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4. TO TL E nd 5 ' u6titfl- Maximum Infomnation Trajectory F'or I YPE OF HLPONT &PE4100 COVERED

An Air,4 6-Air Missile Intercept 4 TIESIS/D1SrPtVM

q~4 7 Ai~ rso ni' L _1

r,( Capt, ,Stephen Wendell Laro/7

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MAXIMUM INFORMATION TRAJECTORY FOR AN

AIR-TO-AIR MISSILE INTERCEPT

APPROVED:

MAXIMUM INFORMATION TRAJECTORY FOR AN

AIR-TO-AIR MISSILE INTERCEPT

BY

STEPHEN WENDELL LARSON, B.S.A.E.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN ENGINEERING

THE UNIVERSITY OF TEXAS AT AUSTIN

December, 1980

ABSTRACT

This thesis presents a method for finding the trajectory to

complete an air-to-air missile intercept which maximizes information.

This is accomplished by formulating a parameter optimization problem

and using a penalty function-Lagrange multiplier method to solve for

the optimal path. The performance index is the trace of the informa-

tion matrix. This Information matrix is derived using an extended

Kalman filter formulation, in cartesian c-oordinates, which makes state

estimates based only on angle measurements. The trace operation on

the information matrix is used because the trace and the integration

operations commute allowing a scalar performance index. Further, re-

duction in the functional form of the performance index is achieved by

weighing the information matrix by the inverse of the measurement power

spectral density. This also avoids numerical difficulties near inter-

cept.

IVI

TABLE OF CONTENTS

Abstract ..... ......................... iv

List of Tables. .......................... vii

List of Figures .. ........................ viii

Nomenclature .. .......................... x

I. INTRODUCTION. ......................... 1

II. GEOMETRY .. ......................... 4

2.1 Inertial Axis System. .. ................ 4

2.2 Missile Axis Systems .. .................. 4

2.3 Target System. ...................... 9

2.4 Launch Geometry. ..................... 9

III. MODELS .................. .......... 13

3.1 Missile Model ............. ......... 13

3.2 Target Model. .......... ............ 21

3.3 Atmospheric Model ......... ........... 21

IV. THE INFORMATION INDEX .................... 22

4.1 Definition .. ...................... 22

4.2 The Dynamical System .. ................. 24

4.3 The State Transition Matrix ....... ........ 28

4.4 Angle Measurement Noise .......... ....... 30

4.5 The Weighting Matrix .. ................. 31

v

4.6 The PerformanceIne. . . . .. .. .. .... 32

V. PROBLEM FORMUJLATION . .. .. .. .. .. .. .. .. . .34

5.1 The Optimal Control Problem. ........... ... 34

5.2 The Parameter Optimization Method. ........... 38

VI. SOLUTION METHOD ................ ....... 41

6.1 Input. .................. ....... 41

6.2 (' ptimization Method. ................... 41

6.3 Function Evaluation. ................... 44

6.4 Numerical Derivatives. .............. ... 45

6.5 Optimization Algorithm ........... ...... 46

VII. RESULTS ............................ 47

7.1 Launch Scenarios ................. ... 47

7.2 Results and Conclusions. ................. 48

vi

LIST OF TABLES

1. Aerodynamic Coefficients, Functions of M ........... 16

2. Aerodynamic Coefficients, C N (a,m) .. ............ 17

3. Intercept Results .................. .... 50

vii

LIST OF FIGURES

1. Missile Axes Systems .......... .................... 7

2. Launch Geometry .......... ..................... 10

3. Intercept Geometry and Measurement Angles .... ......... 11

4. Lift and Drag ......... ....................... ... 18

5. Boresight Angle ......... ...................... . 18

6. Function Evaluation ........ .................... . 44

7. Best Information Horizontal Trajectory, 3000', 00 ...... 52

8. Best Information Vertical Trajectory, 3000', 0 ........ .. 52

9. Best Information Boresight Angle, 3000', 00 .......... 53

10. Best Information Boresight Angle Rate, 3000', 0 ...... 53

11. Best Information Az and El Angles, 3000', 00 .......... 54

12. Best Information Az and El Angle Rates, 3000', 0° ..... .54

13. Best Information Horizontal Trajectory, 3000', 300 ..... 55

14. Best Information Vertical Trajectory, 3000', 300 ...... 55

15. Best Information Boresight Angle, 3000', 30' . ....... .56

16. Best Information Boresight Angle Rate, 3000', 300 ..... .56

17. Best Information Az and El Angles, 3000', 300 . ....... .57

18. Best Information Az and El Angle Rates, 3000', 300 .. ..... 57

19. Best Information Horizontal Trajectory, 7000', 600 .. ..... 58

20. Best Information Vertical Trajectory, 7000', 600 ... ...... 58

21. Best Information Boresight Angle, 7000', 600 .......... .59

viii

22. Best Information Boresight Angle Rate, 7000', 600 ..... . 59

23. Best Information Az and El Angles, 7000', 600 ........ . 60

24. Best Information Az and El Angle Rates, 7000', 600 . . . 60

25. Minimum Time Horizontal Trajectory .... ............ .. 61

26. Minimum Time Vertical Trajectory .... ............. .. 61

27. Minimum Time Boresight Angle .... ............... ... 62

28. Minimum Time Boresight Angle Rate .... ............. ... 62

29. Minimum Time Az and El Angles ....... ............ . 63

30. Minimum Time Az and El Angle Rates ... ............ . 63

ix

NOMENCLATURE

a control parameter vector

a T a , a target inertial acceleration components (ft/sec 2)TX Ty aTZaz, el azimuth and elevation measurement angles (deg)

azB missile normal acceleration (g's)

al,a 2 ,bl,b 2 measurement noise constants

b parameter veitor

g acceleration of gravity (ft/sec2)

time-normalized equations of motion

observation-state relation

h altitude above sea level (ft)

'JjB'B unit vectors along missile body axes

il,,l unit vectors along inertial axes

iWjwkW unit vectors along missile wind axes

k number of equality constraints

m number of constraints

mass (slugs)

n number of control parameters

t time (sec)

tr trace operator

u control vector

x state vector for missile model

x

z measurement vector

AAY launch yaw aspect angle (deg)

BAY launch boresight angle (deg)

C constraint residuals

C axial force coefficient for a = 0

CD drag coefficient due to angle of attack

CD 2 drag coefficient due to angle of attack squareda

CN normal force coefficient due to angle of attack

CN intercept in CN interpolatiou

D drag (lb)

F characteristic matrix for state vector

Fx,Fz missile body axis forces (lb)

G characteristic matrix for the control vector

GC matrix of first derivatives of C with respect to b

G matrix of first derivatives of with respect to b

H matrix of first derivatives of g with respect to X

HH intermediate values in the performance index

I information matrix

J performance index

L lift (lb)

