i

DEVELOPMENT OF A NEUROFUZZY CONTROL SYSTEM FOR

THE GUIDANCE OF AIR TO AIR MISSILES

By

Eng. Ahmed Momtaz Ahmed

A Thesis submitted to

The Faculty of Engineering, Cairo University

In partial fulfillment of the degree of

MASTER OF SCIENCE

in

Mechanical Design and Production

Under the Supervision of

Prof. Galal A. Hassaan

Professor of System Dynamics

Dr. Yasser F. Zeyada

Assistant Professor of System Dynamics

Department of Mechanical Design and Production Faculty of Engineering, Cairo University

Giza, Egypt

2004

ii

ACKNOWLEDGEMENT

The author would like to acknowledge with great thanks Prof. Galal A. Hassaan,

Department of Mechanical Design and Production, Cairo University, for his supervision,

sincere help and valuable advice.

The author would like also to thank Dr. Yasser F. Zeyada, for the great efforts he

offered to prepare this thesis.

Thanks also to the staff members of the Department of Mechanical Design and Production,

Cairo University who offered their help and support.

iii

SUMMARY

Recently, neurofuzzy modeling techniques have been successfully applied to modeling

complex systems, where traditional approaches hardly can reach satisfactory results due

to lack of sufficient domain knowledge. Much research has been done on applications of

neural networks (NNs) for identification and control of dynamic systems. According to

their structures, the NNs can be mainly classified as feedforward neural networks and

recurrent neural networks. It is well known that a feedforward neural network is

capable of approximating any continuous functions closely. However, the feedforward

neural network is a static mapping. Without the aid of tapped delays the feedforward

neural network is unable to represent a dynamic mapping. Although much research has

used the feedfoward neural network with tapped delays to deal with dynamical problem,

the feedforward neural network requires a large number of neurons to represent a

dynamic response in the time domain. Moreover, the weight updates of the feedforward

neural network do not utilize the internal information of the neural network and the

function approximation is sensitive to the training data. On the other hand, recurrent

neural networks (RNNs) have superior capabilities than the feedforward neural

networks, such as dynamic and ability to store information for later use. Since recurrent

neuron has an internal feedback loop, the RNN is a dynamic mapping and demonstrates

good control performance in the presence of uncertainties, which is usually composed of

unpredictable plant parameter variations, external force disturbance, unmodeled and

nonlinear dynamics, in practical application of air-to-air missile.

In recent years, the concept of incorporating fuzzy logic into a neural network has been

grown into a popular research topic. In contrast to the pure neural network or fuzzy

system, the fuzzy neural network (FNN) possesses both their advantages. It combines the

capability of fuzzy reasoning in handling uncertain information and the capability of

artificial neural networks in learning from processes. In missiles controlling branch it is

needed to minimize the output performance errors until fulfilling the required

specifications such as the minimum locking time and the maximum fuse distance.

Nowadays we have different types of missiles using different types of control strategy

iv

some of them uses reference frame unit as they controlled by a remote base or an

aircraft. Consequently controlling such missiles by reference angles ,, with respect

to fixed frame, other type of control strategy doesn’t use reference frame unit (fire and

forget types) such as all heatseeking, active radar missiles. In this paper, both types of

control strategy are discussed. A technique for the preliminary design of a control

system is presented using neurofuzzy systems for a diverse data range of a highly

nonlinear MIMO 5 DOF model (multi-input multi-output 5 degree of freedom AIM 9R

air to air missile). This system is composed of 3-body axes velocities and rotating in both

pitch and yaw direction only as the rolling motion is prevented due to gyroscopic

stabilizers (anti roll system) attached at the rear fins. The model has cross coupling

effects between the longitudinal and lateral motions due to the coupled equations of

motion. MIMO model was divided into two separated SISO models (single-input single-

output) taking into consideration the nonlinear cross coupling effects, then generating

the neural network controller for each separated SISO model (single-input single-output

model), blending the output performance using a 25 rule base Mamdani type fuzzy logic

controller. A Mamdani fuzzy logic controller can deal with the system uncertainty due

to cross coupling effect and affecting gust in the whole system when instantaneously

applying both inputs to the model. The main goal is to fulfill certain specifications within

its limited ranges such as proximity fuse distance and the target locking time during the

whole mission of the missile consequently keeping the error between actual and desired

performance according these tolerances. The first obvious error is due to the AOA

(angle of attack of the missile) and SSA (side slipping angle), resulting in error in the

desired trajectory itself as the (camera) detector is attached to the body frontal area of

the missile that is inclined to the actual body velocity vector. A second error is due to the

error between the desired and actual trajectory of the missile and this is actually due to

the whole system performance (AIM_9R dynamic model and controllers). To minimize

the second error over different regimes of flight, a hierarchical design of multi NN

controllers using a TSK(Takagi Suguno Kang) fuzzy fusion system classifier is

recommended which compute the weight of each controller according to the flight

regime parameters such as (altitude, velocity, required track rate [target-seeker

distance]). Fortunately both types of performance errors are damping each other to

some extent, this will lead to minimize the net output error of the actual performance.

v

The model was established and simulated through (MATLAB-SIMULINK), and the

final performance was examined using FLC fuzzy logic controllers with and without

NNC neural network controllers.

vi

LIST OF CONTENTS

ACKNOWLEDGMENT i

SUMMARY ii

LIST OF CONTENTS vi

ABBREVIATIONS viii

NOMENCLATURE ix

LIST OF FIGURES xi

LIST OF TABLES xv

Chapter 1 GENERAL BACKGROUND ON AIR TO AIR

MISSILES

1.1 Historical Background 1

1.2 Overview 8

1.3 AIM Missiles 13

Chapter 2 LITERATURE REVIEW

2.1 Introduction 32

2.2 Conventional Guidance and Control Design 34

2.3 Neural Net-based Guidance and Control Design 37

2.4 Fuzzy Logic-Based Guidance and Control Design 43

2.5 Gain-Scheduling Guidance and Control Design 49

2.6 Guidance Laws 53

2.7 Advantages over conventional designs 55

2.8 Problem Formulation 56

2.9 Previous Work 56

2.10 Thesis Layout 56

vii

Chapter 3 MODELING OF AIR TO AIR MISSILE

3.1 Introduction 58

3.2 The Rigid –Body Equations 58

3.3 Evaluation of the Angular Momentum 61

3.4 Euler’s Equations of Motion 62

3.5 Orientation and Position of the Missile 63

3.6 The Flight Path Determination 65

3.7 The Orientation of the Missile 66

3.8 The Applied Forces 68

3.9 AIM 9R Model Characteristics 73

3.9.1 Model verification 73

3.9.2 Cross effect representation 74

3.9.3 Gust effect 75

3.10 Summary 76

Chapter 4 NEURAL NETWORK FOR MISSILE GUIDANCE

4.1 Introduction 77

4.2 Theory and Examples 80

4.2.1 Single-input neuron 80

4.2.2 Multiple-input neuron 80

4.2.3 A layer of neurons 81

4.2.4 Multiple layer of neurons 82

4.3 Neural Network Controllers 83

viii

4.3.1 Control with inverse models 83

4.3.2 Control by feedfoward models 86

4.3.3 Control by feedback linearization 87

4.4 Summary 91

Chapter 5 MISSILE GUIDANCE USING FUZZY LOGIC

CONTROL

5.1 Introduction 92

5.2 Applying Fuzzy Inference System for Control 92

5.2.1 Designing membership functions (Fuzzification process) 93

5.2.2 Rule derivation 95

4.2.3 Fu 5.2.3 Fuzzy rule-based inference 97

5.2.4 Response pattern of a simple generic fuzzy control system 100

5.3 Fuzzy Logic Control Application 102

5.4 Comparison between Different Fuzzy Logic Controllers and PD Controller 103

5.5 Low pass Filter 104

5.6 The Effect of Using Neural Network in the First Time Period 106

5.7 Fuzzy Logic Controller Output Signal 107

5.8 Neurofuzzy Structure 107

5.9 The Effect of Flight Regimes on Neurofuzzy Controller Performance 108

5.10 Composite Fuzzy Guidance Law 108

5.11 Summary 120

Chapter 6 A HYBRID NEUROFUZZY CONTROL SYSTEM FOR

MISSILE GUIDANCE

ix

6.1 Introduction 121

6.2 Methodology 122

Chapter 7 DISCUSSIONS AND CONCLUSIONS

7.1 Discussions 128

7.2 Conclusions 130

7.3 Recommendation for Future Work 131

References 132

Appendix A Simulink models

x

ABBREVIATIONS

AAM Air to Air Missile

AFD Arm/Fire Device

AGM Air to Ground Missile

AIM Air Intercept Missile

AOA Angle of Attack

ASM Air to Surface Missile

CLOS Command to Line of Sight

DOF Degree Of Freedom

DSMAC Digital Scene Matching Area Correlation

F.O.R Frame Of Reference (at earth)

FLC Fuzzy Logic Controller

FNN Fuzzy Neural Network

GBU Ground Bomb Unit

GPS Global Positioning System

IR Infra Red

MIMO Multi Input Multi Output

NNC Neural Network Controller

OOLD Out-Of-Line Device

RF Radio Frequency

RNN Recurrent Neural Network

SISO Single Input Single Output

SSA Side Slipping Angle

TASM Tactical Air to Surface Missile

TDD Target Detection Device

TERCOM Inertial and Terrain Contour Matching

TIVS Thermally Initiated Venting System

TSK Tagaki Suguno Kang

USAF United States Air Force

WGU Wing Ground Unit

xi

NOMENCLATURE

ac Acceleration vector of missile mass center (m/s2)

(acx, acy, acz) Scalar components of ac in X, Y, Z directions respectively (m/s2)

A, B, C Moments of inertia about (x, y, z) axes respectively (kg/m2)

CL Lift coefficient

CD Drag coefficient

CN Side force coefficient

Cm Pitching moment coefficient

Cn Yawing moment coefficient

D Product of inertia yz (kg/m2)

E Product of inertia xz (kg/m2)

J Product of inertia xy (kg/m2)

ea, ee, er Mass eccentricities of control surfaces (m)

F Resultant external force vector (N)

(Fx,Fy,Fz) Scalar components of F in X, Y, Z directions respectively (N)

(Fa, Fe, Fr) Generalized control forces for aileron, elevator and rudder deflection

respectively (N)

f Generalized force in Lagrange’s equation (N)

G Resultant external moment vector (N.m)

h Angular momentum vector of the missile (kg.m2/s)

h’ Angular momentum vector of spinning rotor inside the missile (kg.m2/s)

hx, hy, hz Scalar components of h in X, Y, Z directions respectively (kg.m2/s)

h'x, h’y, h’z Scalar components of h’ in X, Y, Z directions respectively (kg.m2/s)

Ha, He, Hr Hinge moments on aileron, elevator and rudder respectively (N.m)

Ia, Ie, Ir Effective moments of inertia of control systems (kg/m2)

L, M, N Scalar components of G (N.m)

ma, me, mr Masses of aileron, elevator and rudder respectively (kg)

P, Q, R Scalar components of ω (rad/s)

xii

Sw Wing surface area (m2)

Vc Absolute velocity of the missile (m/s)

U, V, O Scalar components of Vc in X, Y, Z direction (m/s)

X, Y, Z Missile body axes components in X, Y, Z directions respectively

x’, y’, z’ Fixed frame of reference axes components in X, Y, Z directions respectively

α Angle of attack (rad)

β Side angle (rad)

δa, δe, δr Deflection angles of aileron, elevator and rudder respectively (rad)

ρ Air density (kg/m3)

Ψ, Θ, Φ Euler equation angles (rad)

ω Absolute angular velocity of the missile (rad/s)

xiii

LIST OF FIGURES

Fig. (1.1) Henschel Hs 293 air-to-ship, wireless guided, gliding bomb A 1

Fig. (1.2) Henschel Hs 293 air-to-ship television guided gliding bomb A 2

Fig (1.3) The Ruhrstahl SD 1400 'Fritz X' air-to-ship, wireless guided 6

Fig. (1.4) The Ruhrstahl SD 1400 'Fritz X' air-to-ship, wireless guided 7

Fig. (1.5) Types of ASM missiles 9

Fig. (1.6) Russian AAM 10

Fig. (1.7) TERCOM, DSMAC missiles guidance technique 11

Fig. (1.8) Tomahawk missile 13

Fig. (1.9) Chinese nuclear missiles DONG FENG/JULANG SERIES 13

Fig. (1.10) Missile components 14

Fig. (1.11) Active homing system 15

Fig. (1.12) Semiactive homing system 16

Fig. (1.13) Comparison between AIM 7 and AIM 120 guidance technique 23

Fig. (1.14) AIM 7 air to air missile 26

Fig. (2.1) Supervisory control scheme 38

Fig. (2.2) Hybrid control scheme 39

Fig. (2.3) Model reference control scheme 40

Fig. (2.4) Internal model control scheme 41

Fig. (2.5) Adaptive control scheme 41

Fig. (2.6) Predictive control scheme 42

Fig. (2.7) Model reference control of coupled lateral dynamics 43

Fig. (2.8) Fuzzy PD controller 45

Fig. (2.9) Typical adaptive fuzzy control scheme 46

Fig. (2.10) Fuzzy sliding mode control scheme 46

Fig. (2.11) Fuzzy model-following control scheme 47

Fig. (2.13) A fuzzy terminal guidance system 49

Fig. (2.14) Conventional gain-scheduling control scheme 50

Fig. (2.15) Fuzzy gain-scheduling control scheme 51

xiv

Fig. (2.16) Neural network gain-scheduling PID control scheme 52

Fig. (2.17) Neural-fuzzy gain-scheduling control scheme 53

Fig. (2.18) Integrated control model for the guidance of AIM 9R missile 57

Fig. (3.1) Missile body axes 58

Fig. (3.2) Missile orientation 64

Fig. (3.3) The applied forces 67

Fig. (3.4) Missile model algorithm 71

Fig. (3.5) AIM 9 Nonlinear Model 72

Fig. (3.6) Pitch angle step response for 5 DOF and 3 DOF models 73

Fig. (3.7) Pitch angle dynamic response for 3D, 2D models 73

Fig. (3.8) Cross coupling effect (yawpitch) 74

Fig. (3.9) Mean value of cross coupling effect versus input frequencies 74

Fig. (3.10) Standard deviation of cross coupling effect 75

Fig. (3.11) Gust effect (pitchyaw) 75

Fig. (4.1) Single Input neuron 80

Fig. (4.2) Multiple Input Neuron 81

Fig. (4.3) Layer of (S) Neurons 82

Fig. (4.4) Direct inverse control 84

Fig. (4.5) Direct Inverse Control at steady state flight 84

Fig. (4.6) Direct Inverse Control at climb angle =90 deg 85

Fig. (4.7) Direct Inverse Control at initial velocity =700 m/s 85

Fig. (4.8) Control by Feedforward 86

Fig. (4.9) Control by Feedforward at steady state flight 86

Fig. (4.10) Control by feedback linearization 87

Fig. (4.11) Control by Feedback Linearization at steady state flight 88

Fig. (4.12) Pitch motion using NN controller with combined sine wave input 89

Fig. )4.13) Integrated square error using NN controller with combined sine wave

input

89

Fig. (4.14) Absolute velocity of the missile using NNC with combined sine wave 90

xv

input

Fig. (4.15) Angle of attack using NNC with combined sine wave input 90

Fig. (4.16) Elevator deflection angle using NNC with combined sine wave input 91

Fig. (5.1) The structure of a Fuzzy Logic Control System 93

Fig. (5.2) Triangular membership function 94

Fig. (5.3) Membership function for input (e) 94

Fig. (5.4) Membership function for input (ce) 94

Fig. (5.5) Membership function for output (u) 95

Fig. (5.6) Fuzzy matching process 97

Fig. (5.7) Fuzzy inference process 97

Fig. (5.8) Fuzzy combination process 98

Fig. (5.9) Defuzzification process 99

Fig. (5.10) Rule base calculations 99

Fig. (5.11) Architecture of the generic fuzzy control system 100

Fig. (5.12) Step response of a simple generic fuzzy controller 101

Fig. (5.13) Fuzzy inference system control action 102

Fig. (5.14) Comparison between fuzzy logic and NN controllers 102

Fig. (5.15) Effect of normalized and denormalized factors 103

Fig. (5.16) Comparison between PD and FL controllers with combined sine wave

input

103

Fig. (5.17) Integral square error for PD and FL controllers 104

Fig. (5.18) Dynamic response of a low pass filter 104

Fig. (5.19) Integrated square error for different fuzzy logic controllers using low

pass filter

105

Fig. (5.20) Error rate using filter 105

Fig. (5.21) Error rate using delay only 106

Fig. (5.22) Effect of using NNC for the first 5 seconds 106

Fig. (5.23) Error, error rate and control signal 107

Fig. (5.24) Neurofuzzy structure 107

Fig. (5.25) Two types of fuzzy logic controllers output performance 108

xvi

Fig. (5.26) Composite guidance law 109

Fig. (5.27) Defuzzification process 109

Fig. (5.28) Fusion Fuzzy system output K1 110

Fig. (5.29) Fusion Fuzzy system output K2 110

Fig. (5.30) Fusion Fuzzy system output K3 110

Fig. (5.31) Comparison between FL, PD controllers in pitch motion with combined

sine wave

111

Fig. (5.32) Integrated square error for PD and FL controllers with combined sine

wave

111

Fig. (5.33) Angle of attack using FLC with combined sine wave input 112

Fig. (5.34) Elevator deflection angle using FLC with combined sine wave input 112

Fig. (5.35) Absolute velocity of the missile using FLC with combined sine wave

input

113

Fig. (5.36) Comparison between fuzzy logic controller and PD controller for .3 rad

step input

114

Fig. (5.37) Comparison between FLC and PD controller for a step response 114

Fig. (5.38) Elevator deflection angle for .3 rad step input using FLC 115

Fig. (5.39) Angle of attack for .3 rad step input using FLC 115

Fig. (5.40) Absolute velocity for .3 rad step input using FLC 116

Fig. (5.41) Path angle response using PD, FL controllers with ramp input 117

Fig. (5.42) Comparison between integrated square error for PD, FL controllers with

ramp input

117

Fig. (5.43) Angle of attack using FLC with ramp input 118

Fig. (5.44) Missile velocity using PD, FL controllers with ramp input 118

Fig. (5.45) Elevator deflection angle using PD, FL controllers with ramp input 119

Fig. (6.1) AIM 9 Model with 2 separate controllers 121

Fig. (6.2) Fig. (5.2) NeuroFuzzy structure 122

Fig. (6.3) Target location transformation from fixed frame of reference coordinate

to missile body axes coordinate

123

Fig. (6.4) Actual trajectories for target and missile 124

Fig. (6.5) 3D Plot for missile trajectory in both X, Y axes with respect to time 124

Fig. (6.6) 3D Plot for missile trajectory in both X, Z axes with respect to time 125

Fig. (6.7) Calculation of Maximum fuse distance, and the best triggering point 126

xvii

Fig. (6.8) Error in Z direction due to AOA, control system error 127

Fig. (6.9) Error in Y direction due to SSA, control system error 127

LIST OF TABLES

Table (1.1) Tomahawk specification 12

Table (1.2) Major types (J, L, M, P, R) of AIM 9 missiles 20

Table (1.3) Major types (B, D, E, G, H) of AIM 9 missiles 21

Table (1.4) Data specifications of AIM 7 25

Table (5.1) Control rules for simple generic fuzzy controller 101

Table (7.1) ISE for PD, NN, FL controllers 129

Table (7.2) Comparison between NNC and FLC for the first 5_seconds 129

xviii

ABBRIVIATIONS

A/C Aircraft

AAM Air to Air Missile

AFD Arm/Fire Device

AGM Air to Ground Missile

AIM Air Intercept Missile

AOA Angle of Attack

ASM Air to Surface Missile

C/L Center Line

DOF Degree Of Freedom

DSMAC Digital Scene Matching Area Correlation

F.O.R Frame Of Reference

F.O.R Frame of Reference (at earth)

FLC Fuzzy Logic Controller

FNN Feedforward Neural Network

GBU Ground Bomb Unit

GPS Globe Positioning System

IR Infra Red

MIMO Multi Input Multi Output

NNC Neural Network Controller

OOLD Out-Of-Line Device

RF Radio Frequency

RNN Recurrent Neural Network

SISO Single Input Single Output

SSA Side Slipping Angle

TDD Target Detection Device

TERCOM Inertial and Terrain Contour Matching

TIVS Thermally Initiated Venting System

TSK Tagaki Suguno

WGU Wing Ground Unit

1

Chapter (1)

GENERAL BACKGROUND ON AIR TO AIR MISSILE

1.1 Historical Background

During World War 2 there have been many ideas about guided weapons such as

guided bombs instead of direct projectiles or bombs. The beginning ideas that were to

evolve into the HenschelHs 293 [1] appeared in as early as 1939 {Fig. (1.1)}. In 1940, an

experimental model having the shape of a glider was built. The goal was to develop a

remote-controlled air-to-surface missile against ships. Development proceeded even

though no suitable rocket motors were available. The experimental model used a

standard SC 500 bomb with extra wings and tail unit but no rudder. Finally a

propulsion system was developed, and the liquid rocket was fitted under the main

missile body. An 18-channel radio system was used for control.

Fig. (1.1) HenschelHs 293 air-to-ship, wireless guided, gliding bomb, model A

The missile was designed to be carried under a parent bomber. Warm exhaust air from

the aircraft engines was channeled to the missile to prevent it from freezing up at high

altitudes. Once dropped the Hs 293 would fall for some 90m (295ft) before the rocket

achieved maximum thrust. The parent bomber would continue to fly a pre_designated

course parallel with the target. The Aircraft could visually track the missile with the aid

of red guidance flare in the tail, and control the projectile using a small control box with

a joystick. The actual flight path resembled a series of arcs as corrections were received

and followed.

