On the development of linear difference equation units

On the Development of Linear Difference Equation Units G. SCHNEIDER investigation. The transfer function of the sampler and hold can be described by:

A linear difference equation unit with constant coefficients (designated Ll unit in what follows) is to be understood as a system in which the relation between input x(t) and output yet) is characterized by the following difference equation aoy(t)

T) + . . . + a"y(t - kT) = boX(t) + b1x(t - T) + ... + bhx(t - kT)

xuU) = x[(n

+ l)Ts ~ t <

(n

+ 1 + 1])Ts 0

~ 1)

< 1

(3)

or by Figure 1. A sampler and hold is consequently characterized by the sampling period T, and by the 'phase' 1]. In general we shall use the symbol of Figure 2 as abbreviated notation for a sampler plus hold. For comparison the American

(I)

Ll units of this form are encountered in control systems most frequently as pure dead-time units (2)

T)

(n

n integer,

+ a1y(t -

yet) = x(t -

+ 1])TJ

R x o-----c:=J----1--!-O"'" <>1f---,.---;

or in the general form 1 as compensating networks in sampleddata systems. It will be shown below how Ll units can be

>---~-xa

(approx.)

R

.....

c

I I

_-00-__-..,

I I

Figure 3. Set-up of a sampler with zero-order hold

I I I

~--""":x.

notation has also been included in this figure. A sampler plus hold is approximated by the circuit of Figure 3. The relay of Figure 3 closes for

. . . --Ix

(n

Figure I. Input and output jimction of a sampler with zeroorder hold

+ 1] -

lE)Ts

<

t

<

(n

+ 1] + tE)Ts

0

< E<

1

(4)

Let E be so small that x does not vary appreciably during the interval 4. For this interval Figure 3 represents a delay unit of first order with the amplification constant 1 and the time constant T = RC. If T is small compared with the interval 4, that is RC <{ ETs, the output has practically reached the value of x at the end of this interval 4. After the relay opens this output stays constant because the capacitor cannot discharge. By connecting these sampler plus hold units in series (Figure 4) a form of dead-time unit can be built. The first sampling unit transforms the input x(t) into a stepped function x,,(t). This value is the input to a second unit which, because 1] = t. samples the stepped function x,,(t) in the middle of the steps.

Sampler + hold

== ~ --!Ho(P): l-;-PTS~ Ts;l']

Figure 2. Symbolfor a sampler with zero-order hold

developed by using the ordinary building blocks of an electronic analogue computer. The purpose of this development is twofold: first in order to use Ll units for compensating actual

n sampling units

..

----~

I

x(t)

xa(t-'T-l"s)

Figure 4. Delay line synrhesizer (n odd)

control systems and, secondly, in order to represent Ll units by analogue computers if sampled-data or dead-time systems are to be investigated. Let us start with the description of a sampler with zero-order hold, because its realization is fundamental for the following

Consequently at the output of the second unit appears the function x,,(t - tTs) displaced by the time tTs from the function x,,(t) (Figure 5). By adding another sampling unit with 1] = 0, an additional lag of tTs is produced, and so on. The output of the nth unit delivers a stepped function which is

977

987

G. SCHNEIDER

delayed by !(n - l)Ts on x,,(t). Since xa(t) is an undelayed approximation to the input function, the output of the last unit is an approximation to the input function delayed by the time Td = !(n - I)Ts. By decreasing the time Ts the accuracy of x(t) xa(t}

~

~~

TS

xa (t-2)

V

I

I

I

~ ........

, ,

(A) zn[(n

r"""1

' ,

+ 2)Tsl

(9)

Ya[(n

+ JI)Tsl

for - r ~ JI ~ f + 1; r, f integer; r, f ~ 0

JI,

This means that the polynomial zn(t), which is used in the interval nTs ~ t < (n + I)Ts as an approximation for y(t), is equal to yet) not only at the beginning and at the end of the interval but also at r preceding and f following sampling instants (Figure 6). (B) dVz/dt V shall be continuous in t = nTs for JI = 0, 1 ... , k. This means in turn that in composing the output z(t) of the

I

H

,I

I

+ JI)Tsl =

---! If-

--l I'--'

Ya[(n

by tapping the chain (interpolation). Consequently it is possible to define the following conditions for Zn(I):

