PURDUE UNIVERSITY

SCHOOL OF AERONAUTICS AND ASTRONAUTICS

OPTIMAL COOPERATIVE CONTROL SYNTHESIS

OF ACTIVE DISPLAYS

by

SANJAY GARG and DAVID K. SCHMIDT

JNASA-1771C2 ) CE1IM&I CGGPEEftllV E CON1BOL

SYNTHESIS OF AC1IVE DISPLAY S Pinal Eeport

{Purdue Dniv.) 160 p CSCL 20P

22nd October, 1985

N86-29651

Unclas

G3/74 43235

Wesf Lafayeffe, Indiana 47907
OPTIMAL COOPERATIVE CONTROL SYNTHESIS

OF ACTIVE DISPLAYS

by

SANJAY GARG and DAVID K. SCHMIDT

School of Aeronautics and Astronautics

Purdue University

West Lafayette, IN 47907

Submitted to:

NASA Dryden Flight Research Facility

Ames Research Center

Final Report for grant NAG2-228

22nd October, 1985
11

ACKNOWLEDGMENT

This research was supported by NASA Dryden Research Facility, Ames Research Center under grant NAG2-228. Mr. E. L.

Duke is the technical monitor.
ill

TABLE OF CONTENTS

Page

LIST OF TABLES v

LIST OF FIGURES vi

LIST OF SYMBOLS xl

ABSTRACT xiv

CHAPTER 1 INTRODUCTION 1

CHAPTER 2 ANALYSIS OF DISPLAYS FOR A K/s2 PLANT 8

2

2.1 K/s Plant and Task Descrigtion 8

2.2 Display Quickening For K/s Plant 12

CHAPTER 3 OPTIMAL COOPERATIVE CONTROL/DISPLAY

DESIGN METHODOLOGY 19

3.1 Problem Formulation 19

3.2 Design Objectives 20

3.3 Solution for _u _._ 23

3.4 Solution for u and u 24

3.5 Application to Pilot-in-the-Loop Synthesis....26

3.6 Numerical Solution Algorithm 29

CHAPTER 4 APPLICATION OF DISPLAY DESIGN METHODOLOGY

TO K/s PLANT 34

4.1 Problem Formulation and Synthesis Results 34

4.2 Evaluation of the Display Quickening

Control Laws .38

4.2.1 Mean-Square Analysis 38

4.2.2 Power Spectrum Analysis 42

4.2.3 Frequency Domain Analysis 44

CHAPTER 5 APPLICATION OF COOPERATIVE METHODOLOGY TO A

MORE COMPLEX DISPLAY DESIGN 52

5.1 System Dynamics 52

5.2 Task modeling 54

5.3 Optimal Display Synthesis for F-15 Model 65
iy

Page

5.4 Comparison of the Displays ....68

5.4.1 Time Histories 68

5.4.2 Frequency Domain Analysis 73

5.4.3 RMS Analysis 76

5.4.4 Spectral Analysis 78

CHAPTER 6 SUMMARY 83

REFERENCES 86

APPENDICES

APPENDIX A DERIVATION OF NECESSARY CONDITIONS

FOR CONTROLLER 2 (u ) 89

APPENDIX B THE OPTIMAL CONTROL PILOT MODEL 94

APPENDIX C ANALYSIS RESULTS FOR K/s2 PLANT 102

APPENDIX D DATA AND ANALYSIS RESULTS

FOR F-15 AIRCRAFT 120
LIST OF TABLES

Table Page

2.1 OCM RESULTS FOR VARYING DISPLAY BANDWIDTH 13

2.2 OCM RESULTS FOR. VARYING DISPLAY CONTROL GAINS 17

4.1 OPTIMAL DISPLAY DESIGNS FOR K/s2 PLANT 37

4.2 OCM ANALYSIS RESULTS FOR VARIOUS CASES 39

5.1 PILOT PERCEPTION THRESHOLDS 64

5.2 OCM ANALYSIS RESULTS ....77

APPENDIX TABLES

D.I TRIM CONDITIONS FOR F-15 MODEL 121

D.2 MATRICES A AND B FOR F-15 MODEL 122

v v

D.3 PILOT OBSERVATION MATRICES FOR STATUS DISPLAY....123
vl

LIST OF FIGURES

Figure Page

1.1 Simplified Block Diagrams for

Control/Display Augmentation 3

1.2 Control vs. Display Augmentation 4

2.1 Display for K/s2 Plant 10

2.2 Parametric Performance - Manual Control Rate 14

2.3 Parametric Performance - Manual Control Activity..15

3.1 Block Diagram for Control

and Display Augmentation. 21

3.2 Simplified Pilot Model Block Diagram 28

3.3 Flowchart for Numerical Procedure 31

4.1 Error vs. Manual Control Rate

for Various Displays 40

4.2 Error vs. Manual Control Activity

for Various Displays 41

4.3 Comparison of Human Operator's

Control Spectrum, A(u>) 43

4.4 Block Diagram for Display Augmented Plant 45

4.5 Comparison of Display Augmented Plants, 6(s) 47

4.6 Comparison of Human Describing Functions, P(s)....48

4.7 Comparison of Human's Phase Compensation

(adjusted phase) .. .49

4.8 Comparison of Closed-Loop Frequency Response

for Displays 1,3 and 5 51
vll

Figure Page

5.1(a) a Response to Step Elevator Input

Z (<S 0.01 rads, +ve down) 55

5.1(b) Mach Response to Step Elevator Input 56

5.1(c) 6 Response to Step Elevator Input 57

5.2(a) a Response to Step2Throttle Input

z (6T =1 ft/sec , +ve thrust) 58

5.2(b) Mach Response to Step Throttle Input 59

5.2(c) 9 Response to Step Throttle Input 60

5.3 Conceptual Display Format for Status Display 61

5.4 Conceptual Display Format for Augmented Display...63

5.5(a) x Response to Step Elevator Input

1 (6 = 0.01 rads, +ve down) 69

ec

5.5(b) x,0 Response to Step Elevator Input.... 70

dz

5.6(a) x. Response to Step Throttle Input

(6Tc = 1 ft/sec2, +ve thrust). 71

5.6(b) x,_ Response to Step Throttle Input 72

a/

5.7 Block Diagram for Status Display .74

5.8 Block Diagram for Augmented Display 75

5.9 Comparison of e Power Spectrum 79

clZ

5.10 Comparison of Pilot Commanded Elevator Spectrum...80

5.11 Comparison of Pilot Commanded Throttle Spectrum...81

APPENDIX FIGURES

B.I OCM Block Diagram 95

C.I Control Spectrum for Limiting Case - A 103

C.2 Control Spectrum for Display 1 ....104

C.3 Control Spectrum for Display 2 105
vill

Figure Page

C.4 Control Spectrum for Display 3 106

C.5 Control Spectrum for Display 4 107

C.6 Control Spectrum for Display 5 108

2

C.7 Bode-plots of K/s Plant Plus Display 1 109

2

C.8 Bode-plots of K/s Plant Plus Display 2 110

2

C.9 Bode-plots of K/s Plant Plus Display 3 Ill

2

C.10 Bode-plots of K/s Plant Plus Display 4 112

C.ll Bode-plots of K/s2 Plant Plus Display 5 113

C.12 Pilot Describing Function for Limiting Case-A....114

C.I3 Pilot Describing Function for Display 1 115

C.I4 Pilot Describing Function for Display 2 116

C.15 Pilot Describing Function for Display 3... 117

C.I6 Pilot Describing Function for Display 4 118

C.17 Pilot Describing Function for Display 5 119

D.I Pilot Describing Function P .

~a

for Status Display

 Z.C. 124

D.2 Pilot Describing Function P ,

oe

e

for Status Display ?? 125

D.3 Pilot Describing Function P.

oe

M

for Status Display 126

D.4 Pilot Describing Function P.

for Status Display « 127

D.5 Pilot Describing Function P gT

a

zc
Ix

Figure Page

for Status Display ............................... 128

D.6 Pilot Describing Function P

oT

e

for Status Display .......... " ............ • ....... 129

D.7 Pilot Describing Function P

for Status Display ......... ^ ..................... 130

D.8 Pilot Describing Function P

c±~

for Status Display ............................... 131

D.9 Pilot Describing Function P

oe

a

for Augmented Display .......

 z. ? ..... . ............. 132

D.10 Pilot Describing Function P

oe

for Augmented Display ....... ?! ....... ............ 133

D.ll Pilot Describing Function P

oe

x

for Augmented Display ........ . ................... 134

D.I 2 Pilot Describing Function P.

oe

for Augmented Display ....... . .................... 135

D.13 Pilot Describing Function P „

_6T

a

for Augmented Display .......

 Z.C. ................... 136

D.14 Pilot Describing Function P

*dl for Augmented Display ...... ...................... 137

D.15 Pilot Describing Function P

oT

Xd2

for Augmented Display. .. ......................... 138

D.16 Pilot Describing Function P
Figure Page

for Augmented Display 139

D.17 e Power Spectrum for status Display 140

D.I8 Pilot Commanded Elevator Spectrum

for Status Display 141

D.19 Pilot Commanded Throttle Spectrum

for Status Display 142

D.20 e Power Spectrum for Augmented Display 143

cLZ

D.21 Pilot Commanded Elevator Spectrum

for Augmented Display 144

D.22 Pilot Commanded Throttle Spectrum

for Augmented Display 145
xi

LIST OF SYMBOLS

Symbol Meaning

2

a normal acceleration (ft/sec )

z

A plant matrix

B system control distribution matrix

C output state distribution matrix

 J

D system noise distribution matrix

D(s) display system transfer function

e plant position error

2

e normal accleration tracking error (ft/sec )

3. Z

E .output control distribution matrix

f fractional attention allocation for i observation

g control rate weighting

g , display control gain

G gain matrices

G(s) plant transfer function/matrix

I identity matrix

J cost functional

K plant gain

K ..... control gain matrix

M Mach error (.001 Mach)

M. Kalman filter gain matrix

P solution to Ricatti equation

F(s) human operator transfer function/matrix

£ pitch rate (.01 rads/sec)

q augmented system state vector

q error weighting
xii

Symbol Meaning

Q ................... state weighting matrix

r ................... control weighting

R,F ............ .... .control weighting matrices

T(s)... ..... . ....... closed-loop transfer function

JJ ................... control vector

v .................. motor noise vector

— m

v .................. observation noise vector

V ................... perturbed forward velocity (ft/sec)

V_ .................. measurraent noise intensity

w ................... system noise input vector

W ................... system noise intensity

2

£ ................. ..K/s plant position

x* e 5 = ± = 5 = , 5 s i s .• ..... state vector

x, .................. display variable

y ......... . ..... ....system output vector, measurment vector

a ................... angle of attack (.01 rads)

j*. .................. perception threshold value for i observation

_X ................... augmented system state vector

j( ................ ...augmented system state estimate vector

5 ................... human operator's control input vector

6 .................. elevator control input (.01 rads)

e

 2 6T .................. throttle control input (ft/sec )

M ................... pilot input

*

y .................. optimal pilot input

p ...... ...... ...... motor noise ratio

p .. . . i ............. observation noise ratio for i observation

a .................. root mean square value of x

X

£ ................... state error covariance matrix

£ (u).. ............. power spectral density of x

A

T ..... ..... ....... .. time delay

T .................. display lag time constant
xlil

Symbol Meaning

T neuromotor lag time constant

9 pitch rate (.01 rads)

operations

(") time derivative of (")

E{.} expected value operator

T

(.) transpose of (.)

(.) inverse of (.)

subscripts

1 refers to Controller 1

2 refers to Controller 2

aug. augmented system

A augmented system

c closed loop system, command

d display augmentation system

o nominal values, unaugmented system matrices

p human operator/pilot

v vehicle
xiv

ABSTRACT

A technique is developed that is intended to provide a systematic

approach to synthesizing display augmentation for optimal manual control

in complex, closed-loop tasks. A cooperative control synthesis technique, previously developed to design pilot-optimal control augmentation

for the plant, is extended to incorporate the simultaneous design of

performance enhancing displays. The technique utilizes an optimal control model of the man in the loop. It is applied to the design of a

2

quickening control law for a display and a simple K/s plant, and then.

to an F-15 type aircraft in a multi-channel task. Utilizing the closed

loop modeling and analysis procedures, the results from the display

design algorithm are evaluated and an analytical validation is performed. Experimental validation is recommended for future efforts.
CHAPTER J^

INTRODUCTION

The utility of providing augmented displays to the human operator

as an aid in the closed-loop control of high order dynamical systems is

well known, and discussion of some of the early work done in evaluating

various types of augmented displays can be found in [1]. With the

advent of high performance aircraft, the amount of information to be

processed by the pilot to successfully accomplish the assigned task has

continued to increase. It has, therefore, become more critical to

determine the best informational set and display dynamics needed by the

pilot so as to reduce the pilot's workload and improve performance. For

example, the use of flight directors to achieve this objective has been

analytically evaluated in [2] using closed loop or "pilot - modeling"

techniques, and validated through man-in-the-loop simulation in [3].

