THE HUI‘IAN OPERATOR AS‘ AN
OPTIMAL FILTER’AND OPTIMAL CONTROLLER

The Is for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
RODNEY DAN WIERENGA
1968

Tribal:

This is to certifg that the
thesis entitled

THE HUMAN OPERATOR AS AN
OPTIMAL FILTER AND OPTIMAL CONTROLLER

presented by
Rodney Dan Wierenga
has been accepted towards fulfillment

of the requirements for

Ph- D- degree in EleCt- Engr.

 

Date April 19, 1968

 

0-169

 

ABSTRACT

THE HUMAN OPERATOR AS AN
OPTIMAL FILTER AND OPTIMAL CONTROLLER

by Rodney Dan Wierenga

In this thesis, a mathematical model of the human Operator
in a control task is postulated and its validity is investigated.
In the model that is suggested, an "augmented plant" is used which
contains not only the plant that the human operator is controlling,
but also the forcing functions and the operator's own time delay,

neuromuscular lag, and noise characteristics.

An optimal time-varying linear filter and an optimal time-
varying linear controller are utilized in the model. Consequently,
because of the form of the augmented plant, the model develops
estimates of the state variables of the plant, the forcing func-
tions, and the man's own characteristics. As postulated, the aug-

mented plant can be time varying.

The validity of the model is investigated in terms of a
transfer function match with data available in the literature.
Also, in that the model is capable of prediction, this attribute
is investigated by means of tracking experiments using a sine—wave

forcing function.

RODNEY DAN WIERENGA
A method for ordering the comparative values of displays
in a control task is suggested. This method is investigated by
means of experiments with human operators in a compensatory control
task using two vastly different displays. The experimental results
are matched with the model where the parameters needed for this

match provide the method of ordering.

In general, the results are very good, and although a great
amount of evaluation of other potential advantages of the model is
still to be done, limited application of the model, as it stands, is

justified.

THE HUMAN OPERATOR AS AN

OPTIMAL FILTER AND OPTIMAL CONTROLLER

By

Rodney Dan Wierenga

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1968

.u‘

. E“ .r;_:2
Gil-U”
8’

i. .
/&/g/eff/ a?!

ACKNOWLEDGEMENTS

The author wishes to thank his advisor, Dr. H. E. Koenig
of the Michigan State University Electrical Engineering Department
and Systems Science Group for his guidance and many helpful sugges-
tions during the preparation of this thesis. The substantial assist-
ance given by Dr. T. E. Perfitt, Dr. P. R. Fuller, Mr. G. E. Teach-
man, and Mr. U. Plate of the Lear Siegler, Inc. Instrument Division

is also greatly appreciated.

In addition, the author expresses thanks to the Instrument
Division for their financial support, for their sponsorship of the
work done on this thesis, and for the use of their facilities, in

particular, the analog and digital computers.

Sincere thanks are given to Miss Anna D'Angelo, Mrs. Emily
Lotter, Mrs. Maryann Smith, and Mrs. Linda Richmond for the typing

of the manuscript and its numerous revisions.

Finally, the author wishes to express his appreciation to
his wife, Betty. Without her interest and encouragement this thesis

would not have been possible.

ii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF APPENDICES

1.

INTRODUCTION

1.1 Statement of the Problem

1.2 Manual Control System

1.3 Characteristics of the Human Operator
1.4 Brief Survey of Present Models

1.5 Objectives of the Study
THE MODEL

2.1 Basic Approach
2.2 Structure of the Model
2.3 Solution of the Model

2.4 Basic Features
VALIDATION FOR TIME STATIONARY CONDITIONS

3.1 Experimental Results of Other Investigators
3.2 Model Solution
3.3 Model Validation

3.4 Sensitivity Studies

iii

Page

vi

xi

12

14

14
15
20

24

28

28
32
42

55

TABLE OF CONTENTS (cont)

Page
4. SINE«WAVE TRACKING . . . . . . . . . . . . . . . . . . . 74
4.1 Experiments . . . . . . . . . . . . . . . . . . . . 75

4.2 Tracking Model . . . . . . . . . . . . . . . . . . 80

4.3 Model Validation . . . . . . . . . . . . . . . . . 88
5. DISPLAY EFFECT . . ; . . . . . . . . . . . . . . . . . . 109

5.1 Description of Experiments . . . . . . . . . . . . . 109

5.2 Parameter Evaluation . . . . . . . . . . . . . . . 121
6. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . 135

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . 177

iv

Table

3.1

3.2

5.1

5.2

5.3

LIST OF TABLES

Forcing Function Magnitudes

Parameters for Model Validation
Display Study Forcing Function
Model Parameters for CRT Match

Model Parameters for DVM Match

Page

31
50
115
130

133

Figure

1.

1

.10

.11

.12

.13

LIST OF FIGURES

Manual control system schematic diagram
Compensatory task block diagram

Pursuit task block diagram

Quasi-linear model block diagram

Manual control system basic block diagram
Basic block diagram with an augmented plant

Solution block diagram

Simplified block diagram with an augmented plant .

Augmented plant for time stationary conditions .
Forcing functions
Match for the constant gain plant

Match for the constant gain plant with a highly
trained pilot . . . . . . . . . . .

Match for the integrator plant

Match for the unstable plant

Model results as P converges

A comparison Of the remnant with model noise
Unstable plant match -- human Operator only
Effect of various TN

Effect of the time delay

Effect Of a precision neuromuscular lag

Effect Of a second-order time delay

vi

Page

10
16
17
22
24
35
38

43

44
45
46
49
51
54
56
57
59

60

Figure

3.16
3.17
3.18

3.19

3.20

4.10

4.11

LIST OF FIGURES (cont)

Effect of the measurement noise

Effect Of the cost associated with u

Effect of the forcing function variance

Effect Of the drift

Effect of the forcing function corner frequency

Effect Of the order of the forcing function
bandlimit . . . . . . . . . .

Effect of the off diagonal elements of Q .
Sine-wave tracking block diagram

CRT display used for sine-wave tracking
Controller used for sine-wave tracking
Complete sine-wave tracking run

Partial sine-wave tracking run

Sine-wave tracking with the eyes closed after
6 seconds . . . .

Augmented plant for sine-wave tracking
Block diagram for sine-wave tracking
Mean sine-wave tracking model response
Mean x and 1

Solution for P

Complete sine-wave tracking model run

Partial sine—wave tracking model run

vii

Page

62
63
64
66

67

69

7O
74
76
77
78

79

81
82
82
89
90
91
92

93

4.16

4.19

4.20

4.21

4.22

LIST OF FIGURES (cont)

Solution for i -- with noise
Model tracking with eyes closed after 6 seconds

Solution for P with the eyes closed after
6 seconds

Digital computer printout Of P with the eyes
closed after 6 seconds .

Frequency response with P(10)
Frequency response with P(l)
Frequency response with PC35)
Sine-wave tracking model block diagram
Power spectral density match
Display study experimental equipment
Display study block diagram

CRT for the display study

DVM for the display study
Controller for the display study
Cross-spectral analyzer accuracy
Typical CRT run

Typical DVM run

CRT match -- Operator 1

CRT match -- Operator 2

CRT match -- Operator 3

viii

Page

94

95

96

100
102
104
105
106
108
110
109
111
111
113
117
119
120
122
123

124

Figure

5.14
5.15

5.16

8.10

B.11

LIST OF FIGURES (cont)

CRT match -— operators 1, 2, and 3
DVM match -- Operator 1
DVM match -- Operator 2
DVM match -- Operator 3
DVM match -— operators 1, 2, and 3

Effect of 2nd and 3rd order Pade approximations
Of the time delay

Basic flow chart for time stationary model
solution .

Example problem augmented plant
Expanded example problem block diagram
Input data

Solution for K

Solution for P

Filter and controller model
Constituent idempotents

Filter and controller transfer function
Man-plant transfer function

Man-plant frequency response data
Example problem Bode plot

The basic Operations to find the cross-spectral
density at wn

ix

Page

125
126
127
128

129

132

142

141
143
144
147
149
152
153
154
155
156

165

169

LIST OF FIGURES (cont)

Figure Page
D.2 Frequency shift and filter combined . . . . . . . 170
D.3 Simplified block diagram for ¢xy(wn) . . . . . . 175

LIST OF APPENDICES

Appendix Page
A Man-plant Transfer Function in Terms Of eAt . . . 138
B Solution for the Time Stationary Model . . . . . . 141
C Effects of g-and g-On the Model . . . . . . . . . 166
D Cross-spectral Analyzer . . . . . . . . . . . . . 168

xi

1. INTRODUCTION

1.1 STATEMENT OF THE PROBLEM

In the design of manually controlled systems, it is highly
desirable to have a mathematical description Of what the human Oper-
ator does in performing his tasks. This description must reflect
the effects of the plant on the man. It must be accurate and prac-
tical to generate without having to perform elaborate experiments

each time the plant configuration or parameters are modified.

In general, an adequate model should characterize a "repre-
sentative" Operator, yet with appropriate modification in parameters,
it should be possible to characterize a broad range Of Operators.

It should serve as a ”building block" in the analysis Of a complete

system, or any part Of one.

For a model Of the human Operator to be Of practical value
in design, it must be capable Of handling multi-axis problems,
account for noise, include prediction, and, in many cases, be time-
varying and nonlinear. It must handle the effects Of side task
loading and task difficulty, and reflect the effects Of different

types Of displays and manipulators.

Models presently available are useful only under restricted
conditions. They are for particular very simple plants, such as

YC = —%—, and they are linear, time stationary, not capable of
s

2
prediction, and do not include the effects Of stress and the input-

Output devices. The multi-axis case is handled only by extrapola-
tion from the single axis case. Given a model for the Operator
when controlling one plant, it is difficult to determine the man-

model when controlling another slightly different plant.

Large-scale simulators and human Operators are used at
present to design and evaluate vehicles, control systems, and in-
struments. The simulation system includes a cockpit mock-up with
instruments along with a computer simulation Of the vehicle and
control system. Such a system is necessary whether it is required
to evaluate a whole vehicle or any part Of it -- such as an instru-
ment. From a statistical evaluation point Of view, many Operators
and a large number Of runs are needed along with a great amount Of

data reduction tO evaluate a given design.

1.2 MANUAL CONTROL SYSTEM
1.2.1 The general System

In its broadest sense, a manual control system is one in
which a human Operator attempts to control a plant by varying se-
lected control variables Of the plant given information about the
system state or outputs from various sensor inputs. This function
is shown schematically in Figure 1.1. The control variable inputs
to the plant are provided by manipulators, as for example, a steer-
ing wheel, foot pedals or a control stick, and the sensory inputs
are provided by diSplays which give information about the condition
of the plant and/or about the input forcing functions. The displays
are usually visual, as for example, the visual field Of an auto-

mobile drive Or the altimeter in an airplane, but they might also

3

be of an auditory or tactile: nature [1]. In addition to displayed
information, the Operator can also in many cases sense the condition
or state Of the plant by means Of his vestibular receptors, and the
condition of a manipulator and the limb controlling it by means Of

joint, skin, spindle, and tendon receptors [2,3].

The plant may have external random forcing functions in the
form of disturbances such as a gust on an airplane. The Operator
may also generate his own forcing function, as for example, the
path a pilot desires to follow while approaching the runway when

landing an airplane.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r_— uuuau OPERATOR 1
FORCINO FORCINO
ruucnous l mm... | ,Jggmgs
1 DATA
1; j OUTPUT
I II , II
I.— _ __ _ _ _ _ _ _ _ _J

 

 

 

 

Figure 1.1. Manual control system schematic diagram.

The output characteristics (such as the arm or leg dynam-
ics), the manipulator, and the plant may all be integrally related
so that input-output equations for each cannot be expressed separ-
ately as a no load transfer characteristic. For example, the
control stick of an airplane represents a load on the arm Of the
pilot. This load, in the form Of a reaction force, is sometimes

referred to as "feel." It reflects, for some aircraft, the

4

aerodynamics of the control surface. The ”load” is sensed by the
Operator and may actually aid him in controlling the plant. A
simple spring-mass-damper system cannot, in general, be used to re-
present this reaction force since it is determined by the dynamics

Of the whole system.

A given set Of variables may be sensed by the Operator in
two or more different ways. For example, the angular acceleration
of an airplane may be visually displayed to the pilot by instruments
but he may also sense it directly by means Of his semi-circular
canals. (Note, however, that pilots are trained to selectively
gignore information sensed by their vestibular receptors and thus

this feedback path is not significant.)

The remaining parts of the human Operator as outlined in
Figure 1.1 are lumped into the data processing block. This block
includes such things as the transformation Of the sensed information
into a usable form so that it can be used in the control task (e.g.,
perception). It provides the feedback paths involved in the arm
dynamics, and it includes the compensation that the man provides in
an attempt to cause the man-machine system to perform as desired

(e.g., so that it is stable with minimum error).

1.2.2 Two Important Classes Of Manual Control Systems

Most of the research work in manual control has been re-
stricted to systems that have a single manipulator and a single
visual display showing the error between some input forcing func-
tion and an output Of the plant. This is known as a "compensatory

task." As shown in Figure 1.2, the difference between the forcing

5
function i(t) and the plant output m(t) is displayed to the Operator.

His task is to keep the error e(t) as small as possible using the
manipulator with output c(t). The manipulator is usually construct-
ed so that it reflects no ”feel” Of the plant and Often so that the

dynamics Of the arm and manipulator are minimized.

The compensatory task system is not as synthetic as it
might at first appear. Many tasks in flying an airplane, for exampka

are basically of this type. Such tasks as landing with glide slope

i") e") cm m(t)

ruwcnou . 0'3"" ”I ”an” MAMPULATOR KM" T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.2. Compensatory task block diagram.

and localizer needles, or flying at a constant altitude or pitch
attitude can be considered compensatory tasks in which the Object

is to maintain zero error.

Some work has been done on the multi-axis compensatory task
involving, for example, two diSplays and two input controls to the
plant. There may or may not be cross-coupling in the plant be-
tween the two inputs and the two displays. In general, the models
that have been developed are extrapolations of the models found

adequate for the single axis task.

A second class of manual control problems that have been

considered is the "pursuit task." As shown in Figure 1.3, both the

6

forcing function and the plant output are displayed separately
(usually with the same type Of display or on the same display, e.g.,
a dual beam CRT) where the task is tO keep the difference between
the displayed variables as small as possible by controlling a single
plant input c(t). More information is available tO the Operator in
the pursuit task since he can see the characteristics of the forc-

ing function and the plant output in addition tO the error.

Ht
FORCINO DISPLAY

FUNCTION
E DISPLAY

 

c(t) m(t)

 

 

 

TL

uuuau
J. OPERATOR uAmpuuroa—a PLANT

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.3. Pursuit task block diagram.

1.3 CHARACTERISTICS OF THE HUMAN OPERATOR
The human Operator characteristics can be grouped under the
overlapping headings, "General Characteristics," "Remnant Character—

istics," and "Physiological Characteristics."

1.3.1 General Characteristics
The general characteristics include three important

features as follows:

1) Adaptability ~ When confronted with a new set of cir-

 

cumstances, or a change of environment, the human
Operator will adapt or change his performance so that

he accomplishes his task "better" with time.

2)

3)

7

Learning - All external factors remaining the same, the
human Operator will improve his performance with ex-

perience.

Optimality - It is generally presumed that the human

 

Operator does the "best" he can in a control task as
measured by some cost function. The particular cost
function he uses and the way he accesses it is not
known. Many investigators have hypothesized that in
compensatory tracking tasks the RMS error is minimized

[4,5,6].

1.3.2 Remnant Characteristics

Only part Of the output power spectrum of the human Oper-

ator can be linearly correlated with his input. The part that is

left over is referred to as the "Remnant." This remnant is usually

subdivided into nonstationarity, nonlinearity, and noise.

1)

2)

Nonstationarity - During the execution of a task, the

 

performance Of the human Operator changes with time.
This is not only the case from run-to-run but during
each run itself. Some Of the factors that apparently
contribute to it are attention, motivation, adaption,
and learning. McRuer et a1. [5] state that nonstation-

arity is a major source of the remnant.

Nonlinearity - Such characteristics as sensory thres-

 

holds, controller displacement limits, force limits,
intermittency, and indifference thresholds are known

contributions to nonlinearities in the input-output

3)

8

characteristics of a human Operator. McRuer et a1. [5]
indicate, however, that nonlinearity is not a major

source Of the remnant.

Noise - The amount and source of noise in the human

Operator is not known and its importance is considered
to be small [7]. However, as stated by Pew et al. [8],
"Analysis of the error power spectra establishes the
presence Of a stable source Of noise power in the
Operator's output that has implications for deriving

models of manual tracking performance."

1.3.3 Physiological Characteristics

There are certain physiological characteristics Of the

human operator that have been investigated and, at least by deduc-

tion, have been characterized in terms Of control theory.

1)

2)

Time Delay - The response of a human Operator exhibits

 

a pure time delay that is generally attributed "to
sensor excitation, nerve conduction, computational
lags, and other processing activities in the central
nervous system" [5]. Time delays Of from 0.1 to 0.2

seconds are used [5].

Neuromuscular Lag - The dynamics of the limb of an

 

operator in response to "commands from the brain" is
frequently described by a transfer function with a

lagging characteristic. Descriptions and justifica-
tions for such a model are given in [2,3,5, and 10].

A first-order lag with a time constant Of from 0.1 tO

9

0.6 seconds [5] is the simplest and most used form,

but up to fourth-order models have been defined.

1.4 BRIEF SURVEY OF PRESENT MODELS

There have been many approaches and methods used in an
attempt to model the human Operator in a control task. The tech-
niques used almost invariably reflect the most recently available
tools. As stated by Muckler [11], "those who work in manual control
theory have been very quickly responsive to changes in general con-
trol theory." However, Muckler adds a note of caution, "Bandwagons
can be a useful mode Of travel -- provided the band is going down
the right street." He also points out that there are literally
thousands Of articles that have been published which list and de-
scribe the more significant work that has been done [10 through 14].
In addition, bibliographies of the more significant contributions

have been compiled as given by [15,16].

Some models attempt to simulate on a computer the various
"black boxes" in a conceived structure Of the human Operator. In
some Of these attempts [17,18] such things as sensory thresholds,

rate limits, and dither are included.

There are some who say, because the human Operator has an
intermittent output, he acts like a sampled data system and should
be modeled as such. Although it is generally agreed that the human
Operator does not act like a periodic sampler, useful results have
been Obtained with this concept. Note in particular the work of
Bekey [14,19]. A lack of adequate control theory to handle the

aperiodic problem is in part responsible for the limited work based

on the sampling concept.