M Mach number

P error covariance matrix

penalty function

.xi

Raz Rel azimuth and elevation measurement ranges

S missile reference area (ft2)

optimization weighting matrix

T missile thrust

T 1 inertial to wind axis transformation matrix

T2 wind to body axis transformation matrix

UR, VR,WR inertial relative velocity components (ft/sec)

V matrix of measurement noise power spectral densities

VM missile velocity (ft/sec)

V TxV ,VTz target inertial velocity components (ft/sec)

W missile weight (lb)

weighting matrix

X state vector for dynamical system

XB,YB,ZB missile body axes

XI,YI,ZI inertial axis coordinates (ft)

XM,YM,ZM missile inertial position components (ft)

XRYRZR missile/target relative position coordinates (ft)

XTYTZT target inertial position components (ft)

XW' W, missile wind axes

missile angle of attack (deg)

optimization factors

r optimization factors

Ax perturbation of x

xii

small number

0 inequality constraint residuals

~target acceleration constant (sec- I

P atmospheric density (slug/ft3)

elements of weighting matrix

T dummy variable of integration

normalized time component

performance index as a function of b

state transition matrix

XYU missile wind axis yaw, pitch and roll angles (deg)

missile body axis yaw and pitch angles (deg)

4 equality constraint residuals

boresight angle (deg)

subscripts

0 condition at t=O

f condition at final time

x partial with respect to x

M missile

T target

superscripts

derivative with respect to time

-1 inverse

T transpobe

xiii

SECTION I

INTRODUCTI ON

Many of the modern, tactical, air-to-air missiles employ passive

seekers such as infrared heat seekers in their guidance systems. For

particular geometries where the line-of-sight inertial angle is small,

the proportional navigation system, which dri.es the inertial line-of-

sight angle rate to null, works reasonably well. If the intercept

geometry is more severe, or if the target is accelerating, then obtain-

ing range and range-rate information becomes important, especially to

derive the time-to-go to intercept. Since passive or jammed sensors

may only have angle information, range and range rate can only be ob-

tained by extraction from sophistic,'ted filter design.

In Reference 1, Sammons et al compare various implementations

of the extended Kalman filter for missile onboard operations. This

filter uses angle measurements to estimate relative position, relative

velocity, and target acceleration. Estimates of range and range rate

are physically obtained by having the inertial line-of-sight angle

constantly changing. However, the proportional navigation guidance

law is always attempting to drive the angle rate to zero. Since these

angles are the only source of information, it seems obvious that, when

the angle rate approaches zero, there will be little or no information

to the filter about range and range rate. The guidance law is working

against the estimator by commanding a flight path which denys informa-

tion to the estimator.

The purpose of this thesis is to find and investigate the opti-

mal trajectory which completes a missile intercept such that some mea-

sure of information is maximized. In this way the filter can make the

most accurate state estimate possible. It is not suggested here that

this be the only measure of performance, but that it be an element in

developing a composite performance criteria.

The particular missile under consideration is a highly-maneuver-

able, short-range, bank-to-turn, air-to-air missile capable of 100 nor-

mal g's. The missile model, that is, the aerodynamics, the autopilot,

the guidance law, etc., is completely developed in Reference 2. The

geometry of the intercept and the missile model are discussed in Sec-

tions II and III.

The measure of information chosen is the information matrix.

Measures on the information matrix imply the relative observability in

estimating the state due to measurement functions alone. This means

that the relative observability of the system is obtained only from

the deterministic system model of the dynamics and measurement. The

information matrix and its use as a performance index is developed in

Section IV.

In Section V, the optimal control problem is formulated. The

parameter optimization method is then discussed, and all the con-

straints are derived.

The optimization method used to solve this problem is the pen-

alty function - Lagrange multiplier method discussed by Fletcher in

3

Reference 3. This is a state-of--the-art method f or solving parameter

optimization problems. It is particularly convenient since it requires

very little set up time and can be used with equality constraints and

inequality constraints at the same time. Section VI contains a brief

discussion of this method along with discussions of the function eval-

uations and numerical derivatives.

Finally, the results are presented in Section VII. The various

launch scenarios are described, and some concluding remarks are pre-

sented.

SECTION II

GEOMETRY

This section covers the problem set-up. The various reference

systems and their relationships are discussed, and the launch and in-

tercept geometries are described.

2.1 Inertial Axis System.

The inertial axis system is a standard right-handed system with

the positive Z I axis pointing toward the center of the earth. The

origin is at mean sea level with the missile launch point lying on the

neg ative Z I axis. Because of this, the magnitude of the Z I component

of the respective position vector corresponds to the altitude of the

missile or target.

2.2 Missile Axis Systems.

There are two axis systems associated with the missile - a wind

axis and a body axis system. These two systems share a common origin,

assumed to be at the missile center of gravity.

The wind axis system is a standard right-handed system with the

Xaxis tangent to the flight path and positive forward. The ZWaxis

is perpendicular to the axisin the plane of symmetry, and positive

down if the missile is in normal, level flight. The Y W axis is situ-

ated perpendicular to the X.1-.ZW plane to complete the right-hand sys-

tem. The angular relationship with the inertial system is through

4

5

the velocity yaw angle, X, the velocity pitch angle, y, and the veloc-

ity roll angle, p. This angular relationship is derived in Reference

4 and given by

cosycosx cosysinX -siny i

JW sinisinycosX sinpsinysinX sinicosy (1)

-cospsinX +cosl1cosX

kW coslsinycosX cossinysinX cospcosy k,

+sinjsinX -sinpcosX

The missile body axis system is described by the positive B

axis directed through the nose of the missile. The ZB axis is perpen-

dicular to the KB axis, in the plane of symmetry, and positive down if

the missile is in normal, level flight. The YB axis is situated to

complete the right-hand system. It is assumed that there is no side-

slip angle; therefore, the body axes and the wind axes are related

only through the angle of attack, . This relationship is also derived

in Reference 4 and is given by

i B -Cosa 0 -sin - -t

JB 0 1 0 JW (2)

k B sinct 0 CoakWB

6

The missile body axis system is related to the inertial system

through the body yaw angle P, and the body pitch angle e. These two

angles completely define the orientation of the body longitudinal axis

in inertial space, and are used as the control variables in the opti-

mal control problem. The inertial to body angular relationship is de-

rived by successive rotations about the yaw axis and the pitch axis

and is given by

cosOcosls cosesin -sinO ii

"B -sinP cos 0 (3)

k B sincosp sinesin cose k,

The relationship of the missile axis system to the inertial axis sys-

tem is shown in Figure 1.

The angles cX and 11 can be found from the angles , e, X, and

y. P and 6 are the control angles, and, therefore, known; X and y are

part of the state vector, and, therefore, also known. The solution is

found by transforming from the inertial system through the wind axes

to the body axes. If Equations (1) and (2) are rewritten as

J =Tl1 JIJ -l 72T 2 W (4)

kW k, kB kW

7

-ZI

e XW

Yw

YI

Figure 1. Missile Axis Systems.

