Parameter sweep in iterative analog computer techniques

Emir Humo : Parameters in analog techniques

IN

ITERATIVE

PARAMETER SWEEP ANALOG COMPUTER

77

TECHNIQUES *

by Emir HUMO ** SUMMARY. - - In this paper, parameter sweep problem in iterative analog computing techniques is studied. Some new method, completely automatic, for parameter sweep study is developed. The method possesses a great flexibility and gives good results, regardless of the mathematical model of the system (linear or non-linear). As an example showing all advantages of this methods, a frequency response curve of a non-linear system is obtained. The system under study is driven by a constantamplitude varying cos wave. System output steady-conditions are verified, amplitude is detected and, through track and store units, recorded on the Y axis of a VARIPLOTTER of which the X axis is swept at the same rate that the driving function.

INTRODUCTION Automatic iterative computation is used today when solving different kinds of scientific problems as : optimisation, boundary value problems, curve fitting, serial solution of partial differential equations, simulation of delay-time, parameter sweep, etc. An iterative procedure makes it possible to automate a lot of operations which would have requested a long and tedious work by means of the classical analog techniques, The purpose of this report is to show how a standard analog computer can be used to solve a problem involving iterative procedures, but at the same time, some new method, completely automatic, has been developed which has proved to be very useful in a parameter sweep study. This method is most flexible and gtves good results, regardless of the mathematmal model of the system (linear or non-linear), P A R A M E T E R SWEEP In many cases, we want to know how one of the components of a particular solution is changing (am. plitude, phase, e r r o r ) f o r a corresponding modification of certain conditions in the problem as: an input, a parameter value, or an initial condition. For example, we might be interested in the steady-state position error of a servo as a function of the controller gain, or in the steady-state value of the output amplitude of a linear or non-linear system (filter) as a function of its frequency and/or its amplitude input, W h e n this problem is being solved in a standard analog way, we unconsciously use an iteratlve procedure, i.e. the operator chooses in advance several values of * Manuscript received May 15, 1964. ** Inst. for Nuclear Science (~B. Kidric >> - Beograd.

parameter at the beginning of every OPERATE cycle, adjusting manually the pots which correspond to these parameter values, and reads the results after each run. In this way, we collect point information to plot the requested diagram in steps. The values between these points are normally approximated by linear interpolation. W e can also underline that there is an essential difference between most of the typical iterative applications where the work is done in <>with a magnitude and direction of change in the parameter which depend on the results of the previous runs, and the parameter sweep problem where the procedure goes on in <( open loop >>, i.e. the changes in parameter values are determined in advance. D E S C R I P T I O N OF P R O B L E M A N D

METHOD

In order to generalize, the proposed method will handle a more complicated case where we have a hatmonic function (sin or cos) as an input whose frequency is changed as a parameter, while the basic system is non-linear. W e want to study the absolute value of the amplitude solution of the system, as a function of the input frequency. This is of great importance when registrating the resonance characteristic of ~t filter which is described by its linear or non-linear mathematical model. In fact, for the development of the method, we have to solve four problems : I) Automatical change of the frequency ~ in steps of discrete size, according to a program determined in advance. 2) Verification of steady-state or any other demanded condition. 3) Amplitude detection of the solution in its final state. 4) Storage of the detected amplitude. Those conditions determine the flow diagram of the problem (fig. 1).