10

Others have said that the human Operator can be modeled as
an adaptive mechanism. To quote Shinners [13], "Most of the work
done in this area recognizes the problems involved and provides,
instead Of exact solutions,guidelines, hypotheses, and constraints
to be considered in the formulating Of a representative model." The

work described by Young and Stark [10] is an example.

By far the most tested and applied model is the quasi-linear
describing function model as developed by McRuer et a1. [4,5]. This
model as shown in Figure 1.4 consists Of a "describing function"
and a "remnant." The describing function gives the output of man

that is linearly correlated with the input. The remnant gives the

r- — “CFC-T.
“mum
Illa) I‘ll“) . “I”, “I“,

 

 

.
ozscmamc
I ruucnon 'L “I

_____J

 

 

 

 

 

 

 

Figure 1.4. Quasi-linear model block diagram.

power at those frequencies of the total output power spectrum that
are not given by the describing function. The describing function
and remnant depend on the plant that is being controlled and upon
the input forcing function. McRuer et al. [4,5] base this model on
experiments using several basic types of plants (Kc, KC /s,

Kc /(s-A), Kc /sz, and Kc /[5 (5-1) 3) and have devised basic model

fOrms and a set Of rules tO be used for determining the Operator's

11

describing function. The most complete form Of this model, referred

to as the "precision" model in [5], is

. T ' +1 T ' +1
Y ___ K e-de L“ K“ 1

p. p Tij+1 T'jw+1 (T n+1) [KEY + ENE”— + 1]
K N1 “’N “’

Gain Time Equal. Lag-Lead

 

 

Delay \— 1", J
Neuromuscular
Characteristic
(1.1)

The neuromuscular characteristic has been deduced using
physiological considerations and feedback control theory [2,3]. The
time delay is attributed to sensor excitation time (e.g., the retina
Of the eye), and nerve conduction time. The gain and equalization
are the characteristics which are added by the human Operator to
provide the complete closed loop man-machine system with good sta-

bility and response.

There are simplified versions of the "precision" model in
[5] which are referred to as the "crossover" model, the "first ap-

proximation," and the "second approximation.”

The crossover model is the simplest and as the name implies,
on an Open loop Bode diagram,the model fits the data best at the
frequency where the gain crosses the zero db line. This model in-
cludes the plant that is controlled and is expressed by

-ere
w e
c

YPYC = T (1.2)

12

where

Y is the pilot transfer function
Y is the plant transfer function
w is the crossover frequency

T is the effective time delay

The choice of the parameters to be used in each Of McRuer's models
is an art, where this choice is made using a knowledge of the plant,
the input function, and a set Of rules. These rules are based on
feedback control theory and the results Of the experiments using the

five plants as defined above.

The first application Of modern Optimal control theory to
the manual control problem was done by Thomas [20]. Obermeyer and
Muckler in [21] give a general discussion on manual control in the
context Of modern control theory. Topics such as the Maximum
Principle, Dynamic Programming, and adaptive control systems as
given by Kalman et a1. [28] are briefly described and related to the
manual control problem. More recently, Obermeyer and Muckler [9]
have turned the problem around, concerning themselves with the
"inverse Optimal control problem." That is, assuming that the
operator performs in an Optimal way, what performance index does he
Optimize? They restrict themselves to a quadratic cost function

and as reported in [9], the results so far have been inconclusive.

1.5 OBJECTIVES OF THE STUDY
The primary Objectives Of this research are to postulate

and demonstrate the validity of a mathematical model Of the human

13

Operator in a control task. It is to be shown that this model agrees
with certain data available in the literature, and, in addition,

that it exhibits characteristics heretofore not adequately included
in a model. A method for ordering the relative values Of displays

in a control task is also suggested and evaluated.

2. THE MODEL

2.1 BASIC APPROACH

It is generally agreed that noise is present in the human
Operator in a control task [5,8]. Whether this noise exists at his
input or output, or both, has not been established [13]. Many model-
lers have considered the human Operator as a controller, but none
have given serious attention tO the perception mechanism at the in-
terface between the displayed information and the control Operation.
A model Of this mechanism is required to transform the raw displayed
information into a usable form. It is therefore postulated in this
thesis that in a control task, this transformation is performed in
the presence of noise, as an Optimal time-varying linear filter (a
Kalman filter). It is further postulated that in performing the
control function, man uses the resultant estimates of the state of
the system as an optimal time-varying linear controller. The data
processing block in Figure 1.1 accordingly includes an Optimal fil-

ter and an Optimal controller.

It is assumed, in general, that Optimal estimates are made
0f.flll.°f the state variables of the system including those Of the
plant being controlled, those associated with the forcing functions,
and those associated with the man's own sensor and output character-
istics. (This is not a necessary assumption and further research may
prove this not to be the case, but it is assumed to be valid in this

thesis.)
14

15
Regarding the human Operator as having "Optimal” character-

istics is not tO say that he performs perfectly, but only that he
attempts to Operate that way. The performance Of the model is de-
graded by the filter as influenced by the noise. Therefore, the

performance of the model, in general, is suboptimal as compared to

an Optimal controller by itself.

Modern control theory with state-space techniques [22] are

used to provide the conceptual framework for systematically dealing
with this large scale multi-variable multi-axis system. The model
is therefore formulated as a set Of state equations using parameters
that may vary somewhat from one Operator to another. Consequently,

once the model is programmed, a variety Of solutions are Obtained by

inserting the appropriate parameters.

By formulating the problem in terms Of a state model, it is
not intended to imply that the actual human Operator thinks in terms
of a particular set Of state variables; only that his performance
can be so characterized. A differential equation model implies that
the human Operator is characterizable as a continuous system. This,
however, is not fundamental to the development since the human Oper-
ator can also be regarded as a sampled data system if desired. The
Optimal filter and Optimal controller in this case are characterized

by difference equations.

2.2 STRUCTURE OF THE MODEL

As shown in Figure 2.1, it is postulated that in a control
task, the human operator consists Of a Kalman or Optimal time-varying
linear filter, an Optimal time-varying linear controller, sensor and
output characteristics with noise inputs v and wc , and internal

forcing functions.

16

I HUMAN OPERATOR 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

roncmo roacme I macaw.
w
ruucnous I v ruucnous ‘ runcnous
«ALIA» OPTIIIAL ounur
DISPLAYS stwsons n L" n cournouzn cmuc'rrmsncs MANIPULATORS run

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I !__I ____________ _1

Figure 2.1. Manual control system basic block diagram.

To solve for the Kalman filter and the optimal controller,
the system is rearranged into two overlapping parts as shown in
Figure 2.2. The human Operator part remains the same as in
Figure 2.1, while the augmented plant contains everything but the
filter and controller. With this rearrangement, the filter and con-
troller can be found given all the characteristics Of the augmented
plant. However the model Of the augmented plant must be in the form
of linear, possibly time-varying, differential and algebraic equa-
tions. Statistically defined inputs (wc and/or forcing functions)
are included by "shaping" Of white gaussian noise through a filter
modeled by a set Of linear, possibly time-varying, differential equa-
tions. The "measurement noise" v is assumed to be additive white
gaussian noise and all components Of v are assumed to be present.
These restrictions on v can be relaxed somewhat by using the
techniques given by Bryson and Johansen [23]. In addition, the
linearity restrictions can be removed if the augmented plant can be

quasi-linearized [24,47].
The augmented plant process equations are

i = Ax + Bu + Dw, x(0) = x0 (2.1)

17

 

—"_—__'l

[— Aueucw‘rso PLANT
_ _ _ _ _ _ Poncme
'] FUNCTIONS
'c I

 

 

I noun OPtnAtm I
IaLAIAu OPTIMAL OUTPUT I
J TV CHARACTERISTICS ““"UUTWS "“7

I FILTER CONTROLLER
I II 3 I
l |

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
I
{ seasons 3 DISPLAYS ‘
I I I “_— I Q I
I : ;522$:g:s V I PmI'lzc'r'I'Ozs I

 

 

 

 

 

[___ ___EZZZ—J I

___—___J

Figure 2.2. Basic block diagram with an augmented plant.

and the measurement vector is

where

and

(2.2)

y = Cx + v
E{v(t)} = E{w(t)} = E{XO} = o (2.3)
E{v(t + T) vT(t)} = N5(T) (2.4)
E{w(t + T) wT(t)} = M6(T) (2.5)

x is the augmented plant internal state vector Of

dimension n
u is the optimal controller output of dimension m
y is the measurement vector Of dimension 1
v, w are uncorrelated gaussian white noise vectors

x0 is a gaussian random vector uncorrelated with v
and w
A, B, C, D, M, N are, in general, time-varying matrices

6(T) is the unit impulse function

18

The scalar cost function J to be minimized by both the

filter and by the controller is a quadratic function Of the form,

J = E % (xTQx+uTRu)dt (2.6)

where
E is the expected value Operator

Q is a constant square symmetric positive semi-definite

matrix

R is a constant square symmetric positive definite

matrix

t is the final time

As shown in [25,26] for the discrete case, the Kalman
filtering problem and the Optimum controller problem can be solved

separately.

First, consider the Optimum linear filter as developed by
Kalman [27 through 31]. The Kalman filter, in general, is a
minimum variance estimator. It is an Optimum linear estimator if
the white noise is non-gaussian and is the_optimum estimator if the

noise is gaussian [25]. It will be assumed, however, that the noise

is gaussian.

The cost function that is minimized is E{(x - i)T(x - i)}

where x(t) is the Optimum estimate Of x(t) given the Observations

19

y(r), O :p T :_ t . The estimate Of the plant state vector as

given in [27 through 31] is

i = (A - kc - BR-IBTKJx + Ry, 1(0) = 10
(2.7)
where
R is the Optimum linear filter gain matrix
and
K is the Optimum linear controller gain matrix
As given by Kalman, R is expressed by
k = PCTN'1 (2.8)

where P(t) is the covariance Of [x(t) - x(t)] given y(T),

0

[A

r :_ t, and is determined by solution of the matrix Riccati

equation

P = AP + PAT - PCTN-ICP + DMDT (2.9)

where the initial condition is

P(O) = E {(xo - :20on - iof} (2.10)

20

The output of the Kalman filter is the Optimal linear
estimate of the augmented plant state vector and is used as the
input to the Optimal controller. Given the state vector as input,
the Optimal control problem reduces to the regulator problem. Using
Pontryagin's maximum principle (see for example [32], [33]) the
optimal control is

T

u = - R-IB Ki (2.11)

where K is Obtained as the solution Of

R + KA + ATK - KBR-IBTK + Q = 0 (2.12)

Since x(tf) is arbitrary, the final condition can be taken as

K(t = 0 (2.13)
2.3 SOLUTION OF THE MODEL
2.3.1 General Solutions

The complete system model can be solved on a digital,
analog, or hybrid computer using (2.1) through (2.13). The Optimal
controller gain matrix as a function Of time is first found by solv-

ing (2.12) backwards from K(t = 0 to K(0). Then the following

f)

matrices are solved for simultaneously from t = 0 to t = tf :

l) the Optimal controller gain matrix K(t) defined

by (2.12) using the initial condition K(0)

2) the covariance matrix P(t) defined by (2.4),
(2.5), (2.9), and (2.10)

21
3) the Kalman gain matrix R(t) by (2.8)

4) the Optimal linear estimate of the state vector

i(t) defined by (2.7)
5) the Optimal control law u(t) by (2.11)

6) the augmented plant response y(t) defined by
(2.1) and (2.2)
Note that if the augmented plant is time stationary, then for tf
very large the Optimal controller gain matrix can be taken as a
constant with K(t) = K(0) over the time interval Of interest, say
from t = 0 to t = t'. The time tf is sufficiently large, if,
when (2.12) is solved backwards, the solution converges to a con-

stant before time t' is reached.

If the beginning period Of time, in addition to the end
period Of time, is not significant, the resultant model for the man
is time stationary. In other words, the model is time stationary if
sufficient time is allowed at the beginning for the solution Of P
to converge to a constant, and sufficient time is allowed at the

end so that K remains constant.

For the time stationary case, the man-model can be written

as a transfer function matrix

UC(S) = H(s) Y(s) (2.14)

22

where
Uc(s) is the s-domain output vector of the human operator
H(s) is the s-domain transfer function matrix

Y(s) is the s-domain measurement vector

The case when Uc(s) and U(s) are of dimension one is considered

in Chapter 3.

A block diagram illustrating the solutions of the augmented
plant, the Kalman filter, and Optimal controller is given in
Figure 2.3. As indicated, it is postulated that the human Operator
formulates a model that is nearly the same as the augmented plant
that he is trying to control. The basic difference lies in the in-

clusion Of R and K.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r--- ._. __ ._. __ __ __ __ __.._ __ __
l | I
I SOLUTION. I 1 I
FOR P ANOK l I
+ A A
I + a + x f 'L.0_R*." _u u. 1 I
l ‘ * [1 | I
I - m" I I
c 3 I
I I
e
I T l
I uLuAu PILTn ANO OPTuIAL comm" L. wanna run A

Figure 2.3. Solution block diagram.

The first part Of the model -- the time-varying filter —- uses the
sensed variables y which are corrupted by noise, and Operates on

them in such a way as to produce time-varying estimates Of all of

23

the state variables Of the augmented plant. These estimates are
Optimal estimates (if the noise is gaussian), in the sense that the

variance of the estimation error is minimized.

With this formulation of the problem, compensatory as well
as pursuit displays can be handled. In the compensatory task, the
value of the displayed error as sensed by the human Operator appears
as a single variable in y . In the pursuit task, the forcing
function and plant output, as sensed by the human Operator, each ap-
pear as a variable in the vector y . The difference between them,
or error, is determined in the controller using estimates Of each

variable as given by the filter.

2.3.2 A Simplified Form

The formulation as given above is very general and provides
a framework for including factors in the human Operator about which
very little is known. 80 that the model can be studied in terms of
available data and in terms Of some Of the general characteristics
Of the human operator that have already been established, it is re—
duced to that shown in Figure 2.4. This model is used in the remain-
der Of the study. As indicated, a compensator task is used where
the display characteristics are ignored. The Operator sensor (except
for the noise) and output characteristics are lumped into a time
delay and neuromuscular lag. Also included in the neuromuscular lag
are the manipulator characteristics which are assumed to be inde-
pendent Of the plant. (A detailed investigation using pressure, free-
moving, and spring restrained manipulators is given in [34].) The

noise inputs v and wl are included to represent man's

24

I" — — _‘Ifiuufio $71.?- _'— “—7
r‘ — 7.3.53.5...— ‘i — — .7 — — — ‘1
I ' II I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FORCINO
I I s"""‘° I FUNCTIONS I
A w u
c c
| ‘ I + * ' + ‘ I
KALMAN OPTIMAL Tm: usunowscuun 4 9L.“
FILTER CONTROLLER 1 DELAY L A6 I
I a "I I
I swAPIuO I I

 

 

 

 

 

 

Figure 2.4. Simplified block diagram with an augmented plant.

measurement and output noise, respectively. The noise input w2 ,
as explained more fully in a later section, is included to repre-

sent an internal forcing function.

2.4 BASIC FEATURES
There are many potential advantages Of the model given here
over other models presently available. Some Of these possible ad-

vantages, as indicated below, are investigated in this research.

2.4.1 Precognition

The model is capable of handling the precognition problem
since the filter and controller part of the model (Figure 2.3), con-
tains a model of the plant, the forcing functions, and the man's own
physiological characteristics. The model "knows" the augmented
plant and can predict its performance. Precognition Of a forcing

function is studied in Chapter 4 with sine-wave tracking.

25
2.4.2 Noise

The noise contribution is included in the model by v and
w. Since v and w are white noise, high frequency components are
present and indeed provide the means for introducing high frequency
outputs in the Operator. If required, the form Of these inputs can
be shaped in the augmented plant by combinations Of leading and lag—
ging characteristics. The variables v and w are vectors and can
be used, if desired, to inject noise at several points internal to
the operator as provided by the augmented plant. The amount of noise
and the shaping required to give a good model for the man remain to
be determined. It is noted that the amount and character Of the
noise is very important because it has a significant effect, as
shown later, on the filtering characteristics and thus on the model
that is generated for the man. This, Of course, is in addition to
the noise itself. Such factors as task difficulty and loading,
stress and fatigue, and the effects Of different displays might be
considered in terms Of noise inputs. The effect Of two vastly dif-

ferent displays is considered in this way in Chapter 5.

2.4.3 Nonlinearity

The present state-of—the-art in optimal linear filtering
and control theory extends to nonlinear plants that can be quasi-
linearized. In such cases the first variation Of the nonlinear
model is used. This will not be considered here but solutions with

nonlinear plants are entirely possible.

2.4.4 Nonstationarity

Assuming a stationary augmented plant, nonstationarity Of

26

the man is accounted for by the time-varying filter. This time
variation corresponds to a time-varying estimate Of the augmented

plant state vector. For example, with a step input, the operator

has a preconception of the variance of the step and the other state
variables. This information is contained in the initial value of P

as given by (2.10). Subsequent to the application Of the step, the
variance Of the operator's estimation error of the state vector changes
with time, eventually approaching a steady-state value. Beyond this
time the operator's estimate of the state variables will not improve
substantially. How good the estimates are depends on the variances

of the noise vectors v and w that are present.

Since the sensor and output characteristics are included in
the augmented plant, any time variations of the parameters of the
corresponding model contribute to nonstationarity. Also, the noise
might be time-varying, giving rise to a time-varying filter with no
”steady-state." The observed intermittency characteristics of man

[19] might possibly be included in this way.

2.4.5 Multiple Inputs and Outputs

The number of inputs to the plant to be controlled theo-
retically can be any number (e.g., roll, pitch, and yaw) since uC
is a vector. Similarly, the number of inputs to the operator is
theoretically unlimited since the input y , representing sensed
measurements, is a vector. The coupling between axes of a plant is
automatically included in the model since, as shown in Figure 2.3,
the operator "contains" a model of the plant which, of course, in-

cludes the coupling. The effect of a multi-axis task on single-

27

axis performance might possibly be included by the addition of noise

at appropriate points in the model.

The noise inputs might be of a time—varying nature when
multiple tasks are being performed on a part time basis. Thus, when
another task is taking much of the Operator's time, the noise would
be large; when the Operator is paying more attention to the central
task of interest, the noise would be small. This would give a time-

varying characteristic to the filter and thus to the model Of the

man.
2.5.6 Forcing Function Characteristics

The model includes the forcing function characteristics.
The type of forcing function -- whether generated externally or not,

as for example, band-limited white gaussian noise, or a sine-wave --
is automatically included in the model of the man since it is in the

augmented plant.