8

the relationship between the body axes and the inertial axes is given

by

B= T 2 T 1 JI(5)

Performing the multiplication leads to

T 3=T 2T 1(6)

where

coscacosycosx cosctcosysinX -cosasiny

-sintcoslisinycosX -sincacosiisinysinX -siniacoslicosy

-sincsinplsinX +sintsin~lcosX

sinpisinycosX sinlsinysinX sinlicosy

T 3 -cssn +ojcs (7)

sinctcosycosX sinacocsysinX -sinctsiny

+cosctcoBs1sinYcosX +CosctcoslisinysinX +cosctcosl~cosY

-IcoscsinhdsinX -cosasinlpcosX

9

Equations (3) and (5) must be equal transformations. By setting the

components equal, the equations for p and a are found to be

tni-cosesin( - X) (8tan = ncos-cosecos( -x) siny (8)

tanct = (sinecosy-cosecos(- X ) siny)cos+cosesin(*-X) sin (tae= cosecos ( -X) cosy+s inesiny(9

2.3 Target System.

For this problem, the target is simply assumed to be a point

mass with position and velocity components in inertial space.

2.4 Launch Geometry.

Figure 2 shows a view of the launch geometry. At launch, the

missile is assumed to be on the negative ZI axis with the body axis

system parallel to the inertial axis system. The target initial posi-

tion is determined by the initial range and the off-boresight angle,

BAY, from the missile. Target direction is determined by the initial

aspect angle, AAY. This angle is defined to be the angle between the

target velocity vector and the line of sight between the missile and

target. The missile and target are assumed to be co-altitude and co-

velocity at launch.

After launch the inertial axis system remains stationary. The

target remains at the launch altitude, but the missile is free to ma-

neuver in any direction. Figure 3 shows a representation of the in-

tercept geometry. The relative distances between the missile and tar-

get are also shown. The two inertial angular measurements are the

10

SLine of Sight

7 Missile XB A

AAYI X'roTarget VelocityVector

Ix

z

SZI

Y

Figure 2. Launch Geometry.

1i

Missile Position-ZTarget Position

ZRX

YI/

Figure 3. Intercept Geometry and Measurement Angles.

12

azimuth az and the elevation el of the target with respect to the mis-

sile. These angles are defined by the relative distances as follows

az - tan-l (Y/(

el = tan- (-ZR/(XR + y2 ) )

SECTION III

MODELS

This section defines the missile model and all of its compo-

nents, the target model, and the standard atmosphere.

3.1 Missile Model.

The missile model used is the three degree of freedom version

of the model derived in Reference 2. Here, the missile is assumed to

be able to instantly assume whatever body angles ipand e are commanded

by the parameter optimization routine. Since it is the trajectory

that is being optimized, it is assumed that this trajectory can be

commanded perfectly by an autopilot; therefore there is no autopilot.

Contributions made by the control surface deflections, by sideslip,

and by moments of inertia are assumed to be negligible.

The missile launch takes place at t =0 in accordance with the

geometry in Section 2.4. The missile is not allowed to guide for the

first .4 second, so for this time period the control angles are zero.

At the end of this delay the missile is allowed to assume any position

commanded by the optimization routine.

The differential equations of motion for the missile are as de-

veloped in Reference 4 for flight over a flat earth:

13

14

1 = kM = VMcOsycosX2 = Y = VcosYsinX

= zM = -VMsiny

k 4 = = (Tcosa - D - Wsiny)/m

5= = ((Tsina + L)cosi - WcosY)/VMm

x6 = = (Tsina + L)sinp/(VMmcosy)

where

T = f(t)

W = f(t)

m = W/g

D = f(caVM,Z ) (12)

L = f(a,VM,ZM)

a-f(x,Y, ,e)

= f(x,Y,PG)

g = acceleration due to gravity, assumed constant.

The functions for a and U were discussed in Section 2. The models for

thrust, weight, drag and lift remain to be developed.

3.1.1 Thrust and Weight Models. The missile weighs 165 lb at launch.

The propellent (50 lb) burns at an assumed constant mass flow rate for

2.6 seconds. During the burn, the thrust is assumed to be constant,

4711.5 lb. There is no other contribution to weight loss during the

missile's time of flight. The thrust and weight profiles are

15

described by the expressions

l4711.5 b 0 < t < 2.6 secT 4711. (13)

0 2.6< t

( (165 - 19.23t) lb 0 < t < 2.6 secW = (14)

115 lb 2.6< t

3.1.2 Aerodynamics Model. The components of the missile aerodynamic

forces along the XB and ZB axes are given by the equations

_=-(Ca + ~ cX 2 1 2FX a + Cxa + a P N S

a a 2(15)

FZ -(CN + CN ) 2 pVM SOT

where S is the missile reference area. The atmospheric density P is

calculated using a standard atmosphere model.

The aerodynamic coefficients in Equations (15) are found by

table look-up and linear interpolation. The values are listed in

Tables 1 and 2. These tables and the interpolation routines have been

taken directly from Reference 2. Drag and lift are then calculated

from Equations (15) in accordance with Figure 4.

D =F X cosa - Fz sina

(16)L -- F Cosa + FX sina

...-

16

C14

o % 0 0

,-.0 0-

C- C)

E-4 0

II r*.. 0 0

(n 0 0

00*0,- 0 0 0"

00

" - ~-0

ON L

14'0 ,0 0

00 mm . ,

-4 ,-

enN

o '-t o 0

o -o

• • 0

I, 0. . 0.

0~1 0

* eN 0

17

O ~ 0% 'T --T HM 0n c1Cr -4 0% 00

9 .4 Cr, 11

00 en C4 00" 0.0 0n -4 a% 00

o: Co Cr, -I 4

mr 00 I CN ) C'J 0

C'4 r -4 co o'

C, C'4 '0 0 00, 04

Ifn 0 0 -4 I-T 00~ r0 '.T

H~~ r- Co r IniI

r4 C . 4 1; H '4

ON -4 0P N c, --T '.o

1- 0 - -

co r" -4 c0 '0 . . co -a,

M *n e - n C4 -

en. 0o m- co LCr, ON 0'. Co D 0

Hi 0 4 4 Co zr z00

Hn 0 n '4 N. C

n 0 -4 '4 C' 0 00 ONeq 0n ON H " n '. oN

Cr4 C, 4 ' '4

18

FX

XWD VM

Fz

Figure 4. Lift and Drag.

Line of Sight

T

0 XI

yu

Figure 5. Boresight Angle.