2

The main weakness of the Hs 293A was that the parent bomber had to fly a steady, level

path. Evasive moves to avoid anti-aircraft fire were impossible, even though the Hs 293

outranged most ship-borne anti-aircraft guns. An improved Hs 293D with a television

camera installed in the head of the missile as aiming system was planned but the war

concluded before it could be realized. Also, the problem of icing was never resolved and

thus further propulsion units were designed. The war ended before these plans left the

experimental stage.

The Hs 293 was based on a normal 500 kg (1,102 lb) bomb with wings and fins added

and an engine suspended from the main body. The Hs 293V-4 and C-1 were guided by

radio, like the Fritz-X, but after the Germans found that the Allies were capable of

interfering with the control signals to the missile wire guidance was adopted.

The C-3, C-4, and A-0 versions of the Hs293 relied on wire guidance. During the

bomb's fall, two wired coils on the wing tips unwound, so maintaining the link with the

launch plane and allowing the transmission of electrical pulses for guidance.

A model D {Fig. (1.2)} was also built, which transmitted television images of the target to

the controller. Model H was supplied with an acoustic/magnetic detector to attack

bomber formations.

Fig. (1.2) HenschelHs 293 air-to-ship television_guided gliding bomb, model D (3views)

3

The guidance operators always sat on the starboard right side of the cabin, and

therefore it was always necessary to attack a target from the port side and in the same

movement direction. Another version from Henschel guided bomb was HenschelHs 294

which used the television guidance from launch plane.

In 1941,Professor Wagner conceived and designed the Schmetterling anti-aircraft

missile and submitted it for appraisal and approval to the RLM

(Reichsluftfahrtministerium, phew!) or the State Ministry of Aviation, which rejected

the design because its defensive purpose was considered weak and still unnecessary.

Official support finally came in 1943, when Allied strategic bombing had accelerated the

war to a higher stage, and the Reich was feeling the pain.

The Henschel company, which was also developing several other types of missiles, was

given the go-ahead and development contract to begin production in February 1945.

Experiments and production were interrupted by Allied bombers and there were only

59 experimental launchings, 34 of them failed for different reasons. Schmetterling was a

bomb-shaped projectile with wings and fins, and two rocket boosters were attached to

the outside for take-off.

The external rockets were powered by solid fuel, while the internal motor was driven by

a fuel composed of hydrocarbon and nitric acid. It could accelerate up to 200 m/s and

had an effective range of approximately 8 km with a ceiling of 1000m. A ground

controller would guide the missile through radio signals and track its course visually.

The warhead contained 20 kg of high explosive and was armed with a proximity fuse

called "Fox."

The lion's share of post-World War II literature concerning German guided missiles has

concentrated on the V-1 "Buzz Bomb" and the V-2 stratospheric rocket. However, the

guidance systems of these weapons were so rudimentary that a successful launch was

one which hit a target as vast as London. German science developed several other, more

sophisticated guidance systems for weapons which were used successfully in combat

before either the V-1 or V-2.

4

These first true guided missiles, by current definition, are virtually unknown today.

They were Henschel'sHs 293A and Ruhrstahl's SD1400X, known as "Fritz X." Both

missiles were air-launched from mother bombers and radio-controlled primarily against

maritime targets. It is interesting to note that of the fifteen battleships lost to airpower

(seven in the open sea) one of these, the 41,650-ton Italian flagship, Roma, was sunk by a

Fritz X. Later, the British battleship, Warspite, was hit by a Fritz X and put out of

action for six months. Fritz Xs also sunk the Royal Navy light cruiser, Spartan, heavily

damaged the cruiser USS Savannah, and hit the cruiser USS Philadelphia.

The Henschel program received a great boost when the RLM assigned Prof. Dr. Herbert

A. Wagner to assume control of their missile program. The brilliant Wagner had earlier

been dismissed by Junkers Aircraft as the company found it impossible to reconcile his

intermittent and desultory research methods with its ideas of series production. After

his dismissal, he carried out independent research on an anti-ship ricochet bomb.

In January 1940, Wagner joined Henschel's already distinguished missile research team.

This team of specialists included: ReinhardLahde and Otto Bohlmann (aeronautics),

Wilfried Hell, from Schwarz Propeller Werke (engineering), Josef Schwarzmann

(electronics), and Dr. Hinrici (design study). Theodor Sturm of

StassfurterRundfunkGessellschaft worked closely with the Henschel team in developing

and adapting radio systems.

The first Henschel design, designated Hs 293V-1, was a standard glider bomb bearing

slight resemblance to its Schwarz forerunner. The Hs 293V-1 did not leave the drawing

board.

The second design, the Hs 293V-2 (later FZ21) was built as a model in February 1940

and used in control tests at Karlshagen. It, too, was unarmed and powerless. Note that,

unlike German aircraft prototype 'V' numbers where only one example was built for

each V number, the Henschel missile V numbers referred to design alone and many

examples could have been built.

Further studies led to a new design in July 1940, the Hs 293V-3. The powerless

prototype was test launched from minimum release altitudes of 3,280 feet. Since the

5

RLM was primarily interested in an anti-ship missile, it was found that a pure glide

bomb lacked sufficient speed and weight to penetrate warship hulls.

The first successful test launching occurred two days later against a small barn as a

target. Subsequent tests ended in failure usually for two main reasons. First, the original

white tracking flares were difficult to follow and were replaced with red flares.

Secondly, the electrical system often failed due to design failure particularly with tubes

which were replaced with relays. In 1941, drop tests used a Heinkel He 177A-0 bomber

and later two He 177A-1s were used in the first successful tests.

The A-0 design led to the finalization of the production design, the Hs 293A-1, which

consisted of three pre-production models (V-2, V-3, A-0). One thousand nine hundred

were produced for the test and modification programs which continued into early 1944

(1,700 were A-0s). Early Hs 293 production was carried out at HenschelWerke 111,

Spandau-Hakenfelde but was transferred to Berlin-Schoenfeld in August 1943.

In early tests, bomb-aimers had difficulty following the bomb in flight due to its high

velocity. Also, the electromagnetically activated spoilers tended to jam. Pneumatic

power was tested but was found to develop problems in low temperatures causing the

electromagnetic system to be retained. It was not until early 1942, when a high-speed

wind tunnel had been constructed, that Dr. Kramer's control spoilers were modified and

that a velocity-reducing tail-mounted air brake was perfected.

As issued the Fritz X was the standard SD 100X armor-piercing (AP) bomb with four

centrally-mounted standard cruciform wings giving the bomb aerodynamic pivotal

points for control. The total span across the wings was 4 feet 5 inches. The tail assembly

consisted of four fins similar to those of an orthodox bomb, but were surrounded by the

air brake ring which contained the control spoilers and operating solenoids. The dive

brake was fitted around the tail fins to limit the terminal velocity of the bomb to 600

miles per hour. The dive brake was fabricated of metal but a section was electrically

insulated to serve as a radio antenna. The fuselage and tail measured 10 feet 8* inches

and had a diameter of 1 foot 10 inches.

6

The sudden but arranged capitulation of the Italian fleet to the Allies on 9 September

1943 spurred the Germans to take quick actions against their former ally. The battleship

Italia was damaged and Roma sunk as victims of a new type of air-to-surface weapon,

the Ruhrstahl/Kramer X-1 [1] as shown in Fig. (1.3) or Fritz X. It was a free-falling

bomb guided by the parent aircraft. Usually it was dropped at an altitude of about

6,000m (19,685ft); by the time of detonation it would have had gained a velocity close to

that of sound.

The first tests were conducted in Germany in 1942, and were moved to Italy later. In

Italy, pneumatic power was tested to substitute for the electromagnetic actuation of the

spoilers. However, variations in temperature in different parts of the atmosphere posed

obstacles and the idea had to be dropped. The Allied advance to Italy forced the

Germans to press the Fritz X into increased use. The cruiser USS Savannah was

attacked successfully alongside several naval transports. During a night attack two

British cruisers collided in utmost confusion. Seven days later the Germans scored hits

on the battleship HMS Warspite, which had to be towed to Malta.

Fig. (1.3) The Ruhrstahl SD 1400 'Fritz X' air-to-ship, wireless guided, gliding bomb

German wire-guided missiles belong generally to the X-series. The X-4air-to-air missile

shown in Fig. (1.4) was the first of the series to be developed in June 1943. It was a

finned and winged projectile with a wingspan of 6ft 6in, and powered by a rocket using

a combination of two fuels (T and C stoff). The wire, coiled around spools, was stored in

the wing tips and carried guidance signals to control spoiler tabs and ailerons on the

7

wings to alter direction. The warhead composed of 20 kg of explosive and a acoustic

proximity fuse called "Kranich."

Fig. (1.4) Restored Ruhrstahl X-4 Air-to-Air missile

It was certainly natural and reasonable that a ground-to-ground version of the X-4 [1]

was also developed, as an anti-tank weapon, in addition the successful Panzerfaust. This

new cousin missile was called the X-7. The X-7 was navigated and stabilized by a

gyroscope and guided by wire, like the X-4, but it contained a hollow charge warhead

armed with an impact fuse which could penetrate over 200mm of armor within 1000

meters. It was driven by a two-stage, solid-fuel rocket motor and weighed 10 kg.

In early 1943, work began on the X-4 air-to-air wire-guided missile by Dr. Kramer at

Ruhrstahl. The missile received a development order in the summer of 1943 and was

given the number 8-344 by the RLM, and was developed to give fighters a chance to

down the ever increasing number of Allied bombers from outside of their defensive gun

range.

By the early sixties the sidewinder waqs the principal heatseeker in western service it

was used by USAF, USN. Its early combat record was not spectacular, as the seeker

performance limitations were exacerbated by the poor reliability of the tube electronics

and the inexperience of its users. Kill probabilities were in the tens of percent, very

sensitive to how well the launch aircraft was positioned. Designed to intercept lumbering

bombers, the AIM was ill suited to knife-fights with MIG-17s at low level.

Nevertheless, no less than 28 MIGs were killed for 175 launches between 1965 and 1968,

by USAF F-4C/D aircraft, an aggregate (kill probability) of 16 %.

8

In 1973, Ford began production of an enhanced AIM-9J-L, later registered the AIM-

9N. The November model employed a similar configuration to the AIM-9B, but the main

printed circuits were redesigned to improve seeker performance.

In 1982 AIM-9L was produced which is essentially an improved AIM-9L. The Mile

has improved background rejection, counter-countermeasures capability and a low

smoke motor to reduce the visual signature of the inbound weapon.

1.2 Overview

Missiles are classified according to many specifications such as range, warhead

weight, guidance technique and mission type. Many types of missiles will be shown and

discussed briefly in the following pages. But the focus of this thesis will be the air to air

missiles. Air-to-air missiles have many guidance techniques such as active, semi-active

radar, laser and heatseeking missiles. These types of guidance technique are classified

according to the range of each missile. For example, radar homing missiles will have

long range than the other heatseeking ones according to the capability of the detecting

element itself. In this thesis, the air-to-air heatseeking missiles (SIDEWINDER AIM 9R)

will be discussed. This missile has a large number of produced units all over the world

about 21000 piece that from the USAF inventory. Table (1.1) gives the specifications of

this missile for many versions.

1.2.1 Missiles types

There are many common types of missiles such as AAM, ASM, SAM, TASM cruise

and ballistic missiles. The next section will discuss each type briefly. Each type of

missiles is classified in accordance with guidance techniques, target-missile range,

propulsion and type of the missile warhead. The basic type will be discussed in this

research is the AAM or AIM missiles.

1.2.1.1 ASM missiles

Figure (1.5) shows different types of ASM missiles.

9

Fig (1.5) Types of ASM Russian missiles

1.2.1.2 AAM missiles

Figure (1.6) shows different types of AAM missiles.

10

Fig (1.6) Russian AAM

1.2.1.3 Cruise missiles

Tomahawk is an all-weather submarine or ship-launched land-attack cruise missile.

After launch, a solid propellant propels the missile until a small turbofan engine takes

over for the cruise portion of flight. Tomahawk is a highly survivable weapon. Radar

detection is difficult because of the missile's small cross-section and low altitude flight.

11

Similarly, infrared detection is difficult because the turbofan engine emits little heat.

Systems include Global Positioning System (GPS) receiver; an upgrade of the optical

Digital Scene Matching Area Correlation (DSMAC) system; Time of Arrival (TOA)

control, and improved 402 turbo engines.

The Tomahawk land-attack cruise missile has been used to attack a variety of fixed

targets, including air defense and communications sites, often in high-threat

environments. The land attack version of Tomahawk has inertial and terrain contour

matching (TERCOM) radar guidance {Fig. (1.7)}. The TERCOM radar uses a stored

map reference to compare with the actual terrain to determine the missile's position. If

necessary, a course correction is then made to place the missile on course to the target.

Terminal guidance in the target area is provided by the optical Digital Scene Matching

Area Correlation (DSMAC) system, which compares a stored image of target with the

actual target image.

Fig (1.7) TERCOM, DSMAC missiles guidance technique

The Tomahawk missile {Fig. (1.8), Table (1.1)} provides a long-range, highly survivable,

unmanned land attack weapon system capable of pinpoint accuracy. The Surface Navy’s

deep strike capability resides in the Tomahawk missile system – the proven weapon of

choice for contingency missions.

12

Table (1.1) Tomahawk specifications

Specifications

Primary Function Long-range subsonic cruise missile for attacking land

targets.

Contractor Hughes Missile Systems Co., Tucson, Ariz.

Power Plant Williams International F107-WR-402 cruise turbo-fan

engine; solid-fuel booster

Length 18 feet 3 inches (5.56 meters); with booster: 20 feet 6

inches (6.25 meters)

Weight 2,650 pounds (1192.5 kg); 3,200 pounds (1440 kg) with

booster

Diameter 20.4 inches (51.81 cm)

Wing Span 8 feet 9 inches (2.67 meters)

Range Land attack, conventional warhead: 600 nautical miles

(690 statute miles, 1104 km)

Speed Subsonic - about 550 mph (880 km/h)

Guidance System Inertial and TERCOM

Warheads

Conventional: 400 kg Bullpup, or

Conventional submunitions dispenser with combined

effect bomblets, or

WDU-36 warhead w/ PBXN-107 explosive & FMU-148

fuze, or

200 kt. W-80 nuclear device

Date Deployed 1983

13

Fig (1.8) Tomahawk missile

1.2.1.4 Chinese nuclear missiles

Figure (1.9) shows different types of Chinese nuclear missiles.

Fig. (1.9) Chinese nuclear missiles DONG FENG/JULANG SERIES MISSILES

Chinese nuclear missiles are classified according the range an the warhead weight

most of them are guided either by inertial/fiber optic, strap down inertial system or

GPS.

1.3 AIM Missiles

1.3.1 Missile components

Guided missiles are made up of a series of subassemblies as shown in Fig. (1.10). The

various subassemblies form a major section of the overall missile to operate a missile

14

system, such as guidance, control, armament (warhead and fusing), and propulsion. The

major sections are carefully joined and connected to each other. They form the complete

missile assembly. The arrangement of major sections in the missile assembly varies,

depending on the missile type.

The guidance section is the brain of the missile. It directs its maneuvers and causes the

maneuvers to be executed by the control section. The armament section carries the

explosive charge of the missile, and the fusing and firing system by which the charge is

exploded. The propulsion section provides the force that propels the missile.

Fig. (1.10) Missile components

1.3.1.1 Guidance and control section

The complete missile guidance system includes the electronic sensing systems that

initiate the guidance orders and the control system that carries them out. The elements

for missile guidance and missile control can be housed in the same section of the missile,

or they can be in separate sections.

15

There are a number of basic guidance systems used in guided missiles. Homing-type,

air-launched, guided missiles are currently used. They use radar or infrared homing

systems. A homing guidance system is one in which the missile seeks out the target,

guided by some physical indication from the target itself. Radar reflections or thermal

characteristics of targets are possible physical influences on which homing systems are

based. Homing systems are classified as active, semiactive, and passive.

(a)-Active homing system

In the active homing system as shown in Fig. (1.11), target illumination is supplied by a

component carried in the missile, such as a radar transmitter. The radar signals

transmitted from the missile are reflected off the target back to the receiver in the

missile. These reflected signals give the missile information such as the target's distance

and speed. This information lets the guidance section compute the correct angle of

attack to intercept the target. The control section that receives electronic commands

from the guidance section controls the missile’s angle of attack. Mechanically

manipulated wings, fins, or canard control surfaces are mounted externally on the body

of the weapon. They are actuated by hydraulic, electric, or gas generator power, or

combinations of these to alter the missile's course.

Fig. (1.11) Active homing system

(b)-Semiactive homing system

In the semiactive homing system as shown in Fig (1.12), the missile gets its target

illumination from an external source, such as a transmitter carried in the launching

16

aircraft. The receiver in the missile receives the signals reflected off the target, computes

the information, and sends electronic commands to the control section. The control

section functions in the same manner as previously discussed.

Fig. (1.12) Semiactive homing system

(c)-Passive homing system

In the passive homing system, the directing intelligence is received from the target.

Examples of passive homing include homing on a source of infrared rays (such as the

hot exhaust of jet aircraft) or radar signals (such as those transmitted by ground radar

installations). Like active homing, passive homing is completely independent of the

launching aircraft. The missile receiver receives signals generated by the target and then

the missile control section functions in the same manner as previously discussed.

1.3.1.2 Armament Section

The armament system contains the payload (explosives), fusing, Safety and Arming

(S&A) devices, and target-detecting devices (TDDs).

(a)-Payload

The payload is the element or part of the missile that does what a particular missile is

launched to do. The payload is usually considered the explosive charge, and is carried in

the warhead of the missile. High-explosive warheads used in air-to-air guided missiles

17

contain a rather small explosive charge, generally 10 to 18 pounds of H-6, HBX, or PBX

high explosives. The payload contained in high-explosive warheads used in air-to-

surface guided missiles varies widely, even within specific missile types, depending on

the specific mission. Large payloads, ranging up to 450 pounds, are common. Comp B

and H-6 are typical explosives used in a payload. Most exercise warheads used with

guided missiles are pyrotechnic signaling devices. They signal fuse functioning by a

brilliant flash, by smoke, or both. Exercise warheads frequently contain high explosives,

which vary from live fuses and boosters to self-destruct charges that can contain as

much as 5 pounds of high explosive.

(b)-Fusing

The fusing and firing system is normally located in or next to the missile's warhead

section. It includes those devices and arrangements that cause the missile's payload to

function in proper relation to the target. The system consists of a fuse, a Safety and

Arming (S&A) device, a Target-Detecting Device (TDD), or a combination of these

devices. There are two general types of fuses used in guided missiles—proximity fuses

and contact fuses. Acceleration forces upon missile launching arm both fuses. Arming is

usually delayed until the fuse is subjected to a given level of accelerating force for a

specified amount of time. In the contact fuse, the force of impact closes a firing switch

within the fuse to complete the firing circuit, detonating the warhead. Where proximity

fusing is used, the firing action is very similar to the action of proximity fuses used with

bombs and rockets.

(c)-Safety and arming devices

S&A devices are electromechanical, explosive control devices. They maintain the

explosive train of a fusing system in a safe (unaligned) condition until certain

requirements of acceleration are met after the missile is fired.

(d)-Target detecting devices

TDDs are electronic detecting devices similar to the detecting systems in VT fuses. They

detect the presence of a target and determine the moment of firing. When subjected to

18

the proper target influence, both as to magnitude and change rate, the device sends an

electrical pulse to trigger the firing systems. The firing systems then act to fire an

associated S&A device to initiate detonation of the warhead. Air-to-air guided missiles

are normally fused for a proximity burst by using a TDDwith an S&A device. In some

cases, a contact fuse may be used as a backup. Air-to-surface guided missile fusing

consists of influence (proximity) and/or contact fuses. Multifusing is common in these

missiles.

1.3.1.3 Propulsion section

Guided missiles use some form of jet power for propulsion. There are two basic types of

jet propulsion power plants used in missile propulsion systems—the atmospheric (air-

breathing) jet and the thermal jet propulsion systems. The basic difference between the

two systems is that the atmospheric jet engine depends on the atmosphere to supply the

oxygen necessary to start and sustain burning of the fuel. The thermal jet engine

operates independently of the atmosphere by starting and sustaining combustion with its

own supply of oxygen contained within the missile.

There are three types of atmospheric jet propulsion systems—the turbojet, pulsejet, and

ramjet engines. Of these three systems, only the turbojet engine is currently being used

in Navy air-launched missiles. A typical turbojet engine includes an air intake, a

mechanical compressor driven by a turbine, a combustion chamber, and an exhaust

nozzle. The engine does not require boosting and can begin operation at zero

acceleration.

Thermal jets include solid propellant, liquid propellant, and combined propellant

systems. As an AO, you come in contact with all three systems. The solid propellant and

combined propellant systems are currently being used in some air-launched guided

missiles. The majority of air-launched guided missiles used by the Navy use the solid

propellant rocket motor. They include the double base and multibase smokeless powder

propellants as well as the composite mixtures. Grain configurations vary with the

different missiles. Power characteristics and temperature limitations of the individual

rocket motors also vary. In some guided missiles, different thrust requirements exist

19

during the boost phase as compared to those of the sustaining phase. The dual thrust

rocket motor (DTRM) is a combined system that contains both of these elements in one

motor. The DTRM contains a single propellant grain made of two types of solid

propellant—boost and sustaining. The grain is configured so the propellant meeting the

requirements for the boost phase burns at a faster rate than the propellant for the

sustaining phase. After the boost phase propellant burns itself out, the sustaining

propellant sustains the motor in flight over the designed burning time (range of the

missile).

1.3.2 Sidewinder AIM 9 missiles

Tables (1.2) and (1.3) show the characteristics of major types of AIM 9 missiles.

20

Table (1.2) Characteristics of major types of AIM 9 missiles.