3. VALIDATION FOR TIME STATIONARY CONDITIONS

The objective of this chapter is to validate the model by
showing how well the model results match data generated experimen-
tally by others in a single-axis compensatory task. The experimen-
tal data are in the form of Bode plots for the linearly correlated
portion of the response. Power spectral density plots for the re—

maining power (the remnant) are also considered.

McRuer et a1. state in [5] that nonstationarity is a large
part of the remnant. Their Bode plots, however, are obtained by
time averaging over a complete run and therefore the nonstationarity
during the run is ignored. Assuming the model can prOperly include
this characteristic, an "average" of the time response is required
to Obtain a time stationary Bode plot. It is assumed in the results
presented here that the noise can be selected so that the P matrix
reaches a steady-state condition which corresponds to ”the average"

Bode plot.

3.1 EXPERIMENTAL RESULTS OF OTHER INVESTIGATORS

The data given by McRuer and Krendel in [4], those given by
McRuer, Graham, Krendel and Reisener in [5], and those given by
Wasicho, McRuer and Magdaleno in [41] are used. These data were
selected because they appear to be very reliable. They are in
general agreement with the data of Others (e.g., Elkind in [40]),

and they are in a form that is readily applied to investigate

28

29

the model. Since the data in each case were generated for

the purpose of matching quasi—linear describing functions, they are
given in the form of open loop frequency response plots (Bode plots)
along with power spectral density plots of the remnant. The data
given in [5] are used primarily and the experiments that were per-
formed are described briefly below. The differences between [5] and
the other references are pointed out when the other references are

used.

McRuer et al. in [5] obtained their data using nine dif-
ferent pilots in a compensatory task (see Figure 1.2) with five dif—
ferent plants to control. They used the sum Of ten sine-waves as a
forcing function. The display was a six inch CRT and the manipu-
lator was a hand operated stick with motion restricted to the roll
axis. The stick was a low inertia, low damped spring restrained
device. With a constant gain plant (Yc = l), the display motion was
one inch per six degrees of stick motion. The force gradient was

2.21 oz/deg at a four inch moment arm.

Their data were generated using a cross-spectral analyzer
where the cross-spectral density at each of the ten forcing function
frequencies was measured and averaged over a four minute period. An
initial 10-15 second transient period was allowed for the operator
to reach a stable tracking condition. Prior to taking the data,
each pilot was given a sufficient number of two minute trial runs to
minimize variability due to practice effects. It was assumed that
this point was reached when his RMS errOr reduced to a relatively

constant value from run-tO-run and usually took from ten to twenty

trial runs.

30

The ten forcing function frequencies were uniformly spaced
over the frequencies of interest. Also, the frequencies were picked
so that there would be no overlapping harmonics, and so that there
would be an integral number of cycles over the four minute run. The
sum of sine-waves produces a ”random appearing" input that is easily
generated, and, in addition, the sine-waves are mathematically
tractable in the cross-spectral analyzer. Since the selected forc-
ing function is random appearing,the operator cannot ”learn" it.
McRuer et al. in [5] show that the sum of ten sine-waves that they
selected is gaussian with a Chi-squared test at the 0.05 level. As
shown in [39] as few as five appropriately chosen sine-waves are

nearly gaussian.

The one—sided power spectrum of the sum of sine-waves is

N

eii(w) = nn§1¢:(wn)6(w-wn) (3.1)

where ¢innI is the half—amplitude of the nth sine-wave. The

variance or mean-squared value is

1
1 EFF IiiI“)dw (3.2)

II
Nh—I
Z

Z Iin.) W”

31
Most of the data in [5] were taken with a display variance of

oi = 0.25 in2 (3.4)

Table 3.1 shows the frequencies and magnitudes of the three forcing
functions, denoted by F1.5, F2.5, and F“, used by McRuer et al. in
[5]. Note that in F1.5: F2o5, and F” the first six, seven, and
eight frequencies, reSpectively, are at one power level while the
power at the remaining frequencies are 20 db lower. The subscript
refers to the frequency of the last Sine-wave at the higher power

level.

Table 3.1. Forcing function magnitudes.

 

 

 

¢IImnI - Power db

wn-Rad/Sec F1.5 F2.5 F“

m1 = 0.157 -10.8 -11.5 -12.0
wz = 0.262 -10.8 -11.5 -12.0
03 = 0.393 -10.8 -11.5 -12.0
mg = 0.602 -10.8 -11.5 -12.0
05 = 0.969 -10.8 -11.5 -12.0
06 = 1.49 -10.8 -11.5 -12.0
07 = 2.54 -30.8 -11.5 -12.0
08 = 4.03 -30.8 -31.5 -12.0
mg = 7.57 -30.8 -31.5 -32.0
010 = 13.8 -30.3 -31.5 -32.0

 

 

 

 

32

The experimental data, given in the form of 1 lo ranges

about the mean, are superimposed on the corresponding results ob-

tained from the model and are given in later sections.

3.2 MODEL SOLUTION

As indicated in Chapter 2, a time stationary model results

when:

1)

2)

3)

4)

the augmented plant is time stationary
the noise is time stationary

the initial time period required for the P matrix
to reach an essentially constant value is ignored
so that the filter gain matrix K can be considered

to be a constant

the final time period required for the control matrix
K to reach an essentially constant value (when solved

backwards) is ignored.

Under these conditions, the time stationary model for the

filter and controller parts of the human operator given in (2.7)

and (2.11) become

SE = Ax + Ky (3.5)

u = a; (3..)

33

where

A - kc — BR-IBTK (3.7)

>>
II

c = - R—IBTK (3.8)

A

and A, C, and K are constant matrices.

For a single-axis compensatory task u and y are
scalars. As shown in Appendix A, the transfer function for the man

without the physiological characteristics is

 

A At A
1(5) = CZIe }K (3.9)

The available experimental data are for a complete system, including
the human operator and the plant he is controlling. The transfer
function for the complete man-plant combination, including the
physiological characteristics is obtained by multiplying the trans-
fer function in (3.9) by the transfer functions for the plant and

the physiological characteristics.

Bode plots for the time-stationary model Of the human oper-
ator and plant so obtained are generated by a digital computer pro-
gram as described in detail in Appendix B. The inputs required by
the program are the matrices A, B, C, and D from the augmented plant
as given by (2.1) and (2.2), the matrices M and N which define

the covariances of v and w as given by (2.4) and (2.5), the

34
matrices Q and R from the cost function as given by (2.6), and
the transfer functions for the physiological characteristics and the
plant. The program gives the steady—state optimal controller gain
matrix K , the steady-state Optimal filter gain matrix R , the
state model for the optimal filter and Optimal controller, the
transfer function for the optimal filter and optimal controller,

the transfer function for the man-plant combination, and a Bode

plot for the man-plant combination.

The particular model used for the augmented plant is shown
schematically in Figure 3.1. Three of the five plant transfer func-
tions Yc presented by McRuer et al. in [5] are investigated. They

are:

l) the constant gain plant

Y = l (3.10)
2) the integrator plant
- 1

Y - s (3.11)

3) the first order unstable plant

(3.12)

3S

 

 

   

 

Tm: utunouuscIJL AI
OtLAT LAO
"IE—T " I“)

 

 

 

 

PORCINO
FUNCTION
SNAP! NO

 

 

 

 

Figure 3.1. Augmented plant for time stationary conditions.

The neuromuscular lag is nominally approximated by a first-

order lag

GN(S) = -————- (3.13)

where TN is taken to be 0.3 seconds. The time delay is nominally

approximated by a first-order Padé approximation [35]

T
- —-s+l
GT(s) = -——————- (3.14)

-s+l
where T is taken to be 0.1 seconds. This time delay is assumed
to be due to physiological characteristics only (conduction, etc.)
and as given by McRuer et al. [5] ranges from 0.05 seconds to
0.1 seconds. Ranges of TN from 0.1 to 0.6 seconds and ranges of
r from 0.06 to 0.15 second are considered in Section 3.4. A pre-
cision neuromuscular lag and a second-order time delay are also con-

sidered in Section 3.4.

36

As shown in Figure 3.1, there are three gaussian white
noise inputs. The measurement noise v is a single dimensional in-
put defined by (2.3) and (2.4) while process noise terms wl and
wz form the two dimensional vector w defined by (2.3) and (2.5).

The variables v, wl, and wz are orthogonal with

va(r) = N6(T) = n6(r) (3.15)
“111 0

wa(r) = M6(T) = 6(1) (3.16)
0 “[22

va(r) = 0 (3.17)

McRuer et al. considered one-sided power spectrums where

Oxx(w) = 4 I Rxx(T)COSwTdT (3.18)
o
= 2 I Rxx(T)COSwTdT (3.19)
Since by definition
I 6(T)coswrdr = 1 (3.20)

the one-sided power Spectrums of v and w become

¢VV(0) = Zn (3.21)

37

211111 0
(I) (w) = (3.22)

The forcing functions given in Table 3.1 are approximated
by gaussian white noise band-limited by a third-order lag filter
with corner frequencies centered around the last sine-wave at the

higher power level. For the forcing function F2.5 the shaping is

 

 

(3.23)

Figure 3.2 shows the magnitudes squared of each of the sinusoidal
components of the forcing function F2.5 superimposed on the shaping
function GF . To illustrate the fit of the approximation, the low
frequency portion of CF is made to coincide with the lower fre-
quencies of the given forcing function. The variance of the model
forcing function ngl is adjusted by m11 in (3.16) so that it is
the Same as that used by McRuer et al. in [S] for F2.5. The vari-
ance of ggwl by (3.2) is

I“)

¢w1w1(w)dw

1 ‘ 5;" . . . 2 (3.24)
(19+ III-9+ )(e I)

a

 

 

 

 

¢lz(wfl)-db

38

 

 

 

m11 abc (pg-c2)bc-(aZ-c2)ac+(a2-b2)ab
2 2 2 2 2 2 (3'25)

(3 -b )(a -c )(b -c )
Taking a = 2.4, b = 2.5, and c = 2.6, the variance is

oi = 0.468 ml, (3.26)

and the required value for “11 using (3.4) is
0.25
“11 - 0.468 - 0.534 (3.27)

 

C)J

l1) lClC)
(I) ~ RAD/ SEC

Figure 3.2. Forcing functions.

39

Experimental results in [4] for a constant gain plant show
that the sum of sine-waves with power levels arranged along the
curve of a third-order lag, and, the rectangular arrangement used in
[5] (Table 3.1) both yield essentially the same transfer function
for the Operator. The experimental differences between third-order
shaping and the rectangular shaping for other plants are not given

in [4] and are assumed to be small. (They may not be, however.)

Referring to Figure 3.1, the noise inputs to the measure-
ment y consist of a pure gaussian white noise term v and a
gaussian white noise term w2 shaped by a pure integrator of the

fonn

00(5) = i (3.28)

The noise term v corresponds to measurement noise as generated,
for example, by the retina of the eye. The input w2 shaped by CD
is viewed as a random drift rate or ”searching" which is applied and
estimated by the operator. As utilized, this input is an internal
forcing function injected at the display. Thus, it is postulated
that the human operator supplies this input (e.g., at the input to
his neuromuscular system) in such a way that it appears as a ran-
dom drift rate on the display. The Operator is randomly "testing”
the system by applying a random rate at the output of the system.
(In some respects this input is like "dither," although "dither” is

usually considered to be at one frequency.)

Since the white noise inputs v and wz are orthogonal

by (3.17) for all values Of

call it w25 , is also ort

7.7
r'x
v-I
V
II

Since v and "25

the sum is

(w) =

V+w2$

From (3.21 and (3.22)

(w)

<I>

40

T , the output of the noise shaping,

hogonal to v.

E {v(t+r)w25(t)>

t
E v(t+T) I w2(a)da
o

t
I E V(t+T)w2(a)‘B do
0

J

t
I R (t+T-a)da = O
VW2
0

Indeed, for all

t

(3.

.30)

.31)

32)

are orthogonal, the power spectrum of

1
(bl/V0”) + $2— <I>w2w2 ((1))

m
jw + 22

 

jw

 

(3.

(3.

(3.

33)

34)

35)

41

Normalizing by the variance of the forcing function given by (3.26),

(3.35) yields

0 (w) . m22 2
V+w25 2n Jw + n
-————————— = . (3.36)

2
m
. ..+ 7—5.2
4.27 . (3.37)

Note that(3.37) only depends on the ratio of %- , and as Shown in
Appendix D for a slightly more general case, so does the solution

of the model.

The cost function from (2.6) is

f
tf )

(2.6)

V?

J = E %- (xTQx+uTRu)dt

 

 

O

L J

It is assumed that the mean-squared value of the displayed variable

plus that of u is minimized. Thus,

Q = cc (3.38)

Since u is of order one and with

R = r (3.39)

42

the cost function reduces to

tf
l T 2
J = E 3 (Cx) (Cx) + ru (it (3.40)
o
3.3 MODEL VALIDATION

The general problem is: given models of the plant, the
neuromuscular lag, and the time delay; what noise and cost function
are required to match experimental data? This might be called the
"inverse optimal filter and control problem." More specifically for
the case considered here, the problem is: given the simple models
for the neuromuscular lag and time delay as defined by (3.13) and
(3.14) with TN = 0.3 and T = 0.1, and given the forcing function
defined by (3.23) and (3.27); what values of n, r, and m22 are re-
quired to match the model results to the experimental data for the
plants defined by (3.10), (3.11),and (3.12)? Bode plots showing

selected matches for these three plants are given in Figures 3.3,

3.4, 3.5, and 3.6.

The model plots correspond to the complete man-plant com-
bination with a unity feedback Open—loop transfer function C(jw).

The solid and dashed lines are gain and phase respectively where,

Gain in db = 20 16g10 |G(jw)l (3.41)

and

Phase in degrees = g§(jw) (3.42)

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44

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_me)flfl= hufififimcu

 

out

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45

.ucmaa nouwnwoucfi ecu now gone: .m.m onswwm

.mehflfls buzuflfitm

 

 

 

 

 

 

 

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46

.ucmfim oHnmumcs ecu you noun: .c.m oyswflm

_cwm\o¢m. huzwnowcm

 

 

 

 

 

 

 

 

 

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mm_oo.oummz I :mm.ou-t I mn~c I Doomumz
w.~umox I m.mnmoz I =.mu~oz I ~.ou:¢p I m.ouz»
.NIm.\muu» mu hchm

 

47

Superimposed on the model results are the experimental data
at the ten forcing function frequencies. The ranges shown are t 10
ranges around the mean where the mean is indicated by the symbol.
The data given by the circles (C)) are from [5] while those given by

the triangles (A) in Figure 3.4 are from [41].

The matching was achieved heuristically varying n, r, and
mzz- Formal optimization procedures were not used to select these
parameters primarily because it was desired to obtain a "feel" for
the effect of each parameter, but also because of potential conver-

gence problems and digital computer running time.

Figure 3.3 shows the model results with and without the
drift provided by w2 . (The model without drift is the example
problem solved in detail in Appendix B.) Comparison shows that
drift provides a low frequency gain increase and additional low
frequency phase shift, and consequently provides a better match with

the data.

Figure 3.4 shows how pilot training might be accounted for
by the model. The data from [41] are for one of the nine pilots
used in [5], but after much more training. Detailed comparison of
Figures 3.3 and 3.4 shows that the gain is higher and the phase
shift is less in Figure 3.4. This is accounted for in the model by
simply reducing the measurement noise. (It might be said that his
perception ability was improved.) The noise term needed to match
the data in [5] is NI = 500 while that needed to match the data in
[41] is NI = 2000. As indicated by the parameter variations in

the next section, the gain increases can also be accounted for by

48
decreasing the cost r associated with the control action. The

change in phase shift at mid-frequencies, however, is in the opposite
direction and therefore r is not the appropriate parameter to
change. The remaining parameter of the three available to account
for the difference is mzz , but as indicated above and in the next

section, m22 primarily affects the low frequency characteristics.

Figures 3.5 and 3.6 show the matches for the plants
Yc= 1/5 and Yc = 5/(s-2). The gain matches for both are very
good. The phase match for the plant YC = l/s is good, but not as

good as desired.

It was found that the model results are quite sensitive to
the selected values of the parameters n, r, and m22. In addition,
the results are very sensitive to P when the solution for P is
stopped before it has converged. Figure 3.7 shows the effect on
the results for the plant YC = 5/(s-2) starting at an arbitrary
P. The solution for P is stopped at l, 8, and 16 seconds. At
16 seconds the solution has converged to a point where the conver-
gence requirement in Appendix B is met. Note that there are
drastic differences at low frequencies but very little change occurs
near the cross-over frequency and above. Since the low frequency
gain is high, the convergence of P has very little effect on the

closed loop response characteristics.

It is interesting to note that in the solution of the P
matrix for each of the plants considered, the particular elements
of P associated with the state of the plant that is being con-

trolled do not approach zero for the unstable plant but they do for

49

I IOIXIISJBUDJOI JSUHJ

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UumKQQS ruzwzommd

 

 

   
     

 

 

 

 

 

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CZA no hum-ALE I .NIm.\muu> m. hchE

50

other plants. In other words, for the unstable plant, non-zero
steady—state variances and cross-variances of the plant variables
are present. This means that comparatively poor estimates of the
state of the unstable plant are made while essentially exact esti-
mates are made for the other plants. This intuitively agrees with
what the human operator actually does since the unstable plant is
difficult to control, where according to [5], there are "momentary
losses of control." Consequently, a human operator may well have

relatively large variances on the estimates of the state of the plant.

The values chosen for n, r, and m22 to match the data in
[5] for the three plants are summarized in Table 3.2. There seems

to be little correspondence between the parameters required for the

Table 3.2. Parameters for model validation.

 

 

PLANT n r m22
1 1/500 1/20 0.008
1/5 1/5000 1/300 0.08
5/(5-2) 1/5000 1/2 0.00135

 

 

 

models. However, consideration of the total noise at the input to

the man gives some agreement as explained below.

The normalized noise power spectrum injected at y is
given by (3.37) and is plotted for the three plants in Figure 3.8.
Some of the remnant data in [5] and [7] are given as power spectra

injected in the same way, and are also shown in Figure 3.8. Both

Figure 3.8.

 

A

51

LG I0.0
w -RAD/ SEC

comparison of the remnant with model noise.