19

3.1.3 Seeker Model. The actual seeker and how it works is not model-

ed. However, to simulate the presence of a seeker, a limit is imposed

on the angle between the positive XB axis and the line of sight to the

target. This angle, shown in Figure 5, is called the boresight angle

and is limited to

l 2I < 600. (17)

This limit represents the maximum angle off boresight that the seeker

can track the target. It is a physical limitation; if the boresight

angle were to become greater than 60, the seeker would break lock.

The boresight angle is found by using the definition of a dot

product. From Figure 5, it is seen that

bAl AT! cos= OA AT (18)

where 0 represents the missile center of gravity, and A is the seeker

gimbal pivot point. The inertial axis system shown is actually a par-

allel system through the missile c.g. T represents the target posi-

ion.

A unit vector in the direction A can be defined using the

Euler angles from Equation (3), that is,

OAI6/ A = (iIcos8cos + jlcosesin - klsinS). (19)

20

The vector AT is the relative range vector

T XRiI + YRJ I + ZR kI (20)

where XR, YR' and ZR are the relative ranges in the X1 , YIP and ZI

directions respectively (see Figure 3). The boresight angle is then

found to be

XR cosecos + YR cosesin - ZR sinOcoso XP (21)

(X2 + Y2 + Z2)

3.1.4 Accelerometer Model. The normal acceleration is limited to a

maximum of 100 g's. Missile normal acceleration is modeled by

aZB = Fz / W (22)

where Fz is the component of aerodynamic force along the ZB axis as

defined in Equations (15), and W is the weight from Equation (14).

The limit, for structural reasons, is

BI < 100 g's (23)

21

3.2 Target Model.

The target is considered to be a non-maneuvering, non-acceler-

ating, point mass moving through inertial space. Therefore, the tar-

get model consists of only the three inertial position components

XT = XTr + VT to

Y T + VT t (24)-- o y

zT = zTo

For this problem, the velocity components are as follows:

V = V cos(AAY + BAY)T x T

VT = VT sin(AAY + BAY) (25)

V =0Tz

where VT is the constant target velocity and AAY and BAY are the ini-

tial aspect angle and off-boresight angle, respectively, as defined in

Section 2.4.

3.3 Atmospheric Model.

The atmospheric model is needed to compute the speed of sound

and the density for a given altitude. This model is a standard, con-

stant gravity atmosphere for the troposphere (0 to 36089 ft) and the

stratosphere (36089 to 82021 ft). It is the same model developed in

Reference 2.

SECTION IV

THE INFORMATION MATRIX

In Reference 1, a dynamical system used in an onboard, optimal

filter to estimate relative position, relative velocity, and target

acceleration is presented. In order to measure the observability of

the state estimate which can be made from this dynamical system and

measurement function, the information matrix is used. In this section,

the information matrix and measures associated with the information

matrix are developed as the performance criteria for the optimization

problem.

4.1 Definition.

The information matrix is defined by Reference 5 as

I(t,to) 0 ft T(Tt)H (T)V_ (T)H(T) 4(T,t)dT (26)to

where O(T,t) is the state transition matrix used by the filter. H(T)

is the matrix of partial derivatives of the measurement function with

respect to the state variables evaluated on the nominal path, and

V(T) is the measurement noise power spectral density. Each of these

elements will be discussed in this section.

The information matrix is related to the inverse of the error

covariance matrix P- (t). If it is assumed that there is no process

22

______________

23

noise, for any time period t i to tj, the information matrix is given

by

l(t j t) 1 -1l(t ) T(t opt i ) PO 0 1(to t t ) (27)

where P is the a priori error covariance (Reference 6). If there is

no a priori information, i.e., if P = O,0

I(tJt i) = P "(t.). (28)J

The larger the eigenvalues of the information matrix, the smaller the

eigenvalues of the correspoa ding error covariance matrix will be, and

the more pcecise 'the estimate will b,. This is the basis for using

the informatiin matrix as the perforr.ice index. The trajectory

which maximizes the eigenvalues of the information matrix is also the

trajectory which give-; tbe o st precise estimate.

A ;calar is needed for the performance index. The two opera-

tions available are the trace and determinant. The trace is equal to

the sum of the eigenvalues of a matrix; the determinant is equal to

the product of the eigenvalues. Because of its simplifying charac-

teristics, primarily that the trace and integration operations are

interchangeable, the trace of the information matrix has been chosen.

Since this is an optimization problem solved on a finite word length

computer, a constant weighting matrix is included to normalize the

terms in the trace. This weighting matrix does change the eigen-

values, but since it is a constant, the maximizing has the same effect.

24

The performance index now becomes

J = tr W I (tf~t0 )

= tr W ftf T(t,tf)HT(x(T))V_ (T)H(X(T)) ((T,tf)dT. (29)t0

Since the trace and the integral operations are interchangeable and

the weighting matrix is a constant, J can be rewritten as

J = tf tr(W( T(T, tf)HT (X(T))V- (T)H(X(T))4(T, tf))dT. (30)to

This defines the form of the performance index used for this problem.

The elements of this equation remain to be defined.

4.2 The Dynamical System.

The dynamical system used by the filter (Reference 1) is of the

form

X(t) = F X(t) + G u(t) (31)

z(t) = g(X(t)) + v(t). (32)

The dynamics equation (Equation 31) is linear in rectangular coordi-

nates. X is a nine-state vector of relative position, relative veloc-

ity and target acceleration given by

iI

25

XR [ ' R ZR UR VR WR aTX a T a TZ T (33)

where

XR R = XT(t) - XM~(t)

y R =y R(t) = y T(t) - YM(t)

ZR = z R(t) = z T()-zMtR R Tt) - N~t)(34)

U R =U R(t) =VT (t) -V (t)

VR = V (t) = VT t) - v (t)R R y M

WR = WR (t) = V T z(t) - VM z(t),

and aT. aT. and a are the inertial target acceleration components.Tx Ty Tz

Equation (31) in scalar form is

R =uRR VR

R 'R

tR R aM

UR =a T - N(35

VR a T aM

TXaM

a -X aT y TY

T -a

T z aT,9

26

where X is a target acceleration constant. The control, u(t), is

assumed known, and consists of the three missile inertial acceleration

components, a M a3 and aMz. From Equation (35) it can be seen that

the matrix F from Equation (31) is

F= o 0 1 1 (36)

L 10 i-X ,

and the matrix G is

i'- - r" -

G = 0-I I 0 (37)

0 0 1 0 ,

where each partition is 3x3, I is the identity matrix, and X is a

diagonal matrix of the target acceleration constants.

