The design of a facility for the measurement of human pilot dynamics

THE DESIGN OF A FACILITY FOR THE MEASUREMENT OF HUMAN PILOT DYNAMIC S

by

Lloyd Reid

..

",

...

THE DESIGN OF A F ACILITY FOR THE MEASUREMENT OF HUMAN PILOT DYNAMICS

by

Lloyd Reid

ACKNOWLEDGEMENT

The author wishes to express his thanks to his supervisor, Professor B. Etkin, for his helpful direction in preparing this report.

The author is indebted to the directox ... and staff of UTIAS for making this work possible. Thanks also go to the Royal Canadian Air Force and the Defence Researoh Board of Canada for providing the CF-100 simulator.

Financial support for the work was provided by the National Research Council of Canada and by the United States Air Force under con-tract AF-AFOSR-222-64, the latter rnonitored by the Air Force Office of Scientific Research, Applied Mathematics Branch.

SUMMARY

This report describes the modification of the UTIAS CF-IOO flight simulator and the development of a data analysis technique in order to study human operators in arealistic flight environment. The operator forms part of a closed loop system which may consist of one or two degrees of free-dom. A method of analyzing data obtained from short record runs is presented which is similar to the weU known cross-correlation, cross-power spectral density method.

T ABLE OF CONTENTS

NOTATION 1. INTRObuCTION

1. 1 Some Aspects of Human Operator Studies 1.2 A Brief Survey of Human Operator Studies 2. MATHEMATICAL BASIS FQR DATA REDUCTION

3.

4.

2. 1 Linear Theory

2.2 Application to Nonlinear Closed-Loop Systems SIMULATION

3. 1 A Brief Description of the CF-I00 Simulator 3. 2 Cockpit Environment

3.3 Motion and Visual Cues 3. 4 Simulated Task

3.5 Differences Between Actual and Simulated Flight EQUIPMENT

4. 1 Random Input Signal Generator 4. 2 Visual Display 4. 3 Control System 4.4 Aircraft Dynamics 4. 5 Monitoring Display 4.6 Recording System 4.7 Data Analysis 5. DATA REDUCTION

6. CONCL USIONS AND RECOMMENDA TIONS 6. 1 Aircraft Dynamics

6. 2 Random Input Source 6.3 BataReduction REFERENCES .APPENDIX A APPENDIX B TABLE 1 TABLE 2 FIGURES 1 - 29 Page v 1 1 2 3 4 7 12 12 12 13 13 13 14 14 14 15 16 17 17 17 21 23 23 24 24 25 31 33 34 34

A(j

w)

b (t) B (j

w)

Bw c (t) C(j W) d(t) D(j w) eet) f(t) Fa Fe g(t) G(j w) het) iet) j Kp m -

..

_Mxy(T J 1.' )

..

NOTATION

aircraft transfer function

impulsive response of system represented by B (j

W)

transfer function relating p (t) to n Ct).

input signal bandwidth in cps.

impulsive response of system represented by C(j w) transfer function re lating o(t) to iet) when net)

=

0 impulsive response of system represented by D(j

W )

transfer function relating eet) to net) when iet) = 0

error signal displayed to pilot (degrees of attitude error) x(t) - m

pilot's stick force output in roU (pounds) pilot's stick force output in pitch (pounds)

impulsive response of system represented by G{j w) transfer function relating y(t) to x(t).

y(t) - r

random input to pilot-aircraft system (degrees)

pilot' s D C gain

the mean value of x(t)

.

s

x(t) y(t, +'t') d t where x (t) and y (t) are zero outside -0)

n(t) the pilot renlllant Nxy(T, jw)

cr

-j

w't'

) e . _{Mxy (T,} 't' ) d't' -(X) o (t) pilot output

P (t) aircraft input caused by n (t)

r the mean value of h (t)

R xy ('t') the cross-correlation between x (t) and y (t)

s the Laplace variable

t time in seconds

T record length in seconds

TI time constant of pilot model equalizer lag term (seconds) TL time constant of pilot model equalizer lead term (seconds) T N time constant of pilot model neuro-muscular lag term (secs. )

u, v variables of integration

x (t) input to system represented by G(j

w )

Y (t) output of system represented by G(j w)

~(t) impulsive response of pilot describing function Y(j

w)

Y(j w) pilot describing function

~(

CA) ) Dirac delta function

~a

aileron deflection (degrees)

J

&e \ . elevator deflection (degrees) /:::"a stick deflection in roU (inches)

..

6.e stick deflection in pitCh (inches)

9 pitch angle (degrees)

~

4>

q)xy(jW)

W

)1 (

_»2

( )* ( - )

time delay (seconds) roll angie (degrees)

cross-power spectral density of x (t) and y (t) frequency (radians / second)

variables related to minimizing the integral square of n (t) variables related to minimizing the integral square of p (t) variables altered by D. C. drift in the electronics

• i

1. INT:f~ODUCTION

In any control system involving man, the man represents the most complex and least understood element. This is due to manIs nonlinear nature as a control element and his wide range of adaptability, the very traits which gain him such a dominant position in many control situations .

In order to obtain a better understanding of how humans perform

and. what constitutes a good control system from the operator' s point of view,

engineering approximations have been developed to describe the human oper-ator under certain conditions. This work was initiated during World War II

by Tustin and Phillips. Progress in this field has been limited by the

comp-lexity of the human operator. The most significant investigations so far have been in the area which involves the operator in a single degree of freedom compensatory control situation* with random-appearing visual inputs. In

this case it has been found that the human operator can generally be

repres-ented by a linear differential equation which will account for most of his output.

As a result, most studies have dealt with this case. It should be nbted that

this is an engineering approximation intended to bring out the average effects

of the human operator in the control loop and it does not necessarily ~ead to a ~

point by point agreement between the actual system response and the ~odel

system which has the operator replaced by the differential equation repre-senting the human.

The purpose of the present study is to modify the UTIAS flight

simulator f.or human operator studies and to investigate a means for analyzing the data obtained in human operator studies.

1. 1 Some Aspects of Human Operator Studies

In recent years, most of th.e work in the field of human operator

studies has been done in order to study pilot/ aircraft systems. It is fortunate

that the compensatory control task is approximated by many in-flight situations

such a::; attitude stabilization, landing approach and almost any control task

performed in gusty air. 1. It has been found that the most suitable input signal

to employ while studying human operators is one that appears to be random. This eliminates the added complexity of pattern recognition and prediction which human operators are capable of in situations where random signals àre not employed.

* In a compensatory control system, the operator is shown the system error

only,' th at is, the difference between the desired and actual system output.

