ZzT I A &y

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I
N A S A TECHNICAL NOTE
A TECHNIQUE FOR IDENTIFYING
PILOTDESCRIBINGFUNCTIONS
FROM
ROUTINE FLIGHT-TEST RECORDS
&yRodney C. Wingrove und Frederick G. Edwurds
Ames Reseurch Center
Moffett Field, CuZzT
NATIONAL
AERONAUTICS
ANi)
SPACE
ADMINISTRATION
.I-.-.
. . .~
. 3 .
.
. ..
.
WASHINGTON, D. C.
MAY 1969
TECH LIBRARY KAFB, NM
0131895
NASA T N D-5127
A TECHNIQUE FOR IDENTIFYING PILOT DESCRIBING
FUNCTIONS FROM ROUTINE FLIGHT-TEST RECORDS
By Rodney C. Wingrove and Frederick G. Edwards
Ames Research Center
Moff ett Field, Calif.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale
by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151
CFSTI price $3.00
-
TABLE OF CONTENTS
Page
................................
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
SUMMARY
1
3
..............................
General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
I d e n t i f i c a t i oEnr r oUrs i nSgt a n d a r d
Methods . . . . . . . . . . . . .
Use o f a Time S h i fitInd e n t i f i c a t i o n
................
ANALYSIS OF IDENTIFICATION ERROR . . . . . . . . . . . . . . . . . . .
..........................
General
Analysis
D e t aA
i ln
ed
alysis
..........................
BACKGROUND
..............
Example 1: Comparison of Theory With Experiment .
Time S h i f t .
Example 2: A Method oSf e l e c t i ntgh e
X . . .
3:
F
l
i
g
h
T
t
e
s
R
t
e
s
u
l
t
s
From
Gemini
Example
CONCLUDING REMARKS . . . . . . . . . . . . . . . . . .
........
. . . . . . . .
. . . . . . . .
. . . . . . . .
........
APPENDIX A . TIME DOMAIN ANALYSIS AND COMPUTER PROCESSING . . . . . .
S t a n d aC
r dr o s s - C o r r e l a t i o n
Method . . . . . . . . . . . . . . . . .
Use of a Time S h i fitn
Computer P r o c e s s i n g . . . . . . . . . . . .
R e d u c t i o inInd e n t i f i c a t i o E
n rror
With Time S h i f t . . . . . . . . .
Computer Processing With Data Bias . . . . . . . . . . . . . . . .
APPENDIX B . IDENTIFICATION OF PILOT/CONTROL DESCRIBING FUNCTION . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPLICATIONS AND DISCUSSION
4
4
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7
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I
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"
A TECHNIQUE FOR IDENTIFYING PILOTDESCRIBING
FUNCTIONS FROM ROUTINE FLIGHT-TESTRECORDS
By Rodney C. Wingroveand
F r e d e r i c k G. Edwards
Ames Resesrch Center
SUMMARY
Previousstudieshave
shown t h a t t h e dynamic r e s p o n s e o f t h e p i l o t c a n b e
r e p r e s e n t e d b y a l i n e a r element(describingfunction)and
a remnant term
( o u t p u tn o i s e ) .
The p r e v i o u s work a l s o h a s i n d i c a t e d t h a t t h e r e
i s an e r r o r
in identifying the pilot describing function
from r o u t i n e t r a c k i n g t a s k
recordsbecausetheoutputnoiseofthepilottransfersthroughthecontrol
l o o p ,p r o d u c i n ga nu n d e s i r e dc o r r e l a t i o nw i t hh i si n p u ts i g n a l .T h i sr e p o r t
shows t h a t t h i s c o r r e l a t i o n ,
and t h u s t h e i d e n t i f i c a t i o n e r r o r , c a n b e
reducedbyshiftingtheinputsignalduringthecomputerprocessingan
amount
e q u i v a l e n tt ot h ee f f e c t i v e
time d e i a yo ft h ep i l o t .T h i sr e p o r ti n c l u d e s
a
t h e o r e t i c a l a n a l y s i s o f t h i s t e c h n i q u e andexamples t o i l l u s t r a t e i t s
application.
The t h e o r e t i c a l a n a l y s i s c o n s i d e r s t h e
f a c t t h a t t h e computerprocessing
i s c o n s t r a i n e dt oi d e n t i f yo n l yp h y s i c a l l yr e a l i z a b l es y s t e m s .
With t h i s cons t r a i n t , it i s shown t h a t t h e e r r o r i n i d e n t i f y i n g t h e p i l o t d e s c r i b i n g f u n c t i o n depends on t h e s p e c t r u m o f t h e p i l o t ' s o u t p u t n o i s e ; t h e i d e n t i f i c a t i o n
e r r o r canbe made small i f t h e n o i s e i s n e a r " w h i t e " i n r e l a t i o n t o t h e
sum
o f a l l e f f e c t i v e time d e l a y s t h r o u g h t h e c o n t r o l l o o p ( p i l o t p l u s c o n t r o l l e d
e l e m e n t ) .T h i sr e s u l t
i s s i g n i f i c a n tb e c a u s e ,
i f t h e s ec o n d i t i o n sa r em e t ,
it i s p o s s i b l e t o i d e n t i f y t h e d e s c r i b i n g f u n c t i o n
of t h e p i l o t i n a feedback
s y s t e mt h a t i s e x c i t e d o n l y b y h i s o u t p u t n o i s e .
The i d e n t i f i c a t i o n of s e v e r a l s i m u l a t e d p i l o t models i s i n c l u d e d i n t h i s
s t u d yt oi l l u s t r a t et h i st e c h n i q u e .A l s o ,r e p r e s e n t a t i v ed a t a
from t h e r e t r o fire phase of the
Gemini X f l i g h t h a v e b e e n a n a l y z e d a n d
are p r e s e n t e d t o
demonstratethesuccessfulapplicationofthistechniquewithroutine
spacecraftoperatingrecords.
INTRODUCTION
T h i sr e p o r tc o n s i d e r st h ep r o b l e m
of i d e n t i f y i n g t h e i n p u t - o u t p u t
relat i o n s h i p of t h e p i l o t b y u s e o f m e a s u r e d d a t a f r o m r o u t i n e f l i g h t o p e r a t i o n s
i n which t h e p i l o t p r o v i d e s f e e d b a c k c o n t r o l .
Theproblem i n u s i n g t h e meas u r e di n p u ta n do u t p u td a t ad i r e c t l y
i s t h a t a n ye x t r a n e o u so u t p u tn o i s eb y
t h ep i l o tc a u s e s
an e r r o ri ni d e n t i f i c a t i o n .T h i sp r o b l e m
is s o l v e d i n t h i s
report by the development of
a computerprocessingtechniquethat,undercertainconditions,yields
an estimate r e l a t i v e l y f r e e from i d e n t i f i c a t i o n e r r o r .
Sincetheidentificationoffeedbackcontrolsystems
i s important i n many
fields,thetechniquehaswidesignificance
and a p p l i c a b i l i t y .
The i n p u t - o u t p u t c h a r a c t e r i s t i c o f
a p i l o t must b e r e g a r d e d as random,
nonlinear,anddependent
on t h e t a s k h e
i s t o perform. Many p r e v i o u ss t u d i e s
a
have shown t h a t t h i s type o f r e s p o n s e c a n b e r e p r e s e n t e d a p p r o p r i a t e l y w i t h
q u a s i - l i n e a r s y s t e m modeledby a l i n e a r e l e m e n t ( d e s c r i b i n g f u n c t i o n ) a n d
a
remnantterm(outputnoise).
The p i l o td e s c r i b i n gf u n c t i o n su s u a l l yh a v eb e e n
i d e n t i f i e d from r e c o r d so b t a i n e di ng r o u n d - b a s e ds i m u l a t o r s( r e f .
1) and
2) w h e r e i nc a r e f u l l yc o n t r o l l e de x t e r n a lf o r c i n gf u n c t i o n s
f l i g h tt e s t s( r e f .
a r eu s e dt oe x c i t et h ep i l o t - v e h i c l es y s t e m .
The p i l o td e s c r i b i n gf u n c t i o n s
are measuredbycomparing
the input and output signals of the pilot with the
known f o r c i n gf u n c t i o n .T h i s
method minimizes t h o s e e r r o r s i n i d e n t i f i c a t i o n
due t o a n y c o r r e l a t i o n o f t h e i n p u t s i g n a l w i t h t h e p i l o t ' s o u t p u t n o i s e .
work and summarizes t h e
Reference 3 c o n t a i n s a good r e v i e w o f t h i s p r e v i o u s
measured p i l o t d e s c r i b i n g f u n c t i o n s .
Most o t h e r methods f o r m e a s u r i n g p i l o t d e s c r i b i n g f u n c t i o n s
depend on
random d i s t u r b a n c e s ( e . g . , a e r o d y n a m i c t u r b u l e n c e , p r o p u l s i v e d i s t u r b a n c e ,
e t c . )t oe x c i t et h ep i l o t - v e h i c l es y s t e m .T h e s e
methodscompute
d i r e c t l yt h e
However,
d e s c r i b i n gf u n c t i o n of t h e p i l o t from h i s i n p u t a n do u t p u ts i g n a l s .
t h e r e i s a f u n d a m e n t a ld i f f i c u l t yw i t ht h e s em e t h o d sb e c a u s et h ep i l o t ' so u t p u tn o i s et r a n s m i t t e dt h r o u g ht h ec o n t r o ll o o pp r o d u c e s
an u n d e s i r e d c o r r e l a t i o n between h i s i n p u t and o u t p u t s i g n a l s , t h e r e b y c a u s i n g a n e r r o r i n
4 , t h ee x p e c t e de r r o r
i s analyzedand it was
i d e n t i f i c a t i o n .I nr e f e r e n c e
shown t h a t i f t h e a m p l i t u d e o f t h e p i l o t ' s n o i s e
i s l a r g e , as compared with
theexternaldisturbance,thentheidentificationerror
is unacceptable.
D u r i n gr o u t i n ef l i g h t - t e s to p e r a t i o n s ,t h e r e
a r e n oc a r e f u l l yc o n t r o l l e d
f o r c i n gf u n c t i o n sa n de v e nt h e
random e x t e r n a l d i s t u r b a n c e may b e q u i t e small
so thattheprincipalsystemexcitation
maycome
from t h e p i l o t ' s o u t p u t
n o i s e .T h i sr e p o r t
shows t h a t i n such s i t u a t i o n s i t may s t i l l b ep o s s i b l e ,
u n d e rc e r t a i nr e a s o n a b l ec o n d i t i o n s ,t od e t e r m i n et h ep i l o td e s c r i b i n gf u n c t i o nw i t h o u ti n c u r r i n g
an u n a c c e p t a b l ei d e n t i f i c a t i o ne r r o r .
One r e q u i r e d
( o r p o s s i b l yt h ef e e d b a c kc o n t r o ll o o p )h a v e
a
condition is t h a t t h e p i l o t
time d e l a y . If t h i sc o n d i t i o n i s met, i t i s p o s s i b l et ot a k ea d v a n t a g eo f
t h i sf a c ti nt h ei d e n t i f i c a t i o nd a t ap r o c e s s i n g .I ne f f e c t ,t h ei n p u ts i g n a l
i s s h i f t e d d u r i n g p r o c e s s i n g byan amount e q u a l t o t h e time d e l a y o f t h e
5-7) h a v ec o n s i d e r e dt h eu s eo f
a
p i l o t . A l t h o u g hp r e v i o u ss t u d i e s( r e f s .
time s h i f t i n t h e measurement o f p i l o t d e s c r i b i n g f u n c t i o n s ,
it was a p p a r e n t l y
notobservedthatthistimeshift
would s t r o n g l y i n f l u e n c e t h e e r r o r i n
identification.
a theoreticalanalysisto
show t h a t t h i s t e c h n i q u e
T h i sr e p o r tp r e s e n t s
will r e d u c et h ei d e n t i f i c a t i o ne r r o r .
