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Theses and Dissertations Thesis Collection

1984

Application of modern guidance control theory to a

back-to-turn missile.

Shin, Bohyun

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/19161

T218334

M i\ \l AuaA-JIJ

Monterey, Califo

t SGHOO -

APPLICATION OF MODERN GUIDANCE CONTROL THEORYTO A

BANK-TO-TURN MISSILE

bv

Bohyun Shin

March 19 84

Thesis Advisor: Daniel J. Collins

Approved for public release; distribution unlimited

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Application of Modern Guidance ControlTheory to a Back-To-Turn Missile

5. TVE OF REPORT 4 PERIOD COVEREDEngineer's Thesis;March 19 3 4

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Bohvun Shin

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Naval Postgraduate SchoolMonterey, California 93945

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19. KEY WOROS (Continue on reverse aide // r.ectasary and Identity by block number)

Optimal control, missile guidance, bank-to-turn missile

20. ABSTRACT (Continue on reverse aide it necessary end identity by block number)

In this work, the control laws of a bank-to-turn missile usingan optimal estimator in the terminal guidance phase weredesigned, and the effect of increasing the number of measurementsensors in the missile to generate more information on the statewas investigated. In the design of the control law the modernoptimal control theory with the a quadratic performance index wasused. Implementation of this control law required the use of a

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Kalman filter as the optimal estimator. The extended Kalmanfilter algorithm was utilized in the present stud}* since themeasurement states were non-linear functions of the statevectors (relative distance, velocity and target acceleration).In order to test the effects of the implementation of theincreased measurement sensors, two-, fcur- and six-measurementsensors were assumed to be implemented in the optimal estimatorBy computer analysis, the designed guidance laws were evaluatedand the effect of the implementation of increased measurementsensors was tested. Miss distance was chosen as a performancestandard during the simulation.

The results of the simulation revealed that the designedguidance law was successful within the specified scenarios,the effect of the implementation of increased measurementsensors for the estimator was favorable only in that increasedmeasurement sensors generated more information about the statevectors, but as the Kalman filter algorithm became more complex,the estimator performance was not enhanced to the degreeexpected with increased measurement sensors.

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Application of Modern Guidance Control Theoryto a

Bank-To-Turn Missile

Bohyun ShinMajor, Republic of 'Korea Air Force

S., Republic of Korea Air Force Academy, 1975

Submitted in partial fulfillment of therequirements for the degree of

AERONAUTICAL ENGINEER

from the

NAVAL POSTGRADUATE SCHOOLMarch 19 84

ABSTRACT

In this work, the control laws of a bank- to- turn mis-

sile using and optimal estimator in the terminal guidance

phase were designed, and the effect of increasing the number

of measurement sensors in the missile to generate more

information on the state was investigated. In the design of

the control law the modern optimal control theory with the a

quadratic performance index was used. Implementation of

this control law required the use of a Kalman filter as the

optimal estimator. The extended Kalman filter algorithm was

utilized in the present study since the measurement states

were non-linear functions of the state vectors. In order to

test the effects of the implementation of the increased mea-

surement sensors, two-, four-, and six-measurement sensors

were assumed to be implemented in the optimal estimator. By

computer analysis, the designed guidance laws were evaluated

and the effect of the implementation of increased measurement

sensors was tested.

The results of the simulation revealed that the designed

guidance law was successful within the specified scenarios,

the effect of the implementation of increased measurement

sensors for the estimator was favorable only in that

increased measurement sensors generated more information

about the state vectors.

TABLE OF CONTENTS

I. INTRODUCTION 13

II. KINEMATICS OF A BANK-TO-TURN MISSILE 15

A. ASSUMPTIONS 15

B. MISSILE CONTROL METHOD 16

C. GEOMETRY OF MOTION IN SPACE 17

D. KINEMATICS OF CONTROL \~ECTOR 21

E. CO-ORDINATE TRANSFORMATION 2 7

III. AN OPTIMAL CONTROLLER 30

A. GEOMETRY 30

B. SUMMARIZED DESCRIPTION OF AN OPTIMALCONTROLLER 32

C. APPROXIMATING TIME -TO -GO 5 5

D. PROJECTED-ZERO-CONTROL MISS (PZC MISS)DISTANCE 36

E. SIMULATION RESULTS 59

IV. OPTIMAL ESTIMATOR 54

A. DESCRIPTION 54

B. STATE EQUATIONS 55

C. MEASUREMENT EQUATION 62

D. THE EXTENDED KALMAN FILTER 69

E. INITIALIZATION OF THE KALMAN FILTER 76

1. Initializing The Error CovarianceMatrix, P 77

2. Initializing The Measurement NoiseVariance Matrix, R 79

5

F. EVALUATION OF THE OPTIMAL ESTIMATORPERFORMANCE 80

V. PERFORMANCE EVALUATION OF THE CONTROL SYSTEM ... 120

VI. CONCLUSIONS 150

APPENDIX A: PROGRAM LISTING 152

LIST OF REFERENCES 169

INITIAL DISTRIBUTION LIST 170

LIST OF FIGURES

2.1 MISSILE CONTROL SYSTEMS IN POLAR CO-ORD 17

2.2 GEOMETRY OF RELATIVE MOTION 18

2.5 RELATIVE VELOCITY COMPONENTS IN POLARCO-ORD 22

2.4 KINEMATICS OF CONTROL VECTORS VIEWED FROMREAR OF MISSILE 23

2.5 KINEMATICS OF CONTROL VECTORS VIEWED FROMSIDE OF MISSILE 2 3

3.1 COMPONENTS OF PZC, MISS DISTANCE 31

5.2 SCENARIO- 1 TAIL-CHASE ENGAGEMENT 42

3.5 SCENARIO-2 HEAD-ON ENGAGEMENT 45

5.4 SCENARIO- 5 SIDE-APPROACH ENGAGEMENT 44

3.5 NORMAL ACCELERATION COMMAND VS TIME 4 5

3.6 ROLL RATE COMMAND VS TIME 46

3.7 MISSILE BANK ANGLE VS TIME 47

5.8 NORMAL ACCELERATION COMMAND VS TIME 5!


3.10 MISSILE BANK ANGLE VS TINE 53

4.1 DISCRETE EXTENDED KALMAN FILTER TIMINGDIAGRAM 71

4.2 SYSTEM MODEL AND DISCRETE KALMAN FILTER 73

4.3 DISCRETE KALMAN FILTER INFORMATION FLOWDIAGRAM 75

4.4 ERROR COVARIANCE COMPONENT (Pll) VS TIME .... 84

4.5 ERROR COVARIANCE COMPONENT (P22) VS TIME .... 85

4.6 ERROR COVARIANCE COMPONENT (P55) VS TIME 86


4.S ERROR COVARIANCE COMPONENT (P55) VS TIME 8S


4.10 ERROR COVARIANCE COMPONENT (P7 7) VS TIME 90



4.15 KALMAN GAIN COMPONENT (Gil) VS TIME 93

4.14 KALMAN GAIN COMPONENT (G21) VS TIME 94



4.17 KALMAN GAIN COMPONENT (G51) VS TIME 9 7

4.18 KALMAN GAIN COMPONENT (G61) VS TIME 9 8




4.2 2 STATE ESTIMATE: Rx VS TIME 10 5

4.25 STATE ESTIMATE: Ry VS TIME 104

4.24 STATE ESTIMATE: Rz VS TIME 10 5

4.2 5 STATE ESTIMATE: Vrx VS TIME 10 6

4.26 STATE ESTIMATE: Vry VS TIME 10 7

4.27 STATE ESTIMATE: Vrz VS TINE 108

4.2 8 STATE ESTIMATE: AT x VS TIME 10 9

4.29 STATE ESTIMATE: AT y VS TIME 110

4.50 STATE ESTIMATE: AT z VS TIME Ill

4.51 ESTIMATED TIME -TO- GO VS TIME 112

4.52 NORMAL ACCELERATION COMMAND A'S TIME 115

4.55 MISSILE NORMAL LOADING VS TIME 114


4.5 5 MISSILE ROLL RATE VS TIME 116

4.56 MISSILE BANK ANGLE VS TIME 117

5.1 MISSILE MAXIMUM ACCELERATION VS MEAN MISSDISTANCE 121


5.5 MISSILE MAXIMUM ROLL RATE VS MEAN MISSDISTANCE 12 5

5.4 MISSILE MAXIMUM ROLL RATE VS MEAN MISSDISTANCE 12 6

5.5 MISSILE TIME CONSTANT VS MEAN MISS DISTANCE . . 128

5.6 MISSILE TIME CONSTANT VS MEAN MISS DISTANCE . . 129

5.7 ONE SIGMA ANGLE ERROR VS MEAN MISS DISTANCE . . 150

5.8 ONE SIGMA ANGLE ERROR VS MEAN MISS DISTANCE . . 151

5.9 MISSILE VELOCITY VS MEAN MISS DISTANCE 15 5

5.10 MISSILE VELOCITY VS MEAN MISS DISTANCE 154

5.11 ENGAGEMENT TIME VS MEAN MISS DISTANCE 156

5.12 ENGAGEMENT TIME VS MEAN MISS DISTANCE 15 7


5.14 MISSILE MAXIMUM ROLL RATE VS MEAN MISSDISTANCE 141

5.15 MISSILE TIME CONSTANT VS MEAN MISSDISTANCE 142

5.16 ONE SIGMA. ANGLE ERROR VS MEAN MISSDISTANCE 145

5.17 MISSILE VELOCITY VS MEAN MISS DISTANCE 144

5.18 ENGAGEMENT TIME VS MEAN MISS DISTANCE 145

5.19 ONE SIGMA .ANGLE RATE ERROR VS MEAN MISS DISTANCE 146

10

LIST OF TABLES

4.1 NOMINAL PARAMETERS 31

11

ACKNOWLEDGEMENT

The author wishes to express his sincere appreciation

to Professor Daniel J. Collins, whose assistance and

encouragement contributed immeasurably to this research.

The author also wishes to dedicate this thesis to his

wife, Eunsook. Without her constant support and under-

standing this work would not have been possible.

12

I. INTRODUCTION

The bank-to-turn missile has high lift acceleration in

a direction perpendicular to its wings. For airbreathing

missiles which are required for large stand-off ranges it

also offers the advantage of lower inlet angles of attack

than that of skid-to-turn missiles.

Although the control laws using the modern guidance

control theory are more complex than the normally used

proportional navigation air-to-ground method, they have

great potential for maneuvering targets in air-to-air

engagement situations. The application of optimal control

and estimation theory to bank-to-turn missile configurations

has been tried in order to obtain increased performance. In

the application of the optimal control theory to the bank-

to-turn missile, it is necessary to have information on all

states or an estimate of the state variables that have not

been measured. Since the state information available from

the typical missile sensors is limited, it is necessary to

employ an estimator. The estimator used in this study was

a first order extended Kalman filter.

The present work addressed the design and evaluation of

the optimal state estimator and optimal control laws for

application to a bank-to-turn missile. In the development

of this work, Chapter 2 will describe the kinematics of the

13

missile and formulate the equations of the motion. All

major assumptions are listed. The optimal controller is

described in Chapter 3. The simulation results of the

controller on three scenarios described later are used to

check the suitability of the optimal controller. The

optimal estimator is covered in Chapter 4, the formulation

of the state equation, measurement equations and the

derivations of the elements of Jacobean matrix are covered.

The extended Kalman filter algorithm for non-linear system

is then reviewed. Before the final computer simulation a

discussion is given on the nominal parameters needed in

initializing the Kalman filter. The results of the simula-

tion of estimators with two-; four-, and six-measurement

vectors on a typical scenario are then presented. In

Chapter 5, the simulation results of the control laws imple-

mented the estimator with two-, four- and six -measurement

sensors on three scenarios is described. The detail analy-

sis of the results is developed in order to investigate the

effect of the key variables of the control system on a bank-

to-turn missile. The mean miss distance determined from 50

Monti Carlo runs as a performance standard was used in the

analysis of the results of the control laws as a function of

the key variables. Finally, the conclusion are summarized

in the last Chapter.

14

II . KINEMATICS OF A BANK -TO -TURN MISSI LE

A. ASSUMPTIONS

The geometry and the equations of motion of the missile

will be developed, representing the positions in space,

under the following assumptions:

1. The local geographic coordinate system will be used

as the inertial reference.

2. To simplify the equation of motion the stability

axes are adopted for the missile body.

3. The missile velocity due to thrust is constant.

There is no missile acceleration in velocity due to

thrust

.

4. Each control surface or surfaces is rigid.

5. Relative to the body axes, each control system has

only one degree of freedom. The missile's control

acceleration vector acts normal to the velocity

vector, that is, the dot product of acceleration

vector and velocity vector is zero, also the

acceleration acts through the missile center of

gravity

.

6. The Euler angles t> , 9 and <b will be used to° r v v

describe the orientation of the missile with

respect to inertial space, where <$> and 6r

are

horizontal and vertical flight path angles, and $

is the roll or bank angle.

15

7. The first order lags with time constants r andP

x to a roll rate command P and a normalo c

acceleration command a will be considered forc

the dynamic response of the missile.

B. MISSILE CONTROL METHOD

Before going into the mathematical detail concerning

the motion of a missile in space as a result of a guidance

command, it is helpful to review with the control law for a

bank-to-turn missile. The guidance system detects whether

the missile is flying too much or too high to the left,

right or vertical. The guidance system measures these

deviations or errors and sends signals to the control system

to reduce these errors to aero. The task of the control

system therefore is to maneuver the missile quickly and

efficiently as a result of these signals. In a bank-to-turn

missile system, the guidance angular error detector produces

two signals R and <j> in terms of polar coordinate expres-

sion, showed in Figure 2.1. The same signals can be

expressed in another way, that is, in cartesian coordinate.

The usual method is to regard the <p signal as a command

to roll through an angle a measured from the vertical

and then to maneuver outwards by means of the missile's

elevators. The method of maneuvering the missile is as

follows. The $ command goes as a positive command to one

control surface and a negative command to the other, this

16

MISSILE

TARGET

FIGURE 2.1. MISSILE CONTROL SYSTEM IN POLAR CO-ORD.

causes the missile to roll. The R command goes to

surfaces always as a positive demand, this causes the

missile to accelerate normal to the velocity vector. The

intention is to make the response in roll fast so that the

commands can be applied simultaneously which makes for

simplicity, only a pair of control surfaces are used as

ailerons and elevators at the same time, control is obtained

by means of the separated servos.

C. GEO.METRY OF MOTION IN SPACE

The motion of a missile as a particle may be described

by using coordinate measured with moving axes (relative-

motion analysis). The analysis of motion can be simplified

by using measurements made with respect to a moving coordi-

nate system. In present work, the equations of motion will

17

K

MISSILE NORMAL

FORCE AXIS

7 MISSILELONGITUDINALAXIS

J MISSILE WING AXIS

FIGURE 2.2 GEOMETRY OF RELATIVE MOTION

be described with respect to the initial missile body axes

taken as the inertial axes. After launching, the missile

body axes changes position, but the relative position,

velocity and acceleration are still computed in inertial

axes, that is, in the initial body axes. The current zero

effort miss distance is calculated. This will then be

transformed to values in missile body axes to compute the

amount of the control inputs to the missile using Euler's

transformation.

Figure 2.2 shows the missile and target as points with

respective vector velocity V and V\ . In this analysis,^ m t'

because IV I is assumed to be at least 2 1 V . I , and the1 m

'

' t ! '

angle of attack is assumed to be small, the lead angle <j>

between Vn and the missile longitudinal axis is small.

Let the vector denote the relative position of the target

with respect to the missile. The orientation of the sight

line vector in inertial space can be represented by the

angle y Rand 6

R , as shown in Figure 2.2. The sight

line vector is resolved into three components in inertial

axes as follow s :

Rx = R*cos(6R ) "ccs (y R) = Xt - Xm

Ry = R*cos(9R ) *sin(;jj

R) = Yt - Ym

Rz = - R*sin (cJ;R) t - Zm

(2.1)

(2.2)

(2.3)

Where R is the magnitude of the sight line vector.

19

Define the relative velocity vector

V = \\ - V- r - 1 - in

•4)

Where V. and V are target and missile velocities.

The rate of change of the sight line vector's magnitude

is equal to the component of VR

along R . It can be

expressed as following:

R =R (2.5)

V *R + V *R + V *Rrx x ry >• rz :

(R2

+ R2

+ R2

)v x v z

(2.6)

Where V = \* - Vrx tx mx

V = V+ - Vrv ty mv

V , = V+ - Vrz tz m:

are the relative velocity components in the inertial frame.

In the geometry, the relative angles <Jj

rand 6 R

also

will be expressed by the relative vectors:

R= - tan'

R7

2 2 YTx >

(2.6)

20

. -1lb = - t an (2.7)

IVh ere and pn are the elevation and the azimuth anglesRWWJ ~ y R

of the relative sight line vector in the inertial frame

Then the rate of change of both angles are

Vr9

R R

sine cos;!/ V - sin9 sin^ V + cosG Vv rx v v ryv v rz

(R ~ + R ~ + R )'

~

x y zJ

(2.8)

VTJ

sin^ii V_ v rx v rv

Rcos^ (R " + Ry ) COS

V

(2-9)

Where Vnr and Vn ,are the rates of change in 6 and

Re R-pa v

\p components, is shown in Figure 2.3. These quantities

will be used later in developing the estimator state

equations and will also be used to represent the variables

for the missile seeker. The quantities 6 D , G D ,ib , ip

,K K K K

R and R (corrupted by noise) are the set of measurements

assumed available from the missile seeker.

D. KINEMATICS OF C0NTR01 VECTOR

Figures 2.4 and 2.5 are views from near and side of

miss ile a and <b represent the magnitudes of the

21

- K

FIGURE 2.3 RELATIVE VELOCITY COMPONENTS IN POLAR CO-ORD

22

DEROLLED-Z-AXIS

MISSILE NORMALFORCE AXIS

a cosem • v

ecos 6to v

DEROLLED Y-AXIS

FIGURE 2.4 KINEMATICS OF CONTROL VECTORS VIEWEDFROM REAR OF MISSILE.