Ongoing research in the area of automatic flight test trajectory

controllers [4,5] has demonstrated the usefulness of such controllers,

rather than manual control, for accurately following specified complex

trajectories. These controllers, however, take the pilot out of the control loop, and it is desirable to avoid this. The possibility of doing

so while aiding the pilot with an appropriate display is worth exploring, and that is 'the subject of this work.

In general, the design of active displays for the human controller

requires extensive real time man-in-the-loop simulation to evaluate

various candidate designs. Although simulator validation is always

appropriate, the objective of this study is to develop an analytical

technique to aid in the design of pilot-optimal display augmentation
systems.

In the past, analytical methods have been proposed in which models

of human behaviour were used to identify, investigate and evaluate the

properties of augmented aircraft dynamics. In all these methods, however, the "pilot model analysis" was added a posteriori. The idea that

the augmentation system works in cooperation with the pilot and a technique which considers the augmentation system and the pilot to be two

controllers working in "parallel" was first suggested by Schmidt [7],

This technique was later modified and its application to synthesize control augmentation that directly optimizes pilot opinion rating was

demonstrated for a modern control configured aircraft [8,9], The

cooperative control synthesis technique has the advantage that it incorporates a mathematical model of the pilot behaviour and uses optimal

control theory to synthesize control gains that are pilot-optimal as

modeled.

Since display augmentation, like control augmentation, has to be in

harmony with the pilot's abilities and limitations in order to be

acceptable to him as an aid in accomplishing his task, the cooperative

synthesis technique is considered to provide an appropriate framework

for synthesizing pilot-optimal display augmentation.

Even though the objective of both the control augmentation and the

display augmentation is to aid the pilot, the way this is achieved is

fundamentally different for the two types of augmentation. The differences between these two types of augmentation are best understood by

considering the simplified block diagrams of Figures (1.1) and (1.2).

Figure (l.l(a)) depicts the pilot controlling an unaugmented vehicle.

For this case, the responses (y) observed by the pilot are those being
Unaugmented

Vehicle

Pilot

Control

Law

Vehicle

I

I Augmented Vehicle I

Pilot

(a) Unaugmented Vehicle (b) Control Augmentation

Unaugmented

Vehicle

Pilot

(c) Display Augmentation

r

Control

Law

Vehicle

Pilot

(d) Control and Display Augmentation

Figure 1.1 Simplified Block Diagrams for

Control/Display Augmentation
Augmented

Vehicle

G'(s)

Pilot

P'(s)

(a) Control Augmented Vehicle

O

Unaugmented

Vehicle

G(s)

Pilot

P(s)

Display

DCs)

(b) Display Augmented Vehicle

Figure 1.2 Control vs. Display Augmentation
controlled by the pilot.

Figure (l.l(b)) shows the case of a pilot controlling an augmented

vehicle. Note that the control augmentation changes the effective plant

dynamics being controlled by the pilot, and may be designed so as to aid

the pilot in accomplishing his task and to achieve desired dynamics for

improved handling qualities. Here again, however, the responses (y)

observed by the pilot are still those he is directly controlling.

Figure (l.l(c)) shows a simplified case of display augmentation,

wherein the vehicle responses drive a display observed by the pilot.

Note that any display dynamic augmentation does not change the characteristics of the vehicle dynamics being controlled (i.e. y/u is

unchanged). It only modifies the dynamics being observed by the pilot

(or x,/u ). Here, clearly, though the pilot observes and controls lc,

d p d

(the displayed signal), the controlled variables of interest are still

the vehicle responses y. The case of combined control and display augmentation is shown in Figure (l.l(d)).

Consider the manual control of a vehicle G(s), with control augmentation such that the augmented vehicle is G'(s), as in Fig. (1.2(a)).

The closed-loop transfer function, including the pilot, from some disturbance u to the vehicle output y is

where P'(s) represents the pilot describing function. Next consider the

pilot controlling the vehicle without control augmentation but with

display augmentation, as in Fig. (1.2(b)). The closed-loop transfer

function for this case is

tt.) »
Now if the display dynamics D(s) are chosen, for example, such that

G(s)D(s)=G'(s), then the pilot describing function P(s) of the display

augmentation case (Fig. (1.2(b))) is approximately the same as P'(s) for

the control augmentation case. Then Eqn. (1.2) becomes

U(s) (1'

3)

1 + G'(s)(s) w

Comparing Eqn. (1.3) to Eqn. (1.1), it is apparent that though the stability characteristics in terms of the closed-loop characteristic polynmial are the same for the two cases of augmentation, the closed loop

transfer functions are not. The point here is that the closed-loop system dynamics obtained through control augmentation may be quite different from that obtined through display augmentation.

Chapter 2 of this report provides motivation for display augmentation through analytical evaluation of various display "quickening" control laws, synthesized essentially through trial and error, for a simple

2

K/s plant. A compensatory tracking task is analyzed using an Optimal

Control Model (OCM) [6] of human behavior.

In Chapter 3 a methodology to synthesize pilot-optimal

display/ control laws which is sensitive to the control and information

processing limitations of the human controller is proposed. This methodology is an extension of the cooperative control synthesis technique

previously developed to design pilot-optimal control augmentation

[7,8,9], Though the proposed methodology has been developed so as to be

applicable to simultaneous synthesis of pilot-optimal control augmentation and display augmentation, the present discussion focuses on the

application of the technique to display design only.

The application of the cooperative display design technique to
2

again synthesize display laws for the K/s plant in the tracking task is

discussed in Chapter 4. When compared to the results of Chapter 2, the

results of this application tend to analytically validate the synthesis

procedure. The displays lead to predicted reduction in pilot workload

when evaluated using the OCM. Also the ability of the methodology to

provide task tailoring of the displays is demonstrated.

In Chapter 5 the application of the methodology to high order

dynamical systems in a multi-control task scenario is demonstrated for a

modern aircraft. The chosen model closely represents the unaugmented

longitudinal dynamics of the F-15 aircraft, and the task is that of

tracking a normal acceleration command while regulating Mach number.

The control inputs available to the pilot are the elevator stick and

throttle. Model-based evaluation of the synthesized display again indicates reduction in pilot workload in accomplishing the task.

Finally a summary of the work is presented in Chapter 6 and recommendations made for future research.
CHAPTER 2

ANALYSIS OF DISPLAYS FOR A K/sZ PLANT

Motivation for providing the human controller with augmented

displays is explored through analytical evaluation of various

t

emperically-derived display "quickening" control laws for a simple K/s'

plant. The results of this evaluation agree with the known fact that

the human operator's workload can be significantly reduced and his performance improved by a proper design of the signal being displayed to

him.

o

2_.\_ K/s Plant and Task Description

Consider the K/s plant dynamics as discussed, for example, by

Kleinman et al. in [6]. The system state equations are

X

X

=

"0 l'

0 0

X

X _ -

0

K

(2.1.1)

where x(t) is the plant position and <5(t) is the input from the human

controller.

A velocity disturbance is applied to the plant. It is modeled as a

first order Markov process having a break frequency of 2 rads/sec and is

given by

(2.1.2)

[} = 0.054 in.2 Defining

= -2xj(t) + w(t)

where w(t) has intensity W = 0.217 to give I

the plant position error (e) and the error rate (e) as

e(t) | x2(t):<«>4 x3(t) (2.1.3)

and letting K = 1 in./in., the plant equations (2.1.1) can be combined
with the disturbance equation (2.1.2) to get

x,

1

X

2

.

X

3.

=

-2 0 0"

1 0 1

0 0 0 > •

V

X

2

X-

_ 3_

+

0'

0

I

, m

5(t) +

Y

0

p.

or in concise form

w(t)

X=AX+BU+D W

o o o

(2.1.4)

The human operator's task is to minimize the position error x (t) in the

presence of the disturbance.

Next consider a display exhibiting some transport lag modeled in

the form

x

d

(s)

 '

1

e(s) (2.1.5)

The dynamics of the disturbance, plant, and display can then be written

as

•

X

xd

A 0

0

-1 1

.

T

d \

-_-

X

x

d

+

B

o

0

6 +

D

0

0

w (2.1.6)

The human operator's observations for this system are

yl

 =xd

(2.1.7)

where it is assumed, in accordance with the known abilities of the human

controller, that he is able to reconstruct the displayed-variable rate

by observing the displayed variable itself.

As shown in Fig. (2.1), the human operator's task with the display

is to keep the needle on the display centered to the best of his
10

Figure 2.1 Display for K/s2 Plant
11

abilities, subject to reasonable workload. This performance objective

can be reflected in terms of minimizing the cost

T

J (6) = E{lim| /(x* + g?)dt} (2.1.8)

T-*-08 o

With the above formulation in mind, the performance of the manually

controlled system 'is evaluated using the OCM model for various values of

T . A brief description of the OCM modeling technique is given in

Appendix B. For all the analysis carried out in this section, the

parameters that define the OCM model were set to the following values:

(i) Human operator's observation time delay, T, set to 0.2

seconds

(ii) Observation-noise ratio set at -20 dB for both observations

(iii) Motor noise ratio was set at -25 dB

(iv) The weighting, g, on the control rate in the human

controller's objective function was always adjusted to yield a neuromotor lag time constant,T , of 0.1 sees.

(v) Very low values of thresholds were used for the observations

available to the human,and he was assumed to devote full attention

to the displayed signal.

The OCM model obtained for the above values of the parameters has been

shown to correlate very well with the experimentally observed behaviour

of the human controller [6], especially in single-axis tasks such as the

one being discussed here. The significance of these parameters in

modeling the human operator dynamics is explained in Appendix B.
12

The results for various values of display lag T are presented in

Table 2.1, in terms of the mean square values of the variables of

interest. These results are also plotted in Figures'(2.2) and (2.3) (to

be discussed later) and correspond to the curve marked (T) in these

figures. From these results it is clear that with only error e driving

the display, the pilot's performance improves as the display bandwidth

is increased. As — -»• °° the pilot's performance approaches a limiting

d

case where the error is displayed instantneously to the human, or the

limiting case where there is no display lag. This limiting idealized

case (labeled A in the figures) assumes that the error can be sensed and

displayed without any computational delay or lag, and may not be achievable. The question now is whether the performance (or pilot's workload

situation) can be improved by augmenting the display dynamics.

2

_2_._2_ Display Quickening For K/s Plant

Consider the augmented display dynamics of the form

-1 1

d T , d T . 2 d 3

where g. is a gain to be adjusted. Since x,(t) is the plant velocity

state, the above form of display will provide some lead information to

the human. Note that this form of display dynamics can be written in

general as

X

d

 = adx

d

 + ud

U

d

 = Gd^d (2.2.1)

where y, = C,x is the vector of plant responses available for driving

the display, and u, is the display control law to be determined. In the

d

formulation of display dynamics as above, the choice of a, determines
13

TABLE 2.1: OCM RESULTS FOR VARYING DISPLAY BANDWIDTH

Td

(sec)

0.20

0.10

0.05

0.02

0.01

-> 0

M.S. Error (e)

(in.2)

0.0215

0.0177

0.0157

0.0142

0.0135

0.0131

M.S. Input (6)

(in.2)

2.176

1.543

1.353

1.261

1..223

1.141

•

M.S. Control Rate (<$)

2 2 (in. /sec )

106.91

76.47

67.44

62.9

60.98

54.73
14

.022

.021 —

C>J

o

V0)

ij

raD

cr

co

c

TO

0)

X

2 2 Mean Square Control Rate (in. /sec )

Nomenclature

Q) Only e driving display, T = .20 > 0

(2) T. = .10, g(J = 1, 2, 3, 4, 5, 6; B : gd = 2

= .05, g, = 1, 2, 3, 4, 5, 6; C : g. = 3

T in seconds

.Figure 2.2 Parametric Performance - Manual Control Rate
15

I I I I I I I I I I I i I i I »'l t i r I I I I I I l_l I I I I I I I I I

.6 .8 1.2 1.4 1-6 4.6

Mean Square Control Input (in. )

Nomenclature

Only e driving display, T = .20 > 0

rd = .10, gd = 1, 2, 3, 4, 5, 6; B : g<J

Td = .05, gd = 1, 2, 3, 4, 5, 6; C : gd

T. in seconds

2

3

Figure 2.3 Parametric Performance - Manual Control Activity
16

the bandwidth of the display.