The measurement equation (Equation (32) ) is nonlinear in rectan-

gular coordinates. The measurement z is a two-dimensional angle mea-

surement, g(X) is the measurement function, and v(t) is a zero mean,

white noise process associated with the measurement, with power spectral

density V(t). The measurement function consists of the two inertial

angle measurements for azimuth and elevation (see Figure 3)

g tan-i (YR/XR) (38)g(X(t)) tan_1 (Z R 2 /( R) )jL [R(X

27

The linearized measurement equation is

6z(t) =H(X(t))6X + v(t), (39)

where H(X-(t)) is the partial of g with respect to X evaluated on the

nominal path. The elements of H are

H(~t) I 191X2 0 0 00 0 0 0 (40

H2 (Xt 92 x~ g2 x 0 0 0 0 0 0J(0

where

H 911 R

X az

H YXR

12 x2 R a

2H g2 XRZR 1(41)2 2X R2 R az

1 el

H y R R. 122 g2 x 2 R a

RH az23 92 = 2 e

Xair

28

where

Raz = R az(t) = (XR2 + YR2) (42)

ReI = Rel(t) = (XR2 + YR2 + ZR2) ,

as shown in Figure 3.

4.3 The State Transition Matrix.

Because of the simplicity of the matrix F (Equation (36)),

the state transition matrix has been derived in closed form. The

elementsof D(t,T) are given by (Reference 1)

P 11 0 0 P140 0 D1 7 0 0

0 2 0 0 25 0 0 %8 0

0 0 330 0 @36 0 0 3

0 0 0 0 0 17 0 044 47 (43)

(D(t,T)= 0 0 0 0 4) 0 0 58 0

0 0 0 0 0 (6 0 0 69

0 0 0 0 0 0 %8 0

0 0 0 0 0 0 0 0 499

29

where

11(t,T) = D2 2 (tt) = 3 3 (t,T) = 1

D44 (t,) - D5 5 (t,T) = %66(t,T) = 1

(D4(t,T) = 0 2 5 (t,T) = 4 3 6 (t,T) = t -T = At(44)

(D7(t, T) f- ( 8 8 (tT) = P99 (t,T) = exp(-X At)1

D7(t,=T) = 05 8 (t,T) M 69 (t,T) = -- (exp(-X At) -1)

D17 (t,T) = 028 (t,T) = 39 (t,)= 1 2(exp(-X At) + XAt - 1)

and where X is the target acceleration constant (Reference 1).

Since the information matrix requires O(T,t), the identity for

the inverse of a state transition matrix

O(r,t) = -1 (t,r) (45)

can be used. Hence, the elements of D(T,t) are

(1,t) = D2 2 (T,t) = 03 3 (T,t) = 1

D44(T,t ) - (5 5 (T,t) = 066 (T,t) = 1

P14(T, t) - 025(T,0 = 0235 6(T, = T - t - -At(46)

(D T ~t) = 8 8 (,t) = ( 0 99 (T,t) - exp(XAt)

d4 7 (t ,t) =t) f 6 9 (T,t) = -1 (exp(X At) - 1)

I= cTt 28(T,t) = c 39(T,t) AA(exp(A At) -X At - 1)

, ~ ~ ~ 1 - .... X 2 l" .. Inml .. +- '... .

30

4.4 Angle Measurement Noise.

The matrix V(t) is a 2x matrix representing the power spectral

density of the angle measurements due to certain target-sensor charac-

teristics, that is,

V(t) =L~(t 2 ()

aV1 1(t) = + b

11 ~R2az

V 2() +~ +b . (47)

Rel1

V 1 1 (t) denotes the power spectral density associated with measuring

azimuth and V 2 2 (t) denotes the power spectral density associated with

measuring elevation. The term a is a constant associated with the

received signal power noise. The term b represents the uncertainty

in target position, possibly due to atmospheric refraction of the

signal to the sensor and fading. For this problem,

a - .25 rad 2ft 2sec

b -56.25 X108 rad 2sec (8

are typical values for current sensors.

31

V(t) may be simplified by factoring out a, that is,

S+ c

V(t) = a az 1 (49)

0 R -2el

where c - b/a.

4.5 Weighting Matrix.

The weighting matrix W is a constant, 9 x 9 diagonal matrix

used to give relative, equal values to the terms in the information

matrix. The off-diagonal terms are all zero, while the diagonal terms

are defined as follows:

A -11 14W22 = W3 3 1I

4 =W W ~W -t 2 (50)44 55 66 2 f

W77 =W 8 8 =1W99 AW 3 e-2t f.

wig W2, and W3 weight the position terms, the velocity terms and the

target acceleration terms, respectively. They have been chosen in an

effort to get the maximum numerical value of each of these terms to be

approximately equal to one. Without this relative weighting, the tar-

get acceleration terms completely dominate the other factors. This

becomes obvious in Section 4.6 where the performance index is

32

simplified, and the effects of the weights can be seen.

4.6 The Performance Index.

After multiplying out the terms in Equation (30) and performing

the trace operation inside the integral, J becomes

J = ftf(($11 HHI II + (22 HH2 D22 + (D 33 HH 3 03 3 )W1

to

+ (D1 4 HH 1 1 4 + 025 H 2 02 5

+ D3 6 HH3 03 6)W2

" (017 HH1 017 + D28 HH2 D28 + D39 HH3 ¢3 9 )W3 )dT (51)

where D f D(T,tf), and where

g2 (T) 2 ()

RH=R 1 X1HH1 HHI(r ) = VII(T) + V2 2 (T)

2 ( T ) 2 ( T

SV22(T)

2

HH= HH(T) = 3I()+ V2()(2

g (T)HH 3 = HH 3 X X3

T 11 (T)

The performance index further simplifies to

33

j= Jf tf _ _ _1 1t I+ 2 1 + cR4F12R0az el

L 2 2 2 1WJ*'11 W2 01 4 + 3 '7 dT. (53)

SECTION V

PROBLEM FORMULATION

This section formulates the problem as an optimal control pro-

blem and presents all of the constraints. The problem is then con-

verted to a parameter optimization problem.

5.1 The Optimal Control Problem.

The objective is to find the control which maximizes the infor-

mation, completes the air-to-air intercept (zero miss distance), and

satisfies all of the dynamical, physical, and aerodynamic constraints,

that is,

Maximize Equation (53) (54)

subject to the terminal constraint of zero miss distance and the sys-

tem model developed in Section 3. The dynamical part of this model

takes the form of differential constraints and is nonlinear in the form

x=f(t,x,u) (55)

where the control u consists of the body angles p and 9.

The initial conditions x 0 for the differential equations of the

missile are

34

35

x 1 (O) =

x 2 (O) = 0

x 3 (0) = -h o

(56)x 4 (0) = VMo

x 5 (0) = 0

x 6 (0) = 0

where the values h0 and VM are input values for initial altitude and

velocity.

The terminal constraint of zero miss distance is formulated as

three equality constraints in the form

T- 1 -(XTf - XMf) / XTf

T (tf,xf) T (YT - g / YT (57)

f Tf Mf TfT (Z ~Z ) ZT

3 T f - Zf) T Zf

where the values of T denote the actual constraint residuals for the

current control. The object is to drive these values to zero. Xr YT'

and ZT are the target inertial position components. The subscript f

denotes the condition at the final time.