I

,,

+ I)Tsl,

Ya[(n

2TS ', 3Ts :, 4Ts I' 5Ts ', 6Ts

I

I

(extrapolation), but also future values

Interval of approxirT]ation

10-

, r=3

r=l nTS (n+1)TS(n+f+l)TS

Figure 6. Approximation of yet) by Zn(t)

TS

2Ts

3Ts

4Ts

5Ts

polynomial generator from parts of the polynomial zn(t) they are to be added to each other in such a way that discontinuities cannot appear before the (k + I)th derivative. In order to comply with conditions (A) and (B) it is obvious that zn(t) must be set down as a polynomial of degree

6TS

Figure 5. Delaying of stepped functions

the approximation is increased. At the same time, however, the number of sampling units must be increased to realize a given dead time. It seems promising to decrease the number of sampling units by trying to reconstruct the original function from the stepped function at the output of the delay line synthesizer. In doing so it must be observed, however, that the limits of this process are described by the following theorem developed by Shannon: A function/(t), the Fourier-transform of which is F(jw), is then and only then defined by its values at the points t

=

nTs if F(jw) disappears for

Iwl

>

7T/T,

m = (r

m = degree Zn

We reduce the final determination of zn(t) to finding the compensating polynomial ~n(t) which when added to zn_l(t) yields zn(t) by using the reconstruction formula zn(t) = Zn-l(t)

If in (A) n is replaced by n -

(5)

Zn_l[(n

Consequently the function to be reconstructed must contain only frequencies which are less than half the sampling frequency. This condition is only approximately met by functions which disappear for t < O. The function between two sampling instants nT, ~ t < (n + I)Ts is to be approximated by a polynomial in t

+ i')T,l = fin[(n

Writing I'

nT, ~ t

<

(n

+

I)T,

I)T,l

Ya[(n - 2)TJ. . .

JI

by

+ 1 we get (r + 1) ~ i' ~ f JI

0

for

-r

(11)

(A) can be met if

~ i' ~

p

(12)

nTs let

=

Cn

X

t 'k

IT (t' n= -r

-

flTsl

(13)

in which condition (B) is also met through the factor 1'k. By appropriate choice of the constants C n condition (A) can be satisfied for )' = P + 1

(7)

can use for defining an appropriate polynomial Zn(l) not only past values y,,[(1I -

I -

I and

(10)

T

If we call the output of the delay line synthesizer Ya(t) and the function reconstructed from this output y(t) accordingly, we

Ya(nT s),

=

+ ~n(t)

+ v)Tsl for JI = f + 1 condition

+ v)T,l =

fin(l)

A device which smooths the stepped function in this way is called a polynomial generator. The output of this generator z(t) is not a polynomial but is composed of parts of different polynomials at

y,,[(n

This means that up to

(6)

z(t) = zn(t)

+ f + 1) + k

where

(8)

978

988

Cn

}',,[(n ='

+ I' + I)TJ - Zn_l[(n + P + I)Tsl v + I)"(r + I' + 1)!Ts'''

(14)

Using equations 13 and 14 the compensating polynomial ~,,(t) can be found and hence equations 7 and 10 determine the amplitude of the output z(t) of the polynomial generator. Next we shall show how such a polynomial generator can be constructed. First a linear transfer unit must be found such that it can deliver the compensating polynomial ~,,(t) if the

ON THE DEVELOPMENT Of LINEAR DIffERENCE EQUATION UNITS

which we designate by Ill, r, I', k. In order to do this we start from the well-known fact that the process of sampling derives from a function(I) [Fourier transform = F(jw)] a function

input is chosen as the Dirac function C"O(I - nT,}. According to equation 13 the transfer function C(p) of this unit must apparently be r

C(p)

= if {I k

n

CL

( 15)

(I - ,liT)}

.

If we take an infinite series of Dirac functions

~ (CnnO(1 -

=

(*(1)

1' = - r

(21 )

nTJ

I/ = U '

the Fourier transform of which is

00

') C,/)(I - nTJ

n-;;;; U

(16)

'

F*(j(l)

.

the resulting output according to equations 7 and 10 is given by ~1I(t)

=

,. ~

~

F[j«(I)

+ I'OJJ],

(22)

'- 'l.