The model which has been developed to approximate the human operator in many situations is shown in figure (2). This particular model was taken from reference (2) and is representative of the majority of models de-veloped in the past twenty years. The term e -'t's represents the computation and signal transmission time delay within the operator and is fairly constant for any given operator. 1:'" has been found to be in the range. 1--+.3 seconds. The term K' TL s

+ 1

represents that part of the human operator which

p _{TI s}

_{+ 1}

the operator can vary to suit the control s~tuation. The model is called q.lasi

-linear because the parameters Kp, TL and TI can vary as a function of time. The final block 1 in a crude way represents the human operator's

TN s +1

neuro-muscular actuation system and like 'l: , TN is fairly constant for a given operator. The addition of a signal called the remnant to the output of the linear blocks represents that part of the human operator' s output which cannot be described by a linear operation on its input. The linear portion of the model is called the describing function.

It has been found that the human operator alters his describing function to suit the system input; the manipulator employed, and the particular controlled element. The operator's describing function can also be influenced by less well defined things such as motivation, learning, additional inputs and fatigue and these result in a describing function with short term and long term variations. Thus, although attempts are made to control these influences on the operator, the describing functions found must be thought of as time averaged descriptions ef the operator, the time length depending upon the method of data analysis employed.

1. 2 A Brief Survey of Human Operator Studies

For a fuller picture of this field see references (3-5).

To date there has been a fairly complete study of the effects of the controlled element dynamics and input bandwidth upon the human operator's describing function in the single degree of freedom compensatory tracking task. Some of these projects also included handling qualities investigations. Much of this work is reported in references (1, 2, 6-10) while references (11, 12) consider the special case of nonlinear controlled dynamics. References (13-17)

investigate the time variabIe properties of the human operator describing function to ga in some insight into what an individual's parameter variation amounts to over an extended period of time. With the single degree of freedom case fairly well in hand, interest is now turning to the application of th is straight forward method to multi-loop configurations. 1S- It must be determined whether a more complex model for the piiot is required when the operator's attention must be divided among several inputs. Although the describing function approach

is attractive because it gives systems engineers a simple means 'Of analyzing c'Ontr'Ol systems which include a human 'Operat'Or, it is generally felt th at a m'Ore accurate descripti'On 'Of the human 'Operat'Or can be achieved by p'Ostu-lating a m'Ore c'Omplex n'Onlinear m'Odel. S'Ome 'Of these m'Odels are presented in references (19 - 22).

Several meth'Ods have been empl'Oyed t'O calculate human 'Operat'Or describing functi'Ons . These include m'Odel matching, parallel filter matching and p'Ower spectral density meth'Ods. With the m'Odel matching technique a f'Orm f'Or the human 'Operat'Or describing functi'On is assumed with the system parameters undete.rmined. By matching the m'Odel 'Output t'O the human 'Oper-at'Or 'Output, b'Oth having the same input, these undetermined parameters are ch'Osen t'O minimize the r'O'Ot mean square difference between the 'Outputs. 23 When the parallel filter meth'Od is used, the input t'O the human 'Operat'Or is als'O fed int'O a set 'Of filters c'Onnected in parellel. The 'Outputs 'Of these filters are weighted and summed in such a manne:r as tó produce a signal which

appir'Oximates the human 'Operat'Or's 'Output. 15, 24, 25. With p'Ower spectral density meth'Ods initially n'O assumpti'Ons need be made c'Oncerning the f'Orm 'Of the describing functi'On. By using p'Ower and cr'Oss-p'Ower spectral densities 'Of signals circulating in the system the human 'Operat'Or's frequency resp'Onse is f'Ound and a suitable describing functi'On fitted t'O this data. 3 .,. ,;,F,'!

':"I.~:

Alth'Ough much w'Ork has been carried 'Out in devel'Oping human 'Operat'Or describing functi'Ons the p'Oint has n'OLyet been reached where exten-sive · use 'Of the results is made in designing man-machine systems. Reference (26) dem'Onstrates h'Ow human 'Operat'Or describing functi'Ons may be used in predicting pil'Ot-aircraft dynamic stability and reference (27) describes their

use in predicting an 'Observed instability with the X-15 research aircraft. De-scribing functi'Ons have als'O been used t'O explain the differences in pil'Ot ratings given an aircraft and its gr'Ound based simulat'Or. 28 Thus, the human 'Operat'Or describing functi'On can be applied t'O real sitl!ati'Ons with effect.

2. MATHEMATICAL BASIS FOR DATA REDUCTION

The p'Ower spectral density meth'Od 'Of -dàta reducti'On was finally selected f'Or use in this study because it appeared t'O be the m'Ost general in appr'Oach. Als'O, when the intermediate step 'Of cr'Oss-c'Orrelati'On is utilized, additi'Onal inf'Orm:ati'On c'Oncerning the 'Operat'Or bec'Omes available such as an estimate 'Of the time delay 1:" The p'Ower spectral meth'Od als'O lends itself readily t'O studying "cr'Oss-talk" between channels in a multi-l'O'Op c'Ontr'Ol sys -tem. In this secti'On is presented a meth'Od f'Or finding quasi-linear represen-tati'Ons 'Of system elements which reduces t'O the meth'Od 'Of p'Ower spectral densities f'Or the case 'Of infinite rec'Ord lengths.

2. 1 Linear Theory

Consiáel' a stabie linear system represented by its frequency response function G(j

w).

If the system input is zero for t ~O _{and t 7to and}

variabie for 0 < t <: to then its output will be zero for t < 0 and will die out to a neglig-ible value at t

=

T ~ to say. See figure (3). The use of finite records ensures the existence of Fourier transforms and allows time averages to be replaced by time integrals.

/

. The output y(t) may be represented by y(t)

=

0 for t < 0

co

y(t)

=

1

g(v) x(t-v) dv for t

~o

t

=

~

g(v) x(t-v) dv (since x(t-v)

=

0 for v>t)

Where g(t) is the linear system' s impulsive admittance. Rxy( 'l:) is defined as Rxy ("è ) = lim 1 T~co 2T T ) x(t) y(t

+

l ' ) d t T 2.1

Thus, if x(t) and y(t) are of finite duration th is leads to a useless result. Investigators in the past have avoided this problem by saying that

T

Rxy(

~

)';'

~ ~

x(t) y(t +'t ) d t

where T is the record length. In this report a different approach is taken by defining

co

~

x(t) y(t

+

1:' ) d t 2.2

-co

where to all intents and purposes x(t) and y (t) are zero outside the interval

o

~ t ~ T. Thus 2. 2 al ways exists and Mxy(T, 1: )

=

0 except for -to <: 't' < T.