The s i m u l a t i o na n di d e n t i f i c a t i o no f
s e v e r a l known s y s t e m e l e m e n t s a r e i n c l u d e d t o
compare w i t h t h e t h e o r y
and t o
i l l u s t r a t et h eu s eo ft h i st e c h n i q u e .A l s o ,r e s u l t so b t a i n e d
from t h e r e t r o f i r e phaseofthe
Gemini X m i s s i o n a r e p r e s e n t e d t o d e m o n s t r a t e t h e
applicationofthistechniquetoroutineflight-testrecords.
2
NOTATION
controllerdeflection(outputofpilot)
errorsignal(inputtopilot)
F o u r i e rt r a n s f o r m
of [ ]
e x t e r n a ld i s t u r b a n c e
c o n s t a n tg a i n
numerator terms i nY c ( j o )
i n t e r n a ln o i s e( p i l o tr e m n a n t )
of
c r o s s - c o r r e l a t i o nf u n c t i o n
e ( t ) and c ( t )
c r o s s - c o r r e l a t i o nf u n c t i o no fe ( t )
autocorrelationfunctionof
a u t o c o r r e l a t i o nf u n c t i o n
t
of
and n ( t )
e(t)
n(t)
time, s e c
c o n t r o l l e de l e m e n td e s c r i b i n gf u n c t i o n
m e a s u r e dd e s c r i b i n gf u n c t i o n( i d e a l )
m e a s u r e dd e s c r i b i n gf u n c t i o n( a c t u a l )
p i l o td e s c r i b i n gf u n c t i o n
estimatedpilotdescribingfunction
c1
e x p o n e n t i a ld e c a yf a c t o r ,s e c - l
residual
A
t i m es h i f tu s e dd u r i n ga n a l y s i s ,
time d e l a yi nY c ( j w ) ,
time d e l a yi n
sec
sec
Yp(jw), sec
p o w e rs p e c t r u mo fe ( t )
power s p e c t r u mo fn ( t )
cross-powerspectrumof
e ( t ) and c ( t )
3
Oen ( j w )
cross-powerspectrumof
w
f r er aqdu/esneccy ,
e ( t ) and n ( t )
BACKGROUND
T h i ss e c t i o nd i s c u s s e st h ep i l o t e dc o n t r o ls y s t e me l e m e n t sa n di n d i c a t e s
theerrorinidentifyingthepilotdescribingfunction
from r o u t i n e t r a c k i n g
task r e c o r d s . A computingprocess f o r r e d u c i n g t h i s i d e n t i f i c a t i o n e r r o r
is
t h e no u t l i n e d .T h i sb a c k g r o u n d
material p r e c e d e s a more d e t a i l e da n a l y s i so f
theidentificationerrorpresented
later in the report.
General Remarks
Figure 1 i s a blockdiagramofthepilotin
a compensatorytrackingtask
s o t h a tt h ei n p u te r r o rs i g n a le ( t )
t r y i n gt oc o n t r o lh i so u t p u tc ( t )
PllOI
r-----1
F i g u r e 1.- I d e n t i f i c a t i o n u s i n g s t a n d a r d
Conlrolled system
r-----1
measurementmethods.
i s k e p tn e a rz e r o .G e n e r a l l y ,t h ei n p u t - o u t p u tc h a r a c t e r i s t i c so ft h ep i l o t
time v a r y i n g . However, f o r t h e
must b ec o n s i d e r e d as complex,nonlinear,and
it i s common p r a c t i c e t o assume t h a t h i s c h a r a c t e r i s t i c s
purposesofmodeling,
3 ) . Thismathematical model
canberepresented
by a q u a s i - l i n e a rs y s t e m( r e f .
Yp and t h en o i s es o u r c en .
The element
Yp(jw),
c o n t a i n tsh el i n e a er l e m e n t
which i s c a l l e d t h e p i l o t d e s c r i b i n g f u n c t i o n , '
i s a l i n e a rc o n s t a n t a frequency response dependent
on t h e i n , p t = = e = ( t ) ." . coefficient system with
I T e c h n i c a l l y , Yp(jw) r e p r e s e n t s a random i n p u t d e s c r i b i n g ' f u n c t i o n
ref. 3 ) .
because random, r a t h e r t h a n s i n u s o i d a l , s i g n a l s a r e u s e d h e r e ( s e e
Also, t oa v o i da d d i t i o n a ln o t a t i o n ,
terms such as Y(jw) will b eu s e dt o
representboththetransferfunctionsoflinearsystems
and t h e d e s c r i b i n g
functionsofnonlinearsystems.
4
The term n ( t ) r e p r e s e n t s t h e d i f f e r e n c e b e t w e e n o u t p u t
of t h e p i l o t , c ( t ) ,
a n do u t p u to ft h ed e s c r i b i n gf u n c t i o n
Y ( j w )d r i v e nb ye ( t )
Thus n ( t )
accountsforremnant
terms such as n o n l iPn e a r i t i e s , time v a r i a t i o n s , a n d
additive noise in the output of the pilot.
.
The c o n t r o l l e d s y s t e m i s m a t h e m a t i c a l l y c h a r a c t e r i z e d b y t h e c o n s t a n t
l i n e a er l e m e n t
Yc and t h en o i s es o u r c e
i. The time h i s t o r yi ( t )a c c o u n t s
f o r n o n l i n e a r i t i e s and time v a r i a t i o n s i n t h e c o n t r o l l e d e l e m e n t ,
timev a r y i n g commands, and a l l disturbancesfromaerodynamics,propulsion,
etc.
external to the pilot.
Identification Error Using Standard
,
Methods
Severalmethods(e.g.,
refs. 4-10)havebeenused
t o compute,fromgiven
r e c o r d so fe ( t )a n dc ( t ) ,
a d e s c r i b i n gf u n c t i o nq m ( j w )t h a tr e p r e s e n t st h e
b e s tl i n e a rr e l a t i o n s h i p
between e ( t ) and c ( t ) . Best h e r e means t h a tt h e
integralofthesquaredresidual,/E2(t)dt,
i s minimizedover a g i v e n r e c o r d
i s t h ed i f f e r e n c eb e t w e e nt h ea c t u a lr e c o r dc l t )a n dt h e
length,whereE(t)
outputofthesystem
?m(jw) e x c i t e db ye ( t ) .
The mea.surements Ym(jw) may
d i f f e r somewhat betweenmethodsbecauseeach
method u s e s s l i g h t l y d i f f e r e n t
approximationsand
modelforms
i n computerprocessing.Generally,themeasurementsofvm(jw)
w i l l b en e a rt h ef o l l o w i n gi d e a ld e s c r i b i n gf u n c t i o n
Ym(jW)
t h a tr e p r e s e n t st h eb e s tl i n e a rr e l a t i o n s h i p
between e ( t ) and c ( t ) f o r random
stationary signalsz:
I nt h i se q u a t i o n ,Q e c ( j w )
i s thecross-powerspectrumbetweene(t)and
c ( t ) andQee(w)
i s t h e power d e n s i t ys p e c t r u mo fe ( t ) .I ni d e n t i f y i n gt h e
p i l o td e s c r i b i n gf u n c t i o nw i t ht h e s et y p e so fm e t h o d s ,p r e v i o u ss t u d i e s( e . g . ,
refs. 4 and 9 ) have shown t h a t t h e r e i s a differencebetweenthemeasured
d e s c r i b i n gf u n c t i o n
Ym(jw) and t h ea c t u a dl e s c r i b i n gf u n c t i o nY p ( j w ) .
This
difference,or"identificationerror,"canbe
shown b y d e l i n e a t i n g t h e
componentsofthecross-powerspectrum:@ec(j,)
= YP(jw)@ee(w) + Qen(j,).
Subs t i t u t i n g t h e s e components i n t oe q u a t i o n( 1 )y i e l d s
error
w i l l c o n t r i b u t ea n
Equation (2) shows t h a t any c r o s s - c o r r e l a t i o na e n ( j w )
e r r o ri ni d e n t i f i c a t i o n .
Such a c o r r e l a t i o nd o e se x i s td u r i n gc l o s e d - l o o p
Yc (jw)andthusappears
as a compoc o n t r o lb e c a u s e
n ( t )t r a n s f e r st h r o u g h
n e n to f
e ( t ) . If n ( t ) i s much smaller t h a n i ( t ) , t h e r a t i o
@en(jw)/Oee(w)
w i l l b e small a n dt h em e a s u r e dt r a n s f e rf u n c t i o n
Ym(jw) w i l l b e n e a r t h e t r u e
21f t h e measurementhas t h e c o n s t r a i n t t o i d e n t i f y o n l y p h y s i c a l l y
reali z a b l es y s t e m s ,t h e n ,
as s h a l l b e p o i n t e d o u t
l a t e r , Ym(jw) i s w r i t t e n i n a
s l i g h t l y d i f f e r e n t form.
5
value Yp ( j w ) . However, i f n ( t ) i s much l a r g e rt h a n
i ( t ) , t h er a t i o
Oen(jw)/@ee(W) will b e s i g n i f i c a n t and t h e m e a s u r e d d e s c r i b i n g f u n c t i o n
Y,(jw)
w i l l b ev e r yd i f f e r e n t
fromYp(ju).
For r o u t i n ef l i g h t - t e s tc o n d i i ( t ) , i t i s n e c e s s a r yt of i n d
t i o n s ,w h e r en ( t )
may be much l a r g e r t h a n
means o f r e d u c i n g t h i s e r r o r .
Such a t e c h n i q u e w i l l b eo u t l i n e dn e x t ,
some
Use o f a Time S h i f t i n I d e n t i f i c a t i o n
5-7) h a v ec o n s i d e r e dt h eu s e
of a time
P r e v i o u ss t u d i e s( e . g . ,r e f s .
s h i f t d u r i n g t h e computerprocessingtoaccountfortheeffectivetimedelay
o f t h ep i l o t .T h i st i m es h i f t i n gr e p r e s e n t so n l y
a s l i g h tm o d i f i c a t i o nt ot h e
i d e n t i f i c a t i o n methods i n f i g u r e 1.
Thistime-shiftingtechniqueillustratedinfigure
i n gs t e p si nt h ec o m p u t i n gp r o c e s s .
2 i n v o l v e st h ef o l l o w -
1. The i n p u ts i g n a le ( t )
is s h i f t e dw i t hr e s p e c tt oc ( t )
A , where A i s e q u i v a l e n tt ot h e
time d e l a yo ft h ep i l o t .
byan
amount
2 . The d e s c r i b i n gf u n c t i o nG m ( j u )
from s t e p( 1 ) .
i s d e t e r m i n e du s i n gt h es h i f t e dd a t a
3 . The e s t i m a t e dt r a n s f e rf u n c t i o n
t r a n s f e r f u n c t i o n as
i s determinedfromthemeasured
Controlled system
PI101
r-----l
r""1
Figure 2.- The use of a time shift
A
in identification.