Z-AXIS

a coscbm Y

HORIZONTAL

FIGURE 2.5 KINEMATICS OF CONTROL VECTORS VIEWEDFROM SIDE OF MISSILE.

23

missile acceleration due to the acceleration command (ac)

and the bank angle due to the roll rate command (Pc) each

other. g represents the gravitational effect at the center

of gravity of the missile. As mentioned in control method,

the sign of a is such that it is positive upward with

respect to the wings. Figure 2.5 shows that the accelera-

tion normal to the velocity vector has the components due

to the elevator control input and the effect of the gravity.

The total magnitude of the acceleration normal to the

velocity vector can be represented as

a -cos6m Y - ^*sin6

The velocity and acceleration components for simple

circular motion can be represented as

an

- V-/P

See Reference 1.

Using this definition, the differential equations

describing the rate of change of the flight path angles are

a cos >i - gcos6m v ° v

Vm(2.10)

24

a si nilmV c o s 3

III V(2.11)

Note am ft cosu) - g*cosG is the magnitude of the normal° v

acceleration component to the velocity vector in missile

side plane and am*sincj) is the magnitude of the normal

acceleration component to the velocity vector in the x-y

plane. The acceleration component of the missile velocity

V has only the component of the effect of gravitv in the

missile side plane under the assumption that the missile

velocity is constant. The differential equation is:

V.m g*sm6v

(2.12)

The rates of change of the missile position components

(Xm, Ym, Zm) can be obtained by integrating the components

of missile velocity. From Figure 2.2, the components of

missile velocity (Ymx, Ymy , Vmz) in inertial frame are

given by

(2.15)X = v = V *cos0 -cosUjm mx m v • v

Y = V = V *cos8 *sini|jm my m v r v

; = Vm m;

V *sin6m v

(2.14)

(2.15)

Let the response of the missile to input command in

normal acceleration and roll rate be considered. If a

demand is made on a missile for a transverse acceleration,

25

it is initiated by sending a signal to the appropriate con-

trol surface. The missile acceleration a follows the

demanded acceleration a, in a manner which mav bed

characterized by a frequency u (weather cock freouencvl

and a damping factor P . For simplicity, in present

work, only the first order lags with time constant t and> } 6

p

t are assumed, the equation of the first-order system cano l

be represented as follows:

X(t) + l./x*X(t)) = f(t)

Taking the Laplace transformation on has

X(S) = F(S)/(S+1./T)

with zero initial condition. Applying these to the BTT

missile control system.

t *a + a = aam m c

t *P + P = PP c

(2.16)

(2.17)

Rearranging the differential equation

a = (a - a ) /tm c maP = (P - P )/*c

v c . m p

4) = P

(2.18)

(2.19)

(2 .20)

Where a and P are the acceleration and roll ratec c

commands. Equation (2.20) is by definition. From the

above differential equations the control ratios are

transformed as follows:

a (S) 1

c a

£_ I jJ_ - _ ±_ (2.22)P (S) t S + 1c p

Equations (2.10) through (2.22) constitutes the dynamic

equations describing missile motion in response to the

command inputs ac and Pc . The solution of these

equations provide missile position (Xm , Ym , Zm),

orientation and magnitude of velocity (ib , 9 , Ym) and

orientation and magnitude of control acceleration ( $ , am)

E. CO-ORDINATE TRANSFORMATION

Given that the missile is on guided flight to compute

the magnitudes of the control inputs at any moment, it is

necessary to transform the instantaneous values from the

sensors in the inertial frame to the instantaneous values in

body frame. The transformation from one frame to another

can be accomplished through a transformation matrix. The

transformation matrix wiill be developed.

Define an inertial coordinate system with unit vectors

I, J, K. Also, define a missile body axis coordinate system

27

with unit vectors i, j, k. It is desired to find the trans-

formation between the I, J, K system and the i, j, k system

The I, J, K system can be thought to be oriented so

that I points north, J east, and K down. Similarly, the

i, j, k system has i along the longitudinal axis of the

missile (which by assumption is along the velocity vector

Vm), j out the right wing, and k down.

Consider three successive rotations \b , 8 and ± .yv ' v

The \p rotation is about the inertial z axis, and

transforms to an intermediate axis systems i -,, j -, , k -, .

The 6 rotation is about the i, axis and transforms to anv J 1

axis system i? , j ? , k-,. The last rotation § about the

i 9 axis transforms from flight path axes to missile body

axes. As a result, the total transformation from inertial

to body axes is given by

[3Cev][*v ] = [a]

i

j

K

(2.23)

Where A is the total transformation matrix from inertial to

body frame. The elements of the a matrix from Reference 1

are

:

all = cos(e )*cos(ijj )^ v v

al2 = cos ( 9 ) *sin(ilj )v v

al3 = -sin(6 )v

28

a21

a::

a25

a51

ao

a55

7 =

s i n ( <j> )* s in (

9

y

s i n (

o

) Ln I Ln I

s i n ( i )'•• c o s (

cos(>) -sin( ; j:-co> mi

V V

cos(cp) s sin(ev ) *sin(

cos(i) -cos

!

cos(.<j>) *sin(

cos(i) *cos(I

.

sin( j>)A sm(!) J

sin[ p) -cos :

T ,. I

29

J 1 1 . AN OPTIMAL CONTROLLE R

Although this work is primarily concerned with the

effect of an optimal estimator on the performance of a bank'

to-turn missile, we will first outline the theory of an

optimal controller in this section. The guidance laws

considered here were first developed by Stallard [Ref. 10].

The following is a brief outline of his work.

A. GEOMETRY

Figure 3.1 shows the applicable geometry representing

the Pro

j

ected-Zero- Control miss (PIC miss) distance in the

terminal state. The problem is considered to be three-

dimensional and two major axes system are used:

1. A seeker-oriented axis system for estimation or

measurement, and

2. The three principal axes of the missile for the

control problem,

Xb , Yb , Zb denote the target coordinates relative to the

missile along its principal axes. The body system is repre

sented a time T = bv a set of Eulian angles which

relate it to an earth fixed system. The angle A<J) is the

incremental roll angle from the present missile axes to any

future orientation at time T , shown dashed in Figure 3.1.

The PZC miss represents the total Proj ected- Zero effort -

miss distance and is defined as the miss distance which

30

would occur if no further control was exercised on the

missile at T = Ti . It is composed of a Y- components

(PZCY miss) and a Z- component (PZCZ miss). Under the

assumption of constant velocity the X- component (PZCX miss)

is represented by the distance that the missile lias to fly

at any time, since it is not effected by control, it is

not shown in Figure 3.1.

Y, at time t

Yb- I

PZCZ

a A<J> >h.i

at time t

PZC-miss

FIGURE 3.1 COMPONENTS OF PZC, MISS DISTANCE.

31

B. SUMMARIZED DESCRIPTION OF AN OPTIMAL CONTROLLER

Given a time- varying linear system of the form,

X = F(t)X + G(tJ U (3.1)

Where X is n-component state vector

U is m-component control vector.

An optimal control is one that minimizes a performance

index made up of a quadratic form in the terminal state

plus an integral form of quadratic forms in the control and

the state:

7 ^_ ^ f_ t = T i^

+5

fT;

T

(XTAX + U

TBU)dt (3.2)

Where the S, and A are positive semidefinite matrix and

B is a positive definite matrix. An appropriate choice of

these matrices must be made to obtain acceptable level of

X(Ti),

(X(t) and U(t) .

The guidance law determined in Reference 8 is of the

form

U (a, , P,C L.

T

With the performance index chosen (the weighting on the

state vector was not considered in Reference 8.

32

TiJ =

2< Xb

+ V J

b ^ I Ti+

2(U - U

b )

TB(U - U

b)dt

(3.3)

Where the first term is one-half the required miss distance,

the second term is included in order to eliminate infinite

energy solutions. B is a 2 by 2 positive definite,

symmetric weighting matrix and of the form

1

(3.4)

with 1/B. = (Ti - To)X maximum acceptable value of

2|U.(T)j" . A detailed derivative of the optimal guidance

law is given in Reference 10.

The solutions of the above equation from Reference 8

give the optimal acceleration command at time T between

TO and Ti as

5Tso

3B, + T1 20

T ( >W (3.5)

Where T is the remaining time to 20 until intercept andct q 00 r

M, is the PZC-miss along the missile normal force axis and

the negative si2n is due to definition of the normal

33

acceleration as positive in the upward direction. The

optimal roll rate command at time T between TO and i'i

from Reference 8 is given bv

21a T (Mi )v by'

T + 65 Bgo 2

(3.6)

Where M, is the PAC-miss along the wing axis. Definingby o o o

the roll error:

a j - - 1 |bv

A$ = tanItt^" (5.7)

.An alternate form for P can be obtained bv dividingc °

Equation (5.6) by Equation (3.5), then

/a T (T_£° go

+ 5 B

T + 65 B

tan(A<j>) (5.8)

go

Because the solution of P was based on an assumption ofc t

small missile roll excusions, the tanA^ term in

Equation (5.8) can be replaced by A<J) to yield

7a T T + 5 B,)c go _go V_

a2

T5

+ 65 Bc go 2

Acf> (5.9)

54

The optimal controller is implemented in the present work

using Equations (3.5), (3.7), and (5.9). In computing th<

control input , T£0

Y and Z -component of PZC-miss and

the elements of the weighting matrix are needed.

C. APPROXIMATING TIME-TO- GO

It is necessary for the Time-To-Go and the Projected-

Miss - Dis tance in missile body axes to be specified at time

T to compute the optimum control input. The Time-To-Go

will be defined at first, then the state variables will be

introduced as means for formulating the necessary equations

The target acceleration components of the state variables

will be modeled since it is assumed that any information

about the target acceleration cannot be obtained from the

active seeker of the missile. Then using this target

acceleration model the algorithm for computing the other

state variables will be developed.

The Time-To-Go until intercept, T- , is required for

computing the control input. Let the time to intercept be

defined as the time to minimum range. Under the assumption

that the relative velocity vectors may be considered

constant at its present value until intercept. T can be

found at time, T , as

:

T. =l

TR V—r —

r

(3.10)

55

Where R is the present position of the target relative to

the missile and the factor in brackets in the component of

V along R . An alternate form of T is:

R V + R V + R VT —__

X rXj_

9 ? 9V+V+V„ (3.11)rx rv rz ' ^ J

The negative sign comes from the sign of V as shown in

Figure 2.2 and Equation (2.6).

D. PROJECTED- ZERO- CONTROL MISS (PZC MISS) DISTANCE

The Proj ected- Zero-Control miss distance (PZC-miss) is

defined as the miss distance which would occur if there

were no further control input after time T .

In this work the state vectors are defined as the

following state variables

X = (R , R , R , V , V , V , a. , a^ t sl. )

T— v x y z rx ry r: tx ty t z

J

Where R is the relative missile distance to targetr °

V is the relative missile velocity to target

a is the target acceleration

A model for target acceleration has been assumed. This

is given as follows:

at(T) = a

tQe" a

'

T|a > (5.12)

56

Where a is the intial target acceleration and '- is the

reciprocal of the acceleration time constant. The initial

target acceleration, 4g's, will be assumed for each

scenario, and a = 1/20 for evasive maneuver as indicated

in Reference 10 will be chosen. These values will be used

in the simulation of the optimal controller.

With the assumption that a state estimator is available

to provide estimates of the state vector at current time (tj,

the values of the state variables for no control inputs at

future time, can be obtained easilv using the integration,

i.e.,

\ (t) = Vr(t0) + j* a

t(T)dx

t

to

(t) = R(t0) + f V fx)dTto l

The results of these integrations are

V (t) = V (tO) + l/o, (1 - e" a(t t0)

)at(t0) (3.15)

R(t) = R(t0) + (t - tO)*V (to)

+ [1/a (a(t - tO) - 1 + e

(5.14)

a(t - tO). ,, (tO)

Where tO and t denote current time and future time. It

should be noticed that above integration is based on the

37

assumption of a single physical dimension. If this assump-

tion is to be applied to three-dimensional problem, it must

be concluded that there are no coupling among the X, V, Z

dimension. Let T = t - t , then the components of thego o l

PZC-miss in inertial axes at current time (tO) can be

expressed as

"MX

RX

Vrx

My

M7 Ti

Ry

Rz

+ Tg°

TO

Vrv

Vrz TO

fX

fy

f

tx

tv

tz TO

(3.15)

(l/2)*g*Too J

"7 T1

Where f = f = f = l/a"[aT - 1 + e go] and the lastx y z

L go

term accounts for the free fall effect of gravity in the

vertical axis

.

The components of the PZC-miss in missile body axes for

control input calculation are obtained via the Euler trans-

formation matrix:

38

Kxl"by

.

Mb^.

[*][e„I[*v ] M)

M

" MX

My

M. z .

(3.16)

Where it is assumed that the missile has exact knowledge of

its orientation angles ijj , 8 , c|> , from the accuratev v

inertial measurement units.

The values of optimal control vectors (a and P J ,r v C C

then, are calculated from time to go (T ) , PZC-miss in° CTO'

o

body axes and weighting matrix components (Bl and B2) which

may be determined according to the desired miss distance and

intercepting time. In a later section, the weighting

matrix will be developed and the scenarios will be set up

for computer simulation.

E. SIMULATION RESULTS

The guidance law in this work was tested on a simple

six-degree-of - freedom simulation using only plausible

numbers without reference to any actual missile or design.

Before the computer simulations, the weighting matrix

of the performance index selected in Reference S will be

calculated using the initial state, X(0) , the missile

operating time imposed, T (5 sec) and the constraint input

vectors. It is assumed that the missile has a velocity

advantage of two over that of the target, a zero- and 0.5

39

second-time lag auto pilots for pitch with limits of -3g's,

+20g's and for roll rate with limits of +6.23 rad/sec.

The allowable miss distance which is usually determined

by sizing of the warhead was chosen as 5 meters. Which

makes the first term of the performance index Equation

(3.3) equal to 12.5 meters. The acceleration term and the

roll -rate term in the performance index are set equal to

this respectively (one-half of square mean miss distance),

then

>r (a - a,) ~ dt = 12.5 m'c d

(3.16)

T.

(P ") dt = 12 . 5 m"~ c

(3.17)

The nominal time of flight for the terminal phase is set

equal to 5 seconds. The components of the weighting

matrix B were based on the following assumptions. It is

9

assumed for (4g's) as a mean-square value of (a - a,) and

2(2 rad/sec) as a mean-square roll rate to be acceptable.

Then the components of weighting matrix are:

B1

= 0.003254 sec°

B = 1.25 m"sec

40

The commands a and P are computed every 0.05 sec

and are frozen when the Time-To-Go falls below this interval

In order to simulate the optimal controller appropriate

scenarios were developed below with restriction that the

Time-To- Go was 5 sec and the speed of missile is two times

faster than that of target. Three scenarios; tail -chase,

head-on, side- approach engagements, were chosen as possible

cases in this work.

In case 1, tail-chase engagement, the target flies 100m

above the missile and 2500m uprange with a horizontal east-

ward acceleration of 4g's. Figure 3.3 shows the case 2.

In case 5, side- approach engagement, the target flies 400m

above the missile and 1900m uprange with a horizontal north-

ward acceleration of 4g's at t = as shown in Figure 3.4.

For simplicity, when the scenarios were set up, 1000m/ s was

taken as a missile speed, 500m/s as a target speed in case

1 and case 2, 600m/s as a missile speed, 300m/s as a target

speed in case 3, and 4g's as a target acceleration for all

three cases. When simulation was executing, this first

simulation assumed perfect state feedback without the use of

an optimal estimator.

The simulation results are shown in Figure 3.5 through

Figure 5.10. At first, the optimal controller with no time

lag was tested for three scenarios, as shown in Figures 3.5

and 3.6. For scenario 1, the normal acceleration command

in initial phase is very large, then decreases to zero at

41

Missile

TO

l

T

4a ' S

aTOe

t/20

Taro.— — * S e t 1 r

I

FIGURE 3.2 SCENARIO- 1 TAIL -CHASE ENGAGEMENT

42

T TOt/20

FIGURE 3.3 SCENARIO- 2 HEAD-ON ENGAGEMENT

43

J

-K

a <-p

//\^

rr=500m/s

—*<•—

XT= 1900m

'l

Z = 4 010m1

1

L -*-

V =6 00m/ secm

/

/

/

/

//

/

a /- 4 a • saTO/ °

/=

-t/20C* --p / C* i-ri s-^ CT

/

/

FIGURE 3.4 SCENARIO- 3 SIDE- APPROACH ENGAGEMENT

44

NO TIME LAG CAS;

o-

oin-

\

WOr

i

LEGEND

A- THIL-CHRSE ENGAGEMENT

O = HEAD-ON ENGAGEMENT

\

\

53

NT

Q.O 1.0 2.0 3.0

Time ( sec4.0 5.0

FIGURE 3-5. N0RMRL ACCELERATION COMMAND VS TIME

45

NO TIME LfiG CRSE

oCO"

LEGEND

A- TRIL-CHflSE EN'GRGEMENT

O - HERQ-CN ENGAGEMENT

D- SICE-RPPRCRCH ENGRGEMENT

-3

-11 ! i 1 i ' t t / !

|i r—r—i-

2.0 2.0 4.0 5.0

Time ( sec

)

FIGURE 3-6. ROLL RRTE COMMRND VS TIME

46

NO TIME LRG CASE

a.o

LEGEND

A- TRIL-ChnSE ENGflGEMEf

- HEBD-ON ENGAGEMENT

D- SIDE-P.PPROflCH ENGAGEMENT

TLme ( sec

)

FIGURE 3-7. MISSILE BANK ANGLE VS TIME

4 7

at the final time. For scenario 2, the acceleration is z.

in the initial phase, then increases and converges to zero

finally. For scenario 5, the acceleration starts from zero,

but, increases rapidly up to the maximum acceleration then

freezes for seconds before decreasing. The acceleration

does not converges to zero at the final time. Comparing

these results with the variation of the roll rate command

in Figure 3.6, the opposite trends are observed in the

results of scenarios 1 and 2, i.e., if an acceleration

command in the initial phase is large, a roll rate command

is small. The acceleration command in Equation (5.5) is a

linear function of PIC-miss in of z-direction of the missile

coordinate system. For scenario 1, PAC-miss in the

z-direction is relatively large, this causes a large accel-

eration command and a small roll rate command in initial

phase. For scenario 2, the relative magnitude of the PZC-

miss in the z-direction is small. This cause a large roll

rate command in the intial phase. For scenario 5, the

PZOmiss in y-direction is of the same magnitude as the

relative distance as the Time-To-Go decreases. This means

that a larger acceleration command than the maximum value

allowable in the missile body coordinate system is required

at some of instance time during the missile flying. Thus

the acceleration does not converge to zero at the final time

In the derivation of the equations for control commands, a

small angle approximation was assumed. However, in the

48

scenario 5 this assumption is questionable. The roll ra

command computed as shown in Figure 5.6 shows a sine wave

characteristics which is possibly due to overcorrection.