Subtype AIM-9J AIM-9L AIM9-M AIM-9P AIM-9R

Service USAF Joint Joint USAF USN

Origin AIM-9E AIM-9H AIM-9L AIM-9J/N AIM-9M

Detector PbS InSb InSb InSb Focal

PlaneArray

Cooling Peltier Argon Argon Argon

Dome MgF2 MgF2 MgF2 MgF2 Glass

Reticle speed

(Hz)

100 125 125 100 Focal Plane

Array

Modulation AM FM FM FM Focal

PlaneArray

Tracking rate

(deg/s)

16.5 classified Classified >16.5 Classified

Electronics Hybrid Solid state Solid state Solid state Solid state

Warhead Blast

/fragmentation

Annular BF Annular BF Annular

BF

Annular BF

Fuse Passive-IR IR/Laser IR/Laser IR/Laser IR/Laser

Manufacturer Hercules

/Aerojet

Hercules

/Bermite

MTI/Hercules Hercules

/Aerojet

MTI/Hercules

Type MK.17 MK36 MOD.7,8 MK.36 MOD.9 SR.116 MK.36MOD.9

launcher AERO-111 Common Common Common Common

Length (m) 3 2.9 3 3 2.9

Span (m) 0.55 0.6 0.6 0.55 0.6

Mass (kg) 70 90 90 90 90

Velocity

(Mach)

<3 <3 <3 <3 <3

21

Table (1.3) Characteristics of AIM-9B, 9D, 9E , 9G, 9H

Subtype AIM-9B AIM-9D AIM-9E AIM-9G AIM-9H

Service Joint USN USAF USN USN

Origin NWC AIM-9B AIM-9B AIM-9D AIM-9G

Detector PbS PbS PbS PbS PbS

Cooling Uncooled Nitrogen Peltier Nitrogen Nitrogen

Dome

window

Glass MgF2 MgF2 MgF2 MgF2

Reticle speed

(Hz)

70 125 100 125 125

Modulation AM AM AM AM AM

Track rate

(deg/s)

11.0 12.0 16.5 12.0 >12.0

Electronics Thermionic Thermionic hybrid Thermionic Solid state

Warhead Blast/frag. Continuous

rod

Blast/frag. Continuous

rod

Continuous

rod

Fuse Passive-IR Passive-

IR/HF

Passive-IR Passive-

IR/HF

Passive-

IR/HF

Manufacturer Thiokol Hercules Thiokol Hercules Hercules

Bermite

Type MK.17 MK.36 MK.17 MK.36 MK.36

launcher Aero-111 LAU-7A Aero-111 LAU-7A LAU-7

Length(m) 3 2.8 2.9 2.8 2.8

Span(m) 0.5 0.6 0.5 0.6 0.6

Mass (kg) 70 90 75 90 89

Velocity

(Mach)

<3 <3 <3 <3 <3

22

1.3.2.1 The imaging Sidewinder-AIM-9R

The AIR-9R is the latest production sidewinder, using an imaging seeker which is a

fundamental departure from the established design. Developed by the naval weapons

center, the AIM-9R uses a modified AIM-9R control actuator, while retaining the fuse,

warhead, motor, wings and canards of its predecessor.

The imaging seeker is built around a focal plane array imaging device, analogous to

the CCDs employed in modern television cameras. A focal plane array has a much

greater instantaneous field of view than a reticle seeker (conventional technique), and

stares at the target and its immediate background, tracking the target by means of

contrast lock similar to that employed by TV guided weapons such as Maverick or

GBU-15. In this fashion, the seeker can account for the background contrast and reject

it, while also providing the potential to discriminate between multiple targets and

countermeasures such as flares.

The WGU-19 seeker uses a three gimbal stabilized platform mounting a visible band

focal plane array device, most likely a 256X256 element InSb array, or a higher

resolution PtSi device on a peltier cooled substrate, to provide coverage down to 4

micron band. The video signal produced by the array is then digested and processed by

a software programmable digital image processor, which tracks the targets and feeds

the autopilot with data so that it can send steering commands to the control actuators.

The AIM-9R is the latest naval sidewinder, using the airframe, fuse and motor of the

AIM-9M with a new digital imaging seeker. The new seeker employs a focal plane array

imaging device effective to visual wavelengths, mounted on a gimbaled stabilized

platform. The use of the imaging seeker has provided a vast improvement in target

detection range, off bore-sight angle, rejection of background and ability to selectively

aim for vulnerable areas of the target. Imaging seekers are immune to jamming

techniques effective against reticle seekers.

23

The AIM-9 provides a major increase in target acquisition range over established

subtypes, with much better tracking performance, and the ability to reject both

background terrain and clouds. The total field of view of the seeker is much greater,

allowing acquisition of off-bore-sight and maneuvering targets, while the software

provides for intelligent selection of an aim point when impacting a target, also the

manufacturer claim effective counter-countermeasures capability against known and

postulated jamming or seduction techniques.

1.3.3 AIM-120 AMRAAM Slammer

The AIM-120 advanced medium-range air-to-air missile (AMRAAM) is a new

generation air-to-air missile. It has an all-weather, beyond-visual-range capability and

is scheduled to be operational beyond 2000. AMRAAM is a supersonic, air launched,

aerial intercept, guided missile employing active radar target tracking, proportional

navigation guidance, and active Radio Frequency (RF) target detection. It employs

active, semi-active, and inertial navigational methods of guidance to provide an

autonomous launch and leave capability against single and multiple targets in all

environments as shown in Fig. (1.13).

Fig. (1.13) Comparison between AIM 7 and AIM 120 guidance technique

24

The AMRAAM weighs 340 pounds and uses an advanced solid-fuel rocket motor to

achieve a speed of Mach 4 and a range in excess of 30 miles. In long-range engagements

AMRAAM heads for the target using inertial guidance and receives updated target

information via data link from the launch aircraft. It switches to a self-guiding terminal

mode when the target is within range of its own monopulse radar set. The AIM-120

also has a "home-on-jam" guidance mode to counter electronic jamming. With its

sophisticated avionics, high closing speed, and excellent end-game maneuverability,

chances of escape from AMRAAM are minimal. Upon intercept an active-radar

proximity fuse detonates the 40-pound high-explosive warhead to destroy the target. At

closer ranges AMRAAM guides itself all the way using its own radar, freeing the launch

aircraft to engage other targets.

AMRAAM is a follow-on to the AIM-7 Sparrow missile series {Fig (1.14)} and

Table (1.4). The missile is faster, smaller and lighter, and has improved capabilities

against low-altitude targets. It incorporates an active-radar with an inertial reference

unit and micro-computer system, which makes the missile less dependent upon the fire-

control system of the aircraft. Once the missile closes on a target, its active radar guides

it to intercept. This enables the pilot to aim and fire several missiles simultaneously at

multiple targets. The pilot may then perform evasive maneuvers while the missiles

guide themselves to their targets.

25

Table (1.4) Specification data of AIM 120 (AMRAAM)

Primary Function Air-to-air tactical missile

Contractor Hughes Aircraft Co. and Raytheon Co.

Power Plant High performance (solid fuel)

Length 143.9 inches (3.66 m)

Launch Weight 335 pounds (150.75 kg)

Diameter 7 inches (177.8 mm)

Wingspan 20.7 inches (525.8 mm)

Range <20 miles (<32 Km)

Speed Supersonic < 4 Mach

Guidance System Active radar terminal/inertial midcourse

Warhead Blast fragmentation

Unit Cost $386,000

Relative costs of

AMRAAM

components

Guidance 68%

Control 9%

Fuse 9%

Warhead 2%

Propulsion 6%

Airframe 6%

Date Deployed September 1991

Aircraft platforms

Navy: F-14D and F/A-18

Air Force: F-15 and F-16

NATO: German F-4, British Tornado and Sea Harrier

26

Fig. (1.14) AIM 7 air to air missile

The AIM-120 grew out of a joint agreement, no longer in effect, among the United

States and several NATO nations to develop air-to-air missiles and to share the

production technology. The AMRAAM program was established as a result of Joint

Service Operational Requirement for an Advanced Air-to-Air Tactical Missile needed

in the post-1985 time frame. The AMRAAM program began with a 1975 study which

recommended that future aerial threats be engaged at 3-40 miles of range.

The AIM-120A is a non-reprogrammable missile (requires a hardware change to

upgrade the missile software). The AIM-120B/C is reprogrammable through the missile

umbilical using Common Field-level Memory Reprogramming Equipment (CFMRE).

The AIM-120C has smaller aero surfaces to enable internal carriage on the Air Force

F-22 aircraft. The USAF All-Up-Round (AUR) container houses an internal cable

which enables up to four missiles to be reprogrammed while in the container. USN

containers are not equipped with the cable and must be opened to reprogram the

27

missile. All three AMRAAM variants are currently approved for use on the F-15C/D/E,

F-16C/D, and F/A-18C/D aircraft.

Four wings, four fins (control surfaces), and the wiring harness cover are mounted

externally, providing additional distinguishing features from other similar missiles,

such as AIM-7 Sparrow. The AIM-120C utilizes ‘’clipped’’ wings and fins in order to

meet the internal carriage requirements of the F-22. AMRAAM consists of the

following major sections: Guidance, Armament, Propulsion, and Control. Other

components include a wiring harness, harness cover, Thermally Initiated Venting

System (TIVS), and wing and fin assemblies.

1.3.3.1 Weapons guidance section

Weapons Guidance Unit. The Weapons Guidance Unit (WGU) consists of the radome,

seeker, servo, transmitter-receiver, electronics unit, Inertial Reference Unit, Target

Detection Device (TDD), the harnesses, and frame structure. All units except the TDD

are contained within a sealed structure composed of the pyroceramicradome, titanium

skin sections, and aluminum aft bulkhead. The TDD, RF and video processor, and the

antennas are attached to the aft skin section as a complete testable assembly.

Electronics group functions include radar signal processing, seeker servo control, and

all of the computations performed in the central data processor. The WGU-16B is used

on AIM-120A missiles, the WGU-41/B is used on AIM-120B missiles, and the WGU-

44/B is used on AIM-120C missiles. Guidance sections on AIM-120B and AIM-120C

missiles contain Electronic Erasable Programmable Read Only Memory which allow

reprogramming of the missile software. Missile software versions are denoted by Tape

and Revision Numbers.

1.3.3.2 Weapons propulsion section

Weapons Propulsion Unit. The Weapons Propulsion Unit (WPU)-6/B consists of an

airframe, integral rocket motor, a blast tube and exit cone, and an Arm/Fire Device

(AFD) with a visible safe-arm indicator. The high performance rocket motor utilizes a

reduced smoke, hydroxyl terminated, polybutadiene propellant in a boost sustain

configuration, an asbestos-free insulated case (an integral part of the airframe), and an

28

integral aft closure, blast tube, and nozzle assembly with a removable exit cone to

facilitate control section installation/removal. Wings are attached in wing sockets at the

forward end of the propulsion section. Provisions are included within this section for

mounting the filter rectifier assembly.

1.3.3.3 Weapons control section

Weapons Control Unit. The Weapons Control Unit (WCU)-11/B consists of four

independently controlled electro-mechanical servo actuators, four lithium-aluminum

batteries connected in parallel, and a steel fuselage section that is bolted to the

propulsion section aft skirt. Each actuator consists of a brushless DC motor ballscrew,

an infinite resolution potentiometer directly coupled to the output shaft, and pulse

width modulated control electronics. The output shaft is engaged directly to a squib

actuated lock so that it does not interfere with the fin (control surface) installation and

removal. The wiring harness cover extends from the aft end of the guidance section to

the forward end of the control section. Its primary purpose is to provide protection for

the wiring harness. The main wiring harness electrically connects the umbilical

connector, guidance section, and control section. The wiring harness cover also houses

the TIVS. The TIVS is designed to vent rocket motor pressure in the event the missile is

exposed to a fuel fire. The TIVS consists of an external thermal cord which, when

ignited, triggers an Out-Of-Line Device (OOLD) that ignites a Linear Shape Charge

that weakens the rocket motor, allowing the rocket motor to vent without exploding.

The OOLD prevents the shaped charge from detonating should the booster in the

OOLD inadvertently detonate due to causes such as high impact. The unit has an

additional safety feature that causes it to “reset” within nine to thirteen units of gravity,

such as the acceleration experienced during missile launching. This feature prevents the

system from functioning during missile free flight so that the associated aerodynamic

pressures do not inadvertently enable the TIVS and thereby degrade the missile

performance. An indicator is on the wiring harness cover showing the condition of the

TIVS, either ‘’ENABLE’’ or ‘’DISABLE’’. Only TIVS equipped missiles are deployed

aboard Aircraft Carriers. The WPU-6/B Propulsion Section (with TIVS) meets the fast

cook-off and sympathetic detonation requirements of the IM program and the policy

delineated in OPNAV Instruction (OPNAVINST) 8010.13B. The other requirements

29

(bullet impact, fragment impact, and slow cook-off) have not been met with the current

configuration. However, the WPU-6/B has been granted the appropriate waivers for

shipboard use.

1.3.3.4 Wing and fin assemblies

Wing and fin assemblies provide flight control of the missile. The four wings are

detachable, stationary flight surfaces with ball fasteners to facilitate quick installation

and removal. The four fins provide the movable control surfaces. The AIM 7 is

interchangeable with AIM-120A and AIM-120B missiles. The AIM-120C utilizes

‘’clipped’’ wings and fins in order to meet the internal carriage requirements on ATD

starting in FY98 that will advance the AMRAAM development to provide a full-up

integration. AMCOM will deliver 2 AMRAAM fire units in 2Q FY99 for Marine Corps

operational suitability and effectiveness testing.

1.3.4 Air to air missiles features and techniques

To understand the benefits of applying the control strategy, many topics in that field

are needed to be discussed. These topics are very important in evaluating any type of

missile. The following subsections present these topics.

1.3.4.1 Target acquisition technique

It is classified into many ways to detect and track a target, these types are classified

according to the range of the missiles and target type, such as TV guided missiles, laser,

radar, and infrared ones (IR). All these different technique follow some guidance laws in

each guidance phases to verify the best trajectory of the missile that it should follow to

keep the target in the required dome origin (shifted) to shorten the duration time until

hitting the target.

1.3.4.2 Aim point when impacting the target

It is the best point in the target at which the missile can cause the greatest damage

after hitting it which is usually the center of the target image on the focal plane array on

the missile dome.

30

1.3.4.3 Proximity fuse distance and locking time

Proximity fuse distance is the maximum distance between the target and the seeker

within which the missile can cause great damage to that target. This is determined by

the type of the missile, the type and quantity of the ammunition, the target geometrical

feature and the control strategy of the missile.

The proximity fuse in AIM-9R functions according to the intensity of the IR rays

coming from the target indicating the target-seeker distance.

Locking time is the time required since the missile enters the proximity fuse envelope

until activating the missile ammunition which could be constant or variable time.

Locking time is very beneficial to secure direct and close impact of the target.

1.3.4.4 Jamming and seduction techniques

All countermeasures that have been established to protect aircraft from missiles such

as Electronic Countermeasures are usually attached to the aircraft at the front part. The

theory of this countermeasures is simply the capability of the instrument to guide the

missile wrongly away from the real position of the target which is done electronically,

but simple jamming and seduction techniques are activated either by dropping flares for

heat seeking missile or chaff for the radar missiles.

1.3.4.5 Reticle seeker (conventional technique)

It is used in most air-to-air missiles as a way to determine the location of the target on

the dome window of the missile in such a way using the wave shift and frequency of the

detector rotating element.

The reticle seeker is the most common optical system design employed in conventional

heat seeking missiles. It was invented by the Germans during the latter phase of World

War 2. It provides a means of using a single detector element to produce an error signal

in rectangular coordinates, for a point target somewhere within the cone which

represents the view of the seeker.

The technique is based on the idea of mechanically chopping the light flux which

impinges on a detector, in such a fashion that the characteristics of the chopped light

pulses vary with position of the light source in the field of view. Because the detector

31

produces an electrical signal directly proportional to the impinging light flux, electronic

hardware can be built to extract a positional error signal in X, Y coordinates, suitable

for driving a missile autopilot with traditional linear controllers.

1.3.4.6 Target seeker distance calculation

Target seeker distance can be easily evaluated from the targets area on the focal plane

array through the missile dome window using the similarity and the target pre-

calculated geometry using the following simple relation.

Target-seeker distance (x) is inversely proportional to the square root of the area of

the target on the focal plane array of the missile detecting screen. Eq. (1.1) is used to

determine target-seeker distance.

Tc

arTarget-seeker distance=

(1.1)

Where (ar) represents the target area on the frontal screen of the missile (m2), Tc

represents a constant depending on the target geometry and the projection angle of the

target on the frontal screen of the missile (target orientations).

This thesis consists of six chapters and two appendices:

- Chapter 1 includes appropriate background about different types of missiles.

- Chapter 2 includes literature review, problem formulation, objectives and thesis

layout.

- Chapter 3 describes modeling procedure of air-to-air missiles.

- Chapter 4 discusses different types of neural network models used for control.

- Chapter 5 explains the fuzzy logic technique and how to apply it for control.

- Chapter 6 presents the results of the computer simulations of the proposed

system.

- Chapter 7 discusses the final conclusions and future work.

- Appendix A presents the Simulink models for the air-to-air missile, neural

network controller, fuzzy logic controller and the whole integrated control

system.

- Appendix B presents an example of a gust effect as a system disturbance.

32

Chapter 2

LITERATURE REVIEW

2.1 Introduction

The development and application of most present day systems and control theory

were spurred on by the need to resolve aerospace problems. This is roughly the problem

of analyzing and designing guidance law and flight control systems (autopilot) for

tactical missiles or aircraft. Therefore, it is beneficial to review the development of

systems and control theory. The guidance and control laws used in current tactical

missiles are mainly based on classical control design techniques. These control laws were

developed in the 1950s and have evolved into fairly standard design procedures (Locke,

1955). Earlier guidance techniques worked well for targets that were large and traveled

at lower speeds. However, these techniques are no longer effective against the new

generation targets that are small, fast, and highly maneuverable. For example, when a

ballistic missile reenters the atmosphere after having traveled a long distance, its radar

cross section is relatively small, its speed is high and the remaining time to ground

impact is relatively short. Intercepting targets with these characteristics is a challenge

for present-day guidance and control designs. In addition, the missile-target dynamics

are highly nonlinear partly because the equations of motion are best described in an

inertial system while the aerodynamic forces and moments are best represented in a

missile and target body axis system.

Moreover, unmodeled dynamics or parametric perturbations usually exist in the plant

modeling. Because of the complexity of the nonlinear guidance design problem, prior

approximations or simplifications have generally been required before the analytical

guidance gains can be derived in the traditional approaches (Lin, 1991) [2]. Therefore,

one does not know exactly what the true missile model is, and the missile behavior may

change in unpredictable ways. Consequently, one cannot ensure

optimality of the resulting design. In the last three decades, optimality-based guidance

designs have been considered to be the most effective way for a guided missile engaging

the target (Bryson and Ho 1969; Lin, 1991; Zarchan,1994) [3], [2] and [4]. However, it is

also known from the optimal control theory that a straightforward solution to the

33

optimal trajectory shaping problem leads to a two-point boundary value problem

(Bryson and Ho,1969) [3], which is too complex for real-time onboard implementation.

Based on the reasons given above, advanced control theory must be applied to a

missile guidance and control system to improve its performance. The use of intelligent

control systems has infiltrated the modern world. Specific features of intelligent control

include decision making, adaptation to uncertain media, self organization, planning and

scheduling operations. Very often, no preferred mathematical model is presumed in the

problem formulation, and information is presented in a descriptive manner. Therefore,

it may be the most effective way to solve the above problems. Intelligent control is a

control technology that replaces the human mind in making decisions, planning control

strategies, and learning new functions whenever the environment does not allow or does

not justify the presence of a human operator. Artificial neural networks and fuzzy logic

are two potential tools for use in applications in intelligent control engineering. Artificial

neural networks offer the advantage of performance improvement through learning by

means of parallel and distributed processing. Many neural control schemes with back-

propagation training algorithms, which have been proposed to solve the problems of

identification and control of complex nonlinear systems, exploit the nonlinear mapping

abilities of neural networks (Miller et al., 1991 [5]; Narendra and Parthasarthy, 1990

[6]). Recently, adaptive neural network algorithms have also been used to solve highly

nonlinear flight control problems. A fuzzy logic-based design that can resolve the

weaknesses of conventional approaches has been cited above. The use of fuzzy logic

control is motivated by the need to deal with highly nonlinear flight control and

performance robustness problems. It is well known that fuzzy logic is much closer to

human decision making than traditional logical systems. Fuzzy control based on fuzzy

logic provides a new design paradigm such that a controller can be designed for

complex, ill defined processes without knowledge of quantitative data regarding the

input-output relations, which are otherwise required by conventional approaches

(Mamdani and Assilian, 1975 [7]; Lee, 1990 [8] ;Driankovet al., 1993 [9]). An overview

of neural and fuzzy control designs for dynamic systems was presented

by Dash et al. (1997) [10]. Very few papers have addressed the issue of neural or fuzzy-

based neural guidance and control design. The published literature in this field will be

introduced in this paper. The following sections are intended to provide the reader with

34

a basic, and unified view of the concepts of intelligent control. Many potentially

applicable topologies are well studied. It is hoped that the material presented here will

serve as a useful source of information by providing for solutions for current problems

and future designs in the field of guidance and control engineering.

2.2 Conventional Guidance and Control Design

Tactical missiles are normally guided from shortly after launch until target

interception. The guidance and control system supplies steering commands to

aerodynamic control surfaces or to correct elements of the thrust vector subsystem so as

to point the missile towards its target and make it possible for the weapon to intercept a

maneuvering target.

2.2.1 Guidance

From the viewpoint of a control configuration, guidance is a special type of

compensation network (in fact, a computational algorithm) that is placed in series with a

flight control system (also called autopilot) to accomplish an intercept. Its purpose is to

determine appropriate pursuer flight path dynamics such that some pursuer objective

can be achieved efficiently. For most effective counterattack strategies, different

guidance laws may need to be used to accomplish the mission for the entire trajectory.