Now substitute y(t

+

"'t') from 2. 1 into 2.2 to obtain

2.3

t

:

..

Equation 2. 3 is sirriply a surface integral in (v, t) space, the fntegrand of which vanishes except inside the r_egion S shown in figure (4).

Çhanging the order of integration, equation 2.3 becomes ~oo

g(v)dv )

-<X>

From 2.2 Mxx(T, t" )

=

~oo

x(t) x(t

+

~

)

d t

-<X>

Substituting 2. 5 into 2.4 we obtain

Define

x(t) x(t

+1:'-

v) d t 2.4

2.5

2.6

2.7

Since the integrand is zero outside the range -to<.'t'< T equation 2.7 always exists.

Substituting 2 .. 6 into 2. 7 we obtain

(T '. \)

_F

e -jw'td

",-N_xy ,J w =

1

\.

r

g(v) Mxx(T,

~

-v) d v o

2.8

Multiply g(v) by 1 = e -j W v e j lAl v and change the order of integration in 2. 8

to obtain Nxy(T, j CA)

=

Consider (00 _j fA) v

1

g(v) e dv o <X> ) e -j W (1:' -v) Mxx(T, 1: -v) d't 2.9 -<X>

1

e -j W u M (T u) d u xx ' - 0 2.10

Replace u in 2. 10 by ('t -v) where v is a positive constant. Thus equation 2. 10 becomes

-jw(t'-v)

M (T, t" -v) d~

xx

Substituting 2.11 into 2.9 we obtain

or But Therefore ( g(v) e -j W v N (T, j w) El v

J

xx o ::

r

g(v) e -J

·w

v d v

~ro

g (v) e -j

t.J

v d v :: G (j W ) o G(jw) :: Nxy(T,jw) Nxx(T, j w) reference (29) 2.11

Thus, in theory, it is possible to get an almost exact estimate of G (jw) by the above rnethod using finite length input and output records.

If in the preceding development Mxy(T, ~ ) were replaced by a

time average and to~(l), then ~ _{Mxy(T,1:' )} ---+ Rxy( 1.'), the

cross-correl-T

ation between x (t) and y (t). The final result would then becorne

G (j w) ::

~xy

(j w)

~xx(w)

as shown in reference (29)

2.2 Application to Nonlinear Closed-Loop Systems

Consider the nonlinear servo system shown in figure (5a) where the nonlinear element represents the pilot and the linear element the aircraft system. The input i (t) is turned on at t

=

0 and turned off at t

=

t~. A con-tinuous record of i (t), e (t) and 0 (t) is taken. It is desired to represent the pilot by a linear system Y (j w) plus a remnant term n (t) such that the system

Y (j w) gives a good representation of the average performance of the pilot

over the particular run represented by the recorded variables. See figure (5b). It is obvious that there are an infinite number of pairs Y(j w), n(t) which may be used to represent the pilot since n(t) need only be chosen to

represent the difference in response between the pilot and Y(j w) to e(t). Since figure (5b) is simply a linear system with two inputs, the principal of super-position applies and the system of figure (5b) may be represented by the sum of the two system s of figure (5c).

Two useful methods of selecting Y(j (..1) now appear. One method is to choose Y(j LV) to minimize the integral square value of n(t), the other to minimize the integral square value of p(t). Here p(t) represents that portion of the total signal being fed to the aircraft which results from what we have called the remnant and it may be calculated from a linear operation on n(t). We wiU distinguish between the two by letting Y 1 (j w) correspond to minimizing the integral square value of n(t) and Y 2(j w) correspond to minimizing the

integral square value of p(t). The calculation of Y 1 (j w) is now presented. In general, for the system of figure (5)

co n(t) = 0 (t) -

~ ~

( 't') e (t - 7:) d'r o Therefore co co (co ) n2 (t) dt

=

r

02 (t) dt - 2 o

~

dt

~

-

Q(t)~

(-

..

't )

e. (t

-~)

dt' o o 0 co

+

~

[r

e (t -1: )

'1á

(

t') d

'lr

d t o 2.12 2. 13

In Appendix A the calculus of variations is applied to find the funcl\on

"11

(t')

-which minimizes 2.13. It is shown that ~ 1 (~) must satisfy

f (co ) eet -

t )

e (t - v) d t o co

= \

o(t)

e(t

:

-~)

dt o

Using the notation of equation 2.2, 2. 14 becomes

\~ ~l

(v) Mee{T,

t'

-v) d v

=

Meo(T,1:)

o

2. 14

2.15

Where T >' to is the time at which signals in the system become negligibly smal!. Equation 2.15 may now be manipulated inthe same manner as equation 2.6 to produce the relation,

= Neo(T, jW) _{2. 16}

Nee (T, j

w )

lf in the calculation of Y 1 (j W ) infinite time records are assumed and time

averages are used instead of time integrals it follows that

=

~eo

(j

W )

~ee

(j

~

)

As a result of minimizing the integral square value of n (t) it is found (see Appendix A) that n1 (t) must satisfy

i. e.

r

n 1 (t) e (t -

~

) d t = 0 o

n1 (t) has zero correlation with e Ct).

When calculating Y 2 (j w) one must minimize the integral square of p (t). Now P (t) = ) (00 b (v) n (t - v) d v

o

2.17

where b (t) is the impulsive response

of

1;3 (j.w;) = ' 00 Also, P (t)

=

0 (t) - \ e (v) {

(t

'

-

'

v) d v o .,< .

.

, . 1

-_:...-.._---

"

1

+

A (j (;,)) Y (j w )

y{jiJ}

where e (t) is the impulsive response of C (j

w )

=

-1

+

A (j w) Y (j ti) }

(see figure

5r.

;

2.18

Now P (t) in 2. 18 may be dealt with as n (t) was in equation 2. 12 in order to minimize the integral square value of p (t). This leads to the two results

and \ 00 P2 (t) i (t - 1: ) d t = 0 o Nio(T,jw} N

ii

(T, j!IJ )

=

Y 2 (j

!IJ )

1

+

A (j ..J) Y 2 (j

W )

or, for infini te length ree ords

Now, if 1>io (j

w )

<Pii

(w)

=

00 . \ P2 (t) iet

-or }

dt . = o N, (T, jW)

=

0 1112

This is shown in Appendix B.