A l t h o u g hp r e v i o u ss t u d i e sh a v ec o n s i d e r e dt h i st i m e - s h i f t i n gt e c h n i q u e ,
it was a p p a r e n t l y n o t o b s e r v e d t h a t t h i s t e c h n i q u e
would s t r o n g l y i n f l u e n c e
t h ee r r o r si ni d e n t i f i c a t i o n .T h i sr e p o r t
shows t h a t when t h i st e c h n i q u e i s
usedwith a measurementmethod i n which P m ( j u ) i s c o n s t r a i n e d t o b e
6
physically reali~able,~ then the identification error due to the correlation
o f e ( t ) w i t hn ( t )c a nb er e d u c e d .T h i sr e d u c t i o n
w i l l b e shown i n t h e n e x t
s e c t i o n where t h e i d e n t i f i c a t i o n e r r o r t o b e e x p e c t e d w i t h t h i s
computing
p r o c e s s will beanalyzed.
ANALYSIS OF IDENTIFICATION ERROR
The r e d u c t i o n o f t h e i d e n t i f i c a t i o n e r r o r b y t h e f o r e g o i n g c o m p u t e r
F i r s t , a general
p r o c e s s i n g w i l l b e i l l u s t r a t e d fromtwo p o i n t so fv i e w .
a n a l y s i s will show why t h e time s h i f t X r e d u c e st h ei d e n t i f i c a t i o ne r r o r .
The s e c o n da n a l y s i s w i l l d e v e l o p e q u a t i o n s t o show, i n more d e t a i l , t h e amount
t h e e r r o r i s reduced. The f o l l o w i n ga n a l y s i s i s p r e s e n t e du s i n gt h ef r e q u e n c y
domain. A similar a n a l y s i s i s p r e s e n t e di na p p e n d i x
A u s i n gt h et i m e
domain.
General Analysis
TO i l l u s t r a t et h er e d u c t i o ni ni d e n t i f i c a t i o ne r r o r ,e q u a t i o n( 1 )
r e w r i t t e n as
.d
is
where F [Re, ( T ) ] r e p r e s e n t s t h e
Fouriertransformofthecrossc o r r e l a t i o nf u n c t i o nR e c ( ~ )
and
F[Ree(.)] r e p r e s e n t s t h e F o u r i e r
transformoftheautocorrelation
f u n c t i o n Ree(.).
Representative
curves4ofthemeasuredquantities
R (T)andRee(T)
a r e s k e t c h e di n
ec
f l g u r e3 ( a ) .
The e r r o rc o n t r i b u t i o n
R,,(T),
c o n t a i n e di nR e c ( T ) ,
is also
shown f o r comparison.
Now c o n s i d e r t h o s e measurement
methods t h a t h a v e t h e c o n s t r a i n t o f
p h y s i c arle a l i z a b i l i t yT. h e s e
Figure 3.- Effect of time shift on correlation
methodsusedonlydataforpositive
functions.
v a l u e so f
T, a n d a
, c c o r d i n g l y t, h e
measured
t
r
a
n
s
f
e
r
f
unction is
_____
~3 T h i sc o n s t r a i n t i s i n h e r e n t i n t h e c o m p u t e r p r o c e s s i n g f o r
mosttimeas c r o s s - c o r r e l a t i o n ( r e f s .
4, 8, and 9) ,
domain measurementmethodssuch
orthogonal f i l t e r s ( r e f s . 4 , 5 , and 7), and p a r a m e t e rt r a c k e r s( r e f s .
4, 6 ,
and10).
Most frequency domain measuringmethodsusingcross-spectral
comp u t i n gp r o g r a m s( r e f .1 )u s u a l l y
do n o tc o n t a i nt h i sc o n s t r a i n t .
However,
such a c o n s t r a i n t c o u l d p r o b a b l y b e i n c o r p o r a t e d .
4Thesedata are fromexample 1, which w i l l b e d i s c u s s e d l a t e r .
(a) No tlrne s h l f i ; X = O
"
"
7
and if t h ei n d i v i d u a l
terms (seeeq.
( 2 ) ) are s u b s t i t u t e d f o r R e c ( ~ ) ,
With t h i sc o n s t r a i n t ,o n l yt h a tp o r t i o n
ofRen(T)
(shown by t h e s h a d e d r e g i o n i n f i g . 3 ( a ) ) c o n t r i b u t e s
identification.
f o rp o s i t i v ev a l u e so f
an e r r o r i n
T
Let u sn e x ti n t r o d u c et h e
time s h i f t X as p r e s e n t e di nf i g u r e
2. T h i s
time s h i f t i s a p p l i e d s o t h a t t h e s h i f t e d i n p u t d a t a a r e e ' ( t )
= e(t - X).
The e f f e c t of t h i s time s h i f t i s i l l u s t r a t e d i n f i g u r e 3 ( b )
where t h e f u n c Refel (T)
t i o n sR e l n ( T ) ,R e f c ( T ) ,a n d
r e s u l t i n g from t h e s h i f t e d i n p u t
d a t a a r e p r e s e n t e d . I t i s shown
thattheadditionofthistimeshift
n
Ref C ( ~ ) and
c a u s e st h eq u a n t i t i e s
R e f n (sbtTho
e)i f t b
tehd
ye
amount
X w i t hr e s p e c tt o
Ref e' (T)
NOW it
i s a p p a r e n tt h a tt h ee r r o rc o n t r i b u t i o no fR e f n ( T ) ,
f o r t h ep o s i t i v e
v a l u e so f
T~ i s reducedand
small shaded area
i n c l u d e so n l yt h e
(b) Wlth tlme shlfl A
f iign3
u (r b
e).
The a c t u a l v a l u e
for the error
term is
-
L \ +
.
3.
Figure 3 . - Concluded.
I ng e n e r a l ,R 2 n ( ~ )
will d e c r e a s ef o rp o s i t i v ev a l u e so f
T.
T h e r e f o r e ,n o t e
-T j w
d.r will be
reduced
as X i s
t h atth e r r o cr o n t r i b u t i o n
Ren(T + X)e
SOm
i n c r e a s e d .F u r t h e r ,t h ee r r o rc o n t r i b u t i o n
f ovr a l u e os f
T
g r e a t e trh a n
A.
w i l l b ez e r o
if
Ren(r) i s z e r o
T h i sg e n e r a ld i s c u s s i o nh a sa t t e m p t e dt og i v e
some p h y s i c a l i n s i g h t i n t o
We will now t u r n
why t h e time s h i f t X r e d u c e st h ee r r o ri ni d e n t i f i c a t i o n .
o u r a t t e n t i o n t o a more d e t a i l e d a n a l y s i s t o d e t e r m i n e t h e
amount t h e e r r o r
can be reduced.
8
Detailed Analysis
In t h i s s e c t i o n , w e w i l l d e r i v e f o r m u l a s t h a t show t h e r e d u c t i o n i n
i d e n t i f i c a t i o n e r r o r as a f u n c t i o n o f t h e p r i m a r y v a r i a b l e s w i t h i n t h e c o n t r o l
loop. As n o t e d e a r l i e r , w e w i l l c o n s i d e rt h eu s eo f
a timg s h i f t X i n t h e
i d e n t i f i c a t i o na n dc o n s i d e r
measurementmethods
i n which Ym(jw) i s
constrained to be physically realizabl-.
To m a t h e m a t i c a l l yr e p r e s e n t a m e a s u r e dd e s c r i b i n gf u n c t i o n
Ym(jw) t h a t
is constrainedtobephysicallyrealizable,
we can u t i l i z e t h e r e l a t i o n s h i p
u s e dw i t ht h e
Wiener-Hopf e q u a t i o n( r e f .
11). Using t h i s r e l a t i o n s h i p f o r
p h y s i c a l l yr e a l i z a b l es y s t e m s ,e q u a t i o n( 1 )
is written
I+
where
+
Q e e ( j w )h a sp o l e so rz e r o so n l yi nt h el e f t - h a l fp l a n e
@ e e ( j w )h a sp o l e so rz e r o so n l yi nt h er i g h t - h a l fp l a n e
[
I+
h aps o l eos n l iytnhlee f t - h a lpf l a n e
This f o l l o w st h eu s u a lf o r m ,
which i m p l i e s t h a t t h e d i r e c t t r a n s f o r m o f
a time
f u n c t i o n t h a t i s s t a b l e and z e r o f o r n e g a t i v e t i m e
w i l l have a l l i t s p o l e s i n
t h el e f t - h a l fp l a n e
(LHP).
Now we i n t r o d u c et h e
time s h i f t
X
as i l l u s t r a t e d i n f i g u r e
2 and d e f i n e
t h es h i f t e dd a t a
as e ' ( t ) = e ( t - X). BecauseQelc(jw)
= eXjaQec(jw)and
@ e , e [w)
I
= Qee(w) , we c a n w r i t e t h e m e a s u r e d t r a n s f e r f u n c t i o n a s
A s shown i n f i g u r e 2 , we c a n d e f i n e t h e e s t i m a t e d d e s c r i b i n g f u n c t i o n i n
terms ofthemeasureddescribingfunction,Pp(jw)
= e-'joPm(jw).
And i f we
assume t h a t t h e r e i s no m o d e l i n ge r r o r ,t h a t
i s , qm(jw) = Ym(jw),then a
theoreticalexpressionfortheestimateddescribingfunction
is
(9)
+
9
I n t r o d u c i n gt h ei n d i v i d u a l
terms f o r
of Yp
The i m p u l s er e s p o n s ef u n c t i o n
t h a n a v a l u eo f
-tP and, s o long as
t e r me X j w Y( j w )h a sp o l e so n l yi nt h e
P
t h i sa s s u m p t i o n , we o b t a i n
Q e c ( j u ) (see eq. (2)),
we have
is assumed t o b e z e r o f o r
time less
X i s l e s st h a n
o r e q u a tl o
T
the
P'
LHP.
Simplifyingequation(10)with
The term ~ ~ ~ c(o nws i s)t s of c o n t r i b u t i o n s from two s o u r c e s :i ( t )
and n ( t ) .
The maximum errorcanbedetermined
by assuming i ( t ) = 0 ( r e f . 4 ) . With t h i s
a s s u m p t i o na n du s i n gb a s i cc l o s e d - l o o pr e l a t i o n s h i p s( r e f .l l ) ,l e tu sd e f i n e
time
T h e s ed e f i n i t i o n s
assume t h a t Yc(jw) i s minimum p h a s e( i . e . ,c o n t a i n sn o
d e l a yo rz e r o si nt h e
RHP) . The c a s e i n whichYc(jw)
i s a nonminimum phase
will b e i l l u s t r a t e d a t t h e endof t h i s s e c t i o n .
Y,
Minimum p h a s e . - From t h ef o r e g o i n ga s s u m p t i o n s ,
of pilotedcontrolsituations,
we f i n d t h a t
10
whichcover
a variety
+
where now t h e e r r o r i s c o n v e n i e n t l ye x p r e s s e d as a f u n c t i o no fQ n n ( j w ) ,t h e
[eX j w @nn(j,)]+
+
and
excitation noise source. In equation (15), the terms
+
as shown i n t h e f o l l o w i n g e q u a t i o n :
Qnn(j,)canbeevaluated
The c o n t r i b u t i o n t o t h e e r r o r t e r m i n c l u d e s t h a t p o r t i o n
of R,,(T)
f o rv a l u e s
g r e a t e rt h a n
X (shaded area i n f i g . 4 ) .