First, midcourse guidance refers to the process of guiding a missile that cannot detect its

target when launched; it is primarily an energy management and inertial

instrumentation problem. When a radar seeker is locked onto a target and is providing

reliable tracking data, such as the missile-target relative range, line-of-sight (LOS)

angle, LOS angle rate and bore sight error angle, the guidance strategy in this phase is

called terminal guidance. Steering of

the missile during this period of flight has the most direct effect on the final miss

distance. The steering law should be capable of achieving successful intercept in the

presence of target maneuvers and external and internal disturbances.

2.2.2 Flight control system

The flight control system executes commands issued based on the guidance law with

fidelity during flight. Its function is three-fold: it provides the required missile lateral

35

acceleration characteristics, it stabilizes or damps the bare airframe, and it reduces the

missile performance sensitivity to disturbance inputs over the required flight envelope.

2.2.3 Conventional design methods

The principles behind controlling guided missiles are well known to control engineers.

Since the basic principles were extensively covered by Locke (1955) [11], a large number

of control technologies have been developed to improve missile performance and to

accommodate environmental disturbances. These techniques are mainly based on

classical control theory. Many different guidance laws have been exploited based on

various design concepts over the years. Currently, the most popular terminal guidance

laws defined by Locke (1955) [11] involve LOS guidance, LOS rate guidance, command-

to-line-of-sight(CLOS) guidance (Ha and Chong, 1992) [12] and other advanced

guidance strategies, such as proportional navigation guidance (PNG) (Locke, 1955) [11],

augmented proportional navigation guidance (APNG) (Zarchan, 1994) [13] and optimal

guidance law based on linear quadratic regulator theory (Bryson and Ho, 1969 [3];

Nazaroff, 1976 [14]), linear quadratic Gaussian theory (Potter, 1964 [15]; Price and

Warren, 1973 [16]) or linear exponential Gaussian theory (Speyer et al., 1982). Classical

guidance laws different from these guidance laws were discussed by Lin (1991) [2], and

the performance of various guidance laws was extensively compared. Among the current

techniques, guidance commands proportional to the LOS angle rate are generally used

by most high speed missiles today to correct the missile course in the guidance loop. This

approach is referred to as PNG and is quite successful against nonmaneuvering targets.

While PNG exhibits optimal performance with a constant velocity target, it is not

effective in the presence of target maneuvers and often leads to unacceptable miss

distances. Classical and modern guidance designs were compared by Nesline and

Zarchan (1981) [17]. The midcourse guidance law is usually a form of PNG with

appropriate trajectory-shaping modifications for minimizing energy loss. Among the

mid-course guidance laws, the most effective and simplest one is the explicit guidance

law (Cherry, 1964) [18].The guidance algorithm has the ability to guide the missile to a

desired point in space while controlling the approach angle and minimizing a certain

appropriate cost function. The guidance gains of the explicit guidance law are usually

selected so as to shape the trajectory for the desired attributes (Wang, 1988 [19]; Wang

36

et al., 1993 [20]). Other midcourse guidance laws are theoretically optimal control-based

approaches (Glasson and Mealy, 1983 [21]; Cheng and Gupta, 1986 [22]). These

research efforts have produced many numerical algorithms for open-loop solutions to

problems using digital computers. However, the main disadvantage of these algorithms

is that they generally converges slowly and are not suitable for real-time applications.

Unfortunately, only rarely is it feasible to determine the feedback law for nonlinear

systems which are of any practical significance. The flight control system used in almost

all operational homing missiles today is a three loop autopilot, composed of a rate loop,

an accelerometer, and a synthetic stability loop. Generally, the controller is in a form of

proportional-integral-derivative (PID) parameters, and the control gains are deter-

mined by using classical control theory, such as the root locus method, Bode method or

Nyquist stability criterion (Price and Warren, 1973 [16]; Neslineet al.,1981 [17]; Nesline

and Nesline, 1984 [23]). Modern control theory has been used extensively to design the

flight control system, such as in the linear quadratic techniques (Stallard, 1991 [24]; Lin

et al., 1993 [25]), generalized singular linear quadratic technique (Lin and Lee, 1985)

[26], design technique (Lin, 1994) [27] and feedback linearization (Lin, 1994) [27]. Over

the past three decades, a large number of guidance and control designs have been

extensively reported in the literature. For a survey of modern air-to-air missile guidance

and control technology, the reader is referred to Cloutieret al. (1989) [28]. Owing to

space limitations, only representative ones were cited above. For further studies on

various design approaches that have not been introduced in this section, the reader is

referred to Lin (1991) [2] and Zarchan (1994) [13]. Current highly maneuverable

fighters pose a challenge to contemporary missiles employing classical guidance

techniques to intercept these targets. Guidance laws currently in use on existing and

fielded missiles may be inadequate in battlefield environments. Performance criteria will

probably require application of newly developed theories, which in turn will necessitate

a large computation capability compared to the classical guidance strategy. However,

advances in microprocessors and digital signal processors allow increased use of

onboard computers to perform more sophisticated computation using guidance and

control algorithms.

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2.3 Neural Net-based Guidance and Control Design

The application of neural networks has attracted significant attention in several

disciplines, such as signal processing, identification and control. The success of neural

networks is mainly attributed to their unique features:

(1) Parallel structures with distributed storage and processing of massive amounts of

information.

(2) Learning ability made possible by adjusting the network interconnection weights and

biases based on certain learning algorithms.

The first feature enables neural networks to process large amounts of dimensional

information in real-time (e.g. matrix computations), hundreds of times faster than the

numerically serial computation performed by a computer. The implication of the second

feature is that the nonlinear dynamics of a system can be learned and identified directly

by an artificial neural network. The network can also adapt to changes in the

environment and make decisions despite uncertainty in operating conditions.

2.3.1 Supervisory control

The neural controller in the system is utilized as an inverse system model as shown in

Fig. (2.1). The inverse model is simply cascaded with the controlled system such that the

system produces an identity mapping between the desired response (i.e., the net-work

input r) and controlled system output y. This control scheme is very common in robotics

applications and is appropriate for guidance law and autopilot designs. Success with this

model clearly depends on the fidelity of the inverse model used as the controller

(Napolitano and Kincheloe, 1995 [29]; Guezet al., 1998 [30]). In the terminal guidance

scheme proposed by Lin and Chen (1999) [31], a neural network constructs a specialized

on-line control architecture, which offers a means of synthesizing closed-loop guidance

laws for correcting the guidance command provided by the PNG. The neural network

acts as an inverse controller for the missile airframe. The results show that it can not

only perform very well in terms of tracking performance, but also extend the effective

defensive region. Moreover, based on its feature of adaptively, the neural net-based

guidance scheme has been shown to provide excellent performance robustness. It was

also demonstrated by Cottrell et al. (1996) [32] that using a neuro control scheme of this

type for terminal guidance law synthesis can improve the tracking performance of a

38

kinetic kill vehicle. Hsiao (1998) [33] applied the control scheme to treat the disturbance

rejection problem for the missile seeker. In addition, a fuzzy neural network control

architecture, called the fuzzy cerebellar model articulation controller (fuzzy CMAC),

similar to this scheme, was proposed by Geng and MaCullough (1997) [34] for designing

a missile flight control system. The fuzzy CMAC is able to perform arbitrary function

approximation with high speed learning and excellent approximation accuracy. A

control architecture based on the combination of a neural network and a linear

compensator was presented by Stecket al. (1996) [35] to perform flight control

decoupling. In Zhu and Mickle (1997) [36], a neural network was combined with a linear

time-varying controller to design the missile autopilot.

Fig. (2.1) Supervisory control scheme

2.3.2 Hybrid control

Psaltiset al. (1987) [37] discussed the problems associated with this control structure by

introducing the concepts of generalized and specialized learning of a neural control law.

It was thought that off-line learning of a rough approximation to the desired control law

should be performed first, which is called generalized learning. Then, the neural control

will be capable of driving the plant over the operating range and without instability. A

period of on-line specialized learning can then be used to improve the control provided

by the neural network controller. An alternative is shown in Fig. (2.2), it is possible to

utilize a linear, fixed gain controller in parallel with the neural control law. This fixed

39

gain control law is first chosen to stabilize the plant. The plant is then driven over the

operating range with the neural tracking performance of a kinetic kill vehicle. Hsiao

(1998) [33] applied the control scheme to treat the disturbance rejection problem for the

missile seeker. In network tuned online to improve the control. The guidance law (Lin

and Chen, 1999) [31] and flight control system (Stecket al., 1996) possess a similar

control scheme of this type.

Fig. (2.2) Hybrid control scheme

2.3.3 Model reference control

The two control schemes presented above do not consider the tracking performance. In

this scheme, the desired performance of the closed-loop system is specified through a

stable reference model, which is defined by its input-output pair {r(t), yR (t)}. As shown

in Fig. (2.3), the control system attempts to make the plant output y(t) match the

reference model output asymptotically. In this scheme, the error between the plant and

the reference model outputs is used to adjust the weights of the neural controller. In

papers by Light body and Irwin (1994) [38], Psaltiset al. (1987) [37] discussed the

problems the neural net-based direct model reference adaptive control scheme was

applied to design an autopilot for a bank-to-turn missile. A training structure was

suggested in these papers to remove the need for a generalized learning phase.

40

Techniques were discussed for the back propagation of errors through the plant to the

controller. In particular, dynamic plant Jacobian modeling was proposed for use as a

parallel neural forward model to emulate the plant.

Fig. (2.3) Model reference control scheme

2.3.4 Internal model control (IMC)

In this scheme, the role of the system forward and inverse models is emphasized. As

shown in Fig. (2.4), the system forward and inverse models are used directly as elements

within the feedback loop. The network NN1 is first trained off-line to emulate the

controlled plant dynamics directly. During on-line operation, the error between the

model and the measured plant output is used as a feedback signal and passed to the

neuro-controller NN2 . The effect of NN1 is to subtract the effect of the control signal

from the plant output; i.e., the feedback signal is only the influence due to disturbances.

The IMC plays a role as a feedforward controller. However, it can cancel the influence

due to unmeasured disturbances, which cannot be done by a traditional feed-forward

controller. The IMC has been thoroughly examined and shown to yield stability

robustness (Hunt and Sbarbaro-Hofer, 1991) [39]. This approach can be extended

readily to autopilot designs for nonlinear airframes under external disturbances.

41

Fig. (2.4) Internal model control scheme

2.3.5 Adaptive linear or nonlinear control

The connectionist approach can be used not only in nonlinear control, but also as a part

of a controller for linear plants. The tracking error cost is evaluated according to some

performance index. The result is then used as a basis for adjusting the connection

weights of the neural network. It should be noted that the weights are adjusted on-line

using basic back propagation rather than off-line. The control scheme is shown in Fig.

(2.5)

Fig. (2.5) Adaptive control scheme

2.3.6 Predictive control

Within the realm of optimal and predictive control methods, the receding horizon

technique has been introduced as a natural and computationally feasible feedback law.

In this approach, a neural network provides prediction of future plant response over a

specified horizon. The predictions supplied by the network are then passed on to a

42

numerical optimization routine, which attempts to minimize a specified performance

criteria in the calculation of a suitable control signal (Montague et al., 1991 [40]; Saint-

Donat et al., 1994 [41]). Figure (2.6) shows the complete scheme of a predictive

controller.

Fig. (2.6) Predictive control scheme

2.3.9 Example

A hybrid model reference adaptive control scheme is described here, where a neural

network is placed in parallel with a linear fixed-gain independently regulated autopilot

as shown in Fig. (2.7) (McDowell et al., 1997 [42]). The linear autopilot is chosen so as to

stabilize the plant over the operating range and provide approximate control. The

neural controller is used to enhance the performance of the linear autopilot when

tracking is poor by adjusting its weights. A suitable reference model is chosen to define

the desired closed-loop autopilot responses ¨ Zrefand ¨ Yrefacross the flight envelop.

These outputs are then compared with the actual outputs of the lateral autopilot ¨ Z and

¨ Y to produce an error measurement vector [ezey]T , which is then used in conjunction

with an adaptive rule to adjust the weights of the neural network so that the tracking

error will be minimized. A direct effect of this approach is to suppress the influence

resulting from roll rate coupling.

43

Fig. (2.7) Model reference control of coupled lateral dynamics

2.4 Fuzzy Logic Based Guidance and Control Design

The existing applications of fuzzy control range from micro-controller based systems

in home applications to advanced flight control systems. The main advantages of using

fuzzy are as follows:

(1) It is implemented based on human operator’s expertise which does not lend itself to

being easily expressed in conventional proportional-integral- derivative parameters of

differential equations, but rather in situation/action rules.

(2) For an ill-conditioned or complex plant model, fuzzy control offers ways to

implement simple but robust solutions that cover a wide range of system parameters

and, to some extent, can cope with major disturbances.

The sequence of operations in a fuzzy system can be described in three phases called

fuzzification, inference, and defuzzification shown as in Fig. 11. A fuzzification interface

converts input data into suitable linguistic values that may be viewed as labels of fuzzy

sets. An inference mechanism can infer fuzzy control actions employing fuzzy

implication and the rules of the interface in fuzzy logic. A defuzzification interface yields

a nonfuzzy control action from an inferred fuzzy control action. The knowledge base

involves the control policy for the human expertise and necessary information for the

proper functioning of the fuzzification and defuzzification modules. Fuzzy control was

first introduced and applied in the 1970’s in an attempt to design controllers for systems

that were structurally difficult to model. It is now being used in a large number of

44

domains. Fuzzy algorithms can be found in various fields, such as estimation, decision

making and, especially, automatic control.

2.4.1 Fuzzy proportional integral derivative (PID) control

In this case, fuzzy rules and reasoning are utilized on-line to determine the control

action based on the error signal and its first derivative or difference. The conventional

fuzzy two-term control has two different types: one is fuzzy-proportional-derivative

(fuzzy-PD) control, which generates a control output from the error and change rate of

error, and is a position type control; the other is the fuzzy-proportional integral (fuzzy-

PI) control, which generates an incremental control output from the error and change

rate of error, and is a velocity type control (Driankovet al., 1993) [9]. Figure (2.8) shows

a fuzzy-PD controller with normalization and denormalization processes. In Mizumoto

(1992) [43] and Qiao and Mizumoto (1996) [44], a complete fuzzy-PID controller was

realized using a simplified fuzzy reasoning method. Control schemes of these types can

be easily designed and directly applied to guidance and control system design. In fuzzy

logic terminal guidance design, the LOS angle rate and change of LOS angle rate can be

used as input linguistic variables, and the lateral acceleration command can be used as

the output linguistic variable for the fuzzy guidance scheme (Mishra et al., 1994) [45].

The LOS angle rate and target acceleration can also be used as input linguistic variables

to obtain an alternative fuzzy guidance scheme (Mishra et al., 1994 [45]; Lin et al., 1999

[31]). It has been shown that these fuzzy guidance schemes perform better than

traditional proportional navigation or augmented proportional navigation schemes, i.e.,

smaller miss distance and less acceleration command. A terminal guidance law was

proposed by Leng (1996) [46] using inverse kinematics and fuzzy logic with the LOS

angle and LOS angle rate constituting the input linguistic variables. A complete PID

guidance scheme employing heading and flight path angle errors was proposed by

Gonslaves and Caglayan (1995) [47] to form the basis for fuzzy terminal guidance. The

fuzzy-PD control scheme has also been applied to various missile autopilot designs

(Schroeder and Liu, 1994 [48]; Lin et al., 1998 [49]). Input-output stability analysis of a

fuzzy logic-based missile autopilot was presented by Farinewataet al. (1994) [50]. A

fuzzy logic control for general lateral vehicle guidance designs was investigated by

Hessburg (1993) [51]. In the papers by Zhao et al. (1993) [52] and Ling and Edgar (1992)

45

[53], fuzzy rule-based schemes for gain-scheduling of PID controllers were proposed.

These schemes utilize fuzzy rules and reasoning to determine the PID controller’s

parameters. Based on fuzzy rules, human expertise is easily utilized for PID gain

scheduling.

Fig. (2.8) Fuzzy PD controller

2.4.2 Hybrid fuzzy controller

Fuzzy controllers can have inputs generated by a conventional controller. Typically,

the error is first input to a conventional controller. The conventional controller filters

this signal. The filtered error is then input to the fuzzy system. Since the error signal is

purified, one needs fewer fuzzy sets describing the domain of the error signal. Based on

this specific feature, these types of controllers are robust and need a less complicated

rule base.

2.4.3 Fuzzy adaptive controller

The structure is similar to that of fuzzy PID controllers. However, the shapes of the

input/output membership functions are adjustable and can adapt to instantaneous

error. A typical fuzzy adaptive control scheme is shown as in Fig. (2.9) Since the

membership functions are adaptable, the controller is more robust and more insensitive

to plant parameter variations (Dash and Panda, 1996) [54]. In a paper by Lin and Wang

(1998) [49], an adaptive fuzzy autopilot was developed for bank-to-turn missiles. A self-

organizing fuzzy basis function was proposed as a tuning factor for adaptive control. In

Huang et al. (1994) [55], an adaptive fuzzy system was applied to autopilot design of the

X-29 fighter.

46

Fig. (2.9) Typical adaptive fuzzy control scheme

2.4.4 Fuzzy Sliding Mode Controller (SMC)

Although fuzzy control is very successful, specially for control of non-linear systems,

there is a drawback in the designs of such controllers with respect to performance and

stability. The success of fuzzy controlled plants stems from the fact that they are similar

to the SMC, which is an appropriate robust control method for a specific class of non-

linear systems. The fuzzy SMC as shown in Fig. (2.10) can be applied in the presence of

model uncertainties, parameter fluctuations and disturbances, provided that the upper

bounds of their absolute values are known (Driankovet al., 1993 [9]; Ting et al., 1996

[56]; Palm and Driankov, 1997 [57]).

Fig. (2.10) Fuzzy sliding mode control scheme

2.4.5 Fuzzy model following controller

To have the advantages of a fuzzy logic controller with a desired level of performance, a

fuzzy adaptive controller can be used in a model-following control system as shown in

47

Fig. (2.11) In this scheme, the error between the plant output and the reference model

output is used to adjust the membership functions of the fuzzy controller (Kwong and

Passino,1996) [58].

Fig. (2.11) Fuzzy model-following control scheme

2.4.6 Hierarchical fuzzy controller

In a hierarchical fuzzy controller as shown in Fig. (2.12), the structure is divided into

different levels. The hierarchical controller gives an approximate output at the first

level, which is then modified by the second level rule set. This process is repeated in

succeeding hierarchical levels (Kandel and Langholz, 1994 [59]).

Fig. (2.12) Hierarchical fuzzy control system

48

2.4.7 Optimal control

A fuzzy logic system can be utilized to realize an optimal fuzzy guidance law. In this

approach, exact open-loop optimal control data from the computed optimal time

histories of state and control variables are used to generate fuzzy rules for fuzzy logic

guidance. First, data related to the state and control variables of optimal guidance are

generated using several scenarios of interest. The fuzzy logic guidance law possesses a

neurofuzzy structure. Critical parameters of the membership functions of linguistic

variables are presented in the connecting weights of a neural network. The collected

data are then used to train the network’s weights by using the gradient algorithm or

other numerical optimization algorithms. After training has been performed

successfully, missile trajectories and acceleration commands for the optimal solution

and fuzzy logic guidance solution will be close during actual flight using these scenarios.

This approach can effectively resolve the computational difficulty involved in solving the

two-point boundary-value problem. The problem considered by Bouletet al. (1993) [60]

was that of estimating the trajectory of a maneuvering object using fuzzy rules. The

proposed method uses fuzzy logic algorithms to analyze data obtained from different

sources, such as optimal control and kinematical equations, using values sent by sensors.

2.4.8 Example

Figure (2.13) shows a fuzzy logic oriented architecture employed in a fuzzy terminal

guidance system (Gonsalvs and Caglayan, 1995) [47]. The architecture is duplicated for

both the heading and flight path angle channels. Guidance path errors drive in parallel

with a PD and a PI controller. The results produced by the fuzzy PD/PI controllers

(uPDand uPI,respec-tively) are combined via a fuzzy weighting rule-base. The combined

control utotalis then processed via a gain scheduler to account for variations over the

flight envelope. A fuzzy terminal guidance system can readily achieve satisfactory

performance that equals or exceeds that of conventional guidance approaches with

additional advantages, such as intuitive specification of guidance and control logic, the

capability of rapid prototyping via modification of fuzzy rule-bases and robustness to

sensor noise and failure accommodation.

49

Fig. (2.13) A fuzzy terminal guidance system

It should be noted that fuzzy control systems are essentially nonlinear systems.

Therefore, it is difficult to obtain general results from the analysis and design of

guidance and control systems. Furthermore, knowledge of the aerodynamics of missiles

is normally poor. Therefore, the robustness of the resulting

designs must be evaluated to guarantee stability in spite of variations in aerodynamic

coefficients.

2.5 Gain-Scheduling Guidance and Control Design

Gain scheduling is an old control engineering technique which uses process variables

related to dynamics to compensate for the effect caused by working in different

operating regions. It is an effective way to control systems whose dynamics change with

the operating conditions. It is normally used in the control of nonlinear plants in which

the relation-ship between the plant dynamics and operating conditions is known, and for

which a single linear time-invariant model is insufficient (Rugh, 1991 [61]; Hualin and

Rugh, 1997 [62]; Tan et al., 1997 [63]). This specific feature makes it especially suitable

for guidance and control design problems. Gain-scheduling design involves three main

tasks: partitioning of the operating region into several approximately linear regions,

designing a local controller for each linear region, and interpolation of controller

parameters between the linear regions. The main advantage of gain scheduling is that

controller parameters can be adjusted very quickly in response to changes in the plant

dynamics. It is also simpler to implement than automatic tuning or adaptation.