The remaining constraints are due to physical or aerodynamic

limitations and are formulated as inequality constraints,G(t,x,u)> 0.

36

The limitations are

tf < 8 sec

IaZ B 1< 100 g's

1 600 (58)

Il <_ 200

IyI < 800

The final time is limited to no more than 8 seconds in order to ensure

that the missile still has enough velocity to maneuver. The missile

normal acceleration is limited to 100 g's for structural reasons. The

missile seeker has a maximum angle limit of 600 off boresight, hence

the constraint on boresight angle. Maximum angle of attack is 200, as

limited by the aerodynamics tables. The limit on the flight path

angle is included to keep the equations of motion defined. The value

of x (Equations (11) ) becomes infinite at y = 900. Instead of re-

defining the equations, a state inequality constraint is used to keep

y away from 900 because, it was felt, the final trajectory would not

require Y above 800. Until the converged solution is found, these in-

equality constraints may or may not be satisfied.

During the optimization process, the inequality constraints are

evalutated according to their constraint residuals. For the final

time constraint

01 I - tf / 8. (59)

37

The remaining inequality constraints could be violated at any time

from t 0 to tf* Therefore, the constraint residuals are evaluated as

integral constraints, that is,

2 t -f pin((1 00 -MZ (t)l 3, ) dt

32 - -tf [min((6o - I Q(t) ), ofl 2 dt

to

(60)

04 . - tf [min( ( 2 0 - I ct(t) ), 0 2 dttot 0

Stf E[min((8o - y(t) ), 0 2 dtto

The performance index (Equation (54) ) and the integral con-

straints are formulated as differential equations with zero initial

values and evaluated along with the state differential equations.

This augments the state vector (Equation (11) ) by five: x7 for the

performance index, and x8 through xll for the integral constraint

residuals. The values of the performance index and the integral con-

straint residuals (Equation (60) ) are the values of the augmented

state variables at the final time, that is,

J ff=~

02 x8f '04 x1Of (61)

3 9f 5 11 f

38

5.2 The Parameter Optimization Method.

The preceding forms an optimal control problem to find the con-

trol u(t) which maximizes the performance index. To solve this, the

problem is converted to a parameter optimization problem where the

constraints are handled by penalty functions.

The problem of having a free final time is converted to a fixed,

normalized final time by defining a new independent variable.

T - t / tf (62)

or

t- t Tf

dt f tf dT. (63)

Thus, the equations of motion become

dx dx(64)dt t-dT = f(tfT'xlu)

or

dxdx g(r,x,u,tf) , (65)

where TO f 0, f f 1. Final time is now a parameter to be found along

with the control. The control is now a function of T, the new

39

independent variable.

The continuous control function is approxi~mated by two cubic

splines, one for each control, with unknown parameters, that is,

u =u(a,T) ,(66)

where a is a vector of n unknown parameters

a = [al a2. .. . .. . .. ..a n]T (67)

For each control function these parameters are the values of the con-

trol at T = 0, T = ;j, T = 1, and the slope at each end point. For any

value of a, a cubic spline is fit to the parameters, and the control

can be found for any T between 0 and 1.

The parameters to optimize now are the final time and the para-

meters of the cubic splines which define the control function. They

can be lumped together as one vector of length n + 1

b = tf a]T.- (68)

From Equations (65), (66), and (68), it can be seen that x f X f (b).

Therefore, since the performance index is a function of x f and tfo J

can now be written

J = (b) (69)

40

where O(b) = -Xf.

The problem can now be restated as

Minimize J = O(b) (70)

subject to

T(b) = 0

O(b) > 0 (71)

where

b= ttf a]T

u u(a,T). (72)

SECTION VI

SOLUTION METHOD

This section gives a brief description of the problem solution

method.

6.1 Input.

The input required for this program is initial missile and tar-

get positions and an initial set of parameters, b. The missile and

target are set up as described in Section 2. An initial altitude,

velocity, range, off-boresight angle and aspect angle must be speci-

fied. The control is represented by two cubic splines, one for each

control angle. Each spline function is assumed to have 5 parameters -

the control angles at T = 0, T - , and at T = 1, and the slope at

each end point. Since tf is also a parameter, a total of 11 parame-

ters must be input.

6.2 Optimization Method.

The optimization method used is a penalty function-Lagrange

multiplier method which is completely developed in Reference 3. The

penalty function used is

P(b,$,S) = 4(b) +1 r(bs)T S r(b, ). (73)

Here, b is the n + 1 vector from Equation (69) which contains the

41

42

parameters to be optimized. r(b) is an m vector where m is the total

number of constraints, that is,

r(b,a ) -= 1 k (74)min(Ci(b) -,) k+l m

where

Ck ='1k

C(b) = k k (75)

C =0

Ck+l O1

Cm rn -k

Hence, the first k elements of C(b) are the equality constraint resid-

uals, and elements k+l to m are the inequality constraint residuals.

S is an m x m diagonal weighting matrix with elements ai, where i goes

from 1 to m. The m vector is used to allow convergence without the

necessity of forcing S to infinity.

This is an iterative method where each iteration involves mini-

mizing P(b, ,S) for a fixed a and S using a variable-metric method for

unconstrained optimization. After each iteration, $ and S are varied

in such a way that the parameters tend to the constrained solution.

The aim is to vary a so that, for a constant S, the value of the product

43

B S tends to the vector of Lagrange multipliers for the problem. The

values ai are only increased in case the rate of convergence of the

corresponding Ci(b) to zero is not sufficiently rapid. Convergence is

obtained when ICi(b)j il for l< Im.

The computer subroutine incorporating this method was taken

from the Harwell library and is named VFOlA. To use VFOIA, the user

must supply a routine called VFO1B which evaluates O(b), VO(b), C(b),

and VC(b), where

VO(b) f ( (76)Dbi

. (b)VC (b) = a (bbi (77)

for 1 _ i - (n+l) and 1 < j _ m.

6.3 Function Evaluation.

The function evaluation consists of finding the performance

index J = O(b), and the constraint violations, C(b). These values

are then used to form Equation (73). Figure 6 is a block diagram

showing the elements of the program which handle the function evalua-

tion. Each block represents a subroutine.

Subroutine SG is called from VFOB with the current set of

parameters b. SG is the driving routine which sets all the initial

conditions, calls the integrator, and then returns the values of

44

Input Parametersb

SG SPLINE

RUNGE

DERIV SPLINE

AERO

ATMOS INTRP2

INTRP1

Figure 6. Function Evaluation.

45

and C. SPLINE is called immediately upon entry to SG to evaluate the

coefficients of the cubic spline functions of the control for a fixed

b. The control can then be evalutated explicitly as a function of T.

The integrator RUNGE is a fixed-step, fourth-order Runge-Kutta

method. The system differential equations, the performance index,

and the integral constraints are all integrated simultaneously.