=

(17)

Z(I)

-3

T,

which holds as long asfU) has no discontinuity. If for 0) > ~ws F(jw) 0, the effect of the Shannon theorem 5 becomes apparent according to equation 22 and Figure 8, namely that only in this case can the original function(/) be obtained from f*(I) by suppressing the added sidebands with an ideal low

00

L

I

= -

11 = 0

which is precisely the desired function if the additional condition that z(/) = 0 for I < 0 holds. In order to produce the

l(t ): ~ cn°(t-nTs )

Ya [t+('F+,nJ

+

(f+1)k(r+T +')J Tsm

G(p)

Tsi"l: 0 z(nTs-O)

at l=nTs

z'(nTS-O) at t = nTs

(f+1) 'Ts

z(m)(nTs-O)

(f+1) m TS m

.

at t=nTs

rn! Figure 7. Principle of selling up a polynomial generalor

pass filter. The hold circuit of zeroth order which is used most commonly for this purpose has according to

input function 16, we have to form a function e(/)- according to Figure 7-which has the value c" at the time I = nT" and we have to submit this function to a sampling process. The values of e(l) between instants of sampling are of no importance. According to equation 14 we get such a function e(l) (with the exception of a constant factor) by extracting ,11,,[1 + U + I )TJ from the delay line synthesizer and subtracting from this a function which has the value zlI _I[(n + I' + I)TJ for 1= nT,. In doing so we find that from the Taylor series for Z,, _I(I) at I = nT", there results Zn _l[(n

+ F+

I)TJ

II~

1'·I(nT.)

Z

= 2.

v= o

11 - 1 ,, J.

'

U

+

I)"T,"

Ho(jw)

T,

1

Z" _II"I(nT, -

0)

=

(18) I

zl"I(nT, -

0)

+ F + I )TJ =

n~

~

v= u

U

+

I)'T}' ,, ' zl"I(nT, -

~ ~

-Ws/2

I

w s /2 Ws

(a)

TsIF*(jw)1

L..,:$---"-IL..~----':~W~

-"+"'--+--'1...1

- Ws -ws/2

(b)

F(fllre 8. (a) Spectrum of input (unction o{ sampler; (b) Ipectrum of OUlplll jilllclion of sampler

(19)

and in place of equation 18 we get z,, _I[(n

(23)

jOlT,

Ideal low pass filter~ __

IF(jwli

Since both polynomial Z" _I(I) and its derivatives are continuous at I = nT" it can be said according to equation 7

=

e - j",T ,

a frequency response which approximates only very roughly to that of an ideal low pass filter. This applies even more for a

-Ws

Z,,_IIVI(nTJ

-

0) (20)

I.

Thus the principle of setting up the polynomial generator is given by Figure 7. An example, which is to be treated later, will discuss for instance how the values z(I'I(nT, - 0) can be derived at the time I = IITs from the transfer unit C(p) without differentiation . Now we want to try to compare the different approximations,

hold circuit of IIIth order (Ill = I, 2, 3 . .. ) in which the values of the function between instants of sampling are extrapolated from a polynomial of IIIth order based on the last 111 + I sampled values (Figure 9). For comparison with this, the frequency response curves of our methods of approximation will now be derived. In order to do this we find in Figure 10 an equivalent circuit diagram for the polynomial generator of Figure 7, in the notation of which the desired frequency response is given by [Half'h(p)

979

989

= Z(p) / Y*(pl]p =jw

(24)

G. SCHNEIDER

It can easily be shown that H

for which not all poles of H mrrk(p) lie in the left half of the

_( ) _ u",k

p -

G(p) G z (e 1',P)

p plane] have been omitted. It can be seen that for m :s;; 5

(25)

there is no stable method with k ) . /~ - y{ -t -•• ~rlI

where Gz
~

2 (which means that the

_ ()

Hm r rk

p

I .

z{t)

•

TSi"1=O

IHI

Lt!.