50

2.5.1 Conventional Gain-Scheduling (CGS)

A schematic diagram of a CGS control system is shown in Fig. (2.14). As can be seen,

the controller parameters are changed in an open-loop fashion based on measurements

of the operating conditions of the plant. A gain-scheduled control system can, thus, be

viewed as a feedback control system in which the feedback gains are adjusted using

feedforward compensation (Tan et al., 1997) [63]. Gain scheduled autopilot designs for

tactical missiles have been proposed by Balas and Packard (1992) [64] and Eberhardt

and Wise (1992) [65]. An approach to gain scheduling of linear dynamic controllers has

been considered for a pitch-axis autopilot design problem. In this application, the linear

controllers are designed for distinct operating conditions using H- methods (Nichols et

al., 1993 [66]). A gain scheduling eigenstructure assignment technique has also been

used in autopilot design (Piou and Sobel, 1996) [67].

Fig. (2.14) Conventional gain-scheduling control scheme

2.5.2 Fuzzy Gain-Scheduling (FGS)

The main drawback of CGS is that the parameter change may be rather abrupt across

the boundaries of the region, which may result in unacceptable or even unstable

performance. Another problem is that accurate linear time-invariant models at various

operating points may be difficult, if not impossible, to obtain. As a solution to these

problems, FGS has been proposed, which utilizes a fuzzy reasoning technique to

determine the controller parameters (Takagi and Sugeno, 1985) [68]. For this approach,

human expertise in the linear control design and CGS are represented by means of fuzzy

51

rules, and a fuzzy inference mechanism is used to interpolate the controller parameters

in the transition regions (Ling and Edgar, 1992 [53]; Tan et al., 1997 [63]). Figure (2.15)

shows the fuzzy gain-scheduled control scheme. The Takagi-Sugeno fuzzy models

provide an effective representation of complex nonlinear systems in terms of fuzzy sets

and fuzzy reasoning applied to a set of linear input-output submodels. Based on each

models, fuzzy gain-scheduling controllers can be obtained by means of linear matrix

inequality methods (Zhao et al., 1996) [69]. An H_ gain-scheduling technique using

fuzzy rules was also proposed by Yang et al. (1996) [70] to ensure stability and

performance robustness. The FGS technique has been used in missile guidance design

(Hessburg, 1993 [51]; Lin et al., 1999 [31]) and aircraft flight control design (Gonsalves

and Zacharias, 1994 [71]; Wang and Zhang, 1997 [72]; Adams et al., 1992 [73]). A

robust fuzzy gain scheduler has also been designed for autopilot control of an aircraft

(Tanaka and Aizawa, 1992) [74]. In a paper by Pedrycz and Peters (1997) [75] a

controller of this type was applied for attitude control of a satellite.

Fig. (2.15) Fuzzy gain-scheduling control scheme

2.5.3 Neural Network Gain-Scheduling (NNGS)

NNGS can incorporate the learning ability into gain-scheduling control (Tan et al.,

1997) [63]. The training example consists of operating variables and control gains

obtained at various operating points and their corresponding desired outputs. The main

advantage of NNGS is that it avoids the need to manually design a scheduling program

52

or determine a suitable inferencing system. A representative neural gain scheduling PID

control scheme is shown in Fig. (2.16). In Chai et al. (1996) [76], an on-line approach to

gain-scheduling control of a nonlinear plant was proposed. The method consists of a

partitioning algorithm used to partition the plant’s operating space into several regions,

a mechanism that designs a linear controller for each region, and a radial basis function

neural network for on-line interpolation of the controller parameters of the different

regions. A neural controller design technique for multiple-input multiple-output

nonlinear plants was presented by Maia and Resende (1997) [77]. This technique is

based on linearization of a nonlinear plant model at different operating points. Then a

global nonlinear controller is obtained by interpolating or scheduling the gains of the

local operating designs. The neural gain-scheduling technique has been used in various

fields, such as hydroelectric generation (Liang and Hsu, 1994) [78], process control

(Cavalieri and Mirabella, 1996) [79], robotic manipulators (Wang et al., 1994) [80] and

aircraft flight control systems (Chu et al., 1996; Jonckheereet al., 1997).

Fig. (2.16) Neural network gain-scheduling PID control scheme

2.5.4 Neural-Fuzzy Gain-Scheduling (NFGS)

NFGS is implemented using a neural-fuzzy net-work that seeks to integrate the

representational power of a fuzzy inferencing system and the learning and function

approximation abilities of a neural net-work to produce a gain-scheduling system (Tan

et al., 1997 [63]; Tomescu and VanLandingham, 1997 [81]). As in NNGS, interpolation

53

of the controller parameters is adaptively learned by a neural-fuzzy network. Unlike to

FGS, the fuzzy rules and membership functions can be refined using learning and

training data. In contrast to NNGS, NFGS provides a more meaningful interpretation of

the network; in addition, expert knowledge can be incorporated into the fuzzy rules and

membership functions. The control scheme is shown in Fig. (2.17).

Fig. (2.17) Neural-fuzzy gain-scheduling control scheme

2.6 Guidance Laws

Song and Zhang [82] applied a passive homing missile’s variable structure

proportional navigation with terminal angular constraint. The objective was to design a

variable structure proportional navigation law for the passive homing missile so that the

missile can hit the target with a minimum miss-distance and desired impact attitude

angle within the required overload when it flies at a lower trajectory.

Choi and Chaw [83] applied a nonlinear adaptive guidance considering target

uncertainties and control loop dynamics. The fourth order state equation for integrated

guidance and control loop is formulated considering target uncertainties in control loop

dynamics. The state equation is further changed into the normal form by nonlinear

54

coordinate transformation. A nonlinear adaptive guidance law was proposed to

compensate the uncertainties in both target acceleration and control loop dynamics. The

proposed law adopted the sliding mode control approach with adaptation for unknown

bound of uncertainties.

Innocenti [84] applied a nonlinear guidance technique for agile missiles. This technique

presents new approaches to the guidance of agile missiles. They are based on nonlinear

discontinuous control techniques applied to the generation of guidance laws capable of

taking advantage of the vehicle’s post-stall capabilities. Agility and maneuverability

requirements imply a higher bandwidth and robustness for the guidance loop, which are

addressed by a variable structure controller format. Formal stability considerations are

presented, and the guidance structures are validated using nonlinear simulation.

Anderson and Burkhalter [85] applied a design of a guided missile interceptor using a

genetic algorithm. An apportioned genetic algorithm was used to manipulate a solid

rocket design code, an aerodynamic design code and a three-loop autopilot to produce

guided missile interceptor designs capable of accurately engaging a high-speed /high-

altitude target. Definition of the optimization problem required 29 design variables and

four primary goals were established to assess the performance of the interceptor

designs. Design goals included the following: minimize miss distance, minimize intercept

time, minimize takeoff weight, and minimize loading.

Zhou and Chundi [86] applied motion tracking sliding-mode guidance of a homing

missile with bearings-only measurements. Motion tracking sliding-mode control theory

was utilized to design a homing-missile guidance law, which induces missile target line of

sight angular-velocity-rate to track an arbitrary given signal. An adaptive two-step filter

was used to estimate missile-target relative distance, relative velocity, and target

acceleration involved in the guidance law during the whole guidance period.

Recently, neurofuzzy modeling techniques have been successfully applied to modeling

complex systems, where traditional approaches hardly can reach satisfactory results due

to the lack of sufficient domain knowledge. Much research has been done on

55

applications of neural networks (NNs) for identification and control of dynamic systems

[87]. According to their structures, the NNs can be mainly classified as feedforward

neural networks and recurrent neural networks [87]. It is well known that a

feedforward neural network is capable of approximating any continuous functions

closely. However, the feedforward neural network is a static mapping. Without the aid

of tapped delays the feedforward neural network is unable to represent a dynamic

mapping. Although much research has used the feedfoward neural network with tapped

delays to deal with dynamical problem. The feedforward neural network requires a

large number of neurons to represent a dynamic response in the time domain [88].

Moreover, the weight updates of the feedforward neural network do not utilize the

internal information of the neural network and the function approximation is sensitive

to the training data.

2.7 Advantages over conventional designs

(1) Fuzzy guidance and control provides a new design paradigm such that a control

mechanism based on expertise can be designed for complex, ill-defined flight dynamics

without knowledge of quantitative data regarding the input-output relations, which are

required by conventional approaches. A fuzzy logic control scheme can produce a

higher degree of automation and offers ways to implement simple but robust solutions

that cover a wide range of aerodynamic parameters and can cope with major external

disturbances.

(2) Artificial Neural networks constitute a promising new generation of information

processing systems that demonstrate the ability to learn, recall, and generalize from

training patterns or data. This specific feature offers the advantage of performance

improvement for ill-defined flight dynamics through learning by means of parallel and

distributed processing. Rapid adaptation to environment change makes them

appropriate for guidance and control systems because they can cope with aerodynamic

changes during flight.

2.8 Problem Formulation

Missile guidance control system is a complicated work due to the missile nonlinear

model that contains a cross coupling effect between its lateral and longitudinal motions.

56

The missile also is subjected to a continuous disturbance (gust effect) while its speed

reaches more than 5 Mach in some types. In spite of all these circumstances the miss

distance between the missile and the target should not be more than 5 meters to secure a

great damage to the target. In order to achieve such requirements a neuro-fuzzy control

system will be used in the present work.

2.9 Main Objective of the Thesis

This technique will be applied to AIM 9R as a passive homing missile using the

simplest guidance law that is CLOS (command to line of sight) to eliminate any errors

developed from different guidance laws to easily evaluate the applied control strategy.

The miss distance and minimum locking time should be calculated according to this

strategy that should satisfy the proximity fuse distance and acceptable minimum locking

time.

2.10 Thesis Layout

In this thesis, the concept of incorporating fuzzy logic into a neural network has

been discussed. In contrast to the pure neural network or fuzzy system, the fuzzy neural

network (FNN) possesses both their advantages. It combines the capability of fuzzy

reasoning in handling uncertain information and the capability of artificial neural

networks in learning from processes Fig. (2.18). In missiles controlling branch it is

needed to minimize the output performance errors until fulfilling the required

specifications such as the minimum locking time and the maximum fuse distance.

Nowadays there are different types of missiles using different types of control strategy

some of them uses reference frame unit as they are controlled by a remote base or an

aircraft. Consequently controlling such missiles by reference angles ,, with respect

to the fixed frame of reference. Another type of control strategies doesn’t use reference

frame unit such as all heatseeking and active radar missiles (passive and active homing

types). In this thesis, both types of control strategies are discussed. It presented a

technique for the preliminary design of a control system using neurofuzzy systems for a

diverse data range of a highly nonlinear MIMO 5_DOF model (AIM 9R air to air

missile) which is composed of 3_body axes velocities and rotating in both pitch and yaw

direction only as the rolling motion is prevented due to gyroscopic stabilizers (anti roll

57

system) attached at the rear fins. The model has cross coupling effects between the

longitudinal and lateral motions due to the coupled equations of motion. MIMO model

was divided into two separated SISO models taking into consideration the nonlinear

cross coupling effects, then generating the neural network controller for each separated

SISO model, blending the output performance using a 25 rule a Mamdani fuzzy logic

controller for each SISO model to be able to deal with the system uncertainty due to

cross coupling effect and affecting gust in the whole system when instantaneously

applying both inputs to the model. The main goal is to fulfill certain specifications within

its limited ranges such as the proximity fuse distance and the min target locking time

during the whole mission of the missile.

Fig. (2.18) Integrated control model for the guidance of AIM 9R missile

58

Chapter (3)

MATHEMATICAL MODELING OF AN AIR TO AIR MISSILE

3.1 Introduction

The first step in this research work is the development of a mathematical model of

AIM 9R air-to-air missile. There are a number of simple models commonly used for

dynamic analysis depending on the type of analysis to be performed. In this chapter it is

started by developing a comprehensive model sophisticated enough to be used in general

dynamic analysis. The model provides the ability of implementing various modern

control systems such as neural network and fuzzy logic control. This chapter is

organized as following:

3.2 The Rigid –Body Equations

The rigid-body equations of motion are derived from the first simple principles.

Let m in Fig. (3.1) be an element of mass of the missile, having velocity v relative to

fixed axes, and let F be the resultant force that acts on it.

Fig. (3.1) Missile body axes

59

Newton’s second law then gives the equation of motion of m as:

F= mvd

dt (3.1)

Summing Eq. (2.1) for all elements m in the missile to give:

F= mvd

dt=

dt

d v m (3.2)

The quantity F is a summation of all the forces that act upon all the elements.

The internal forces are those exerted by one element upon another occurring in equal

and opposite pairs, by Newton’s third law, and hence contribute nothing to the

summation. Thus F=F is the resultant external force acting on the missile.

The absolute velocity of m is

v = vc+ rd

dt (3.3)

wherevcis the velocity of the mass center. Thus,

v m= (vc + rd

dt) m (3.4)

v m=mvc +dt

d r m (3.5)

Since C is the mass center, then r m=0,so that

v m=mvc (2.6)

Where m is the total mass of the missile.

Thus Eq. (3.6) becomes:

F=mvcd

dt (3.7)

This equation relates the external force on the missile to the motion of its mass center.

The relationship between the external moment and the rotation of the missile is needed.

60

It is obtained from a consideration of the moment of momentum. The moment of

momentum of m is by definition h=rv m.

dt

d( h)=

dt

d(rv) m (3.8)

dt

d( h) =

rd

dtv m + r

vd

dt m (3.9)

Now from Eq. (3.3):

rd

dt=v-vc (3.10)

Also,

rvd

dt m=r F= G (2.11)

Eq. (3.11) follows from Eq. (3.1)and introduces G, the moment of F about C. Eq.

(3.9) then becomes

G=dt

d( h)-(v- vc) v m (3.12)

Since vv=0, Eq. (3.12) becomes

G=dt

d( h)+ vcv m (3.13)

Eq. (3.13) is now summed for all elements, as was Eq. (3.1)and becomes

G=dt

d ( h) + vc v m (3.14)

By an argument similar to that for F, G is shown to be the resultant

external moment about C, denoted G. h is called the moment of momentum, or

angular momentum of the missile and is denoted by h. Using Eq. (3.3) and noting that

vcvc=0 then Eq. (3.14) is reduced to as follows:

61

G= hddt (3.15)

3.3 Evaluation of the Angular Momentum (h)

From the definition of (h)

h= ( h) = (rv) m (3.16)

Let the angular velocity of the missile be

=Pi + Qj + Rk (3.17)

Where P, Q, R are the scalar components of respectively, and i, j, k are unit vectors

in the directions of x, y, z.

Now the velocity of a point in a rotating rigid body is given by

v= vc + r (3.18)

thus,

h= r ( vc + r) m (3.19)

h= rvc m+ r ( r) m (3.20)

Since vc is constant with respect to the summation, and since r m=0, then the first

sum in Eq. (2.20) is ( r m)vc=0.

The second sum is conveniently expanded by the rule for a vector triple product.

r ( r) = r2 –r( r) (3.21)

62

Since the vector r has the components (x, y, z) then Eq. (3.20) finally becomes

22 2( ) m (Px Qy Rz) myx z h r (3.22)

The scalar components of Eq. (3.22) are:

hx=P ( y2+z2) m-Q xy m - R xz m (3.23)

hy=-P xy m + Q (x2+z2) m - R yz m (3.24)

hz=-P xz m-Q yz m+R (x2+y2) m (3.25)

The summations that occur in these equations are the moments and products of inertia

of the missile (A, B, C, D, E , J), then the equations can be written in matrix form as:

x

y

z

h A J E P

h J B D Q

E D C Rh

(3.26)

where

A=2 2(y z ) m

B=2 2(x z ) m

C= 2 2(x y ) m

D= yz m

E= xz m

J= xy m

3.4 Euler’s Equation of Motion

It is noted from Eq. (3.15) that it is required the derivatives with respect to time of Eq.

(3.26). If axes of reference are non-rotating, then it is clear that, as the missile rotates,

63

the moments and products of inertia will vary, and derivatives of A, B,C, D, E, J will

appear in the equations, as well as derivatives of P, Q, R. This is the most undesirable

situation. It can be avoided, however, if the frame of reference Cxyz is fixed to the

missile and move with it, where A, B, C, D, E, J will be constants.

The derivative of a vector A, referred to a frame of reference rotating with angular

velocity is

Ad

dt=

A

t + A (3.27)

where

A Ax

tt i +

Ay

tj +

Az

tk (3.28)

The vector equations of motion Eq. (3.7), (3.15) become when referred to the frame of

reference C x y z, fixed to the missile (body axes of the missile).

Fx=m(U. + QO- RV) (3.29)

Fy=m(V. + RU - PO) (3.30)

Fz=m(O.+ PV - QU) (3.31)

dxdt

dydt

ddt z

L R Q h

M R P h

N Q P h

(3.32)

64

3.5 Orientation and Position of the Missile

Since the frame of reference adopted for the equations of motion is fixed to the

missile, and moves with it the position and orientation of the missile can’t be described

relative to it.

For this purpose an earth fixed frame of reference Ox’y’z’ is introduced as shown in

Fig. (3.2). Let oz’ be taken vertically downward, and Ox’ horizontal in the vertical plane

containing the initial velocity vector of the mass center. The origin O is assumed to

coincide with C at t=0.

The orientation of the missile is then given by a series of three consecutive rotations,

whose order is important. The missile is imagined first to be oriented to that its axes are

parallel to x’y’z’. It is then in the position Cx1y1z1. The following rotations are then

applied.

1. A rotation about cz1, carrying the axes to Cx2y2z2 (bringing Cx to its final

azimuth).

2. A rotation about cy2, carrying the axes to Cx3y3z3 (bringing Cx to its final

elevation).

3. A rotation about cx3, carrying the axes to their final position Cxyz (giving the final

angle of bank to the wings).

65

Fig. (3.2) Missile orientation around fixed frame of reference

3.6 The Flight Path Determination

It is recommended to have the coordinates of the flight path relative to the fixed

frame x’y’z’. To do this the velocity components in the direction x1,y1,z1 are required.

Let these be denoted U1,V1,O1, and similarly let subscripts 2, 3 denote components in the

directions of (x2,y2,z2) and (x3,y3,z3) respectively. Therefore:

dt

dx'=U1 (3.33)

dt

dy'=V1 (3.34)

dt

dz'=O1 (3.35)

here

U1=U2cos -V2sin (3.36)

V1=U2sin +V2cos (3.37)

O1=O2 (3.38)

66

U2=U3cos +O3sin (3.39)

V2=V3 (3.40)

O2=-U3sin +O3cos (3.41)

and

U3=U (3.42)

V3=Vcos -Osin (3.43)

O3=Vsin +Ocos (3.44)

Combining Eqs. (3.36) through (3.44) gives:

dt

dx'=Ucos cos +V(sin sin cos -cos sin )+O(cos sin cos + sin sin )

(3.45)

dt

dy'=Ucos sin +V(sin sin sin +cos cos )+O(cos sin sin -sin cos )

(3.46)

dt

dz'=-Usin +Vsin cos +Ocos cos (3.47)

The coordinates (x’,y’,z’) are then given by the integral of Eqs. (3.45), (3.46) and (3.70).

Integrating these equations in other than special simple cases is obviously a formidable

problem, but can be easily solved using MATLAB (simulink editor).

67

3.7 The Orientation of the Missile

The orientation of the missile would be presented in terms of angular velocity

components (P, Q, R). Let (i, j, k) be unit vectors, with subscripts 1,2,3 denoting

directions (x1,y1,z1),etc.

Let the missile experience, in time t, an infinitesimal rotation from the position

defined by , , to that corresponding to ( + ), ( + ), ( + ). The

vector. representing this rotation is given approximately by:

i1+ j3+ k2 (3.48)

Then the angular velocity is exactly:

=

t 0lim

t

= ϕ.

i1 + ϴ . j3 + ψ.

K2 (3.49)

If the components of given in Eqn. (3.48) be projected onto Cxyz, and remembering

that =Pi + Q j + R k, obtaining the following relations:

P= ϕ.- ψ.

sin (3.50)

Q= ϴ .cos + ψ.

cos sin (3.51)

R= ψ.cos cos - ϴ

. sin (3.52)

68

In order to find the angles ( , , ) in terms of (P, Q, R), the above equations must be

solved. If they are regarded as algebraic equations in (ϕ. ,ϴ

., ψ.

) then:

ϕ. = P + Qsin tan +Rcos tan ϴ (3.53)

ϴ .

= Qcos -Rsin (3.54)

ψ. = (Qsin +Rcos )sec ϴ (3.55)

Using Eqn’s. (2.56), (2.57), (2.58), ( , , ) are given by:

= t

0

(P + Qsin tan + Rcos tan )dt (3.56)

= t

0

(Qcos -Rsin )dt (3.57)

= t

0

(Qsin +Rcos )sec dt (3.58)

However it can be solved analytically only in simplified forms, or using MATLAB.

3.8 The Applied Forces and Moments

The equations of motion derived before contain the following forces and moments:

1-The external forces acting on the missile (Fx, Fy, Fz).

2-The external couples acting on the missile (L, M ,N).

3-The control surface hinge moments (He, Hr, Ha).

4-The generalized control forces (Fe,Fr,Fa).

The forces and moments in 1, 2, 3 above are the external actions on the missile .They

are of two kinds gravitational and aerodynamics ,and are in general determined by

69

orientation, configuration, and motion of the missile : i.e, by the variables ( , , ),

( e, r, a), (U,V,O), (P,Q,R).The generalized control forces, on the other hand are

internal forces.

The external forces are as following:

From Fig. (3.3) the components of the weights in the directions of the axes are found to

be.

Xg=-mg sin (3.59)

Yg=mg cos sin (3.60)

Zg=mg cos cos (3.61)

Fig. (3.3) Weight forces acting on the missile during flight

With the aerodynamic forces (including propulsive forces) denoted by (X, Y, Z), the

resultant external forces are:-

Fx=X- mg sin (3.62)

Fy=Y+ mg cos sin (3.63)

Fz=Z+ mg cos cos (3.64)

70

Since gravity does not contribute to the moments about the mass center, then (L, M, N)

are entirely aerodynamic. The effect of mass unbalance of the control surface in

producing a gravitational hinge moment will be assumed small and neglected, so that the

hinge moments (He, Hr, Ha) are also purely aerodynamic.