Y2 (j

w )

'1

+

A (j

w )

y

'

~

Ü

w )

,

'0 then _{, \} Now e (t)

=

r

b₂ (v) i (t - v) d v

+

~oo

d₂ (v) n2 (t - v) d v o 0 2.19 .t • • 2.20 • \ I ' 2.21 , } 2.22

where d

2(t) is the impulsive response of

- A(jlA> )

(see figure 5)

=

Multiplying 2. 22 through by i (t -

't' )

and integrating with respect to t we

ob-tain

00 00 00

~

i(t -

t' )

e(t) dt =

~

i(t -

~

) d t

l

b2 (v) i(t - v) d v

+

r

Hl -"I:)dl

r

_{d 2 (v)n2 (I - v) dv}

CD <X>

or Mie(T,

'r )

=

j

b2 (v) Mii (T,

'r

-v)dv

+

J

_{d 2}

~v)Min2(T,

't" -v) dv 2.23

Equation 2. 23 may be manipulated in the same manner as equation 2.6 to ob-tain

2.24

From equation 2.21, Nin2 (T, jw)

=

0, and noting that

1

substituting 2. 25 into 2. 20 leads to

Nio (T, j w) Nie (T, j~) , equation 2.24 reduces to 2.25 2.26

.

.

or) for the case of infinite lengtn. records

=

\,

~io

(j w)

~ie

(j w)

In general the describing functions Y 1 (j W ) and Y 2 (j tIJ ) will depend upon the type of input used and its amplitude characteristics in the time range O---'T. Thus) if short record lengths are employed the Y1_{s may}

be expected to vary to some extent from one test to the next due to variations in the inpvt characteristics. However) if an input with a Gaussian amplitude distribution is employed along with record lengths in the order of 2 minutes it has been found that a Gaussian input describing function for the pilot is ob-tained. 10 It should be noted that the above analyses when applied to short

r-record runs must include the sVJitching-on and switching-off transients) the

•

need for including the transients diminishing as the record lengths increase. •

Since the manner in which the pilot responds is dependent up on the input form) these transients might represent a slightly diffel'ent pilot response from that expected with a random input. If the finite length record was obtained by truncating a longer record a similar small er!Ï'or would be introduced by the initial and final segments of the record. The error introduced in the ca1cu-lation of the pilot describing function by power spectral means is due to the

, use of finite length records where the exact theory assumes infinite length records.

Since Y 1 (j

W )

and Y 2 (j

w )

are obtained by employing different criteria it is not expected that they are gene rally identical. Reference (30) shows that for the case of infinite length records

A ( - j

W)

q;

n 2 n2 (w)

1+A(-jw)Y2( - j w ) epee(W)

Because of the criteria used in developing Y 1 (j ei) ) and Y 2 (j

w )

it would appear that Y 1 (j

w )

is well suited for use in investigating the pilot as a single element (i. e. pilot opinion studies» while Y 2 (j W ) is suited for pilot studies intended to produce describing functions for use in piloted systems analysis. Another situation arises if, as has been postulated) the pilot be-haves more or less as a linear element whose output is summed with a ran-dom nois'e generator. 3') 20 If the human operator is actually operating in this manner then the remnant would be represented by . the output of the noise generator and hence would be uncorrelated with the system input i (t). Hence in this case) the best description of the pilot's linear portion would be ob-tained by employing Y 2 (j

uJ )

since it is based on zero correlation between

remnant and system input. It should be noted that the measurement of Y 1 (j u) corresponds to the describing function found by employing model matching techniques while Y2(jW) has been employed in the past when croSs -power spectral density methods were used. Thus the present literature appears to contain describing functions based on two different optimizing criteria.

3. SIMULATION

3.1 A Brief Description of the CF-100 Simulator

A fuller description of the facility is contained in reference (31). The aircraft simulated is a twin-engined interceptor.

The simulation facility consists of a CF-100 Mark IV B Opera-tional Flight and Tactics Trainer with a cockpit equipped either with actual air-craft hardware or simulated parts which externallythe aircrew cannot

dis-tinquis.h from the aircraft parts and a 60 cycle A. C. electromechanical analog

computer coupled with an instructor's console to provide the dynamics. The cockpit is of the fixed-base type with no external visual cues provided. The mathematical model for the simulator's flight system is comprised of a set of nonlinear differential equations capable of simulating aircraft operating con-ditions such as landings, takeoffs, combat and tactical maneuvers and a

variety of emergency situations.

3. 2 Cockpit Environment

In order to achieve arealistic simulation several important factors in cockpit design must be considered. The instruments presented to the pilot should correspond to normal flight instruments. 32 When a pilot feels lfat homer! in the simulator the mental extrapolation required of the pi lot to envision the simulated flight as actual flight is reduced. The use of normal instruments has been adhered to in the UTIAS simulator. Likewise, the presence of simulated aerodynamic and engine noises helps to create the correct atmosphere as weU as mask unwanted simulator noises. These sounds are incorporated in the CF-100 simulator, the sound level being set by the experimenter. Added realism is achieved through the use of instrument panel vibrators. Finally, the control stick movement and feel must be up to the

appropriate aircraft standard.. This is accomplished in the present system

by an electromechanical computing system coupled to the stick by a cable and pulley system.

3.3 Motion and Visual Cues

Since the present simulator facility does not incorporate any motion or external visual cues, the tasks presented to the pilot must be ones which are not primarily dependent upon these cues. It has been found that the more stressful the situation and the more difficult an aircraft is to fly the more the pilot depends upon motion cues. 28 Also, motion cues are

valuable when evaluating aircraft flying under simulated turbulence conditions. 32 The need for external visual cues is also very dependent upon the task, i. e.

they are an important factor when simulating landings but not very necessary when simulating high altitude flight. Thus our facility is best suited to simu-late an aircraft with good handling qualities flying at high altitudes.

3. 4 Simulated Task

Previous simulator studies have shown that more consistent and controlled experiments result when the task is clearly defined and presented to the pilot prior to flight. Additional realism is achieved if the simulator test is conducted in a manner similar to a real flight test program. The task to be performed by the pilot during the present simulation program is the control of small perturbations in pitch and roll attitude. In order to isolate the pilot from the laboratory environment and to minimize the ne,ed for ex-ternal visual cues the cockpit canopy has been 1?lacked out. 'Thus the pilot task is presented as a night mission. Added incentive to',:track the attitude errors may be provided if the pilot is told that the aircraft is to act as a platform for reconnaissance hardware which is only effective when attitude errors are negligible.