The complete system model is contained in, or called by DERIV.

For every time T, DERIV finds the control by calling SPLINE and evalu-

ates the differential equations. There are 11 differential equations:

6 for the state equations, 1 for the performance index, and 4 for the

integral constraints. The aerodynamics tables required to evaluate

drag and lift are contained in AERO. ATMOS, the standard atmosphere

model, must be called to find the Mach number and the atmospheric den-

sity for the instantaneous altitude and velocity. The aerodynamic

tables (Tables 1 and 2) are functions of Mach or of Mach and angle of

attack. The routines used to linearly interpolate the aerodynamic co-

efficient tables are INTRPI and INTRP2, respectively.

6.4 Numerical Derivatives.

To obtain Vo(b) and VC(b) as required for VFO1A, numerical de-

rivatives are evaluated by the method of central differences. For any

function G(x), the numerical derivative with respect to the kt h vari-

able, xk, is found by the formula

.G G(xk + AXk) - G(xk -A xk) (78)xk

46

where

f XkC: , EX k =(79)

C2, X .< .

6.5 Optimization Algorithm.

The algorithm used for constrained, numerical optimization in

VFOA is stated as follows:

1. Guess b.

2. Set initial values of S, .

3. Minimize P(b, ,S) with respect to b.

a. Obtain and C by calling function evaluation routine.

b. Obtain V and VC by calling numerical derivative

routine.

c. Evaluate P(b,$,S) and ;P(b, ,S)/ab.

d. Use variable metric method for unconstrained minimiza-

t ion.

4. If the final conditions are set to desired accuracy, go

to 6.

5. Vary 0, and S if necessary; to to 2.

6. Return to executive program.

IV

SECTION VII

RESULTS

In this section the launch scenarios are described, and the

results are discussed.

7.1 Launch Scenarios.

Three different launch scenarios have been chosen for the com-

putation of optimal information trajectories. Each intercept is ini-

tiated at 10,000 ft altitude with the missile and target co-altitude

and co-speed at .9 Mach. The scenarios differ in the launch range

and aspect angle only, as the boresight angle is zero for all engage-

ments. The engagements are initiated at 3000 ft, 00 aspect; 3000 ft,

30* aspect; and at 7000 ft, 600 aspect. The target for each intercept

is flying straight and level and nonaccelerating.

In order to test the system model and for comparison purposes,

one additional intercept has been run, that of minimum final time.

This is done simply by replacing the performance index with

J = tf (80)

Everything else remains the same. The launch for this intercept

occurs at 3000 ft and 300 aspect angle.

47

48

7.2 Results and Conclusions.

Since the measurements for the dynamical system are angle mea-

surements only, it is expected that to increase the information the

missile needs to fly a path which keeps these angles changing. The

optimal trajectories found with this method are consistent with this

expectation. During the intercept, the missile "fishtails" and "por-

poises" to the maximum extent possible without violating any con-

straints. The angle rates for the boresight angle and for the mea-

surement angles are kept high during the entire intercept. The bore-

sight angle tries to go from limit to limit.

Figures 8 through 24 present the results in graphical form for

each of the intercept scenarios. The missile launch point is where

X= 0, Y,= 0, and Z= -10,000 ft. Since nothing can happen for the

first .4 seconds, the plots show only the time period for .4 St S tf

The figures are grouped by intercept, each group containing all the

results for each scenario. The first two figures in each group show

the actual intercept and target trajectories projected onto the YIY

and the X1-Z1 planes respectively. For the horizontal trajectories,

the viewer is above the X.I-Y 1 plane looking down; for the vertical

trajectories, the viewer is on the positive Y I axis side of the X-

plane. The next four figures are for the boresight angle, the bore-

sight angle rate, the azimuth and elevation (measurement) angles, and

the azimuth and elevation angle rates, all plotted versus time.

The next 6 figures (Figures 25 through 30) present the same in-

formation for the minimum time trajectory. These are shown primarily

for comparison with the best information trajectories, but they also

49

demonstrate the validity of the system model and the optimization pro-

cess. The minimum time results are just what you would expect. The

trajectory is an easy, but direct turn onto an intercept path which

approaches a straight line. There is no variation in elevation. An

abrupt, bard turn would induce more drag and, thus, make a longer in-

tercept. For contrast, the best information trajectory is also shown

on the minimum time trajectory plots.

Table 3 lists the solved intercept problems by engagements.

Each problem has converged satisfactorily, and the miss distances are

all within 1 foot of the point mass target. The only inequality con-

straint which acts on the converged solution is the maximum boresight

angle. The values of the performance index are also listed, but the

only ones that can be compared are, the ones with the same initial geo-

metry. For the 3000 ft, 30' launch, the performance index for the

maximum information trajectory is much higher than for the minimum

time trajectory, as it should be.

Note the high number of function evaluations and the amount of

computer time required for the best information solutions. Because of

this, it would be impossible to use this optimization process in an

onboard computer for a guidance scheme. If something like this were

desirable, however, it may be possible to streamline the process in

some way. This is a possible area for further research. For example,

the aerodynamics tables could be replaced by some approximating func-

tion, eliminating the table look-up and interpolation process.

Another possibility is to formulate an empirical formula for

the guidance law which would approximate the best information

50

0n

0. E-4 '-

0 0 o L

E4 L4w

411

0 0H

o -4 '. -

41 uT 0 LM 044 ci C 0 0

A-4

Cd V 0 0 0

41 C. CC-44 03

C4'. V- C4 0 0

z4 w-

100

04. 1-4

0004. V 000

4 4. 0 0; r4

LU -W 0 0D 0:C:, C', I

M e '

51

trajectory. The plots for the three intercepts show a definite simi-

larity between the scenarios. On each intercept the missile appears

to be attempting to do the same thing, limited only by its requirement

to complete the intercept. If, after running more trajectories, this

is found to be the case, then an empirical formula might be found.

Another possible area for future research on this project is to

find out how the trajectories would differ if some information about

the state is removed. For example, if Win Equation (53) is set to

zero, the best information trajectory without target acceleration in-

formation would be found. This is because the target acceleration

terms in the state transition matrix would be blanked out. If, it is

found that the optimal trajectory is satisfactory without the target

acceleration information, then this information would not have to be

estimated by the onboard filter for the guidance law.

b- L

52

c-10

-4

MISSILE

:0 1bo. 00 2b. 00 300. 00 ,0O. 00 50o. 60 0;;. 0 0 0oo. o0'300. 00X INERTIAL AXIS FT v10'

Figure 7. Best Information Horizontal Trajectory,

3000 Feet, 0 Degrees.

a

MISSILE

U-.

Cn-,

x

c-

rEi.

c 00 600.00 200.o00 3bO.00 40.00 500.00 600.00 7bO.0O ,oo. 00X INERTIAL AXIS FT w10'

Figure 8. Best Information Vertical Trajectory,


53

a,

x5.