Finally, the equations of motion of the missile are as follows:

1-Force equations

Using Eqn’s (3.29), (3.30), (3.31), (3.62), (3.63) and (3.64), force equations are given by:

X- mg sin = m(U. + QO – RV) (3.65)

Y+ mg cos sin = m(V. + RU – PO) (3.66)

Z+ mg cos cos = m(W. + PV- QU) (3.67)

2-Moment equations

Using Eqn’s (3.23-3.25) and (3.26), moment equations are given by:

L=AP.- ER

.+ QR (C-B) - E PQ (3.68)

M=BQ.

+ RP (A-C) + E( 2P - 2

R ) (3.69)

N=-EP.

+ CR. + PQ (B-A)+E QR (3.70)

3-Missile orientation equations

ϕ. = P + Qsin tan +Rcos tan ϴ (3.71)

ϴ .

= Qcos -Rsin (3.72)

ψ. = (Qsin +Rcos )sec ϴ (3.73)

71

4-Flight path equations

dt

dx'=Ucos cos +V(sin sin cos -cos sin )+W(cos sin cos +sin sin )

(3.74)

dt

dy'=Ucos sin +V(sin sin sin +cos cos )+W(cos sin sin -sin cos )

(3.75)

dt

dz'= - U sin + Vsin cos + Wcos cos

(3.76)

For AIM canard configuration, further simplification may be made since the xy (as well

as xz) is also a plane of symmetry (i.e., E=0) and moment of inertia about the y axis is

generally equal to that about the z axis (i.e., B=C), and so due to antiroll gyroscopic

stabilizers rolling about x axis will be mostly prevented so angular velocity about x axis

P will almost equal to zero, hence the resultant simplified equations are:

M=BQ. (3.77)

Where Q. is angular acceleration about y axes and M is given by:

M= M + M

+ M '' + M '

' + M '' (3.78)

N=CR. (3.79)

Where R. is angular acceleration about z axes and N is given by:

N= N + N

+'N

' +'N

' +'N

' (3.80)

72

The mathematical model derived herein is built in Matlab / Simulink environment to

facilitate various computer simulations. Figure (3.4) shows the computation algorithm of

the missile model while Fig. (3.5) shows the missile detailed model in simulink editor.

(3.4). Appendix A includes some detailed models describing the building process.

Fig. (3.4) Missile model algorithm

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Fig. (3.5) AIM 9 Mathematical Modeling

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3.9 AIM 9R Model Characteristics

3.9.1 Model verification

The model was established and examined using (Simulink-MATLAB) simulations.

The output performance was then verified with another simplified model (3 DOF

Model) expressing only the longitudinal motion [89].

Figure (A1) shows 3 DOF model in Simulink model appendix A. This model will be used

in AIM 9R missile model verification. Figures (3.6) and (3.7) show the verification

process using Step and frequency response respectively.

Fig. (3.6) Pitch angle step response for 5 DOF and 3 DOF models

Fig.

(3.7) Pitch angle dynamic response for 5 DOF and 3 DOF models, linearization was

performed at frequency =.3 (rad/s) as an example

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Figure (3.6) and (3.7) indicate that AIM 9R missile model that was represented by 5

DOF model is verifying 3 DOF missile model with some errors due to the cross coupling

effect in 5 DOF model rather than 3 DOF model. Also the 5 DOF is highly nonlinear

more than the simplified 3 DOF model.

3.9.2 Cross effect representation

As mentioned before the model has a considerable cross coupling effect between its

longitudinal and lateral axes. This effect can be expressed through Fig. (3.8), (3.9) and

(3.10) using some quantitative measure such as mean value and standard deviation of

the cross coupling effect.

Fig. (3.8) Cross coupling effect (yaw on pitch)

Fig. (3.9) Mean value of the cross coupling effect versus input frequencies at a

constant amplitude of .1(rad) rudder deflection as an example

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Fig. (3.10) Standard deviation of cross coupling effect (yaw on pitch)

Figures (3.9) and (3.10) illustrate the cross coupling effect on longitudinal motion that

reaches the maximum value at rudder deflection input frequency =0.1 rad/s then decays

sharply until reaching rudder deflection input frequency =0.5 rad/s then continuing to

decay gradually according to the frequency increase.

3.9.3 Wind gust effect (model disturbance)

To determine the gust effect it is needed to examine the whole system without gust

effect and then with gust effect as shown in Fig. (3.11). This is done by applying a

Dryden gust through the pitch angular velocity Q and angle of attack .

Fig. (3.11) Gust effect (pitch motion)

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3.10 Summary

(a) Derivation of AIM 9R missile model including the aerodynamic and

gravitational forces and moments was implemented.

(b) Derivation of missile orientation and flight path equations was implemented.

(c) AIM 9 R model was verified using 3 DOF model for step and sine wave inputs.

(d) Cross coupling effect between longitudinal and lateral motions was examined

and illustrated using mean value and standard deviation methods. Resulting

that the maximum cross coupling effect on longitudinal motion is at rudder

deflection frequency =0.1 rad/s.

(e) Wind gust effect was illustrated and examined according to Dryden gust model.

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Chapter (4)

NEURAL NETWORK FOR MISSILE GUIDANCE

4.1 Introduction

As it was discussed in the previous chapter the AIM 9R missile model was built and

verified with another simplified one. Now the main objective is to use such model with

the most suitable controllers to drive this missile in the right trajectory that allow the

missile to hit the target and cause a great damage to its structure.

This chapter discusses the capabilities of the neural network, different types of neural

network controllers and how to apply these types of controllers to the AIM 9R missile

model in spite of the existing cross coupling effect between the longitudinal and lateral

motions and wind gust (system disturbance). The development of the neural network

passes through many phases during the last century and will be illustrated briefly as

follows.

Scientists have just begun to understand how biological neural networks operate. It

is generally understood that all biological neural functions, including memory, are

stored in the neurons and in the connections between them. Learning is viewed as the

establishment of new connections between neurons or the modification of existing

connections.

The neurons considered here are not biological. They are extremely simple

abstractions of biological neurons, realized as elements in program or perhaps as

circuits made of silicon. Networks of these artificial neurons do not have a fraction of the

power of the human brain, but they can be trained to perform useful functions.

Concepts and their accompanying mathematics are not sufficient for a technology to

mature unless there is some way to implement the system. For instant, the mathematics

necessary for the reconstruction of images from computer-aided tomography (CAT)

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scans was known many years before the availability of the high-speed computers and

efficient algorithms finally made it practical to implement a useful CAT system.

The history of neural networks has progressed through both conceptual innovations

and implementation developments. These advancements, however, seem to have

occurred in fits and starts rather than by steady evaluation.

Some of the background work for the field of neural networks occurred in the late

19th and early 20th centuries. This consisted primarily of interdisciplinary work in

physics, psychology and neurophysiology by such scientists as Hermann von Helmholtz,

Ernst Mach and Ivan Pavlov. This early work emphasized general theories of learning,

vision, conditioning, etc., and did not include specific mathematical models of neuron

operation.

The modern view of neural networks began in the 1940s with the work of Warren

McCulloch and Walter Pitts, who showed that the networks of artificial neurons could,

in principle, compute any arithmetic or logical function. Their work is often

acknowledged as the origin of the neural network field [90].

McCulloch and Pitts were followed by Donald Hebb, who proposed that classical

conditioning (as discovered by Pavlov) is present because of the properties of individual

neurons. Donald Hebb proposed a mechanism for learning in biological neurons [90].

The first practical application of artificial neural networks came in the late 1950s,

with the invention of the perceptron network and associated learning rule by Frank

Rosenblatt. Rosenblatt and his colleagues built a perceptron network and demonstrated

its ability to perform pattern recognition. This early success generated a great deal of

interest in neural network research. Unfortunately, it was later shown that the basic

perceptron network only could solve a limited class of problems [91].

At about the same time, Bernard Widrow and Ted Hoff introduced a new learning

algorithm and used it to train adaptive linear neural networks, which were similar in

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structure and capability to Rosenblatt’s perceptron. The Widrow –Hoff learning rule is

still in use today [92].

Unfortunately, both Rosenblatt’s and Widrow’s networks suffered from the same

inherent limitations. Rosenblatt and Widrow were aware of these limitations and

proposed new networks that would overcome them. However, they were not able to

successfully modify their learning algorithms to train the more complex networks [93].

Many people, influenced by Minsky and Papert, believed that further research was a

dead end. This, combined with the fact that there were no powerful digital computers on

which to experiment, caused many researchers to leave the field. For a decade neural

network research was suspended [94].

Some important work, however, continued during the 1970s. In 1972 Teuvo

Kohonenan James Anderson independently and separately developed new neural

networks that could act as memories. Stephen Grossberg was also very active during

this period in the investigation of self-organizing networks [95].

Interest in neural networks had faltered during 1960s because of the lack of new

ideas and powerful computers with which to experiment. During the 1980s both of these

impediments were overcome, and research in neural networks increased dramatically.

New personal computers and workstations, which rapidly grew in capability, become

available. In addition, important new concepts were introduced as follows.

Two new concepts were most responsible for the rebirth of neural networks. The first

was the use of statistical mechanics to explain the operation of a certain class of

recurrent network, which could be used as an associative memory. This was described in

a seminal paper by physicist John Hopfield [96].

The second key development of the 1980s was the backpropagation algorithm for

training multilayer perceptron networks, which was independently discovered by

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several different researchers. The most influential publication of the backpropagation

algorithm was by David Rumelhart and James McClelland [97], [98].

Well, so much for the history of neural networks to this date. The real question is,

“what will happen in the next twenty years?” Will neural networks take a permanent

place as a mathematical /engineering tool, or will they fade away as have so many

promising technologies? At present, the answer seems to be that neural network will not

only have their day but will have a permanent place, not as a solution to every problem,

but as a tool to be used in appropriate situations.

4.2 Theory and Applications

4.2.1 Single-input neuron

A single-input neuron is shown in Fig. (4.1). The scalar input P is multiplied by the

scalar weight W to form WP, one of the terms that are sent to the summer. The other

input, 1, is multiplied by a bias b and then passed to the summer. The summer output n,

often referred to as the net input, goes into a transfer function f, which produces the

scalar neuron output a.

Fig. (4.1) Single Input neuron

The neuron output is calculated as:

a= (wp+b) (4.1)

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4.2.2 Multiple-input neuron

Typically, a neuron has more than one input. A neuron with R inputs is shown in

Fig. (4.2). The individual inputs P1, P2, …PRare each weighted by correspondingelements

W1,1,W1,2,…..W1,Rof the weighting matrix W.

Fig. (4.2) Multiple Input Neuron

The neuron has a bias b, which is summed with the weighted input to form the net input

n:

n= W1,1 P1 +W1,2 P2+ …..W1,R PR+b (4.2)

This expression can be written in matrix form :

n= WP+b, (4.3)

where the matrix W for the single neuron case has only one row.

Now the neuron output can be written as:

a= (WP+b) (4.4)

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4.2.3 A layer of neurons

A single-layer networks of S neorons is shown in Fig. (4.3). Note that each of the R

inputs is connected to each of the neurons and that the weight matrix now has S rows.

Fig. (4.3) Layer of S Neurons

The layer includes the weight matrix, the summers, the bias vector b, the transfer

function boxes and the output vector a.

Each element of the input vector p is connected to each neuron through the weight

matrix W. Each neuron has a bias b, a summer, a transfer function f and an output a.

The outputs from the output vector aare given by.

a= (WP+b) (4.5)

4.2.4 Multiple layers of neurons

Now consider a network with several layers. Each layer has its own weight matrix W,

its own bias vector b, a net input n and an output vector a. It is needed to introduce

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some additional notation to distinguish between these layers. Superscripts will be used to

identify the layers. Specifically, number of the layer will be appended as a superscript to

the names for each of these variables.

Now some different types of neural network kits are prepared to be applied to

different control systems. These neural networks depend mainly on recurrent neural

networks such as inverse models, feedforward, feedback linearization…etc [17].

4.3 Neural network controllers

4.3.1 Control with inverse models

Conceptually, the most fundamental neural network based controllers are probably

those using the ‘inverse’ of the process as the controller as shown in Fig. (4.4). The

simplest concept is called direct inverse control.

The principle of which is that if the process can be described by:

y(t+1)=g(y(t),….,y(t-n+1),u(t),…,u(t-m)) (4.6)

where

y(t-1)……y(t-n+1) are the past values of output signal

u(t-1)……u(t-m) are the past values of control signal

a network is trained as the inverse of the process:

u(t)=g-1(y(t+1),y(t),….,y(t-n+1),u(t-1),u(t-m)) (4.7)

The inverse model is subsequently applied as the controller for the process by inserting

the desired output, the reference r(t+1), instead of the output y(t+1). Figures (4.5)

through Fig. (4.7) represent the dynamic response of this model without any disturbance

effect.

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Fig. (4.4) Direct inverse control

Fig. (4.5) Control by Direct Inverse Control at steady state flight

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Fig. (4.6) Control by Direct Inverse Control at climb angle =90 deg

Fig. (3.7) Control by Direct Inverse Control at initial velocity =700 m/s

It is clear from Fig. (4.5), (4.6) and (4.7) that the control signal is varying according to

the initial conditions of the launched missile. So in case of training off line all these

initial conditions should be taken into consideration otherwise training should be

performed on line. Training on line requires small sample time and very fast training

algorithm with a very fast microprocessor. In mean while if the contribution of this

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neural network controller is for few seconds in the missile mission it will be prefered if

the training was done off line as it will be discussed later.

4.3.2 Control by feedforward model

Using inverse models for feedback control leads to a dead-beat type of control, which is

unsuitable in many cases. If a PID controller has already been tuned for stabilizing the

process, an inverse model can be used for providing a feedforward signal directly from

the reference model as shown in Fig. (4.8). The dynamic response of this model is shown

in Fig. (4.9)

Fig. (4.8) Control by Feedforward

Fig. (4.9) Control by Feedforward model at steady state flight

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4.3.3 Control by feedback linearization

Feedback linearization is a common method for controlling certain classes of

nonlinear processes. This method provides a simple example of a discrete input-output

linearizing controller that is based on a neural network model of the process. The

discrete feedback linearization structure is given as follows.

y’ (t)=f(y(t-1),….,y(t-n),u(t-2),…u(t-m))+g(y(t-1),…,y(t-n),u(t-2),…,u(t-m))u(t-1) (4.8)

fand g are two separate networks. A feedback linearization controller {Fig. (4.10)} is

obtained by calculating the controls according to:

u(t)=w(t)-f(y(t),…,y(t-n+1),u(t-1),…,u(t-m+1))/g(y(t),…,y(t-n+1),u(t-1),…,u(t-m+1)) (4.9)

Selecting the virtual control, w(t), as an appropriate linear combination of past outputs

plus the references enables an arbitrary assignment of the closed-loop poles. Figure

(4.11) represent the dynamic response of this model without any disturbance effect.

Fig. (4.10) Control by feedback linearization

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Fig. (4.11) Control by feedback Linearization at steady state flight

Finally, it is obvious from Fig. (4.9), (4.10) and (4.11) that the inverse model has a great

advantage in controlling the AIM 9R missile model with the best performance rather than

feedback linearization or feedforward models. The feedforward model has bad performance

compared with the inverse one beside it has a very high frequency control signal that can not

match a practical missile actuator band width.

Finally, when inverse control technique is applied to the AIM 9R missile model in

chapter 2 the missile response can be examined with a sine wave input as follows:

-Figure (4.12) shows the missile pitch motion using NN controller (desired and actual ptch

angle).

-Figure (4.13) shows the integral square error using NN controller that is relatively high when

used for the whole mission.

-Figure (4.14) shows the missile absolute velocity of the missile using NN controller during the

whole mission varying sharply with the elevator deflection.

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-Figure (4.15) shows the missile angle of attack using NN controller indicating the maximum

angle of attack = 9 that maintain the low angle of attack aerodynamic missile model used in

the present work.

-Figure (4.16) shows the missile elevator deflection angle using NN controller according to the

reference input (reference pitch angle).

Fig. (4.12) Pitch motion using NN controller with sine wave input

Fig. (4.13) Integral square error using NN controller with sine wave input

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Fig. (4.14) Missile absolute velocity using NN controller with sine wave input

Fig. (4.15) Missile angle of attack using NN controller with sine wave input

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Fig. (4.16) Elevator deflection angle using NN controller with sine wave input

4.4 Summary

(a) Explaining the basic structures of the common neural network.

(b) Using different neural network model for control.

(c) Selecting the best performance neural network controller .

(d) Applying the neural network controller after training to the AIM 9R missile

model.

(e) Evaluating the performance of neural network controller and the complete

system performance.

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Chapter (5)

MISSILE GUIDANCE USING FUZZY LOGIC CONTROL

5.1 Introduction

The term fuzzy logic has been used in two different senses. It is thus important to clarify

the distinctions between these two different usages of the term. In a narrow sense, fuzzy

logic refers to a logical system that generalizes classical two-valued logic for reasoning

under uncertainty. In a broad sense, fuzzy logic refers to all of the theories and

technologies that employ fuzzy sets, which are classes with unsharp boundaries.

For instance, the concept of “warm room temperature” may be expressed as an

interval (e.g., [70 C or78 C] in classical set theory. However, the concept does not have

a well-defined natural boundary. A representation of the concept closer to human

interpretation is to allow a gradual transition from “not warm” to “warm”. In order to

achieve this, the notion of membership in a set needs to become a matter of degree. This

is the essence of fuzzy sets.

The power of the fuzzy logic appears when there is a need to control a nonlinear

model. Also the fuzzy logic enables control engineers to easily implement control

strategies used by human operators.

In the present work a real application of missile guidance used a fuzzy logic controller

to deal with the nonlinearity in the system and any other uncertainty such as cross

coupling effect between lateral and longitudinal motions and the gust effect.

5.2 Applying Fuzzy Inference System for Control

The fuzzy inference technique can be very useful in control engineering. A standard

control system would utilize a numerical input and produce numerical output and so

should a fuzzy controller. The knowledge base contains the set of inference rules chosen

to achieve the control objectives and the parameters of the fuzzy systems used to define

the data manipulation in the fuzzification, inference engine, and defuzzification

processes as shown in Fig. (5.1). The input to the fuzzification process is the measured or

estimated variable that appears in the antecedent part of the if-then rule. This input

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variable has associated linguistic values to describe it. Each linguistic value is defined by

a membership function, parameterized by data from the knowledge base .In the

inference engine the decision –making logic is conducted, inferring control laws from the

input variables through fuzzy implication. The final step is the defuzzification process

where a crisp control command is determined based on the inferred fuzzy control law

[99, 100].

Fig. (5.1) The structure of a Fuzzy Logic Control System

5.2.1 Designing membership functions (Fuzzification process)

Designing membership functions for a fuzzy set is a very important step in the

fuzzification process. Type number of membership function is affected by many factors

such as the type of the dynamic model, the nonlinearity of the model and the degree of

the affecting disturbance (uncertainty of the system). A membership function can be

designed in three ways:

(a) Interview those who are familiar with the underlying concept and later adjust it

based on a tuning strategy.

(b) Construct it automatically from data.

(c) Learn it based on feedback from the system performance.

The first approach was the main approach used by fuzzy logic researchers until 1980.

Due to lack of systematic tuning strategies in those days, most fuzzy systems were tuned

through a trial and error process. Figure (5.2) shows one of the simplest type of

membership functions which is a triangular type membership function. There are many

other types such as Guassian and trapezoid.

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Fig. (5.2) Triangular membership function

In this work the triangular membership function was chosen for all the used fuzzy sets

(error, change in error, control signal). Figures (5.3), (5.4) and (5.5) show the

membership functions for the error signal, the change in error signal, and the control

signal respectively.

Fig. (5.3) Membership function for error input (e)

Fig. (5.4) Membership function for error rate input (ce)

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Fig. (5.5) Membership function for control signal output (u)

5.2.2 Rule Derivation

The knowledge of the relationship between inputs and outputs of fuzzy controllers

are expressed as a collection of if-then rules to form what so called the rule-base of the

fuzzy controller. Four methods, possibly used in combination, are mainly used to

generate the rule-base of a fuzzy controller.

1-Expert experience and control engineering judgment [99, 100, 101].

In this approach the knowledge and experience of the designer are used to manually

construct the rule-base of the controller according to observation of the system.

2-Observation of an operator’s control action [102, 103, 104].

Many practical control tasks are difficult to describe by an explicit mathematical model.

However, they are successfully performed by skilled human operators. The input/output

data can be used to construct fuzzy control rules to mimic the behavior of human

operator.

3-Fuzzy model of the plant [105, 106].

A fuzzy model of the plant can be thought of as a linguistic description of the

dynamic characteristics of plant using fuzzy logic and inference. A set of subsequent

fuzzy control rules can be designed based on the fuzzy model of the plant to be

controlled.

4-Learning approaches [107, 108].

Motivated by the need of a systematic method to generate and modify fuzzy rule-bases,

much research is being conducted on developing learning approaches. This technique

begins with the self-organizing controllers which consist of two levels of fuzzy rule bases.

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The first rule base is the standard control fuzzy rule base. The second level contains a

fuzzy rule base consisting of meta-rules, which attempt to assess the performance of the

closed loop control system and subsequently used to modify the standard rule base.

Learning approaches based on evaluation theory, such as genetic algorithms, may have

a promising potential towards the derivation of fuzzy rule bases. Also, neural network

can be used to form a hybrid neurofuzzy control system that is capable of self-tuning.

4.2.2.1 Structure of fuzzy rules

A fuzzy rule is the basic unit for capturing knowledge in many fuzzy systems. A fuzzy

rule has two components: an if-part (also referred to as the antecedent) and a then-rule

(also referred to as the consequent):

IF <antecedent> THEN <consequent>

The antecedent describes a condition, and the consequent describes a conclusion that

can be drawn when the condition holds.

The structure of a fuzzy rule is identical to that of a conventional rule in artificial

intelligence. The main difference lies in the content of the rule antecedent-the antecedent

of the fuzzy describes an elastic condition (a condition that can be satisfied to a degree)

while the antecedent of a conventional rule describes a rigid condition (a condition that

is either satisfied or dissatisfied). For instance, consider the two rules below:

R1: IF the annual income of a person is greater than 120K, THEN the person is rich.