With the same OCM parameters as above, an analysis was performed

for two values of T : .10 and .05 sees. These values were chosen so

a

that the display dynamics are near the bandwidth of the human controller

as modeled. For each of these values, g, is varied from 1 to 6. The

d

results are presented in Table 2.2 and are also plotted in Figures (2.2)

and (2.3) so as to compare them with the previous results. In the two

figures, the curve marked (2) corresponds to T = .10 sees and that >^X Q

marked (3j to T, = .05 sees.

^-s a

Fig. (2.2) is a plot of mean square error (e) vs. mean square

•

manual control rate (6) for the various display cases discussed above.

Fig. (2.3) is a plot of mean square error vs. the mean square manual

control input (6). From these two figures it is clear that the mean

square control input and control rate both decrease as the display control gain g, is increased. The mean square error initially decreases as

g, is increased, and then increases beyond a certain value of g, that

depends on the choice of display bandwidth.

Note that earlier work [10,11] has shown that the human operator's

workload is directly related to his mean-square control rate. This means

that it should be possible in this case to improve performance (of which

mean square error is a measure), while at the same time decreasing workload. Moreover, the results indicate that for a given display bandwidth

there is an optimal choice of display control gains. For example, point

C in Figures (2.2) and (2.3) is such an optimal display for T = .05

sees, and for this case the performance is slightly better than even the

idealized limiting case at point A.
17

TABLE 2.2: OCM RESULTS FOR VARYING DISPLAY CONTROL GAINS

(a) T. = .10 sees

8d

1 •

2

3

4

5

6

M.S. Error (e)

(in.2)

0.0144

0.0138

0.0143

0.0157

0.0175

0.0195

M.S. Input (<5)

(in.2)

1 1.113

0.733

0.486

0.339

0.248

0.187

M.S. Control Rate (<S)

2 2

(in. /sec )

54.75

35.92

23.71

16.52

12.05

9.01

(b) T = .05 sees

8d

1

2

3

4

5

6

M.S. Error (e)

(in.2)

0.014

0.013

0.0127

0.0128

0.0131

0.0136

M.S. Input (6)

(in.2)

1.175

0.968

0.789

0.639

0.521

0.427

M.S. Control Rate (6)

• 2 2 (in. /sec )

58.06

47.47

38.49

30.97

25.15

20.46
18

It would then appear desirable to develop a systematic approach to

display augmentation which will make it possible to directly synthesize

the optimal display gains without having to resort to trial and error.

In the following chapter an extension of the optimal cooperative control

synthesis technique [7-9] is proposed as a methodology to accomplish

this.
19

CHAPTER 3_

OPTIMAL COOPERATIVE CONTROL/PISPLAY DESIGN METHODOLOGY

The problem formulation for the cooperative control/display design

methodology is presented and the necessary conditions for optimality are

derived in detail. Application of the methodology to pilot-in-the-loop

synthesis is demonstrated and various special cases of augmentation are

discussed.

3^.1_ Problem Formulation

In this section the mathematical formulation of the cooperative

control synthesis technique is presented. Necessary conditions for the

simultaneous optimality of the display and control augmentation systems

are developed in the sections that follow. The procedure followed here

is very similar to that of [8,9].

Consider a dynamic system acted upon by two controllers, and

described by the linear time invariant set of first order differential

equations

Bloul +B2ou

2

+Dow

_ _ , _ n

_ _ _ _

with xeR , u1 eR , u~ eR and w a zero-mean Gaussian white noise process

with intensity W. The two controls represent two physically independent

controllers, and in section (3.5) it is shown how the controller 1 (ui..)

can be made to closely approximate the OCM model of human behaviour.

The display dynamics are assumed to be of the form

xd = Vd + Bdoud

- "d

with x.eR , u.eR , and u is the display quickening controller. The
20

objective is to find the optimal cooperative controllers 1 and 2

(u. and u_) along with the optimal display control law u,.

Controller 1 (u.) is assumed to have noisy observations available

for feedback given by

C, x + C.-x, + C u, + v lo dl d u d y

(3.1.3)

where v is also a zero-mean Gaussian white noise process with intensity

V . This controller will be shown to include state estimation.

y

The augmentation controller u_ and the display control law u, are

assumed to have noise-free system outputs y~ and y,, respectively,

available for feedback, where

(3.1.4)

Note that the above formulation does not allow feedback of the display

states ~x. to the augmentation controller ^L. Finally, these two latter

controllers are constrained to have the direct output feedback form

= C2ox; yd ~ Cd

X

_

x

d

U

2

 =

u

G

2

C

2oX

G

d

C

d

(3.1.5)

The interaction between the different controllers is shown in the block

diagram of Figure (3.1).

3_.2_ Design Objectives

Controller 1 is to be optimal with respect to the cost

Jl = E{lim /(xQloxfxqidxd+uR1u1+uF1u2)dt} (3.2.1)

in the presence of the action of control inputs u_ and u,. Here E{"}
21

Figure 3.1 Block Diagram for Control and Display Augmentation
22

indicates the expected value operator and the weighting matrices are

Q. > 0, 0. , > 0, R. > 0, F. > 0. Conversely, Controller 2 (vL) and the

display control law "u, are to be optimal with respect to the cost

T

(3'2

'

2)

in the presence of the control action TT. . The weighting matrices are

Q2o > 0, Q2d > 0, R2 > 0, F2 > 0, F2d > 0.

Augmenting the system dynamics (3.1.1) with the display dynamics

(3.1.2), the state-space description of this augmented system is

X

x

d

+

~

A

o °"

0

B

2o

0

A

d

x

_*d

r ° ~ "

2 + LV

U

d

 +

B

lo

0

D

o

0

w (3.2.3)

Defining x = COL (x, x.), (3.2.3) can be written in a compact form with

appropriate definitions for the matrices as

X = AX + B^ + B2u2 +

The measurments can similarly be written as

Cuud

+ Dw (3.2.4)

(3.2.5)

The two cost functions can then be expressed in terms of the augmented state vector "x as

J, = E{lim ^ j

T+» c

T

7. = E{lim ^ f/~

T

u)dt}

(3.2.6)
23

where the weighting matrices Q and Q2 are appropriately defined. (Note

that this formulation is formally that for a multi-player non-zero sum

game, and we seek a Nash solution [22]).

_3 .J. Solution for u.

In the presence of the action of control inputs u. and u,, as given

by (3.1.5), the dynamics of the augmented system (3.2.4) are

X = Aaugx + Blul

where

Aaug (A + B2G

2

C

2

 + BdG

d

C

d

)

(Cl +CuG

d

C

d>

 (3'3

'

2)

and the performance index J becomes

J. - E{llm /(I(Q.GF G,C,)7+ uR^ )dt} (3.3.3)

T-*» o i^^i^ ^ 11 1

Equations (3.3.1) and (3.3.3), in the case of uncorrelated process

and measurement noises (w and v ) and for V > 0 (i.e. V - positive

definite) , describe the standard non-singular linear quadratic Gaussian

regulator problem for controller "u.. . The optimal controller is known

[12] to have the fora

Uj = KjX (3.3.4)

_A

where "x is

 tne minimum mean-square estimate of the system state vector

")<. The gain matrix K is given by

K = -R^B^P (3.3.5)

with P > 0 and symmetric, the solution of the algebraic Ricatti equation
24'

A

g

P + PAaug + <Q!+CF1G2C2> - PB

The dynamics of the Kalman state estimator are

= 0 (3.3.6)

A x

augA uj + M1(y1 - C X)

where the Kalman filter gain matrix M is given by

M.

T -1 z c1 v L

aug y

with E (> 0) the solution of the algebraic Ricatti equation

aug

T T

+ E A + DWD -

aug aug y aug

E = 0

(3.3.7)

(3.3.8)

(3.3.9)

_3 .4^ Solution for u_ and "u ,

The optimal controller u. as derived above has the form

Uj = Kjlc; "x = A^+ Mj^ (3.4.1)

where A — (A +B1K1-M,C ). Then in the presence of this control

action u1 , the system dynamics (3.2.4, 3.3.4, 3.3.7) can be written in

terms of the augmented state vector q — COL (x» x) as

M1C1

 Al

q + o M

l

C

u

_ TD Qud + [o MJ

w

(3.4.2)

which can further be written in a compact form, with appropriate definitions of matrices, as

The intensity of the process w' is W

+ BjUd + D'w'

W 0

0 V

(3.4.3)

The index of performance to be minimized by u? and u, then becomes

, T

J2 = E{lim Y J(qQ'q + U2

F

2

U

2

 + udF

2dud

)dt} (3.4.4)
25

with

Q' *

0

The design objective can then be stated as that of finding the optimal

controller u« and optimal display control u, which minimize the cost J_

as given by (3.4.4), subject to (3.4.3).

Proceeding in a way as detailed in Appendix A, it can be shown that

the gains G. and G, which correspond to the simultaneous optimality of

the two controllers u? and u, are given by

and

(3.4.5)

(3.4.6)

GTM = ~Fo [B- OjHL

i. i L

rT

2

0

([C2 0] L

r T

C

2

0

-F

-1

2d

~y

M.C

1 u

T

HL

C

T

C

d

0

([Cd 0] L

C

T

C

d

0

— — T Here, L = E{—q — q } satisfies the relation

A L + LAT + D'W'D'1 = 0

c c

(3.4.7)

and H satisfies

AH + HA +0= 0

c c v

where the following definitions have been used

A -

c =

Aaug BlKl

MlCaug Al '

(3.4.8)

C

2

G

2

F

2

G

2

C

2

+CdG

d

F

2dGd

C

d

0

The solutions (3.4.5) and (3.4.6) are derived from the gradient condi-
26

tions

3G,

0]L 0]HL } = 0 (3.4.9)

and

OvJ A7

3GT

 = 2<F

2d Gd

t Cd 01L

d

„!

C

d

0

+

'

B

d

M.C

1 u

T

HL

C

T

0

} = o (3.4.10)

respectively, as shown in Appendix A.

Equations (3.4.7) to (3.4.10) then give the four necessary conditions for the simultaneous optimality of the display and control augmentation for the task, as defined by the performance index J given that

Controller 1 is of the form stated in (3.4.1).

_3_.5_ Application to Pilot-in-the-Loop Synthesis

In the preceding sections, a dual performance optimization problem

was discussed and the necessary conditions for the optimality of the

various controllers, and the expressions for the resulting gain

matrices, were derived. The association of Controller 2 (u_) with plant

augmentation, and of display control (u,) with the display augmentation

should be apparent in the above formulation. In this section it will be

shown that Controller 1 (u-) can be made to closely approximate the OCM

model of human behaviour by an appropriate choice of the relevant parameters. In this manner the cooperative methodology, as developed above,

can be used to do "pilot in the loop" synthesis of the display/control

augmentation design.

Consider the Optimal Control Model as discussed in Appendix B.
27

A — __

iji ^ _jj _jp

With u — U (t) and x — [x , u (t)], the dynamics of the augmented system in Equation (B.A) are

x (t) = AQx (t) +. BlQuo(t) + DQw(t) (3.5.1)

with A , B. , and D as in (3.1.1).

The observations available to the pilot, as given by (B.2) can be

rewritten as

y (t) = C x (t-T) + v(t-r) (3.5.2)

P op

where C = [C | d] , and the pilot's cost function (B. 3) becomes

T

J (U*) = E{lim Y / (*IVo

 + ^G

^o)dt} (3.5.3)

P -

10 o P P

Comparing the above formulation for \i with that for Controller 1

(u.) as in Section (3.3) we notice that Equations (3.5.1) to (3.5.3)

have the same form as Eqns (3.3.1) to (3.3.3), but for the simplification that the time delay, T, has been eliminated in the observations for

u. . The absence of the time delay simplifies the dynamic order of the^

pilot model by eliminating the linear predictor in the control structure

(B.17, B.18). The motor noise (v in (B.12)) is however accounted for

m

in this formulation in that it may appear as an additional disturbance

in Eqn. (3.3.1). The structure of the pilot model as represented by

Controller 1 (71.) is as shown in Figure (3.2).