78

Annales de l'Association ]nternationale pour le calcul analogique

N o 2 - - A v r i l 1965

in the OPERATE mode during the RESET cycle, and integrate with a constant rate the value --dx ~/Tx~ for T~ second. The output of the integrator (11) will be changed by an amount A ~0 at the end of the RESET cyde. An initial condition for the variable ~o is set normally on the integrator ( 1 1 ) a t the beginning of the process, but as the iteration procedure starts with a RESET cycle, rite initial value ~1 set before the first iteration must be lowered by an amount Zx~o to reach the desired value o~ at the beginning of the parameter sweep investigation.

to arm , ato , or X-y 6 . . I f C " " ' - v - - q [U 7pto, 7 7 ter "~~.-" a ~ o ~

r~ "w Ir'~"~.l 1~:"..... I J ~ ~ I ~ I--'N..r V'-dr..s to pen to ~nteyt-~'"~ ...... ~ " " ~ " "-- ptbtter°fZ-Y LIs qOeiect~on ] [J"/'- ['1"/Fig. 1. l~irst of all, it is dear that an iterative procedure is most suitable to satisfy all the required conditions, In automatic iterative computation, the notion of parameter has a new significance. Here, the parameter is a number which is constant during any single OPERATE cycle, but which can be changed between two successive OPERATE cycles. In classic analog computation, a parameter is represented by the position of a potentiometer and upon demand we change its value manually between the different runs. But, in iterative procedure, the parameter value is changed by the machine itself; this means that the parameter can be regarded as a computer variable, therefore, as an amplifier output voltage.

Every change of the input frequency initiates a transient on the system output. The characteristics of this one depend on the formulation and the parameters of the system. Most generally, we are interested in the graph of the absolute value of the amplitude in steadystate condition versus input frequency. Consequently, during every iteration, the system amplitude has to reach the steady-state conditions. This can always be achieved with a suitable choice of the OPERATE cycle duration. A system with a strong damping will have a short To, but :~ system with a long transient time will require a sufficient long To for the verification of the steady-state conditions.

This conclusion immediately shows a way to realize the change of frequency. The simplest way to change the value of frequency in a suitable manner is by means of an integrator (fig. 1). During every OPERATE cycle, the value of frequency is a constant, so the integrator 11 must be in the HOLD mode, simultaneously, the generator of harmonic frequency and rite analog model of the system must be in the OPERATE mode (see table I). This table shows the change of the integrator modes during all the iteratlve computation.

When rite time dependent steady-state conditions ate achieved, a circuit detecting the maximum value of the amplitude is switched on (comparator 34 - fig. 5). This circuit can be switched on at any instant during the OPERATE cycle. This will depend on the information to detect; for example : the max. value of the amphtude is steady-state condition or perhaps the max. value of the amplitude during the transient.

If we want to change the frequency .o by a discrete step &~ in every iteration, the integrator (11) must be

A storage device, made of a track and store unit, is used to feed the pen of an X-Y plotter while its arm follows the o signal. Therefore, the whole graph will remain shifted backward by one IC-OP cycle.

TABLE I Mode of the machine

RESET

OPERATE

HOLD

1-fit iteration t=

i-th ~teratton

n-fit iteration

0

A02, A05, All, A22, A23, A46

RESET

OPERATE

RESET

OPERATE

RESET

OPERATE

A14, A45

A06

A14, A45

A06

A14, A45

A06

A02, A03 A22, A23 A46

A02, A03 A22, A23 A46

A02, A03 A22, A23 A46

A02, A03 A22, A23 A46

A02, A03 A22, A23 A46

A02, A03 A22, A23 A46

All, A06

A14, A45

All, A06

A14, A45

A l l , A06

A14, A45

All

All

All

Emir Humo : Pm'ameters m analog techniques In this way, we obtain a discrete approximation of the curve [A=(~0)l~t.~. This curve is discontinuous, However, when taking a sufficiently small frequency increment A o~ at every iteration, the quality of the approximation increases to reach at the limit case:

79

If the equation (2) Is differentiated with respect to tune, and taking into account equation (1), the result is:

x + ,Oo=X+ tax3 =

lira ]A=(o)[ Btep = IAx(o~)[ oonunuo,,~ & o --+ O

G cos ~ot

where ,00, h, G, are constant and x = nF. This equation in the theory of a non-linear system is known as Duffling's equations. However, in real physical systems, dissipation is always present. In our system, the dissipation is due to the ohmic resistance of the coil and also to the current and hysteresis losses in the iron core. So, the mathematical model of the

where we get the continuous curve IAx(~o)l ~o,Un,o,~

Example: As an example, we take an electrical system consisting of the serial combination of one capacitance and an inductance which is non-linear (fig. 2a).

system will include a term x : ~xO,)l"

[l

7~ + k:c + O~o~X+ k~x8

F

Esin,at

i~

_

'~,' ~,,'

o)

of this equation is the discontinuous amplitude jump when the driving function frequency is varied. This becomes clear when analizmg the IA~(o0l characteristic of the output.