The simulation results for a 0.5 second tine lage are

shown in Figure 5.8 and Figure 5.9. The time lag causes

much larger acceleration control commands. This includes

overcorrections, so that the control commands do not

converge in the final phase. The other characteristics

follow the explanations given in the analysis of the cases

with no time lag

.

For all three scenarios the miss distances were obtained

in the simulation of both cases, no- and 0.5 second-time

lag. The miss distances are below 0.5m for all three

scenarios in the case of no-lag. In the no lag case the

control laws gave excellent results. In the lag case, the

miss distances are 5.9m for tail-chase engagement, 0.8m for

head-on engagement, and 52.9m for side-approach engagement.

The miss distance of the side-approach engagement with a

different maximum acceleration limit fo 25g's results in a

miss distance of less than 5m. This higher maximum acceler-

ation limit will be used in the scenario 5 during the simula-

tions for testing the performances of whole control system

in Chapter 5.

The previous results are summarized as follows : (1) the

tracking of a target is very dependent of the missile and

target geometry, (2) the capabilities of the missile

49

(maximum acceleartion and maximum roll rate limits an J time

constant) in maneuvering against the target is very important

factors, (3) higher maximum accelerations are required in

any side approach engagement, and (4) the assumption of

small AcJ> in the optimal control law is a limitation in the

general case. Nevertheless, the control laws developed

from the optimal control theory for both the no lag and the

0.5 second-time lag cases yielded successful results for

the specified scenarios. Since the optimal controller with

0.5 second-time lage auto pilots resulted in a miss distance

of approximately 5m with complete state variables feedback,

it is expected that the inclusion of an optimal estimator for

these particular scenarios will results in still larger miss

dis tances

.

50

TIME LAG ( .55EC] CRSE

CM

td~

.rfSSSSB^3£E3Ea

J2&

f

Q

!

LEGEND ^A= TF1IL-CHR5E ENGAGEMENT

O - HERD-ON ENGAGEMENT- 3IDE-RPPR0RCH ENGRGEMENT

2.0 3.0

Time ( sec

FIGURE 3-8. NQRMRL RCCELERRTIQN COMMRND VS TIME

51

TIME LAG ( .5SEC) CRSE

o

LEGEND

A- TRIL-CHRSE ENGAGEMENT0- HEPC-CN ENGAGEMENT

D- SICE-APPROACH ENGAGEMENT

-3££^^x

a

i

o~1 ' r

3.0

-! 1 r

5.00.0 1.0 2.0 4.0

Time ( sec

)

FIGURE] 3-9. ROLL RATE COMMAND VS TIME

TIME LfiG ( .55EC] CnSE

o -

o.-r

I

LEGEND

A- TRIL-CHRSE ENGAGEMENT- HERD-GN ENGRGEMENT

D- SIDE-RPPRORCH ENGRGEMENT

0.0 1.0 2.0 3.0

Time ( sec

)

4.0 5.0

FIGURE 3-10. MISSILE BP.NK RNGLE VS TIME

53

I V . OPTIMAL ESTIMATOR

A. DESCRIPTION

Without full state variable feedback it is necessary to

estimate the state variables. This chapter describes the

Kalman filters used in the state estimation. These filters

were designed for respectively: (a) two measured states,

(b) four measured states, and (c) six measured states.

All measurements have noise superimposed on them. It is

assumed that the nature of the noise source is known

completely

.

The Kalman filter is the optimal estimator of the

states. Since the measurements are non-linear, it was

necessary to use the extended Kalman filter theory. A

first order extended filter was assumed in this work. In

developing the filter algorithm, the non-linear measurement

equation is linearized about the most recent state estimate

and then the Kalman filter algorithm is applied. This is a

extended Kalman filter algorithm. The results of the

extended Kalman filter will yield suboptimal state estima-

tion .

In practice, the missile will have sensors of various

types that provide inputs which are related in some fashion

to the states to be estimated. The main emphasis in the

present work will be on active seeker which is assumed to

54

provide the optimal estimator with measurements of I of-

sight angles for the case of the missile having two measure-

ment sensors. For the case of four measurements one has two

of line-of -sight angles, range and time rate change of

range. For the case of six -measurements one has two of

line- of - sight angles, two of time rate change of the line-

of -sight angles, range and its time rate change. When the

measurement equations and the Jacobean matrix are developed

the six -measurement case will be derived since the other

cases have the same kind but less number of measurement

s ensors

.

We will first consider the state equations and then the

measurement equations. This will be followed by a discus-

sion of the extended Kalman filter. Measurement error

effects on the initialization of the Kalman filter will be

analyzed. Finally, the estimator for the measurement cases

will be simulated in order to test the effects of the imple-

mentation of the different measurment vectors on the

extended Kalman filter.

B. STATE EQUATIONS

The state vector is limited for convenience to the

following state variables:

TX = (R , R , R , V , v , V , a. , a. . a. J- K x ' y ' z rx' ry rz ' tx ' ty' tz ;

(4.1)

55

Where X, = R , X_ = R , X_ = R are the components1 X Z y 3 Z

r

relative position, X, = V , X^ = V , X .= V lie com

L 4 rx* 5 ry rz

ponents of relative velocity, and X 7= a«. v , X„

X9

= atz

tx' "8 "ty

the components of target acceleration.

In order to develop the state equations, dynamic equa-

tions of target motion are needed. The target model

selected for tracking applications must be sufficiently

simple to permit ready implementation in weapons system for

which computation time is at a premium yet sufficiently

sophisticated to provide satisfactory tracking accuracy.

To meet these requirements, the model described by Singer

[Ref. 10] will be used in this work. The model will be

presented for a single spatial dimension in order to

enable accurate tracking performance estimates to be made

for a variety of sensor measurement. If the targets under

consideration normally move at constant velocity, the

accelerations due to turns or evasive maneuvers may be

viewed as perturbations upon the constant velocity trajec-

tory. The target acceleration a(t) therefore will be

termed the target maneuver variable, in a single physical

dimension. The target maneuver capability can be satis-

factorily specified by two quantities: the variance or

magnitude of the target maneuver and the time constant,

or duration, of the target maneuver. Hence the target

acceleration, namely, the target maneuver is correlated

56

in time. A typical representative model of the target

acceleration as a correlation function, can be express

as follows

:

l (t) = E[at(t)a

t(t + t)] = a

a t

ma > (4.2)

Where them

is the variance of the target acceleration

and a is the reciprocal of the maneuver time constant.

2The variance a ~ of the model will be approximated using

the following formula which represents the acceleration

probability density suggested by Singer [Ref. 10]:

maxm [1 + 4P - P n ]L max J (4.5)

where a is a maximum acceleration rate, P is themax ' max

probability of maneuvering at a , and P n is thev ' & max '

probability of no target maneuvering. And as mention in

previous chapter, a = 1/20 for an evasive maneuver will

be used in this work.

The above process can be modeled as the output of a low

pass filter driven by white noise and it can be described

by the equation

at

= - at

+ W (4.4)

57

where a (t) ,the correlation function of the white no

input sat is 1 1 es

•

2(t) = 2ao

25(x)

w ^J m v J 14.5)

The acceleration model above will be applied to each

X, Y, Z dimension with no cross coupling, the system state

equations can be written then as:

X=FX+a + W

000100000000010000!0000 1000000000100000000010000000001

RX

i

Ry

R_

Vrx

V_ry

=

Vrz

?tx

fty

atz

X

y

Rx

Ry

i

R^

Vrx

- amx

Vry

+ " a.ty

+

Vrz

- ayz

a.tx

wX

ty y

K 2

I

w00000000z cz z

I

(4.7)

The Equation (4.7) is a linear state equation with the

assumption that the forcing function a vector, i.e., missile

acceleration, can be precisely measured and also resolved

into X, Y, Z components in the inertial frame.

58

Many sensors have a constant data rate, sampling t

data every T seconds. If the above system equation is

represented in discrete equations. The appropriate system

equations of motion may be expressed by

X(K + 1) = :(T)X(K) + B(K) + u(K) 14.8)

where v(T) is the target state transition matrix, B (K)

the deterministic forcing input vector which is composed of

first and second integration and U(K) the inhomogeneous

driving input due to white noise. Since it is assumed that

the missile acceleration is a known deterministic forcing

function, the system equation of motion can be obtained by

direct integration in each single dimension with E|W(t)|=0

The desired integration in discrete form yields

R(tO+T)

V (tO+T)r^

aT(tO+T)

1 T l/ot2(-l + aT + e"

aT) R(tO)

1 l/a(l-e"aT

)

aT

-//tO+T

tOam

(t ^ dtdt

c tO+TJ a„ (t)dtto

m

V (tO)r

at(tO)

X(K + 1) = * (t , a ) X(K) + a (K) (4-9)— mWhere K + 1 = (K + 1)T = t + T, K = KT = tO .

59

Next, if the driving input U('t) vector is considered,

this input is not a sampled version of the continuous tii

white noise input W(t) • Since W(t) is white noise,

U(k) also should be a discrete time white noise sequence

that is, E[U(K)U(K+i-)] = for i + . Then as shown

in Reference 10, Li and W are related by

u(k:, CK+1)

J $

KT[K + l)t - t| W(t)dt (4.10)

Where U(K) is a white noise sequence with covariance

matrix

E[U(K)U(K)T

Q (4.11)

Following the above derivation, it can be easily

expanded to the case of three independent dimensions in X,

Y, Z. For a = a = a = a ,and a = a = a = a' x y z x y z

since the model of target acceleration is for one dimension,

the transition matrix is

TI f1

I

I £9 I (4.12)

60

Where I and are 5 by 3 identify and null matri

f, = 1/a (aT - 1 .+ e

f\ = l/a(l - e"aT

)

- : r

f- = e-aT

The forcing acceleration vector is

//(K + 1)T

KT KX

(K + 1)TIf a

KTmy

(t)

(t)

(Kr r

l)Tamz

KT

(K + 1)T

KT mx

(K + 1)T

KT my

(t)

(t)

(K + 1)Ta

KTmz

dxdt

dxdt

(t) dxdt

dx

dr

(t) dx

(4.15)

61

The covariance matrix of white noise U(K)' is

Q

Qn l Q 12l Q13

I

Q 12I Q 22

I Q25

I

Qi-i Q ?-i Q«i

15 :o

(4.14)

Where

Qn = (a2/a

4) [1 - e"

2aT+ 2aT + (2/3) (aT)

3- 2 (aT)

- 4aTe"aT

]

-aTQ 12

= (o7a ) [e"-li

+ 1 - 2eai

+ 2aTea

- 2aT + (aT)"]

2 2 -2aT -raTQ13

= (a /a ) [1 " e" al

- 2aTeal

]

Q 22= (a/a") [4e

2,2,,, -aT -2aT+ 2j - e . aT]

Q 25= (a

2/a) [e"

2aT+ 1 - 2e"

aT]

Q- = a2[l - e-

2aT]

C. MEASUREMENT EQUATION

In the case of the estimator with six -measurement sen

sors the present study assumes that the active seekers of

missile give the measurements of the sightline angles

62

GR , i|>

R , the time rate change of the sight line angles

8 ,, , '<J n , the relative ranae R , and the time rate change

of the relative range R contaminated by the Gaussian white

noise. Then the equations of measurements in discrete time

form are

Z(K) = h(x,K) + Y(K)I 1.15)

h(x)

where Z_(K) is a sequence of measurement vector contaminated

by white noise V(K) and

elevation angle

elevation angle rate

azimuth angle

azimuth angle rate

range

range rate

The missile seekers actually measure the data about the

seeker axes. Thus, throughout this development the assump-

tion is made that the missile possesses an inertial measure

ment unit that accurately specifies the missile's orienta-

tion in space so that a transformation from seeker to

inertial coordinates can be made.

Referring to Figures 2.2 and 2.5, the six-measurement

vectors can be written

3r

q" r

—

r r

R

R

63

Rtan

R^

X v

V . r R V + R R Vx z rx v z ry

2 ?

(R + R )Vvx v r:

R (R+R+RUR+Rv x v z ^ v x y

2.1/2

\b = tan+ r

-

Ry

RX

_ m

ru

r R cos

R V + R Vv rx x ry

(R " + R )*• x y

J

R= (R2

+ R2

+ R")*• x y z

R(V

r.R)/R

R V + R V + R Vx rx y ry z r

( R2

+ R2

+ R2

)

l ' Z

K x y zJ

but Rx

= Xx

, Ry

- X2

, Rz

= X- , Vrx

= X4

, Vry

= X. ,

v = x,rz t>

64

The measurement vector can be expressed in terms of s

vector

h(x) =

hl

1

h2

h-3

=

h4

h5

h6

tanX,

_____ T_T7

X-Ax-X. + X9

2X_X, +

[ fx-,2

+ X2)X.,X,

1 j 4 2 _ _ ' 1 2 j b} 7 7 •? /? 7 71/9

' xr + V + x3 > <

xi

T X:"J

tanX2

1

X1X2X4

+ X1X2X5

t\ - x2' + x

3

z)

1/z (X^ + X,")

? ?(X - + x/ * x,-)

2 , 1/2

xnx, + x x_ + x_x,14 2 5 _ 6

? ?

(xr + x, + x_-)1/2

(4.16)

The measurement noise covariance matrix for the active

seeker can be written

E[V(K)V(K) T] = R =

a Q

a'2

o

a,2

a*2

a R2

0a65

?

(4.17)

R

Where the diagonal elements are the variances of th i di-

vidual measured quantities.

As shown in Equation (4.16), the measurement vectors are

the non-linear functions of state vectors. It is impractical

to implement in non- linear form because the computation of

gain K and error covariance matrix P in update process

in extended Kalman filter algorithm is not possible. To

simplify this computation, and to implement the extended

Kalman filter, the Jacobean or matrix of partial derivatives

H will be determined, i.e.,

3h.

H. •

1"! 9X(4.18)

The Jacobean will be a 6 by 9 matrix because h is a vector

with six elements and X is a vector with nine elements.

Performing the operation, the components of the matrix H

are

H = (-X X ) / (D D )

44 12 1

= (X X )/ IE D )

45 12 1

= H = H46 47 4 E

= (X )/(D )

51 1 C

i

= (X )/(D )

_ *. 2. L

I= (X )/(D

)

53 3 C

= H =049

66

OH = rt = J H5 5 5 6 57

j Hbb

3 V: =

61

c 2

63

6 5

66

f (D X ) - (D x )]/ (D )L 4 2 1 z1

J

[ (D x ) - (D x ) ]/ (D )

5 2 2 J

[{D X ) - (DX)]/(D )

k 6 2 3 -J 36

= (X )/(D )

1 C

= (X^)/(D )

(X )/(DJ3 C

K = H = H =067 68 69

2 2 2 1/2C = (X + X f X )

C 1 2 3

2 2 1/2C » (X + X )

1 1 2

2 2 2 2E = X X |(X +X ) (2D D -X -:< (X +X )7(3E *D )

1 1 3 l U 6 01 1 2 5n-J 1 C

.22 2222 > 22E = X X f(X *X ) (2D D -X -X (X + X ) J ( 3 C +C )

2 23^56 012 1 d 6 J 1C.2 2 2 2

= JX (X * X ) X (X + X ) J (D +2D )2K146 25 6 J 1

11( XI>,( D D 1

= ( X X ) / I D D )

12 2 3 1

= ("D )/ (C )

13 1 C

= H = H = H = H = H

1U 15 16 17 13 19=

67

5 3h = (E )/(D D )

: i i c 1

5 3

H = ( E ) / ( B D )

2 2 2 C 1

24

(S )/(D D )

3 C 1

2 3

(X X ) / ( D D )13 3 1

(X X )/(D D )

2 3 1

^ *j

27

( D X ) / ( E >

1 3

28=

33

39

2 2

(-X )/ (X + X )

2 1 2

2 2(X )/(X + X )

1 1 2

OH = H34 35

= H = K36 37

=

H =038

,1=

(V /(DC °1 »

i\2

a 3

3 4(D )/(D

nD )

3 2[XXX |X - X j/ (D D )

L1 2 3 4 5 J

1

E = X (X4 2 4

2 2 2 2'X ) f-D D f X (D +2D )1b

L1 11 o J1

= X (X - X ) \-D C + X <D + 2D ")]- 14 5^01 21 J

68

D. THE EXTENDED KALMA.N FILTER

Optimal estimators that minimize the estimation error,

can be divided into two processes, the one is the updating

process to estimate the state vector at the current time,

based upon all past measurements, the other is the extra-

polating process to estimate the state at some future time.

Optimal estimation procedure for linear state equations and

non-linear measurement in the form of discrete time will be

described referring to Reference 6. Figure 4.1, a timing

diagram shows the flow of the various quantities involved

in the discrete optimal filter equations.