It is worth mentioning here that, though the simplified pilot model

is used in the synthesis procedure presented in this report, the complete model (with predictor, etc.) is used to evaluate these designs.

Moreover, at each iteration of the synthesis process, the parameters

(e.g. noise intensities) in the simplified model are updated to yield
2

8

ra QJ

> in

S_ -t-

OJ O ^ Z JOo

fOQ

.

c s-

(O tU E •!->

01

OJ <J

O E •^ ra O)

O) oc

i— (O

(O -Q

c s-

i_ 3 0) •!-> +J W x ••— UJ O

CO • h00

(TJuo<u T3Oo<•o(Ua

.

5

•Hen0) V4300 •HPn
29

results that are consistent with the complete OCM model. It has been

shown in [9] and [16] that by selecting proper noise intensities for the

control noise v and the measurement noise v , the simplified model may

yield the same human operator dynamics as the complete OCM model.

_3^6_ Numerical Solution Algorithm

For the case of only control augmentation (!!_) being designed, the

problem reduces to one discussed extensively by Innocenti in [8,9] as

the Linear Optimal Cooperative Regulator (LOCR) problem. With the

display-related terms removed from the problem formulation, the necessary conditions for optimality as derived above reduce to those in

[8,9].

For the case of only display control law being designed, the necessary conditions simplify, with the terms related to Controller 2 (1L)

removed from the problem formulation. The block diagram for the controller interaction is the same as in Figure (3.1) but without Controller 2. A brief description of the solution algorithm, alongwith a

flow chart for the computer program written to implement the display

synthesis procedure, is provided here. Since the algorithm is very

similar to that in [8,9], the reader is referred to the references for a

more detailed description.

The problem formulation for the case of display augmentaion only

can be summarized as follows:

Given the linear time invariant system

•

"X = A~x + B^ + Bdvid + Dw

?i

 = c i X+ C

u « d + % (3-

6

-

1}
30

yd ~ dx

we wish to determine the optimal controllers u. and "u. that minimize

respectively

T

Jl = E{lim Y / (X^lc + u^RjU^dt} (3.6.2)

and

I

 AT rn __ rp rp __

J2 = E{lim Y /(X Q2X+Ul

R2ul+udF2dud^dt^ (3.6.3)

T-*» o

subject to the constraint

ud = Gdyd (3.6.4)

In Equations (3.6.1) to (3.6.4) all the vectors and matrices are as

defined in Section (3.2).

The numerical procedure solves for the display control law (u"j)

which satisfies the necessary conditions of optimality for the cost J_

(3.4.7 to 3.4.10), given that the corresponding Controller 1 (IT..) is

optimal with respect to the cost J (or satisfies 3.3.5 to 3.3.9). The

algorithm for the solution procedure is summarized in Fig. (3.3) and a

step-by-step description of the algorithm is given in the following.

Step-1. This step consists of selecting a starting display control

law "u, to initialize the numerical optimization procedure. A reasonable choice (as is shown in later application) is to,select G, such

d

that the display variables x,, as given by (3.1.2), closely approximate those observations of the pilot which are of primary importance

for accomplishing the assigned task.

Another important choice to be made, before starting the iterative
31

©

Starting Display

Control Gains, G

LQG sol. for u ,

Current G,

d

©

no s- Convergence X, Yes

Conjugate

Gradient search,

with Updated u

Figure 3.3 Flowchart for Numerical Procedure
32

procedure, is that for the intensities of the noise processes w and

v . These should be chosen such that the optimal Controller 1, for

the starting display augmentation, closely represents the OCM pilot

model.

Step-2. Solve for the optimal Controller 1 (u.), as detailed in

Section (3.3), for the initial display augmentation, and compare with

the corresponding OCM pilot model to confirm that the choice for the

noise intensities is appropriate.

Step-3. Form the closed loop system, as in (3.4.3) and (A.4 of

Appendix A), for the current display augmentation (u,) and the

corresponding optimal Controller 1. Solve the Lyapunov equations

(3.4.7) and (3.4.8) for the matrices L and H, respectively. Obtain

3J"2

the cost J_ as in (A.6) and the gradient -r~— as in (3.4.10). L a(j.

a

Step-4. Next a conjugate gradient search procedure is used to

upgrade the display control law u,. This consists of making incremental changes in the gain matrix G in the negative gradient direction

till a value of G, is reached for which the cost J« shows an increase

d L

as compared to its value over the previous step. Note that at each

incremental step the controller "u. is updated to be the LQG controller for the current display law "u,.

d

3J2

Step-5. Calculate -r—- for the display control gains G, obtained at

dvTj d d

the end of Step 4 and check whether the necessary condition (3.4.10)

for J- to be minimum is satisfied. If yes, then go to Step 6 other-
33

wise go back to Step 3.

Step-6. Check for the convergence of the entire iterative process.

This is done by evaluating the difference between the display control

gains G obtained at the end of Step 5 and those at the start of Step

3. If this difference is less than some bound e, then proceed to

Step 7, otherwise go back to Step 3 and repeat the gradient search

procedure.

Step-7. The Controller u^ and the display control law "u"d at the end

of Step 6 are the optimal solution for the problem formulated as

above. But we further require that the Controller 1 be an approximate

representation of the human behaviour as modeled by the optimal

control- theoretic model. Therefore the control gains and closed loop

statistics obtained at the end of Step 6 are compared with the

results of the evaluation of the corresponding display augmented system using the OCM model. If a reasonable agreement is obtained

between the two results, then the display control law obtained is

-*v

pilot-optimal and the iteration process is stopped. Otherwise the

noise intensities "v and w are updated and the iteration process

started anew from Step 1.
34

CHAPTER 4_

APPLICATION OF DISPLAY DESIGN METHODOLOGY TO K/s2 PLANT

Using the algorithm discussed in Chapter 3, various display augmen

_ 2

tation control laws, u will be synthesized for the K/s plant. The

control laws are evaluated, using the full-order OCM model, in terms of

the mean square values of the variables of interest, and the power spec

trum of the human operator's control input. Detailed frequency domain

analysis is also carried out for the various displays.

jV._l_ Problem Formulation and_ Synthesis Results

2

The dynamics of the K/s plant augmented with the display may be

expressed in the form

:."

X

*d

A 0

o

0 3d

r -|

X

xd

+

B

o

0

6 + "

d +

D

o

0

w (4.1.1)

with A , B , D and the intensity W of the white noise process w as

defined in Section (2.1), and B = 1. Here the display dynamics are as

in (2.2.1), and u is the display control law to be determined.

As discussed in Chapter 3, Controller u. in the cooperative problem

•

formulation is analogous to the control rate "u of the OCM. So defining

A •

u — 6, the dynamics of Eqn. (4.1.1) can be written as

(4.1.2)

where the control noise term, v , is the human's "motor noise". The

m

X

.

x

d

•

6

=

"A o B "

0 0

0 a, 0

a

0 0 0_

X

x

d

5

• •

+

o"

0

J

- -

u

l

 +

"o"

1

o

u

d

 +

•

D 0

o

0 0

i

0 rp

n

w

V

m

- -
35

outputs available for feedback to Controller 1 (u ) are the same as the

human's observations. Therefore for the display augmented case

where v is the human's observation noise. To be consistent with the

analysis of Chapter 2, the responses driving the display are chosen to

be

7d = fx2, x3]

T (4.1.4)

where x (t) and x_(t) are the position error and the plant velocity,

respectively, and are as defined in Section (2.1). Then the display

controller, to be designed, has the form

•

Xd ' adxd + ud

ud

 = Vd (A. 1.5)

with G = [g.~, gjol* The human operator's objective for the displayaugmented system is to regulate the displayed variable x . Thus the

controller u minimizes the cost function J., where

T

J, = E{lim i /(*? + r.u^)dt} (4.1.6)

1 T~» T o d l l

Here the choice of r depends on the desired neuro-muscular-lag time

constant T of the human as modeled. For the purposes of the display

n

2

design for the K/s plant, r was continually adjusted to yield T I 0.1

sees.

The objective in the display design is to help the human operator

regulate the tracking error e (= x ), in the presence of the velocity

disturbance, and with minimal workload. This objective can be formulated as that of finding the optimal display control law u of the form
36

(4.1.5) which minimizes the cost function J where

T

J2 = E{lim Y /(°ee

2

 + r2u* + f^u^dt} (4.1.7)

o

Equations (4.1.2) to (4.1.7) then define the problem formulation

2

for the display design for the K/s plant, within the framework of the

cooperative methodology. Using the algorithm discussed earlier, the

optimal display control gains, G , are determined for various values of

d

relative weighting on the error in cost function J . Here relative error

<le

weighting is defined as the ratio — which is varied by changing q

r

2 ' e

and/or r . The results are presented in Table 4.1. Note that in

(4.1.7), f needs to be positive definite in order to get a finite

optimal solution to the problem. However, since the display control

does not reflect any measure of energy, the weighting f mav be chosen

ia

to achieve some selected overall display gain, or sensitivity. For the

results presented in this section, simply a constant value of foj = .001

2d

is used, and note that the display sensitivity

xd

is not constant.

ss

Since the algorithm that synthesizes the optimal gains is iterative, a starting display gain matrix has to be specified. For the

results presented in Table 4.1, G = [20, 0] is used for the starting

dislay control law. (Note from Section (2.1), the starting gains are

such that the displayed variable x,. is just lagged error). Also, for

comparison with the results of Section (2.2), a, = -20 sec was chosen

d

(this corresponds to T = 0.05 sees). The weighting r in J., and the

variances for the noise processes v and v are constantly updated in

y m

the iteration process in order to make the controller u closely approximate the OCM model with the parameters selected as in Section (2.1).
37

TABLE 4.1: OPTIMAL DISPLAY DESIGNS FOR K/s2 PLANT

.05 sees , Gd

 = [8d2'

 8d3]

Case

1.

2.

3.

4.

5.

qe

1

2

2

4

4

r

2

2xlO~4

2xlO~4

lxlO~4

A

1x10

5x1 0~5

Optimal G

[36.6, 11.9]

[49.2, 13.0]

[64.4, 16.3]

[88.1, 17.3]

[118.2, 22.6]
38

From the results presented in Table 4.1 it is noted that as the

relative weighting on the error is increased, the display sensitivity

(

TJ gj-O becomes larger. This is because a low value for f„, was chosen

d dZ 2d

as discussed earlier. But, as the analysis in the following section

will show, it is the relative position and velocity gain magnitudes in

the display which are of most interest, rather than the absolute values

of the display gains.

A

% 2. Evaluation of thj; Display Quickening Control Laws

The display gains obtained in Section (A.I) are now to be evaluated

using the full-order human operator model, with the same parameters in

the model as given in Section (2.1). For the purposes of comparison,

the initial display case (G, = [20, 0]) and the limiting case of no

d

display lag are also evaluated.

The display designs of Section (4.1) are referred to as Designs 1

to 5, as defined in Table 4.1. The limiting case of no display lag is

referred to as "A", the initial display case as "B" and the case of best

performance obtained through trial and error in Section (2.2) as "C".

b_*^'l_ Mean-Square Analysis

The OCM analysis results, in terms of the mean-square values of the

tracking error, the human's control rate, and control input for the

various cases of display augmentation mentioned above, are listed in

Table 4.2. These results are also plotted in Figures (4.1) and (4.2).

Fig. (4.1) is a plot of mean-square error vs. mean-square control rate,

and Fig. (4.2) is a plot of mean-square error vs. mean-square control

input. The scales are chosen to be compatible with Figs. (2.2) and

(2.3).
39

TABLE 4.2: OCM ANALYSIS RESULTS FOR VARIOUS CASES

Case

1.

2.

3.

4.

5.

A.

B.

C.