52

the model results as given by (3.37), and the data from [5] and [7]
are normalized by the variance of the forcing function. The dashed
line is the remnant data for the plant Yc = l as adapted in [7]
from [40]. The data indicated by the key of the figure are from
selected runs for the plants YC = l/s and YC = 5/(s-2) as re-
ported in [5]. The data for the plant Yc = l are taken from the
average of several runs while those for the other two plants are

from single runs.

As shown in Figure 3.8, there is general agreement between
the experimentally measured remnant and the noise used in the model.
However, only the noise and remnant for the plant YC = 1 agree
closely. According to [5], the plant YC = l is the only one where
noise is the majority of the remnant, where the significant part of
the remnant for the other plants is nonstationarity. Thus, it might
be concluded that since the noise for the plant Yc-= 1 matches the
remnant data best, a noise input that yields the desired Bode plot
is not closely related to the remnant when nonstationarity is im-
portant. Consequently, the noise needed to obtain a steady-state
P which will give the desired Bode plot is not the noise which will
be equivalent to the remnant. On the other hand, the forcing func-
tion used in the model is not identical to that used to generate the
data (Figure 3.2), and may have a significant effect on the noise
required to generate the model transfer function. Also, the remnant
is on the order of 5 percent of the total power and it may not be
accurate. Nevertheless, it appears as though the noise and remnant

are not equivalent. More experimental data are required, however,

S3

with forcing functions that can be exactly included in the model to

sufficiently support this conclusion.

Though it is theoretically possible to use the exact ex-
pression for the ten sine-waves as the model forcing function, such
a representation requires twenty state variables and is not feasible
with the present computer prOgram. (The maximum number of state
variables allowed is seven.) Also, if the exact expressions were
used, it is possible that the P matrix would not converge,even in
the four minute run time considered in [5]. (The sum of sine-waves
would "appear random" in the model although the estimates of each
would be improving with time.) To obtain a time stationary trans-
fer function, it would be necessary, as with experimental studies,
to use a ratio of cross-spectral densities (see Appendix 0). Under
these conditions, the noise used to generate a Bode plot to match
experimental data along with the nonstationarity that would be
present might be closer to the remnant. Further experimental and

model studies are required.

The addition of drift provides the low frequency phase
shift that McRuer et al. in [5] attribute to the neuromuscular
system (see Section 1.5). Also, the lightly damped second-order
characteristic at high frequency, attributed to the neuromuscular
system in [5], is present in the models for all the plants con-
sidered here. (see Figures 3.3 through 3.6). The model results
along with the data from [5] for the human operator alone (without
the plant) for the plant Yc = 5/(s-2) are shown in Figure 3.9.

The model exhibits the same characteristics attributed to the

54

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55
neuromuscular system in [S] as given in (1.1). However, since the

neuromuscular system is assumed to be representable here by a first-
order lag, it can be concluded that the gain and phase character-
istics come from the controller and filter, and ngt_from the neuro-
muscular system. Since the very high frequency gain does not match
well, a second-order neuromuscular characteristic may give better
results, but it should be more heavily damped than advocated in [5].
On the other hand, the forcing function used in the model has the
greatest deviation from the experimental function at high frequencies

(Figure 3.2), and may give rise to the discrepancy.

3.4 SENSITIVITY STUDIES

The effects of changes in the parameters of the system are
now studied to determine the sensitivity of the results to these
changes. Some of the results are compared with experimental data.
Unless otherwise indicated, the plant is Yc = 1 without drift.
Caution should be used in extrapolating these results to other

plants since the effects are not necessarily the same.

Figures 3.10 and 3.11 show the effect of the neuromuscular
time constant TN and the time delay I . Note that the complete

gain curves move vertically with changes in both TN and T ,

but the phase shift is altered only at high frequencies.

Let it now be assumed that the precision neuromuscular
characteristic given in [5] is better than the first-order lag
(contrary to some of the statements in Section 3.3) and will im-

prove the match. This characteristic as expressed by (1.1) is

S6

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57

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58

 

0.3 1
2
——+ 1 _2— 0_.2_4_
0-05 (0'1 5+1) [(16.5)2 16.5 s + 1]

(3.43)

 

The model results with this neuromuscular characteristic are shown
in Figure 3.12 along with the results using the simple first-order
characteristic. The forcing function in both cases is gaussian
white noise shaped by a second-order filter with corner frequencies
at 2.4 and 2.6 rad/sec rather than the third-order filter used in
all previous studies. This was done because the computer program
is limited to a state vector consisting of seven elements. To make
the cross—over frequencies the same, a value of RIp = 1000 was
used with the precision model while a value of RIS = 100 was used
with the simple model. It is evident by consideration of the fit in
Figure 3.3 that, in general, the model with the first-order lag

neuromuscular system gives the best results.

Figure 3.13 shows the effect of a more accurate approxi-
mation of the time delay. The first-order approximation of the time

delay used in all studies thus far is

-TS ; -s + 2/r

W (3.44)

A more accurate second-order approximation is

- ._...42
e T5 = E : + 4;:32 (3.45)

 

59

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60

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61

Comparison of Figure 3.13 with the data given in Figure 3.3 shows
that the high frequency gain and phase characteristics are improved

somewhat by the more accurate approximation of the time delay.

Figure 3.14 shows the effect of changing the measurement
noise n. The smaller the noise, the higher the overall gain. Also,
the smaller the noise, the smaller the phase shift. Controlled ex-
periments given by Bearne and Kahn [44] on the addition of noise to

the display variable confirm this result.

Figure 3.15 shows the effect of changes in the cost func-
tion associated with the control. Since the cost function can be
written as

t

f
(xT g~x + u2)dt (3.46)

La
ll
IT)

(see Appendix C), this plot essentially shows the relative effects
of changes in the weightings associated with x and u. Referring
to Figure 3.15, the results are unaffected as %- becomes very

large, i.e., the effect of the cost on u becomes negligible.

Figure 3.16 shows the effect of changes in the forcing

function variance. As given by (3.26), the variance is

2 -
oi - 0.468 In11 (3.28)

and, with variances of 0.1, 0.25, and 0.5, the corresponding values

for m1 are 0.214, 0.534, and 1.068. Since a change in the

l

62
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63

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64

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own—m I oomu~z
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6S

amplitude of the inputs changes the results, so that superposition

does not hold,a nonlinear characteristic of the model is evident.

Figure 3.17 shows the effect of drift for the plant
Yc = 1. A higher drift gives a higher gain at low frequencies and

increased phase shift at mid-frequencies.

Figure 3.18 shows the effect of the forcing function band—
width on the plant YC = 1/s holding the variances of the forcing

functions constant at
0% = 0.534 (3.47)

The data points shown are the mean values of the experimental data
reported by McRuer et al. in [5] using the three forcing functions
F1.5, F2.5, and F“ shown in Table 3.1. The i 10 ranges for the
data with the F2.5 input are given in Figure 3.5. The i lo
ranges for the data with the F1.5 and F“ forcing functions are
essentially the same although some of the low frequency ranges are
somewhat larger. The gain curves fit quite well but there seems to
be greater effect on the model than on the data. The fits of the
phase shift curves are good but not as close. Nevertheless, the

general trend of the changes in gain and phase shift are present.

Figure 3.18 also shows that there is a marked difference
in low frequency phase shift for the three forcing functions. In
fact, at w = 0.01 the phase shift with F“ is almost -180 degrees,
while for F1.5 and F2.5 it is greater than zero. It was found

that the low frequency phase shift is very sensitive to the

66

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68

parameters n, r, and m22 . This sensitivity may account for the
much greater low frequency variability present in the data for some
of the plants in [5] which has been attributed to "indifference" on

the part of the operator.

Figure 3.19 compares the responses using first, second,
and third order bandlimited white noise forcing functions with con-

stant variance. The bandlimiting functions considered are

 

 

 

 

GF1(s) (_§_.+ 1) (3.48)
2.5
1
Gp2(s) (254 + 1)(256 + 1) (3.49)
6133(5) 1 (3.50)

 

(3%Z'* 1)(2?s * 1)(5%3'+ 1)

It is shown in [4], for Yc = 1, that the rectangular forcing func-
tion (Figure 3.2) produces essentially the same Operator response as
the third-order characteristic. Note that changes in the form of
the bandlimiting has a significant effect on the gain and phase
shift. The characteristics indicated in Figure 3.19 are consistent

with the experimental results given in [4] and [5].

Figure 3.20 shows the effect of changes in the form of the

cost function of x . The nominal cost function used in all

69

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71

analyses so far is

T xTCTCx (3.51)

X
D
X
II

= x —1 0 1 0 0 x (3.52)

 

 

(xI - X312 (3.53)

where for the plant Yc = 1 (Appendix B), x1 is the man's output

and x3 is the forcing function. This cost function is altered to

 

 

the form
F 1 0 a 0 0
O 0 0 0 0
xTan = xT a 0 1 0 0 x (3.54)
0 0 0 0 0
0 0 O 0 0
L. _J
= 2 2
x1 + 2a x1 x3 + x3 (3.55)
where a = -0.8, -0.3, + 0.3. The results shown in Figure 3.20

indicate that vast change in the response occurs with changes in a.
It is interesting to note for the case when a = 0 that the best

thing the operator can do is nothing. This can be explained by

72

considering the complete cost function where

 

 

 

r tf
J = 134% (xTQx + ru2)dt (3.56)
k 0
r tf n
= E 6-:- (xi— + x3 + ru2)dt$ (3.57)
t 0 J

Since nothing can be done about the forcing function x3 , the best

value for both x1 and u is zero for all t. Since
x1(0) E u(O) E 0 (3.58)

this amounts to no control action by the human operator -- a trans-

fer function of zero.

It is stated in [5] that the plant gain has very little
effect on the man-plant transfer function. If the gain of the plant
in the model is changed, there is a significant change in the result-
ant Bode plot. However, if the cost function associated with u is
changed by an amount proportional to the square of the change in
plant gain, the model solution remains the same. This is true since
the augmented plant (Figure 3.1) is linear and the plant gain can
always be moved back to the input u. Therefore, the plant gain

will appear as a coefficient in the matrix B in the state model of

73
the augmented plant [see (2.1)]. If BR-lBT remains unchanged,

the solutions for K in (2.12) and for x in (2.7) remain unchanged.

In addition, from (2.11)

T

Bu = -BR-1B Ki (3.59)

indicating that the solution for x in (2.1) would also be un-
changed. Therefore, if BR-IBT is held constant with changes in
plant gain, the model results will be unchanged. This implies that
man changes his cost function R with plant gain. This also im-
plies that the values for R required to match the data are directly

related to the plant gains used to obtain the data.

4. SINE-WAVE TRACKING

One of the important features of the model of the human
operator presented here is that it includes precognition. That is,
it accounts for man's ability to predict the characteristics of a
forcing function, for example, and to take advantage of this knowl-
edge, so that a better or higher level of control is accomplished.
This characteristic of man is demonstrated by using a sine-wave
forcing function in a compensatory task as shown in Figure 4.1.

Once the Operator has tracked the sine-wave for a short period of
time, he will have essentially zero time delay. Furthermore, he
can close his eyes, eliminating the error signal, and still continue

to track.

 

 

 

 

 

 

Slut-WAVE A
GENERATOR DISPLIY MAN CONTROLLII PLANT v

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1. Sine-wave tracking block diagram.

In this chapter, the results of a sine-wave tracking
experiment are given and related to the results of the model.
These results are also related to the experimental work reported
by Pew et al. in [8]. In addition, tracking with the "eyes closed"

is considered both experimentally and with the model.

74

75
4.1 EXPERIMENTS

Experiments were performed with the aid of an analog com-
puter using the system shown in Figure 4.1. The sine-wave forcing
function had a frequency of 7.159 rad/sec and a half-amplitude of
1.414 volts. A five-inch CRT (cathode ray tube) display was used
(Figure 4.2) with the gain set at 2 cm/volt. The controller was a
hand Operated force sensing device (Figure 4.3), Operated around
its pitch axis, with an output Of 0.003 volts/gram at a 3 inch mo-
ment arm. The displacement Of the stick was approximately 0.005

mm/gram. A two minute run length was used.

A single subject was trained until his mean-squared error
and integrated-absolute error reached a relatively consistent level
from run tO run. Typical recordings after training are shown in
Figures 4.4 and 4.5. The measured mean-squared error Of the run
shown in Figure 4.4, normalized by the measured mean-squared error
if the Operator did nothing, is .083/1. The measured normalized
integrated-absolute error is 2.32/8.78 = 0.265. The results Of a
corresponding experiment reported in [8] yielded a normalized
integrated-absolute error Of approximately 0.32. Thus, the results

are reasonably close.

The top traces in both Figures 4.4 and 4.5 are the forcing
function sine-wave i(t). The middle traces are the output m(t), and
the lower traces are the difference or error e(t). Figure 4.4 shows
a complete run while Figure 4.5 gives just the first part of a run
with a greatly expanded time scale. Note that, in general, i(t)

and m(t) are in phase, indicating that precognition is present. At

76

 

CRT display used for sine-wave tracking.

Figure 4.2.

77

 

Controller used for sine-wave tracking.

Figure 4.3.

78

 

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1" I‘ . I

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“ I

20 40 60 80 I00 I20

t~SECONDS

Figure 4.4. Complete sine—wave tracking run.

IZIMHHHHHHH
HHHHH I III"

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WVIUUVVVVVVIVVV

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80

time zero, however, there is a delay of approximately 0.15 seconds
before the subject starts to track, indicating no prediction ini-
tially. Also note that m(t) and e(t) contain frequencies (both
higher and lower) other than the forcing function frequency. This
illustrates the presence of noise. Figure 4.4 shows that the out-
put m(t) is almost always of smaller amplitude than i(t), implying
that the open loop gain is not very high. The first part of e(t)
appears to consist primarily of the forcing function frequency. The
amplitude appears to start out large and essentially damp out after

about 4 seconds.

Figure 4.6 shows the results when, after tracking for
six seconds, the operator closes his eyes. Thus, for t > 6, the
operator performs as an open loop system without an error signal.
The results show, however, that the operator continues to track
the sine-wave, but with increased error. In a few runs, m(t) and
i(t) were as much as 180 degrees out of phase after the eyes were
closed. The operator apparently operates from a "built-in" source

of information ”learned" during pervious operations.

4.2 TRACKING MODEL

4.2.1 State Model of the Augmented Plant
The augmented plant employed in this study of sine-wave
tracking is shown in Figure 4.7. This block diagram can be ex-

panded as shown in Figure 4.8.

81

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W W‘V‘
Figure 4.7. Augmented plant for sine-wave tracking.
+
“ 3. _.
7
-us
Figure 4.8. Block diagram for sine-wave tracking.

From Figure 4.8 the state model process equations are

 

 

x1"‘ ”0
“(0
.x3 = 0
x,+ 0
L0

-1/TN l/TN 2/T

0

0

'LO

0

0
a

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_J

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FX 1“

x2

X3 4'

X4

X5

 

 

 

 

F0“

0

 

 

W

O

 

(4.1)

83

and the measurement is
X11

x2
y = (1 0 -1 0 0) x3 + v (4.2)

X4

 

 

x5

4.2.2 Model Parameters
As in Chapter 3, the neuromuscular lag time constant and

the time delay are, respectively

T = 0.3 (4.3)

T = 0.1 (4.4)

The sine-wave forcing function has a frequency

8
II

7.159 (4 S)

and half amplitude

9.)
H

1.414 (4 6)

A gaussian white noise source w shaped by a first-order

lag

65(5) = “a

S‘HD
a

(4.7)

is applied at the input to the neuromuscular system. In addition,
a gaussian white noise input v uncorrelated with w is added to
the measurement y. Since v and w are white noise, their co-

variances are

E {v(t + 1)v(t)} n6(r) (4.8)

E {w(t + T)w(t)} m6(r) (4.9)

84

As in previous models, the cost function matrices are

q

I.1 0 -1 O O
0 O 0 O O

Q: -1 0 1 o o (4.10)

 

 

R = r (4-11)

where Q has been chosen so that e2 is minimized.

The parameters n, m, wa’ and r in (4.7), (4.8), (4.9), and
(4.11) are selected so that the model gives results consistent, in-
sofar as possible, with the experimental results. As described in

more detail below, the values selected for these parameters are

n = 1/300 (4.12)
m = 0.25 (4.13)
ma = 4 (4.14)
r = 1/50 (4.15)

4.2.3 Covariance Matrix

The covariance of (x-x) as given by (2.9) is

P = AP + PAT - PCTN-ICP + DMDT (2.9)

where from (2.10) the initial condition with i = O is

o
P(O) = E {XOXZ} (2.10)

Upon substitution of the appropriate matrices from (4.1), (4.2),

(4.8), and (4.9) into (2.9), the covariance of (x-x) becomes

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86
In the time solutions for P that follow, it is assumed

that, after training, the Operator has non-zero initial estimates of
the covariance of the forcing function only. By (3.3) and (4.6),
the initial variances of XI and x2 are 1. Since x1 and x2

are orthogonal, the cross-variances are zero. Hence

r- 1

1 O 0 0 0
0 l O 0 0

P(O) = O 0 0 0 0 (4.17)

 

 

In view of the assumption that
P15(0) = P25C0) = P35(0) = Pu5(0) = P55(0) = 0 (4-18)

it follows from (4.16) that the time solutions for these elements of
P are identically zero for t >0. Also, since P is symmetric, the

unknowns in P for t 3_0 reduce to the 10 elements: p11, p12,

P13, Plk’ P22, P23, PZH: P33, P39, and Pan-

4.2.4 Optimal Control Law

The optimal control law as given by (3.8) is

c = —R'IBTK (3.8)

Since R-1 and BT are known, the control law is known once (2.12) and
(2.13) are solved for K. Both K and C can be found using the
computer program described in Appendix B. For the parameters de-

fined above, this program yields

87

 

 

 

 

[.5 ‘1 r‘ '~
011 3.735
C112 5..5€37

cT = c13 = -4.388 (4.19)
Clu -1.952
C15 -1.753
_ _, L _

4.2.5 Filter Model
From (2.7), (2.8), and (2.11), the state model for the

filter is

T 1

x = Ax + PC N- (y-Cx) + Bu (4.20)

where it is assumed that
i = o (4.21)

With this assumption, it is implied that the operator's initial

estimate of the augmented plant state vector is zero.