The discrete system equation whose state at time kt is

denoted by simply X(K) , where U(K) is a zero mean,

white sequence of covariance Q(K) , is

X(K) = $ X(K - 1) , + B(K - 1) + U(K - 1) (4.21)

where is a transition matrix

E[U(K)] (4.22)

E[U(i)U(j)T

] = Q5i;j

(4.23)

The measurements are the non- linear function of the system

state variables, corrupted by uncorrelated white noise of

covariance R(K) . The measurement equation is written as

69

where

Z(.K) = h (X,K) + V(K)

E[U(K)] =

E[V(i)V(j)T

] = Reij

(.4.24

(4.25J

(4.26)

The initial conditions are

A AE[X(0)] = X(0) ECCX(O) - Xn )(X(0) - Xn )] = P~

QJ Vii , -0 (4.27)

E(WkVjT

) = uncorrelated (4.2S)

The Kalman filter algorithm is to minimize the estima-

A.tion error, i.e., error covariance. If X denotes an esti

mate of state vector, X , and X is the mean value of the

state vector, the error covariance matrix is defined as

P = E[(X - X)(X - X)1

] (symmetric) (4.29)

The development of the extended Kalman filter algorithm with

a few definitions is essentially more application of the

expectation operator E. We will consider the updating

process then the extrapolating process. The update equa-

tions are used to incorprate the latest measurement in the

estimate and in the covariance. After we obtain the updated

covariance matrix we will then consider the optimum choice

of Kalman gain and derive the equation for the esti-

mation of state. Let X(K-1)(-) denote an estimate of X

70

H(K-1),R(K-1)

X(K-1)(-)

v(K-Z),QfK

P(K-1)(-)

X(K-1)(+)

; ( K - 1 ) , Q (' K -1

)

P(K-1)(+)

extrapolat in:

process

H(K) ,R(K)

XfK) !-

)

P(K)(-)

upda

X(K)O)

CK),Q(K

>

P(K)(+)

ting

process

t(K-l)

measurement

process

t(K)

measurement

proces s

FIGURE 4.1 DISCRETE EXTENDED KALMAN FILTER TIMING DIAGRAM

71

at time k-1 , given measurements up to and includi .

K-2 , P(K-1)(-) denote the error covariance, K(.K-l)

denote Kalman gain and Hf.lc-1) denote the Jaccobean matrix

at time K-1 . We will seek an optimal observer in the

presence of the state excitation noise and in the presence

of the observation noise.

P(K-1)(+) E{[I - K(K-1)H(K-1)]X(K-1)[X(K-1) T(-) (I-K

(K-1)H(K-1)T)+Y(K-1)

TK(K-1)

T]

+ K(K-1)V(K-1)[X(K-1) (-)T(I-K(K-1)H(K-1)

T)

+ V(K-1)TK(K-1)

T]}

4 .30)

Rearranging Equation (4.50) using the definitions

Equations (4.26) and (4.29) and the assumption that the

measurement errors are uncorrel ated , then one has

P(K-1)(+) = [I-K(K-1)H(K-1)]P(K-1) (-)[I-K(K-1)H(K-1) T]

(4.31)

+ K(K-1)R(K-1)K(K-1) T

This equation updates the error covariance matrix.

Then we still need an optimal gain K. We will choose to

minimize the diagonal elements of the covariance matrix,

i.e.,

J(K-l) = E[X(K-1)( +)

TXV(K-1)(>)] = trace [P(K-l)O)] (4.32)

mathematical model

s vs tern

—I

u ( K - 1

)

X,

b ( K - 1>

L_.

i x„r+i

1 *k-i

K-lD e 1 a v j

measu remen

JL.

Discrete Kalman filter

+

i(K-l) f+ >

I

Delav >

TBBMK53E3.

.

K

*K(-)

*K-1HK

uo

FIGURE 4.2 SYSTEM MODEL AND DISCRETE KALMAX FILTER.

73

Introducing a bit of matrix calculus, one has

~ [trace A B AT

] = 2AB 4.32

aK(K-l)[l-K(K-l)H(K-l) J = -H(K-l)

T(4.34)

Applying these formulas to Equation (4.31) and solving

for K(K-l),

T rK(K-l) - P(K-l) (-)H(K-l)

A

[HfK-l)P(K-l) (-)H(K-l) l

+R(K-l) ]~ 1

(4.55)

This is the Kalman Gain Matrix.

Rearranging Equation (4.31)

P(K-1)C+) = [l-K(K-l)H(K-l)]?(X-l) (-1) (4.36)

Then the update state estimate at time K-l is equal to

the extrapolated state estimate at time K-2 plus a term

which weights the measurement residual via gain K(K-l) .

1((K-1)( +) ='X(K-1)(-) + K(K-l)[Z(K-l)-h(k-l),

/X(K-l)(-))]

(4.37)

We need to consider also the propagation of the estimate

X(+) between measurements and the propagation of the

covariance matrix P(K-1)(+) between measurements. Propa-

gation of the estimate is straight forward and involves only

74

HK-1

RK-1 QvK-l

rK-l

-!- I. ,ll. t , ,,,.,- . . , - I — |,

compute P,._ , ( + ) J

EQ ( 4 - 2 7

)

t K

HK

--,-

compute !'r

EQ(4- 25)

Reasonab ! e n e sslChecks

.

..- .j

compute k'

EOT 4- 26)

Update

Estimate

h:'^ x

I:ompute X,, ( + ) i Reasonableness \-»m>

Checks

;'K-1

x r (-)—K

setK

->

h(K)compute X v f -

)

EQf4- 24)

FIGURE 4.3 DISCRETE KALMAN FILTER INFORMATION FLOW DIAGRAM

75

the application of the state transition matrix and t]

forcing terms

.

X(K)C-) = I>(K-1)/

X(K-1H + )+ U(K-.l) (4.33)

iy the definition (4.29), solving for P(K)(-)

P(K)(-) = <?(K-1)P(K-1) ( + )i(K-l)T

+ Q(K-l) (4.59)

Figure 4.2 illustrates the above developed equations in

block diagram form which contains a system model, measure-

ment process and the Kalman filter to be implemented, and

Figure 4.5 shows a simplified computer flow diagram of the

discrete Kalman filter.

Equations (4.55) through (4.59) constitute the recursive

formulas for implementing the extended Kalman filter

algorithm. The process is initializing by providing values

x(0)(-) and p(0)(-) . In order for the extended Kalman

filter to be simulated, the next section will describe the

initialization of the Kalman filter.

E. INITIALIZATION OF THE KALMAN FILTER

The one objective of this work is to test the perform-

ance of BTT missile control system having a Kalman filter as

an optimal estimator as a function of different measurement

vectors. All values for initialization of former worker in

this work (Ref. 14) will be used for computer simulation.

76

1 • I nit iali zing the Error Cov ari ance Matrix , P

Let a , a and a define the standard deviati np v a

of relative position estimate, of relative velocity esti-

mate, and of target acceleration estimate. The initial

values of error covari ance matrix p , will be chosen as

follows. The first six elements of the initial state

vector are determined by

UO) = X(0) + V(0)

Where the measurement vector X is the true measurement

vector X plus V caused by measurement uncertainties.

The expected mean square of initial covariance matrix

components. With the assumption that the initial

covariance matrix is diagonal with the diagonal elements

reflecting the uncertainties associated with the initial

state vector estimates, the components of the covariance

matrix are

E[V1

] = E[V2

Z] = E[V

5-]

E[V/J = E[V, ] = E[V/]v

The last three elements of the state vector, acceleration

components, are set to zeros

x7

= x8

= x9

-

2. Initializing the Measurement Noise Variance Matrix R

Measurement errors tor the active seeker

assumed to be due to thermal noise, gimbal angle

error, environmental noise, and glint. The values of one

sigma errors in each measurement will be used in present

study. Because in the case of angle measurement, the envi-

ronmental noise is proportional to the square of range to

target and glint errors the wander of the apparent target

centroid as seen by the seeker as varing inversely with

range to target. The errors caused bv noise are generallv

a highly complex function of the target geometry, target

radar cross section, radar receiver, signal-to-noise ratio

and etc. For simplicity, in present study, the constant one

sigma errors (average values) in each measured vector are

assumed to initialize the Kalman filter. Referring to

Reference 15, the one sigma values of the average errors

for measurement of angle, angle rate, range and range rate

of the present active seeker are tabulated below.

(1) angle measurement errors

(2) angle rate measurement errors

(5) range measurement errors

(4) range rate measurement errors

Before the simulation runs is undertaken, it will be

briefly discussed the manner in which the Kalman filter on

board the missile is initialized at the start of the

0.15 -0 .6 deg

0.5 -2.0 deg/sec

5 meters

6 m/sec

79

engagement based on sensor data from the launch aircra :t

.

The launch aircraft's sensors are assumed capable of leter-

mining , within some prescribed accuracy, the relative

position and relative velocity of the target with respect

to the missile at the instant of launch. This information

is sued to initialize the first six elements of the filter's

state vector. However, the launch aircraft is assumed to

be able to provide no information regarding target

acceleration. Thus the three elements of the initial state

vector are set to zero, finally, the complete list of

nominal parameters for initialization is provided in Table

4.1.

F. EVALUATION OF THE OPTIMAL ESTIMATOR PERFORMANCE

The performance of the state estimator with two-

measurement vectors (case 1) , with four-measurement vectors

(case 2) , and with six-measurement vectors (case 5) , was

simulated to evaluate the effect of the possible implementa-

tion of the more measurement sensors on the bank-to-turn

missile for typical scenario (tail chase engagement case).

The error covariance and the Kalman gain components

selected, and the states estimated, were computed as shown

in Figure 4.4 through Figure 4.30. The control laws imple-

mented with the estimator also were tested to evaluate the

performance of the control system of the BTT missile as

shown in Figures 4.51 through 4.55.

30

TABLE 4-1

NOMINAL PARAMETERS

3 s cn c

2 3 r -

3 cr.s :

: :

-

T j • 3

a6'

Gq 1 . 1

e,u4 J . 5 1 : :/ .

•

; . 5

3 J. -. .-.

]

t r r

d or. : s i g m = s j.a r.t

ifir. 5-: r i '. •. • i. j. - _

6 n i s s

1

1 s i c 1 1 rats

risna ccr. start

- i r " e en s t a r.t

vi:- i =si 1 ;

- oraxi rc urn

normal 2cc.

9 missile iraxisnua)

re II rats

10 filler frocess n c is

p a ra in = t = i

a target r m s ace.

c tare?-: .-nan uver

tan d width

11 ccnticll^r cost

function d a r ?. sir- 1 ' :

T -. .{

a

o . m / s c

.) . 5

J . 3 o C

20 7' s

0.3 - 2.0

z .

6.23 r a a /s 5 c 1 - 1

^ g ' s

1/2 s~c

i «;nq:.t en a

o n

0.003254 si

e

1 . 2 -j 7. 5 :• c

81

Figures 4.4 to 4.12 represent the variations of t

error covariances of each state variable. Figures 4.

through 4.7 show the three covariances for the relative

eositions. i.e., P, n , P~ , P,, . Only the covariance' 1122 33

of the two measurement case varies substantially. Thus

one would expect that the gain components G-, -, , G?

, , G--.

would show only variations in the two measurement case as

are shown in Figures 4.15 throueh 4.15. The trend noted in

the set of graphs (4.4, 4.5, 4.5) and in the corresponding

Kalman gains is also observed in the other covariance

components. Thus the covariance components for the velocity

(Figures 4.7, 4.S and 4.9) should larger variation is

reflecting in the corresponding Kalman gains of 4-16, 4-17,

4-18. The covariances of the acceleration components show

a similar development but there appears to be less variation

in a. and a„_ acceleration covariances. In the case ofty tz

a t the variation of the Kalman oains are restricted to atz °

much narrower range. The error covariance and the Kalman

gain components in all three cases eventually converge to

very small values. Figure 4.22 through Figure 4.30 show the

results of the state estimations of the three estimators.

Figures 4.22, 4.25 and 4.24 show the variations of the

estimated relative position (R , R , R ). Figures 4.25,

4.26 and 4.27, the variations of the estimated relative

velocity (V ,, V , V ). Figures 4.28, 4.29 and 4.50 theJ rx ' rv j-z

variations of the estimated target acceleration (a. ,

a , a ) . The trends of the curves representing the

estimated states are similar except for Figure 4.25 which

shows a wide variation in the estimated V component ofi A.

the twr o measurement case from the true state values.

Generally, the estimated states of the estimator with

two- and four-measurement are close to the true states,

in spite of the wide variation of error covariance compo-

nents of two-measurement case. The estimator with six-

measurement vectors generates the estimated states which

show almost same characteristics as the estimator with

four-measurement vectors, but usually underestimated the

states in comparison with the generated states of the

estimator with four-measurement vectors. Figure 4.32 and

Figure 4.54 show the curves of the control commands com-

puted using the estimated states. The results of the six-

measurement case have a wholly different form from the

curves representing the computed control commands in the

other cases. The normal acceleration commands computed

using the estimted states with the s ix -measurement vectors

initially have smaller values, but finally reach maximum

value as shown in Figure 4.52. Also as shown in Figure

4.54, the roll rate commands run from the minimum limit to

the maximum limit. The large variations of the both control

commands of case 5 during a short time interval result in

83

IRIL-CMRSl ENGAGEMENT CRSE

LEGEND

A - 2-MEHSUREMENT VECTOR CRSE

D - 4-MEfiSUREMENT VECTOR CRSE

O - B-NERSUREMENT VECTOR CRSE

Q.O 1.0 2.0 3.0

Time ( sec 1

4.0 5.0

FIGURE 4-4. ERROR GQVRRIRNCE COMPONENT CP n '

VS TIME

84

TRIL-CHRSE ENGAGEMENT CR:

CO

o -i

a-

CM

Q_

ce: lo-

ci;-

en

o

DLO'

o -1

a

LEGEND

A - 2-MEflSUREMENT VECTOR CASE

D - 4-MEflSUREMEMT VECTOR CfiSE

O - S-MEFISUREMENT VECTOR CfiSE A *\

A \A \i k

t i1 1

1 r

a. a 1.0

—j—i—i—i—i—|—i—i—-—i—

j

2.0 3.0 4.0

Time ( sec J

5.0

FIGURE 4-5. ERROR COVRRIflNCE COMPONENT (P 22'

VS TIME

85

7RIL-CHRSE ENGAGEMENT Cf

a

LEGEND

A - 2-MERSUREMENT VECTOR CASE

- 4-MERSUREMENT VECTOR CRSE #\O - 6-MERSUREMENT VECTOR CRSE / \

i r1 1 1

i i 11 r I r

2,0 3.0 4.0

Time ( sec 1

zizz^mzcB-5.0

FIGURE 4-5. ERROR CQVRRIRNCE COMPONENT tP^]

86 VS TIME

TRIL-CHR5E ENGAGEMENT OR:

LEj'ENO

A - 2-MERSUREMENT VECTOR CRSE

CTGR CrSE

O - B-MEHSUREMENT VECTOR CRSE

~i 1 1 1

1

1 1 1 r

1

1 1 1 1

1

r

CO 1.0 2.0 3.0

XLme

(

sec

4.0 5.0

FIGURE 4-7. ERROR CQVRRIflNCE COMPONENTS,

VS TIME

TAIL-CHASE ENGAGEMENT CRSE

CD-t

LEGEND

A - 2-:-'L"SUREMEN7 VECTOR CRSE

„V\.

-MEASUREMENT VECTOR CRSE J?\

2.0 3.0

Time ( sec

]

4.0

FIGURE 4-3. ERROR COVRRIflNCE COMPONENT

88

55J

VS TIME

TRIL-CHR5E ENGAGEMENT CflSt

aa.

LEGEND

A - 2-flER-SUREMENT VECTOR CRSE

D - ^-MEASUREMENT VECTOR CRSE

G - 5-MERSUREMENT VECTOR CRSE

~»1 r-

4.0

-i 1

1

1 r

5.0q.o 1.0 2.G 3.0

Tims ( sec

]

FIGURE 4-9. ERROR GOVRRIRNCE CQMPQNENTC' 66

J

VS TIME

89

THIL-CHRSl ENGAGEMENT CRS1

LEGEND

A - 2-MEflSUREKENT VECTOR CF.EE

C - ^-MEASUREMENT VECTCR CR5E

O - 6-MEflSUREMENT VECTOR CASE

FIGURE 4-10. ERROR CQVRRIflNCE COMPONENT (P77

VS TIME

90

TRIL-CrlfiSE ENGRGEMENT CH3E

LEGEND

A - 2-MERSUREMENT VECTOR CHSE

C - 4-MEfiSUREMENT VECTOR Zrz:

C = 6-MEnSUREKENT VECTOR CH3:

FIGURE 4-11. ERROR CQVRRIRNCE COMPONENT (P^)

V5 TIME

91

TRIL-CHR5E ENGAGEMENT CRSE

\

a-U3

enen

Q_

E—

CDen

a

CD

a.

LEGEND

A - 2-MEflSUREMENT VECTOR CRSE

D - 4-nEflSUREriENT VECTOR CH5E

—

|

1 I I 1

1

:1 1 1

1

1 1 1 1

1

1 1

2.0 3.0 4.0 5.

Q

Time ( sec

)

0.0 1.0

FIGURE 4-12. ERROR C0VRRIRNCE COMPONENT (Pgg

)

VS TIME

92

IRIL-CHflSE ENGAGEMENT OR:

a 4

LEGEND

A - 2-MERSUREKENT VECTOR CRS;

- 4-MERSUREMENT VECTOR CASE

- 6-MERSUREMENT 'ItlZTZR CRSc

FIGURE 4-13. KRLMRN GRIN COMPONENT ( G, 1) V5 TIME

TRIL-CHF13E ENGRGEMENT CRSE

oCM

aCO

l:oe:;e

a - 2-mersurement vector cpsi

- 4-mersurement vector crse

o - 6-mersurement vector crse

A

,*^~AsEHnssss^EE rsHis^r^'rrerrrrr—rcr

(XI

C3 3 '

a 1

aCO-CO

aoCO

I

a.

a

1.0 2.0

Time ( sec

A

—,1 1 1 1

11 1 1 r—|

r

3.0 1.0 5.0

FIGURE; 4-14. KRLMRN GRIN COMPONENT (G21

) VS TIME

94

ItiIL-CHRSl ENGRGEMENT CH5;

aa

er.C3in

LEGEND

A - 2-MERSUREMENT VECTOR CRSE

D - 4-MEHSUREMENT VECTOR CASE

O - 6-MERSUREMENT VECTOR CRSE h£ &

a.o 1.0

T 1 1 1 p2.0 3.0 4.

