is II

RECOHP II USERS' PROGRAM NO. 1102

PROGRAM TITLE: 125-R FIRST ORDER SYSTE1~i (FLOATING POINT) PROGRAM CLASSIFICATION: Subrou tine

AUTHOR:

PURPOSE:

DATE:

w.

Wellman

Servomechanisms, Inc.

Research Division Goleta, California To solve the equation,

c - r ⁼

'T

(dr/dt)

for r, where c is given as a function of time.

11 M£W 1961

Published by

RECOUP Users' Library at

AUTONETICS INDUSTRIAL PRODUCTS

A DIVISIOn OF NORTH AMERICAN AVIATION, INC.

3400

E. 10th Street, Long Beach

5,

Calif.

DI!')ClAIMER

Although It is assumed that all the precautions have been

!aken to check out. t~is program thoroughly, no responsibility

IS taken .by the originator of this program for any erroneous

~esul!s, misconceptions, or mi:;rcpre:;entations that may appear

In thiS I?rogram. .Furthermore, no responsibility is taken by Autonetlc~ Industrial Products for the correct reoroductions uf this program. No warranty, express or irTiplied, i:; cxtend;;'d by the use or application of the program.

1ZSR Pa e 1

RESE~QCH D IV I S ION • SER\(';r.~ECH.A.r.; IS~,·S. INC ..

---=-::;.a_=_..::...,

PURPOSE:

FIRST ORDER SYSTEM (floating point)

To solve the equation, c - r = i (dr /dt)

for r, where c is given as a function of time.

METHOD:

The following recursion relation is used:

r

=

^{A ( c - c 1)} + B ( r 1-c 1)

+

c

n n n- n- n- n

where: ^A= .2...(I_€-~t/·r) ^B

=

€-~t/T

L1 t . ,

giving the. value of r at the n'th instant in terms of c at the n'th instant and.

the previous rand c at the (n-l)'th instant. Since all storages are upgraded internally, it is only necessary to enter the routine with successive CiS,

exiting with the successive r's, if the interval between n an.d 'n-l is unchanged.

The routine assumes that c varies linearly with time between points, so that it is only necessary to choose the points 0'1 c so that c is sufficiently well represented as a series of straight lines.

RESTRICTIONS:

T:>O

CALLING SEQUENCE:

TRA (location of this routine); exit is to next half-word. On entry, A and R should contain c ; on exit, A and R will contain r .

n n

Pa e 2

RESEARCH DIVISIOf\ • SERVOMECHANISMS, INC.---~~t.;;;..-=--

. PARAMETER STORAGE:

The following locations must be filled a8 indicat~d with the correct initial conditions before this subroutine is used. The rand c values are upgraded automatically; time i8 assumed to increase by an amount ~t

each time through the routine.

Lo + 30 r Lo

+

32 c

Lo

+

34 At Lo + 36 ^T

In addition, the following are calculated and made available in the location.

shown, but need not be filled with initial conditions.

Lo + 42 c-r Lo

+ _

44 dr/dt

New r 1 or c 1 values can be stored in the-indicated locations at any time,

n- n-

resulting in a discontinuitYi thus discontinuities in either. r or c can be solved for accurately.

A t may be changed a8 desired to provide larger or smaller steps in time, to show more or le88 detail, or to suit the nature of the input function' c. T may also be changed in a step-wise fa8hion, thus yielding an approxi- mate solution to many non-linear equations.

ROUTINE REQUIRED:

Exponential (AN -044) in 1050; called for by TRA instruction in (Location of this routine

+

46)

STORAGE REQUIRED:

100(8) sectors, plus use of both high speed loops.

REMARKS:

Thelolution i. "exact" in that the recursion relationship will give the exact solution (except for round-off errors) if c varies linearly between

' . 10 0 4 A F< f v • 2 60

125R Pa e 3

RESEARCH DIVISIO\ . SER'v:CME(H,IH.IS~,45o I I \ ; C . - - - 1 I I : - - -

points and T is a constant. Non-linear and time-varying solutions can be approximated by choosing A t small enough so that ^T

is

essentially constant over each computation interval. The constants A and B in the recursion relationships 'are computed with an accuracy within O. 5 x 10 .. 10

• TIME:

0.8 seconds to 80lve for r and evaluate A and B.

n

0.2 seconds to solve for r (no change in ~ t or , ).

n CHECKOUT:

This routine generated correct solutions to:

(1) simple step response: r

=

_8(S+ ¹ ₁₎

=

^{1- €}

-tiT

1

=

(2) non-linear sY8tem step response: r =~~---

a(1+ s+ ta)

FILE UNITERMS:

t

l+t

First Order System, Differential. Equation. Time Constant

~---~--~---~ ~, l{j 0 4'" fl.f'" 2 6 0

.

