Effect of random sampling interval on sampled data model of human operator in compensatory trackin

Bekey, G. A.

Biddle, J. M.

Jacobson, A. J.

English

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THE EFFECT OF A RANDOM SAMPLING INTERVAL ON A 
SAMPLED-DATA MODEL O F  THE HUMAN OPERATOR 
G. A. Bekey 
J. M. Biddle 
A .  J. Jacobson 
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This paper w a s  prepared for presentation at the USC-NASA Conference on 
Manual Control, held March 1 ,  2 and 3, 1967 in Los Angeles ,  California.  
https://ntrs.nasa.gov/search.jsp?R=19680017493 2020-03-12T10:48:04+00:00Z
C. A. Bekey, J, M. Bidcl le  
University o f  Southern Cciifornia,  Lo;; lhgelcs, Califorfiia 
A. J. J S ~ C O ~ S O ~ ~  
Xughec Aircraft. Cmpmy, Culver City, Cal i fomla  
Thcr.e are a nudoer of proposed mc?themst:ics.l models of the human operator i n  a 
eompenmtwy  tracklng fur.ct.5.on. 
6.elzy, htwe been r e l a t i v e l y  eaey t o  incmporate  i n  most of these moCels. 
tb.2 re!mizYlt,, or those frequencies i n  the human operator's output which ere not 
l j nea r ly  mi-relate.? tr i th t h e  input, has never been easy t o  -include i n  a satisPying, 
organic f'nsh-ion i n  a m y  of the  nodels. 
been advnnced for  D number of reasons one of which is t h a t  such aodels ~ 5 3 . 1  produec 
naturall;/ frequenqies not present i n  the  i.nput,due to  the presence of tfie saqpler. 
S o m  fcsturea of a  ham^ operator, such as time 
However, 
I n  recent  years, smplo_d.-d.ata moacls have 
Histor ical ly ,  sampled-data models of the  himan operator have assunied a per iodic  
smpfzjr. 
chz.racterist ic fea tures  have beeii wzli  described. 
mch u s w y l c r  t h m  the  w t p d t  spectra w i l l  h e w  "humFs" containing d i r e c t  an3 
revzrsed inia&;c.s of the i n p u t  spectra  centered about i n t e g r a l  values of the sampling 
frequenc3. 
vs.itic of sampling freqaency ra ther  l i k e  a deep val ley between tw.;, h i l l s .  
The behrivlor of such snuqiers  has been studiecl extensively and certajn 
For instcnce, if t h e  model conta-ins 
I n  additioz,  there  is a null point  i n  the  output sp rc t r a  et each in t eg re l  
In  AFf?DL--T2-65-15, (Reference 1) it i s  s t a t ed  that careful ly  8nal.yzed ,-ecords 
of ac tue l  liuzian operator outpats f a i l e d  t o  show the humped spectr.a charac tc r i s t ic  
of a period!c simpler.. 
due t o  t i m e  varying parmeters ,  such a n  gains, t i m e  constants and t i m e  delays 
within the human operator t ransfer  function. If these parameters may vary, how 
wou1.d the  behavior of a human operator model. be affected i f ,  asswing it contained 
In  t h a t  report, tbe authors theorized t h a t  t h e  remnant is 
a sampler, the sampling in t e rva l  were a l l m e d  t o  vary i n  some fashior,? 
is an investigatio.1 of a sampled-data hmm operator m o d e l  i n  which t he  s m p l i a g  
interval varies randomly about EORE mean value. 
Our experiment 
-_.- 
%his work w8s supporte3. Ir, part  by the National Aeronautics a d  Space Admini.stratio3 
wider Grsnt NASA-MCE? 05-018-022 
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EXPERIXDITAL PLAN 
We designed the experiment based on a simple analog computer simulation of a 
sampled-data model of the  human operator i n  a one dimensional compensatory track3.ng 
loop with a constant gain. Two types of data hold, 8 zero order and a f irst  
order were investigated, and d i f f e ren t  mounts of randomness were used for a 
parametric study. 
the sampling in t e rva l  w a s  made t o  vary i n  a Gaussian fashion about the  mean interval .  
The amount of randomness was increased i n  three s teps  t o  a maximum variat ion of 
1 0  = 17% of 8, mean .33 sec. sampling interval .  
with a first order hold, and with two successive amounts of randolamess with a 
maximum 1 d variat ion of 12.5$ of a mean . 3  sec sampling interval .  
parmeters  were held f ixed for EL given c8se. 
of f i v e  essent ia l ly  non-harmonic s i n e  waves. 
model vas a l s o  constructed a n d m  simultaneously with the sampled-data model. 
model was an approxiinate one, including a gain and l ag  terns, but  without a t i m e  delay. 
This simplification is  j u s t i f i a b l e  i n  view of the use t o  which the model WES put, as 
explained be1a.r. 