TLme( sec 1

7r

5.0

FIGURE 4-15. KRLMflN GRIN COMPONENT (

G

311 VS TIME

95

TfllL-CHRSE ENGAGEMENT CRSE

a c00

ato'

LEGEND

A - 2- N!ERSiJRE*ZNT VECTOR CRSE

D - 4-MERSUREMENT VECTOR CRSE

O - 6-MEflSUREKENT VECTOR CRSE

a.

a

-i 1 1 1]

1 1 1 11 i r—i r

1.0 2.0 3.0 4.0

Time ( sec 1

i—'—

'

5.0

FIGURE 4-16. KflLMflN GRIN COMPONENT (

G

41) VS TIME

96

TRIL-CHR5E ENGAGEMENT CnSE

m 6

CD

oain

oo-ai

oO -I

inrv

LEGEND

- 4-flEHSUREMENT VECTOR CP.SE

- 6-MERSUREMENT VECTOR CASE

i

a.

a

L.O 2.0 3.0

Tome ( sac

4.0 5.0

FIGURE 4-17. KRLMRN GRIN COMPONENT (G51

J VS TIME

97

TAIL-CHASE rwcfi

i T

LEGEND

A - 2~MEftSUREf1ENT VECTOR CASE

- i-MERSUREMENT VECTOR CRSE

O - 6-MEHSUREMENT VECTOR CRSE

CO 1.0 2.0 3.0

Time ( sec J

4.0 5.0

FIGURE 4-L8. KALMAN GAIN COMPONENT [GB1

3 VS TIME

98

iril-chrse engagement crss

aCM

I

oa

a - 2-MEflsUREMENT vector chse '

D - 4-MERSURET1ENT VECTOR CRSE

O - 6-MERSUROIENT VECTOR CRSE

i r r r -r 1 1 1 11

r

a.

a

1.0 2.0 3.0

TLme ( sec

4.0 5.0

FIGURE 4-19. KRLMflN GRIN CGriP ia71

V5 TIME

99

en nlC3

GD'CD

oo

LEGEND

A - 2-tfEfiSUREMENT VECTOR CfiSE

D - 4-flERSUREMENT VECTOR C,g5l V

G - 6-flEHSUREMENT VECTOR CRSE V/*

i1—'"

a.a

—t

1

r } r——

•

r 1r

1r r f r*

1.0 2.0 3.0

Tims

(

sgc )

-i—|—i—

-

5.04.0

riGURE 4-20. KRLMfiN GAIN COMPONENT (

G

fl1] V5 TIME

100

TKTI -PUOCr PTMPRPF'MP\IT HOC!iniL Urtnou LlNbnbQllLiNi Oho!

:^%

' r- p >i n

A - 2-MERSUREMENT VECTOR CASE

D - 4-tlEHSUREPOT VECTOR CfiSE

O - 6-MEP.SUREMENT VECTOR CASE

-i 1 1 1 1 11 r i

1 1 r -i 1 11

5.02.0 3.0

Tome I sec

]

4.0

FIGURE 4-21. KRLMRN GRIN COMPONENT (

G

gi) VS TIME

101

the worst miss distance, 15.4m. The control commands he

other cases behave almost like characteristics repres

that the control commands are increasing smoothly :

>

zero to its maximum control command, after two or three

seconds then slowly decreasing to converging values. The

relatively small variations of the both control commands

during a short time interval comparing with the control

commands of case 5 result in 9.7, miss distance in case 1

and 8.7m, miss distance in case 2. From above results, the

estimator with four-measurement vectors gives the best

result, the estimator with six-measurement vectors the

worst result and the estimator with two-measurement vectors

almost same result as the of four-measurement case comparing

with the result of the estimator with six-measurement vectors

In original measurement equation, for case 1, the

missile has two-measurement vectors which give the informa-

tion about the relative angles, 6 n and \b n , was assumed.& ' R R

These two -measurement vectors are non- linear functions of

three relative distance components, X-,, X_ , X_ . For case 2,

the missile has four-measurement vectors which give the

information about above two angles plus relative range and

time rate change of relative range which are also non- linear

functions of the components of six state vectors, X-, , X-,,

X, , X,, X r and X, . For case 3, eventhough the missile3 4 5 6

has six-measurement sensors are assumed, the measured

102

TRIL-CMRSl engagement crse

om 'ag.00

-̂ -.

-~f-.- ---

- -

S**i• -.-.

- -I

-^•*-I- -IO -

- hf-i

O -1 ^CD - '~t

oo-^

a

CO

CD

©

,Q 4

Oa

om

od.

LEGEND

+ - TRUE Rx

A- ESTIMATED RX12:

C- ESTIMATED Rx(4:

G - ESTIMATED \:S

*—-* T"t

n r r-

Q.a 1.0 2.0 3.0

Time ( sec

)

4.0 5.0

FIGURE 4-22. STRTE ESTIMRTE: RxVS TIME

10

IRIL-CHRSE ENGRGEMENT CR3E

-rv* rt

2^wb4^

:j

^̂ 4\ a

Gk + Gj

. 7?iRv

A - ESTIMATED RY (2)

D- ESTIMATED RY(4)

O - ESTIMATED RyCS)

©©

2.0 3.0

TLme ( sec

]

4.0 5.

a

FIGURE 4-23. SIHIl E5TIMRTE: RyVS TIME

104

TRIL-C^ C SE ENGRGEMENT CR5E

c —CO

oa'

o

CO

LCD

CD

Q^

I

LEGEND

A- ESTIMRTED Rr (2)

D - EJ37r^F.TEID R (-)

- ESTIMATED rO:S;

jmgmPMA,

24? ^sg^

iiiiiii ii iuii

CJCM

I i i r

a.

a

L.a 2.0 3.0

TLme ( sec

4.0 5.0

FIGURE 4-24. STRTE ESTIMATE: RzVS TIME

10 5

TRIL-CHRSE ENGAGEMENT CR!

2.0 3.0

TLme ( sec

FIGURE 4-25. STATE ESTIMRTE; VRX

75 TIME

106

IRIL.-CHRSE ENGRGEMENT Ci

R"1

CD-

C3 H

a j

4T -

MX2 -

to

co gff Ac -i A rreP

£=4'

COI

mi

C3 —

|

I

° 1

\ 1

% %V«L

tv

%

LEGEND4-- TRUE V

Ry

A- EISTlr-TETD VRr C2)

n- E5T: Mn7Ej 7^(4)

O- ESTIMATED Vov (SJ

irtTtS

4-

a?/

-i^oj

~~1—'—'—'—'—1—'—'

—

'—'—1

—

'

2.0 3.0 4.0

TLme ( sec

)

0.0 1.0 5.0

RIGURE 4-25. S7RIE ESTIMRTE: VRY

VS TINE

107

TRIL-CHRSE ENGAGEMENT C^zi.

tv.

o —

LCtoLNU

+ - TRUE VRZ

a - estimated v^ 2

D - ESTIMATED 4

- ESTIMATED V'RZ

5

-r-—

i

1 r-

1.0 2.0 3.0

Time [ sec

.0

FIGURE 4-27. STATE ESTIMRTE: V 9T VS TI

108

inlL~uhh;Dt tiNbribQriLiNi unbL

^3Si

3.0

ILme i sec

)

FIGURE 4-28. STRTE ESTIMRTE:: au VS TIMI

109

IRIL-CHnSE ENGRGEMENT C

a -

onmi

o-^ J

i

_r

—

r

j —

i

:

t—'—I—

Q.Q 1-0

O - E2TIMRTEH all 6

3

~1—'—'—'—'—i—'

—

r

2.0 3.0

Time ( sec

)

-t—i—|

—

t •i

5.04.0

FIGURE 4-29. SIRTE ESTIMATE: an V5 TIME

110

IrliL LnhbL LiNbnbLl iLlNl brio!

LEGENDi <r

A - ESTIMATED 0^(2]- ESTIMATED o^M)

C - ESTIMRTED o~,i5;

2.0 3.0

Time ( sec

FIGURl 4-3G. sthtt TIMRTE: Gtz VS TIME

111

TRIL-CHR3E ENGAGEMENT Gn:

A -

-

O -

LEGEND

true: time-to-go

estimated time-t0-g0(2)

estimated time-t0-gch4]

estimated time-ta-gq(6j

2.0 3.0

Time ( sec

)

FIGURE 4-31. ESTIMRTED TIME-TQ-GQ VS TIME

112

TRIL-GHRS: ENGAGEMENT GnSI

(0

CDQ-

uv

LEGEND

A - 2- s"E."SU-;rE

MENT CP5E!

- 4-MEflSUREMENT CRSE

O - B-MEnSURENENT CRSE

r^=gi

<£V?ibd

/ |L/\4

I f

d

5

fam .

aa. .

ad. 1

1r—r——

i

[

11 1 1 : 1 1 p

p-—

j

1 r-

Q.O 1.0 2.0 3.0

TLme ( sec

]

1.0 5.0

FIGURG 4-32. NGRKfiL RCGGLERnllGN COMMAND VS TIME

113

lniL~LnnbL :NGflGD1E:NT CF I JL_,

(M

OL.O-

00 acr,^-*

£

ain'

LEGEND

A - 2-NERSUREMENT 2"5^

D - 4-MEF15UREMENT CRSE

- B-MEaSUREMENT CRSE

rg^gS*5

a. ^13—i

1 1i 1

1 1 1 1]

1' ' '

1 1 r-

Q.C 1.0 2.0 3.0

Time ( sec J

—I1 1

1

1

i [ 1r

4.0 5.0

FIGURE 4-33. MISSILE NORMAL LOADING VS TIME

114

tril-chrse: engrgcment cr:

LE3ENQ

C - 4-MERSUREMENT CRSE

2.0 3.0

TLme ( sec

FIGURE 4-34. ROLL RRIE: CQMMRND VS Til

115

TRIL-CKRSE ENGRGEMENT CRSS

LEGEND

a - 2-mef1surement crse

- 4-mersljre v e:n7 case

o - 6-mefisurement crse

1

a .

ft

^^Esjgf] A

a.

a

L.O 2.0 3.0

Time ( sec

]

4.0 5.0

FIGURE 4-35. MISSILE ROLL RAIL VS TIME

116

tqti -puqcj: FNQflRFMFMT pci n 1 1_. ijnnjL i . in •.:nul i il. i n i uno c.

LEGEND

n - 4—MERSI REMENT CP c:r

q „ g-f-lEfisuREMENT CP.S"

iume l sec J

FIGURE 4-35. MISSILE BfiNK RNGLE VS TIME

117

vectors using the six-measurement sensors contain

state vectors which are exactly same kinds and n u

the measurable state vectors using the four measure

sensors in case 2. As the angle rates, 5} ^ 5

included to the estimator as the measurement vectors in

case 5, the both angle rates are non-linear functions of

only six-state vectors, X., X , X-., X , X- and X .

The results of the estimator with four-measurement vectors

and with six-measurement vectors show the almost same

variations of the error covariance and the Kalman gain com-

ponents of the six measurement case as of the four measure-

ment case as shown in Figures 4.4 through 4.21.

The Kalman filter in this work is a first order filter.

Thus the components of the Jacobean matrix contain only the

first order term, i.e., all higher order terms are neglected

As the order of the system is increased one would expect a

more complex system to response differently. The above

reasons affect to the underestimation of the states of six

measurement case as shown in Figures 4.22 through 4.30.

From the above considerations it can be concluded that as

the increased number of measurement vectors is implemented

to the extended Kalman filter algorithm, the result of the

system may not be enhanced up to the expected degree due to

the increased complexity of the system. The system with the

118

first order filter as a optimal estimator may con

to make less desirable results due ro the effec / .

lectins of the higher order terms in the filter alsoritl

119

v - PERFORMAN C E H VA LUATION OF THE CONTROL 'EM

This section will discuss the performance of t le control

system implemented in a bank-to-turn missile for each scen-

ario. The maximum normal acceleration, the maximum roll

rate and the time constant are the key parameters which

reflect the missile capabilities, the noises to measured

vectors are the parameters of the environmental effect, and

auxiliary variables of some importance are then total engage-

ment time and missile velocity. The mean miss distance

determined from the 50 Monti Carlo runs was used as a

performance standard for the estimators of tuo- (case 1) ,

four- (case 2) and six-measurement vectors (case 3) on each

scenario. The simulation results are shown in Figures 5.1

through 5.19. The sensitivities to variations of the para-

meters will be discussed below. Since the simulation results

show essentially the same characteristics for the tail -chase

engagement (scenario 1) and for the head-on engagement

(scenario 2). The performance on these two scenarios will

be discussed together for each parameter. The side-

approach engagement (scenario 5) will be discussed sepa-

rately.

Figures 5.1 and 5.2 show the miss distance variation as

a function of missile maximum normal acceleration. For

scenario 1, case 1 shows a mean miss distance of 9 . Sm at

120

o J2

s =

CD

"

-lJ

CO

._>

en £

OH

oo

5.0

:^IL-CHR5l engagement cf .

x=0 .05 { 1/ssc ) aR=3(m] CT&=6(m/sec

= o- =. 1 5 ( deq ] =0.5 sec;

a. "a," . D i jeq/ssc ; '^vc-^S.l:9 -J i IhA

Enaaqement Tlme=5(sec

Missile Velocity = 100G (rn/sec)

LE3END

A= 2-MEflSUREMENT VEC1 i

a = 4- m e=sl:r: v e:>

;: vectors

O - 6-MERSUREMENT VECTORS

10.0 15.0

aaox

l 9 S

20.0 25.0

FIGURE 5-1 MISSILE MAXIMUM ACCELERATION VS

MEAN MISS DISTANCE

121

"J!

%i

£ =>

CD

0)

c

*JCO

. J>

OCO •

CO°

CO

o

10.0

:rd-on engagement case

x-0.05(l/sec) crR=3(m] cr£=6(m/sec

=.15 (dea) t=0.5(sec)

= cr,= . b ( deq/sec )

1flX= 6 . 28 ( red/sac

)

LnGccement iLme=^lsecJ

MLssLLe VeLocLtu = IOCS (^/sac

LEGEND

A = 2-MEflSUREMENT VECTORS

n - 4-^E c SE c::-'E',: vectors

O = 6-MEflSUREMENT VECTORS

15.0 20.0 25.0

°.«<S'«'

FIGURE 5-2. MISSILE MRXIMUM ACCELERATION VS

MEAN MISS: DISTANCE

122

above log's, case 2.9m at above 15g's and case 5, L5.

above 15g's. Each case also shows that as ti axi tit

of the normal acceleration decreases, the mean ;

:

in case 1 is Jess sensitive than in the other two cases.

For scenario 2, the result of case 1 shows the minimum mean

miss distance is 10m and the result of case 2 is 17m at

a = 21cr r

s respectively, the result of case 3 is 52m atmax a v ' '

a = 17g's. The trends of the curves show that the meanm ax ^

miss distances in case 1 and case 2 vary in same manner,

which rises sharply as the maximum acceleration decreases.

On the other hand, the mean miss distances in case 5, vary

smoothlv above a = 13g's . One may conclude from themax °

analysis as follows: for scenario 1, the mean miss distance

can be decreased since more information on the state vectors

can be obtained from the increased measurement vectors in

estimator for low maximum acceleration. In order to get the

mean miss distance below 20m for all three cases, one needs

a of over llg's. For scenario 2, as the measurement statemax ° '

vectors are increased in estimator, the mean miss distance

does not decrease, possibly due to the effect of neglecting

the higher order terms in the extended filter. As the mag-

nitude of the values of the missile and target geometry

components becomes larger, the effect of the linearization

becomes more pronounced. a = over 19g's is required tov max ° 1

123

to obtain the mean miss distance below 15m in case 1 a

case 2. In case 5, the mean miss distance does not

below 20m.

Figures 5.5 and 5.4 the variations of the mean miss

distances as missile maximum roil rate varies. [n scenario

1, for case 1 and case 2, a mean miss distance of below 10m

can be obtained when P is greater than 5 rad/sec.m a x °

case 5, one has a mean miss distance of below 16m with

P greater than 6 rad/sec. The trend of the mean missmax to

distances is almost constant over certain range in values

of P . The variation of the mean miss distance belowmax

this range is a tenth of the magnitude comparing with the

variation of the mean miss distances as a function of

maximum acceleration. It is concluded that for scenario 1,

the mean miss distance in case 1 and case 2 is not effected

by the roll rate if the roll rate is above 5 rad/sec. In

case 5, higher roll rate limit can reduce the mean miss

distance. For scenario 2, the mean miss distances of case

1 and case 2 are like the trends for scenario 1. However,

the mean miss distance of case 5 has a minimum of 26m at

P =4 rad/sec, then increases up to 36m then decreasesmax ' r

slightly. This trend ma}- be due to the effect of the roll

rate lag, which is caused by the increased maximum roll

rate and by the underest imat ions of the state variables in

case 3. The other characteristics, i.e., the mean miss

distance decreases with increased maximum roll rate limit

124

TriIL-GH-51 ENG :NT Cr.5l

CD

o-

£ a'— -

0)

acc*}en

.->Q

ca"

x-0.05(l/SEC) crR-3(m) o-

R=6(m/sec)

c =ct =. 15 (dec ) "C

=Q .5 (sec)9 3

a ,=.5(daa/3ec) ^ qy=Û lq s J

Encaaement TLme*"5(sec)

Missile VeLocLtu = 1000(m/sec

legend

£ = 2-ÊR5UREîNT VECTORS- 4-MERSUREMENT VECTORS

0= 6-MERSUREMENT /ECT

-] 1 1 11

[

1 1 1 1

1

1 1 1 1

1

1 1 1•' ,

•:

11

1

1

0.0 2.0 4.0 5.0 8.0 10.0

P (rad/sec)max

FIGURE 5-3. MISSILE MAXIMUM ROLL RnTE VS

MEAN MISS DISTANCE

125

HERD-ON ENGRGEMENT CRSJ

CD

CH

(D

O

CO

W 'I

CO S

aC3-I

x-0.05( 1/SEC) c- D=3[m] o-p-5(m/s8c

».-*•" lo( sec

rpx= ^u Lq s J

Enaaqemeni Tlme=*5(sec]

MLssLle VeLocltu - lOCGlm/sec)

-A-

LEGEND

A - 2-M I=5urtc:Mc::iT /ECTGRS— (i.MracjipfMF; — UPPTP, 1? ;l_l t i I Li i^J[\u iCiN 1 < i iL _O - 5-MEnSUREflENT VECTCRS

•A- -A

:.0 5.0

pM>cd/sec]a.t 10.

o

figure 5-4. missile: mrximum ROLL RRTE vs

MEAN MISS DISTANCE

126

and for scenario ! , case 2 shows less near miss di lc

than case Z, but for scenario 2, the opposite tren s,

can be explained with the same reasons as those of Eirst

ana Is /is

.