- M.S. Error (e)

(in.2)

0.014

0.0132

0.0131

0.01272

0.01269

0.0131

0.0157

0.0127

M.S. Input (6)

(in.2)

0.389

• 0.492

0.514

0.650

0.665

1.141

1.353

0.789

•

M.S. Control Rate (6)

2 2

(in. /sec )

18.64

23.72

24.78

31.58

32.40

54.73

67.44

38.49
40

.62-1

o

l_l

lJ

rt

cr

C

ta

.OL8

.017

Nomenclature

1 : G. = [36.6, 11.9]

d

_ 2 : G » [49.2, 13.0)

• U

3 : G = [64.4, 16.3]

d

4 : G, = [88.1, 17.3]

d

- 5 : G, = [118.2, 22.6]

d

A : No Display-Lag

B : Gd = [20, 0]

C : G, = [20, 3]

d

Beg. Display

Increasing Weight on Error

rA**r^

B

-tag

1 1 1 i i I i i i i I \ i i i I i i i i I i i i i I > i i i I ' ''

15 20 25 30 35 40 45 50 55 6O 65" 7O

2 2

Mean Square Control Rate (in. / sec )

Figure 4.1 Error vs. Manual Control Rate for Various Displays
41

^

vj

•

c

•HError (

rt

3

CT

C

1

.022

,011

.ol

.019

.016

.017

.015

.014

.013

ni?

; Nomenclature

:

 1 : G , = [36.6, 11.91

d

r~ 2 : Gd = [49.2, 13.0] ' A : No Display-Lag

:

 3 : G = [64.4, 16.3] B : G = [20, 0]

Q Q

'. 4 : G, = [88.1, 17.3] C : G = [20, 3]

Q d

~ 5 : G, = [118.2, 22.6]

d

—

- Beg- Display — "2^»X

/ls

 Increasing Weight on Error , ^\

1

^^>^

 r ^st iiTrial

 ' Etroy"*°

^—^^ X/ A

I i I i 1 I i I I 1 i I I I 1 i i i i 1 i ; I i 1 1 i i 1 1 i I i i \ I I I i 1 i i i i 1 j i i i \ I I / I

.3 .4 .6 .7 . .6 .3 1 1-1 1-2 1.3 . 1-4

2 .

Mean Square Control Input (in. )

Figure 4.2 Error vs. Manual Control Activity for Various Displays
42

From these results it is clear that as the relative weighting on

the error in the cost function J is increased, the optimal cooperative

display design methodology leads-to display gains which give improved

performance at the expense of increased control activity. It is noted

that for all the 5 cases obtained using the cooperative methodology, the

final optimal display gains are such that the performance is significantly improved as compared to the starting display (B), and at the same

•

time the "workload" (5) and the control effort (6) are considerably

reduced. If the weighting on the error is high (Cases 4 and 5), performance comparable to the limiting case (A) and the best performance case

(C) is obtained, along with reduced workload and control effort. Moreover it appears that for the selected display bandwidth (a^ = -20

d

sec ), tracking performance (rms error) better than that of Case 5 cannot be obtained. Increasing the weight on error in the cost function J

any further would only have the effect of leading to a display design

requiring higher control effort without any noticeable improvement in

performance.

4.^2_.^. Power Spectrum Analysis

Using the OCM modeling technique, the power spectra of the human's

control inputs and the system responses can also be estimated. Note

that the mean-square value a of a zero mean process x is related to the

area under its power spectral density £ (co), or

X

oo

a

2

 = 1 / Z (co)do) (4.2.1)

o

Figure (4.3) provides a comparison of the power spectra of the human

operator's control input (6) for the displays 1, 3, and 5, and the lira-
43

-40 i I 1 I I I I I

Frequency (rads/sec)

Figure 4.3 Comparison of Human Operator's Control Spectrum,
44

iting case A. The plots in Figure (4.3) show that the control power

required is lower for all the optimally synthesized displays when compared to the idealized Case A with no display lag. Moreover, the peak

value increases as the weighting on the error in the cost J is

increased, corresponding to the human's task becomes increasingly more

difficult from display design 1 to design 5. The power spectral densities of the human's control input 6 for all the displays are shown in

the Figures (C.I) to (C.6) in Appendix C. These figures include: 1) the

total control input power (in dB), 2) the portion of the control correlated with the command driving the system, and 3) the human operator's

remnant, or the uncorrelated portion of the control input.

Since the mean-square value of the errors for the display cases 1

to 5 and the limitintg Case A were not significantly different, (see

Table 4.2), the power, spectral densities of the errors for these cases

would be expected to be similar. This was indeed found to be the case

and so the error power spectral density plots are not included in the

report.

.4_«.2_..3 Frequency Domain Analysis

The human operator model can also be represented in the frequency

domain in terms of the transfer matrix between the human's observations

and his control inputs [6,13]. Since the Bode characteristics of the

plant and the display are known, frequency-domain analysis of the

manually-controlled display-augmented systems can be carried out, once

the human describing functions are estimated from the model.

With the human operator so represented, the blodk diagram for the

task is shown in Fig. (4.4). The display-augmented system is defined as
4

5

c

cfl •O0)c0) O. eno

14-1E(0p00COoo

-a-•

•a-01t*300
46

j

-JdT

 _ — r — <S(s)

and the human describing function is

PCS)

For the displays 1, 3, 5, and A, G(s) and P(s) are compared in Figures

(4.5) and (4.6), respectively. (The Bode-plots of G(s) and P(s) for the

cases 1 to 5 and the Case A are shown in Appendix C.) From these frequency responses in Fig. (4. 5) and Fig. (4. 6), the open-loop Bode-plots

may also be determined (not shown). It appears that open-loop magnitude

crossover (u> such that P(.iu)) G(JUJ) | = 0 dR) is approximately constant

at between 2-2.5 rads/sec for all these cases, consistent with the command signal bandwidth (break at 2 rads/sec). Also, as the weighting on

error increases (Case 1 -> Case 5) , the sensitivity (gain) of the total

plant plus display increases. From Fig. (4.5) it is clear that all the

synthesized displays provide additional lead between 1 and 10 rads/sec.

As the weighting on the error in the cost J« is increased (Case 1 ->

Case 5), this phase lead provided by the synthsized display gains

decreases.

The human operator's phase compensation in Fig. (4.6) is more

easily interpreted by looking at this phase compensation with the time

delay (T) of 0.2 seconds removed. This adjusted phase is then given by:

pc(u>) = $ P(jo>) + 57.3 T u> (4.2.2)

A comparison of this adjusted phase for the various display cases is

provided in Figure (4.7). From these plots it appears that the human

would need to generate much higher phase lead for Case A as compared to

all the optimal display augmentation cases. For Cases 1-5, the required
47

100

1 10

Frequency (rads/sec)

100

Figure 4.5 Comparison of Display Augmented Plants, G(s)
48

pa

•o

0)

TJ

c

60

40

30

20

10

0

T10

-20

10

10

Frequency (rads/sec)

Figure 4.6 Comparison of Human Describing Functions, P(s)
49

60

0)

P

CO

rt

80

60 -

40 -

20

0 -

-20 -

-40 -

-60 -

-go

1

Frequency (rads/see)

10

Figure 4.7 Comparison of Human's Phase Compensation

(adjusted phase)
50

phase lead increases with the increase of error weighting in the cost

function J^. Thus, using the phase lead as a measure of workload, the

trends in Figure (4.7) indicate that the display augmentation reduces

the human's workload. Moreover, the plots predict increasing workload

as the error weighting (q ) is increased in the synthesis of the

display.

With C(s) and P(s) known, the closed loop transfer function,

x( s)

^

s^ = /c\ » is easily obtained, through algebraic manipulations using

X \ S /

c

the block diagram of Fig. (4. A), to be

- - P(s) De(s)

x(s)

-

 _

 ---

 _

- -

 s _ • (4.2.3 _ .)

1 - - P(s) [D (s) + s D.(s)]

2 e

o fi

Here D (s) — - and D.(s) — - , x (s) is the commanded positio

e - s - ad x - s - ad c

and x(s) is the plant position state. Closed-loop frequency response

for Cases 1,3 and 5 of display augmentation is compared in Fig. (4.8).

The magnitude and phase plots in Fig. (4.8) indicate that as the error

weighting (q ) is increased, the synthesized display is such that tracking performance is improved.

Hence, both by comparing these results from the display augmentation synthesis technique with the results of Chapter 2, and through

additional closed-loop analysis, one can conclude that this synthesis

technique appears to lead to desirable results.
51

s- \

cc

T?

N.^

CJ

-a

3

4J

T-l

c

eo

TO

/^^

00

0)

o

x^

<u

(0

to

f-«

p->

U

-5

-10

-15

— on zu

0

,

-30

-60

-90

-i?n

— ». ~H 1 *u J m W^LV^ J ~ ^-TT*^J*r..-S.T^— ,

• '— ^^^

— v-

<;

-* /^\

<Sx> rJ^J

X

--^ /T)

N N v~Ai/

•y»N

— -.\ N

\\xv

: \'Vx -

\<:--

X% "v.

- \

~ . i i i i i i i i 1 \ i i i i t i l '-

1 1 K

^"'-•^x-

 v /^r\

v-^CD

>-^

- -4 PCs) DCs ) ©

/

 '

 ^

^

xfs) s^ \\v^

~_

 x

c

(

s

> 1 - -L P(S) [DgCs) + s D.(s)] %

. 1 ... 1 . 1 1 _L 1 1 1 III '

1

Frequency (rads/sec)

Figure A;8 Comparison of Closed-Loop Frequency Response

for Displays 1,3 and 5
52

CHAPTER _5

APPLICATION OF COOPERATIVE METHODOLOGY TO A

MORE COMPLEX DISPLAY DESIGN

In this chapter, an example application of the optimal cooperative

methodology to a higher-order system in a multi-input task is presented.

The intent is to demonstrate the application of the cooperative methodology to complex systems and the design presented here should not be

considered a finalized design. The chosen plant to be controlled

represents the unaugmented longitudinal dynamics of an aircraft, and the

task is that of tracking a normal-acceleration command while regulating

Mach number. The control inputs available to the pilot are the elevator

stick and throttle, and the pilot may be assumed to be controlling the

aircraft remotely from the ground, for example. The chosen model and

the task formulation are discussed in detail.

The task is first described and modeled, using an initial unaugmented status display. Then an optimal display augmentation control law

is synthesized using the proposed algorithm. The pilot's performance

for the unaugmented and the augmented display systems is then evaluated

and compared. Based on these evaluations, improvement in performance and

reduction in the pilot workload is predicted for the optimally augmented

display.

^._1_ System Dynamics

The dynamics are those for an F-15 type aircraft without any flight

control augmentation, but including a Mach sensor (with lag), a thrust

lag modeling the engine, and a fast first order actuator for the eleva-
53

tor control. (This model is from Ref.[21], but without the command augmentation system.) The vehicle state vector is

3<T = [V,o, q, 9, 6 , M, 6T] - the perturbed forward velocity (ft/sec),

angle of attack (.01 rads) , pitch rate (.01 rads/sec) , pitch attitude

(.01 rads), the elevator deflection (.01 rads), sensed Mach number (.001

2

Mach), and the thrust acceleration (ft/sec ).

m

The pilot control vector is 6 = [6 , <ST ] - the commanded elevaec c

tor deflection and the commanded thrust acceleration. The state-space

representation of the model is

x = A x + B 6 (5.1.1)

V V V V

where the matrices A and B are as given in Appendix D. The states are

the perturbed values about the trim conditions also listed in Appendix

D.

The pilot's assumed objective here is to track a normal acceleration command (a ) while regulating Mach number to the best of his abilzc

ity. Rather than the deterministic normal acceleration command (a )

z c

used for the pushup-pullover maneuver in previous studies [4,5], the

command signal to be tracked is generated, for the purpose of display

control law synthesis, by using a second order Markov process with a

break frequency of 1 rad/sec and a damping ratio of 0.7. With

—T A *

x — [a , a ], the command system is represented in the state variable

c = zc zc

form as

x =

-1 -1.4

x + w (5.1.2)

The intensity of,the white noise process w is chosen to be W = 181 so
54

o

as to yield the rms command of a =8 ft/sec (or 0.25 g's).

3

zc

Defining x — [x , x ], the vehicle dynamics augmented with the

o = c v

command system are written in the state-space form as

x = A x +B6+D w (5.1.3)

o o o o o c

with appropriate definitions of A , B and D . The time response of the

variables a , M and 9 is obtained for unit step control inputs. The

time histories for step elevator input are shown in Figures (5.1 (a)-

(c)) and those for step throttle input in Figures (5.2 (a)-(c)).

1-1

 Task

 Modeling

For the initial unaugmented status display, the pilot's observations are assumed to be the normal acceleration command signal, the a

z

tracking error (e = a -a ), the sensed (lagged) Mach number and pitch

Si 7* ZC Z

attitude. It will again be assumed that the pilot can reconstruct the

rates of the displayed signals. Then the pilot's observation vector,

7

T = [a , a , e , e , M, M, 9, 9], is of the form

p zc zc' az' az' ' '

7 = C "x + E "6 (5.2.1)

p o o

with C and E as given in Appendix D. A conceptual display format for

this status display is shown in Fig. (5.3).

