0500 0·1000 0·2000 0·5000 1·0000
0·760 0·761 0·763 0·765 0·769 0·778 0·787 0·801 0·827 0·851 0·877 0·912 0·936
8·23 x 105 3·27 x 105 96700 3850015300 4560 1830 740 227 94·1 39·6 12·9 5·61
757
767
J. F. COALES AND P. J. LAWRENCE
The tables of the parameters of the optimum filter show that as fl approaches zero, Y1 and Yz approach 1 and ,3
the non-dimensional parameters defined in equation 5, so two ae 2 K2(J2 further non-dimensional parameters equal to A2 and A 2a 3
e
could be defined. In fact, the parameters we have used are E -
Q;
J
~=3 cx=2 p= 1
QI
~ 0·8
2ae 2/J('l. + 1)2 A2(f/ + 1)( 'l.2 + /'J)
~tO'6
Ql
v =0·16
w
E 0.4 "U
These have the advantage of being quantities which have easily understood physical significances, E being the ratio of R.M.S. error to the R.M.S. value of the input signal, and fl being the ratio of the R.M.S. value of the input signal to the R.M.S. power at the motor input multiplied by the motor gain. Tables have been prepared by the authors, relating fl and E for given values of 'l., /'J, p and v. The tables were prepared using EDSAC 11, the computer at the Cambridge University Mathematical Laboratory. The value of ~ appropriate to a given value of fl is calculated by a process of repeated inverse linear interpolation between two initial estimates. The integrals in equations 8 and 9 are calculated by setting up and solving the set of linear equations which gives the integral as one of the unknowns (see Laning and Battin2). Having calculated ~, the transfer function Ho(p) is known, from the corresponding open loop transfer function =Hc(p)Hm(p) can be calculated as Co
v =0·36
eT
_ JA2a3(tJ + 1)('l.2 + /J) It 2K2(J2f/('l. + 1)2
HcCp)H",(p)
'·0
.~ 0·2 c;; E
o
Z
1-1-
Inverse of normalized power constraint
Figure 3. Relationship of normalized error E plotted against It, the inverse of normalized power constraint, for different values of normalized inverse noise/signal ratio
respectively and Y3 and Y4 are a complex pair with equal real and imaginary parts (i.e. a filter giving 1/21i2 damping factor). It can also be shown that the limiting value of is
e
vfJ fJ + 1
a --o
a1 - v (for v = 0 this expression reduces to 'l.) but this limit is not approached over the useful range of ft. Graphs such as Figure 3 could be used as design charts in order to simplify the interpretation of the tables. For example, it is easy to pick out on a graph the point beyond which an increase in power will not lead to any significant reduction of error. Ideally, the remainder of the tabulated information, i.e. the parameters of the optimum transfer function, could also be presented graphically. This has not been done, since the wide ranges of values recurring and the changes between real and complex values of the y's make interpretation of such graphs rather difficult. The method of interpolation between tabulated values of the parameters has been considered and the authors have attempted to ensure that linear interpolation in (1., fJ and It is sufficiently accurate. Linear interpolation in v1/2 also appears to be sufficient. Some checks of interpolated values against values calculated by the EDSAC have been undertaken; these have been satisfactory.
+ Cl e..a
cl!.. 1 a For the purpose of synthesizing Hc(p), the transfer function of the compensating network, it is convenient for the denominator of the above expression to be factorized into the form
The tables include the values of G, 0, Y1' Y2' Y3 and Y4 for each entry. In some cases the denominator is only a cubic in pia or one of the Y's may equal in which case the entry may appear to be incomplete. Tables 1 and 2 are typical tables, it is hoped to publish the complete set of tables elsewhere.
e,
The authors wish to thank the Director of the Cambridge Uni1"ersity Mathematical Laboratory for permission to use EDSAC Ilfor these calculations. They would also like to record their appreciation of preliminary work done on this subject by A. R. M. Notol1, J. F. Barretf and J. K. Bargh.