The variation o f the mean miss distances with the time

cosntant as a parameter is shown in Figures 5.5 and 5.6.

For both scenarios, one has a flat plateau for low values of

the time constant and a sharp rise with increasing calues of

the time constant, i.e., for scenario 1, Figure 5.5 shows

the curves of the mean miss distances slightly increase up

to the time constant of 0.9 second, for scenario 2, the

curves up to 0.5 second are plateau, then for both scenarios

the curves rise sharply. For scenario 1, the mean miss

distance of case 2 is less than that of case 1 up to 0.5

second time lag, but above 0.5 second lag, the results show

an opposite trend. Also over this time lag, the curves of

case 3 for both scenarios are not sensitive to the variation

of parameter up to the time constant of 0.7 second for

scenario 1, 0.9 second for scenario 2. The general trend of

the curves, i.e., as the time constant becomes longer, the

miss distance becomes larger, is due to slow system response

From the figures showing the results, the time lag of less

than 0.5 second is necessary to obtain the mean miss

distance of the required order of magnitude.

Figures 5.7 and 5.8 show the variations of the mean miss

distances as a function of one sigma angle noise. For

1

ItiiL LritioL LNbribLriLlNi Ul

5i

a-

a -t

to

0)°J

CO

._> a -

a d^CO

CO

._>

2Z C -I

cD-

od-

oo

x=*0.05 ( 1/secl CT=3(m) crb

=6(m/sec]

a = j = »15(dea/ a -^Cta's)3 w ,11 CX w

<7.=

c-, = .5 (dec/sec J PveY= 5 . 28 (rad/ssi

.nqaqement 1 Lme=b Isec

J

Le Velocity - 1000(m/sec

LEGEND

A - Z-^ 5;^:'-:::.: VECTORS

a- 4-MEflSUREMENT vectors

* 6_?1EF!SI!REMENT /ECTORS

~i 1 r-

0.0 C.2 0.4 0.5 0.8

Time Constcnt (sec

1.0 1.2

C T nRE] 5-5. MISSILE TIME CONSTANT VS

MEAN MISS DISTANCE

128

HEflD-ON ENGAGEMENT CRSE

CDo

'

in

3

s^_, —

fOa>

or-

c-J

.->

o.3

03 CD0)

.-) CM

CD

a

x-O .05 ( 1/sec 1 crR=3(m) or^=6(m/sec]

a -a =.15(dea) a =20(q'3

LEGEND

A - 2-MEflSUREMENT VECTORS

D - 4-MEflSUREflENT VECTORS

O - o-MERSUREMENT VECT: : 5

i

'

0.2—I—i—i—:—i—|—r—i—i—:—;—i—i—i

—

;—|—

r

0.4 0.6 C.3 1.0

Time Constcnt (sec)

0.0 1.2 1.4

FIGURE 5-5. MISSILE TIME C0NSTRNT V5

MEAN MISS DISTANCE

129

TRIL-CHRSE ENGAGEMENT CR3E

o

o•

ic -

co

.-Ja

"3I

CO

03

oCD

t r

0.0

x=u . 05 ( 1/sec j o-o=3 1 m ) cr£~6(m/sec]

-c-0.5(sec) a =20(a's)

er.-<x.-.5( deq/sec ) P M^.=5 .23!rac/sec)

Eaaaemsn.L TLme = Sisecl

Velocity - 1000(m/sec)

LEGEND

A = 2-MEflSUREMENI VECTORS= 4-flEHSUREMENT VECTORS

O = B-MERSUREMENT VECTORS

-j—r—i1 1

1 r

0.2 0.3

"'—1

—

r

0.4—I—1—1—1—1—1—

1

0.5 0.60.1

,rzLe Lrror la =a— 9 3)

de<

FIGURE 5-7 ONE SIGMR RNGLE ERROR VS

MEAN MISS DISTANCE

130

HERD-ON engagement case

o03

£ O

03

CI

oCO

co ?-.J

= 0„05[ I/sec J crR=3(m] a6=5-m/sec]

^=Q.5i sec

)

a =2maxnu f n 'ol

4 Vj O ^

.5 ( daq/sec

)

DMfiX

=° ' 28 ( rac

Eqaaerr ent i l m 9 = q̂ ( sec

)

1 iLSSlo e Ve LccLty = 1000 (m/ sec

L. Li CL.I U

A - 2-MER5UREMENT VECTORS

a = 4-mersurement vectors

- 6-ilERSUREMENT /ECT0R5

—1—

'

0.50.0 0.1 0.2

RngLe0.3

Error (

a

0.4

(deg

0.5

FIGURE 5-8. ONE SIGMA ANGLE ERROR VS

MEAN MISS DISTANCE

131

scenario 1, as shown in Figure 5.", the m . am mi

tance is obtai for the six measurement cas<

miss distance being . Lest for the two me ;

case. Above about the angle error of 0.5 degree the mi:

distance is not a strong function of the an L< error. For

small error values, i.e., less than (K2 degree, the curves

seem to be approaching each other. the scenario 2, the

miss distance is a strong function of the angle error. For

the two scenarios (Figure 5.7 and 5.8), the two measurement

case lias similar results up to an error of 0.4 degree. All

cases show a strong rise in the miss distance with increas-

ing error. The four measurement case seems pasticularly

sensitive

.

The variation of the miss distances as a function of

missile velocity are shown in Figures 5.9 and 5.10. During

simulation the initial stand off distance was increased to

keep the engagement time the same. One important aspect of

Figure 5.9 is that represents an opposite trend of the

vatiations in the mean miss distance to the general trend

of the variation in the mean miss distance. Under the

missile velocity of 700m/s, the mean miss distance is less

than 10m for case 3, from 15m to 12.5m for case 2, from 15.7m

to 14.5m for case 1. However, the curves for scenario 2

show the usual trend, i.e., the mean miss distance of case 1

is the smallest and that of case 5 is the largest among the

mean miss distances of all three cases at a value of the

132

73 Ti -pupep FMQfiQF ,ri

CD-CD

X=0.05( 1/secl CT =3i^] oo=5(m/sec'R

t=u . b I sec J

; a _=.15(deq -20(q's)

a- = a. = .bideq/8 -J

sec J

-fix=6.28(rad/sec)

Lnaaqsmeni Ui"ne = ~ ( sec ^

:.

o

§°-.J CD-

00M

CO

CO

._)

21 aL

a. !

co-

co

LEGEND

a = 2-mersurement vectors

a = 4- me-e ,j^::/e

,

.t /ect jr

O = 6-MERSUREMENT VECTORS

400.0 6G0.0 800.0 1000.0

MlssLle Velocity (m/sec)

—

i

1200.0

FIGURE 5-9. MISSILE VELOCITY VS

MEAN MISS DISTANCE

153

-e c:D-on en : : ,

2::

x»0.05(l/sec

CD

gs'Hco

CD '

CO

._)QCOM

|

CO 1

.-J

o-i

d-i

oCD.

o-R=Jlm

R

=201 .:'-

Enccqement TLme=5(sec)

/.=«ec 0 . 5 ( se

o- =cr^= .5 ( deg/sec ) PMRY=6.28(

cr.=

a-.= .15(dsq] a

dud'

EGENO

A - 2-MERSUREMENT VECTORS

- 4-MERSUREilENT SECTORS

O - 6-MERSUREMENT V

,A-

A"•A- -A-

-A

1—i—>

—

—i—i—'—i—'—i—i—i

—

r"—i—'—r—'

—

4G0.0 6C0.0 3C0.G 1C00.0

Miss Lie Velocliu (m/secJ

1200.

o

FIGURE 5-10. MISSILE VELOCITY VS

MEAN HISS DISTANCE

154

given parameter. For both scenarios, the general trend

the curves slowly increases as the missile velocit;

increases, except for one curve of case 1. It la-

rized the foregoing, considerations as follows: As the

values of the missile and target geometry components become

smaller, i.e., relatively less effect of neglecting the

higher order terms in the first order filter and less

contamination of the measured vectors to noises, the mean

miss distance can be enhanced with the estimator of the six-

measurement vectors. As the estimator algorithm becomes

simple, the trend of the mean miss distances is less sensi-

tive to the target geometry.

Figures 5.11 and 5.12 represent the effect on the mean

miss distance of the total engagement time. For simulation

the initial stand off distance was increased to keep the

constant missile velocity. The general behavior of the miss

distances of all three cases slowly decrease as the engage-

ment time increases. For case 1, the mean miss distance

slowly decreases or is almost constant at below 10m. In the

other two cases, there is more variation in the miss distance

but the general trend is still downward with increasing time

to go. Note that as time to go increases for a given T,

one has more characteristic time interval available reflected

in T /t . The mean miss distance therefore should be lessgo

as the total engagement time increases. This fact is veri-

fied from both figures

.

155

TfllL-CHRSF] ENG " ZNT Cfi:

a

'J2

O -tm

o

s=.

COM

.J

CO

co

o

ad-

a -

a

x=0.05 t 1/sec J cR= 3(rn ] o-^=6(m/sec) t=0. 5 (sec)

<j ( deq/sec

)

3 ~ ^'

a.=a.= . 15 ( deq) a =2Q(q's)e p ^j max ^

Missile Velocltu = 1000(m/sec]

LEGEND

A = 2-MERSURENENT /EC7QRS

D- 4-MERSUREMENT VECTORS

= B-MERSUREMENT VEC

-i r

4.0 7.0 10.0

Engagement Time (sec)

FIGURE 5-11. :ngrgement time vs

1EAN MISS DISTANCE

136

h: cu-QN ENGAGEMENT CnSE!

.=0.051 1/sec) aR=3(rn] a^=5(m/sec] *-0.5(sec)

a "o- = „5(dea/sec) PMR =6.28 rad/sec]9 w

15(deq) on i >

-<c.U I Q s

Missile Velocity = 1 000 Im/sec

J

£ aa •

LEGEND

a = 2-mehsurement /ECTORS

= 1-MERSUREMENT VECTORS

O = 6-MER5UREMEMT VECTORS

oc

._;oa)°.

a-

oa.

4.0 6.0 3.G

Enqaaament Time (sec)

10.0 12.0

I CURE 5-12, ENGAGEMENT TIME V5

MEAN TOS DISTANCE

13

Figures 5.13 through 5.18 >], the variation

in miss distances of ill three cases for side-

engagement (scenario 3) as a function : each para r.

The large miss distances resulting '

- m this, r:

d 3

motivated its separate consideration, General trends of

curves are similar to the other scenarios but with higher

miss distance. This scenario is a strong test of the

system. In this scenario, the control laws are subject to

the error due to the limitation of the small angle assump-

tion and the higher maximum acceleration capability is

required due to the large variation of the PZC-miss in

Y- and Z-direction. Thus there are some unexpected trends.

Figure 5.15 shows the variation of the mean miss

distances for each case as the a increases. Althoughmax &

the missile velocity is less than that of scenarios 1 and 2,

the maximum allowable acceleration is increased. As shown

in figure, the mean miss distance for case 1 is 11m,

for case 2 is 10m, for case 5 is 15.5m only at a = 23g's- max

The figure shows the miss distance variations in this

scenario are more sensitive to the a than in the othermax

two scenarios. The mean miss distance of below 40m can be

obtained at a = over 19g's, i.e., more a ismax & max

required than in scenarios 1 and 2. This trend of the mean

miss distances can be explained with the larger variation of

138

the state vector in Y-di r .tion ivhi :] .-;'

Le In control :o land on .

As shown in ; Lgure 5.14, the variation?

distances of case 1 and 2 as a function of paconstant above P =4 rad/sec. That of case 3, ho rev,max ' '

continuously decreases up to P =6 rad/sec, thee, imax

almost constant. For scenario 3 one needs a i urn re 1 .

rate of about 5 or 6 rad/sec to have an effective BTT

missile, which is higher than in scenarios 1 and 2.

Figure 5.15 shows the variations of the mean miss dist

as a function of time constant. Up to the time constant of

0.7 second, the mean miss distance is less 20m. For all

three cases, comparing with the results for scenarios 1 and

2, the figure shows the curves of the mean miss distances

are less sensitive in this scenario. The target accelera-

tion vector lies in same direction as the missile velocity

as shown in Figure 5.4. Because the components of the

relative velocity are major components in computing the

control commands, the more correct control commands mav be

computed due to the small uncertainty in M,p

and Mbz

components. This results in smooth variation of the mean

miss distance curve. The trend of the miss distances as

the one sigma angle error increases, shows some different

characteristics from the trend for scenarios 1 and 2 in

Figure 5.16. The mean miss distances for all cases

increase continuously as the noise input increases up to th<

139

OC3

W._)

OC*P

._>to

oO-l

aa.

15.0

S;CE-PPPPC C CP ENGAGEMENT CRSE

x-0.05(l/sec) crR-3(m) ^-6 -/sec)

a -a =.15(deq] t=0.5i sec]

o-, = 7.= . b ( deg/ssc J P M c>,= S . 28 ( rca'/ssc

Engagement Tlme~b(sec)

MLssLle Velccltu = 600(m/s8c)

LESEND

A = 2-MEflSUREMENT :::3R5

a - 4-mersurement vect .

O - 6-MERSUREMENT VECTORS

20.0

: lasmax w

25.0

FIGURE 5-13. MISSILE MAXIMUM ACCELERATION VS

MEAN MISS DISTANCE

140

CD-CO

CD-I

CD

CJ

-J CD-

CO

to o 1

•-> CD5~ CO

J

aCD

O.Q

SIDE-flPPROnCH ENGAGEMENT CRSE

x=Q.G5il/SEC) <jR=3(m) dA=5(m/sec

a =c- =

. 15 (deal ^=0.5(sec]9 v-'

^=.5 (cleg/sec) aMflX

=23

Inqaaement TLme^olsecl

ssLLe VeLccLtL; = 600(m/sec)

LEGEND

A - 2-nERSUREM£NT VECTORS

D = 4-MEflSUREilENT VECTORS

O - 6-tlEflSUREMENT VECTORS

2.0 4.0 6.0 8.0 10.0

P (rad/sec)max

FIGURE 5-14. MISSILE MAXIMUM ROLL RATE VS

MEAN MISS DISTANCE

141

SIDE-APPROACH ENGAGEMENT CASE

x=0 .Ob i i/sec ] CTD=3(m] o-a=6 ( m/sec ]

ao'

o!*>

CD

l_

->CO

._)

o3

to "-

0') n._> fM

ar.

a = a =. 15 ( deq

9 ^

cr.=

cr .= .5 I ^9g/s9!9 rD »J

max 3

=6.28(rad/sec)

Enaacement T :

^rne="-D ( sac )

Missile Velocity = 600(m/sec]

LEG L'

a= 2-!iEnsu^£ v£: 1 : VECTORS

D- 4-HEaSUREMENT VECTORS

O = 6-MERSUREMENT VECTORS

")—i—i—i—i—|—i—i—i—i—|—i—i—i—i—1—i—i—i—i—|—i—i—i—i—|—i—i—i—:—]—i—i—i—!—|—r~

0.0 0.2 0.4 0.6 o'.S 1.0 1.2 1.4

Time Constant (sec)

FIGURE 5-15. MISSILE TIME CONSTANT VS

EAN MISS DISTANCE

SIDE-APPROACH ENGAGEMENT Cf E

aCD

'

o33

CO

CD

c

*->

03

._>

CDa

-

w Si

CD-CM

x=G.05( 1/sec) CTR="3(m] c^5 : :n/s8c!

or.-o-," = 5 (dea/sec 1 PMqY= 6 . 28 ( rad/sec )

t=0.5( sec

)

a ~23(q'<flax _/

Eqaqemerrt Time = 5 i sec

)

Missile Velccllu = 600(m/sec)

LEGEND

a- 2-riEfi5UREtfENT vec

D = 4-MERSUREMENT VECTORS

o- 5-:-'::;:.-i:"::.: vectors

t 11

1

1 r

a. a o.i

-i—|—

i

0.50.2 0.3

Angle Error I o- =c ) (deql

a . 4 0.5

9

FIGURE 5-16= ONE SIGMA ANGLE ERROR VS

MEAN MISS DISTANCE

14

side-approach engagement cpse

o

— d jonCD J

IIa

a

od

0.05(l/sec] aR=3(m) a

ft-6(m/ssc] -t=0.5isec

3 5)

sg/sec j P..,cX= 5 .. 22 ( rad/sec )

1 ^ f H eg i =23 i q's )max -3

Engagement TL-ne = b l sec 1

LEGEND

A- 2-MERSUREMENT VICTORS

- 4-MERSUREMENT /ECTORS

- 6-MERSUREMENT VECTORS

4C0.0

"i r-

600. 80G.0

MLssLLe Velocity (rn/sec

10C0.0

FIGURE 5-17, MISSILE. VELOCITY VS

MEAN MISS DISTANCE

144

SIDE-RPPROflCH ENGAGEMENT GPSE

X-0. 05(1 /sec] crR=3(m] a^6(m/sec) -=0.5 (seel

, ,—r!D

U)

u

D-Jw

._)Qa

CO r-i

W a._> *r

6.0 8.0

Engagement Time (sec;

10.0

FIGURE 5-18 ENGAGEMENT TIME VS

MEAN MISS DISTANCE

145

o-

C3-1to

oto

CD o

ca

CO

CO fCO

OHCM

O-t

x=0 .05 ( 1/ssc J crR=3(m] cr^=6(m/sec J t-0.5(s8c"]

a "a -. 5( dea/sec ) P Vq Y "5 .23 ( rad/sec j

3 w I In A

a- = CT-= . 15 ( dea ) EnqGqement Tlrne =5 ( sec )

LEGEND

A - T^iL-CHnSE: ENGAGEMENT

C - HERD-QN ENGAGEMENT

D- SIDE- RPPRORCH ENRGHGEMENT

0.3 Q.a

RngLe Rats Error (dsg/sec

FIGURE 5-19. ONE SIGMR ANGLE RRTE ERROR VS

MEAN MISS DISTANCE

146

one sigma error of about 0.4 degree. \bove the one sigma

error of ''>. \ 8 degree , case 1 d then rises :

case 2 continues decreasing slowly, case 5 increase

continuously. The variations of the mean miss distances in

scenario 1 are almost plateau and in scenario 2 are almo I

constant up to the one sigma angle error of about 0.4 degree

then rises sharply as shown in Figures 5.7 and 5.S. rhis

reason can be deduced from the fact that as the one sigma

angle error becomes large, the same magnitude of the two

state vectors in different direction, i.e., relative

distance in X-direction and relative in Y-direction, having

the same order of magnitude ma}' be contaminated to the noise,

on the other hand, the other scenarios, only the components

in the X-direction being relatively large are contaminated

to the noise. From this fact, it can be concluded that the

missile in a side -engagement case is more sensitive to the

angle error than the other scenario.