3. 5 Differences Between Actual and Simulated Flight

In general, when considering aircraft configurations rated as satisfactory, results obtained in fixed and moving-base simulators and in actual flight correlate well. Also, depending up on the task and dynamics,

the simulators may be easier or more difficult to fly than the actual aircraft. 28 When pilot describing functions found from flight and simulator data are com-pared it is generally found that significant differences do exist. 33 For ex-ample it is found that the gain adopted by the pilot in actual flight may be as low as one half the gain adopted in the simulator. 34 These differences are attributed to two factors, the pilot' s mental attitude as he faces the same task in two different situations and a lack o'f fidelity in the simulator due to such limitations as lack of motion cues. Thus caution must be exercised when applying pilot describing functions found from si:m'uLawr studies. It appears th at describing functions measured in such studies may best be used to compare various flight conditions rather than to define an exact pilot describing

4. EQUIPMENT

This section describes the various volved in obtaining a pilot' s describing function. representation of these sub-systems.

4. 1 Random Input Signal Generator

sub-systems which are in-Figure (6) is a block diagram

The ideal input for the type of analysis to be employed is one with a flat power density spectrum of Gaussian noise over the bandwidth of interestJ with a sharp cutoff outside this band. However J since no such

generator was immediately available it was decided to carry out the initial system checks with a less than ideal input spectrum to make use of available equipment. The input system finally used consisted of a Scott Type 811A Random Noise Generator whose output is filtered through one or two of the simulator's spare servo units. This is shown in figure (7). The'variation of the 60 cycle component in the noise generator output activates the system and any resulting high frequency components are filtered out by the servo units. A power spectrum of the input system is shown in figure (21). A typical time record of this input signal is shown in figure (8). Since the greatest amount of power in the pilot's output was expected to be in the range 0-1 cps. it was felt that this input would be satisfactory for the present

application. This bandwidth is also typical of the fr:equ!3Rcies commonly en-countered by fighter aircraft flying through turbulence.

4. 2 Visual Display

The purpose of the display system is to present to the pilot an unmodified indication of the system error signal in two degrees of freedom. The simulator's artificial horizon waq, first employed but it was found that because of its mechanical nature its dynamics altered the error signals sig-nificantly. The original mechanical display system was therefore replaced by a cathode ray tube display system as shown in figures (9J 11). The

dis-play is approximately 30 inches from the pilót's face. The disdis-play gives an inside-out indication of the aircraft attitude errors i. e. in the display the aircraft is stationary while the horizon moves.

On the cathode ray tube horizontal plates is a 60 cycle signal to produce the horizon line. On the vertical plates there are two signalsJ one

is a D. C. signal proportional to the error in pitchJ the other is a 60 cycl~

signal proportional to the error in roll. As aresult the aircraft signals

combine to produce a straight line Lissajous figure which.causes-the~horizon to roll in response to the roU error, signal while the error in pitch signal causes the display to pi tch up or down. The system schematic is shown in figure (12).

Since the angular error signals are read by linear potentiometers the pitch error is displayed as inches of deflection per degree of pitch and the

roll error is displayed as tan- 1 (ron error). However, the present application of the simulator is restricted to small perturbations and thus the difference between the angle and Us tangent will be small. A second effect of the cathode ray tube disp18.y is to displéiY pitch error. when a roU error is indicated, in

a manner which differs from a true artificial horizon display. With an aircraft flying at a constant angle of roU wUh Hs nose on the horizon any change in its pitch attitude is indicated on a true artificial horizon by the perpendicular distance from the centre of the display to the horizon bar. However, with the prese.nt cathode ray tube display this pitch attitude is shown as the vertical distance from the centre of the display to the horizon bar. Thus. with this display a trained pilot would read the pitch information as the true pitch error times the eosine of the indicated roll error. This error only enters in the two degrees of freedom case and is small for smaU angles of roU. These differ-ences are shown in figure (10).

The gains in the display systern were chosen to produce the same display as the origina.l artificial horizon; in pitch. 044" /degree, in roU the indicated angular error equals th.e true error at 00 _and

:!-

_{7. 5}0 _{• Linearity}

checks on the system indicated ~ 2% linearity in true roU error over the range ! 300 _and ~- _{1% linearity in displayed pitch error over the range! 150 , which}

is the complete pitch display range.

4. 3 Control System

The simulator i.s flown with a centraHy located stick to which is connected an electro-mechanical artificial feel systern. Since the present simulation is being carried out at constant Mach number, alti.tude, etc. the system acts as a second order mechanical system. Although the control sys-tem does exhibit sorne nonlinear characteristics, the most noticeable being a 1. 5 pound breakout force in both pitch and roll, it was felt th at a good representation of the system could be achieved by a linear G'.nalysis. Linear approximations of the control system dynamics were found by means of a Fourier analysis described in references (36. 37).

The approximate Laplace transfer functions considering pilot stick force as input are

Se 13.2 degrees/pound

=

s2 +5s

+

12.5 in pitch Fe and ~a 59.0

=

degrees / pound in roU. Fa s2+11.9s

+

52. 9

lf pilot stick displacement is considered to be the system ,input then the above transfer functions are replaced by simple gearing ratios

-b

e _{= 3. 1} Ae

degrees / inch in pitch

and

~a

1.7

= degrees /inch in roll.

Aa

The stick grip is seen in figure (11). The effective stick length for pitch control is 22.5 inches and for rol! control 8.75 inches.

4. 4 Aircraft Dynamics

The simulator has been modified to provide two uncoupled de-grees of freedom, either singly or together. The pilot task desired was a simple pitch/rol! attitude control problem about a steady reference flight condition at constant altitude. In order to relieve the pilot of the task of maintaining th is trimmed reference flight condition while controlling distur-bances in attitude, all the simulator variables not connected with the at~itude

task were set to the desired reference conditions and frozen at them. Thus the pilot's task is reduced to an attitude control problem in an aircraft whose autopilot system controls all system variables except smal! perturbations in pitch and rol!. The simulation should be representative of this situation as long as the pilot does not allow large attitude errors to remain over extended periods of time. The reference flight condition simulator settings are listed in Table (1).

References (36, 37) describe the modifications employed to obtain the desired uncoupled degrees of freedom. Simplified schematics of the resulting J3ystems are shown in figures (13, 14). In order to account for non-ideal behavior of the computing elements the aircraft system transfer functions were experimentally found by a Fourier analysis. The resulting transfer functions are

S

₌

14.5 _{in pitch}

~e

s2 +9s+8.15 and

,

5.5

=

in rol!

~a

s (.025s + I)

4. 5 Monitoring Display

In order to monitor pilot performance during test runs it was felt that the experimenter should be provided with a display of pilot response to the input signals, an indication of tI:e RMS value of e (t) and a voice link wi th the pilot.