_: ___ .a_-....-..-:.._

- -

SM/I, Research Division, Digital Computing Facility Page

--L

^of..L

Program No .. 12.5

R

^Title _F,~ iZ:

Q

R. Q~&.

S")!iZ::f:,..

Programmed by: Date

-

~

-Il-' (

Loc'n. Cm'd -Addr. Contents Accumulator b Remarks

·0.0

fF·S·T ; "'."1.0 .

C·T·V ^.;2.·0.0

· I fS'h'~ ⁱ

7 ·7-1., ·j·o

T·Ft·A

2-1:..7.·7 ..

^·0 .. -., 2.l..

.~ F·e·A _{F·~·B ·~·o}

_{. 5'=6}

_~ ^~ _I ^{... tt ~}

..

_~~ .. c

;1 .)( ^~ * _-

·3 F·D>J ^·~o·O -~ ('1- ) ( , -to -" )

~,-,

&A.

_1---

·7·tJD

~

.'\

_~ 'O·ooo.o

\

-·0·0 o.o.o.~.t) ) A·~ll

..

e··)

·S

^{fi-·O·O 000·0.0}

,

-·oc ⁽⁾

^o~-t)·1

11

-6

It·o .() O·(';)·{).(')·o 'l3 ~"SB ... ·0·0 oo~.~.o

·7

F·C·A, ·5-"6·0 1. 4:-hc.,..c If ^{K <. 2,,,.,0}

rr:R.·A --

^.]§'I)

..

_{. ',0 F·C.A :!t~ .0} ^I(

,.

F·s· ... r

:~ -:2.·0 <'0

1

· I F·s·S

l1·'·'·2.·0

^~.~

..

^{Reaur~o V"}

:F·",·P .0·&1·0 Al~-tCo\ ./

. 2- 1='. S·T 7·'_·7 ·D.fJ ^{1a din •} (

F.:.C,S

'·7·'·0;(;) ..

^~.

\

3

FAP

7·'·b·"',0

^{~1!:2 ~D -"-0 J}

.f.nj.p

. t)·().o 8 ^{BloCf) -ro) I}

r

^.

~

Un ^{7·'·7·0·0}

^{Al~·_~_'\-Bt rft-(~}

F·~·T ·~·~·o £ • III: - ,.. I

·S iF·!)·" 7·~~ ^~ ;! ⁾

F·S·T . . ~:~.c~

.f., F-e·A ~2·0 It!

,

"

F·5·6 ^·..,·a·()

^{A: - ...}

^....

^J

---

.",

^{'F·S·7 ·3·0·{)}

"'0

~;

if·R.·A

...Q...e - - -. ~1- IT t- ^~

..

I,

^{.2.,.{) .... l).O} O.()·O·o.o ~ _ A-t/y

-0·0 000-0·0

,

• f .i-.Q.Q iOO·Q·()·O

1

-.O·D O·O·O·O()

1/ --

a.

^,A·PI)

:7·2..7·,·0

:C·T·L. ·3·0·0 L,

·3 :ST·(j ·lry·D e..J<.1f

F·e·S

,j·7·b·¥·O_

^{- 6. of.}

~

,F·D·V 7·j·,·hD

1'" - A.+: / '1'

F·s:r _'·7-'·lf

₀

.s

~·'5·B

'·7·7·0·0

^{olJ l A~/,..)}

f-r.i'·E ^.7-'.0 .... ""'.,

~k~

..

.tc.

··k

P,C·A

7·7·'·'t·{)

^{' J '}

. . . . . ..

.

'r-R·A

^.~.,()

·7 F·S.T

·3·0.0 !"\ b v """ tfI_

,,·«·A

^{0.0.0.0·1 )}

~

-

,

SM/I, Research Division, Digital Computing Facility Page

--.3:.

^of

^"3,

Program No .. I 2. S -(~ T i tl e

Programmed by: Date

_. S-tl .. "

Loc'n Cm'd .Addr. Contents Accumulat'or b

-

Remarks

·3·0 i\ .'

^I'~

.

\

.-1')

^\

•• ^J

^{' J}

. .

^'

^. If

^{/T \\ , _1 I T I A ....}

·2 ^"

r\

_~ ^r:.,...

.

~ .::. t ) I.l r::. I

T-, ""

^{j "}

, \

·3 _if

^I

· ^\

\ -

, _{·4 1\}

_A~

·

·5 - \-

^J

--

If

J

,

. _.b

_i\

.,-. _/

~ ,f

. ., _J - ^-;

j 17

~'

. ·1J.f) ,:+.'1.0

00·0.0.0

i\

^E -A<f:I-r

·

I :

^-.0.0

(;:J.o·o·oo

· ^. ,

^{i - -}~t.~q (::>0.0.0 .0

, ·

'I

-.O.CI 0.0.0.0.1.