Figure 5.17 shows the variations of the mean miss

distances as the missile velocity increases as a parameter

(i.e., holding the total engagement time of 5 seconds and

changing the relative position of the target). Up to the

missile velocity of 700m/s the curves of the mean miss

distances are plateau, after that velocity the curves rise

sharply. Figure 5.S shows the variations of the mean miss

147

distances as a function of the total engagement time I.e.,

holding the constant velocities of the missile

target ana changing the relative position of the target;

scenario) .

figure 5. IS shows the mean miss distances rise sharply

over the engagement time of 6 seconds. This is somewhat

unexpected and may be an artifact of the implementation

procedure. In other scenarios the mean miss distance as a

function of the missile velocity increases slightly as the

missile velocity increases and that as a function of the

total engagement time decreases slowly with increased

engagement time. from above analysis one has a conclusion

that the computed control commands should converge in the

final time as the computed commands of the two- four-

measurement case shown in Figure 3.32 and Figure 5.54 to

obtain the low mean miss distance. However, it can be

expected in this scenario that the computed control commands

do not converge possibly due to the violation of the small

angle assumption in deriving the equations of the control

commands (I.e., in this scenario the M. component is

large in Equation (5.7), that causes the large mean miss

distances). However, in the case of relatively small magni-

tude of the missile and target geometry components, the

error caused by the violation of small angle assumption is

relatively small comparing with the errors caused by the/ it o >

contaminated of the measured vectors to noises,

14 8

limitations of the control inputs, an I he estimation

tSic state vectors using first -order filter. Ph ;

verified with the results chat the mean miss d:

11m for case 1, 10m for case 2, 14m for case 5 at the

missile velocity of 600m/s and the engagement time of 5

seconds, as shown in Figures 5.17 and 5.18.

The final Figure 5.19 shows the variations of the .

miss distances for case 5 of all three scenarios as a func-

tion of the one sigma angle rate error. Ihe result repre-

sents that the variations of the mean miss distances due to

increased input values of one sigma angle rate error are

almost constant for scenarios 1 and 2. For scenario 5, the

mean miss distance varies at constant rate of inclination.

This slope of the curve is relatively small comparing with

the slopes of the mean miss distance curves as the other

parameters change. From this analysis, it may be concluded

that the exposure of the measured vectors to the angle rate

error noise is neg legible in the case of present study.

149

VI . CONCLUSIONS

A simple but: effective biased guidance lav; and t ree

extended Kalman filter equations have been deri ed from :

optimal control theory for a bank- to- turn missile with zero-

lag and with 0.5 second time lag autopilots for pitch accel-

eration and roll state.

The equations of motion are linearized around the present

orientation of the missile with the assumption of small

future roll angles. During computer simulation, the optimal

control laws and estimators were tested separately invoking

the separation theorem. In the simulation of the system

three optimal estimators were used, i.e., estimator

with two measure:.! cut vectors, with four-measurement vectors

and with six-measurement vectors. The system has been eval-

uated for three scenarios, tail chase engagement, head on

engagement and side approach engagement.

From the analyses of the simulation results in Chapters

3, 4 and 5, it can be concluded as follows: The control law

was successfully simulated for a hypothetical bank-to-turn

missile with pitch acceleration and roll rate autopilot

within the imposed limits on three scenarios. Miss distances

with zero- lag were negligible for all three scenarios

(below 0.5m) . For the time lag case one had miss distance

of 5.9m for scenario 1, 0.8m for scenario 2 and 32.9m, with

150

0.5 second lag and with arnnx= 20g's. By Li irposi ig a

higher maximum g of 2 3 on the scenario 3, the miss

distance was reduced to below 5m miss distance.

expect a high g requirement for a side engagement situa

tion as in the case of scenario 5. It is evident that the

missile's tracking ability against the target is very

sensitive to the missile and target geometry, the missile's

manuevering capability is also a very important factor.

Optimal control laws applied to the side approach engage-

ment place a severe strain on the system from the viewpoint

of the small angle approximation. Increasing the number of

measurements did not result in increased performance in the

sense of smaller miss distances for the missile except in

isolated cases. This may be due to the increased com-

plexity of the system as the number of measurements was

augmented. Further, in this stud}' only first order extended

filters were implemented. It is possible that better

performance could be obtained at the expense of increased

computational load, by using a second order extended Kalman

filter.

151

APPENDIX A: PROGRAM LISTING

This appendix provides listing of the computer prog]

used in the present study. Only one program for the six

measurement case is provided. Except for a few differences

in the subroutine filter the programs for the A./o other

cases are identical to the program included here.

152

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10 CCKTI2JUEVIC C0NTISU3

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120 ccsrikuE13C CCSTINU3

B (1,1E ( 2 , 2R t3,3a (4,4S 15,5a ( c , o

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154

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Q (5,6) =Q22C (9,9) = *33

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Q (1,4) =Q (1,4) + F12C(2,d)=QU,d)+F1.<:2 (3,6) =Q (3,o) +F12

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140 CCXTINUEISC CCSTI2J02

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SIART CASH LOC£

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SCICE: LCCE

C5EINZ IXTEGSATICS VARIABLES

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156

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V5Z=VT2-V£Z

aix=Axxo*EExs i-c~ft i 2 = A £ : 3*33 X? (-C : 1*1)4TZ=ATZ0*CEX2 (-CTZ*T)

X2 ( 1) = BXX2(2) =B'iX2 13) =32X 2 { 4 ) = V ?. ;(

X2 i 5) = VBYX2 (6) =vh :

X2 (7) = ATXX 2 ( 8 ) = A T X

X2 <9) = ATZ

1GC=-{5X*VHX+EX*VEY + EZ*VHZ) / (7SX**2+ Vra**2+'/aZ**2)

£C£;J. MEASD3H:iE5I VEC1C:

S 1 = 33 2ST (SX**2+SY** 2)S = D SQ3 1 ( £ 2 ** 2 -£':'- * 2 + SZ* * 2

)

3ET= (3 X*VEX+S i*VRY*SZ*VaZ)/STHTS=-DATAN2 (SZ,S1)P£IS=DA2AN2 ( 3 : .£>)T£TSD= ( (-CSIM (TKIS)*CCOS (PS IS) *SX*VB X) - (3 3 IN (THIS) *DSIN(PS

*RY)+ (DCOE (TH1SJ>S2*V5Z) ) / (S*S)ES13L= ( f-ESIN (Sill) *Si*VRX) + (LCC3 (PS IS) *ST*VaY)

) / (5*S* 333 3BEIS=3BDI£D= (SX*VHX+ £ i*SE Y+SZ* VSZ) /S

Z I 1) =IfiIS 03LE [il (N))*SZ1Z 1 2) - r HT £ E + C E L E ( « 2 (

v

.i )

- 3 3 2

Z ( 3 ) =? 31 5 +-D 3 1 E i . 2 ;'..))* S Z 3

13) * 3 ": • Y

(THIS) )

Z (4.2(S))*SZ5Z (5) = 2EIS fCsL: (

Z (6) =8DI£E+D3LE (h2 (il) ) *SZ6

IHITIAIIZE SIA1E T«£C10?.

IF (N-GI. 1) 30 TO 750

CALL LHCSK(ISEEE,Sn,-6,l 1 )

X |1)=RX+E ELE ( JiK ( 1) ) *SrOSX (2) =R X+ C ELS ('.'.>

( 2 ) ) * SPOSX (2)=aZ+LEL£ (Vifl (3 )

* S20S

75 i

7. |4)=V HX+3313 (Si -. (U)i ( 5 ) = 7 B X + C 3 L £ ("»

'* (5X (6) =VHZ+CDLE | vi • (6X (7) =0 .

X (8) = 5.X(S}=0.CCKIINUE

SVELv : i.

7 3L

LI. Li: L ,

-

C12 = DC0£ ilHlS) *0:C13=-DSIK [TH .. !

XIC3 = D 11*X(1)+C12*X(2)+D1 : X(3)

157

GENERATE GuI^ANCE CCf.MASDS

CALL 3 DICE (X,IGC2 ,AC ,PC, CI)

IF (AC. GT.+AaAX) AC = A•

If (AC. LT. -AMIS) JC--.''.:!';

IF (PC GT.+t HA :<) PC=E KA !

I? |EC. LT . -::::.:;; [C=-EKA <

I? ITSPIN.IE- J. ) r:"=^CI r ( TA C C . L E . . ) A fi = A C

T0?.2 VABIA31 ? C a E LO -

P H ID=?

AAA [MlEE2 («)B A N K (

N

TG1 (>!)

1 G 2 ( :i

)

1 <N =

U2 (N) =

U3 (.,) =

U« (») =

:: 5 i.s )

=

U6(J! =07 |!li =Gt(.N) =U9 |N) =

El ()•') =

E2 (N) =E3<N) =E-4 ;:,} =

H5 (N) =• [K]

2 j (N) =

EeiM) =

2SJS) =E 1 1=05P22=?5E33=DSP<44 = QoE55=0SP6c=0SE77=05PS8=DSE99=D5PI (N) =

£ 2 |!l) =

? 3t!,;i

=

EU (N) =

to (N) =

FS |N> =PS (V) -G1 1

G21G3 1

G 4 i ( :<

)

g 5 i ( y

)

LIKE 3 If

HI*57.3

= AC= £ C

J"= ehid

=TGC=TGC2ax?.Y

7 3 Y

v a za: :

lis!

X (6)

lie7

!

x I s

)

Catsai i

;ai; -

:

2

(1

(K.

.:• I ( P (

6

C31 I? (7caifP(d

P1 1i -

P33P<4 a

F55?6c?7 7

F8 8?99=gai:j (1,i=GAIN (2,1= GAIN (3,

-

=GAIN [4 , 1

=GAIN (5,

'

LINE E;

T . XEE I

S

=*5

, 1) )

;r);

fT? MCASJ D

K = CTHTD=?HT*57.3?£ID = ? SI^57 .3PHID=PHI*57WHITE [5, 1C]WHITE (6 , .0)HHITE (6, 2C)

) GC TC 250

"i

3I,£,XLCS -IGG-TG02 .AC.AH. EC, EM- PiT, EX,SY,SZ, /BX, 7SS,7RZ, ATX,AIY,ATZI,X(1) ,J (2) ,1 (3) ,.1(4) ,X{5) ,X{6) ,X('

rzX(7) ,x{8) ,

-,: ;j)

158

102025C

30C3C•4

3dC

U0!

50

iSi:: (6, 2C) r,E11,E22,?33,=>i |,E55,?6 6,£77,E88FC5MAT(/1CF13.3)F C £ a AT ( 1 f 1 3 . 2 )

CONTINUE

15 IL.LT .LESI NT)L^J

TC 3 50

BBIT2(6,3C) :

DC 300 1= 1,9KB I IE (6, i*C) 3AIK <I # 1) ,SAIS<I , 1) ,GAI'J U, 1) , GAI>i (I, 1) , GAIN (I, 1)

FORMAT (//S7X, 'GAI2>F C E

v:, : [52X,6F13.5)

CCJiTIIi US

93

K A IH (T =, £ 3 . . ) V)

CHECK FOB I S A 1 I C K

STATE VECTOR)

IF (SDT .GE.G. .CS .:gc -LZ . . 3) GC TC 900IF |IGO. GT.iT) GC IC UOOCT=TGOisicp= i

CCNTI?IUE

INTEGBAIE TRAJECTORY ONE 5TEr 8HEAD

CALL INTEG (X,£)

IF (ISTOP .GT.O) GC TC 9 10IF (N. SE. :. .MX1 CC IC 300

EXECUTE SALMAN Fill Ft (UPCATi

CSII FILTER (X,E,E,Z,GAIN)

GC 13 70C

IEESIHAL EBINT A .\ C E LOT

CCSTIttUEt=y{1)CX=Y (1 1) -2(5)DX=I (12 -Y 6CZ=i|1j| -Y (7)CM=DSQi?T |EX**2+CY**2+DZ** 2)XfISS (III)=DX

ZMISS (II1)=DZCMISS (iiiHdmSRIIE ( 6 , £ C) I , E X , C Y , Z Z , D M .1 IIFCfiMAT (// SX, ( I= ' ,F 10. J , 5 .< ,

' DX = ' ,F 10 .3. 5 X ,' D Y= ' ,71 J.3,5X

ic,

, DZ= , ,F1C.3,5X, , CISS = , ,F10.3,5X,'CASE=' / I7)

IF (IPLOT-NE. 1) GC TO 950

CALL P LO TG (TT , E 1 , N , 1 , 1 , 1 ,

*,11,0„.0.,0.,C.,5.,5.)CALL ?LOTG(ri,U1 / N,2,1,Q,

* ,11,3.,0.,().,C..3. I5.)CALL PLCTG (TT <Ez,S,1,1, 1,

*,11,O.,0.,0.,C.,3.,5.)CALL PLCTG (TT . Uz. S , 2 , 1, ,

* , 1 1 , . , . , J . , J . , d . , 5 .

)

CALL PLCTG iTT. E3 ,N ,1 ,1,1 ,

*. 11,.;. ,0..3.,fi.,3. r 5.)CALL PLOTG (TT . C3 . N ,2 , 1 , ,

* , 11 , . ,0.,G.,0.,5.,5.)CALL PLOTG (TI , E4 ,H , 1 , 1, 1 ,

*, 16,0. , C-,0. ,C.,5. ,5.)CALL PLCTG (TT ,0U,S, 2, 1, 0,

»,16,0. r 0- r J-,C.,D.,5.)CALL PLOTG(ri .E5 1 >, 1,1, 1 ,

*,16,0-,G..O.,C..D.,5.1 „CALL PLCTG (TT . C5 , .\ , 2 ,1 , 3 ,

* , 1 b , . ,0.,0.,0.,o.,5.)CALL PL01G(TT,E6 ,N',1 ,1, 1 ,

TIME |SEC)

TIBE (SEC)

lit I [2F.C)

TIME (SZC)

II HI (SZC)

TIKE (SEC)

TIME (SZC)

TIME (SZC)

TIME (SEC)

TIME (SZC)

TIME (SZC)

',10,'

',10,'

1, 1 C ,

'

', 10,'

',10,'

',10,'

',10,'

',10,'

',10,'

',13,'

',10,'

RX (METERS)

SX (METERS)

Tl (METERS)

RY (METERS)

?.Z (METERS)

RZ (METERS)

VRX (METSHS/SEC) '

VRX (M3TEHS/SEC) '

VRY C-iZTZRS/SEC) '

VRY (MSTS2S/SEC) '

VRZ (METERS/SEC) '

159

*,16,0.,C.,0.,C.,5.,5.)CALI ?LOIG (?T . 06 t S, I , 1, 0,

call ? L a r g i t r . :• 1

.

:. , i , i , i

,

*, 13,0- r 0.,J.,C.,3.,5.)C.iU ?L01G(TT,C7,N', 2,1,0,

*,19,0.,0.,0.,0.,5.,5-)CALL ?LOTG (Tl . •

fK,1 ,1, 1 ,

* , 1 9 , . , . , . , fl . , 5 . , 5 .

)

CALL PLJIG (II , Cc ,:. , - , I, J ,

*,1S,0.,0.,0.,i.,,5.,5.)CALL ? L ,: 1 G (rT,£9 z N, 1,1,1,

*,1S,iJ.,3-,U-,C-<D., .

C A Li. -: _c: l- i'i .'.',:,.,!,. ,

*,19,0. ,0.,0.,C.,i.,5.)CALL ?LC lG CI, 2 , «,1 ,1,1 ,

*,0.,0. ,0.,0. ,5..=.)CALL PL^IG (T'I,E2 ,N,1 ,1, 1 ,

*,C.,0.,0.,u-,5.-'d.)CALL ?LCTG(TT , Ei,!.', 1,1, 1,

*,C-,0.f0.,0. ,5. ,5.)

c a L ls

? l c i g ( i

:

,: y , .. , i , i , l

,

*,C.,0. ,0.,0. ,:.,5.)call ?l^; 3(T r

r .t5 £::,i ,1, 1

,

*,0.,0.,0.,U.,5.,b.)call plotg (ii ,E6,5,1 ,1,1 ,

*,0.,0.,0.,0.,5.,S.)CALL PI,G3G(T'I,E7,S,1,1,1,

*f 0. f 0.,0.,0. f S..£.)CALL PLO TG (T T , f 6 - S , 1 , 1 , 1 ,

* , Yj . , ^ • , O . , o . , _ . f ^ »i

call ; l : : g , r : . : r , ;. , i , i , i

,

CALL ? LC T G (T T , A ; C <i , 1 , 1 , 1

* ,' Mosa m a_c_l:: >1ic (.1/

CALL PLC! G (T T , 1 I. li , 11 , 2 , 1 ,

* ,' NOaa . L ACCELEi ftllCJi (."!/

call ?lc:g ,n ,eec ,:; ,•

. , , !

*, BCLL 52TE I BAL/SEC) ', 19CALL PLOTG (TT , E23 ,N ,2,1 ,0

*, 'EXLL BAIL [fiAE/S£C)',19CALL ?LCIG (TT , E ? .. .- .N ,1, 1,

*, 'LANK A SGLE |CEG) ', 16,0.

CALL ?lcig(::,ig:,::, - ,i,i*, 'TIME IC GO I SEC) ' ,1b, 5.CALL ?L01G(TT ,TG1 ,N,2,1 .0

*

.