I t i s s e e nt h a tt h i sc o n t r i b u t i o nt o
t h ee r r o rt e r m
w i l l bereduced as X
i s i n c r e a s e d . (However, t h i st h e o reticalderivationholdsonlyfor
t h o s ev a l u e so f
X less t h a n o r
Error
to
equal
contrlbutlon
Tp. 1
-
A
-7
Figure
4i n d i c a t e s
those
(15a)
Equation
c o n d i t i o n s u n d e r which t h e i d e n t i f i c a t i o ne r r o r w i l l be small. For
L A i
nstancen
, o t et h a t
i f X is posi4.- Reduction of error contribution w i t h
a.. .
t i v e and i f n ( t ) i s nwo hi si et e
-Tju
(Rnn(T) impulse
i s an
a t T = 0 ) , then
w i l l b en oe r r o ri ni d e n t i f i c a t i o n .
dT i s z e r oa n dt h e r e
AI^
r
"
S,mRnn (T
+
More g e n e r a l l y , t h e i d e n t i f i c a t i o n e r r o r
w i l l b ez e r o
if
where
T h i sr e s u l ta p p e a r st oh a v es i g n i f i c a n c ef o r
many a p p l i c a t i o n s . The most
i m p o r t a n tp o i n t i s t h a t when t h e s e c o n d i t i o n s a r e
met, a d e s c r i b i n g f u n c t i o n
YP(!w) w i t h i n a feedback system can theoretically be measured with the system
e x c l t e do n l y by t h e i n t e r n a l n o i s e n ( t ) .
I n most r e a l i s t i c s i t u a t i o n s , Rnn(-c) will n o t b e i d e n t i c a l l y z e r o f o r
v a l u e so f
T > A.
W
e will n e x t show,however,
t h a tt h ei d e n t i f i c a t i o ne r r o r
canbereduced,and
i n some c a s e s b e made q u i t e small, w i t h more r e a l i s t i c
formsofRnn(T).Forexample,assume
t h a tt h en o i s en ( t )t a k e st h e
form
= Ke - a I I which, f o r small a , would benarrow-band(nonwhite)noise.
Rn,
T h l sf o r ma g r e e sq u i t ew e l lw i t h
some experimentalmeasurementsofthepilot
remnant.(Forinstance,thisexponentialformwith
~1 = 5 s e c -a
' g r e e sw i t h
t h e measured n ( t )i nr e f e r e n c e s
3 and 1 2 . ) With t h i s form, we can e v a l u a t e
t h ec o n s t a n tf a c t o ro fe q u a t i o n
(15) as
and a r r i v e a t
The e r r o r term on t h e r i g h t
s i d e o f t h e e q u a t i o n i s a f u n c t i o n of t h e magnit u d eo ft h ec o n s t a n tf a c t o r
e -ax. As 1 i n c r e a s e s and i f a i s l a r g(en e a r
w h i t en o i s e ) , t h e nt p ( j w )
2 Yp(jw).
Conversely,
i f A = 0 , t h e nt h er e s u l t
i s i d e n t i c a lt ot h a t
shown i nr e f e r e n c e 4: vp(jw) = - l / Y c ( j u ) .
Yc
Nonminimum phase.- Let u sd e f i n et h e
nonminimum phase terms as
t o r e p r e s e n t anypure time d e l a yi nY c ( j u )
andN,(ju)
t or e p r e s e n t
any RHP zerosinYc(jw).
T h e n ,b yi n c l u d i n gt h e s et e r m s , i ne q u a t i o n s( 1 2 )t o
( 1 4 ) , t h ee s t i m a t e dd e s c r i b i n gf u n c t i o n
becomes
Note t h a t ,i nt h i sc a s e ,
i f Rnn(T) = 0 f o r
> Tee, t h e n Y (jw)
need
P
X, i s r e q u i r e d i n t h e a n a l y s i s ) i n
nothave a timedelay(andnotimedelay,
orderthattheidentificationerrorbezero.
Inthis
b ez e r o i f
more g e n e r a l c a s e , t h e i d e n t i f i c a t i o n e r r o r i n e q u a t i o n ( 1 8 )
will
where
A < T
- P
The i d e n t i f i c a t i o n e r r o r c a n b e
made small i f t h e a u t o c o r r e l a t i o n f u n c t i o n o f
i s n e g l i g i b l ef o rv a l u e so f
T g r e a t e rt h a nt h e
sum
t h ei n t e r n a ln o i s eR n n ( ~ )
o tf h e
time d e l a y s X + T
W
e can a l s on o t et h a t h et e r mN i ( - j m ) / N z ( j u )
> C'
r e p r e s e n t a t i v eo f a Padeapproximation t o a time delay.Thus,any
i n Yc(jw) will t e n d t o a c t
as an a d d i t i o n a le f f e c t i v et i m ed e l a ya n d
furtherreducetheidentificationerror.
is
RHP z e r o s
will
T h i sa n a l y s i so ft h ei d e n t i f i c a t i o ne r r o ri n d i c a t e st h a tt h ei n t e r n a l
n o i s en ( t )
neednotbe
a h i n d r a n c et oi d e n t i f i c a t i o n ,b u tr a t h e r
it will a i d
o f feedbackcontrolsystems
i f t h ec o n d i t i o n so fe q u a intheidentification
t i o n (19) a r em e t .T h i sa n a l y s i sa l s o
may h a v ea p p l i c a t i o ni n
many o t h e r
as biology,economics,andchemicalprocesses.Althoughthese
f i e l d ss u c h
otherapplicationsarenotconsideredinthisreport,they
do c o n t a i n t i m e
d e l a y s and some o f t h e measurementscan
b e made o n l y w i t h t h e n o i s e i n t r o d u c e d
w i t h i nt h e s es y s t e m st ob ei d e n t i f i e d .
12
APPLICATIONS AND DISCUSSION
The u s e o f t h e c o m p u t i n g t e c h n i q u e o u t l i n e d i n t h i s r e p o r t
w i l l be
illustrated through the identification of
two e x a m p l e s u s i n g s i m u l a t i o n d a t a
andoneexample
u s i n ga c t u a lf l i g h td a t a .
Each example w i l l i l l u s t r a t e a d i f f e r e n tp o i n t .
With example 1, t h e f o r e g o i n g t h e o r e t i c a l r e s u l t s
will b e comp a r e dw i t he x p e r i m e n t a lr e s u l t s .
With example 2 , a method for s e l e c t i n g t h e
time s h i f t X w i l l b ei l l u s t r a t e d .
With
example
3 , a na p p l i c a t i o nu s i n g
actualflightrecordsfrom
Gemini X w i l l b e i l l u s t r a t e d .
The s i m u l a t e d s y s t e m s f o r t h e
f i r s t two-examples are shown i n f i g u r e 5 .
The dynamics f o rt h e s ee x a m p l e s
were s i m u l a t e d on a d i g i t a l computer. The
output of a random n o i s e program was a p p r o p r i a t e l y f i l t e r e d t o o b t a i n t h e
d e s i r e ds p e c t r u mo fn ( t ) .
The r e s u l t i n g dynamicrecordsofe(t)andc(t)
wereprocessgdusingthe
method d e s c r i b e di na p p e n d i x
A. The e x p e r i m e n t a l l y
determined Yp(jw) t ob ep r e s e n t e df o rt h e s es i m u l a t e de x a m p l e sr e p r e s e n t st h e
a v e r a g ev a l u e so b t a i n e d
from 1 2 separate20-secondruns.
.
.
"7-
d
I r""-
'
I
1
"
"
"
( b l Example 2
Figure 5.- System examples used to illustrate identification technique.
13
Example 1:
Comparison o f Theory With Experiment
To i l l u s t r a t e t h e t h e o r y , t h e s y s t e m i n f i g u r e S ( a )
was simulatedandan
i d e n t i f i c a t i o n was made on t h e known model. The p i l o t model a n dc o n t r o l l e d
elementwere
Yp(jw) = 4e -OS3jwandYc(jw)
= l/jw.
The measurements were made
= 0 , a n dw i t ht h eo n l ye x c i t a t i o nb e i n gt h e
w i t hn oe x t e r n a ld i s t u r b a n c e ,i ( t )
i n t e r n a ln o i s es o u r c e ,n ( t ) .
Two formsofthenoisespectrumwereconsidered:
conan n ( t ) w i t h a s p e c t r u m t h a t a p p r o x i m a t e s w h i t e n o i s e t o i l l u s t r a t e t h e
d i t i o n fromequation(16)for
no i d e n t i f i c a t i o n e r r o r , andwith a spectrum
whose a u t o c o r r e l a t i o nf u n c t i o n i s Rnn(.r) = e - a ' . r ' t o i l l u s t r a t e t h e t h e o r e t i c a l r e s u l t s f r o me q u a t i o n( 1 7 )f o ra ne x p e c t e di d e n t i f i c a t i o ne r r o r .
I n t e r n a lw h i t en o i s e . -F o rt h i sc a s e ,t h ee x c i t a t i o ns o u r c en ( t )
had a
s p e c t r u mn e a rw h i t en o i s e .
The time s h i f t u s e d i n t h e
computerprocessing was
t a k e n a t A = 0 . 2 sec.Theseconditions
meet t h o s es p e c i f i e df o re q u a t i o n( 1 6 ) .
will i d e n t i f y t h e a c t u a l
According t o e q u a t i o n( 1 6 ) ,t h ee s t i m a t i o nt e c h n i q u e
system, yp (J-w) = 4 e - ~ . 3 j w
A
I
I
Figure 6 (a) presents the experimentally determined magnitude
qP (jw)
andphaseangle
{ ? ( j w ) as f u n c t i o n so ff r e q u e n c y .A l s o
shown f o r comparison
P
a r e -themagnitude
Yp ( j w ) I andphase
ang l e { Y P (jw)oftheactualsystem.
The e s t i m a t e d a m p l i t u d e ( E ( j w )
P
v a r i e s 2 0 . 5 dB a b o u t h ea c t u a l
v a l u ef o rf r e q u e n c i e st oa b o u t
9 r a d / s e c and t h e p h a s e a n g l e
.( t P ( j w )
i s w i t h i n k 0 . 5 " o ft h ea c t u a lv a l u e .
Thesedifferencesappeartobewithin
t h ee x p e r i m e n t a la c c u r a c i e so ft h e
s i m u l a t i o nT
. h e s er e s u l t s u b s t a n tiatethetheoreticalconclusionthat
it i s p o s s i b l e t o i d e n t i f y t h e
I
d e s c r i b i n g f u n c t i o n of a systemthat
i s e x c i t e d by n o i s en ( t )i n t r o d u c e d
.I
withinthesystem.
1
1
I
IO
Frequency
w
, rad / sec
I n t e r n a ln o n w h i t en o i s e . -T h i s
c a s eu s e st h e
same c o n t r o le l e m e n t s
and t h e same v a l u e A = 0 . 2 s e c as i n
t h ep r e v i o u sc a s e .
However, t h e
assumed n o i s e s p e c t r u m h a s
a more
r e a l i s t i c form
Figure 6.- Identification of example 1;
X = 0.2 sec.
A
Yp(jw) = 4e
.
= e- 5 ' T ' .