For the augmented display system, the pilot's observations are to

be

 XJ.» XH?' a anc* ®' a*on8 with the associated rates. (Note that

compared to the status display case, e and M are replaced by the

cl Z

"intelligent" display variables .x,, and x, 0 , while a and 6 are maindl a.2 zc

tained in the pilot's observation vector). Therefore the outputs available for feedback by the pilot are
55

o

<D

tn

3.S -

3 -

Z.5

2

1.5

1

0

-.5

\7

4 5

Time (sec)

S 10

Figure 5.1(a) a Response to Step Elevator Input

(6e £= 0.01 rads, +ve down)
56

•5 3 0)

o

o

0

-1

t ! ' ! t I ' ' »titttti!!it;i! i ! : i I » t I i ! : »:»«:::: ! i ; i ; : ; ! :

4 5

Time (sec)

a s

Figure 5.1(b) Mach Response to Step Elevator Input
57

n

3 4

Time (sec)

Figure 5.1(c)

 9 Response to Step Elevator Input
58

o

V

CO

-.01

-.02

-:05

-.04

-.05

-.OB

-,07

-.08

: 1 : ; ! I ; ; ; : ! ! ; ! ! ; I : ; • ; '; ; : j I ; ; ; !

12. 3 4 5 6

Time (sec)

j !;;;•:!!::.•:: :

789 1

Figure 5.2(a) *z Response to Step Throttle Input

(6Tc= lft/(sec ), +ve ththrust r )
59

1,2

1 -

.8 -

JC

u

«

£ .6 -

c

o

c .

.4 -

,2 -

3 4 5

Time (sec)

Figure 5.2(b) Mach Response to Step Throttle Input
.60

,09

.08

.07

.05

w

-O

re

c

•

O

.03

,02

,01 -

0

. L. j.J .* * I — — I I I T j~r __i _t * t i i _ i i. ? ? ! i r t r

'4 5

Time (sec)

a 10

Figure 5.2(c) 8 Response to Step Throttle Input
61

o

<8

IS

r— 1

O

o

o

*

ow

^0

30

10

ID

0

-10

-20

-30

-40

-CA

_

—

—

j

_

__

—

—

—

—

-

, /^^

' TeQZ

a

zD — *•

xi ,, .. i •OM '

-

—

—

-

_

—_

-

—

~

^~

—

—

-

-'^

40

30

10

uo

0

-10

-20

-30

-40

-SO

(ft/sec")

Attitude 6

(Markings at 1°)

CRT

2

(Assumed Markings at 0.5 ft/sec and 0.5 M)

Pilot's objective: (a) Keep Q over the a bar

zc

(b) Keep <] within + (15) M

Figure 5.3 Conceptual Display Format for Status Display
62

_ ' • • * " T

y

p

 = ^Xdl' xdl' Xd2' X

d2'

 a

zc'

 azc' 9

'

9

^

A conceptual display format for the augmented system is shown in Fig.

(5.4).

Reasonable pilot modeling parameters for this analysis are as follows :

(i) Pilot's observation time delay, T, set to 0.2 seconds.

(ii) Observation noise ratio set at -12 dB for all observations.

(iii) Motor noise ratio set at -12 dR for both control channels.

(iv) The weighting on the control rates in the pilot's cost function adjusted to yield a neuromotor lag time constant of 0.2 sees

in the elevator channel and 5 sees in the throttle channel.

(v) The pilot's fractional attention allocation (f ) was set at

0.3 for each of the observations a , e and M, and 0.1 for 9.

zc az

(vi) The pilot's indifference thresholds for the various observations are set to values listed in Table 5.1.

Though parametric performance analysis is not too sensitive to many

of these variables, the values of observation and motor noise ratios are

chosen to be higher than in Chapter 2 in order to be consistent with the

pilot behaviour observed in real-time simulations of complex tasks.

Also the pilot's neuromotor lag time constant for the throttle channel

is set at 5.0 sees to reflect the fact that the pilot changes the throttle setting very slowly, since the engine response is slow. Finally,

from Fig. (5.3), it is seen that though the pilot, observes a , e and

zc 0 z
63

50 DW -

so -f

^\

w

 - - AJ 30 — *fH ~"

c

•5 - 3

 20 —

>> • I

TO

'ct ^

en • -

•* /% '-

-0 0 —

v-x

S-ioX I

-

•c -20 — c -

re

«T3

 --30 ~

X -

-40 —

»

I

^Ii TL^*' .. (TT^ " i 0 '

"?x« i

I

-

— HO

=- 30

—— 'l^\Ox

— -ID

:

 °zc

— 0

I (ft/sec )

5- -10

—

— -20

—

Z_ ^ -^un

— -40

-50

 -3U

CRT

(Assumed Markings at 0.5 display units for x , and xj^')

dl d2

Pilot's objective: (a) Keep Q at 0 marking for x

(b) Keep <] at 0 marking for x

Attitude 6"

(Markings' at 1°)

Figure 5.A Conceptual Display Format for Augmented Display
64

TABLE 5.1 PILOT PERCEPTION THRESHOLDS

Observation

a (ft/sec2)

zc

eaz(ft/sec2)

M(.001 Mach)

9 (.01 rads)

x,. (display units)

dl

x (display units)

Threshold

0.5

0.5

0.5

0.2

0.5

0.5

Rate Threshold

1.0

1.0

1.0

0.4

1.0

1.0
65

M (a , x and x for the case of augmented display - Fig. (5. A)) from

Z C Q. 1 Q. Z.

the same physical display, he still needs to scan their values

separately. So we will consider his attention to be divided equally

between the three primary observations, with attitude information only

secondary to the task. The choice of the values for pilot's thresholds

is based on the assumed display size and minimum resolution in the conceptual display format of Figs. (5.3) and (5.4). The thresholds for the

rates are twice the thresholds for the corresponding displayed variable,

for all the. pilot's observations.

For the unaugmented status display, the pilot's objective function

is taken to be

T

J ("3) = E{liml / (e2 + .0001M2 + .05(5T )2

P ~O

 T

 o az C

(5.2.2)

which reflects the pilot's emphasis on tracking the a command, and also

z

is consistent with the units of a and M in the dynamic model. This

objective corresponds to the pilot attempting to keep the square, in the

display format of Fig. (5.3), over the a bar, while at the same time

zc

maintaining the Mach needle within + (15). Note in this case the

pilot's commanded thrust ( 6T ) is explicitly included in the cost objective. This was found to be necessary to keep the commanded thrust within

limits deliverable by the engine. The weightings in (5.2.2) wet'e determined after a few trials of the modeling procedure. Finally, in

(5.2.2), g and g» are chosen to be consistent with the assumed pilot's

neuromotor lag time constants for the two control channels.

_5..3_ Optimal Display Synthesis for F-15 Model
66

For designing display augmentation for the aircraft discussed

above, 'the basic display dynamics may be taken to be of the form

or

with

xdl

• =

-20 0 "I

0 -20

X

dl

X

d2

+

"l 0"

0 1

"dl"

xd

 = Vd + V'd (5.3.1)

"

 Cd*o

(5.3.2)

where x is as defined in (5.1.3). In (5.3.2), y is chosen by the

designer, and the display gains G are to be synthesized so as to be

optimal for the task to be performed. For the display design presented

in this section, y, is chosen to be

-T

tazc, azc, V, a, q. 9, « M] (5.3.3)

which means that all the vehicle states as well as the command states

are available for driving the display. Note that in (5.3.1), the

display bandwidth is chosen to be faster than the pilot's dynamics.

Also two display states (along with two display control laws) are

chosen.

•

Defining Controller 1 to be the pilot's inputs,or "u ^ T, the

dynamics of (5.1.3), augmented with the display system and pilot's control inputs can be written as

X

0

x

d

-$

=

"A o B "

0 0

0 A, 0

a

0 0 0 _

X

o

x

d

"6

+

"o"

0

I

U

l

 +

0

Bd

p .

u

d

 +

D0 0'

0 0

0 I

w£

. "!.

(5.3.4)
67 •

where the two-dimensional white-noise process v is the pilot's motor

m

noise in the two control channels.

With the augmented display (see Fig. (5.4)), the pilot's task is

that of regulating the display states x and x10. This can be modeled

d 1 d 2

as finding the optimal Controller 1 ("u ) which minimizes the cost function J given by

= E{lim i /(x^ + x2

2 + 0.05(5Tc)

2

(5.3.5)

The objective function J0 to be minimized by the display control

law u is, however, taken to be

T

J

2

 = Jp(T) + E{llm

 T /'OOOOKu^j + u2

2)dt} (5.3.6)

Note here that J ("6) is as in (5.2.2), or the pilot's objective with the

status display. The weights on the display controls (u., and u,~) are

dl d2

chosen to be very small initially, at least, so that their contribution

to J is negligible as compared to J (6). The choice of J« as above is

consistent with our objective of finding a display control law which

helps the pilot perform his overall task of tracking the a command

z

while regulating the Mach number.

In order to initialize the numerical solution technique, an initial

guess for the display control gains is made such that x corresponds to

-e (i.e. a -a ) and x „ to 0.01 M in the steady state. Recalling

diZ Z ZC *" £1 £

Eqns.(5.3.2) and (5.3.3), then, the starting display control gains are

-0.082 -32.21 0 0 -5.096 0

G [

-20 0 -O.C

d "00 0 00 0 0 00 0 0. 2

start

(5.3.7)

For this choice of G , J (u.) for the initial display law closely
68

represents J (5) of the original status-display case.

The optimal display control gains determined from the algorithm

converged to

A ["^26.08 -6.329 -1.475 -38.05 -12.06 -4.019 4.923 3.219"

Gd = -18.72 -16.74 0.8258 -13.44 -36.74 -15.40 33.06 -0.202

(5.3.8)

In the synthesis process for the optimal display, the weightings g

and §9 in the cost function J^u.) and the variances for the noise

processes v and v are constantly updated so as to make the controller

u closely approximate the full-order pilot model.

_5_.j4 Comparison of the Displays

The pilot's performance, as predicted by model-based evaluation, is

compared for the two cases - the status display and the "optimal"

2

display augmentation synthesized as above. As in the case of K/s

plant, the criteria used for comparison are - rms values of the variables of interest, control and error spectra, and frequency domain

analysis. Time histories for step control inputs are also provided to

gain a better understanding of the significance of the "intelligent"

display variables x and x _. For brevity, the status display is

referred to as Display A and the optimal display as Display B in the

following discussion.

_5.4_ .J_ Time Histories

For Display B, the time histories for the two display variables xdl

and x,- for a unit step elevator input are shown in Figs. (5.5 (a)-(b))

and those for a unit throttle input are shown in Figs. (5.6 (a)-(b)).

Significant coupling is evident in this case, especially for elevator
69

CO

4->

T-l

e

a.

CO

0

4 5

Time (sec)

10

Figure 5.5(a) x Response to Step Elevator Input

(6e£= 0.01 rads, +ve down)
70

10 -

c

3

n

r-l

Cu

5 6

•o

I i i i I i ! i i ! ! ; , ! : * : i ii!i:ii!s::ii:: : : ! : : i : !:;:;::;: :

3 4 5

Time (sec)

a

Figure 5.5(b) x._ Response to Step Elevator Input
71

c

3

CO

a

CO

-.03

-.06

-.09

5-.12

-.15

-.18

-.21

ii!i!itit!888i!;>8lli t I i ! : t i : •! :

2 3 4 5

Time (sec)

10

Figure 5.6(a) x Response to Step Throttle Input

dl

(<5T.= lft/(sec2), +ve thrust)
72

,014

,012

~ .01 CO

4-1

•H

c

3

3 .003

CL

v:

<006

,004

.002

0

I ' < ! : ii!;tttri;::;ii::::: :

4 5

Time (sec)

8 9 10

Figure 5.6(b) x Response to Step Throttle Input

d2
73

commands. From these time responses, it would appear that the strategy

for the pilot might be to attempt to "control" x with the throttle and

x,. with the elevator. This is due to the fact that x10 is essentially

d 2 d i

uncontrollable from throttle inputs. For a step normal acceleration command, there will be an instantaneous change in both xjt and x.,, (as is

dl d2

*

apparent from the optimal display gains G in (5.3.9)). In order to follow the command, then, the pilot's strategy would appear to be to apply

the elevator to regulate x and simultaneously use the throttle to null

out the effects of the elevator command and the normal acceleration command on x . The ultimate success of this strategy as well as comparison

with that for the case of status display (Display A), of course,

requires additional analysis and simulation.