Ts

Figure 10. Equivalent circuit for polynomial generator of Figure 7

second derivative of z must be continuous). The first stable method is given by mrl'k = 6 0 3 2, which case has been added to the table. Several frequency response curves which a re derived from Table 1 a nd equation 26 are presented in Figure 11. Figure 1I (a) is related to all the methods which are characterized by I' = k = O. This means, according to Figure 6, that except for the value of y at the end of the interval of approximation 110 further future values a re necessary. As can be seen, the approximation to the ideal low-pass filter is not improved as the order m of the approximating polynomial is increased . The most advantageous cases are probably given by m = 2 or 3. This is true also if except for the value of y at the beginning of the interval of approximation ol1/y future values are used (r = k = 0), because by interchanging of rand ,; the frequency response curve of IHI stays the same while / H changes sign only. This can also be seen from Figure 11 (b) in which the characteristics of all methods with m = 5, k = 0 have been compiled. Apparently from these the one (5 220) with r = I' is outstanding. This method consequently uses the same number of past and future values. Particularly it has the feature that its phase frequency response curve disappears for all values of w. For the cases (3 I I 0) and (I 000) this can be said simi larly. Finally Figure 11 Cc) refers to the methods

+90 0

W

ws /2

Ws

Figure 9. Frequency response of (extrapolating) holding rircuits

z transformation. In this way we find that the general representation of Hmrrk takes the following form

(I -

H",rrJ.:Cp) = e - PT ,)1n+ l Ts pTs a",(pT s)'" + am_1(pTs)'n-l + . . . + a o X -:-b",-:(e---'P=-lffl''-':-)-="':-+--:-b"'- :::"l(:-e-::p"""ii']''-:,)-="::'-""1-+-.-.-.-+---'-bc-o

(26)

Table I shows all values of a. and b. for all approximations which use polynomials of not more than fifth order (m :s;; 5). All methods which lead to unstable polynomial generators [i.e.

Table I bo = 0)

(a 5 = a 6 =

f

111

k

----------------~-------------------------------~------------------------------- ------

0

o

0

I

o

0

I

o

3 3

2

0

I

I

6

3

0 0

2

0 0 0

3

-3

I

I

3

-I

3 2

0

I

2

0 0

3

0 0 0 0

12 12 12 12 12

18 6 -6 -18 -9

4

0 0 0 0 0

120 60

I I

60 60 60 60 60 60 60

2

180

2 2

3

4 4 4 4 4

I

2

5 5 5 5 5 5 5

0

0 I 2 3 4 2 3

6

0

3

3

2 I

0 I

o

I

2 2

I -I

3

3

2 2

3

I

6

-I

3

I

2 12

11

3

-I -I

-I I

11 2

-3

105 15 -15 15 105

50 - 5

12 -3 2

-3

-72

-3 33

5 -50 2 -6

-180

66

- 9

-60 - 120 -24

I

12 12 12 9

3

60 60 60 60 60

12 36

980

990

24 48

12

96

78

6

ON THE DEVELOPMENT OF LINEAR DIFFERENCE EQUATION UNITS

I.!i

l!i!. TS IHI I Ts for

IH,OOO!!! 0

IWI > Ws 1000 2100 3200 4300 5400

~ ~

" ~

+ SO·

0-05 ()o05 0-07 0-10

c.>s/2

_'____w

" O-!!:

2100

(a) -SO·

3200 4300 5400

w CJ.> s /2

Ws

IHI

l!:!.

1;

5040

IHllTs lor IH s220 ;: 0

Iwl >ws 5400 5310 5220 5130 5040

5130

+SO·

" 0-15 " 0-04 ~ ()O03 ~ 0-04 " 0-15

W

( b) 5310

-SO·

5400 W

ws/2

Ws

~

IHI

Is IH S031 I :: IH 4021 1 5121

+SOo

IHISOl2 • IH sl21

ws/2

t

IH,0321 :: I HmI I

5031

IHII Ts for

w

Iwl> Ws 3011 4021 5031 5121 6032

(C)

~

4021

0-021

'\

~ 0-014 ~

0-014 ~ 0-014 ~ 0-006

-SOo

CJ.>

e»s/2

Ws

Figure 11_ Frequency response-of polynomial generators

981

991

\

,

30"

"-

.... 5121

G. SCHNEIDER

which guarantee the continuity of the first and second derivatives of Z(t) (k = 1 or 2 respectively). These methods in their effect come rather close to the ideal low pass filter. Particularly the almost complete suppression of side bands should be noticed. Of these methods the case 3 0 1 1 will be taken as an example and the respective polynomial generator investigated more closely. From equation 15 follows for mrfk = 3 0 I I G(p)