Information Plotting Some of the information in the tables can be conveniently presented graphically as plots of fl against E for fixed values of the other parameters. A typical plot is shown in Figure 3. Some features of the graphs are of interest. As ft approaches zero (i.e. as the power available increases), E approaches a limit dependent on)l. Interpreted physically, this implies that the following error which occurs is due almost entirely to the noise added to the signal to be followed and not to the power limitation. An explicit formula for the limiting value of E in terms of 1", 'l. and tJ is easily obtained but it is too complicated to be of interest. Conversely, as ft becomes large, E approaches unity. This is because as the power limitation becomes severe, the output vanishes so the error becomes equal to the input. It will also be noticed that variation with p is combined to the region of transition between the approaches to the limits mentioned above, and is comparatively small.
O~~~-L~~~~~-LLU~~-L~~~~. 0·0001 0'001 0·01 0·1
References BARRETT, l. F. Ph.D. Thesis, University of Cambridge, 1959 lAMES, H. N., NICHOLS, N. B. and PHILLIPS, R. S. The Theory of Servomechanisms. 1947. New York; McGraw-Hill 3 LANING, T. H. and BATTIN, R. H. Random Processes in Automatic Control. 1956. New York; McGraw-Hill , NEWTON, G. C Compensation of feedback control systems subject to saturation. J. Franklin Inst. 254 (1952) 281,391 5 NEWTO:--J, G. C. GOULD, L. A. and KAISER, l. F. Analytical Design of Linear Feedback Controls. 1957. New York; Wiley 6 TSIEN, H. S. £i1liineering Cybernetics. 1954. New York; McGrawHill 7 WIENER, N. Extrapolation, Interpolation and Smoothillli of Stationary Time-Series. 1949. Cambridge, Mass.; Technology Press 8 LAWRENCE, P. l. (to be published) 1
2
758
« 768
CHARTS AND TABLES FOR THE OPTIMIZATION OF AUTOMATIC CONTROL SYSTEMS WITH RANDOM INPUTS
Summary K
Although it is nearly twenty years since the Wiener-Kolmogorov theory of optimization of automatic control systems was first published, it has not been generally used because of the large amount of calculation required to solve a particular problem. With the use of non-dimensional parameters a small number of charts and tables can be prepared which enable the optimum design of a linear automatic control system of limited power output to be determined by a very simple procedure. Non-dimensional parameters relating to the performance of the system are defined. The relationships between these and the parameters of the input are calculated, using the Wiener-Kolmogorov theory. The optimum filter transfer function (closed loop) is calculated and hence the roots of the open loop transfer function are obtained. These relationships have been applied to a control system with
fixed parts having a transfer function of the form -C· -·--b with input
pp
.
A2(W 2
.
signal assumed to have a spectral density (
0
ur
+ )
+x2a 2)
.
+ a 2)( w· 2 + I'-alio ") wIth
white noise added. Charts are obtained which relate mean squared error to mean output power for different values of the input parameters. In designing a system the required mean output power is determined from these charts and then the required poles and zeros of the open loop transfer function can be read from tables of which samples are included. It is shown that this optimization procedure is quite general for linear systems with power constraint, provided the spectral density can be expressed approximately in the form given, which is believed to be the case for most commonly occurring random processes.
Sommaire se fixe certaines parties, ayant une fonction de transfert de la forme
Bien qu'il se so it ecoule pres de vingt ans depuis que la theorie de Wiener-Kolmogorov de l'optimisation des systemes de commande automatique a ete publiee pour la premiere fois, elle n'a pas ete utilisee de far;on generale en raison de la gran de quantite de calculs necessaires habitueIIement pour resoudre un probleme particulier. On a trouve qu'en utilisant des parametres dans dimensions il est possible de preparer un petit nombre de graphiques et de tableaux, qui permettent de determiner, par un pro cede simple, la conception optimale d'un systeme de commande automatique lineaire a puissance de sortie limitee. Dans le rapport on definit les parametres sans dimensions relatifs a la performance du systeme. On calcule les relations entre ces parametres et ceux de l'entree, pour le systeme optimise, en utilisant la theorie de Wiener-Kolmogorov. On calcule egalement la fonction de transfert du filtre optimal (chaine fermee), et on tire les racines de la fonction de transfert en chaine ouverte. Ces relations ont ete appliquees a un systeme de controle dont on
pep
K
+ b)'
et dont le signal d'entree est suppose avoir une densite
+ x 2a 2 ) + a2)(w2 + f3 2a2) a quoi s'ajoute le bruit blanc.