By substitution and simplification (4.20) becomes

 

 

 

 

 

 

 

 

 

 

Q, 0 us 0 o 0 £1 0 Fh"p”
i; -u' o o o a £2 0 ”‘2 ’ ’23
d . .
_ _ - ha-Pn - ~
3? f3 0 0 l/TN l/TN 2/TN :3 . 1n" u 9% (y - xl 9 x3) (4. 22)
x, o o o -wa o x. o p‘“ ’ P3“
x5 0 0 0 0 ~2/1J ;5 2/t _P‘S ‘ p35j
_ d h- _ _1 b J

88

This model is the same as the state model for the augmented plant,
except for the last term. As shown, the last term involves the
measurement y , the estimate of y which is (x1 - x3), the

noise term n , and elements of the covariance of (x - 2).

4.3 MODEL VALIDATION

Time solutions for the complete sine-wave tracking model
as described above, were realized on an analog computer. Although
there are minor accuracy problems with the analog computer in this
application, the results shown in Figures 4.9 through 4.16 provide

a good "feel" for the solutions.

Figure 4.9 shows the forcing function i(t), the Operator's
output m(t), and the error e(t), when no noise is applied.
Since there is no noise, these are the mean values of the response.
The amplitude of m(t) is smaller than i(t) and indicates, as
for a real human operator (Figure 4.2), an open loop gain that is
not very high. The m(t) trace exhibits an initial time delay of
approximately 0.1 seconds. After the first cycle, however, there

is very little time delay.

Figure 4.10 shows the mean state vector superimposed on

the mean estimated state vector. After a half cycle, the two are

MAM/\MMMMM
_-;;\/V\/vvvvvvvvvvvw

 

.ZAAAAAAAAAAAAAAA
-vvvvvvvvvvvvvvv‘

° sssssss

 

Figure 4.9. Mean sine-wave track oooooooooooo

mmmmm
mmmmm

AAA/\AAAAAAAAAA
VVVVVVVVVVVVVV\

.mmmmm
dummmmv

 

v“ v ‘1 via—7'—

 

G ‘9
O
-.J.

AAAAAAAAAAAAAAA
XVVVVVVW' VVVVVVW

91

 

 

 

 

 

 

 

 

 

 

 

 

 

4 8 8 IO
'"SECONDS

Figure 4.11. Solution for P.

92

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Win

 

O 20 4O 60 80 IOO |20
P'SECONDS
2
I
e 0
-|
-2

Figure 4.12. Complete sine-wave tracking model run.

.MMAMMMAMM
:VVVVVVVVVVVVVW‘

 

M AMA WW
:,:Vv/\/\VV\ WW WWW WV

 

t~scconos
2...
I
' 0W
-gu

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VMWVVVVVVAVVVVVVA

MAAAAAAAAAAAAA
MVVAVUVVVVVVVAVA

A AAAAAA A
-\VUVVVAVVVVV VVAAV

0'? L<Eo>- <,2:>—
k“:

x)
' .9 1' ..
v o—fi_—!

 

 

AVVM WAAAAAAAAAAWAAWA.
4V VVV WVVVVVV

“AAA AAAAAAAAAAAAA
ITAA WAAAA AAAAAAAAA

:H ‘ HA. I“
1 ”AV! vvvyvv vvv

 

 

96

 

 

|-. eves CLOSED

 

 

 

 

 

 

9"SECOWS

Figure 4.16. Solution for P with the eyes closed after 6 seconds.

97
essentially the same. The mean estimate of the noise given by x”
is zero (except for computer error) after the same period of time.

The values for x5 and x5 are the same starting from time zero

since the variance of (XS-x5) is identically zero for all t .

Figure 4.11 shows the solutions for the elements of P.
As indicated, the solutions reach nearly steady-state conditions
in less than two seconds. Using the digital computer solution
described in Appendix B, however, the P matrix solution at 2
seconds is
Fl.292E-02 9.467E-04 9.846E-03 1.5725-02 0.0
9.467E-04 9.983E-03 -3.684E-03 2.079E-03 0.0

P = 9.846E-O3 -3.684E-03 3.871E-02 7.934E-02 0.0

 

 

(4.23)
1.572E-02 2.079E-03 7.9348—02 3.47OE-01 0.0
L__0.0 0.0 0.0 0.0 0.0_
and at 10 seconds it is
r n

2.669E-03 .198E-O4 2.17SE—03 3.219E-03 0.0

I
H

-1.198E-O4 2.632E-03 -l.2655-03 1.012E-04 0.0

I
H

P = 2.17SE-03 .265E-03 3.154E-02 7.029E-02 0.0 (4.24)

3.219E-03 1.012E-04 7.029B-02 3.317E-01 0.0

 

 

Lo.o 0.0 0.0 0.0 0.0

(The convergence criterion defined in Appendix B is met at t = 10
seconds.) Considerable reductions in the elements of P occur that
are not evident in Figure 4.11. These changes could have a large
effect on the solution for x, and because of the large dynamic
range dictated, it is the primary reason for accuracy problems with

the analog computer.

98

Figure 4.12 shows a two-minute run corresponding to the
two-minute run of the real human Operator shown in Figure 4.4. The
Ineasured mean-squared error divided by the measured mean-squared
error, if no control were applied, is 0.094/1. This agrees reason-
ably well with the value of 0.083/1 obtained for the real human
operator. In addition, the normalized measured integrated—absolute
error is 0.265/0.936 = 0.283. This also agrees quite well with the
value of 0.265 obtained for the run shown in Figure 4.4 and with

the value of 0.32 obtained by Pew et al. in [8].

Figure 4.13 is a partial run of the analog solution showing
the first 14 seconds when the noise terms are included. Visual
comparison with Figure 4.5 indicates that "generally" the same fre-
quencies and characteristics are present. Note that, as in Figure
4.5, the forcing function appears to be predominant in e(t) for
the first 4 seconds. The initial time delay, as in the case when

no noise is applied, is approximately 0.1 second.

Figure 4.14 shows the solutions for the estimates of the
state vector when noise is included. The solutions for £1 and
i, are quite close to the desired sine-waves while the solutions
for x3, x“, and x5 are "noisy." The solutions for £1 and £2
take about four seconds to reach the forcing function amplitude.

Note that this is longer than the corresponding time of one second

for the mean values in Figure 4.10.

Figure 4.15 shows the solution when the "eyes” of the model
are closed at six seconds by setting C = 0 in the solution for

both P and x. Note that after the "eyes” are closed the model

99
<:ontinues to track, although the error has obviously increased.
‘Figure 4.16 shows the solution for the covariance of(x-x) for the
same length of time. Note that p33, p34, and P44 increase
rapidly to higher levels. A digital computer solution for P
(Figure 4.17) shows that the other elements of P do not increase.
This is contrary to what the human operator would do after a long
period of time since after some unknown period of time he "loses"

his estimates of the system parameters. This could be accounted for

 

by including a parameter estimator block, in addition to the filter
and controller. Nevertheless, for the time period of 14 seconds
considered here, the results of Figure 4.15 for the model are close

to those in Figure 4.6 for the real human Operator.

Figure 4.18 is a Bode plot of the steady-state man-plant
transfer function while tracking the sine-wave as obtained from
the model. Note the high gain rise and sharp spike in the phase
angle in the neighborhood of the forcing function frequency. In-
5pection of the printout for the plot shows that the forcing func-
tion frequency is located between the high and low gain spikes

where at the sine-wave frequency

81001159) = 18.88 db 0.36 degrees (4.25)

Since the phase shift at the forcing function frequency is nearly

zero, there is essentially no time delay. This result agrees with

 

the general characteristics in Figure 4.5 and with the statements of
Pew et al. in [8], that no time delay is observed when a sine-wave

is tracked.

T = 0.0 SECONDS
PO
1.000EOOC 0.0
0.0 l.UCCE+00
0.0 0.0
0.0 0.0
0.0 0.0
T = 1.012 SECONDS
P
1.5575-02 3.824E-03
3.8248-03 2.576t-02
1.067E-02 -8.4C4E-03
1.9185-02 7.024E-03
0.0 0.0
T = ~ 2.012 ShCONDS
P .
1.2925-02 9.467E-04
9.467E-04 9.9835-03
9.8465-03 -3.6845-03
1.5725-02 2.079E-03
0.0_ 0.0
T 8 3.012 SECONDS
P
8.4235-03 -1.063E-03
“1.0635-03 7.8315-03
7.167E-03 -4.325E-03
100095-02 “505365-04
0.0 0.0
T 8 4.012 SECONDS
p
,50596E-03- ’ZOOBOE‘O“
-2.0805-04 6.9665-03
4.5405-03 -3.261£-03
6.751E-03 3.999e-04
0.0 0.0
Figure 4.17.

100

C>C<DC>O
no 0 on
C30<3C>O

100675-02
“01455-02
709895-02
0.0

908466-03

-3.6845-03
3.871E-02
70934E-02
0.0

7.1675-03

-4.325E-03
3.687E-CZ
7.604E-02
0.0

4.5‘0E‘03

-3.261E-03
3.4305-02
7.2955‘02
0.0

OCDC>C¢3
to o 00
O‘DCDCJO

109185-02
7.0266-03
709895-02
3.516E-01
0.0

1.5725-02
2.0795-03
7.9365-02
3.4708-01
0.0

100096-02

-5.536&-0§
7.6645-02
3.399E-01
0.0

6.7515-03
30999E-0‘
7.2966-02
30360E-01
0.0

OCDCDOID
oo o 00
OGDCDOCD

0.0
0.0
0.0
0.0

Digital computer printout of P with
the eyes closed after 6 seconds.

p

ABOVE MATRIX 15 SS SOLUTION FOR P TC HITHIN 0.011

T a

5.076E-03
30715E-U‘
3.870E-03
6.177E-03
0.0

4.5565-03

-9.926E-05
30417E'03
5.100E-03
0.0

T a

4.100E-03

‘1.b95E-04
1.690E-04
3. 5 TIE-C5
0.0

I a

4.282E-03
20629E'04
-10594E‘05
-8.157E-07
0.0

I a

6.571E-03
4.196E-05
‘03316’06
1.5595-07
0.0

5.022

7.047.

8.071

8.616

SECONDS

30715E-0“
50099E’03

-1.971E-03
902696-04
0.0

101

3.870E-03

-1.9TIE-C3
3.320E-02
7.227E-02
0.0

6.002 SECONDS

SECUNOS

-9.926E-05
“oObbE’O3

-2.199E-03

-4.115E-04
0.0

SECONDS

-l.595E-O4
4o§22£’03

-50126t-04

-70588E-05
0.0

SECONDS

206295-04
4.339E-03
-1.879E-05
" 101,045-06
0.0

SECONDS

4.196E-05
4.049E-03
‘095‘5'07
3.023E-08
0.0

Figure 4.17.

3.477E-03

-201995’03
3.857E-02
8.3715-02
0.0

1.69CE-04
-SOIZCE'CQ
2.252E-C1
.202695-01
0.0

-10594E-CS

-1.879E-05

.2.271E-01
2.271E-01
0.0

4.331E-06
4.9548‘67
2.271E-01
2.271E-01
0.0

6.1776-03
9.26QE-04
7oZZ7E’OZ
3.353E-01
0.0

5.1cce403
“40115E-0‘

8.371E-02.

305906-01
0.0

305776-05

-70688E-05_

2.2696-01
5.0COE-01
0.0

’80157E'C7

-101446-06
2.2715-01
5.000E-01
0.0

1.559E-07
3.023E-08
2.2716-01
5.0005-01
0.0

(Con't).

CDCCDCDO
o oo o
(DOCDCDO

0.0
0.0
0.0
0.0
0.0

102

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103

Consider now the effects of inaccuracies in P on the
response of the model. The results given above correspond to P(tf)
as determined by the convergence requirement given by (8.7). The
solution for P ended at a time of 10 seconds. The results when
the solution for P is terminated at 1 second are shown in Figure
4.19. The results when P is terminated at 35 seconds are shown
in Figure 4.20. (This latter result is obtained by changing the
convergence requirement on P from 0.01% to 0.0001%.) Comparison
of these figures indicates that, as the solution for P converges,

the gain and phase characteristics become better—and-better "tuned"

to the forcing function frequency.

The gains and phase angles at the forcing function fre-

quency at 1 second and 35 seconds are, respectively

G (j7.159)

1 18.97 db 0.06 degrees (4.26)

G35(j7.159) 18.46 db 0-74 degrees (4,27)

Comparison of (4.25), (4.26), and (4.27) along with Figures 4.18,
4.19, and 4.20 shows that the gain and phase angle at the forcing

function frequency remain essentially the same as P converges.

Consider now the power spectrum of the error signal e.
If C(jw) represents the complete man-plant transfer function, the
closed-loop block diagram of the sine-wave tracking model can be

drawn as shown in Figure 4.21. The power spectrum of e is

104

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105

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‘E

 

 

 

 

 

 

 

 

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end: - m~.ou_~z - omuee - oomuez
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106

 

 

 

 

 

 

 

 

(3('(0) 2 ‘” 2
0 - ———l¢——- ;_fi a
ee (w) 1+G(Jw) ¢VV (w) + (jw+wa) (TNjw-I-l) (1+6 (jw)) (DWWOD)
1 2
1+G(jm) 11(“) (4.28)
and the variance is
2 _ __1_. .
oe — 2" [ ¢ee(w)dw (4 29)
o
W
“'0
.7373; i
O
O +
V GUullTJIwHIP W

 

 

 

 

 

 

 

 

Figure 4.21. Sine-wave tracking model block diagram.

The power spectrums of i, v, and w from (4.5), (4.6), (3.21),

(3.22),(4.12), and (4.13) are

¢ii(w) = n(1.414)2 6(w-7.159) (4.30)
0 (w) = -35- (4 31)
vv 300 °

¢ww(w) = 0.5 (4.32)

The transfer function for the man—plant combination for the

0.01% accuracy requirement on P (as plotted in Figure 4.18) is

-8o.39[jb+7.410] [(jd)2—.000092(je)-400] [(ju)2+0.0603(ju)+49.92131]

50w)=
[0.3(jw)+l] [jm+20] [jwe4.292] [(jm)2+52.43(jw)+1160.7] [(je)2+0.01144823()u).51.33896]

(4.33)

107

Using the values of ma and TN defined by (4.3) and (4.14), the

variance of e by numerical integration is

0.07238 + 0.0104 (4.34)

0.8278 (4.35)

The power spectrum from (4.28), normalized by this variance, is
plotted in Figure 4.22 along with data points adapted from [8].

(The data in [8] are in terms of e, and were adapted by first multi-
plying ¢éé(m) by l/w2 , and then normalizing by the area under
¢éé(w)/m2 obtained by graphical approximation.) There is good
general agreement, although a sharper high frequency break in the

model results would improve the match.
In general, it can be said that in view of

1) visual comparison of the human operator time responses

with the time solutions of the model

2) the closeness of the error power spectrum of the model

with the experimental data in [8]

3) the fact that the model time delay at the forcing func-

tion frequency is essentially zero

4) the model tracks with its "eyes closed"

the model comes very close to what the human operator does.

108

.coume xuwmcow Hmuuoomm aozom .NN.v whamfim

auumxo¢c. ruzwzowcu

 

 

   

 

 

 

 

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5. DISPLAY EFFECT

The effects of changes in the form Of display on the trans—
fer function of the human Operator are considered in this Chapter.
Both experimental results and model results are considered. The
combination of parameters required to match the model results with
the experimental results provide insight into possible ways of

evaluating displays.

5.1 DESCRIPTION OF EXPERIMENTS

Experiments were performed using the Applied Dynamics
AD-256 analog computer and associated equipment shown in Figure
5.1. A simple "pitch axis" compensatory task as shown in Figure
5.2 was used wherein the difference between the forcing function

and the plant output is displayed to the Operator.

 

 

 

I Q . c _
FORCIN“ "u” ‘N
ruucnou '< >'—‘ 0|SPUY OPERATO a co umouu PLANT

Figure 5.2. Display study block diagram.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Two vastly different displays are considered. The first is
the S-inch CRT (cathode ray tube) display shown in Figure 5.3, and

the second is the DVM (digital voltmeter) display shown in Figure 5.4.

Since with the DVM, the Operator must read the sign Of the displayed

109

110

’A .

.
A
A
C
I
O
n
C
C
. .

a..
.u.»

o
.

.ucoamflsco Hmucesfinomxo xOSHm amammflo

H

m OHDMflm

0.00.1.1.

1.:

 

4.

111

 

 

Figure 5.3. CRT for the display study.

 

 

Figure 5.4. DVM for the display study.

112

voltage as well as read its numerical value, it is much more diffi-

cult to use.

With the CRT display, a vertical display motion was used
with a gain of 2 cm/volt. A large dot with a diameter of approxi-
mately 3 mm was used as the moving index. The zero position was
indicated but no numerical scale was provided. The background grid
of the CRT, however, was visible. The Operator was expected to
judge the error (including sign) by the relative distance between
the dot and the zero position. A "fly to" sign convention was
utilized where, in effect, the zero position is flown to the dot.
For example, if the dot is above the zero position, a pulling action

Of the controller is required.

The DVM gain was 10 volts/volt. The display used provides
up to 1000 readings/sec with no sign change, and up to 100 readings/sec
with a sign change. Consequently, the maximum time delay was 0.01
second. This is small compared with the minimum time delay of
0.1 second used for the human Operator and, therefore, has an in-
significant influence on the experiment. As shown in Figure 5.4
with a typical reading, the sign and two digits are displayed. In
terms of the CRT display, a plus voltage on the DVM corresponds to
the dot above the zero position. Accordingly, a plus voltage on the

DVM required a pulling action Of the controller.

The controller employed in the study is the three-axis
"force sensing stick” shown in Figure 5.5. (The same controller
is shown in Figure 4.3 without the handle.) It is considered a

force sensor because approximately i 1 mm of horizontal motion at

113

 

Figure 5.5. Controller for the display study.

114

the center of pressure is the maximum travel required in the experi-
ments. As stated above, the task was a pitch axis task, so only the
pitch axis of the stick was used. The gain was 2.49 volts/deg and

the force gradient was 1.62 lbs/deg.