= !l'{t 2(t

-

T s}}

= .; - 2~s = P

P

T } ( T6 ):1 pp ,

( T2

different, since z"(t) is made up of two components, of which ZI" is continuous at I = nTs' but Z2" has a discontinuity at this point. In order to get z"(nT, - 0) at t = nTs it is necessary to tap point P in Figure 12. For the same reason an additional tapping at P is necessary for forming z"'(nTs - 0). By considering that the part of Figure 13 shown dotted cancels itself in effect (which is true not only in this special case) one gets the final form of the polynomial generator 3 0 1 I. This generator can be transformed immediately into analogue computer circuitry. In order to show the capabilities of this method Figure 14 represents the output z(t) of the polynomial generator which is caused by the special input function

)2)

P s

(27)

Insertion of this transfer function into Figure 7 makes it e(l)

, / ' e 1lt( ll

l~ I

y(/)

=

1-

Cri.(l - I o)

cos [wCt - (0) - 4>1

for

I ~ 10 (29)

a(l)

a(I) ..

Ts i TJ:O

a(nTS-O)at 1 =nTs

Figure 12. Set-up of a sampler with summing hold

\

....

~

Ts z'(t)

z (t) Figure 13. Polynomial generator in Ihe case mrPk = 30 1 1

necessary to build a sampler with a summing hold (series connection of a sampler and an integrator). Since the transfer function of this unit for 'rJ = U can be described by aCt)

=

a[(n -

I)Tsl

+ e(nTs)

for

nTs:-S;; t

<

(n

+

I)T. (28)

it is obvious that it is equivalent to the positive feedback of two samplers with normal hold circuits (Figure 12). From a circuit of this form a(nTs - 0) can also be taken at the time I = nTs, that is the value which the output aCt) had in the past interval (n - I)Ts:-S;; I < nTs. With this we find from Figures 7 and 12 the polynomial generator 30 1 1 which is shown in Figure 13. It is not difficult to form z(nTs - 0) = z(nTs) and z'(nTs - 0) = z'(nTs) since z(t) and z'(t) are continuous at t = nTs. With z"(nT, - 0) and z"'(nT, - 0) things are

In this representation the cases

wo/(I).

=

1/10,

1/4,

1/2·5

(30)

have been considered. It can be seen that z differs markedly from y only in the last case. It should be observed that already the cases w o/(/). ~ 1/2 cannot be approximated satisfactorily by any polynomial generator, no matter how complicated. Polynomial generators have their useful application in all cases when it is necessary to connect discrete values by a continuous curve, for instance: (1) for mapping the results of a digital computer (2) for changing sampled-data systems into continuous systems (if a certain dead time is permissible) (3) for representing dead-time units on an analogue computer.

982

992

ON THE DEVELOPMENT OF LINEAR DIffERENCE EQUATION UNITS I Wo ,Wo

o

I 1 I

=10

T,

3•

4,

5T,

6T,

7,

8 ,

Figure 14. Approximation by the method mrfk = 301 1

Now we shall turn our attention to the development of general ~ units, the transfer function of which is described by equation 1. By putting ao = 1, which does not restrict the generalization, one finds by using the Laplace transformation Y( ) =

p

j, units is reduced to the treatment of dead-time units shown earlier. We are primarily interested in using ~ units for compensating sampled-data systems when T = Ts is the sampling period. If the compensating unit is included in the system as usual in the way shown in Figure 16, its development can be attained comparatively easily. In this case the input function of the compensating unit is a stepped function and this is also true for all input functions of the dead-time units of Figure 15. Consequently by observing T = Ts and Figure 5 the dead-time units can be built by the series connection of two samplers with zero-order hold. In conclusion let us summarize the most important possibilities of applying general ~ units of this easily realized form: (a) Compensating units for sampled-data systems instead of the much more cumbersome compensation based on digital techniques. (b) Compensating units for originally continuous systems, after these have been changed into 'artificial' sampled-data systems by adding a sampler with hold. In this way better results can often be attained than with continuous compensation, particularly with systems with dead time. (c) Investigation of sampled-data systems by analogue computers, particularly for finding useful compensating units.