, ~,

b) Fig. 2,

If initially co is chosen much lower than o0 and then increased, the value of I&(o)l increases as shown by figure 2b When m reaches on, Iax(,o)l jumps to a

The system is driven by a constant-amplitude varying frequency sin wave. The non-linear phenomenon is associated with the saturation of an iron-core inductor, The m~tantaneous value of the current <> in the coil as a function of the total instantaneous flux F in the core is described by the equation :

J

= - -N

=

F +

aF 3

smaller value as shown by the dotted line. A further increase in ~o leads to a slow reduction in IAx(~)[. Now, if co is decreased, ]Ax(co)[ follows the lower segment of the curve at the point ~0 = t0a. Another jurn of A~(m) from the lower to the u er branch p and { remains I stable for lower frequencies. pP happens

(1)

Lo The circuit ~quation for figure 2a can be written : z dF + N -E sin o t (2)

C

= A cos ot.

The analog model of the system, with all necessary circuits, is shown in figure 3. The solution of this problem, obtained by the proposed method on the standard analog computer TR-48, is shown in figure 4. The discrete step of co is A ,o = o12.

dt

GENERATOR OF HARMONIC FREQUENCY ................................................................................ L~ . . . . . . . . . -10

t_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i ~ 10

PARAMETER

SWEEP

t

. . . . . . ANALOG

MODEL

>~,

~ 1

......... .

.

.

.

.

.

.

.

.

.

.

.

~

.

.

.

._

~l'.t/

$7 ~roso Ito R of OPERATEeol, AO21A031A221 A231 A46 '~ f toRE~ETcod ] All, Ao6PTEcoif ERA [ /"351J [[ toofRESETcuff "

"[toarm 'Iof x-y

~ .

L

OF SYSTEM

RV2or

100 -o- , ~ -

$2

117

I

2 -~

X-"7 I M I

-

'v/I

~-~I

X-'7.l

M I

kTI

I I

. . . . . . . . . . . . . . . . . . . . . . .

,,qo

t" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

-,",.-----/ otAO6 ~op.erdtdto OPERATE bus L col?of A%,A45

_."

'3

° -

'+,lO.----.J

'to RESETeoiTof

L

. . . . . . . .

[

1

L . . . . . . . . . . . . . . . . .

1

5

/

~. . . . . . . . . . . . . . .

~ .j

-*

U"d

__1~."

~

~

I!

~

I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MODE CONTROL

115 /

'

I

I

I ~

AMPLITUDE DETECTION

Fig. 3.

I I

. . . . .

l ,

!

k . . . . . . . . . . . . . . . . . . .

STORAGE

.1

x-y

totter

80

Annales de l'Assocmtion internationale pour le calcul analogique

N O 2 - - Avril 1965

lAx(~) Ima x /"

°18

qs

)

q3

,

012

\ .....

k

Fig. 4. CIRCUITS DESCRIPTION

The simple calculation shows that the lengths of the RESET and OPERATE cycles are:

lV[ode Control Circuit.

a + 1

For the purposes of iterative computation, it is necessat'}, to cycle the computer mode automatically between RESET and OPERATE. To obtain this aim, the circults which involve one integrator (46), two pots, one comparator (35) and a function switch ($1), are used (fig. 5). For this method, it is very important that the duration of RESET and OPERATE cycle can be adjusted over a wide range. The iteration is started and stopped by a manual function switch. The circuit is very simple and can be used on machines without the repetitive operation facility, to RESET coi/' of RV

ZOV

S1

Tr =

b

=

I

a.b

or, for requested values of Tr and To the pot-settings a and b : T~ T~ + To a -b = To To . To Amplitude Detection Circuits. The detection of the maximum value of amplitude in the desired state is made by means of the circuit in figure 6.