For
t h i sc a s e ,t h et h e o r y( e q .( 1 7 ) )
p r e d i c t st h ef o l l o w i n ge s t i m a t e d
d e s c r i b i n gf u n c t i o n :
I
error
"
"
(T)
-0.2jw
- 0 . 3 j w - 0.37(jw + 4e - 0 . 3 j w
)e
L_
14
Rnn
T h i st h e o r e t i c a lv a l u eo f
qp(5w) i s p r e s e n t e d i n f i g u r e
6 ( b ) a l o n gw i t h
t h eq P ( j w )o b t a i n e df r o mt h ee x p e n m e n t a ld a t a .A l s o
shown f o r comparison
are t h e d e s c r i b i n g f u n c t i o n s o f t h e
-.TP(iw),Theory
a c t u a ls y s t e m
Yp(jw)and
t h en e g a t i v e
-T P ( j w ) , Experimenl
i n v etcrohsofeen t r o l leelde m e n t
20 m
/
"
"_
Yp(iw). Actual
e s t i m atthe ed
<P
al
2
l0-
value,
actual
:1/Yc(jw).
The e x p e rdi emrei vnet adl l y
Yp(jw) i n t h i s f i g u r e i s c l o s e t o t h a t
p r e d i c ttbhehyede o r y .
We can see
t h a tf from
i g u r et h i s
magnitude
(jw)] i s about 4 dB below
the
I Yp ( j w) , a t t h e
I vp
I
lowerfrequenciesandtends
to give
o f a p p e a r a nI c et h e
( s l o p e = 20 dB/decade) a t t h e h i g h e r
f r e q u e n c i e sO
. v e r a l lt,h e s t i m a t e d
magnitudetendstoward
I l/Yc(jw) as
"
w 3 -50e q( 1up7ar)tb
ei. d
y
o inc t e d
The e s t i :<Fa
agrees
however,
angle, phase mated
a,
"
"
"
-I
q u i t ew e l lw i t ht h ea c t u a lv a l u e .
a
5L
lead I
'
I
0
0
-
-100 L
.I
IO
I
Frequency ,
w
,
rod /sec
I f a time s h i f t were n o t u s e d i n
t h i s example, t h a t i s , i f X = 0 , t h e n
t h ee s t i m a t e dd e s c r i b i n gf u n c t i o n
would b e Yp(jw) = -l/Yc(jw) , shown by
6(b).
I t is intert h el i n ei nf i g u r e
e s t i n gt on o t et h a tw i t h
X = 0, the
v a l u eo ft h ec o n s t a n tf a c t o ri nt h e
errortermofequation(17)
is
Figure 6.- Concluded.
-ah
e
= 1. The v a l u eo ft h i sc o n s t a n t
f a c t o rw i t h
X = 0 . 2 s e c , as shown i n
-ah
= 0.37.
Thereequation (20), i s e
X = 0 . 2 secthemagnitude
f o r e ,f o r
oftheerrorterm
was reduced
approximately 63 p e r c e n t .
2
q
L
l
Figure 7 i l l u s t r a t e s t h e e f f e c t
of X on t h ei d e n t i f i c a t i o ne r r o r .
The control system and spectrum of the
ex
a
r ec ittha et i osame
n n oas
i s e i n $he s o u rprevlous
c e(T)
, =e
case.
-51T1
0
-
2 <>n
a 4
Yp(jw),Experimental
data,
"
"
"
I
Frequency,
i nf i g u r e7 ( a )f o r
10
rod / sec
X = . 0 5 sec and X = . 3 s e c
( a ) Describing
functions
for
Figure 7.- Effect of
example I; R,,,,(T)
w,
x
on identification;
= e-51TI.
a r e shown
X = 0 . 0 5 and 0 . 3
s eA
c .l s o
shown f o r comparison
are
Y (jwjandthe
t h ea c t u a ls y s t e m
n e g a t i v e i n v e r s e of ?&e c o n t r o l l e d
element - l / Y c ( j w ) .
These
experiment a l d a t ai l l u s t r a t et h ee f f e c to f
X
p r e d i c t e db yt h et h e o r y
ofequat i o( 1
n7).
For small v a l uoe fs
X, i n
15
,
15
:;""_""""
CaL
IO
t h i s case X = 0.05 SFC, t h ee s t i m a t e d
d e s c r i b i n g f u n c t i o n Yp ( j w) t e n d s
toward
-l/Yc(jw).
For
larger
values
o f X , i n t h i s case X = 0 . 3 s e c ,t h e
e s t i m a t e d e s c r i b i n fgu n c t i o n
is
"
"
0
0
0
n e aatrh
cetesruyasl tdeems c r i b i n g
function.
0
0
pclEr e s e n t e d
are
0
I
I
I
0
I
I
I
"
"
"
0
T
l
4'
<P
+-
-mm
-50-
7Y"Q-"""-.-
I
0
m
m
0
S
0
a
data
Experimental
i nf i g u r e7 ( b )f o rs e v e r a lv a l u e so f
I
I
I
X.
These r e sf auorlret s
value
one
of
frequency,
w = 1 r a d / s e c .I nt h i s
I
I
I
f i g u r e ,t h ee x p e r i m e n t a ld a t a
are comparedwiththevaluefortheactual
-18").
system ( l Y p l x 1 2 dB and 4 Yp
comparison
This
the
t ihlal u
t strates
identificationerrordecreases,
as p r e t ot h e o r y b
, y dicted
o fa v a l u e
Yp , Actuol
X = 0.3 s e c , which i s e q u at ol
T .
,
.
Yp , Theory
(The t h e o r y , as n opt er d
eviously,Pis
,Y.p . Experiment
v a loi ndfloyr
X 5 T ~ . ) The e r r o r
in lqPl
v a l u eos f
0
.I
I
I
I
.2
.3
.4
A , sec
T i m es h i f t ,
( b ) ldenliflcotionerrorfor
w
I
.5
= I rod/sec
,
;
i s s e e nt oi n c r e a s ef o r
h much l a r g e trh a n
T h i s iesxb pet o
e cbt e d
cause
c a n n o tp r o p e r l y modelYp(jw)
T
,P
Yp(jw)
with
X > T
P ' T h e s de a t a - i n d i c a t teh atth e
minimum e r r o r i n
yPl o c c u r s n e a r
I
F i g u r e 7 . - Concluded.
v a l u eo f
X
M T
P'
p h a s e a n g l e { Yp
v e r ys m a l l
at X M T
A s shown by t h i s example,thevalueof
n e atrhtei mde e l a tym
o i n i m i zteheer r oird e n t i f i c a t i o n .
,
a
The e r r o r i n t h e
is alsoseentobe
X shouldbe
We s h a l l now r e c a l l an i n t e r e s t i n g p o i n t p r e v i o u s l y d i s c u s s e d ( w i t h
e q .( 2 ) )t h a tc a n
now b er e i n f o r c e dw i t he x p e r i m e n t a ld a t a .T h a t
i s , t h em i n i mization o f t h e i d e n t i f i c a t i o n e r r o r ,
as we have shown above, i s n o t e q u i v a l e n t
t of i n d i n gt h e
minimum v a l u e f o r t h e s q u a r e d r e s i d u a l
~ ~ ( i nt t e) g r a t e do v e r a
The f o l l o w i n gt a b l ep r e s e n t st h ee x p e r i m e n t a l l yd e r i v e d
a givenrunlength.
v a l u e so ft h e
f i t t e r m/ E 2 ( t ) d t ,n o r / c 2 ( t ) d t , as
m a l i z e dw i t hr e s p e c tt o
Time s h i f t X ,
Normalized f i t term
a function X.
T h i st a b l e
shows t h a t
sec
JE2(t)dt/'c2Ct)dtthe
minimum v a l u ef o rt h e
f i t term i s
0
0.02
n o t a t X = T = 0 . 3 s e c ,b u tr a t h e r
.1
.19
i s a t X = 0.' This i s t ob ee x p e c t e d
.2
.25
becausewith
X = 0 then
.3
.27
i s essenYp(jw) = -l/Yc(jw)andthere
.4
.28
.31
t pi aelrclfoyercr te l a t i o n
between
.5
e ( t ) and c ( t ) .
16
The f a c t t h a t t h e minimum v a l u ef o r f i t term / s 2 ( t ) d ta p p e a r s a t X = 0
can l e a d t o an e r r o n e o u s i n t e r p r e t a t i o n i n s e l e c t i n g t h e b e s t v a l u e f o r
X.
For instance, i n p r e v i o u ss t u d i e s( r e f s .5 - 7 ) ,
X was s e l e c t e db yu s i n gt h a t
value whichgave t h eb e s tc o r r e l a t i o nb e t w e e n
e ( t ) and c ( t ) ( i . e . , t h em i n i mum v a l u ef o rJ s 2 ( t ) d t ) T
. h i sp r e v i o u s
method o fs e l e c t i n g
X i s unsatisf a c t o r y , however,because
as we h a v ej u s tn o t e d ,
i f n ( t ) >> i ( t ) , t h e nt h e
b e s tc o r r e l a t i o n
i s w i t h X = 0 and, i n t h i sc a s e ,q P ( j w )
= -l/Yc(jw).
An
a l t e r n a t e method o fs e l e c t i n g
Example 2:
w i l l b e i l l u s t r a t e d bythefollowingexample.
X
A Method o f S e l e c t i n g t h e
Time S h i f t
The previousexamplepointedoutthatthe
time s h i f t X s h o u l db en e a r
t h e time d e l a y T~ t o m i n i m i z et h ei d e n t i f i c a t i o ne r r o r .
The time d e l a y
may beapproximately known i n some s i t u a t i o n s ( e . g . ,
r e f . 3) b u t , i n g e n e r a l ,
i t s v a l u e will b e unknown and w i l l depend on t h e p a r t i c u l a r p i l o t i n g t a s k .
T~
This example i l l u s t r a t e s onemethod
o fs e l e c t i n g
X and w i l l c o n s i d e r
i d e n t i f i c a t i o n o f thesystem shown i nf i g u r e5 ( b ) .
The e x c i t a t i o nn o i s e
= e - 5 1 T 1 and i ( t ) = 0 ;
s o u r c e i s t h e same a su s e di nt h ep r e v i o u sc a s e ,R n n ( ~ )
however, d i f f e r e n t forms f o rt h ep i l o t
c o n t r o l l e de l e m e n t
modelYp(jw)
= 2(jw
+ l ) e - O a 5 j w and
1
Yc(jw) =
werechosen
t od e m o n s t r a t et h ei d e n 1)
t i f i c a t i o n o f more complexdynamics.
We will assume i n t h i s i l l u s t r a t i o n t h a t
Yp(jw) i s unknown. The o b j e c t i v e w i l l b et oe s t i m a t et h ev a l u eo f
and
TP
t h e nu s et h i sv a l u ef o r
A t oo b t a i n a b e s te s t i m a t e f o r q P ( j u ) .
jw(jw
+
was used t oe s t i m a t e
For t h i s example,thefollowingprocedure
t h u ss,e l e c t
X.
1.
P l o tt h ee s t i m a t e dd e s c r i b i n gf u n c t i o nf o r
2.
Determine a t r a n s f e rf u n c t i o nt h a tf i t st h ep l o t ,t h a t
-TpJ w
, etc.
Yp(jw) M (K1 + K2jo)e
3.
Note t h ev a l u eo fe s t i m a t e d
4.
Repeatsteps
1 through 3 u n t i l a v a l u eo f
T~
i s obtained.
thestimated
T~
a s e l e c t e dv a l u eo f
T
1'
and
X.
is,
from s t e p 2 .