S ta r t ing  with the  zero order hold and a purely periodic sampler, 
The experiment w a s  then repeated 
A l l  other model 
The model input consisted of the sun 
A lixietir, continuous human operator 
This 
Output data was recorded and used as follows. The continuous m o d e l  m t p v t  was 
recorded and power spec t r a l  malyses  were performed on the two purely periodic rur,s. 
Althoueh useful f o r  conparison, t h i s  signal 1.~8s not of major Interest. The smpled 
model output also vas recorded and. power spec t ra l  analyses pcrformed mainly f o r  
cmip3clson purposes. 
increased was the y i r m r y  objective, anothcr s igna l  w a s  used f o r  the reasom given 
below. This s ignal  was the  difference of continuous and sampled model outputs, as 
shown 3n Figure 1, The time constant and gain -(K” and K’) of the continuous mode1 
were adjusted so tha t  i t s  frequency response matched t h a t  of the sarripled nodel as 
closeljj as possible i n  the low frequency or input region. 
when compared to the sampled model output, enhances the higher frequency remnmt 
area and allows scal ing i n  the  da ta  reduction t o  decrease the  e f f ec t  of dats. chsrlnel 
noise. Due to t h i s  fact,’ our resu1ts:are’based on spec t ra l  analysis  of t h i s  signal. 
Finall) ,  t he  sampling in t e rva l  s t a t i s t i c s  were collected t o  confirm the experimental 
d i s t r ibu t ion  used and were checked by e. ehi-squared test t o  show t h a t  they were 
closely Gaussian. 
Although the  behavior of t h i s  model as the  randmncsr; is 
This difference signal, 
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The block diagren (Figure 1) shows the  general arrangement of t he  analog 
simulatlon. 
The hold on the  sampled model, however, must be supplied with a random pulse 
t r a i n ,  and the  remaining blocks i n  the  diagram represent t he  hardware t h a t  generated 
it. 
a fixed mean. 
conveniently and independently adjust ing both t h e  m e a n  and the  var ia t ion of the 
in t e rva l  between samples. 
!@e m o d e l s  and the  input sine-wave generators are self expanatory. 
The in t e rva l  between pulses w a s  required t o  vary i n  a Gaussian fashion about 
Our method f o r  accomplishing t h i s  afforded us the  capabi l i ty  of 
A t r i ang le  wave o s c i l l a t o r  w a s  used t o  provide a stable and syrmxnetrical wave 
form with a l i n e a r  slope and equal pos i t ive  and negative voltage excursions. A t  
each pos i t ive  and negative peak of t h e  t r i ang le  wave a sample of the s ignal  from 
t h e  Gaussian noise generator w a s  taken and held. 
wave w a s  then compared with t h i s  sample of the  noise i n  a cornparator. 
of t h i s  comparator was operated on by a pulse forming network t o  give a t r a i n  of 
clean rectangular samples with a fixed "on" time and a random variat ion about a 
mean sampling in t e rva l  of one half t h e  period of t he  t r i ang le  wave. 
of the in t e rva l  var ia t ion were Gaussian and t h e i r  variance was  controlled by the  
r e l a t i v e  amplitude of the r m 6 .  noise voltage t o  the  peak-to-peak voltage of the 
t r iangle  wave. 
to-peak voltage t o  prevent any embarrassing extraneous pulses. 
l i m i t  yoltage t o  r m s .  noice voltage w a s  chosen suff ic ier i t ly  la rge  as t o  have l i t t l e  
e f f ec t  on the  in t e rva l  s t a t i s t i c s .  
The running slope of the t r iangle  
The output 
The s t a t i s t i c s  
The noise voltage w a s  l imited t o  a value no greater  than the  peak- 
The r a t i o  of the  
RESULTS AND DISCUSSION 
Figure 2 shows the  parer spectrum of the  output of the sampled-data model using 
a zero order hold. 
s ide  of the  3 cps sampling frequency. 
a t  3 cps c lear ly  displays the  expected charac te r i s t ics  of a sampled da ta  model. 
In  Figure 3 the  spectra  of the  difference s ignal  i s  sham for the  same set of 
experimental conditions. There i s  approximately a 10 db enchancement of the  
sampling frequency area over Figure 2, the  sampled m o d e l  output. 
a t  3 cps i s  evident. 
d i r e c t  and reversed images of the  input frequencies. 
quiet  region betveen the input and s a p l i n g  frequency regions. 
Note the  input frequencies and the "hwnped"spectra on e i t h e r  
This graph, with i t s  strong "valley" t r e n d  
Again t h e  n u l l  