Ir-

a

c.a

00 030 1.0 2. 40 3:20 4.00 4:30 so3 6: 40

TIME SEC

Figure 9. Best Information Boresight Angle,


C5 c

zcc

0.~v00 Iso 1:60 2:40 3:20 4:00 4:00 5:s 6: 40TIME SEC

Figure 10. Best Information Boresight Angle Rate,


54

L

LIl

-~j

Z;

- LEVATIN8

C,

0.00 so1 1,50 2. 40 3. 20 4100 4100 560 6.40TIME SEC

Figure 11. Best Information Az and El Angles,


aa

-0 - ELOOT

--- COU

0

cz J-

.Jw cc,/

'r0"L 0 0.30 1.600 2:40 3'.20 4.00 4.30 5: so 6.40TIME SEC

Figure 12. Best Information Az and El Angle Rates,


55

O,

-" "- - T IME

o:o0 1 O.0 o 26o.00 360.00 4Wo.o0 sbo.cc Gba.cc 7bo.o 00 oo0.ccX INERTIAL AXIS FT 010'



a

,~.

U-

a

a

L 00o 1 O.o00 2bo.00 3bo.o00 4bo.o00 sbO.o 00 Q. 00 7bQ. Q G 0X INERTIAL AXIS FT w1O'



• | .... .

56

0S="m.

"Jc,.c;

LU

U a03~

Za.

W.

10

oa0 0: 0o 0: so 3 so 2: 40 3:20 4:0 0 4 3o 5. so 6. 40

TIME SEC



uaLa

Ca

Lawc-

CO

-cI

"'C

0.00 0. o 160 240o 3:20 400 so0 5,60 6. 40TIME SEC



57

ca d.IAcI

L I

Z a- -- - ftear. \

d

0.00 0.30 1.60 2. 40 3.20 4.00 4.30 5'60 6.40TIME SEC

Figure 17. Best Information Az and El Angles,


a

d... DOT

csa

ILl

ca

Cc

z -% 1-.

0:0 0" soo 1o s"o 2. 40 3" 20 4.0 oo 0 5:e s" o 6.60,TIME SEC

Figure 18. Best Information Az and El Angle Rates,


58

.8

UZ

"I.

. .

7wa

0. 0 2 . 00 40. GO 4O. 00 Ii. 0 00 .M 0 1k2. 00 1"40. O0 '0 IO0X INERTIAL AXIS FT M10'



0

-NISSLE

:~ ~ ~ ~ U 0 0 .. R#I

x:x

Lugg

L.00 i000 40. 0 .0 O 3b . 0 0 00 . o 12 o0 140.00 1S0.0DX INERTIAL AXIS FT wlO'



ai

LII-a

59

0a.

U

000

Ui -

0

0.000 0 .00 .00 D.0 Do0 .00 .00 $.00TIME SEC



0a

(JO

uc

ca

uic

z

C. I03

0.00 1. 00 2.00 K.00 4.00 5.00 6.00 7.00 @.00TINE SEC



60

Uja

cc d

"INUl- LVTO

C!a,

0:0 1.a20 .0 4 o 5 o 60 :0 90

-JO'3 J

ai

LLiJ~a

CL

00 1.00 2.00 3:00 4.00 5:.00 6. 00 7.00 3.00TIME SEC

Figure 2. Best Information Az and El Angl aes,


61

%I %

Crd

Cal

.... HI TI MEIG.- -o -.... MA IW wO

0

o00o bo.oo0 2b.o 00 A .o00 4bo.o 00 0. Do sbo.o00 7b.oo0 800.00X INERlTIAL AXIS FT 110 a

Figure 25. Minimum Time Horizontal Trajectory.

Y

.... T IMET

U,.-

a.

N- •Ncc a

'aPAX NNRII XI Tm~

L.

0. 00 1 bO. 00 200. 00 300. 00 40.00 A00 bO. 00 700. 00 Oba. 00X INERTIAL AXIS FT w10'

Figure 2. Minimum Time Horional Trajectory.

62

c'iaCa

U

-1

cc.;.

cna

10

@0:00 0: 40 :a*.s 1:20 1:s 2: 00 2: 40 2:00 3120TIME SEC

Figure 27. Minimum Time Boresight Angle.

cn c

LU

I.-

- C

0.00 040 0.30 I.2 1.160 200 2.0 2 '.0 .20TIME SEC

Figure 28. Minimum Time Boresight Angle Rate.

63

I~If

Ui5WH-J

c"

0.00o a:40 0:3 1.20 1'.0 2. 00 2K.40 2. 0 0:2TIME SEC

Figure 29. Minimum Time Az and El Angles.

AlDOEtDO

-L OS

LaJ-0

I.-

-j

0.00 0. 40 0 0 1':20 1:60 2:00 2: 40 2.30 3:20TIME SEC

Figure 30. Min~imum Time Az and El Angle Rates.

64

REFERENCES

1. Sammons, J.M., Balakrishnan, S., Speyer, J.L., and Hull, D.G.,

"Development and Comparison of Optimal Filters," U.S. Armament

Laboratory, AFATL-TR-79-87, October, 1979.

2. Smith, R.C., "A Six-Degree of Freedom Computer Simulation of an

Air-to-Air Missile Intercept," M.S. Thesis, The University of

Texas at Austin, May, 1979.

3. Fletcher, R., " An Ideal Penalty Function for Constrained Opti-

mization," J. Inst. Maths Applics. 15, 319-342, 1975.

4. Miele, A., Flight Mechanics, Theory of Flight Paths, Volume 1,

Addison-Wesley Publishing Company, Inc., Reading, Massachusetts,

1962.

5. Jazwinski, A.H., Stochastic Processes and Filtering Theory,

Academic Press, New York, New York, 1970.

6. Maybeck, P.S., Stochastic Models, Estimation and Control,

Volume 1, Academic Press, New York, New York, 1979.

65

VITA

Stephen Wendell Larson was born in Ames, Iowa, on October 15,

1948, the sonl of Kenneth W. and Alice M. Larson. After graduation

from Thomas Jefferson High School, Dallas, Texas, in 1966, he entered

Purdue University. He received the degree of Bachelor of Science in

Aerospace Engineering from the University of Texas at Arlington in

May, 1971. He then entered Officer Training School and was commis-

sioned in the United States Air Force in September, 1971. Following

completion of Undergraduate Pilot Training in October, 1972, he was

assigned as a T-33 pilot at Tyndall Air Force Base, Florida. In July,

1976, he was stationed at Langley Air Force Base, Virginia, as an

F-106 pilot. He entered the Graduate School of the University of Texas

in August, 1979, sponsored by the Air Force Institute of Technology.

Permanent Address: 3970 Cobblestone CircleDallas, Texas

This thesis was typed by Ann Jessen.

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