A2(W 2
spectrale (w 2
Les relations entre les parametres sans dimensions peuvent etre tirees des graphiques qui lient I'erreur quadratique moyenne a la limitation de la puissance moyenne de sortie. Dans le projet d'un systeme la puissance moyenne de sortie desiree est determinee d'apres ces graphiques et ensuite on lit sur des tableaux, dont on donne quelques exemples, les poles et les zeros recherches de la fonction de transfert en chaine ouverte. On montre que ce procede d'optimisation est tout a fait general pour les systemes lineaires a puissance limitee pourvu que la densite spectra le puisse etre exprimee approximativement dans la forme donnee, ce que I'on croit etre le cas pour les processus aleatoires les plus communement rencontn!s.
Zusammenfassung Obzwar seit der ersten Veroffentlichung der Optimierungstheorie flir Regelungsanordnungen von Wiener-Kolmogorov fast 20 lahre vergangen sind, so ist diese Theorie wegen des zur Losung einer bestimmten Aufgabe anfallenden Rechenaufwandes bisher nicht allgemein angewandt worden. Bei Verwendung normierter Parameter findet man jedoch, daLl einige Diagramme und Tabellen aufgestellt werden konnen, welche die optimale Bemessung einer linearen Regelungsanordnung mit begrenztem Leistungsausgang in sehr einfacher Weise ermoglichen. Im Beitrag werden die das Verhalten der Anordnung beschreibenden normierten Parameter definiert. lhre Verkniipfung mit den Parametern der Eingangsgr6f3e flir eine optimierte Anordnung wird noch der Theorie von Wiener-Kolmogorov bestimmt. .. Weiter werden die optimale Filter-Ubergangs~~nktion (geschlossener Kreis) berechnet und daraus die Wurzeln der Ubergangsfunktion des offenen Kreises ermittelt. Diese Beziehungen werden auf eine ruhende Regelungsanordnung mit einem Frequenzgang der Form
pep
K
+ b) angewandt, wobei eine EingangsgrOf.le mit einer spektralen
Dichte
+ x 2a 2 ) + a2)(w 2 + f3 2a2)
A2(W 2
(w 2
mit zusatzlichen weil.len Rauschen angenommen ist. Die Beziehungen zwischen den normierten Parametern lassen sich in Diagrammen darstellen, welche das mittlere Fehlerquadrat der optimierten Anordnung mit der mittleren Leistungsbegrenzung der AusgangsgroLle in Beziehung setzen. Beim Entwurf einer Anordnung werden die mittlere Ausgangsleistung aus den Diagrammen bestimmt und darauf die benotigten Pole und Nullstellen des Frequenzganges des offenen Kreises aus den Tabellen abgelesen. Es wird gezeigt, daf3 dieses Optimierungsverfahren ziemlich allgemein flir lineare Systeme mit Randbedingungen flir die Leistung anwendbar ist, wenn die spektrale Dichte etwa mit der angegebenen Form iibereinstimmt.
DISCUSSION E.
BLANDHOL
(Norway)
The attempt to bring statistical methods down to the practical stage is a very important step in this field of automatic control. It is quite true that statistical methods have not yet been much applied to process control problems. I think that another reason for this, not mentioned in the paper. is that processes are usually very slow, and thus extremely long averaging times are needed to obtain a sufficient accuracy in the statistical estimates. Very often, however, there exists an upper practical limit to the possible averaging time, and this time may not give the desired accuracy . There also exist practical difficulties due to drift in the measuring equipment, possible non-statiomirity of the signals, etc.
Newton's use of a power constraint in the statistical sense means that the system is allowed to operate in its non-linear range only a small fraction of the time, thus forcing the system to be approximately linear and amenable to mathematical treatment, but this is not the best solution. A limiter characteristic should be incorporated in the power source dynamics, and the system should be allowed to operate at maximum available power as much as it likes. To my knowledge, no sufficiently general solution exists to this important non-linear problem. Further efforts should, in my opinion, be directed more towards the evaluation of charts and tables for an exact or approximate solution to the non-linear case than to the restricted class of systems considered by the authors. although their tables will of course be very useful in a number of cases.