One can eas i ly  pick out the  peaks which correspond t o  the 
Note a l so  the r e l a t ive ly  
The next th ree  f igures  (Figures 4 through 6) show t he  same output, zero order 
hold and difference signsl,  f o r  increasing amounts of sampling in t e rva l  randomness. 
Histograms of the  sampling in te rva l  ( A  T) s t a t i s t i c s  are p lo t ted  for  each case on 
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the  corresponding spectrograph. 
that the  d is t r ibu t ions  are Gaussian. 
Chi-squared tests on these substantiated the  claim 
These f igures  show the  diminution of t he  peaks 
i n  the  sampling region as the  randomness increases. 
and the  points  where spectral peaks would be i n  the  periodic case are marked on each 
f igure.  
however, i s  the"f i l1ing in"  of the  val ley a t  the  sempling frequency and also of the . 
normally quie t  region between the  input frequencies and the  sampling area. 
A T (1 Q) = 12$ case (Figure 5) we note a 5 - 10 db decrease i n  the peaks, approzimately 
a 7 db increase i n  average power l e v e l  a t  the val ley and a considerable increase i n  
the  noise i n  the formerly quie t  region. 
ness and in$igure 6 (A T, (1 Q) = 174) it is hard t o  f ind  any resemblance t o  the 
spectrum of a sampler, even with the  general area marked. 
obtained for  the case of model containing a first order hold (Figures 7 through 9). 
In these cases the  sampling frequency is 3.3 cps and two degrees of sampling in t e rva l  
randomess, A T (1 Q) = 6.5% and 12.5$, are presented i n  terms of the difference signal.  
The average sampling frequency 
With a A T (1 Q) of only 7$ we can see a d e f i n i t e  decrease. More s igni f icant  
For the  
This t rend continues with increasing randoni- 
Similar r e s u l t s  were 
CONCLUSIONS 
O u r  experimental r e s u l t s  show that :  
(1) It mi.ght be impossible t o  determine the presence of an i n t e r n a l  sampler 
by examining the human operator 's  output spectra if a random sampler is 
assumed. 
an unreasonable amount of randomness i n  the sampling interval .  
Noteworthy is the  f a c t  t h a t  t h i s  assumption does not require 
(2) Assuming a randomly varying sampler our r e s u l t s  are compatible with both 
those presented in  Reference 1 and those obtained from earlier sampled- 
data models. 
(3) This then demonstrates the  f e a s i b i l i t y  of using sampled-data m o d e l s  
for the  human operator, provided only t h a t  the sampling interval  include 
a reasonable amount of randomness. 
(4) Finally, our r e s u l t s  suggest t h a t  attempts a t  enhancing the r zman t  
portion of t he  human operator 's  output and improvement of the  s igna l  
processing and power spec t ra l  analysis  wil l -Pe important i.n fur ther  deter-  
mination of the  va l id i ty  of sampled-data models f o r  the  human operator, 
AFXAS OF POSSIBLE EXTENSION 
We have worked with a random variat ion i n  the sampling interval .  Furthermore 
we used noise with a Gaussian probabili ty d is t r ibu t ion  t o  control the randomness. 
It is not  pa r t i cu la r ly  evident t h a t  t h i s  would be the  "best" dis t r ibu t ion  t o  use 
i f  one w e r e  spec i f ica l ly  t ry ing  t o  match the  model output spectrum t o  ac tua l  human 
operator output spectra. In f ac t ,  it does not seem unlikely t h a t  some s o r t  of 
determinis t ic  "cr i ter ion function" could be used t o  determine the sampling in t e rva l s  
(in conjunction with some degree of randomness). 
the realm of model matching and form the  b a s i s  f o r  possible f'uture work. 
A l l  these p o s s i b i l i t i e s  are In 
ACKNOWLEDGEMENTS 
W e  would l i k e  t o  grateful ly  acknowledge the  assis tance and advice of 
M r .  M. J. Merritt and M r .  J. C. Maloney both of the  USC Elec t r ica l  Ehgineering 
Department. The spec t ra l  analyses were computed on equipment belonging t o  the 
USC Computer Sciences Laboratory. 
REFEREDICE 
Mc h e r ,  D . ,  E. Krendel, D. Graham, and W. Reisener, "EZuman P i l o t  Dynamics i n  
Compensatory Systems", AFFDL-TR-65-15, A i r  Force Systems Command WADC, Ju ly  1965. 
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The effect of a random sampling interval on a sampled-data model of the human operator

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