Bibliography JANSSEN, G. Discontinuous low-frequency delay line with continuously variable delay. Nature, Lond. 169 (1952) PHILBRICK, G. Bucket-Brigade Time Delay: Palimpsest on the Electronic Analog Art. 1955. Philbrick Researches, Inc. RAGAZZINl, J.R. and FRANKLIN, G.F. Sampled Data Control Systems. 1958. New York; McGraw-HiII CHESTNUT, H ., DABuL, A. and LEIBY, D. Analog computer study of sampled data systems. Trans. Amer. Inst. elect. Engrs (1959) (Pt II)

+ . . . + b k e - kTp X( )= D(e-Tp)X( ) b +b 0 le 1 + ale-pT + ... + ake kTp p P -pT

(31) Figure 15 is the representation of a circuit consisting of deadtime units which has the transfer function D(e- Tp ) described by equation 31. Consequently the development of general

F(!{ure 15. Representation of D(e- Tp )

Comp~nsat ion

unit

Basic unit

dl)

dt)

+

Figure 16. Compensation of a sampled-data system

983

993

G. SCHNEIDER

Summary This paper is concerned with the development of basic units which can be described by linear difference equations with constant coefficients. Starting from the simulation of dead-time units with a delay line synthesizer the problem is considered of how a function can be reconstructed from a series of its values which are equally spaced in time. It is shown that this can be done satisfactorily by polynomial approximation if 'future' values of the function are available. In the case of simulating dead-time units these values can be attained by tapping the delay line synthesizer. Various reconstruction methods

are discussed and the respective polynomial generators are described. These generators can also be used for mapping the output of digital computers or to change sampled-data systems into continuous systems if an additional dead time is permissible. Finally a comparatively simple method for building general difference equation units is shown, which can be used for compensating sampled-data systems. In this way it is also possible to represent systems of this kind fully on an analogue computer.

Sommaire Le present rapport traite la constitution de cellules de transfert qu'on peut decrire au moyen d'equations Iineaires aux differences it coefficients constants. Commen~ant par la simulation des elements it temps mort au moyen d'une chaine d'echantillonnage, on se heurte au probleme de la reconstruction de la variation d'une fonction it partir d'une suite equidistante de ses valeurs de fonction . On montre que cela peut se faire de fa~on tres satisfaisante par une approximation polynomiale, pour autant qu'on dispose des valeurs ' futures' de la fonction, que I'on peut obtenir dans le cas de la simulation des elements it temps mort par prise sur la chaine d'echantillonnage. On discute un certain nombre de procedes de

reconstruction de ce type, et on indique les polynomes generateurs correspondants. Ces derniers peuvent entre autres, trouver des applications pour I'inscription des resultats des calculateurs numeriques, ou pour la transformation des systemes d'echantillonnage en systemes continus, si on tient compte en plus d'un certain temps mort. Finalement, on indique une possibilite relativement simple pour la constitution des termes generaux de l'equation aux differences, qui trouvent des applications comme termes correct ifs des systemes d'echantillonnage. De cette fa~on, on reussit aussi it simuler de tels systemes dans toute leur etendue sur un calculateur analogique.

Zusammenfassung Der vorliegende Beitrag behandelt den Aufbau von Obertragungsgliedern, die durch lineare Differenzengleichungen mit konstanten Koeffizienten beschrieben werden konnen. Beginnend mit der Nachbildung von Totzeitgliedern mittels einer Abtaster-Kette stellt sich das Problem, wie der Verlauf einer Funktion aus einer aquidistanten Folge ihrer Funktionswerte rekonstruiert werden kann . Es wird gezeigt, daB dies in sehr befriedigender Weise durch Polynom-Approximation moglich ist, sofern 'zukiinftige' Werte der Funktion zur VerfUgung stehen, die man sich im Falle der Nachbildung von Totzeitgliedern durch Anzapfung der Abtaster-Kette verschaffen kann. Eine Anzahl von Rekonstruktionsverfahren

dieser Art wird diskutiert und die zugehorigen Polynom-Generatoren angegeben. Letztere konnen u.a. auch Verwendung finden zur Aufzeichnung der Ergebnisse von Digitalrechnern bzw. zur Verwandlung von Abtast-Systemen in kontinuierliche Systeme, sofern man zusatzlich eine gewisse Totzeit in Kauf nimmt. Schlief31ich wird eine relativ einfache Moglichkeit zum Aufbau allgemeiner Differenzengleichungsglieder gezeigt, wie sie als Korrekturglieder in Abtast-Systemen Verwendung finden. Auf diese Weise gelingt es auch, solche Systeme in vollem Umfang auf einem Analogrechner nachzubilden.

984

994