X
a p p r o x i m a t e l ye q u a lt o
The e s t i m a t e d d e s c r i b i n g i u n c t i o n s f o r
example 2 a r e p r e s e n t e d i n
f i g u r e 8 f o r X = 0 . 2 and 0 . 4 s e c .A l s o
shown forcomparison i s t h ed e s c r i b X = 0 . 2 and
i n gf u n c t i o nf o r- l / Y c ( j w ) T
. h e s ee x p e r i m e n t a ld a t af o r
0 . 4 s e cf o l l o wt h et r e n d sp r e d i c t e db yt h e o r y .
The c u r v e f o r t h e h i g h v a l u e
of X ( a s compared t ot h ec u r v ef o rt h e
lowervaluesof
X ) t e n d s away from
-l/Yc(jw).
I t was f o u n df o rt h e s ed a t at h a t
any v a l u eo f
X between A = 0 . 3
s e c and 0 . 8 s e c r e s u l t e d i n a p p r o x i m a t e l y t h e
same d e s c r i b i n g f u n c t i o n as
shown f o r X = 0 . 4 s e c .T h i se s t i m a t e dd e s c r i b i n gf u n c t i o nc a nb e - a p p r o x i -TpJ w
form Yp(jw) NN (K1 + K2jw)e
matedby a t r a n s f e r f u n c t i o n o f t h e
. The
"
17
estimated T
v a l u efsr o m
fitting
P
this transfer function to the plots
are p r e s e n t e d i n f i g ( e . g . ,f i g .8 )
u r e 9 f o rs e v e r a lv a l u e so f
X.
It
i s s e e nt h a t
X i s e q u a lt ot h e
at X w 0.5
e s t i m a t e d time d e l a y , T
P’
sec. T h e r e f o r e , X = 0 . 5 s e cs h o u l d
be selected for use in this
example
I t i s seen
i d e n t i f i c a t i o na n a l y s i s .
that for this
example t h e method
works well f o r e s t i m a t i n g t h e a c t u a l
T
50 -
I
I
IO
I
Frequency ,
w
, rad /sec
Figure 8.- Identification of example 2.
P’
F i g u r e 10compares t h ee s t i m a t e d
d e s c r i b i n gf u n c t i o nu s i n g
X = 0.5
secwiththeactualdescribingfunct i o nf o r example 2 . Both t h e magnitudeandthephaseangleofthe
e s t i m a t e dd e s c r i b i n gf u n c t i o na r e
seen to be near these values for the
a c t u a ls y s t e m F
. o rt h i sc a s e w
, ith
X = 0 . 5 s e c ,t h et h e o r yo fe q u a t i o n ( 1 7 )p r e d i c t st h a tt h ei d e n t i f i will b e
cationerrorinmagnitude
e -ah -- 0.08. The e x p e r i m e n t adl a t a
shown i n f i g u r e 10 a p p e a r t o b e
w i t h i nt h i s8 - p e r c e n ti d e n t i f i c a t i o n
e r r o r as p r e d i c t e d b y t h e t h e o r y .
/
? ~ J w ~
X ,:
/
5 sec
/
/
t
/
50
-
I /’
/
.2
1
0
.2
I
1
I
I
.4
.6
T l m e shlft, X . sec
I
I
.8
I .o
-100 L
IO
.I
Frequency ,
Figure 9.- Comparison of estimated
example 2.
18
T~
with A ;
w
, rad / s e c
Figure 10.- Comparison of
example 2.
with Y
P
.
P’
Example 3:
F l i g h t Test R e s u l t s From Gemini X
F l i g h t d a t a fromGemini X were a n a l y z e d t o i l l u s t r a t e t h e a p p l i c a t i o n o f
t h i si d e n t i f i c a t i o nt e c h n i q u e .I na n a l y z i n gf l i g h t - t e s td a t a ,
it i s b e s t t o
select a s e c t i o n o f t h e r e c o r d t h a t c o n t a i n s d i s t u r b a n c e s e x t e r n a l t o t h e
p i l o t . As n o t e d e a r l i e r i nt h ed i s c u s s i o nf o l l o w i n ge q u a t i o n
(2), external
d i s t u r b a n c e s w i l l t e n dt or e d u c et h ee r r o ri ni d e n t i f i c a t i o n .
The r e t r o f i r e
maneuver i s a c a s e i n which e x t e r n a l d i s t u r b a n c e s are i n t r o d u c e d d u e t o t h e
unsymmetric r i p p l e f i r i n g o f t h e f o u r r e t r o r o c k e t s .
The r e l a t i o n s h i p o f t h e
p i l o t c o n t r o l t a s k , t h e j e t c o n t r o ls y s t e m ,a n dt h ed i s t u r b a n c e sd u r i n gr e t r o f i r e i s i l l u s t r a t e ds c h e m a t i c a l l yi nf i g u r e
11.
r-----1
PllOi
F i g u r e1 1 . -P i l o td e s c r i b i n gf u n c t i o na n df l i g h tc o n t r o ls y s t e m ;e x a m p l e
3.
During r e t r o f i r e , t h e p i l o t c o n t r o l s t h e a t t i t u d e a b o u t e a c h o f t h e t h r e e
axes.There
i s no c o n t r o lc o u p l i n gb e t w e e nt h e s ea x e s ,a n dt h ep i l o ta p p e a r s
t o t r e a t them a s t h r e e s e p a r a t e
tasks.
O f t h et h r e ea x e s ,t h ec o n t r o la b o u t
t h e yaw a x i s c o n t a i n e d t h e b e s t c o n s i s tentcorrelationbetweenattitude
d e v i a t i o n s , e ( t ) , and c o n t r o ls t i c k
d e f l e c t i o n s ,c ( t ) .
A time h i s t o r yo f
t h e r e c o r d e d yaw c o n t r o l d a t a i s p r e s e n t e di nf i g u r e
1 2 . Thesecontrol
data w i l l be used to illustrate the
measurementof
the pilot describing
"I I--2 sec
functionduringtheretrofireofthe
Gemini s p a c e c r a f t (. I ts h o u l db e
emphasized t h a t t h i s was a normal
r e t r o f i r e maneuverand t h e a s t r o n a u t
hadnopriorknowledgeofthis
i d e n t i f i c a t i o na t t e m p t . )
F i g u r e 12.- Time h i s t o r y of yaw c o n t r o l task
duringretrofire.
The p i l o t d e s c r i b i n g f u n c t i o n
obtainedforthedataoffigure
1 2 are
19
-Gp(I~l,
X=O
p r e sienC
n_otufe1
fridv3
g e.usr e
magnitude, Yp (jw)
I
1 , and phase angle,
{ q P ( j u ) , a r e p r e s e n t e d as f u n c t i o n s
<
f r e qf ouorefn c y
X = 0 and 0.6 s e c .
Also
shown
f
o
r
comparison
is the
“”’
d e s c r i b i n gf u n c t i o n 5f o r- l / Y c ( j w ) .
this
The os if g n i f i c a n c e
was
p r e vn ioot ue sdl y .
The
p r et hd ei cotrsy
for that
Xe s=t i0m, a t ehde
I
d e sf u
c rnicbtiinogn
9 (jw) w i l l t e n d
10 -
-3
L
”
”
”
”
_
P
line
m
D
._
;-10-
: /
/
-2oL
I
P
toward-l/Yc(jw)
as i l l u s t r a t e di n
f i g u r e1 3 .
However, f o rt h i sf l i g h t
s i t u a t i o n ,P p ( j w )d o e s
not coincide
r u m
”
0
u 3
-50
- l e/w
Y
x
b iaectc(hjtaw
luys) e
d i s teuxrtbearnncael s
the
lo
firing
otfh ree t r o r o c k e t s
c o n t r o ls y s t e m .
”
”
- I00
I
.I
Frequency,
Y
, rad /sec
of
due t o
and j e t
Figure 13.- Identification of pilot describing
function; example 3.
For X = 0.6 s e c ,t h ee s t i m a t e d
d e s c r i b i n gf u n c t i o nv p ( j w t) e n d s
away from t h ec u r v eo -f l / Y c ( j w ) .
Any v a l u eo f
X from
about
X = 0.3 t o
0 . 7 s e cr e s u l t e di na p p r o x i m a t e l yt h e
same d e s c r i b i n g f u n c t i o n as shown f o r
X = 0 . 6 set. Thisestimateddescribingfunctioncanbeapproximated
by a
- T jw
a constang
t a i na n d
a t i m ed e l a y ,
Yp(jw) w Ke
.
t r a n s f e rf u n c t i o nw i t h
1.0 -
.8
-
”
a,
.6-
v,
a
-
u
0
,E
.4-
W
I
0
.2
I
I
.4
.6
Tlme shlft, X , sec
Figure 14.- Comparison of estimated
X ; example 3.
-
I
I
.8
~~
I
T~
.o
with
As n o t e d p r e v i o u s l y , t h e v a l u e
of A t h a t w i l l m i n i m i z et h ei d e n t i f i c a t i o n e r r o r i s dependent on t h e
effectivetimedelayofthepilot,
F o rt h e s ed a t a ,t h ep r o c e d u r e
k s c r i b e d p r e v i o u s l y was used t o
X.
determine -rp a n dt ,h u ss,e l e c t
With t h i sp r . o c e d u r e , T~ was e s t i matedby f i t t i n g t h e t r a n s f e r f u n c tiontotheplotsforseveralvalues
of A . These r e s u l t sa r ei l l u s t r a t e d
i n f i g u r e 1 4 where t h e e s t i m a t e d
TP
as a f u n c t i o n
v a l u e sa r ep r e s e n t e d
I t i s s e e tnh a t
X i s equal
of X .
totheestimatedtimedelay,
TP’
a t X z 0 . 6 s e c .T h e r e f o r e ,
X = 0.6
s e c was s e l e c t e d f o r u s e i n t h i s
i d e n t i f i c a t i o na n a l y s i s .
One p r o m i s i n g f e a t u r e i n a n a l y z is that the estiingtheflightdata
mated d e s c r i b i n g f u n c t i o n s a r e
20
I
~
5The d e s c r i b i n gf u n c t i o nf o rt h ej e tc o n t r o ls y s t e m
e s t i m a t e df r o mt h ef l i g h td a t a .
Yc(jw) was
~
r e l a t i v e l yi n s e n s i t i v et ot h ee x a c tv a l u eo ft h e
time s h i f t , A .
F o rt h e
e s t i m a t e dp i l o td e s c r i b i n gf u n c t i o n s( e . g . ,f i g .1 3 ) ,t h ep l o t s
are approxim a t e l yt h e same f o rv a l u e so f
X fromabout0.3
t o 0 . 7 sec ( t h ee s t i m a t e d
T
remainedthe same a t about 0 . 6 s e c ) . I t a p p e a r s t h a t t h e e x a c t v a l u e u s e d f o g
A i s n o t c r i t i c a l i nt h i sa p p l i c a t i o no ft h ei d e n t i f i c a t i o nt e c h n i q u e .
The e s t i m a t e dd e s c r i b i n gf u n c t i o nf o r
X = 0 . 6 sec ( f i g .1 3 )r e p r e s e n t s
a c o n s t a n tg a i ns y s t e mw i t h
an e f f e c t i v e time d e l a y .T h i sr e s u l t ,a l t h o u g h
notdirectlycomparabletotheresults
f r o mp r e v i o u ss t u d i e s ,a p p e a r sr e a s o n a b l e .F o ri n s t a n c e ,w i t h
a r a t e command system,whichapproximatesthecontrol
1 has shown t h a t t h e p i l o t d e s c r i b i n g
systemusedinthissituation,reference
f u n c t i o n w i l l b e e s s e n t i a l l y a c o n s t a n tg a i ns y s t e mw i t h
a time d e l a y . The
v a l u e T~ f o rt h et h r e e - a x i sf l i g h td a t a
i s h i g h e rt h a nt h ev a l u ef r o mt h e
1. However, o t h e rs t u d i e ss u c h
as r e f e r e n c e1 3
s i n g l e - a x i sd a t ai nr e f e r e n c e
have a l s o shown h i g h e rv a l u e s of -rP when t h e p i l o t i s i n v o l v e di nt h e
comp l e t e t a s k o f m o n i t o r i n gt h ei n s t r u m e n tp a n e la n dc o n t r o l l i n ga b o u tt h r e e
s e p a r a t ea x e s .