R2: IF the annual income of a person is high, THEN the person is rich.

Where high is a fuzzy set defined be membership function. The rule R1 is a conventional

one, because its condition is rigid. In contrast R2 is a fuzzy rule because its condition can

be satisfied to a degree for those people whose income lies in the boundary of the fuzzy

set High representing high annual income. The consequent of fuzzy rules can be

classified into two categories:

(a) Crisp consequent: IF….THEN y=a

Where a isnonfuzzy numeric value or symbolic value.

(b) Fuzzy consequent: IF….THEN y is A

Where A is a fuzzy set.

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4.2.3 Fuzzy rule-based inference

The algorithm of fuzzy rule-based inference consists of three basic steps and an

additional optional step.

(1) Fuzzy matching: Calculate the degree to which the input data match the

condition of the fuzzy rules. Figure (5.6) shows the degree of matching for two

rules using two different inputs through conjunctive conditions.

Fig. (5.6) Fuzzy matching process

(2) Inference: Calculate the rule’s conclusion based on its matching degree. Figure

(5.7) shows the clipping method for fuzzy inference for the first rule at matching

degree =0.3.

Fig. (5.7) Fuzzy inference process

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(3) Combination: Combine the conclusion inferred by all fuzzy rules into a final

conclusion. Figure (5.8) shows the combining fuzzy conclusions inferred by the

clipping method.

Fig. (5.8) Fuzzy combination process

(4) (Optional) Defuzzification: for applications that need a crisp output (e.g., in

control system), an additional step is used to convert a fuzzy conclusion into a

crisp one. Many ways can be used for such a purpose such as MOA (Maximum of

area) and COA (Center of area). The centroid (or COA) defuzzification method

calculates the weighted average of a fuzzy set as shown in Fig. (5.9). The result of

applying COA defuzzification to a fuzzy conclusion can be expressed by the

formula:

A i ii

A ii

( )y yy

( )y

(5.1)

Where A is the membership of area A.

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Fig. (5.9) Defuzzification process

In the present work these steps are applied to the inputs of the fuzzy logic

controller (system error, change in error) the control action (control signal) is then

generated at the defuzzification process. Figure (5.10) represent 25 rule-based

inference of Mamdani type fuzzy logic controller and final defuzzification step.

Fig. (5.10) Rule base calculations

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5.2.4 Response pattern of a simple generic fuzzy control system

The simple control strategy of the fuzzy control system can be developed as shown in

Fig. (5.11), maps the error, e=yd–y into the control action u. At the heart of this scheme

is a fuzzy logic control algorithm that operates in discrete time steps of period T and

maps the normalized values of error, en (t), and the change in error, cen(t), defined

respectively as

en(t)=noe e(t) (5.2)

cen(t)=noce (e(t)-e(t-T)) (5.3)

wherenoeand noceare the corresponding normalization factors, into changes in the

control action un (t) via rules of the form

If en (t) is P and cen(t) is N then un(t) is Z.

Where P, N and Z are short for positive, negative and zero, defined as fuzzy sets over the

normalized domains and definition of the relevant variables as shown in Fig.(5.12) and

Table (5.1).

Fig. (5.11) Architecture of the generic fuzzy control system

In reference to Fig. (5.11) the computed value of ( u) is subsequently denormalized and

integrated (added to the past value of (u) to form the present value of (u)) as follows:

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u nu(t) u(t T) (t)de u

(5.4)

wherede u is the corresponding denormalization factor.

Fig. (5.12) Step response of a simple generic fuzzy controller

Table (5.1) Control rules for a simple generic fuzzy controller

Region 1 2 3 4

IF error is P N N P

AND error rate is N N P P

THEN control signal (δu) is Z N Z P

In the same way the 25 rule-based Mamdani type fuzzy logic controller was established

and simulated through MATLAB then generating the final control action as shown in

Fig. (5.13). The main effort was done in tuning the fuzzy logic controller normalized and

denormalized factors to get the best output performance of the whole system.

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Fig. (5.13) Fuzzy inference system control action

5.3 Fuzzy Logic Control Application

Figure (5.7) shows the response of the missile in terms of its pitch angle when

controlling using a neural network and fuzzy logic controllers. It is significant from Fig.

(5.14) that the output performance of the Fuzzy Logic Controller is better than the

Neural Network Controller output performance.

Fig. (5.14) Comparison between Fuzzy logic controller and NN controller

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5.4 Comparison between Different Fuzzy Logic Controllers and PD

Controller

Figure (5.15) and (5.16) show the effect of using different normalized and denormalized

factors on the performance of fuzzy logic controller for pitch motion. Figure (5.17)

shows the difference of using a simple control law (PD controller)and the fuzzy logic

controller on pitch motion.

Fig. (5.15) Effect of normalized and denormalized factors

Fig. (5.16) Integral square error using PD and FL controllers for pitch motion

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Fig. (5.17) Comparison between PD and FL controllers with combined sine wave input

for pitch motion

5.5 Low pass Filter

It is required to apply a low pass filter to blend the fuzzy logic controller input

signals (e, e.). The dynamic performance of that filter is shown in Fig. (5.18) and (5.19).

The low pass filter is Butterworth type of cutoff frequency = 11 (rad/s).

Fig. (5.18) Dynamic response of a low pass filter

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Fig. (5.19) Integral square error for different Fuzzy logic controllers using low pass

filter

Figure (5.20) shows the error and error rate for the pitch motion using a low pass filter

that gives a smooth curvature expressing the real rate of increase and decrease of the

actual error, while Fig. (5.21) shows the error and error rate using the regular delay that

gives a bad display of error rate.

Fig. (5.20) Pitch error rate using filter

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Fig. (5.21) Pitch error rate using delay only

5.6 Effect of Using Neural Network in the First Time Period

It is clear now that using fuzzy logic controller is better than PD controller or NN

controller in this application. But for the first time period (first 5-seconds) it is obvious

from Fig. (5.22) that using a neural network controller with fuzzy logic controller is

better than using fuzzy logic controller only, as the integrated square error when using

NN controller is less than that when using fuzzy logic controller alone.

Fig. (5.22) Effect of using NNC for the first 5 seconds

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5.7 Fuzzy Logic Controller Output Signal

The output signal from the fuzzy logic controller is shown in Fig. (5.23) according

to the error (between the desired and the actual) and rate of error (e ,e ) with sine wave

input.

Fig. (5.23) Error, error rate and control signal as functions of time due to a sine wave

input to a fuzzy logic controller

5.8 Neurofuzzy Structure

Figure (5.24) shows the internal structure of the NeuroFuzzy controller

implemented through the Simulink editor.

Fig. (5.24) Neurofuzzy structure

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5.9 The Effect of Flight Regimes on Neurofuzzy Controller Performance

Figure (5.25) shows the effect of flight regime (velocity) on the output performance of

the Fuzzy logic and Neural Network controllers.

Fig. (5.25) Two types of fuzzy logic controllers output performance

5.10 Composite Fuzzy Guidance Law

The justification for selecting any one of the guidance laws from this set is based on the

following heuristic analysis. When the target is far away, proportional navigation

guidance law (low level controller) yields moderate acceleration commands. However, as

the range becomes small, the acceleration commands will become very large. Thus,

under small range conditions, alternate guidance laws that use bounded acceleration

commands must be found. Since bang-bang guidance law (high level controller)

intercepts targets using bounded acceleration at every time instant, this guidance law is

ideal for use under small range conditions. The augmented proportional navigation

(medium level controller) can serve as the “bridge” guidance scheme between

proportional navigation and bang-bang law. Figure (5.26) shows the composite guidance

law with the selected parameters K1, K2, K3 calculated from Fuzzy inference system.

110

Fig. (5.26) Composite guidance law

Eighteen fuzzy inference rules are used in the composite fuzzy inference system to

calculate K1, K2, K3 as shown in Fig. (5.27) using the ruled-base inference principle as

discussed in the beginning of this chapter.

Fig. (5.27) Defuzzification process

111

Figures (5.28), (5.29) and (5.30) show the values of K1, K2, K3 according the Fuzzy

inference system inputs (Range, Range rate). Thus calculating the contribution of each

controller in the final control action.

Fig. (5.28) Fusion Fuzzy system output K1

Fig. (5.29) Fusion Fuzzy system output K2

Fig. (5.30) Fusion Fuzzy system output K3

112

Finally, the missile response to various inputs, as simulated using MATLAB and

missile model of chapter 2 when applying fuzzy logic controller of chapter 4 is as

follows:

(a) Sine wave input for pitch motion

- Figure (5.31) shows the missile pitch motion (desired and actual) reflecting the

effect of using fuzzy logic controller rather than PD controller.

- Figure (5.32) shows the integral square error for PD and FL controller indicating

that FL controller is obviously better than PD controller as the integral square

error for FL controller is less than the integral square error of the PD controller.

- Figure (5.33) shows the missile angle of attack using NN controller indicating the

maximum angle of attack = 9 that maintain the low angle of attack aerodynamic

missile model used in the present work.

- Figure (5.34) shows the missile elevator deflection angle using FLC according to

sine wave input as a reference input indicating that the maximum deflection of

the elevator is 21 and the nominal deflection is 9 .

- Figure (5.35) shows the missile absolute velocity increasing from 250 up to 550

m/s according to sine wave input for 40 s mission interval.

Fig. (5.31) Comparison between PD and FL controllers with sine wave input for pitch

motion

113

Fig. (5.32) Integrated square error for PD and FL controllers with wave input for pitch

motion

Fig. (5.33) Angle of attack using FLC with sine wave input

114

Fig. (5.34) Elevator deflection angle using FLC with sine wave input

Fig. (5.35) Absolute velocity of the missile using FLC with sine wave input

(b) Step input for pitch motion

- Figure (5.36) shows the missile pitch motion (desired and actual) reflecting the

effect of using fuzzy logic controller rather than PD controller.

115

- Figure (5.37) shows the integral square error for PD and FL controller indicating

that FL controller is obviously better than PD controller as the integral square

error for FL controller is less than the integral square error of the PD controller.

- Figure (5.38) shows the missile angle of attack using NN controller indicating the

maximum angle of attack = 7 that maintain the low angle of attack aerodynamic

missile model used in the present work.

- Figure (5.39) shows the missile elevator deflection angle using FLC according to

sine wave input as a reference input indicating that the maximum deflection of

the elevator is 13 and the nominal deflection is .45 .

- Figure (5.40) shows the missile absolute velocity increasing from 250 up to

830m/s according to sine wave input for 40 s mission interval.

Fig. (5.36) Comparison between the step responses of the missile due to fuzzy logic and

PD controllers for 0.3 rad step input for the pitch motion

116

Fig. (5.37) Comparison between FLC and PD controller errors for 0.3 rad step input for

the pitch motion

Fig. (5.38) Angle of attack response for 0.3 rad step input using FLC

117

Fig. (5.39) Elevator deflection angle response for 0.3 step rad step input using FLC

Fig. (5.40) Absolute missile velocity response for 0.3 rad step input using FLC

(c) Ramp input for pitch motion

- Figure (5.41) shows the missile pitch motion (desired and actual) reflecting the

effect of using fuzzy logic controller rather than PD controller.

118

- Figure (5.42) shows the integral square error for PD and FL controller indicating

that FL controller is obviously better than PD controller as the integral square

error for FL controller is less than the integral square error of the PD controller.

- Figure (5.43) shows the missile angle of attack using NN controller indicating the

maximum angle of attack = 5 that maintain the low angle of attack aerodynamic

missile model used in the present work.

- Figure (5.44) shows the missile elevator deflection angle using FLC according to

sine wave input as a reference input indicating that the maximum deflection of

the elevator is 7 and the nominal deflection is 1.2 .

- Figure (5.45) shows the missile absolute velocity increasing from 250 up to 750

m/s according to sine wave input for 40 s mission interval.

Fig. (5.41) Path angle response using PD, FL controllers with ramp input

119

Fig. (5.42) Comparison between the integral square error for PD, FL controllers with

ramp input

Fig. (5.43) Angle of attack response using FLC with ramp input

120

Fig. (5.44) Elevator deflection angle response using PD, FL controllers with ramp input

Fig. (5.45) Missile velocity response using FLC with ramp input

121

5.11 Summary (a) Applying fuzzy logic controller to AIM 9 R missile model and rule-base

derivation.

(b) Using low pass filter in order to blend the pitch error signal express the rate of

error properly.

(c) Evaluation of fuzzy logic controller performance compared with PD and neural

network controllers indicating that the FLC is better than using PD or NNC for

the whole mission.

(d) In order to decrease the error in the first time period, a superposition between

FLC and NNC should be used specially in the first time period.

(e) Development of a neurofuzzy structure using both types of controllers (FLC and

NNC).

(f) Multi-controllers and multi-guidance laws should be used to maintain the highest

performance of the controllers for all regimes of flight.

122

Chapter (6)

A HYBRID NEUROFUZZY CONTROL SYSTEM FOR MISSILE

GUIDANCE

6.1 Introduction

In this chapter an integrated control system for a multi-input multi-output air-to-air

missile based on both neural network and fuzzy logic controllers is presented. The

proposed control system is shown in Fig. (6.1) using MATLAB where the reference

inputs that represent the angles between the target position on focal plane array and the

missile C.G or (the angles between the tracking camera axes and missile body axes). A

superposition between the NN and FL controller’s outputs is shown in Fig. (6.1) where

the final control signals represent the elevator and rudder deflections. The main goal is

to control an air-to-air missile to follow the desired trajectory according to the actual

missile track rate with minimum error between desired and actual trajectories to secure

a minimum proximity fuse distance between the missile and the target (aircraft) to

achieve the required impact against the target. This control system is exposed to a

significant uncertainty due to cross coupling effect between inputs and due to the gust

affecting the missile actual performance.

Fig. (6.1) AIM 9 Model with 2 separate controllers

123

6.2 Methodology

Eventually, this project can be briefly explained in the following seven steps:

1.Deal with the (multi-input multi-output) model as two (single-input single-output)

models.

2. Train neural network controllers for the AIM model separately using applied kits.

3. Apply (25 rule MAMDANI fuzzy logic controllers) to both NN controllers models

after connecting them to the actual (multi-input multi-output) model as FLC will

enhance the output performance in case of existing gust or cross coupling effect as it is

the best way to deal with large uncertainty without having instability in the dynamic

system as shown in Fig (6.2).

Fig. (6.2) NeuroFuzzy structure

124

4.Simulation and Results

The proposed control system is established and simulated using MATLAB/SIMULINK

software. Appendix A includes block diagrams for MATLAB/SIMULINK dynamic

models of the AIM 9R missile. The reference input comes from the angle between the

moving camera and missile axis. The results of simulation are examined for step, ramp,

sine wave before applying the actual input data from the camera location. The results

are accepted in accordance with the data sheet of this missile referred to Tables (1.2)

and (1.3).

5- In order to track the target according to its location of the target on missile focal

plane array (missile dome window) it is required to transfer target location according to

a fixed frame of reference to its location according to missile body axes. This can be

done by transferring the target location to the missile body axes at fixed frame of

reference origin then transferring this location to the actual missile body axes as shown

in Fig. (6.3).

Fig. (6.3) Target location transformation from fixed frame of reference coordinate

to missile body axes coordinate

6-To plot the target and seeker flight actual paths it is required to calculate the aircraft

orientation angles then integrate Eq.’s (3.50), (3.51) and (3.52) using

MATLAB/SIMULINK editor to get the aircraft trajectory. Consequently the missile

125

trajectory from the intensity and the position of the target on the plane focal array at the

dome of the missile is calculated. Both aircraft and missile trajectories are shown in Figs

(6.2), (6.4) and (6.6).

Fig (6.4) Actual trajectories for target and missile

Fig. (6.5) 3D Plot for missile trajectory in both X,Y axes with respect to time

126

Fig. (6.6) 3D Plot for missile trajectory in both X, Z axes with respect to time

6- The maximum fuse distance can be calculated from the target-seeker difference

distance using Fig. (6.7) then verified according to the amount and type of missile

fragmentation. As the amount of kinetic energy absorbed by the target through the

fragments impulsion is inversely proportional to the miss distance between the aircraft

ad the missile. The ammunition type used in the missile affect also the amount of energy

absorbed by the target in the moment of collapse. Eq. (6.1) expresses the effect of the

ammunition type used in the missile and target-seeker miss distance.

tc.s

c.rc.r

LK.E. .E .

r (6.1)

K.E. = kinetic energy absorbed by the target in the moment of collapse (joule)

= fragmentation efficiency

127

c.rE = total energy of the fragments (joule)

tc.sL = average target cross section length (m)

c.rr = target-seeker miss distance (m)

Locking time is the shown in Fig. (6.7) and equals to half of the maximum locking time.

Fig. (6.7) Calculation of maximum fuse distance according to simulation data

7-Evaluating the integrated control system output error between the desired and actual

flight paths. It is required to calculate the angle of attack error, side slipping angle error

that results from the false desired trajectory coming from the inclination of the missile

axial axes to the missile actual velocity vector. Thus, subtracting these errors from the

total error gives the integrated control system errors as shown in Figs. (6.8) and (6.9).

128

Fig. (6.8) Total error in Z direction due to AOA and control system errors

Fig. (6.9) Total error in Y direction due to SSA and control system errors

129

Chapter (7)

DISCUSSIONS AND CONCLUSIONS

7.1 Discussions

In the present research work an initial research effort on nonlinear control design

methods for air-to-air missiles autopilots was conducted. Two different autopilots logics

were discussed: nonlinear fuzzy logic controller and neural network controller (inverse

model technique). The performance of these autopilots was demonstrated in nonlinear

simulations. Due to the wide variation in the flight conditions encountered by the

changing center of gravity location during flight, altitude, variable missile velocity and

the target-seeker distance that judge the type of applied guidance law, old linear

autopilots require gain scheduling with respect to Mach number, altitude, center of

gravity location of the missile. In the present thesis to cope with the system nonlinearity

fuzzy logic controller is used supported with neural network controller in the first time

period. But for the wide variation in the flight conditions fuzzy fusion system is used to

apply the adequate autopilot and the corresponding guidance law in accordance with the

flight conditions.

In the present aerodynamic model of AIM 9 missile has been established. Then PD,

neural network, fuzzy logic and combined controllers were applied separately and

respectively. The performance has been measured for different inputs.

The combination of fuzzy logic and neural network controllers is examined also to get

the best performance of the whole system especially with the affecting gust and the

existing cross coupling effect.

The simulation results in chapter 4, 5 show the performance of the dynamic system

for the three different types of controllers (PD, fuzzy logic and neural network) for

different types of inputs (step input, ramp input, combined sine wave). The final results

130

were statically evaluated by using the ISE (integrated square error) as shown in Table

(7.1).

Table (7.1) ISE for PD, NN, FL controllers

Controller type Step input Ramp input Combined sine wave input

PD controller .00388 .00633 .112

NNC --- --- 1.75

FLC .00318 .00580 .066

It is clear form Table (7.1) that the fuzzy logic performance controller has the best

performance compared with the PD and NN controllers as it has the smallest values of

ISE for different types of inputs.

The combination of fuzzy logic and neural network controllers has an advantage of

reducing the net errors in the first 5_seconds of the mission but for the rest of the

mission the performance decays rapidly. Table (7.2) explains the feature of the

combined and fuzzy logic controllers.

Table (7.2) Comparison between combined (NNC and FLC) and FLC only

Type of controller ISE

after 1

second

ISE

after 2

second

ISE

after 3

second

ISE

after 4

second

ISE

after 5

second

ISE

at the end of

the mission

Combined controller .009 .011 .020 .027 .035 .112

FLC .017 .020 .032 .034 .035 .066

It is noticed from Table (7.2) that the performance using both FLC and NNC in the

first 5 seconds time interval of the mission is better than the performance using FLC

only. As the fuzzy logic controller command is initially increasing slowly to reach the

desired value according to the incremental control signal value, but for the rest of the

mission it is recommended to use FLC only as using NNC may loose some stability

against the uncertainty even when used with FLC on the contrary when using FLC

131

alone. As it is robust enough when used alone to stabilize the whole system against the

existing uncertainty (cross coupling effect, gust). Consequently it is recommended to use

both types of controllers NNC, FLC in the first 5_seconds and using the FLC alone for

the rest of the mission.

Due to different regimes and wide input data range the Mamdani Fusion Fuzzy

System Classifier will enhance the output performance using multi controllers for

different regimes. Finally it is recommended that the locking time for triggering should

not exceed 0.065 seconds, the maximum proximity fuse distance should not be less than 5

meter. These specifications are according to the dynamic model data used in the

simulation.

7.2 Conclusion

(a) In order to fulfill the required specifications of maximum proximity fuse

distance and target locking time, a hybrid neurofuzzy control system should be

used in the first 5 seconds of the mission while fuzzy logic controller should be

used alone the rest of the mission.

(b) To enhance the performance of the missile during its whole mission it is

required to use multi guidance laws and multi neurofuzzy control systems to

cope with the uncertainty of the missile model according to flight conditions and

the target-seeker distance.

(c) To apply different guidance laws along the whole mission it is required to use

variable missile body axes, that means the origin of the missile body axes will be

shifted according to the target-seeker distance and applicable guidance law.

(d) To verify the maximum damage to the target during collapse it is required to

use variable locking time that activate the proximity fuse at the closest distance

to the target. This can be done by activating the proximity fuse at the moment

that (miss _ dis tan ce)t

= the smallest value= 0

132

7.3 Recommendations for Future Work

1-Exprimental application of the control techniques in the present work to verify the

results that has been reached from the computer simulations.

2-Optimization of the guidance system of the air-to-air missile according to the target

maneuver data (predicted path, velocity and acceleration). This will require multiple

types of guidance techniques verifying the best performance at any target-missile

regime.

133

REFERENCES

[1] http://vectorsite.tripod.com/avbomb2.html

[2] Lin, C. F. Modern Navigation, Guidance, and Control Processing. Prentice-Hall,

Englewood Cliffs, NJ, U.S.A., 1991.