Accordingly provision was made for a monitoring station. The error signal seen by the pilot and his output are displayed on a dual beam oscilloscope. The experimenter and the pilot can communicate through the simulator's VHF radio system. An RMS Gomputer was constructed using spare simulator components and this is used to monitor the RMS value of e(t). A schematic of this computer is shown in figure (15). This system operates by integrating the error squared over a 22 second period at 34 second intervals, resetting itself at the end of each 22 second integrating period. A lamp indicates to the experimenter the period over which the com-puter is integrating and the estimate of the RMS error is displayed on a

voltmeter. Since only a rough estimate of pilot performance was desired the

system was not developed to a high degree of accuracy.

4. 6 Recording System

The records of i(t), o(t) and e(t) for each piloted "flight" are taken on a multi-track Ampex SP - 300 FM tape recorder. Since the siI'I1u-lator systems are A. C., and D. C. signals were needed for analysis pur-poses, a set of D. C. ener_gized potentiometers.of 1% linearity was installed, paralleling the A. C. system. Figure (16) shows the pitch record system, the roll system is identical. The resistance values were chosen to prevent loading effects . ,The resulting gains for the various recording systems are listed in Table (2) along with the sign conventions employed.

4.7 Data Analysis

The calculation of the M xy and Rxy described in,section (2 ) requires three basic operations; signal time delay, multiplication and inte-gration. Since we had ready access to an analog computer it was decided to begin by analyzing our data by analog methods, later switching to digital methods if necessary.

The generation of the pure time delay for our random signals was first attempted on the analog computer. Since the Laplace transfer function for a time delay network is e -"t-' s where ~ is the value ~f the time delay, the following approximation can be }ls_ed for small values of the

--rs

e =

s2 't'" 2 6 s~ + 12 s21;'2 + 6st" +12

This is called the second order Pad: approximation and it can be patched on the analog computer. It was found that insufficient time delays and band-width made this system unsuitable for our application. The time delays used to obtain the results in this report were therefore produced by a correlator unit on loan from the U. of T. Electrical Engineering Dept .. This unit is described in reference (38). The correlator is essentially a two track FM tape recorder with a variabIe length tape loop between two of the four heads employed. The unit has a flat frequency response over the range 0-50 cps. and unity gain.

A diagram of the circuit used to calculate M xy and Rxy is given in figure (17a). This circuit contains a mean removal section in order to eliminate the effects of D. C. drift in the time delay unit. To see the effect on the calculated values of M xy and Rxy of removing the D. C. levels consider the signals x(t) and ·y(t) which are of T seconds duration with mean values respectively mand r. Let f(t)

₌

x(t)

=

0 h.(t)

₌

y(t)

=

0 -m

-

r for 0

<.-<T

otherwise for O<t<T otherwise 4. 1 4.2

Thus f(t) and het) have zero mean·s in the time range 0 - T. Consider the desired convolution integral.

\ x(t) y(t

+

1:' ) dt -0) And Rxy( 't' )

.

_T

1 T = ) x(t) y(t +'t) d t o T

=

~

x(t -'t;') y(t) d t T

~

x(t) y(t

+ '( )

_{dt, M xy (T,?: )} o 4.3 4.4 T = ) x(t) y(t +'t') d t o

.1>

For the case where T ~1:' ~ 0 4.4 may be revyritten by using

4. 1 and 4. 2 as T

S

x(t - 't' ) y(t) dt

=

T . T T-~ '

ç

f(t - 'l:') h(t) dt + m

S

h(t) dt + r ) f(t) dt + mr(T -"t' ) o o ~ 0 4.5

. For the case where - T<~", 0 4. 3 may be rewritten as

T T T

+t'

T

\ x(t) y(t' +

t')

dt

=

ç

f(t - t') h (t) dt + m \ h{t)dt + r

~

f(t) dt + mr (T +1=)

o 0 o - t '

4.6

Now consider the problem of calculating R (t'). lf x(t) and

y (t) were infinite stochastic signals then Rxy(

t'

)----? mr?s t' increases. 29

Thus in order to get a .good estimate of Rxy(

1:

»T should be selected so that

the convolution integrals 4.3 and 4.4 approach the value mrT when ~ is

still much less that T. lf this is done then the terms in equations 4. 5 and 4. 6

which contain the time integ-rals of f(t) and h(t) over (T

-11:'1 )

seconds of their domain

of existence wilLgenerally be negUgibly-·small and m r (T -I~

I)

approximated '. > •

by (m r T) over the useful range of ~ (i. e. that range of t:'" over which Rxy( t')

is variable). In th is case a good estimate of Rxy(1:") may be obtained if

ç

f(t - 't') h(t) dt. mand rare known. It should be noted that for T<oo the m

o

and r used will only be estimates of the infinite signal means.

Now conside:t the signals x*(t) and y*(t) which appear at the

output of the time delay unit. In general they will have means m* and r* which

differ from mand r due to

b.

C. drift in the electronics. .

Equations 4.1 and 4.2 are how replaced by

f(t)

₌

x*(t) - m* for O<t<T

=

0 otherwise

h(t)

₌

y*(t) - r* for O<t<:T

=

0 otherwise

The output produced by the circuit of figure (17a) is I

T

S

x* (t -'t ) y* (t) dt - m* r* T

o

which becomes, if the above rnentioned choice of T is made

T

T

\ x* (t -

t' )

y* (t) d t - rn * r* T

=

~

f (t - "t" ) h (t) d t

o 0

over the useful range of

't .

If _{M xy is to be calculated,}

1:

must range from - T ~ T.

For long record lengths this .could not be done on the present equipment since

1:

max. was

i"

20 seconds. In the case of short record lengths the time inte-grals of f (t) and h (t) in equations 4.5 and 4. 6 would nót be negligible over ·the fuU range of 1:' . _{and hence the ,:,alue of M xy} calculat~~ by the anal~g circuit

would be suscephble to D. C. drIft effects unless auxlllary c.alculatlOns of m*,

r* and the partial time range integrals of f (t) and h (t) werecalculated. Thus

the evaluation of M xy seems to be best suited to digital methods where D. C. drift is not a critical factor.

. g)xy (j _{W ) and N xy (T,}

j~)

are found by performing sine and

cosine transformatlOns on Rxy (1:" ) _{and M xy (T,} 1:). Before the

transforma-tions can be performed the range of ~ over which Rxy ( 1:') _{and M xy (T,} t )

are significantly different frorn. a constant must be determined. Let the

maxi-mum value of 1: in this range be

t"o.

_{In the case of M xy (T,}

1:').

"to

<

T because Mxy(T, 1:') = 0 for

l't'I>T.

In the case of Rxy(-Z:), Rxy(1:" ) may be

considered as consisting of two parts, a variable part existing from -.~ 0 to

7:

0 and a constant part of value m r existing from

1:'

= -

00 to

J::

= +

00.

Consider the calculation of

~xy(j

W )

00 .