F u r t h e ra n a l y s i so ft h i sf l i g h td a t a
i s p r e s e n t e di na p p e n d i x
B . This
a p p e n d i x i l l u s t r a t e s how t h e d e s c r i b i n g f u n c t i o n o f t h e p i l o t / c o n t r o l
combinat i o n c a nb ei d e n t i f i e du s i n gt h et e c h n i q u eo u t l i n e di nt h i sr e p o r t .T h i s
i l l u s t r a t i o n i s i n t e r e s t i n gb e c a u s e i t p r e s e n t s t h e i d e n t i f i c a t i o n o f
an
i t s own i n t e r n a l n o i s e
unknown s y s t e m( i . e . ,p i l o t / c o n t r o ls y s t e m )u s i n go n l y
sourceforexcitation.
CONCLUDING REMARKS
T h i sr e p o r th a s
shown t h a t i n m e a s u r i n g p i l o t d e s c r i b i n g f u n c t i o n s t h e
i d e n t i f i c a t i o n e r r o r due t o t h e c o r r e l a t i o n o f t h e i n p u t s i g n a l w i t h t h e
pilotoutputnoise
canbereduced
by s h i f t i n g t h e i n p u t d a t a d u r i n g t h e
p u t e r p r o c e s s i n g by an amount e q u i v a l e n t t o t h e p i l o t t i m e d e l a y .
com-
Both t h e o r y and experimentaldatahave
shown t h a t t h e i d e n t i f i c a t i o n
e r r o r canbe made small i f t h e a u t o c o r r e l a t i o n f u n c t i o n ,
Rnn(T), o f t h e
i n t e r n a ln o i s es o u r c e( p i l o tr e m n a n t )
i s n e g l i g i b l ef o r
T
g r e a t e rt h a nt h e
sum o f a l l e f f e c t i v et i m ed e l a y st h r o u g ht h ec o n t r o ll o o p( p i l o tp l u sc o n t r o l l e de l e m e n t ) .T h i sf i n d i n gh a ss i g n i f i c a n c ei ng e n e r a ls y s t e m si d e n t i f i c a t i o nb e c a u s e , when t h e s e c o n d i t i o n s are met, i t i s p o s s i b l e t o measure t h e
d e s c r i b i n gf u n c t i o no f
a systemwithfeedbackusingonly
i t s own i n t e r n a l
noisesourceforexcitation.
Representativedataselected
from t h e r e t r o f i r e p o r t i o n o f t h e
f l i g h t w e r ea n a l y z e du s i n gt h et e c h n i q u eo u t l i n e di nt h i sr e p o r t .T h e s e
resultsdemonstratethefeasibilityofidentifyingthepilotdescribing
functionfromroutineflight-testrecords.
Gemini X
Ames ResearchCenter
NationalAeronauticsandSpaceAdministration
M o f f e t tF i e l d ,
C a l i f . , 94035,Jan.15,1969
125-19-01-42-00-21
21
APPENDIX A
TIME DOMAIN ANALYSIS AND COMPUTER PROCESSING
we w i l l first u s e time domain a n a l y s i s t o o u t l i n e t h e
I nt h i sa p p e n d i x ,
s t a n d a r dc r o s s - c o r r e l a t i o n
method ( r e f s . 4 and 8 ) . W
e s h a l lt h e ni n t r o d u c e
X and t h e c o m p u t e rp r o c e s s i n ge q u a t i o n su s e df o rt h er e s u l t s
t h et i m es h i f t
i nt h i sr e p o r t .
The r e d u c t i o ni ni d e n t i f i c a t i o ne r r o rd u et o
X w i l l then
b ep o i n t e do u tu s i n gt h e s e
time domain e q u a t i o n s . And, f i n a l l y , we w i l l
discuss a modificationofthese
c o m p u t e rp r o c e s s i n ge q u a t i o n st oa c c o u n tf o r
any d a t a b i a s .
Standard Cross-Correlation
Method
Let u s c o n s i d e r a n a l y s i s i n t h e
time domain i n which t h e l i n e a r i n p u t o u t p u t r e l a t i o n s h i p can b e e x p r e s s e d i n
terms o f a c o n v o l u t i o n i n t e g r a l
is, assumed t o bezero
The hp(T) i s t h e p i l o t i m p u l s e r e s p o n s e f u n c t i o n t h a t
T > t, ( i . e . , a
f o r T < 0 ( i . e . , hp ( T ) i s a r e a ls y s t e m )a n da l s oz e r of o r
f i n i t e memory time, b). A s i m p l ed i s c r e t ea p p r o x i m a t i o no fe q u a t i o n( A l ) ,
t oa l l o wd i g i t a lc o m p u t a t i o n ,
is
c (k) = A t
f
hp (m)e (k - m) + n (k)
m= 0
The s e t of e q u a t i o n s (A2) canbe
where A t i s t h ed i s c r e t es a m p l i n gt i m e .
w r i t t e n i n v e c t o r - m a t r i x form as
where
e(ko - 1)
+
. . .
e(ko
.. .
e(K
1)
E = A
'
22
-
(A2 1
hp
--
An e s t i m a t e o f
f ormu 1a
c =
-
h
“I?
c a nt h e nb e
n =
-
made, u s i n gs t a n d a r d
l e a s t s q u a r e s , by t h e
We s h o u l d p o i n t o u t t h a t t h e m a t r i x t o b e i n v e r t e d ,
ETE , c o n t a i n st e r m s
t h a tr e p r e s e n td i s c r e t e
m e a s u r e m e n t so ft h ea u t o c o r r e l a t i o nf u n c t i o nR e e ( ~ ) ,
and t h a tt h ev e c t o r
ET c c o n t a i n st e r m st h a tr e p r e s e n td i s c r e t e
measurements
o tfh ec r o s s - c o r r e l a t i o nf u n c t i o nR e c ( T ) .
F o r i n s t a n c e t, h ev e c t o r
ET,
can
b ew r i t t e ni nt e r m so ft h ec r o s s - c o r r e l a t i o nf u n c t i o na s
/
L
k=ko
T
Ec =
23
Use of a Time S h i f t i n Computer P r o c e s s i n g
The time s h i f t X i s i n t r o d u c e di n t ot h ec o m p u t e rp r o c e s s i n gb ys h i f t i n g
a d i s c r e t e number o ft i m es h i f t s
L , where X = L A t .
t h ei n p u td a t ae ( k )
The l i n e a r p i l o t model is t h e n e x p r e s s e d as
c(k) = A t
f
(A6 1
hp(L + m)e(k - m - L) + n ( k )
m= o
Thisformassumesthattheimpulseresponse
hp i s z e r of o rt i m el e s st h a n
A.
This form a l s o assumes a memory time o f t m = h + M A t .
For t h er e s u l t s
i nt h i sr e p o r t ,
M = 9 and A t = 0 . 0 5 s e c (. L a r g e rv a l u e so f
M were a l s o
t r i e d w i t h no s i g n i f i c a n t changes i n t h e r e s u l t s . )
Using t h el e a s t - s q u a r e sf o r m u l a t i o n ,t h ei m p u l s er e s p o n s ef u n c t i o no ft h e
p i l o t was determined by the following
matrix i n v e r s i o n on a d i g i t a l c o m p u t e r :
-
f
eZ(k-L)
...
e(k-L)e(k-1-L)
e(k-1-L)e(k-L)
e(k-L)c(k)
e(k-L)e(k-M-L)
K
e2(k-1-L)
.
. .
e(k-M-L)e(k-1-L)
,
..
k=ko
L
k=ko
c
k=ko
2
f
1
k=ko
k=ko
K
f
k=ko
f
d
k=ko
e(k-1-L)e(k-M-L)
e(k-1-L)c(k)
= (At)-
k=ko
e(k-M-L)e(k-L)
k=ko
e2 (k-M-L)
f
e(k-M-L)c(k)
k=ko
L
This time domain s o l u t i o n was f u r t h e r t r a n s f o r m e d i n t o t h e f r e q u e n c y
u s i n gt h ef o l l o w i n ga p p r o x i m a t i o nf o rt h eF o u r i e rt r a n s f o r m :
1
domain
M
A
Yp(jw) = e
-L A t j w A t
ReductioninIdentification
G p ( L + m)e -m A t j w
E r r o r With Time S h i f t
I no r d e rt o
show, u s i n g time domain a n a l y s i s , t h a t t h e t i m e s h i f t
r e d u c e st h ei d e n t i f i c a t i o ne r r o r ,
we can write e q u a t i o n (A4) as
h
kp =
kp
-
+ [ETE] - ET,-
error
24
X
The i d e n t i f i c a t i o n e r r o r , shown above i n v e c t o r form, i s due t o t h e c o r r e l a t i o no fe ( k )w i t h
n (k)
The t e r m si nt h ev e c t o r
ETn c a nb er e g a r d e d
as
d i s c r e t ev a l u e s of t h ec r o s s - c o r r e l a t i o nf u n c t i o n
Re;(.).
.
T
E n =
M
f
e ( k - M)n(k)
k=ko
If R e n ( ~ )i s nonzero f o r T > 0 , t h e nt h et e r m s
z e r o and t h e r e w i l l b e an i d e n t i f i c a t i o n e r r o r .
Ren(m), m 2 0 , w i l l b e non-
Now, i n t r o d u c i n g a d i s c r e t e number of t i m e s h i f t s
equation (A7), t h ev e c t o r
ETn
- becomes
-
L, such as u s e di n
e (k - L)n (k)
<=ko
f
e ( k - 1 - L)n(k)
k=ko
T
E n =
M
e(k - M
-
L)n(k)
k=ko
-J I
We can see t h a t t h i s u s e o f t h e
time s h i f t removes t h et e r m s
Ren(m),
0 2 m < L , andadds t h e t e r m s
Ren(m), M < m < M + L.
The terms t h a t are
25
added. g e n e r a l l y are smaller t h a n t h e terms removed; t h u s , t h e u s e o f
a time
s h i f t L, o r t h ee q u i v a l e n t
X , r e d u c e st h ei d e n t i f i c a t i o ne r r o r .
(Time s h i f t i n g w i l l n o t s i g n i f i c a n t l y a l t e r t h em a t r i x
ETE.)
F u r t h e rn o t et h a tt h e
m 2 L.
i d e n t i f i c a t i o n e r r o r w i l l b ez e r o i f Ren(m) i s z e r o f o r v a l u e s o f
Computer P r o c e s s i n g With Data Bias
The d a t ac ( k )
and e ( k )o b t a i n e df r o mr o u t i n ef l i g h t
tests w i l l usually
c o n t a i n some t y p e o f l o n g - t e r m v a r i a t i o n a b o u t
which t h e s h o r t - p e r i o d dynamics
12.)
a r et ob ee s t i m a t e d .( S e e ,f o ri n s t a n c e ,t h ef l i g h t - t e s td a t ai nf i g .