A46

.

_

HOLO

-C

.

to OPERATE c o i ? o f A46

~'.~.-~ "Ax,[-"~I"

to RESET bus

~'~ "

~R

a +

T

37,

"1

.

- -_ SU _ to OPERATE bus

-qO

+10 ¢- - -

_o

!

II

1

Fig. 6.

' IU(t)

_

T~. ~

reser-°pe~

V \

steodZ stateperiod transient state

per, od

Fig. 5.

increases, holds maximum value when the input signal decreases, and varies lower than the stored one. Let us suppose that the stored value is A~ The outz"

put Axlof the amplifier 42 is applied to the input of the amplifier 43, and the output of this amplifier will

E m i r H u m o : Parameters m a n a l o g t e c h n i q u e s

be positive as long as (A%) >

(A,2).

81

Reeommandations

The diode is

non-conducting and integrator 45 holds voltage A , .

allX

Auteurs de textes

N o w , let us suppose that Axz becomes larger than A x ,

destin6s h (~tre publi6s dans les Annales

the output voltage of amplifier 43 becomes negative and the diode starts to conduct.

de l'Assoeiation internationale

Because the diode is connected into the input 10 of the integrator 45, this one has a high gain and the output voltage of integrator 45 increases rapidly. However, negative feedback loop through amplifier 42 reduces the input voltage of the integrator to zero when Ax reaches Ax .

pour le ealeul analogique.

1

',

Let us say, also, that the circuit i n this configuration can only hold the positive m a x i m u m of the input voltage, CONCLUSION

t. Textes. Les m~moires seront publi6s dans la langue choisie par Ieurs attteurs. Les textes destines k la revue <~les Annales ~ (Ivi6molres, notes, questtons ou r~ponses) doivent &re ~crit~ sur une seule /ace de la feuille. Une m~me feuille ne peut contenir que le texte partiel ou global d'un seul article. Les textes sont remis ~t la r~daction, dactylographi~ts en double exemplatre, en laissant une marge blanche de 5 cm minimum ~ gauche de la feuille. Les titres de chapitres et de paragraphes apparaItront nette~eat ddtach~s du texte. 2. Equations.

This process is a completely automatic one. The method is not restricted to linear systems, but can also be successfully applied on non-linear systems,

Les gquations math~matiques ins,~r~es dans le texte seront clairement formul~es, de telle faqon que nul doute ne puis~e subsister en ce qui concerne les notations et symboles utilis~s.

A very important characteristic is the great flexibility consisting of the possibility to change the duration of the O P E R A T E cycle. This one depends on the parameter of the system (the system is strongly or loosely damped). The amplitude detection circuit can be switched on at any m o m e n t d u r i n g the O P E R A T E

3. Figures.

cycle, depending whether we want to detect the amplitude in the transient or the steady state conditions. For example, for plotting maximum-maximorum amplitude versus frequency diagram, we do n o t need an automatic switch between the analog model of the system and the amplitude detection circuit, Acknowledgement. I wish to thank Mr. W i l l y Claes, of Electronic Associates Inc., European Division, Brussels, for his help and many invaluable advices.

REFERENCES W.J. CUNNINGHAM: Introduction to Non-linear Analysis. McGraw-Hill Book Company - 1958

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G H A N N A U E R : Automatic Iterative Operation of the TR-48

Analog Computer. Princeton, New Jersey, U.S A. & P. RAZE: Specml c,rcuits, E.C.C. Report No. 55 - Brussels, March 1962.

J P WAHA

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