'TIMS TC GO (SEC) ' , 16, 0.CALL ?LC1G(TI,G1 1 ,N,1.1 / 1

*, 'G (2, 2) ' ,6,0. ,0.,0. ,0. , 5CALL ?L01G (II ,G2 1 ,li , 1.1 , 1

* ,' G < 5

,

2 ) ',6,G.,C.,0.,0.,5CALL PLOT G (TT,G3 1,ii, 1,1, 1

*, 'G (8,2 ' ,b,0. ,C. ,C. .0. ,5.,D.)CALL ?LOTG(TT,G1 1 ,N , 1,1, 1, 'TIKE

* , ' G (3 , 1) ' ,5 , C . , C . ,0. ,0- , 5 . ,5 . )

ciai iszz)

TIME |S EC)

PI 81 (SEC)

TI3E (SEC)

TIKE (SEC)

i : ;•:•

naiis •

i i

riKE (SEC)

TIKE (SEC)

TI3E (SEC)

TIME (SEC)

TIKE

ri:

(SEC)

(SEC)

IS EC)

(SEC)

(slc;

I '

i

s c * *

:

(SEC',30

XML I '-J

I J. .. I U-vEC^*,) • ,30TIM

. , C . , J . ,

"TI2E (SEC0. ,C.,0.,0, ' i

i

:' ei

J . , C . , . , 5 .

'TIKE (SEC. , . , . , 5

' PIKE (GLC) . , c . , d . , 5

1 ™ - '« - / I 7 £

l TI?.E (SEC,5.)rial (sec

CALL PLCTG(TT,G5 1 ,"J, 1,1 , 1 ,' TIME**.\9S6 l}11'<?.0. ,C..Q. ,0. ,S.,5.)CALL PLOT (0. ,U . ,?99)

(SEC

1 C , ' 73 z ; )'

10, ' \:/.v

' ' :-

10,' ATX ( iETEHS/SEC **2)

'

TO, ' ATY {::£IL=S/JEC~ J'.0 '

10, ' '."•

(! ET2aS/SEC**2) '

10, ' ATZ ,'.:: 3S/ sec i ^2) •

. -.i: (as i!Eas/ss ;**2)

10 , 'SU2T (P11)

IC, »SGHT(P22)

10, 'SQRT(?33) ' ,9

ic , • sqst (?uu) «

,

10, SQST (?55) ' ,

' C

' S QST ( P7 7 ) »

,

'SCjHI(?S8) '

•SQSI (i?9y) •

10,

10,

1C,

IC,

,10

; i

:

/-

ir3.), 10

, 10

, 10

, 10

, 10

. , . , . , 5 .

* f J • t \j » j b

5.)

,5. !

95 C CCHTINCE

i CCaPQTE KISS DISIANCE STATISTICS

17 (HCASS.EQ. 1) GC TC 975CALL STAT (XaiSS , :!C AS E ,XB AS,XR£S)CALL SIA1 iiJlLL;,'.GA5:: ,:-;^, fSCS)CALL STAT (2.V;ISS, '..CAS E,Z3A5, 3KS)CaLLSIAl(Dai£S,NCASE,DBAo,DSl!S)S SITE (b, ij_

H£ITE(6j6C) XE8E,XEKS«HITE(6,"/C) YE A 5, 'iR S3«EITE(6»60) ZEJ»B,ZEKSWRI1E(6,SC1 DEAE.E5KS

55 FCE-MAT (//13X, • KISS CISTA^iCL STATISTICS'/)6C FCEMAT (/Si,' X-CCMECSENT' ,5X, ' XEAN=' , SB. 2,5X,

'

aHS=',F8.2)

160

7C Foaa.s r f/sx, 1 y-cc^ic:;30 .- ' • : !/ 5X, ' :-^:c::^-:. IT

FCEaAX (/5.(, ' :c:.;: . :i .

'97 5 CCSTINUESIC?

, 3X, »KEAI, . ,

' »£ANa 2 , 5 .<

,

?.as = '

161

50

103

SCBHOOTISfi jCIDE (X ,TGO, AC, =C,CT)I2EIICIT 6EAL j

E [1-r ,

DIMENSION (9)CCEaOK/TfiSEE/ 2;i,a2;,A23,A31,a32,A3 3,31,3;

= 9.3

MZ = X (S)TGC=- ( FX*VHX + FY 'FY + EZIF (TGO.Li.O.) GC IC

) / ( VEX >*2 + VSY* '2 t-V3Z**2)

F0= (CT*TGC-T. +E2XE (-CT*TGC) )/ |(

ZEKX= 3 X+ V EX*~GC+ •

ZEKY=P. K+ liEX*TGC + AIY*FQZEJiZ=RZ+VEZ*TGC + ATZ* EO-. 5 '1* (TGC

Z EZ = A3 1 *ZE ax + A 3 2*Z£*Y + A 3 3 *Z E3 ZP 1 = 3.* TGC/ (3.*E1 *-IGO**3)ac=-?i *5

DEHI=DA?AS2 (ZEY, -ZF.Z)F 2 = 7 . * AC * AC *TGC* ( 1GO **3 * 3 .* S 1 )

F3= (AC*AC*TGO**5) (c.3.*32)PC=F2*DPEI/F3GC TO IOCCONTINUEAC=0.p C = .

CCS1INUE

£5C

162

55

60

65

7C

75

3UE3C3IHELIC

COMMONG = : .

:iur -,: =

THTC=0FSI0=CVC=DEI.: 2=t ...

•£C = u^I

7 XC = V0r::= : j

7ZC=-V

riSS E?iT2G (EZIK,a:IT 52 A L * € (i! - H , 0- Z

)

lOi DJSIM ( 16) , EEOUT { 16) , Zl 1 ( 1 )

,• r»C/ SC,EC,ATXO, EKO ,aizo ,ctx

16

MS

, cx

;

r OT , .

* C £1 N ( I H 1 )

(?S 10)

NUHE2ICAI IN TEGS A TIC N ( •08 ESI RUi ; j - SOI

DCT=D1HIESIv = ::

PHIA .

v.=

EH=I? (

1* I

DSCDECCZCDECDECegoZ iCD ECDECDECEECDEC- v p

;

dec:DEC!eec<i? (

IE (

DCIF I

if!DE1DE2

GC"USECE2USEGCUSECZ2USEGCUS?USEDEI

75 .11=1,4SIN (1)

= p£,IN I :

Mil)

IS?I A CUT (

UT(UT(UT (

UT

C. G

a:i) =

5!

(7)UT t 6

]T{9)DT(10)UT (11

1213141516INC.

la i .

I. J

1

.

: (A.I

'Ail*-Q*

' '. * D

V * D

A T X& r :

= AT

!f'»!

)=1

!=C SIN (1 5)=C E I S ( it)]

5 ItHI) -G*CCO£ ITHT)) /V!r: : ij /{v*ccos r... :

;

IIHT)1KT) *~C23 (?SI)IHT) *DSIN (PS

..:i

-ctx*?)XE <-CTY*T )

£XP(-CTZ*S)

75ai.* 1

.

21.

i=6TO1 = 3

HiTO1 = D

Eg.- ^

EC.= D;= D.E275EG= D5 E

75

rQ3= e«= 0.= J

.

GI.T. J

-;;G

2)4)i:<'(

CUT

(1.

; _ :_ i cr

: i n 5 ;- ( ? c - 1 .- ) /ts e i k

To =8 A IGC TC 60GC TC 65GC TC 7

I) »CT

c i = u .

TO1=31=<N(i:

UT (I) * 2- *CTE2 (I) + IS511*.25

i) *;.*ciI + CSE1

Dlfl) *C12 (I) +USE1)/b.D E 1 ( I ) + C S E

1

IF(TSPIN.IE.O„) DEIS |15)=?CIE ITACC. LZ.O . ) CEIN ( 16) =AC

TiiT=D2IN (2)P£I=DEIN <3)v=czi:i i-oVX=V*QCOE (THT) *DCCS (ESI)V 1= 7 * D CC S |T HT ) * CS IK (ESI)VZ=-V*DSI5 (THT)

165

B < 1] =D EI S j 1

-( JC + VXO*DT)

5 (2} = QEJ -1 Y C + V Y

B {2) -- - ( Z C i- V ZO *Cc ('») = ,' i- . xoE ( 5 ) = V { - 1 \

5 (6) =V2-

B (6) =0.2 |9)=0.RcTU] N

ESC

164

3

1001 5 C

i

. i ...

'

{ A , X , 9 , S

I 5 J 1 = 1 , i

XX (I) = AX |I)-e (J)CCKTINU2

iX IRAPGLAIE

CAL

EX2HAPGLA1E :CVASIAfiC2

CALL VMOLFF (?,A1,S,9,9,9 ,9,? AT, 9, 12)

CALL /MDLFF (A ,5*1,9,3,9 ,9,9, A EAT, ; ,I3)

DC 15 J 1=1,9CC 100 0=1,3? ? 1 1 . J ) = A £A T 1 1 .01 + C 1 1 . J )?? [I, J) = A£AT (I ,0CONTINUEcc :i xi:j ue

E C 8 a H >.:,z .i : X A 1 s I C ES

X1 = XX ( 1)( 2

)

x.; = xx;5):c u = x x ;

-•i

X5 = XX 5)X 6 = X X ( 6 i

D =DS0 3T (X1**2+X2**2+X3* .

.... .-/: (11**2+X2**2)E2=X1»XU *X2*XS *X3*X6E1=XU+X6E2=X5+ X6E3=XU-X5EU=D1*D1+2.*D*CC3=X1*X3D<J=X2*X3D 5 = ( X 1 * XC6=X2*E3D7=X1*E3

* (2. *E*C*E1*E1*E 1- (X1*X1*EH-X2*X2*£2) * (3 . *D 1*D 1 + D*D)» (2. * C * D * E 1 * C 1*S 2- ( X 1*X1*E1+X2*X2*E2) * (j.*D1*=0 1+D*C)1*E1+X2*X2*E2)* (D*D-X3*X3)

1 m *X 1*21-o* c *c 1 *i

-d*d*di*di+x2*x2*eiO

DC 250 1=1,6CC 200 0=1,9

1 1, J) =0.20 CCSTISUE25C ccxir.i us

H (1,1) =Xa (1,2) =xH (1,3 =-a < 2 , i ) = dK (2,2) =DK (2,3) =DH (2,4) =XH (2,3 = XH ( u, 6) =Da |3,1}=-H (3,2) =XH (<.,1) =CH (4,2) =£a (4,3) =(

a (u,5) =|

H (5, 1) = X

H (5,2) =X

1*X3/ |E1*C*D)2*X3/ (C1*E*C)C 1/ (D*C)2 / I i D * * 5 3) )

J))

2/ ((0**5) * (C1«/((C**3l*(C15/( D**5) *D1)1 *Sl *X3/ (C*C*D*D1

)

2*X2*X2/ (l*C*D*0 1)1*X3/ |E*D*C)X2/ (d*C1)1/ (D 1 *C1)fc/ ((E**3) MCI"'; ) )

// ( (0**3) * (C1**4J )

X1*X2*J3*E3)/((D**3-X1*X2)/ IC*E1*01)X 1 * X 2 ) / (

G ' D 1 - D !)

1/D

* |E1**2) )

16 5

5035C

:0

U5C

5CC

55C

50065C

H (5, 3) =X3/DH(c,1)=(C*0*XU-D2*X1)/(D*H<6,2)=(£*D*X5-E2*X2)/(C*D*DH 1 6 , 3 ) =

, E *

.

-.. |

I . • ) = A 1 / J

H (6,5) = X2/DU (6 , S) = X3/D

CO 350 1=1,9OC 30 j J=1,bHI II, J] =H |J,I)

" J :;

E

compute kalman saih matrix

CAII / ;iu iff ( ??,::, ,.:

, ;,o , j ,i ,?h:, ^ ,i-)

CALL VXUL5F (H ,£HI , 6 , S,6 ,6 ,9 ,H EET,6,I 5)

DC «50 1=1.6C C 4 G J = 1 , oi li, j) =;:;hi (i ,j) f s u ,0)CCKTIMUECCNXIbJ UE

INVERT Y » A T S

I

i

CALL LISV1F(X,6,6,XI!>7,0, *ORK,IER) '

CALL VflULFF {PHI, YIJiV ,9,5 , 6,9 ,£,G,9,I6)

UPDATE SI ATE VECICS

KH(1)=-DA1AS2<X3,E1)HH<2)=({X1*X1*X3*XU)+(X2*X2*X3*X5) + (E1*E1 :!=X3*

I / i 1 -

'

I

HK | j| =DA1AN2 (X2 , XI)HH(aj = ({-X1*X2*X4)+| i * X 2 'X5))/(G*D1*£1)HH (5) =CH S ( 6 ) = D 2 / C

DO 5 CO 1 = 1,6C2 (I)=Z (I)-HH (I)CONTINUE

call vhuiff (G , :

:

DC 550 1=1,9X (I)=XX (I) *-DX |I)

[SUE

,1,9,6,DX,9,I7)

UFEATE CCVARIAKCE

CALL VHUIFF(G,H,9,6,S,9,6 ,GH,9,I8)

DC £50 1=1,9CC 6 00 J =1,9P <I,Jj =DEI (ICCSTINOEC S TIN UE

i)-GH |l ,0)

CALL VMOIF? (F,EP,S,9,9,9 ,'

RF.TURHEKE

166

IJiFLIC

V=£AT*;. = c * : r

s 1 = DEXE2=C£X

V 11 = 7/V 1 2 = 7 /

m )= < ,'

I 2 2 -- / •

7 2i=V/V33 = V

Q 11 = 7 1

£12=V1; 15= ; 1

C22=V2Q23=V2C32=VJ

TIN; PI! C|

C,DT,f 1,F2,i 3,Q I 1 , * 12 ,Q 1 3 , J.II SEAI^E (A-H ,0-2)

*2

Ff-fl)F {

- 2 . * A

)

(C**3j

(C**2)

1 * [1.-S2+2.*A+ 1 2 . / 3 . ; * ( A * * 3 ) - 2 .-

2 * { H 2 + 1 . -.

•- I + 2 . * A * S 1 - 2 . * A +A * A

)

J* [ 1.-52- 2- *A* HI)'2* {t * • 1 - ;

- - - 2 + 2 . * \ )

13* |£2+1 .-2.*£1)3* [ 1.-52}

:1)

ml-EKE

l.+E1)/(C**2)-t 1)/C

167

IOC

20i

/*

/*

SCEaOOTIX I ST SI (X,M, AVGIMPLICIT 3 E .M J c

tA - .-

,0-

£ E JiS 10 S .((50)

sue=o.DC ICO 1=1,

M

so2=sa«+x (i)CCSTIN DESVG=5U 2/ 8K£ t) .".

i

£ C 2 20 1=1,3I)-.A vG) **2

C C :. : . I E

3 M S =0 £ S U iJ 2 / A li]

EETUfiNEND

.5YSIN OD *

16S

•

LIST OF REFERENCES

1 . uriam, :

. L , tics , IVi] ;,

19 71.

2 . Etkin , B . , Dynamics of Atmos ph eric Fligh t , W i 1 ey , 1

5. Garnell, P. and Last, D. J., Guided Weapon ControlSystems , Pergamon Press, 1977.

4. Ogata, K . , .'Modern Control Engineering, Prentice-Hall,Tl QC 197 .

5. Br>' son, A. E. and Ho, Y. C., Applied Optimal ControlSystems , Wiley- Interscience , 19 7 2.

6. Gelb, A., Applie d Optimal Estimation,MIT Press, 19

7. Jazwinski, A.H., Stochast ic Processes and Filter! /

1

.

Theory , Ac ademi c Press, 1 970.

8. Stallard, D. V. , An Approach to Optimal Guidance for a

Bank- to-Turn Missile Guidance Systems, Af atl-Tr- 7929

,

February 19 79.

9. Naval Surface Weapon Center Technical Report Tr-34 \ 5

,

Development of an Adaptive Kalman Targ et Trackin gFilter and Predictor for Fire Control Applications

,

Clark, B. L. , March 197 7.

10. Singer, R. A., "Estimating Optimal Tracking FilterPerformance for Manned Maneuvering Targets," IEEETransactions on Aerospace and Electronic Systems

,

Vol. Aes-6, no. 4, July 19 70.

11. Mcgehee, R. M. , "Bank-To -Turn Auto-Pilot Technology,"National Aeros pace and Electronics Conference , Vo 1 . 2

,

197S, pp. 683-6 9 6.

12. Mchee, R. M., "Bank-To -Turn Technology," AIAA Guidanceand Control Conference , August 6-8, 1979, pp. 415-422.

15. Howard, D . , P

r

edicting Targ et -Caused Er ror s in Rad a r

Measurements , Naval Research Laboratory, E as con, 1976

.

14. Ohlmeyer, E. J., Appli catio n o f Optimal Estimation o nControl Con cepts to a Bank-To-Turn Missile, Naval Post'graduate School, Master Thesis, October 1982.

169

[NITIAL riON LIST

M . ('

Defense Technical Information Center 2

Cameron StationAlexandria, Virginia 2 2 5]';

Library, Code 0142 2

Naval Postgraduate SchoolMon t e rev , Cali forn i a 9 59 4 3

Department Chairman, Code 6' 1

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 9 3945

Professor D. J. Collins, Code 67Co 2

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 9 5945

Professor H. A. Titus, Code 62Ts 2

X:: vr a I P os tgradua he I

Monterey, California 95943

Shin, Hvunchul 5

224-17 (20Tong 6 BAN)Chun-ho-1 dong , Kang-dong kuSeoul , Korea

An, Byunggul 1

SMC -2816Naval Postgraduate SchoolMonterey, California 93943

Lee, Cheongkoo 1

SMC ?2 7S1Naval Postgraduate SchoolMonterey, California 95943

Chang, Jinhwa 1

SMC #2391Naval Postgraduate SchoolMonterey, California 95945.

170

5 c: *\

I - t ' • rv

, U 4

Shin

Application of modernguidance control theoryto a back-to-turn mis-sile.

sep e? /yra

207604

ThesisS4737cl

ShinApplication of modern

guidance control theoryto a back-to-turn mis-sile.

&.*Ct«6

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