5_.^._2. Frequency Domain Analysis

The block diagram representations for the pilot's task formulated

as above are shown in Figures (5.7) and (5.8) for Displays A and B,

respectively. Note that in each case the pilot has four observations

and two controls available to him. (The rate observations, as stated

earlier, are reconstructed by the pilot from the displayed signals.) So

the frequncy-domain representation of the pilot consists of a 2 x A

transfer matrix. For Display A the pilot representation is

PCs)

P . P . p. p. oe oe oe oe

a e M ~5~

zc az

P P P P

' 5T 6T _6T _6T

a e M 0

zc az

(5.A.I)

and for Display B

PCs) =

P * P . P . P.

oe oe 6e oe

a x_, x._ ~Q~

zc dl d2

P P P P

5T 6T 6T 5T

a x,

zc o "d2

 r

(5.A.2)
7

4

Ou

in

•HQCO3«J 4JCO

(-1

60

cd00

iHoacu900
7

5

<>B

p.

CO<U60

" eCO00 •HOOoiHpaoo0) HCO
76

Here P is the pilot describing function fron the normal acceleration

oe

a

zc

command observation a to the commanded elevator input 5 . The other

zc

 v ec

elements of (5.A.I) and (5.4.2) are similarly defined. The Bode-plots

for the eight pilot describing functions for Display A are presented in

Figures (D.I) to (D.8) and those for Display B in Figures (D.9) to

(D.16) in Appendix D, for reference.

With the pilot represented by eight transfer functions, it is not

straightforward to define a meaningful measure of pilot workload in the

frequency-domain. Since the vehicle dynamics and the display dynamics

are known and the various pilot describing functions are available as

7

above, additional frequency-domain analysis, as in the case of the K/s"

plant, may in principal be performed using the block diagrams of Figures

(5.6) and (5,7). However, the results presented in Chapter 4 showed that

the performance and workload information obtained from statistical and

spectral analysis of the model-based predictions is consistent with that

derived from the other frequency-domain analysis. Therefore, results

from statistical and spectral analysis will be used to evaluate and compare the two Displays A and B.

.5 ..4.3. RMS Analysis

Using the complete human operator model, closed-loop analysis was

performed for both the status display (A) and the optimal augmented

display (B). The rms values for the variables of interest for the two

displays are shown in Table 5.2. From these results it is clear that

there is a slight improvement in a tracking performance for the case

z

with display augmentation, and Mach number regulation is much improved.

Both the pilot control inputs and control rates are lower for the
77

TABLE 5.2 OCM ANALYSIS RESULTS

(a) Performance Measures

Classification

A: Status Display

B: Optimal Display

e

az

ft/sec2

7.009

6.659

•

e

az

11.920

9.349

M

.001 Mach

39.11

•11.32

e

.01 rads

8.45

3.36

(b) Workload Measures

Classification

A: Status Display

B: Optimal Display

$

ec

.01 rads

2.95

1.78

•

6

ec

13.21

8.06

6T

c

ft/sec2

2.566

0.819

«5T

c

0.1742

0.1228
78

optimal display case, indicating that the designed display augmentation

should lead to reduction in pilot workload.

_5.4^A_ Spectral Analysis

The power spectra of the a tracking error and both the control

z

inputs for the Displys A and B are shown in Figures (D.l) to (D.6) in

the Appendix D. These plots show the total power (in dB), the portion

correlated with the command driving the system, and the uncorrelated

part, or the contribution due to the pilot's remnant.

The a tracking error power spectrum for the Displays A and B is

compared in Figure (5.9). This power for Display B (augmented case) is

slightly higher than Display A at low frequencies, but is less than

Display A for frequencies above 0.7 rads/sec. Even though the rms

tracking error with the optimal display (B) is not too much lower than

that with the status display (A), the augmentation appears to reduce the

errors at the more-demanding higher frequencies.

The pilot's commanded elevator power spectrum and commanded thrust

power spectrum for Displays A and B are compared in Figures (5.10) and

(5.11), respectively. From these figures it is clear that the power for

both pilot control inputs is much lower at all frequencies for the

2

optimal display case. As was the case of K/s plant, this is an indication that the pilot's workload in accomplishing the task should be

reduced.

Thus, from this initial example application, it appears that the

suggested display augmentation synthesis technique can lead to desirable

results when applied to manual control of complex dynamics. As further

experience is gained in applying the technique, even better performance
79

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CJ

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00

(HP)

ZB
80

D.

W

CO

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(X

CO

T3

0)

•u

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<u

60

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o

01

Cfl

o

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0)

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tr

<u

a

cu

a.

to

w

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rH

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(O Ol CO TCN

(SP) 7 '
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Page Intentionally Left Blank
83

CHAPTER fi_

SUMMARY

The objective of this effort was to develop an analytical technique

to aid in the design of display augmentation. The main differences

between display augmentation and control augmentation were pointed out

using simplified block diagrams. Although different from augmenting

plant dynamics, it is usually possible to improve human operator performance in manual-control tasks by providing the human with an augmented

display.

Model-based evaluations were performed for various emperically2

derived display "quickening" control laws for a simple K/s plant in a

compensatory tracking task. The results of this analysis, in terms of

mean square values of tracking error and manual control activity and

rate, showed that significant reduction in human operator workload and

improvement in performance is possible through a proper design of the

signal being displayed to the human.

The cooperative control synthesis technique previously developed to

design pilot-optimal control augmentation was then extended to incorporate the simultaneous design of pilot-optimal display augmentation.

The problem formulation for the cooperative synthesis technique was discussed in detail and the necessary conditons for optimality were

derived. A numerical algorithm for applying the methodology to perform

pilot-in-the-loop synthesis of optimal display augmentation control laws

was then presented.

The application of the methodology to a simple system was demonstrated by synthesizing various display augmentation control laws for
84

2

the K/s- plant. By varying the weighting parameters in the cost function, it was shown that this methodology has the potential of providing

a systematic approach to design of task tailored display augmentation.

The synthesized display designs were evaluated, using the full-order

human operator model, in terms of the mean square values of the variahles of interest; and the power spectrum of the human operator's control

input. The evaluation results predicted a reduction in human operator

workload and improvement in tracking error performance with the augmented displays. Moreover, when compared to the emperically-derived

display "quickening" control laws, the results tended to validate the

cooperative approach to display design.

Detailed frequency domain analysis was also performed for the synthesized displays, using the frequency-domain representation of the

human operator model. All the augmented displays were shown to provide

the human operator with some lead information, reducing the required

human operator phase compensation from the case with no display augmentation. A comparison of the closed-loop frequency responses for the

various synthesized displays indicated that the cooperative methodology

does lead to desired results.

An initial application of the methodology to high-order system

dynamics with a multi-control task was then demonstrated, using a

normal-acceleration tracking task and an F-15 type aircraft model. The

problem formulation was discussed in detail and a preliminary display

design was synthesized using the cooperative technique. Analytical

methods for evaluating the synthesized display for the complex task were

again employed, and the results of the analysis were compared with those

for an unaugmented status display. Based on the statistical analysis
85

results, improvement in performance and pilot workload is predicted for

the synthesized optimally augmented display.

Since the model-based evaluation of the display designs predicts

promising gains in pilot workload, it is suggested that these results be

validated through real-time, man-in-the-loop simulation. For the display

design synthesized for the aircraft model, further insight needs to be

gained into the significance of the two display variables. Finally, the

application of the cooperative methodology to display design for complex

systems should be further explored, and systems considered for which

other methods have been used to synthesize the display, to provide

further comparison and validation.
86

REFERENCES

[1] Poulton, "Tracking Skill and Manual Control," Academic Press,

1974.

[2] Hoffman, W.C., Kleinman, D.L., and Young, L.R., "Display/Control

Requirements for Automated VTOL Aircraft," NASA-CR 158905, ASITR-76-39.

[3] Korn, J. , Gully, Sol W. , and Kleinman, D.L., "Validation of an

Advanced Cockpit Display Design Methodology via

Workload/Monitoring Tradeoff Analysis," 18th Annual Conference on

Manual Control, Dayton, Ohio, June 1982.

[4] Duke, E.L., Jones, F.P., and Roncoli, R.B., "Development of a

Flight Test Maneuver Autopilot for a Highly Maneuverable Aircraft," AIAA Paper 83-0061, Jan. 1983.

[5] Walker, R.A., Gupta, N.K. , Duke, F..L., and Patterson, B.,

"Developments in Flight Test Trajectory Control," AIAA Paper 84-

0240, Jan. 1984.

[6] Kleinman, D.L., Baron, S., and Levison, W.H., "An Optimal Control

Model of Human Response," Parts I and II, Automatica, vol. 6, pp.

357-383, 1970.

[7] Schmidt, O.K., "Optimal Flight Control Synthesis via Pilot Modeling," AIAA Journal of Guidance and Control, July-August 1979.
87

[8] Schmidt, O.K., and Innocent!, M., "Pilot Optimal Multivariable

Control Synthesis by Output Feedback," NASA CR-16312, July 1981.

[9] Innocenti, M., "Cooperative Pilot-Optimal Augmentation System Synthesis for Complex Flight Vehicles," Ph.D. Thesis, Purdue University, West Lafayette, Indiana, May 1983.

S

[10] Schmidt, D.K., "On the Use of the OCX's Ouadratic Objective Function as a Pilot Rating Metric," 17th Annual Conference on Manual

Control, Los Angeles, CA, June 1981.

[11] Wierwille, Walter W., and Connor, Sydney, A., "Evaluation of 20

Workload Measures Using a Psychomotor Task in a Moving-Base Aircraft Simulator," Human Factors, Vol. 25, pp. 1-16, 1983.

[12] Kwakernaak, H., and Sivan, R., "Linear Optimal Control Systems,"

Wiley-Interscience, 1972.

[13] Bacon, B.J., and Schmidt, O.K., "An Optimal Control Approach to

Pilot/Vehicle Analysis and the Neal-Smith Criteria," NASA Contractor Report 170416, April 1984.

[14] McRuer, D., Graham, D., Krendel, E., and Reisener, W. Jr., "Human

Pilot Dynamics in Compensatory Systems," AFFDL-TR-65-15, July

1965.

[15] Curry, R.E., Hoffman, W.C., and Young, L.R., "Pilot Modelling for

Manned Simulation, Vpls. I and II, AFFDL-TR-76-124, Dec. 1976.

[16] Phatak, Anil V., "Investigation of Alternate Human Operator

Optimal Control Model Structures," Air Force RMRL, contract
88

#F3'3615-78-C-0501.

[17] Noble, B., Daniel, J.W., "Applied Linear Algebra," Prentice-Hall

Inc., 1977.

[18] Joshi, S.M., "Design of Optimal Partial State Feedback Controllers

for Linear Systems in Stochastie Environments," Paper 6D-4, IEEE,

Souteascon, Charlotte, N.C., 1975.

[19] Neal, T.P., and Smith, R.E., "An In-flight Investigation to

Develop Control System Design Critera for Fighter Airplanes,"

AFFDL-TR-70-74, Vol. I, Dec. 1790.

[20] Anderson, M.R., and Schmidt, O.K., "Closed-loop, Pilot/Vehicle

Analysis of the Approach and Lending Task," AIAA Paper No. 85-

1851-CP, 1985 Guidance and Control Conference, Snowmass, Colorado.

[21] Walker, R.A., and Gupta, N.K., "Flight Test Trajectory Control

Analysis," NASA Contractor Report 170395, Feb. 1983

[22] Pavassilopoulos, G.P. , Medanic, J.V., Cruz, J.B. Jr., "On the

existence of Nash Strategies and Solutions to Coupled Riccati

Equations in Linear-Quadratic Games", JOTA, Vol. 28, No. 1, May

1979.

[23] McRuer, D.T., and Krendel, E.S., "Dynamic Response of Human Operators", Wright Air Dev. Center WADC TR 56-524, Wright-Patterson Air

Force Base, Ohio, Oct. 1957.