The plant was a simple constant gain with a transfer function

of

Y = 1 (5.1)

The forcing function was the sum of ten sine-waves with fre-
quencies and amplitudes as given in Table 5.1. The frequencies are
the same as those used in [5] (See Table 3.1). The amplitude charac-
teristic corresponds to that of a pure second-order lag with a corner

frequency at 2.54 rad/sec. The shaping transfer function is

 

(5.2)

The variance of the forcing function in terms Of inches of display

motion was

012 = 1.7 inZ/Hz (5.3)

The transfer function of the human Operator is determined
from the cross-spectral density between the forcing function and the
error, and that between the forcing function and the controller out~
put. (The assumption is made that the display and controller trans-
fer functions are part of the operator, i.e., the transfer function

is between e and c in Figure 5.2.) The Operator transfer func-

115

tion as given in [43] is

(5.4)

where wn is the frequency at which the cross-spectral density is
measured. Taking advantage of the fact that the forcing function
consists of the sum of sine-waves, the cross-spectrums at each of
these frequencies is easily obtained as shown in Appendix D. Given
the cross-spectrums, the values for the transfer function at the ten
forcing function frequencies were obtained by (5.4). The real and
imaginary parts of (5.4) provide the information for the gain and

phase plots.

Table 5.1. Display study forcing function.

 

 

wn - Rad/sec 012(mn)-db
i in inches
w} = 0.157 —2.11
wz = 0.262 -2.17
w3 = 0.393 -2.28
we = 0.602 -2.56
ms = 0.969 -3.26
mg = 1.490 -4.64
w7 = 2.540 '8-10
we = 4.030 -l3.03
wg = 7.570 -21.00
O10 = 13.800 -3l.70

 

 

 

 

116

With the analog computer equipment available, it was possi-

ble to evaluate the cross-spectrums of 0. (w ) and 0. (w ) at
1e n 1c n
“6: my, mg, mg, and “10 at the same time the subject was performing.
The time histories of e and c were recorded using the FM tape
recorder shown in Figure 5.1. After the run, these recordings were
used to find the cross-spectrums of 4. (w ) and 4. (m ) at the
1e n 1c n

remaining frequencies w1, w2, O3, mg, and ms. The majority Of the

computer patching shown in Figure 5.1 is the cross-spectral analyzer.

Figure 5.6 illustrates the accuracy of the cross-spectral
analyzer when the human Operator, display, and stick were simulated

on the computer with

Y = ——4—-—— (5.5)

The solid and dashed lines are respectively the theoretical gain

and phase from (5.5). The data points are the measured results using
the spectral analyzer. At all of the measurement frequencies, the
gain is within 2 db and the phase is within 10 degrees. Hence, for

the purposes of this study, the analyzer results are quite good.

Three human Operators were used. These Operators are en-
gineers without formal pilot training. Prior to taking data, they
were trained to use the two displays on 120 second practice runs.
The choice of which display to use was made on a non-systematic
basis. The practice runs were accomplished over a one week period
with up to 10 runs a day. The Operators were instructed to minimize
the displayed error and proficiency was measured during the run by

their mean-squared error where

117

.xomhsoom Homxamcm Haywoomm1mm090 .o.m Ouswwm

 

 

 

 
  

 

 

 

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Tau» mm ._.zcgm

 

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2:6 .

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M M

. AV 1'.
...... 8.
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S 9
1. mm
I’ . iv v N
.0... 4.0
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3 t
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a. .W

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0 Dm.hN\rm.ummz 0

MSE = — e2dt (5.6)

0

Once the MSE reached a consistent low level (after 20-30 runs per

display), the Operator was considered to be trained.

Each recorded experiment extended over a 240 second period.
Up to eight runs were made per eight hour day with two or three in
one sitting. The choice of display again was non-systematic. Ten

runs for each operator with each display were made.

The gain and phase angle of the transfer function at the
ten frequencies for each run and each pilot were computed using
(5.4). The mean and unbiased standard deviation (10 value) for
each gain and phase angle for each Operator and both displays were
then computed and plotted. In addition, the combined mean and un-
biased standard deviation for each gain and phase angle for the

three pilots were computed and plotted for both displays.

The time history of a typical CRT run is given in Figure
5.7, and that for a typical DVM run is given in Figure 5.8. The
top traces in both are the ”random appearing" forcing function. The
middle traces are the controller outputs. The bottom traces are the
error or differences between the respective top traces. The ampli-
tudes of the error with the DVM display are obviously much larger

and generally indicate that the DVM is more difficult to control.

 

 

 

119

1 1&1}... k l.
H 111» ‘11'1 1',
4O 80 ‘ IZO ISO 200 240

l-SECONDS

Figure 5.7. Typical CRT run.

120

<l-

b

l
1

go 111l1ll‘1 I‘ll ’ 1“ J1 'l
3. l l" w

 

1. 11

GNOLTS
O'

 

1
9

 

O 40 80 IZO l60
1~SECONDS

A

8~VOLTS

 

 

Figure 5.8. Typical DVM run.

121
5.2 PARAMETER EVALUATION

The results of the experiments are given in Figures 5.9
through 5.16. At each of the ten frequencies, the circle «3) indicates
the mean value and the bars indicate the i l 0 band around the mean.
Figures 5.9, 5.10, and 5.11 are the results for Operators l, 2, and 3
respectively when using the CRT display, while Figure 5.12 shows the re-
sults for the three operators combined using the CRT display. Figures
5.13 through 5.16 are the corresponding results using the DVM display.
In general, with the DVM, the gains are lower and the phase angles are

more negative.

White noise shaped by

 

05(5) = 1 (5.7)
__S._+l ___s_+1
2.44 2.64

is used as the model forcing function. For this shaping the variance

of the forcing function is

2
oi = 0.625 11111 (5.8)

From (5.3) and (5.8), the white noise input power spectral density
required to make the model forcing function variance the same as

that used in the experiments is

 

“11 = 0.625 (5'9)

= 2.72 (5.10)

With the experimental forcing being the sum of sine-waves with a

power spectrum distributed according tO a second-order lag (See (5.2)

122

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130

and Table 5.1], the model input approximation is much better than

that used in Chapter 3 (see Figure 3.2).

Consider first the match of the model with the CRT data.
Detailed comparison of the experimental data reveals that there are
slight differences between the gains and phase angles from Operator
to operator. The parameters needed to match these data correspond-
ingly are slightly different. The parameters selected to match the
Operator results individually and combined are summarized in Table
5.2. The match as indicated in Figures 5.9 through 5.12, in each

case, is very good.

Table 5.2. Model parameters for CRT match.

 

 

 

OPERATOR NUMBER
PARAMETER
1 2 3 1,2,&3
l/n 70 70 70 70
l/r 200 200 50 150
ml, 0.4 0.8 0.5 0.8
r 0.1 0.1 0.1 0.1

 

 

 

 

 

 

 

In Chapter 3, where a match with the data in [5] is accom-
plished, the time delay 1 is not considered to be a matching
parameter. With the DVM display, however, the only parameter that
can be changed to improve the match of the high frequency phase
angle, in particular, is the time delay. For longer time delays,
the corner frequency of the first order Padé approximation is
well back in the region of the data and the order Of the approxi-

mation significantly influences the results. To reduce this error,

131

a second-order approximation is used (for both CRT and DVM matches).

A comparison with

T = 0-4 (5.11)

is given in Figure 5.17. The high frequency rise is very dependent

on the order of the approximation and, as indicated by the DVM data

in Figures 5.13 through 5.16, the rise is also present in the data.

(A higher order approximation would probably improve the match in
Figure 3.9.) Since in some cases severe convergence problems develop-
ed in the root extraction routine, even with the corners in the Padé
approximation wide apart, a higher than second-order approximation

could not be used.

Consider next the match of the model with the DVM data.
As with the CRT data there are differences in the results for the
three Operators. Also, the run-to-run variability of the DVM data
for each Operator is greater as indicated by the larger i 10 bands
in the data, especially at high frequencies. The large high fre-
quency variability is probably due to the fact that the forcing
function amplitude at these frequencies is very small and conse-
quently almost ignored by the Operators. The matches are good but

not as good as those with the CRT.

Comparison of the results for Operators l and 2 given in
Figures 5.13 and 5.14 with the results for Operator 3 in Figure 5.15
indicates that Operator 3 has a considerably different phase angle
at high frequencies. This implies a much shorter time delay for

Operator 3. Investigation of the "flying” technique of each

132

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133

Operator reveals that Operators l and 2 used a "proportional control"
while the third used a kind of "dither" control. The first two made
corrections based on the sign and size Of the displayed voltage

while the third built a substantially constant frequency dither into
his control. He looked at only the sign of the voltage and at-
tempted tO produce + and - signs on the display as rapidly as he
could and still have what he considered good control. This tech-
nique as compared with the proportional technique produced less high
frequency phase shift. Apparently, Operators l and 2 needed more
time to think about the magnitude of the displayed voltage and con-

sequently produced a longer time delay.

The parameters selected for matching the DVM data are

summarized in Table 5.3. Comparison of Tables 5.2 and 5.3 shows

Table 5.3. Model parameters for DVM match.

 

 

 

 

OPERATOR NUMBER
PARAMETER
1 2 3 1,2,63
l/n 8 5 5 5
l/r 200 200 50 150
11122 0.4 0.8 0.5 0.8
1 0.5 0.5 0.1 0.37

 

 

 

 

 

 

that a difference in measurement noise and in the time delay
accounts for the difference in the displays. The noise would
correspond to reading difficulty while the time delay would corres-
pond tO the amount of mental processing needed by the Operator.

The results with Operator 3 indicate that a great deal depends

134
on the Operator's technique. Instruction and/or training could

minimize this effect.

In conclusion, an ordering of the two displays has been
accomplished where greater noise and time delay are associated with
the DVM display. Other displays could be classified in the same
way and the comparative usefulness Of each in a control task

evaluated.

6. CONCLUSIONS AND RECOMMENDATIONS

The model, as postulated in Chapter 2, is very general
and, on an a priori basis, is capable of handling many Of the
characteristics of the human Operator in a control task. As de-
scribed in Chapter 2, these characteristics are

l) precognition

2) noise

3) nonlinearity

4) nonstationarity

5) multiple inputs and outputs

6) forcing function characteristics

A great deal of detailed investigation and verification by means of
experimental data is needed for each one Of these areas. In Chapters
3, 4, and 5 the following topics, covering a cross-section of the
above areas, are investigated:

1) a time stationary model

2) sine-wave tracking

3) a display study

In Chapter 3, the validity Of the model under time sta-
tionary conditions is studied by comparing the model results with
those found experimentally. As shown in Figures 3.3, 3.5, and 3.6,
the results agree quite well with the experimental data. The re-
sults with the plants Yc = l and YC = 5/(5—2) agree exceptionally

13S

136
well. The gain match with the plant Yc = 1/5 is very good while
the phase angle match at mid-frequencies is not as good as desired.
The difference could be attributed to either the fact that the
model forcing function is not exactly the same as that used to ob-
tain the eXperimental data, or possibly due to the approximations
used for the neuromuscular lag and/or time delay. Further investi-
gation into the exact form of the neuromuscular lag is required.
More experimental work should be done using forcing functions that
are more amenable to analysis. Also, the time delay approximation

should be improved.

As discussed in Chapter 3, it is assumed that a noise
characteristic can be selected that provides, by means of the model,
a time stationary transfer function which matches the experimental
data. These data are Obtained by averaging and the nonstationary
characteristics, which are known tO be present [5], are eliminated.
If the nonstationary characteristics Of the model were taken into
account, the correspondence Of'a "time averaged model" may improve
the results for all plants, especially for the plant Yc = l/s (for
both the transfer functions and the remnant). A study should be
made in which the nonstationary characteristics are taken into
account, possibly in the time domain or with double Fourier trans-

forms.

The sine-wave tracking results, in Chapter 4, show that
the model indeed performs very much like the real human Operator.
A comparison of Figures 4.4 and 4.5 with Figures 4.12 and 4.13

shows the time responses are visually quite similar. Comparison

137

Of the power spectrum of the model with that found experimentally
by Pew et al. in [8] (see Figure 4.22), shows that the two are much

alike.

The model tracks a sine-wave with essentially zero phase
angle as given by (4.25). Also, the model continues to track a
sine-wave even after its "eyes" are closed. Both indicate that the
model predicts and takes into account precognition. (This cannot

be done by any other model presented so far in the Open literature.)

The model time response, after the "eyes" are closed, is
in general agreement with the human Operator time response (see
Figures 4.6 and 4.15). After some unknown period Of time, however,
the model results will not correspond to those of a real human
operator. This occurs because the human Operator probably "loses"
his estimates of the system parameters. (It is assumed that the
man knows these parameters in the filter and controller.) A more
complete model of the human Operator, as indicated in [21], would
probably include a ”parameter estimator," in addition to the filter

and controller and should be investigated.

The display study in Chapter 5 shows that an ordering Of
the relative values of displays is entirely feasible using the
model. It is shown that the change in measurement noise and time
delay that are needed to match the model results with the experi-
mental results can be used to quantify the relative values of the
displays. (Two vastly different displays are used in the study.)
Further study is needed to determine the sensitivity Of this tech-

nique to display design.

APPENDIX A

A

At
e

MAN—PLANT TRANSFER FUNCTION IN TERMS OF

The state model for the human Operator, excluding the phys—

iological characteristics, as given by (3-5) and (3-6) is

i = Ai+ky (A.1)
u = ER (A.2)
The

where for the purpose at hand u and y are taken as scalars.

general solution for x(t) is
t
i(t) = eAtfio + eACt'T)ky(T)dr (A 3)
o
where the initial conditions are
i0 = i(o) (4.4)
are set to

TO develop a transfer function, the initial conditions

zero, whereupon the Laplace transforms of (A.2) and (A.3) yield
01(5) (A15)

Z{eAt} 1211(5) (A.6)

UCS)

i(s)

138

139

Substitution of (A.6) into (A.5) gives the transfer function

 

0(5) _ - ;Z‘ At . A 7
Y(S)—C {e}K (.)
Since it is assumed that there are no repeated elgenvalues, eAt
can be written as the sum Of first order terms of the form
A A t
A t A t
eAt = Ale 1 + Aze 2 + --- Ane n (A.8)
where, in general, (X1, AZ, ---, An) are the complex elgenvalues of
A and (A1, A2, ---, An) are the complex constitutent idempotents

of A. Taking the Laplace transform of (A.8) and combining the re-

sult with (A.7) gives

 

 

CA k CA k 5A R
9131 = 1 + 2 + --- + n (A.9)
Y(s) s-Al s-AZ s-An

Noting that the CAiR are scalars, (A.9) can be rearranged into

 

n-1 n-2
u;s; = 00(015 + C2s + --- + an) (A 10)
Y1s) (s-1111s-1211s-13) --- (s-An)
where the constants (C0, C1, ---, Cn) are scalars.

The expression given by (A.10) is the transfer function
for the human Operator without his physiological characteristics.
The transfer function for the physiological characteristics and the

plant are specified with a combined transfer function, in general,

140

in the form of a ratio of the product of quadratic factors. Thus,
the required Open loop transfer function for the complete man-plant

combination becomes

C (C sn-1+C sn-2
0 1 2

(5-1,)(s-12)(s-13) (S-lnllszwlwd (52+cms+dml

+ + Cn)(s +als+bl) (s +aks+bk )
C(s)

 

(A.11)

APPENDIX B

SOLUTION FOR THE TIME STATIONARY MODEL

The time stationary model is Obtained using a digital com-
puter. The computer yields a time stationary state model and trans-
fer function for the filter and controller, a transfer function for
the man-plant combination, and a Bode plot for the man-plant com-
bination. The solution is programmed in Fortran and solved using
the IBM System/360, Model 50 digital computer located in the
Analytical Engineering Department of the LSI Instrument Division.
The Bode plots are made with a Calcomp plotter. Figure 8.1 is a

flow chart showing the basic blocks Of computation that are used.

An example problem is given to illustrate the solution
for the time stationary model. The augmented plant is given in

Figure 8.2 where, as in Chapter 3, simple models of the neuromuscu-

 

 

 

 

 

 

 

 

 

   

 

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Figure 8.2. Example problem augmented plant.

lar lag and time delay are used. As indicated, a first-order lag

is used for the neuromuscular lag with a time constant of TN = 0.3.

141

142

Read and
Write
Input Data

1

Solve For
Steady-State K

11

Solve For
Steady-State P

1

Solve For
State Model
Of
Filter and Controller

A

1

Find Eigenvalues
and
Constituent Idempotents
of
Filter and Controller

1

Rearrange State
MOdel to get its
Transfer Function

1

Combine Filter and
Controller Transfer Functions
with the
Physiological Characteristics
and Plant Transfer Functions

1

Make
Bode Plot

Figure 8.1. Basic flow chart for time stationary model solution.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

143

A first order Padé approximation of the time delay (see 35) is used
with a delay time of T = 0.1 seconds. A gaussian white noise input
wl, colored by a third—order lag with corner frequencies Of 2.4, 2.5,
and 2.6 rad/sec is used as the forcing fUnction. The plant under con—
trol is a simple constant gain plant with Yc = l. The measurement

noise v is assumed to be white gaussian noise.

The blocks as shown in Figure 8.2 are expanded into separate
elements in Figure 8.3. Included for purposes of illustration, are
noise inputs w2, w3, and w. at various points in the model which
would correspond to noise in the neuromuscular system or noise at

various points involved with the time delay.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8.3. Expanded example problem block diagram.

Figures 8.4 through 8.11 show the computer printout for the
example problem. Figure 8.4 gives the three Optional title cards
that can be used along with the general form of the system model of
the augmented plant, the covariance matrices of the noise inputs, and
the cost function. The dimensions of x, u, w, and y and the numeri-
cal values for the coefficient matrices A, B, C, and D Of the aug-

mented plant are also given in the same general format in Figure 8.4.

144

-MANUAL CONTROL STUDY
TRANSFER FUNCTION IN A SINGLE AXIS COMRENSATOR TASK
PLANT IS YC=I
TN'0.3 - TAU90.I - HDI=2.4 - HD2=2.5 - ND3=2.6
NI‘SDD - RI=20 - M2?=D.534

JANUARY 5. I968

SYSTEM MODEL
DXDT = AX + BU + OH

Y 8 CX 0 V

CDVARIANCE MATRICES
C(IUToHTITI 8 M DELTAITAUI

CIVI 8 N DELTAITAUI

COST FUNCTION

E 10 TO TF INT IXTOX + UTRUI DTI

DIMENSION OF N

DIMENSION OF U

I.OODE+00

DIMENSION OF H

4.0005100

DIMENSION OF Y

I o OOOFOOO

Figure 8.4. Input data.