'C'

CPxy(j

w)

= _1_

~

e -Jw Rxy (

~

) d

'L

2"'rr --00

+

=

1 21T 1 2TT 00 .

:t-~

e -JIÀi m r

d't

-00

7

₀

~

cos (

w't")

(Rxy('l:') - m r) d't -"t_o

+

t:~ " .•

j

2".

'to

\ sin (W'l') (Rxy(t') - m r)

d~

-t"o

Thus, the D. C. levels in the signals x(t) and y(t) lead to a spike at A)

=

0 in the

I

Re

(~xy(j

w) )

I .

The sine and cosine transformations are performed by .generatmga signal proportional to (Rx ('t) - m r) in the analo.g computer and multiplying it by sin (

c.Jt' )

and cos

r

w't ) as generated by an oscillator patched on the computer, the product being integrated. The

(Rxy( ~ ) - mr) .signal is produced by a curve-follower which follows a plot of (Rxy( 't') - mr) vs. 1: traced out in conducting ink. Each point on the

<Pxy(j

w)

curve requires a new oscillator setting. The circuit used for this operation is shown in figure (17b).

In order to check the accuracy of the correlation process used, the autocorrelation of a square wave was computed. The frequency of the in-put was 5 cps. and the inin-puts to the multipliers were :±- 60 v. This test will indicate the best results we can expect from the system since the inputs to the multipliers were large. The computed correlation shown in figure (18) indicates that excellent results may be expected if signal levels into the multi-pliers are large. However, our random signals have frequent zero level crossings and thus our experimental accuracy will not be this good.

In order to check the Fourier transforming process the cosine transform of a linear function was computed. The excellent results of the transform are. shown in fi:gure(Hl).~ As was .the case with correlation com-putations, the Fourier transform results would be expected to deteriorate if the function to be transformed spent much time near the zero axis.

Thus it appears that any variability in the reduced pilot data will be largely due to the performance of the multipliers employed.

5. DATA REDUCTION

Two separate sets of data were used to test the system. One set was obtained with a graduate student, R. H. Klein, as pilot (subject AL the second set with the author as pilot (subject B). Subject A has had two years experience with a Boeing 707 simulator and approximately 50 hours of light airèraft ~lyir?:g time. .. The author had no previous flight experience. No training program was carried out as it was not considered to be necessary since the runs were being undertaken purely as a system check. Both runs were made using the single degree of freedom pitch mode.

A section of a typical set of pilot records is shown in figure (8). The autocorrelation and power spectral density curves for the input signal are

. .given in figures (20, 21). It should be noted the 0 (t) recorded is the output

of the control system, in this case be' and that this is sirnply related to the actual pilot output only if the pilot is considered to be apositioning device. When the pilot is considered as a force producer the transfer functions of

section (:4 . .3) allow one to rernove the effects of the controller dynamics by

considering the controller to be a linear element in series wfth the pilot.

The transferfunction Y2(jtJ) was calculated for each subject

using the power spectral density technique along with the assumption of approxi-matelyinfinite record lengths. This assumption proved to be quite good since 6 minute record lengths. were used and it was found that the maximum time delays required were

!

15 seconds. At first time delay increments of O. 1 seconds were employed, but it was found th at except near values of

7:

= 0 time delay incrernents of 0.2 seconds were sufficient to define the Rxy(-z:') curves.

Figures (22 - 27) represent the full set of data used in calculating Y 2(j

t..J).

It was found that the data points obtained for Rxy( t' ) and

~xy(j

IJ,) ) were repeatable to within

!

4%

_{of full scale, thus the (Rx} (1:) - m r) values were considered to be negligibly small when they reached

5%

of full scale. This resulted in

I

'r

I

max.= 15 seconds. An estimate of the expected accuracy of the results assuming ideal cornputing elements may be obtained by using an approximate estimation of the Rxx( 1:') normalized standard error presented

in reference (39).

1

[

R xx (0)

Rxx (~)

J

where Bw is the bandwidth of x (t) and T is the record length. It must be

noted that this will only give an estimate of the error in the Rxy{ 1:' ) since no

similar forrnula has yet been developed for cross-correlations. In our

sys-tem with Bw - l·cps. and T

=

360 seconds

1

60

[

]WhiCh indicates that

for

r

=

0 we can exp ect an error in Rxx (1:") of less than 1.7% of Rxx{

r.)

for 2/3 of the time due to the finite record lengths employed. The final amplitude and phase curves for both subjects are presented in figures (26 - 29). These results are good up to 4 rad. / sec. because .the input spectrum had little power above 4 rad. /sec .. During tne test runs it was found that both subjects were unable to reduce the RMS error level significantly below its level with no pilot present. This was attributed to the lags produced by the aircraft dynamics and

contr,ol system coupled with an input with power at frequencies too high for the

presént application. ' Thus it was not surprising-when it came time to fit

de-scribing function curves to the data of figures (26 - 29) to find that at least a second order lead had to be included to obtain a reasonable fit.

Describing functions were fit to the data by hand calculations .

:~hrst an approximate fit was made to the amplitude ratio data then this was

/modified by considering the phase plot. This process was carried out

/

/ through several cycles until a reasonable fit of both amplitude and phase data

was obtained. The describing functions were Subject A ,

~

1. ~ (10s

+

1) (s

+

4) degrees = e (5 s

+

1) (s

+

20) degree Subject B ~·e _26(s

₊

_{1. 5)2 degrees} = e (2 s

+

1) (s

+

20) degree

It was decided that nothing would be. gained by using a higher' order lead term.

The pilot was considered as a position producer because the large extra

amounts of lag introduced by the control system force transfer function made it impossibleto fit the pilot phase data with a second order system. As it

turned out no information concerning-the pilot's,pure time delay was found.

This. again is because of the second order describing function employed. It

may be assumed that the subjects were acting in a highly nonlinear fashion as

they were required to generate second orde~ lead terms. 1

6. CONCLUSIONS AND RECOMMENDATIONS

6. 1 Aircraft Dynamics

With the present system the aircraft dynamics cannot be altered in any simple manner to any desired form. This is a great handicap when dealing with pilot describing functions because the form of the describing function depends directly upon the dynamics controlled. Also, if other sys-tems such as the control system, visual display etc. are to be studied with a fixed input form, provision mus,t be made for adjustment of aircraft dynamics in order to achieve an overall system in which the pilot can operate as linearly as possible.

If the tasks to be simulated continue to be the simple attitude problem described in this report, a small D. C. analog computer with 20 - 40 operational amplifiers CQuld be used to provide aircraft dynamics and to moni-tor various pilot performance criteria. With proper interface equipment this auxiliary analog computer could be connected up to the cockpit instruments and the A. C. analog system thus leading to a more versatile .. simulation facility more in keeping with its present research role.