[24] Levison, W.H., Kleinman, D.L., and Baron, S., " A Model for Human

Controller Remnant", IEEE, Trans. Man-Machine Syst., Vol. 10, Dec.1969
89

APPENDIX A'

DERIVATION OF NECESSARY CONDITIONS FOR CONTROLLER ^ (u )

As stated in section (3.A), the system dynamics in the presence of

the control action u Can be written in terms of the augmented state

vector q as '

A q + B u9 + B u, + D w 1 2. i da

(A.I)

where q, A,, B_, B,, and D are as defined in section (3.4). The out1 L d

puts y_ and y. are given in terms of q by

J2 = [C2 OJq" = C^q (A. 2)

yd = [Cd 0]q = C^q

and the index of performance J_ becomes

E{lim /(qTQ q (A.3)

with Q again as defined in section (3.4).

The closed loop system under the action of Controller 2 (u-) and

the display control law u,, with the control laws as given by (3.1.5),

can then be written in a compact form as

q = A q + D w (A.4)

where

A =

c

aug

M.C

1 aug

B

l

K

l"

A

l

with A , C and A, as defined in Chapter 3.

aug' aug 1

The system given by (A.4) is identified as a linear stochastic time
90

invariant system driven by a white noise process w . From linear stochastic system theory [12], in the steady state case the system state

covariance matrix L is obtained as the solution to the following

Lyapunov equation

T ' ' 'T

A L + LA +,.D W D = 0

c c

' _' T

where W is the intensity of the process w and L=E[q q ].

Using the trace operator [17], J. can be expressed as

(A.5)

where

Q - Q +

tr[QL]

T T T T

C

2

G

2

F

2

G

2

C

2

 + CdG

d

F

2dGd

C

d °

0

(A.6)

(A.5) and (A.6) can now be treated as a constrained parameter

optimization problem [18]. Defining H as a symmetric matrix of Lagrange

multipliers, the cost J_ can be augmented with the constraint (A.5) to

get

J, = tr[QL] + tr[H(A L + LAT + D W D T)]

L C c

(A.7)

The necessary conditions fo.r optimality applied to (A.7) yield the

following relations

A*H HAc + Q = 0

t e i

D W D

^3G2,

(A.8)

(A.9)

(A.10)

(A.ll)
91

Equations (A.10) and (A.11) need further elaboration because the

augmentation gains G. and the display control gains G, are embedded in

/ d

the dynamics of the closed loop system.

(a)

In J?, the terms containing G_ are Q and A . Therefore

3J

where the trace identity tr[HA L]=tr[HLA ] has been made use of.

Further note that in A , G. appears both in A as well as in A . But

c 2. aug l

A. is part of the optimal Controller 1 (u.), as is apparent from

(3.4.1), and since we are looking for a Nash solution, Controller 1 is

to be considered fixed in the minimization process for J_. Thus the

only gains that may be adjusted in Controller 2 are those in A where

A = A + B_G0C0 + B.G.C,. Then

aug 2 2 2 d d d

Noting that

T T T

C

2

G

2

B

2

0

0

0

=

C

T

C

2

0

0]

(A.13) can be simplified to get

tr{HL

0

0]} 0]HL

0

(A.13)

(A.14)

(A.15)

Also from the definition of Q we have
92

tr[QL) - - trl

3G,

tr {

T T

C

2

G

2

F

2

G

2

C

2

0 0

L}

0

G

2

F

2

G

2

[C2 °3L}

2F0G0[C0 0]L

0

(A.16)

Combining (A. 15) and ('A. 16), the condition (A. 10) can be written in the

expanded form as

C

T

C

2

0

+ [B^ 0]HL

'4

0

} = 0 (A.17)

(b)

In J_, the terms containing G, are Q and A . Therefore as before

3J2 3 T

—^ = •;:£- {trfQLj + 2tr[HLAM}

oG, 3G, C

a a

(A.18)

Again noting that in A , A. and M. are part of the optimal controller 1 (u.), the only free gains G, to be adjusted are those appearinge in A and C . Then

 aug aug

-5!- tr[HLA?] = •£- tr {HL

3Gd c 3Gd

B

d

G

d

C

d

M

l

C

u

G

d

C

d

(A.19)

The matrix on the right hand side of (A.19) can be written as

BdGdcd

M

l

C

u

G

d

C

d Vu

G

d

[Cd 0] (A.20)
93

which gives

B

d

G

d

C

d °

M.C G,C, 0 1 u d d } =

"

B

d "

M

l

C

u

T

HL

T

C

d

0

tr{HL

Further, similar to the procedure used to get (A.16), we have

(A.21)

9G

tr[QL] ,[C, 0]L (A.22)

Combining (A.21) and (A.22), (A.11) can be written in the expanded form

as

-3GT=2{F 2d Gd

[ C d° l L

d

'4

0

+

"V

M

l

C

u

T

HL

c

T

C

d

0

(A.23)

(A.8), (A.9), (A.17) and (A.23) together give the necessary conditions for the simultaneously optimality of the augmentation controller

u« and the display control law u,.
94

APPENDIX £

THE OPTIMAL CONTROL PILOT MODEL

The optimal control approach to human operator modeling was

developed by Kleinman, Baron and Levison in the early 1970's. Since

that time it has been used successfully to model manual pilot behaviour

in a variety of complex tasks. A brief discussion of the actual model.

is given here as it forms the basis for the synthesis and analysis of

display augmentation designs presented in this report. More detailed •

discussion of the structure of the model can be found in [6] and [13],

The model is based on the assumption that a "well-motivated" human

pilot adjusts his gains and compensation for the vehicle and task such

that an objective function, J , is minimized, subject to human limitations. Typically a piloting task, aircraft and display dynamics are

represented in the block diagram shown in Figure (B.I) and by the time

invariant differential equation

•

"x(t) = Ax(t) + bu (t) + Dw(t) (B.I)

P

where the A matrix can be an agregation of task, plant and control system dynamics. The vehicle states are represented by the vector lc(t)

while u (t) is the pilot input (assumed to be scalar here for the purposes of explanation - multi-input tasks are easily accomodated within

the framework of the model) and w(t) is a "white" noise disturbance with

intensity W.

An output vector y (t) represents those variables which the pilot

can observe, either through the cockpit display, out the windows or by

the "seat of his pants". His observations are given by the relation
9

5

e00 •HOooPQPQ(U360
96

yp(t) = Cx(t-T) + dup(t-T) + vy(t-r) (B.2)

where T is a pure delay associated with the inherent delay in the

pilot's perception, and v is a vector white noise process with intensity V which models the imperfection in the pilot's observations.

The pilot's task is reflected in the minimization of a quadratic

performance index of the form '

T n

J (p*) = E{lim -p / ( Z q.x.2 + ru2 + gu2)dt} (B.3)

P T>OO

 i o i=l * * p p

subject to pilot observations y (a) for time o < t and with cost functional weightings q > 0, r > 0, and g > 0.

•

Inclusion of the control rate, u , in the cost function naturally

leads to a first order lag in the pilot's control law analogous to the

neuromotor lag of the McRuer crossover pilot model [14], The time constant of the lag, furthermore, may be adjusted through variation of the

control-rate weighting constant, g.

By defining a new state vector as

X

T

(t) = [xT(t), up(t)],

•

the augmented system can be represented as (u (t) = u (t))

X(t) = AQx(t) + bQUo(t) + DQw(t) (B.4)

with

A

o -p " 0 01 '

 b°

The minimizing control law is

o

with

U*(t) = -lx(t) (B.5)
97

1 = - bTK (B.6)

g o o

where x(t) is the current estimate of the state x(fc)f

unique positive definite solution of the Ricatti Equation

A

TK + K A + Q - - K b bTK =0 (B.7)

o o o p o gooo o

with Q = diag(q , r) . Expanding the optimal control law as

(B.8)

where P.(t) is the current estimate of the system state x (t), and letting

T = -i- (B.9)

by iterating the control-rate weighting, g, the equivalent lag time constant, T = -= - , can be adjusted to a desired value. The control law

n+1

can then be written as

where

T u (t) + u (t) - u (t) (B.10)

n p p

W*(t) = -lgp(t) (B.ll)

with

1

e

From the human operator viewpoint, an exact control input is not

possible. This uncertainty is modeled by the addition of motor noise v

in the control equation, or

np up(t) = (t) + vm(t) (

where v (t) is a Gaussian "white" noise source with intensity, V . The

m m
98

controller gains previously calculated are assumed to remain the same in

the presence of motor noise. This assumption reduces the human operator

model to sub-optimal control behaviour.

The current state estimate is derived from the combination of a

Kalman state estimator and a least mean square predictor. The system is

rewritten as '

X

T

(t) = txT(t), up(t)]

X(t) = AlX(t) +

7p(t) = vy(t-T), - N(0, V

(B.13)

where

A b

n

. b.

[C | d] ,

n

D 0

The disturbance noise has the form w.(t) = fw (t), v (t)] with covaril m

ance matrix

W16(t-o) (B.14)

fw o "

Wl - 0 V L

 m.

The Kalman filter generates an estimate, x(t) of the model states

x"(t) from

X (t-T) = Axx(t-T) +

+ ECv'^y (t) - (B.15)
99

where the error covariance matrix, E , is the solution of

A1Z1 + E1A1 + D1W1D1 " EICIV~ICIEI = °

The least mean squared predictor is governed by the equation

Finally, the current state estimate is found by combining the Kalman

state estimate and the predicted state through the relation

_ _ V-2-

P(t) = ?(t) + e [x(t-x) - C(t-r)] (B.18)

In order to apply the optimal control model, it is necessary to

know the various human response parameters T,T ,V ,V introduced above.

Published data in manual control [14] indicates that typical values for

the effective time delay are T=0.15-0.25 sees. The neuromotor lag time

constant T is of the order of T =0.1-0.6 sees with T 10.1 being typical

[23]. It should be noted that results reported in [14] indicate that T

n

varies inversely with forcing function bandwidth.

Investigation of the properties of human controller remnant [24]

have shown that the sources of remnant can be modeled on the basis of

constant noise to signal ratios. Thus the observation noise associated

with the i observed variable y, has an intensity

(vPii = pyi E{yi} (B'19)

where the constant p is based on the human controller's attention

being limited to the i observation only. Similarly, the motor-noise

v (t) is modeled to have an intensity given by

m

Vm = pu E{U2} . (B.20)

The constants p . and p are referred to as observation noise ratio and

yi u
100

motor-noise ratio respectively, and are generally specified in terms of

their decibel values. Model matching analyses have shown values of

p. =0.01i: (-20 dB) and p =0.003* (-25 dB) to provide a good fit with

experimental data for manual control tasks involving single-axis tracking. These ratios have much higher values for complex multi-control

tasks. Since the noise intensities are specified as ratios of the closed

loop intensities of the corresponding signals, an iterative process is

necessary to determine the optimal control model for which the relations

(B.19) and (B.20) are satisfied.

Studies of human perception abilities have shown that if a quantity

y is displayed explicitly to the human operator, he can extract the

•

rate of change of that quantity, y, . Thus the rates of the displayed

quantities should be included in the pilot's observation vector y .

Furthermore, if the human controller has more than one explicitly

»

displayed observation available to him, his attention is divided between

the various displays. The error induced due to this division of attention is modeled by modifying the observation noise corresponding to the

i displayed variable as

<Vn

 =

 -

 (

where (V°).. is as given by Eqn. (B.19). Here f is the fraction of

total attention allocated to the i observed variable and has limiting

values, at no attention (f =0) and full attention (f =1) [15], The values

of f are generally determined on the basis of the importance of that

particular observation in successfully accomplishing the task as modeled

by (B.3).
101

If a particular signal y is very small in magnitude, the human

controller may not be able to detect its non-zero value (visual threshold) or may choose not to react to the small perturbations (indifference thresholds). This aspect of the human controller is modeled by

modifying the observation noise intensity corresponding to the i

displayed variable as '

"

 (B'22)

(N(ayi' al»

2

where a . = E{y. } and N is the describing function of a dead-zone ele-

/

ment [6]. (V ).. is as obtained from (B.21) and a. is the value of the

threshold for the i observation. Pyscophysical studies have shown that

these thresholds typically correspond to values of 0.05 deg and 0.15

deg/sec at the pilot's eye. Also, a reasonable value for these thresholds is the minimum resolution marking on the display.

Finally, combining (B.19), (B.21), and (B.22), the intensity of the

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APPENDIX D_

DATA AND ANALYSIS RESULTS FOR F-15 AIRCRAFT
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