A
-303DOE*OO 6.660E600
0.0 -20000F*01
0.0 0.0
0.0 0.0
0.0 0.0
B
-3.310E+00
2.0006001
0.0
0.0
0.0
C
-100006‘00 0.0
D
0.0 53.330E+00
0.0 Z.DDOE+DI
0.0 0.0
0.0 0.0
20600F+00 0.0
M
0.0 0.0
0.0 503408-01
O.” 000
0.0 0.0
0.0 0.0
NI IINVERSE OF N1
5.000E002
0
1.0005000 0.0
0.0 0.0
'1.000E+DD 40.0
0.0 0.0
0.0 0.0

R1 IINVERSE OF R1

2.0005001

Figure 8.4.

145

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146

ADDITIONAL NUMERATOP FACTORS

COEF OF COEF OF CONSTANT
5.5 S TERMS
0.0 '100006+00 200006.01

ADDITIONAL DENDMINATDR FACTORS

COEF OF COEF 0F CONSTANT
5.5 5 TERMS
0.0 1.0005000 2.0006001
0.0 3.000E-Ol 1.000E+OO

Figure 8.4. (Con't).

D

TMIN 8

0.0100 SECONDS

147

BEGINNING OF SOLUTION FOR STEADY STATE K

K0

K

0.0

OiDCDOCD
o o oo o
0.3CDDCD

T 3

90791F'02
202105-02
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2.321E-02
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0.200

0.400

0.600

0.800

SECONDS

CIOTDCDD
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SFCONOS

7.710E-07
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SECONDS
2.321E-0?
60‘215-03

-20564E-02

-6.136E-03

-1.154E-03

SECONDS

2.321E-02

6.421E-03‘

-205006'02
’601376’03
’101566‘03

Figure 8.5.

-100436-“1
-2.397E-O7
101106-01
1.900E-02
2.590E-03

-l.083E-01
-2.541E-02
101665-01
2.284E-02
400895-03

-1.083E-01
-2.5445-02
1.168E-01
2.329F-02
4.660E-03

-1.083E-01

.'20544E‘02

1.169E-01
2.344E-02
4.639F-03

313C292)
o O o
OCDCD3£3

-I.776E-02
-50073E-03
1.9OOE-02
6.732E-03
7.768F-06

-2.001F-02
-b.IOOE-03
2028‘8’0?
70°‘25'03
2.168E-03

-?.012F-07
-6.136E-03
203295-02
901‘16'03
3.193F-03

'20 0125-0?
-6.137E-03
2034‘E’OZ
90699E‘O3
3.8675-03

Solution for K.

0.0
0.0
0.0
0.0
0.0

‘2.307F‘01
'7oang‘O‘
2.59OF-01
7.768F-04
10‘36E-0‘

-3.343F-03
-10126F-01
4.089F-03
2.168F-03
7.912E-04

-3.616F-03
-10154F'03
6.460E-03
3.193F-O3
1.682E-03

-304206‘01
f101566'03
6.639E-O1
3.967E-03
20‘9QE-03

T 8

1.010E-01
ZOSZIE'OZ
-I.003E-01
-2.012E-07
-3.420E-01

T a

1.010E-01
2.321E-D?
-I.083F-01
'2.012E-02
'3.620E-03

T I

1.010E-01
200216-02
'1.081E-01
-ZOOIZE'OZ
-3.QZOE-03

T a

1.010F-01
2.321E-07
-10083E‘01
“200125-02
-3.620E-03

ABOVE MATRIX

1.000

1.700

1.600

1.410

SECONDS

200715-02
60621E‘0‘
-205446‘02
-601S7E-03
-IOES6E-03

SECONDS

203215-02
6.4216-03
-2.5§¢E-02
-6.137Eé03
-I.156E-03

SECONDS

2.321E-02
bo‘ZIE'OD
-2.564E-02
-6.137E-03
-IOISOE-03

SECONDS

201716-02
604216-03

“205645-02.

'601375'03
-1.1566-O3

Figure 8.5.

148

‘IOOBBE‘OI
-705“E-02
I.‘6°E‘Ol
2.350E-0?
6.732E-03

-10083E-Ol
'205405‘07
IOIOQE-01
2.352E-02
‘01785'03

-1.083E-01
“205048-07
101695‘01
203535-02
4.8006-03

-100836'01
-205‘4F-02
1.169F-01
203545-02
4.805E-03

IS SS SOLUTION FOR K TD

(Con't).

“700175-02
-60131F'03
2.3506-02
9.959E-03
OOZOIE-03

-2.012E-02
-6.137F-03
2.3525-0?
1.008E-02
‘QSTOE-OS

-2.012E-02

-6.137E-03'

20‘535‘02
1.013E-02
005865-03

-2.012E-OZ
-6.137F-03

2.356E-02

1.0165-02
6.610E-03

HITHIN 0.12

'30‘7”E’03
-I.156F-03
‘0732F-01
4.761E-03
3.1006-03

'30‘20E’01
'10156F‘OS
‘0778E’03
-§.§76F-03
3.608E-03

-3.420E-03
'101566'01
4.800F-03
b.5866-03
3.717E-03

-3.420F-03
-10156F-03
‘0805E'0‘
‘06106-01
301716-03

DTMIN 8

0.0025.

SECONDS

149

BEGINNING OF SOLUTION FOR STEAOY STATE P

PD
0.0
0.0
0.0
0.0
0.0

33333 33333
on o 00 0‘00 0 0
33333 3'3333

33333
0 00 00
33333

SECONDS

0.050 sscnuos'

33333
0 o. o 0
33333

SECONDS

33333
00 o 00
33333

0.292 SECONDS

33333
0 o 00 o
3-3333

0.437 SECONDS

Figure 8.6.

0.0
0 0

5.000E-01'

100005-01
1.000E-01

0.0
0.0
3.507F-02
3.2486-02
1.247E-02

0.0
0.0
1.828F-02
‘0359F‘02
108485-02

0.0
0.0
I.¢77€-07
3.386E-02
7.9766-02

0.0
0.0
103085-02
2.97‘E-02
306035-02

0.0
0.0
1.000F-01

5.0005-01'

100006-01

0.0
0.0
3.2‘8E'02
3.8266-01

1.1316-01'

0.0
0.0

4.359F-02'

2.4‘nF-01
1.476F-01

0.0
0.0
2.9745702
1.018F-01
loBSbE‘O]

Solution for P.

0
-O.

0.0
0.0
1.000E-01
1.000E-Dl
5.0006-01

0.0
0.0
1.2475-0?
1.13lE-OI
5.284F-Ol

0.0
0.0
10848E’07
1.476%-01
So?84F’01

0.0
0.0
7.976E-0’

1.7625-01
600826-“,

.0
0
3.603F-02

108566‘01
‘0962E‘01

33333
33333

33333
3333.)

33333
00 o no

33333

33333
0 o. 0 0

33333

0.652

0.972

1.112

1.227

1.4i2

150

- SECONDS
n.” 0.0
0.0 0.0
0.0 1.270F-02
000 7.9616-n2
000 301°4E-02
SECONDS
0.0 0.0
0.0 0.0
0.0 1.272F-OZ
0.0 2.943F-02
SFCONDS
O.” O.”
0.0 0.0
0.0 1.243F-02
0.0 2.040F-02
000 .307025'02
SECONDS
0.0 0.0
0.0 0.0
0.0 1.730E-02
0.0 .2.fl36E-02
0.0 3.704E-07
SECONDS
0.0 0.0
0.0 000
no” 1.238F-02
0.0 7.835E-O7
0.0 3.705E-07
SECONDS
0.0 0.0
0.0 0.0
0.0 10238F'02
0.0 2.835E’02
0.0 3.704E-02
Figure 8.6. (Con't).

0.0
0.0
2.061F-0?
909846-02
108336-01

.0
.0
.0435-02
07015-07
1.781E-01

0
0
2
9

0.0
0.0
2.040F-02
9.258F‘02
10729E'Ol

0.0
0.0
208366-02
QOZ‘RE-OZ
1.7285-01

0.0

0.0

203356-02
9.2‘56’02
1.7?8E-01

1.727E-Ol

0.0
0.0
30794F-02
108335-01
50912F-01

0.0
0.0
3.000F-02

1.781E-01
50697F-01

0.0
0.0
3.7026-02
107795-01
5.6265-01

1.72RE-01
5.624F-01

0.0
0.0
3.704E-D2
1.7275-01
5.623E-01

T 2 1.627 SECONDS

33333
00 0 00
33333
33333
0 on 0 0
33333

T 8 1.657 SECONDS

0.0

3333
3333

ABOVE MATRIX IS SS SOLUTION FOR P TO HITHIN 0.012

Figure 3.6.

151

0.0
0.0
1.2386-02
ZOEB‘E-02
307006-02

(Con't).

0.0
0.0
2.8346-02
9.237E-02
10727E-01

0.0
0.0
2.834F‘02
002376‘02
1.727F‘Ol

0.0
0.0
3.7045-07

1.727F‘fi1
5.623F-01

0.0
0.0
3070‘5’07
10727F‘01

5.623E-01

152

AH
5.10.5.00 1.007F+01 -o.asvs+0n -1.7120+00 -7.n12F-01
-5.120£+01 -4.0455+01 5.976F+01 2.720F+01 4.692£+00
6.190F+OO 0.0 -8.S9OF+00 ?.400§+00 0.0
1.417ro01 0.0 -1;417F+n1 -7.500£+00 ?.SOOF+OO
1.3526+01 0.0 -1.85?E+01 0.0 -2.6OOF+OO
CH
-2.560£+00 -1.023£+00 2.963E+00 1.IISE+OO 2.146F-01
KH
0.0
0.0
6.1°OE+00
1.4175001
l.8526+01
11111 -- EIGENVALUFS
-2.1045+00 1.2ane+00
-2.%04F+OO -l.238E+00
-?.061E+01 1.5309+01
-Z.OhlE+01 -1.5?6E+01
-3.llZE¢OO 0.0
SOLUTION FOR EIGENVALUES CONVFRGEO
CHARACTERISTIC PDLYNDMIAL
1.000F+00 4.09se+01 1.0005+03 5.9966003 1.400F+04
1.4076000

Figure 8.7. Filter and controller model.

153

c.e
c.r
0.9
v.0
o.c

«cuwsoo..
~onunw—.~
«oowr>~.¢
«cum—pn.s
~c1mo<r.~-

~c1usoo.~1
enlwmm—o—I
«c1urrs.on
mcuuuso.pn
Acnucemow

nr1u~¢s..
.cuuows.s
uchcemor
.c1wcrm.¢
.olw~¢n.r

«(numerouo
.cuwovsosn
~t1u¢0r.r1
pc1w¢bmoea
~c1u~¢n.c1

uo1ucm~.N
—CIwmoo.>1
COOWCNO.N
~01w—c0..
co+um—o.~

«eluccc.~
~c1wmec.~
«o-um~o.c
Ncawcoe.~1
rc1u-—.no

ACIQ¢ooou
~c1wmoc.~
rc1wmmc.¢
«cameoc.~1
melwwwuono

.ouumnhom
—c1ucao.m
cc.w¢—m..u
«eth~s.¢1
(coupcM.—1

~coun~p.m
.coucoc.m
cc.w¢—n.~u
«rune—s..-
cc¢u~c«.~o

0.0
coo
Coo
Cot
Cot

uOIwmo—ou
ACIwNnmoc
«Clwcomom
ucuwmb—.c
~cuu——v.~1

~01wmo—.~1
«Olwmcm.01
«Ciucomom1
~01wmhuoc1
uc1m—uro—

ucnwswc.—1
~cuwhcc.~1
—c1u~«m.o
Nc1w¢~>.ou
~r1wrnv.o

—CIuhNo.~
weluhcc.~
pcumnnw.¢1
mcuwc—~.m
.cnumnm.on

.mucooomEowfl accouwumcou

wosmocw.on
.ouw~n~.m

ocowo——._n
~c1u——r.co
oc+u~——.~1

«c1m—h¢.¢
«Ciwpnmom
«01mm000—
~CIwh~co~1
Nc1u0¢0.~

helm—hwom
«alwhmwor
NCImmOO.—
—CIm-Oom1
NCIuorcoN

«c1u0h0.01
pcauommow
~C1uoccor
—CIunno.~
~C1umrwov

wonucsn.¢1
pcuuomm.n
_couoo¢.m
penumno.m
ucnumcwow

c.r
c.c
0.:
o.c
0.0

uclumwho—
—c1wncc.—
wcuuceo.~
co+wro~.~
ucnuc~¢.¢1

paiwmwboul
~Clu~¢c.~|
NCluc¢¢osl
oc+mro~.~1
~€1u1m¢.c

—c1u¢cc.—
ccowucc.u1
CC+wCNso~l

~c1w¢oo.oJ

cc.m~mo.—1

~c1u¢cc.~1
Co+uhoc.—
cc¢uom~._
— Ouccowo
CCOw—¢¢.—

~o1uocu.u1
—onw0mr.h

ooomcco.~1
~01mcc0.—1
ccoueor.~1

~cumc~o.c
_cnwcom.c
_ouu—r~.~
ucuuoc¢.~
ucoumcn.mn

—euw¢~c.o
”cuuoom.c

.—c1w—mh.~

~CIucc¢ow
wolwwo~.m1

~01ucmworl

_—c1mcc..¢c

ccowmwe.—
uonw¢ce.~n
coowmwo.—

—OIwcmm.m1
_Cluoc~.wl
catwrnc.~
~01w¢oc.—1
cooumwco—

.w.m opswflm

—c1w>mo.—1
—c1w¢r¢.~1
NCIuhoo.OI
unluo~0.>1
~C0wn0£oN

—onwsro.—
acuu¢m¢.—
~c1usec.c
~cnuc~c.r
pcuwwcc.n1

Nelmm~o.m1
—01ww~«.~
—01w¢—m.c
malwh—c.~
pelwwwoot

NOIummo.n
~01wr~r.~l
~CIwc—w.¢1

.cnu»...ua.

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mu=d<>2wc—u m¢

NOthtcoo
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wolwownoc
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pelwc—c.—I
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—c1wa>~.¢

~c1w¢os.m1

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netwOF—oc

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.oumcmo.mu
~c1u~_0.m
_cuwmco.mu

uccwomr.—
~01Ln0—.«
—ovucmc.m1
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—01wm0N.~1
~01wexroul
~01w¢——.OI
co¢wc¢uo~l
~01w~fic.m

~c1w~o~.~
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~01uc——.o
ccowc¢~.~
~01w—s«.m1

~01worm..-
«clwram.c
cc.wsws.~
—eluwcc.c
oo+w->.—

uonwcrr.—

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00¢whmp.~l
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co+w—-.~1

uo1mcnw.m
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ocean—o.n
~61w0—~.~
oo+uhco.m

~01w—Cm.0|
~c1uocw.¢1
—O1woct.ul
pCIwGOR.rI
~01uokron

uOqueM.01
~C1wccn.¢1
—01wccc.—I
uCIwonr.m1
—CIumhm.m

palm—rc.m
—01umrm.m
roouorR.—I
~C1w~rmow
(cowuer.—l

—c1m—no.m
netwcmr.¢
co+momm.—1
donw_nm.~
ccou—¢M.—I

awcac uzdm Z— thwaaiwc~ hzth—bmzcu

154

0111 -- HHEQF TF 15 SUM OF 8111/15 - LA1111 TERHS
2.311F+00 -4.501F+OO
2.311F+OO 4.003F+00
1.653F+01 -l.628F+00
1.6S3E+01 1.628E+00
109818-01 000

THE TF AS A RATIO OF POLYNOMIALS IS AS FOLLOWS:

C1011 C111SIN-11 + C121SIN-21 + --- + C1N1 1

DIVIDED BY
IS'- LA1111 IS - LA1211 --- 1S - LA1N11
C101 -- CONSTANT TERN OF TRANSFER FUNCTION

3.048EOOI 0.0

C111 -- NUMFRATOR POLYNOMIAL COEFFICIFNTS
1.000E+00 0.0
3.233F+01 1.190F-06
30014E*02 6.3“4E-06
1.101E+03 1.7895-04
1.660E+03 -9.516E-05

COMPLFTE NUN COEF BEGINNING WITH COEF OF S1N-11

-3.eane+01 -4.7445+0é 1.3zns+04 1.065E+05 0.407e+0e
1.7786006
COMPLETE DENORINATOR 001091710 FACTORS

1.000E+00 6.6086+00 6.8416+00
1.000E+00 0.123F+01 6.600E+02

0.0 1.000EOOO 3.IIZE+OO
0.0 1.000E+00 Z.000E+01
0.0 30000E'01 100005000

Figure 8.9. Filter and controller transfer function.

155

oLANT IS YC=1
TN=0.3 - TAU=0.1 - HDI=2.6 - HD?=2.5 - H03=?.6
NI=500 - RI=70 - "22:0.536
TRANSFER FUNCTION DATA
JANUARY 5. 1968

COEFFICIENTS - FORWARD LOOP NUMERATOR

-3.a405+01 -4.7445+02 1.128F406 1.0655+05 8.467F+OS
102786.06 '

FACTORS - FORWARD lOOP NUNFRATOR

1.000E+00 8.905E+00 2.491E+01
1.0DOF+00 -10678F"04 -4.000F+O7
0.0 1.000E+00 3.332E*00
0.0 0.0 -30848£+01

FACTORS - FORHARD LOOP DENONINATOR

I.OOOE+OO 4.608FOOO 6.84IE*OO
1.0006600 4.1235+01 6.609F402
.O , 1.0005000 3.1125+OO
.0 IoOOOE‘OO 200°06’01
.0 3.000F-01 1.0005+00

333

Figure 8.10. Man-plant transfer function.

FREQUENCYIRPSI

100005-02
1.03RF-07
1.076E-02
1.117F-02
1.1S0E-O7
lo’n7F-02
1.267E-02
107°4F-n2
1.341E-O?
1.303E-O2
104455-02
1.5005-0?
1.556F-02
106146-02
106156-”?
1.7386-02
loqan-GP
1.071F—0?
100416-02
2.0166-02
ZonRQF-n2
201685-02
202495“02
2.3348-02
2.421E-02
2.512F-02
2.606E'02
2.704F-OZ
20R06E-02
ZoquF-O?
300205-07
301336-02
3o’gIE’02
3.373E-02
3.5006-02
306015-02
3.767E-02
3.0095-02
4.0555-02
“O’GRE-OZ
“03655-02
405296-02
006°9F'02
4.076E-02
SOOSQE-OZ
50248F-02
5.445E-02
506506-02

Figure 8.11.