There may b e a b i a s o r d r i f ti ne ( k )b e c a u s eo fi n s t r u m e n t a t i o ne r r o r s
or
b e c a u s et h ep i l o t
may b ec o n t r o l l i n ga b o u t
some nonzerovalue.There
may be
b i a si nc ( k )b e c a u s eo fi n s t r u m e n t a t i o ne r r o r s
o r b e c a u s et h ep i l o t
must h o l d
i f t h e v e h i c l e i s o u to ft r i m ) .
some n o n z e r o o f f s e t w i t h t h e c o n t r o l l e r ( e . g . ,
Intheanalysisoftheflight-testrecordsforthisreport,
a b i a s t e r m and a
d r i f t t e r m wereadded t o t h e f o r e g o i n g f o r m u l a t i o n
so that
c ( k ) = bo + b l k A t + A t
c
hp(L + m)e(k - L - m) + n ( k )
m= o
n
h
The e s t i m a t e dc o n s t a n tb i a s
term b oa n de s t i m a t e dd r i f t
term b l ,a l o n g
w i t ht h ee s t i m a t e di m p u l s er e s p o n s ef u n c t i o n
G,(m) , were determinedbythe
l e a s t - s q u a r e ss o l u t i o n .T h i s
new f o r m u l a t i o nr e q u i r e dt h ei n v e r s i o no f
an
M + 3 s q u a r em a t r i xi n s t e a do ft h e
M + 1 s q u a r e m a t r i x shown p r e v i o u s l y i n
equation (A7).
26
APPENDIX B
IDENTIFICATION OF THE PILOT/CONTROL DESCRIBING FUNCTION
The d e s c r i b i n g f u n c t i o n o f t h e p i l o t / c o n t r o l c o m b i n a t i o n h a s b e e n u s e d
e x t e n s i v e l y( e . g . ,
r e f . 1) t o a n a l y z e t h e c l o s e d - l o o p
dynamicsofpiloted
s y s t e m s .T h i sa p p e n d i xi l l u s t r a t e st h eu s eo ft h et e c h n i q u eo u t l i n e di nt h i s
r e p o r tt oi d e n t i f yt h ep i l o t / c o n t r o ld e s c r i b i n gf u n c t i o n .T h i si l l u s t r a t i o n
w i l l u s e t h e Gemini d a t a p r e s e n t e d p r e v i o u s l y i n f i g u r e
12.
The d e s c r i b i n gf u n c t i o nq X ( j u )t ob ei d e n t i f i e d
t ot h ep i l o t
and t h ec o n t r o ls y s t e mi nf i g u r e
15.
i s shown i n r e l a t i o n s h i p
T h i sd e s c r i b i n gf u n c t i o n
PllOt
r-----l
I
I
I
L - - - - - J
h
Estlmote Y,
"
"
"
7
F i g u r e 1 5 . - T h et e c h n i q u ef o ri d e n t i f y i n gt h ep i l o t / c o n t r o ld e s c r i b i n gf u n c t i o n
1
was measuredbetween
t h ea t t i t u d ee r r o rs i g n a le ( t )
and t h e a t t i t u d e r a t e
s i g n a l$ ( t ) .
The Bode p l o t so b t a i n e d
from t h e s ed a t aa r ep r e s e n t e di n
f i g u r e1 6 .C u r v e so ft h ee s t i m a t e dd e s c r i b i n gf u n c t i o n
Yx(jw) a r e shown f o r
h = 0 and 0 . 7 s e c . Also shown f o r comparison i s t h e Bode p l o tf o rt h en e g a t i v ei n v e r s eo ft h ef e e d b a c kp a t h ,
-jw.
The t h e o r yp r e d i c t st h a t ,f o r
X = 0,
t h ee s t i m a t e dd e s c r i b i n gf u n c t i o n
will i d e n t i f y - j wb e c a u s e ,i nt h i s
case,
t- h. e- r ea r e no d i s t u r b a n c e se x t e." r n a...
lt o.t h e
measured
d e s c r i b i n gf u n c t i o n .
As
.
1It i s i n t e r e s t i n g t o o b s e r v e t h a t
the identi.ficationofthe
pilot/controldescribingfunctionrequiredonly
a s i n g l e c h a n n e l of r e c o r d e d
d a t a . The o u t p u ts i g n a lu s e di nt h e
computerprocessing was t h er e c o r d e d yaw
r a t e ,$ ( t ) .
The i n p u ts i g n a l was a l s od e t e r m i n e df r o mt h i s
same recorded
d a t a as
i - . . _ - .
e(t) =
~
~
-s
t
$(T)dT + b i a s + d r i f t
0
where t h e unknown b i a s and d r i f t a r e a c c o u n t e d f o r b y t h e
appendix A.
method shown i n
27
shown i n f i g u r e 1 6 , t h e c u r v e f o r
X = 0 e s s e n t i a l l yc o i n c i d e sw i t ht h e
c u r v feo r
-jw.
F o ri n c r e a s i n gv a l u e so f
X, the
estimateddescribingfunctiontends
- j w . Any
away from t h ec u r v eo f
v a l u eo f
A between 0 . 4 and 1.0 sec
resultedinapproximatelythe
same
Bode p l o t as shown f o r X = 0 . 7 s e c .
The e s t i m a t e d d e s c r i b i n g f u n c t i o n
can
beapproximatedby
a describingfunca c o n s t a n tg a i n
K,
and a
t i o nw i t h
,.
Figure 16.- Identification of pilot/control
describing function.
/
1.0 -
/
/
time d e l a y -rx; Yx(jw) = x , - T x j w
The e s t i m a t e dv a l u e sf o r
T~
are
p r e s e n t e di nf i g u r e
17 f o r s e v e r a l
X.
W
e can s e teh a t
X
v a l u eosf
e q u a tl ot h ee s t i m a t e d
T~
at
X M 0 . 7s e c ,
s o X = 0 . 7 s e c was
selectedforthisanalysis.
is
,.
From t h ee s t i m a t eY x ( j w ) ,
we
c a nd e t e r m i n et h ed e s c r i b i n gf u n c t i o n
forpi!o;/controlledelementcombinat i o n , YpYc(jw). The d e s c r i b i n gf u n c ,.
t i o n beY,
can
combined
the with
i n t e g r a t i o n , l / j w , as shown i n
foi g
bt1o
u
t a5r i,en
.8-
qPqc (jw)
"
W
.6
/
-
-
73
W
For t h e r e s u l t s from f i g u r e1 6 ,t h e
is
estimateddescribingfunction
/
0
i
jw
/
c
.4 -
/
W
-0.7jw
/
qX(ju) = 1.3e
/
i.P?C (jw)
0
.2
.4
.6
Tlme shtft, X . sec
Figure 17.- Comparison of estimated
o fc o n t r o l l e de l e m e n t st h ep i l o t
Px ( j w>
=
.a
T~
w
so that
1.3e-0.7jw
jw
I .o
Tr he isasup lpter ae rass o n a b l e .
From
p r e v i o u ss t u d i e ss u c h
as r e f e r e n c e 1,
with a .
it hasbeen shown t h a t f o r a v a r i e t y
e-~xjw
X
w i l l control so t h a t Y Y (jw)m
PC
j w
The a c t u a lv a l u e
Thisform i s t h e same as f o u n di nt h ea c t u a lf l i g h tr e s u l t s .
K,,
of 1 . 3r a d / s e c )
i s lowerthanpref o rt h eg a i n( ac r o s s o v e rf r e q u e n c y ,
dicted in reference 1 andthevaluefortheeffective
time d e l a y (-rXM 0 . 7 s e c )
28
I II
I
II I
I I 111
i s h i g h e rt h a np r e d i c t e di nr e f e r e n c e
1. Again, i t is r e a s o n a b l et oe x p e c t
( s e er e f s .
10and13)
thatthesedifferencescanbeattributedtothefact
thatinreference
1, t h e p i l o t was c o n t r o l l i n g o n l y a s i m p l e s i n g l e - a x i s t a s k ,
whereas,fortheactualflightdata,thepilot
was c o n t r o l l i n g a b o u t t h r e e
axesandmonitoringthecompleteinstrumentpanel.
29
REFERENCES
1.
2.
Newell,Fred
D . ; and P i e t r z a k ,P a u l
E.:
I n - F l i g h t Measurement of Human
.n
, o3
.,
May-June 1968,
Response C h a r a c t e r i s t i c s . J . A i r c r a f t , v o l 5
pp.
277-284.
3.
Henry R . :
A Review o fQ u a s i - L i n e a rP i l o t
McRuer, Duane T . ; andJex,
HFE-8,
Models. I E E E T r a n s a c t i o n s on Human F a c t o r si nE l e c t r o n i c s ,v o l .
no. 3,Sept.1967,pp.231-249.
4.
Elkind,Jerome
I . : F u r t h e rS t u d i e s o f M u l t i p l eR e g r e s s i o nA n a l y s i so f
Human P i l o t Dynamic Responses; A Comparison o f AnalysisTechniquesand
EvaluationofTime-VaryingMeasurements.
ASD-TDR-63-618, March 1964.
5.
Elkind,Jerome
I . ; S t a r r , Edward A . ; Green,David M . ; andDarley, D .
L u c i l l e :E v a l u a t i o no f
a Technique f o r DeterminingTime-Invariantand
Time-Variant Dynamic C h a r a c t e r i s t i c s of Human P i l o t s . NASA TN D-1897,
1963.
6.
Todosiev, E . P . ; Rose, R . E . ; Bekey, G . A . ;
TrackingPerformanceinUncoupledandCoupled
CR- 532 , 1966.
7.
Wierwille, Walter W . ; and Gagne, G i l b e r t A . :
A Theory f o r t h e Optimal
D e t e r m i n i s t i c C h a r a c t e r i z a t i o n o f t h e Time-Varying Dynamics o f t h e
Human O p e r a t o r . NASA CR-170, 1965.
8.
i nt h e
T a y l o r , Lawrence W . , J r . : A Comparison o f Human ResponseModeling
Time andFrequency
Domains. Proceedings o f t h e 3 r d
Annual NASAUniversity Conference on Manual C o n t r o l . NASA SP-144,1967,
pp.
137-153.
9.
J. B.:
Determination o f SystemCharacterGoodman, T . P . ; andReswick,
i s t i c s from Normal OperatingRecords.Transactions
o f t h e ASME,
v o l . 78,
pp.
259-268,
Feb.
1956.
and Williams, H. L . :
Human
Two-Axis Systems. NASA
10.
Adams, James J . ; Bergeron, Hugh P . ; andHurt,George
J., Jr.:
TransferFunctionsinMulti-AxisandMulti-LoopControlSystems.
NASA TN D-3305, 1966.
11.
Newton, George C . ; Gould,Leonard
A . ; andKaiser,James
Design of L i n e a r Feedback Controls.JohnWileyandSons,
30
F.:
Human
Analytical
N . Y . , 1957.
12. McDonnell, J. D.; and Jex, H. R.: A "Critical" Tracking Task f o r ManMachine Research Relatedto the Operators Effective Time Delay.
NASA CR-674, 1967.
13. Clement, W. F.; Jex, H. R . ; and G r a h a m , D.: Application of a System
Analysis Theory f o r Manual Control Displays to Aircraft Instrument
Landing. Paper presented at 4th Annual NASA-University Conference on
Manual Control, Univ.of Mich., March 1968.
NASA-Langley, 1969
-5 A-
~
~
~
2993
"~
31