6. 2 Random Input Source

The random noise generator employed in this study was not de-signed for low frequency work and hence there is no gu'arantee that its low frequency s.pectrum amplitude has a Gaussian probability distribution. Also, there is nQ simple way to alter the input spectrum to meet the demands of

various pilot tasks. This was primarily the cause of the difficulty the subjects encountered in controlling the RMS error during the reported test runs. Future experiments could be improved if a proper low frequency Gaussian noise gene-rator were employed along with a set of filters for spectrum shaping.

6. 3 Data Reduction

The method of data analysis employed in the report appears to be satisfactory for obtaining pilot describing functions. An accuracy check involving thecalculation of a describing function for a known nonlinearity was not possible with the input system employed. The present analog set-up h9wever, is very time consuming, requiring 21 hours of computer time to produce a complete set of data as illustrated in figures (22 - 27). All the steps required in obtaining a describing function however, are well suited to digital c"omputer techniques. Thus the next logical step is to convert our analog re-cordsto digital form and have the complete analysis process carried out on the U. of T. IBM 7094 digital computer. This will also eliminate D. C. drift

problems in the analysis ~uipment and "allow a comparison between describing functions obtained usin~ _{Ylxy and Nxy . The digital facility could also be used}

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"Human Pilot Dynamic Response in Flight and Simulator. "

WADC TR-57-520, 1958

. "Power Spectral Analysis of Some -Air-plane Response Quantities Obtained During OperaÜonal Training Missions of a Fighter Airplane. "

NASA TN D-366, 1960

"Determination of Transfer Functions for the CF-I00 Simulator Longitudinal Systems. "

M.A.Sc. Thesis, U. of T., 1965

"Transfer Functions for the UTlAS Flight Simulator in the Single Degree of Freedom Uolling Mode. "

38. F.G. Mason 39. J. S. Bendat 40. R.F. Vomaska M. Sadoff F. J. Drinkwater III 41. C. B. Westbrook 42. G. E. Cooper 43. F. A. Muckley J.E. Nygaard L.l. O'Kelly A. C. Williams Jr. 44. B. P. Brown H.l. Johnson R. G. Mungall 45. L. W. Taylor R.E. Day 46. M. Sadoff N. M. McFadden D. R. Heinle 47. L. G. Summers A. A. Burrows

"The Design of an Analog Correlation Computer. "

University of Toronto, Dept. of Electrical Engineering, Research Report No. 32, 1964

"Random Process Theory and Applications. rr

Measurement Analysis Corp.

May 1964

"Tte Effect of Lateral-Directional Control Coupling on Pilot Control of an .Airplane as Determined. in Flight and in a Fixed-Base Flight Simulator. rr

NASA TN D-1141. 1961

"Simulation in Modern Aero-Space Vehicle

pesign. rr

AGARD Rep. 366, 1961

"The V se of Piloted Flight Sim ulators in

Take-·off and Landing Research. " AGARD Rep. 430, 1963

"Psychological Variables in the Design of Flight Simulators for Training. "

W ADC T R -56 -3 6 9, 1959

"Simulator Motion Effects on a Pilot's Ability to Perform a Precise Longitudinal

Flying Task. "

NASA TN D-367, 1960

"Flight Controllability Limits and Related Human Transfer Functions as Determined From Simulator and Flight Test. "

NASA TN D-746, 1961

"A Study of Longitudinal Control Problems at Low and Negative Damping and Stability

with Emphasis on Effects of Motion Cues. " NASA TN D-348, 1961

"Human Tracking Performance Vnder

Transverse Accelerations . " NASA CR-21, 1964

48. L. A. Clousing 49. M. Sadoff 50. H. A. Kuehnel 51. J. 1. Elkind D. L. Darley 52. J. F. Kaiser R. K. Angell 53. P. Jespers P. T. Chu 54. 55. 56. A. Fettwers G. T. Morgan J. S. Cook III A. Chapanis M. W. Lund J.E. Gibson'

"Simulator Requirements Deduced From Comparisons of Pilots Performance in Ground Simulators and in Aircraft. "

ICAS Paper No. 64-554, 1964

"A Study of a Pilot' s Ability to Control During Simulated Stability Augmentation System Failures. "

NASATND-1552, 1962

"In-Flight Measurement of the Time Re-quired for a Pilot to Respond to an Aircraft Disturbance. "

NASA TN D-221, 1960

"The Statistical Properties of Signals and Measurements of Simple Manual Control System. "

ASD-TDR-63-85, 1963

"New Techniques and Equipment for Correlation Computation. "

Electronic Systems Laboratory,

MIT, 7668 - TM - 2, Dec. 1957

"A New Method For Computing Correlation Functions . "

Paper Ifpesentated at the International Symposium on Information Theory,

Brussels, Sept. 3 - 7, 1962

"The Human Pilot. "

Norair, AE-6-4-111, 1954.

"Human Engineering Guide to Equipment Design. 11

McGraw HilI Book Co. Ltd.,

New York, 1963

"Nonlinear Automatic Control. " McGraw HilI Book Co. Ltd.,

APPENDIX A CD The Minimization of

~

n2 (t) d t 0 ) let I = ) n 2 (t) d t o therefore I

=

r

02 (t)dt - 2

r

dt

r

0

(t)~( ~)

e (t -

f')

d~

o 0 0 0 )

+

~

o

[ r

e (t -

't )

'% (

1: )

d

t" ]

2 d t A-1 let

~

('t,)

=

~

1

(~)

+

À

z ( ' t ) A-2 where

0?J1

(1:') is the value of ~('I: ) which minimizes I, À is an arbitrary small constant. and z (

7: )

is an arbitrary smooth function.

Thus I = I (

~

) has a minimum at " = O. Therefore "dI( " ) - 0 d À À= 0 Substituting A:-2 into A-1 and differentiating with respect to À obtain

dl ( " )

₌

dJ...

therefore

=

0

=

À

=

0 CD 0 ) - 2

)

dt

~

0 (t) z (

't )

e (t -

7:)

d'r 0 0 +2

"i

co

[CO

i

e (t -

'r )

z: ( 'r ) d "t'

)2

dt +2

[dt

[r

e(t-t)'?I!(1:')d't'][re (t-V)Z(V)dV]

CD 0 ) - 2 ) dt ~ 0 (t) z ('t) e (t -;: ) d "r o 0

+

2 fdt [fe(t

-'t)~('t)d't

][\e(t - v)z(v) dv ] o 0 0 A-3