1 - - . _ . _~.--

r\

·2. ·f·O·D ()O·O.o.o _1..-

c-r

·

^{-.0·0 OOOO!)}

·3

r·oe

^{rJ.~~o.~ .0 }}

·

~ _{..,'_.-..} ,:-.0.0 ^o~.C,.o/)

V

·

^.If k-.O.C> 0. :~().C.O

!\ ^;'::

t'l"

,-.0.') ,-., ,,.., "... r, •.

, .

·

... .,.'.1._ ... ' •. , ... . J

.~ "'·0.0 ^{~··,:.t ..} ·~,.~.,::., I

-.0.0 D.C.C.O.O

V

6 IF·2.I.

' . ·2.;0·0

j. IT-RA- l'O' s:~.o r\ ^l:" ~

.11 ...

·1 ^p.2»£,

'7.

7·'·~

(.)

^~ • AlJc_Jc.

~;rR

·4b·' ..

"

.-~C C·T·l ^{. bOO} IT-IR.·A 7·7·'·0·0

·

^{.J +-·00 .00.0·0.0 '3(2J,g}

-.0..0 00.0·3.0 .'2,

+.,

·2

'i.2..,.o.,

_'/~J

-q·o

'1-2·'·0·1

·3 ^1+·4'()

^{0·0.0 ·0·0 1/7. !}

-·o·t>

^0.0.0.00

·

^{,It +·0·0} IO·S:S:Af·C

II

fa' -·l..·b 6·0·4.·7·0

· ^S

^i+.t}

^{:z. IS:}

l·S"·~·1 IJ&f ,

·

^{-.~ J..} ~.2·S.l..'

. · '" ^+./.1

17·'"}·'"j·7·1 .,~

1\

_f:l"'~ ^{" ,}

. ,+-7·, ,·7·7·7·1 ^\

~7 +·o.() 0·0.0.0.0 ^~,..~

J

-.D·C o·!')·C;·c>·t:> I

I~MI

I, Research Division, Digital Computing Facility P a g e i o f l Program No ..

'25"- S

Title

Programmed by: Date

S-'-ll -,

I

-

Loc'n Cm'd .Addr. Contents Accumulator b Remarks

.,~ F·~·1" ·~·O·D ^~ -w-

e8

^-,

Foe·A.

·:to.o

^{") - At/1-}

--

-

---_.-

. t _·X·A·R ID,O·() ·()O

--

~.----.---

·A·L.~S 0.0.0·'·0 CX-"_ ^'I .cP·~8

-- ^{.. -}

.2-

·C;·"A

17·7·"·5""·'

^,

~A'D'D ·571.0 '3Ci'~e· e-.p -+'!t

- . -

. '

.:3 rr-PL

·0·;)..·0 ,...:.... C 1(p ~ -3 -- x> 2-)

..

~ ~.

c.'·V ^·7·DtJ

^I~ 1Ift/,. '" 'I A C~eu1A.t

...

8 ..

clir-e,.11,

t .~

L4--'O·D

. ().1o.(J 6of~ G38 ^c::.)\P "''It.

rr-m.,

·()'70 ^~ e AD ulfo ~. -~:~ ^1-110 ~ ~d 8&.~_

·5 IX·A~ ^00·0·00' ,

I "

IA·R.S OOlJD·t;:

-•• ''1''

^(Fo

^{I •.. .2-1.}

^{\L > ~ ... e}

.b

_5T·O

,7·'J·7·..,·{j

vs~ ~dA.1 .!,.~~..I.

It'» PR- ^·).'10

"c.

t

+.

cAI.:ul .. "t~ 8

.7

_~C.·T·L

·s-oo

IAoO'D ·7·7·0 \( s-I

·7.{)

^t1'Jp.R,.

_7·1·7·']0

~

IA·DD

17· 7·/:'·S'·()

'I",

!

.,

mPR 7·7·7':J 0 ^I..

-

-~'."'-

--.-

" AD·D 1·7t, ·'2,.,0 "~

,

... _ •..

-

.'2,. MP·R ?'7·7,'·rJ

"

lLf'ln

7·7·'

.~() ^{112. ,}

·3

ImPR

7·7·,·,·1)

^Ie

^_XI,,!

^{+ K'/~~_+}

^K!', Fs-r ,·7·/:;·' .()

te-.* .

.I.j F.C·A 7·71-·h·O ^~

A-D-D '7·7·~·1 0

l ,

^{-I: -}11/1')( ''1ot,]

.~ ^,.'

·s

^{IC·,.L .~.> t).o}

f. F·S;,- ·(j·J..fIJ

e

·

^{.l, -:';r.V .,.0.0}

I . ·T-R·A 'J. ']·7·CIo'IJ

"

.,

^{.,...OD J;. 2,.. I · 0·1 II ::J}

-·,·0

J.j.2,..,.{)./

f ^{. · . · · .}

.

~ '"

..

. . . ^'

· ·

· .

,

.