156

FORWARD

CAINIOB1

1.314F101
1.314F+nl
1.314E+01
1.3IAE+OI
10314F§nl
1.314F+Ol
1.314E+01
1.314F+01
1.314E+OI
1.314F001
1.314F+01
1.314F+01
1.314E+01
IOOIQF§OI
l.3146+01
1.314E¢01
1.314F+nl
1.314F+OI
1.314E+01
I.314F*01
IQBIQFTOI
1.3146+Ol
1.314F+01
1.314E+01
10316F+01
1.314F+01
1.314F001
103146001
1.314F+01
10314F+01
1.314F+01
10314E+01
1.314F+01
1.314E+01
1.314F+01
1.314E+01
1.314E+01
1.314E+01
1.316E+Dl
l.314€+01
1031£E+01
1.314E+01
1.314E+01
1.314F+01
1.3IAE+01
1.314E+nl
1.314E+01
10314E+OI

LOOP
PHASFIDFGI

-4.77sF-01
-4043gF-nl
-406OZE-OI.
-4077QF-OI

-4oq‘4F-01
‘5.139F‘01
-s.3325-01
-s.613F-01
-5.74OF-OI
-5.906F-01
-6.179E-01
-60411F-OI
-60651E-OI
-609n1F-01
-T.160E-01
-7047QF-nI
-70708F'OI
-7.QQTE-Ol
-R.?07€-01
'8.6ORE-OI
-80012F-0‘
“907675-01
‘90 61. SF-OI
-90975F-01
-1.03SE+00
‘1.074F+OO
'10114F‘00
-l.156E+00
-1.199E+00
-l.244E+OO
-10?QIF+00
-1.13OF+OO
-10390F§00
-1.442F+00
-1.4°6E+00
-l.502F+00
-10610F000
-10671F+00
-1.734E+00
-10799F000
-10866E+00
-10936F+00
-2.000F+00
-2.084F+00
-2.162F+00
'20244F+00
'20328F+00
’20‘15F+00

Man-plant frequency response data.

5.9026-02
6.002E-02
6.310F-02
6.793E-0?
7.312F-02
1.5875-02
703716-02
8.107E-02
80673E'02
8.701F-02
9.1215-02
0.019E-02
100195-01
1.057F-01
1.097F-01
1.1385-01
1.100E-01
1.2?5F-01
102115-01
1.318E-01
10368E‘nl
100195-01
1.4725-01
1.528F~01
1.585E-01
106455-01
1.706E-01
1.770F-01
108075-01
1.006E-01
1.977F-01
2.091E-01
20129F’01
ZQ’OOE‘OI
2.291F-01
20377F’OI
20‘665‘01
205596-01
206556-01
2.7556-01
2.RSRE-01
2.965F-O1
3.0766-01
3.1926-01
3.312E-01
3.436E-01
30565F‘OI
306996-01
3.837E-01
3.9RZE-01
0.131E-01

Figure 8.11.

157

1.314F‘01 ‘2.506F+OO
1.310E+OI -7.6OOF+00
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1.311E*Ol -7.5665*00
1.311E*01 -7.847E*00
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1.310F+01 -8.763F+OO
1.310E+01 -9.091F+OO
1.310F+01 “9.432F+OO
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1.309F+01 -1.015E+01
1.309E+01 '1.053E+01
1.308E+01 -1.093E+01
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103028001 -10577F+01
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(Con't).

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Figure 8.11.

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7.5005900
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(Con't).

301346.00
3.251F+00
3.173E+00
1.500F+On
3.6SIF+OO
3.757F+00
3.909F+00
4.0565+00
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Figure 8.11.

159

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Figure 8.11.

160

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(Con't).

161

The matrix M that must be read into the program is de-

fined by

u(t+r) u(t) T

E = M6(T) (8.1)
W(t+T) w(t)
Since u is known, the covariance terms involving u are zero,
and since u is single dimensional, the first row and column of M
in (8.1) are zero. For simplicity all components of w are
assumed to be zero except for the forcing function wl. As given

by (3.27), the variance of WI is 0.534, yielding the complete

matrix M as shown in Figure 8.4.

Since v is a scalar, the covariance of v is of dimen-

sion one with
1

As shown in Figure 8.4, the value chosen for n-1 is

n'1 = 500 (3.3)

With the assumption that the operator simply minimizes the
integral-squared error between the forcing function x3 and the
plant output x1, and that he also minimizes the integral-squared
value of the control u, the cost function matrices Q and R
become those shown in Figure 8.4. The value selected for the single

dimensional R—1 is

162
Finally, given in Figure 8.4, is the s-domain transfer
function representing the physiological characteristics of the man

and the plant, as required in Appendix A.

Figure 8.5 shows the printouts of the time solution for
K where (2.12) is solved backwards starting with K(tf) = 0. The

solution is continued until

 

 

[.2 ....)1- .2. a-..“
1.1 13 111. 13 . < 0.001 (B 4)
Z l '—
i,j kij “3
or
2 <
i’j kij(t) __0.0000001 (B 5)

 

 

Figure 8.6 shows the printouts of the time solution for P

with an arbitrarily selected initial condition of

P(O) = 0 0 .5 .1 .1 (8.6)

 

 

The solution is assumed to have converged when

 

.Z.[P (t)| - .Zijp.. (t-0.2) I
.1’3 11. 1’ *3 1 :_0.0001 (3.7)

, Z ‘
Pij(t)

1,]

 

163

01‘

Z

1 3 1 0.0000001 (8.8)

 

 

Pij(t)

In both the solution for K and P, the Runge—Kutta

integration technique is used (36).

Figure 8.7 shows the solutions for the matrices A, C, and
K for the state model of the filter and controller as given by
(3.7), (3.8), and (2.8). Also shown are the eigenvalues and charac-

teristic polynomial of A.

Figure 8.8 shows the constituent idempotents corresponding
to the solution eAt. The characteristic polynomial and the con-
stituent idempotents are found using the conjoint algorithm (37)
and the eigenvalues of the characteristic polynomial are obtained

using Barstow's method (38).

Figure 8.9 shows the transfer function for the filter and
controller in two forms and then shows the numerator and denominator
of the complete transfer function including the physiological charac-
teristics and the plant. The first form for the filter and con—

troller is

 

 

81 82 En
6(5) = s-LA1 + s-LA2 + '" I 571K; (8'9)
and the second is
C0(C1$n-1 + Czsn_2 + --- + C“)
G(s) = (8.10)

(s-LA1)(s-LA2) -—- (s-LAn)

164

The eigenvalues LAi are given in Figure 8.7 and the 81 and Ci
are given in Figure 8.9. (The first columns are real parts and the
second columns are imaginary parts.) The complete transfer function
including the physiological characteristics and the plant is shown
next where the numerator is given as a polynomial and the denominator
is given as quadratic factors. The first column of the quadratic
factors is the coefficient of 52, the second the coefficient of s,

and the third the constant term.

Figure 8.10 shows the input data to the Bode plot sub-
routine. The first three lines are the titles as given in Figure
8.4. The forward loop numerator coefficients correspond to the
complete system numerator in polynomial form as given in Figure
8.9. Shown next are the quadratic factors for this numerator, again
starting with the coefficient of 52, followed by a repeat of the
complete denominator quadratic factors as given in Figure 8.9.
Figure 8.11 shows the frequency, gain, and phase angle of the com-
plete man-plant transfer function. Since the feedback is unity, the
forward loop response that is printed out is the complete open loop
response. A plot of this data as provided by the Calcomp plotter is
given in Figure 8.12. The match of this result with experimental

data is discussed in Chapter 3 in connection with Figure 3.3.

165

1 101X118338030) BSBHd

.uon mwom Eofinoum mamemxm .NH.m opsmfim

_uwm\o¢m. huzwzowmm

 

 

 

 

 

mmzommmm puzmaommu moo; Dmmzmou
3mm.onmmz I omn_m I Down—z
m.mumoz I m.mumoz I v.mu~o3 1 _.ou:¢h s m.onzh
_nu» mm hchm

 

“@531. wL. 0. - 5.1.3.0. ml. .. p . 5%.}. .LL 1. Vfirfiwt - m m 1:...
w m

1 ......... w

wmczm
Z—to
”Av Lfi.
0 m
m m
. 11:1: .
m. . m
m .X. m
I
I”

a ll .
wgv ll L m
m m
p. ..
.o m
0 0
o o
m. .0
w m
m; m
m m
va LTD”
m mmumid m

NIUO

190)

APPENDIX C

M

EFFECTS OF fi-AND Q ON THE MODEL

F
If N, as defined by the covariance of the measurement

noise v given by (2.4), can be written as
N = nI (C.1)

where n is a constant scalar and I is the unit matrix, then the
time solution for the man—plant combination depends only on the ratio

M/n, where M is a matrix, and not on M and N individually.

Upon examination of the equations given by (2.1) through
(2.13) that define the complete man-plant combination, it is ob-
served that M and N appear in the matrix Riccati equation for P
as given in (2.9), N (and P appear in the Kalman gain matrix for
k given in (2.8), and 8 appears in the model for the estimated
state vector given in (2.7). Thus, if F depends on the ratio M/n

only, then the complete model depends on the ratio only.

The dependence of K on the ratio M/n can be shown by
first substituting the inverse of (C.l) into (2.8) and (2.9), and

dividing (2.9) through by n. Thus

3=A3+3A -ECICB+D—D (C.2)
n n n n n

166

167

,2 = SGT (c.3)
Next, let
p* = :— (0.4)
and since n is a constant
13* = E (0.5)
By substitution
13* = AP* + P*AT - P*CTICP* + 1}ng (C.6)
12 = P*CT (0.7)

The solution for P* depends on the ratio g-only and therefore so

does K and the complete solution for the man-plant combination.

A similar argument can be applied to the solution for the

Optimal control gain matrix K where with R of the form
R = r1 (C.8)

the solution for K depends on the ratio Q/r only, and therefore,

so does the solution for the complete man-plant combination.

APPENDIX D
CROSS—SPECTRAL ANALYZER

As described in detail in [5, 19, 43, 45, and 46], the
techniques for finding the cross-spectral density of two stationary
random functions of time are well known. The detailed analysis of
the accuracies involved when the techniques are mechanized with
realizable components, however, is extremely complex. The funda-
mental basis and the methods used in the display study in Chapter 5

are briefly outlined here.

The basic Operations for finding the cross-spectral density
at the frequency wn are performed as shown in Figure 0.1. The
two stationary random time functions x(t) and y(t) are frequency
shifted using the sine and cosine of wn’ filtered with a very nar-
row bandwidth filter, multiplied, and averaged over a long period

of time.

Consider the operations performed on x(t) in Figure 0.1.
The multiplications by cos wnt and sin wnt provide the real and
imaginary parts, respectively, of a frequency shift where in the time

domain

x2(t) x(t) [coswnt - j sinwnt]

(D.1)

-jw t
x(t) e 1‘

168

169

 

 

 

3%;
" 52

-$1N anI 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

£X1111 ‘XAII
/ . ‘ flf'aoxymn)
' 2w 1 + 1 T
“I . &x1(t) TL-L—v
' §
12%
w :0
X3111
11) 9° .
8
I T ____.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\ ‘ 1' ()dt
1Cy.01 ‘

 

 

 

 

 

 

 

Figure 0.1.

 

\\JL~m(O 4]

 

The basic operations to find the

cross-spectral density at wn'

and the frequency shift is

X2(w)

The complete frequency

by wn.

m -jwnt -jwt
x(t)e e dt

—@

x(w+wn) (0.2)

domain representation is shifted to the "left"

The narrow bandwidth filter, centered around w = 0, that

follows the frequency shift in Figure 0.1 yields

X1(w)

X2(w) for -wo < w < mo

0 elsewhere

170

and
X1(w) = X(w+wn) for -w0 < w < wo
(D.3)
= 0 elsewhere
By a change of variables
X1(w—wn) = X(w) for -w0+wn < w < wo+mn
(0.4)
= 0 elsewhere
For mo sufficiently small, both (D.3) and (D.4) give
me) = x (mm)
(0.5)
X1(w) = O for w # wn

The operation in (D.4) is a bandpass filtering of X(m) over the
interval (~w0+wn, w0+wn) as indicated in Figure 0.2. Since both
(D.3) and (D.4) give (D.5), the shift of the narrow-band filter to
the right is equivalent to shifting X(w) to the left and filtering

with a filter centered at w = 0.

HOW»)?

 

 

Figure 8.2. Frequency shift and filter combined.

If h(t) represent the impulse response of the combined

operations shown in Figure D.2, convolution gives

(1)

x1(t) = h(T)X(t-T)dT (D.6)

—€D

171
Y1Ct) = J h(r)y(t-T)dr

The cross—correlation of the complex quantities x1(t) and

Y1(t) [42] is

(O)

E (X1(t+o)y1(t))

R
x1Y1

00

E I h(T)x(t+o-T)dT I h*(a)y(t-a)da

- -m

(D.7)

(D.8)

(D.9)

Since the integrands are square integrable, the order of integration

can be interchanged [42] giving

Rx1y1(o) = J h(T) J h*(a)ny(o+a-T)dadt

— _m

'The cross-spectral density is

m -jwo
¢ 0 = e R 0 do
X1Y1( ) X1Y1( )

which, by substitution of (D.10) yields

00 _jwo co co
¢x1y1(w) = J e J h(T) J h*(a)RXy(o+a-T)dadrdo

— — -m

_ “ij jwa -jw(0+a-T)
- h(r)e h(-a)e e ny(o+o-T)dodadr

— -w —

(D.10)

(0.11)

(0.12)

(D.13)

172

chwmc-jmxycw (0.14)

1)
X1Y1(w)

2

(bx), (w) (D.15)

 

‘HUM

The expected value of the complex quantities x1(t) and

y1(t) at the same time t is

l1—1

8 {x1(t)yf(t)} = 2” J ¢x1y1(w)dw (D.16)

Substitution of (D.15) gives

 

2
E {X1(t)yf(t)} = €%—[ lH(jw) ¢xy(w)dw (D.17)

With 00 in Figure (D.2) sufficiently small, application of the
mean value theorem yields

“0

E {x1(t)y’f(t)} = T ¢xy(0n) (D.18)

'The time average in Figure 0.1, with the assumption that the
functions are ergodic, corresponds to the ensemble average on the

left side of (D.18). In expanded form, (D.18) becomes

8 { [Re x1(t) +j Im x1(t)] [Re h(t)-3'1m y1(t)]}

w
= 73— [Re ¢xy(wn) + j Im 0xy(wn):] (0.19)

giving

0.1
712 Re ¢xy(‘*’n) = E {Re 211(1) Re Y1“) + m x1(t) Im h(t)}

(D.20)

173

(L)

-9-Im 0

fl xy(wn) = E (_Im x1(t)Re y1(t) - Re x1(t) 1m yl(t)‘>

(0.21)

It is desired to have a running time that is not too long a
time for a human operator to properly perform his task. This places
a lower bound on the filter bandwidth and an upper bound on the
averaging time. As stated by Blackman and Tukey in [45], the filter
bandpass should be "several to many times" wider than the reciprocal
of the run time. Also, to obtain a good average, the run time
should include several periods of each wn' A second-order lag with
a corner frequency of 0.0707 rad/sec and a damping ratio of 0.707
were selected for the filter. A four minute run time was used.
Thus, the filter bandpass of 0.184 [=2(0-0707)(1-3)] rad/sec iS
many times larger than 0.0042(=l/240). Based on Table 5.1, the
number of cycles per run, for the 10 frequencies, ranges from 6 to
527 allowing a reasonable number of periods at each frequency for
averaging. The frequency spacing between the sine-waves in i(t)
from Table 5.1 ranges from 0.105 to 6.23 rad/sec. With the filter
selected, the main lobes of power [19] centered around each wn are
within mu 1 0.041. Since 2(0.041) is less than the minimum separa-

tion of 0.105, there is no overlapping of these main lobes of power.

The cross—spectral den51t1es ¢ie(wn) and ¢ic(wn) are
required in (5.4). Since the forcing function i(t) is composed
of the sum of sine-waves, the same sine-waves can be used for the

frequency shifting. Consequently, setting

x(t) = i(t) (D~22)

174
the steady-state real and imaginary parts of x1(t) can be com-

puted for the ten forcing function frequencies. In the time domain,
the impulse response of the ideal low-pass filter is

sinwot

_—7fi?—— (D.23)

hp(t) =

giving by convolution

 

” sinwo(t-T) .
Re x1(t) = wE-r) [COS 111;] 1(T)dr (0.24)

..00

sinwo(t-T)

u(t-T)

2

a COSw TCOSw T + --- + a cos w T + ---
1 n 1 n n

—CD
+ alocoswnrc050101:] d1

(0.25)
The constants an are the half amplitudes of the sine-waves in
i(t). Since mo is chosen smaller than the sums and differences
of mu and mi for wni‘wi, the terms involving COSmnTCOSwiT,

wn # mi are zero. The term containing coszwnr yields

an m sinw0(t-r)

7:. (t-T)

-00

 

Re x1(t) coszwnrdr (0.26)

00

an sinwot 1
= —- —— [— + 00520) (t-T)]d1’ (0.27)
n T n

-00

N

-—“—()r-1- -3 (028)
‘ n '" 2 ‘ 2 '

175

In a similar manner, the imaginary part of x1(t) is
found with the coswnt term in (0.24) replaced by sinwnr. In this
case also, all terms containing sinwnrcoswir for wn # mi, are

zero. The remaining term gives the result,

Sinw0(t-T)

 

Im x1(t) = “(t_T) an51nwn1coswnr d1 (0.29)
an m sinwor 1 2

= —n— - ___—TI” ['5] (S In wnT)dT (D. 30)

= 0 (D.31)

The block diagram at one frequency, after the above sim-

plifications and assumptions are made, is shown in Figure 0.3. The

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

COSmnI
1 . ..."...
. “"" M [(1201 'F—-*
0
10:0
w.) w 0.134 T 2 ‘s'w°xy(“’n)
—** C I _{ [(101 ~——-0
0
10:0
SlNunI
Figure D.3. Simplified block diagram for ®Xy(wn).
input is either
y(t) = e(t) (0.32)
or
y(t) = c(t) (0.33)

176

which give, respectively, ¢ie(wn) and ¢ic(wn)'

As indicated in [46], prewhitening of y(t) can be used
to improve the scaling of the analyzer -- particularly at high fre-
quencies. Given the ratio

¢ic(wn)

0.
1e n

prewhitening in both e(t) and c(t) cancel, leaving the transfer

function unchanged. Prewhitening with a transfer function

. _ jw + 1
Gw(3w) — 20L“) + 20] (D.35)

was used.

10.

11.

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