[3] Bryson, A. E., Jr. and Y. C. Ho Applied Optimal Control. Blaisdell, Waltham, MA,

U.S.A., 1969.

[4] Zarchan, P. Tactical and Strategic Missile Guidance, 2nd Ed. AIAA, Inc., Washington,

D.C., U.S.A., 1994.

[5] Miller, W. T., R. S. Sutton, and P. J. Werbos Neural Networks for Control. MIT Press,

Cambridge, MA, U.S.A., 1991.

[6] Narendra, K. S. and K. Parthasarthy Identification and control of dynamical systems

using neural networks. IEEE Trans. Neural Networks, 1990, pp.4-27.

[7] Mamdani, E. H. and S. Assilian An experiment in linguistic synthesis with a fuzzy logic

controller. Int. J. Man Machine Studies, 1975, 7(1), pp.1-13.

[8] Lee, C. C. Fuzzy logic in control systems: fuzzy logic controller part I. IEEE Trans. Syst.

Man and Cyb., 1990, 20(2), pp. 404-418.

[9] Driankov, D., H. Hellendoorn, and M. ReinfrankAn Introduction to Fuzzy Control.

Springer, Berlin, Germany, 1993.

[10] Dash, P. K., S. K. Panda, T. H. Lee and J. X. Xu Fuzzy and neural controllers for

dynamic systems: an overview. Proc. Int. Conf. Power Electronics, Drives and Energy

Systems, Singapore, 1997.

[11] Locke, A. S. Guidance. D. Van Nostrand Co., Princeton, NJ, U.S.A. 1955

[12] Ha, I. and S. Chong Design of a CLOS guidance law via feedback linerization. IEEE

Trans. Aero.Electr. Syst., 1992, 28(1), 51-63.

[13] Zarchan, P. Tactical and Strategic Missile Guidance, 2nd Ed. AIAA, Inc., Washington,

D.C., U.S.A. 1994.

[14] Nazaroff, G. J. An optimal terminal guidance law. IEEE Trans. Automat. Contr., 1976,

21(6), 407-408.

[15] Potter, J. E. A Guidance Navigation Separation Theorem. AIAA, Washington, D.C.,

U.S.A. 1964, AIAA Paper 64-653

134

[16] Price, C. F. and R. S. Warren Performance Evaluation of Homing Guidance Laws for

Tactical Missiles. TASC Tech. Rept. TR-170-4, The Analytic Sciences Co., Reading,

MA,U.S.A. 1973.

[17] Nesline, F. W. and P. Zarchan A new look at classical vs. modern homing missile

guidance. AIAA J. Guid. Contr., 1981, 4(1), 78-85.

[18] Cherry, G. W. A General Explicit, Optimizing Guidance Law for Rocket-Propellant

Spacecraft., AIAA, Washington, D.C., U.S.A. 1964, AIAA Paper 64-638

[19] Wang, K. Optimal control and estimation for grazing angle problem. Proc. American

Control Conf., Atlanta, GA, U.S.A. 1988.

[20] Wang, Q., C. F. Lin, and C. N. D’Souza Optimality-Based Midcourse Guidance.,

Washington, D.C., U.S.A., 1993, AIAA Paper 93-3893

[21] Glasson, D. P. and G. L. Mealy Optimal Guidance for Beyond Visual Range Missiles.

AFATL-TR-83-89, USAF, Eglin AFB, FL, U.S.A., 1983.

[22] Cheng, V. H. L. and N. K. Gupta Advanced midcourse guidance for air-to-air missiles.

J. Guid. and Contr., 1986, 9(2), 135-142.

[23] Nesline, F. W. and M. L. Nesline How autopilot requirements constrain the

aerodynamic design of homing missiles. Proc. American Contr. Conf., San Diego, CA,

U.S.A., 1984.

[24] Stallard, D. V. An Approach to Autopilot Design for Homing Interceptor Missiles.,

AIAA, Washington, D.C., U.S.A., 1991, AIAA Paper 91-2612.

[25] Lin, C. F., J. Cloutier, and J. Evers Missile autopilot design using a generalized

Hamiltonian formulation. Proc. IEEE 1st Conf. Aero. Contr. Syst., Westlake Village,

CA, U.S.A., 1993.

[26] Lin, C. F. and S. P. Lee Robust missile autopilot design using a generalized singular

optimal control technique. J. Guid., Contr., and Dyna., 1985, 8(4), 498-507.

[27] Lin, C. F. Advanced Control System Design. Prentice-Hall, Englewood Cliffs, NJ,

U.S.A., 1994.

[28] Cloutier, J. R., J. H. Evers, and J. J. Feeley Assessment of air-to-air missile guidance

and control technology. IEEE Contr. Syst. Mag., 1989, 9(6), 27-34.

[29] Napolitano, M. R. and M. Kincheloe On-line learning neural-network controllers for

autopilot systems. J. Guid., Contr., and Dyna, 1995, 33(6), 1008-1015.

135

[30] Guez, A., J. L. Eilbert, and M. Kam Neural network architecture for control. IEEE

Contr. Syst. Mag., 1998, 8(2), 22-25.

[31] Lin, C. L. and Y. Y. Chen. Design of advanced guidance law against high speed

attacking target. Proc. Natl. Sci. Counc. ROC(A), 1999, 23(1), 60-74.

[32] Cottrell, R. G., T. L. Vincent, and S. H. Sadati Minimizing interceptor size using neural

networks for terminal guidance law synthesis. J. Guid., Contr., and Dyna., 1996, 19(3),

557-562.

[33] Hsiao, Y. H. Adaptive Feedforward Control for Disturbance Torque Rejection in

Seeker Stabilizing Loop. M.S. Thesis. Feng Chia University, Taichung, Taiwan,

R.O.C., 1998.

[34] Geng, Z. J. and C. L. MaCullough Missile control using fuzzy cerebellar model

arithmetic computer neural networks. J. Guid., Contr. and Dyna, 1997, 20(3), 557-565.

[35] Steck, J. E., K. Rokhsaz, and S. P. Shue Linear and neural network feedback for flight

control decoupling. IEEE Contr. Syst. Mag., 1996, 16(4), 22-30.

[36] Zhu, J. J. and M. C. Mickle Missile autopilot design using a new linear time-varying

control technique. J. Guid., Contr., and Dyna, 1997, 20(1), 150-157.

[37] Psaltis, D., A. Sideris, and A. Yamamura Neural con-trollers. Proc. 1st Int. Conf.

Neural Networks, San Diego, CA, U.S.A., 1987.

[38] Lightbody, G. and G. W. Irwin Neural model reference adaptive control and

application to a BTT-CLOS guidance system. Proc. IEEE Int. Conf. Neural Networks,

Orlando, FL, U.S.A., 1994.

[39] Hunt, K. J. and D. Sbarbaro Hofer Neural networks for nonlinear internal model

control. IEE Proc. Pt. D, 1991, 138(5), 431-438.

[40] Montague, G. A., M. J. Willis, M. T. Tham, and A. J. Morris Artificial neural networks

based multivariable predictive control. Proc. IEE 2nd Int. Conf. Artificial Neural

Networks, Bournemouth, U.K., 1991.

[41] Saint-Donat, J. N. Bhat, and T. J. McAvoy Neural net based model predictive control.

In: Advances in Intelligent Control, Chap. 8. C.J. Harris Ed. Taylor and Francis, 1994.

[42] McDowell, D. M., G. W. Irwin, and G. McConnell Hybrid neural adaptive control for

bank-to-turn missiles. IEEE Trans. Contr. Syst. Tech., 1997, 5(3), 297-308.

[43] Mizumoto, M. Realization of PID controllers by fuzzy control methods. IEEE Int. Conf.

Fuzzy Syst., Piscataway, NJ, U.S.A., 1992.

136

[44] Qiao, Q. Z. and M. Mizumoto PID type fuzzy controller and parameters adaptive

method. Fuzzy Sets and Systems, 1996, 78, 23-25.

[45] Mishra, S. K., I. G. Sarma, and K. N. Swamy Performance evaluation of two Fuzzy-

logic-based homing guidance schemes. J. Guid., Contr., and Dyna., 1994, 17(6), 1389-

1391.

[46] Leng, G. Missile guidance algorithm design using inverse kinematics and fuzzy logic.

Fuzzy Sets and Systems, 1996, 79, 287-295.

[47] Gonsalves, P. G. and A. K. Caglayan Fuzzy logic PID controller for missile terminal

guidance. Proc. 1995 IEEE Int. Symp.Intell.Contr., Monterey, CA, U.S.A., 1995.

[48] Schroeder, W. K. and K. Liu An Appropriate Application of Fuzzy Logic: A Missile

autopilot for dual control implementation. 1994 IEEE Int. Symp.Intell.Contr.,

Columbus, OH, U.S.A., 1994.

[49] Lin, C. K. and S. D. Wang A self-organizing fuzzy control approach for bank-to-turn

missiles. Fuzzy Sets and Systems, 1998, 96, 281-306.

[50] Farinewata, S. S., D. Pirovolou, and G. J. Vachtsevanos An input-output stability

analysis of a fuzzy controller for a missile autopilot’s yaw axis. Proc. 3rd IEEE Conf.

Fuzzy Syst., Orlando, FL, U.S.A., 1994.

[51] Hessburg, T. Fuzzy logic control for lateral vehicle guidance. Proc. 2nd IEEE Conf.

Contr. Appl., Vancouver, BC, Canada, 1993.

[52] Zhao, Z. Y., M. Tomizuka, and S. Isaka Fuzzy gain scheduling of PID controllers.

IEEE Trans. Syst., Man, Cyb., 1993, 23(5), 1392-1398.

[53] Ling, C. and T. F. Edgar A new fuzzy gain scheduling algorithm for process control.

Proc. American Contr. Conf., Chicago, IL, U.S.A., 1992.

[54] Dash, P. K. and S. K. Panda Gain-scheduling adaptive control strategies for HVDC

systems using fuzzy logic. Proc. Int. Conf. Power Electronics, Drives and Energy

Systems, New Delhi, India, 1996.

[55] Huang, C., J. Tylock, S. Engel, and J. Whitson Comparison of Neural-Network-Based,

Fuzzy-Logic-Based, and Numerical Nonlinear Inverse Flight Controls, AIAA,

Washington, D.C., U.S.A., 1994, AIAA Paper 94-3645.

[56] Ting, C. S., T. H. S. Li, and F. C. Kung An approach to systematic design of the fuzzy

control system. Fuzzy Sets and Systems, 1996, 77, 151-166.

137

[57] Palm, R. and D. Driankov Stability of fuzzy gain schedulers: sliding-mode based

analysis. Proc. 6th IEEE Int. Conf. Fuzzy Systems, Barcelona, Catalonia, Spain, 1997.

[58] Kwong, W. A. and K. M. PassinoDynamically focused fuzzy learning control. IEEE

Trans. Syst., Man, Cyb., 1996, 26(1), 53-74.

[59] Kandel, A. and G. Langholz Fuzzy Control Systems. CRC Press, Boca Raton, FL,

U.S.A, 1994.

[60] Boulet, V., E. Druon, D. Willaeys, and P. Vanheeghe Target estimation using fuzzy

logic. Proc. 1993 IEEE Int. Conf. Syst., Man and Cyb., Piscataway, NJ, U.S.A, 1993.

[61] Rugh, W. J. Analytical framework for gain scheduling. IEEE Contr. Syst. Mag., 1991,

11(1), 79-84.

[62] Hualin, T. and W. J. Rugh Overtaking optimal control and gain scheduling. Proc.

American Contr., Conf., Albuquerque, NM, U.S.A, 1997.

[63] Tan, S., C. C. Hang, and J. S. Chai Gain scheduling: from conventional to neurofuzzy.

Automatica, 1997, 33(3), 411-419.

[64] Balas, G. J. and A. K. Packard Design of robust time varying controllers for missile

autopilot. First IEEE Conf. Contr. Appl., Dayton, OH, U.S.A, 1992.

[65] Eberhardt, R. and K. A. Wise Automated gain schedules for missile autopilots using

robustness theory. First IEEE Conf. Contr. Appl., Dayton, OH, U.S.A, 1992.

[66] Nichols, R. A., R. T. Reichert, and W. J. Rugh Gain scheduling for controllers: a

flight control example. IEEE Tans. Contr. Syst. Tech., 1993, 1(2), 69-79.

[67] Piou, J. E. and K. M. Sobel Application of gain scheduling eigenstructure assignment

to flight control design. Proc. 1996 IEEE Int. Conf. Contr. Appl., Dearborn, MI,

U.S.A, 1996.

[68] Takagi, T. and M. Sugeno Fuzzy identification of systems and its applications to

modeling and control. IEEE Trans. Syst., Man, Cyb., 1985, 15(1), 116-132.

[69] Zhao, J., V. Wertz, and R. Gorez Fuzzy gain scheduling controllers based on fuzzy

models. Proc. 5th IEEE Int. Conf. Fuzzy Syst., New Orleans, LA, U.S.A, 1996.

[70] Yang, C. D., T. M. Kuo, and H. C. Tai gain scheduling using fuzzy rules. Proc. 35th

Conf. Dec. Contr., Kobe, Japan, 1996.

[71] Gonsalves, P. G. and G. L. Zacharias Fuzzy logic gain scheduling for flight control.

Proc. 3rd IEEE Conf. Fuzzy Syst., Orlando, FL, U.S.A, 1994.

138

[72] Wang, J. and W. Zhang A dynamic back propagation algorithm with application to

gain scheduled aircraft flight control system design. Proc. Intell. Infor.Syst., Los

Alamitos, CA, U.S.A, 1997.

[73] Adams, R. J., A. G. Sparks, and S. S. Banda A Gain scheduled multivariable design for

a manual flight control 1system. First IEEE Conf. Contr. Appl., Dayton, OH, U.S.A,

1992.

[74] Tanaka, T. and Y. Aizawa A Robust Gain Scheduler Interpolated into Multiple Models

by Membership Functions., Washington, D.C., U.S.A, 1992, AIAA Paper 92-4553.

[75] Pedrycz, W. and J. F. Peters Hierachical fuzzy controllers: Fuzzy gain scheduling. 1997

IEEE Int. Conf. Syst. Man, Cyb., Orlando, FL, U.S.A, 1997.

[76] Chai, J. S., S. Tan, and C. C. Hang Gain scheduling control of nonlinear plant Using

RBF neural network. Proc. IEEE Int. Symp.Intell.Contr., Dearborn, MI, U.S.A, 1996.

[77] Maia, C. A. and P. Resende Neural control of MIMO nonlinear plants: a gain-

scheduling approach. Proc. 12th IEEE Int. Symp.Intell.Contr., Istanbul, Turkey, 1997.

[78] Liang, R. H. and Y. Y. Hsu Scheduling of hydroelectric generations using artificial

neural networks. IEE Proc. Gener.Transm.Distrib., 1994, 141(5), 452-458.

[79] Cavalieri, S. and O. Mirabella Neural networks for process scheduling in real-time

communication systems. IEEE Trans. Neural Networks, 1996, 7(5), 1272-1285.

[80] Wang, Q., D. R. Broome, and A. R. Greig Intelligent gain-scheduling using neural

networks for robotic manipulators. Workshop on Neural Network Applications and

Tools, Liverpool, U.K., (1994).

[81] Tomescu, B. and H. F. VanLandinghamNeurofuzzymultimodel control using Sugeno

inference and Kohonen tuning in parameter space. 1997 IEEE Int. Conf. Syst., Man,

Cyb., Orlando, FL, U.S.A., (1997).

[82] Song, J.M. and Zhang, T.Q. “Passive homing missile’s variable structure proportional

navigation with terminal angular constraint” Aeronaut, 2001, vol.14, pp.83-87.

[83] Chio, J.Y. andChaw, D.K. “Nonlinear adaptive guidance considering target

uncertainties and control loop dynamics” IEEE Trans.,vol. 1 pp.506-511, 2001

[84] Innocenti, M. “nonlinear guidance technique for agile missiles” Control Eng., 2001, vol

9, pp.1131-1144.

139

[85] Anderson, M.B. and Burkhalter, J.E. “Design of a guided missile interceptor using a

genetic algorithm” Spacecr Rockets , 2001, vol 38, pp.28-35.

[86] Zhou, Di and Mu. Chundi “Motion tracking sliding mode guidance of a homing missile

with bearings-only measurements” Trans JpnSoc Aeronaut Space Sci, 2000, vol. 43,pp.

130-136.

[87] Narendra K. S. and Parthasarathy K., ”Identification and control of dynamical systems

using neural networks,”IEEE Trans. Neural Networks, 1990, vol. 1,pp.4-27.

[88] Park Y. M., Choi M. S., and Lee K. Y., ”An optimal tracking neuro-controller for

nonlinear dynamic systems,” IEEE Trans. Neural Networks, 1996, vol. 7, pp.1099-1110.

[89] Hamdy A., and El Desoky S., ” Development of the preliminary design of the missile

command guidance and testing “ M. Sc.Thesis, Cairo university, 1997

[90] Ku C. C. and Lee K. Y.,”Diagonal recurrent neural networks for dynamic systems

control, ” IEEE Trans. Neural Networks, 1995, vol. 6,pp.144-156.

[91] Brdys M. A. and Kulawski G. J., ”Dynamic neural controllers for induction motor,”

IEEE Trans. Neural Networks, 1999,vol. 10,pp. 340-355.

[92] Widrow B., Hoff M.E., ”Adaptive switching circuits,”1960 IRE WESCON convention

Record, New York: IRE Part 4, 1960, pp. 96-104.

[93] Rosenblatt F., ”The perceptron: A probabilistic model for information storage and

organization in the brain,” Psychological Review, 1958, vol. 65, pp. 386-408.

[94] Minsky M. and Papert S., Perceptrons, Cambridge, MA: MIT Press, 1969.

[95] Kohonen T., ”Correlation matrix memories,” IEEE Transactionson Computers, 1972,

vol. 21,pp. 353-359.

[96] Hopfield J. J., ”Neural networks and physical systems with emergent collective

computational properties, ” Proceedings of the National Academy of Science, 1982, vol.

79, pp. 2554-2558.

[97] Rumelhart D. E., Hinton G. E. and Williams R. J., ”Learning representations by back-

propagating errors,” Nature, 1986, vol. 323, pp. 533-536.

[98] Chow T. W. S. and Fang Y.,”A recurrent neural-network-based real-time learning

control strategy applying to nonlinear systems with unknown dynamics,” IEEE Trans.

Indust.Electron., 1998, vol. 45,pp. 151-161.

[99] Takagi T. and Sugeno M., “Fuzzy identification of systems and its applications to

modeling and control,” IEEETans, Syst., Man, Cybern., 1985, vol. 15, pp. 116-132.

140

[100] Sugeno M. and Yasukawa T.,”A fuzzy-logic-based approach to qualitative

modeling,”IEEE Trans. Fuzzy Syst., 1993, vol. 1, pp. 7-31.

[101] Wang L. X., Adaptive Fuzzy System and control: Design and Stability Analysis.

Englewood Cliffs, NJ: Prentice-Hall, 1994.

[102] Dickerson J. A. and Kosko B., ”Fuzzy function approximation with ellipsoidal

rules,”IEEE Trans. Syst., Man, Cybern., 1996, vol. 24, pp. 542-560.

[103] Cherkassky V., Gehring D., and Mulier F., “Comparison of adaptive methods for

function estimation from samples,” IEEE Trans. Neural Networks, 1996, vol. 7, pp.

969-984.

[104] Chen S. and billings S. A., “Neural networks for nonlinear dynamic system modeling

and identification,” Int. J. Control, 1992, vol.56, pp. 319-346.

[105] Choi J., Van Landingham H. F., and Bingulac S., “ A constructive approach for

nonlinear system identification using multilayer perceptrons,” IEEE Trans. Syst.,

Man, Cybern. B, 1996, vol. 26, pp. 307-312.

[106] Mendel J. M. and Mouzouris G. C.,”Designing fuzzy logic systems,” IEEE Trans.

Circuits Syst., 1997vol. 44, pp. 885-895.

[107] Sugeno M., “Theory of Fuzzy Integrals and Its Applications,” Doctoral Thesis, Tokyo

Institute of Technology, 1974.

[108] Adlassnig K. P., “Fuzzy set theory in medical diagnosis, “ IEEE Trans. Syst., Man,

Cybern., 1986vol. 16, pp, pp. 260-265.

[109] Menon P. K. and Iragavarapu V. R., “Blended Homing Guidance Law Using Fuzzy

Logic,” Optimal Synthesis Inc., 1998

[110] Lu-Ping Tsao and Ching-Shaow Lin, “A new optimal guidance law for short-range

homing missiles,” Department of system engineering, Taiwan, R.O.C., 2000

[111] Narendra K. S. and Parthasarathy K., ”Identification and control of dynamical

systems using neural networks, ”IEEE Trans. Neural Networks, 1990, vol. 1, pp.4-

27.

[112] Ku C. C. and Lee K. Y.,”Diagonal recurrent neural networks for dynamic systems

control, ” IEEE Trans. Neural Networks, 1995, vol. 6, pp.44-156.

[113] Takagi T. and Sugeno M., “Fuzzy identification of systems and its applications to

modeling and control,” IEEETans, Syst., Man, Cybern., 1985, vol. 15, pp. 116-132.

141

[114] Menon P. K. and Iragavarapu V. R., “Blended Homing Guidance Law Using Fuzzy

Logic,” Optimal Synthesis Inc., 1998

[115] Lu-Ping Tsao and Ching-Shaow Lin, “A new optimal guidance law for short range

homing missiles, ” Department of system engineering, Taiwan, R.O.C., 2000

[116] Hosny, A.M., Zeyada, Y.F., Hassaan, G.A., “Development of a neurofuzzy control

system for the guidance of air to air missile” MDP 8, Cairo, 2004, vol 1, pp. 465-475.

142

Appendix A

Integrated Control System

143

M 9R Aerodynamic Model

144