759
769
J. F. COALES AND P. J. LAWRENCE
The power spectrum (Il" (w) chosen by the authors has a specific form depending on four parameters. I would like to ask the authors if they have considered the possibility of using the first few terms of some series expansion for the power spectrum, for instance terms representing low-pass filtered or band-pass filtered white noise, or if they have found their expression to be the simplest one and yet sufficiently versatile. When fitting an estimated power spectrum to the form chosen by the authors four parameters have to be determined . The estimate will always have a statistical uncertainty due to the finite ensemble or time average used. It would be interesting to know how accurately each of the parameters can be determined for a given uncertainty in the power spectrum estimate. Work along these lines has just been started at the Automatic Control Laboratory in Trondheim, Norway. In particular, we will try to determine how the parameters of the optimum system function are influenced by the statistical uncertainty in the estimates of the various signals and disturbances involved. Finally, I would like to ask the authors if they use any specific graphical or numerical method for the practical determination of the four parameters in the power spectrum.
A. N.
POKROYSKI
it is difficult to obtain sufficiently long records to give sufficient accuracy in the statistical estimates is a very real one. However, the results of the work described in the paper show that the parameters of the optimum system change only slowly with the input parameters and therefore high accuracy in the estimates is not likely to be required. In fact, the results lead me to conclude that an optimum system based on guessed spectral densities is likely to be better than one designed for step inputs . I entirely agree with Dr. Blandhol that a limiter characteristic should be incorporated in the power source dynamics in order to give the best solution and, in fact, this work was started in order to get a comparison between our relay system with predicted change-over! and the optimum linear system. The result of this comparison was that the best relay system with predicted change-over will give the same mean square error as the optimum linear system for one-third of the maximum output torque, and it has the added advantage that it is to some extent self-adjusting, since a change in the statistical power of the input does not require a change in the parameters of the system, whereas the parameters of the optimum linear system are different for different input powers. Unfortunately, it is not easy to compare the relay system with the optimum linear system on the basis of output powers, since this depends on the form of the input signal, because the relay system will hunt in periods when the input signal is constant for any appreciable time. The power spectrum
(U.S.S.R.)
In recent years methods have been developed for the synthesis of optimum impulse systems and continuously-acting systems with random disturbances. When the random disturbances are stationary comparatively simple solutions are obtained which find ever greater application in engineering practice . However, in order to bring the problem to a practical conclusion, it is usually necessary (except in a few trivial cases) to make very laborious calculations. This particularly applies to the calculation of impulse systems and it hinders utilization of the available results in engineering problems. Moreover, as the calculations become more complicated, the probability of error increases. It is therefore advisable to tabulate the most laborious part of the calculation and solution of integral equation used for the synthesis of continuously acting systems and to solve systems of linear, algebraic equations for the synthesis of impulse systems. Bearing in mind the known formulation of the problem of Zadeh and Ragazzini for continuously-acting systems, or that of V. P. Perov and K. 10hnson for impulse systems, we should apparently limit ourselves to examination of useful 'regular' signals in the form of polynomials of not higher than second or third power and steadystate random interference with correlation functions of the form R(T) =
(J2
e-o;(T) {cos {Jr
+ X ~ sin fJH}
Allowing for all possible combinations of parameters of the correlation function and memory time of the system, this will cover a considerable number of problems encountered in engineering practice. The author of this communication has carried out considerable preparatory work for setting up the necessary tables. It is proposed to make the calculations on universal digital computers.
References COALES, 1. F. and NOToN, A . R. M. An On-Off Servo Mechanism with Predicted Change-over. Proc. I.E.E. Pt. B (1956) 449-460 2 BURT, E. G . C. Self-optimising Servo-systems with Random Inputs. Cambridge Seminar on Non-linear Control Problems. 1954 3 BURT, E. G. C. Self-optimising Systems. 1956. Heidelberg, Fachtagung Regelungstechnik
1
1. F. COALES, in reply. Dr. Blandhol's point that statistical methods have not yet been much applied to process control problems because
760
770