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Development of Fuzzy Logic LQR Control Integration for Full Mission Multistage Aerial Refueling Autopilot

A Dissertation Submitted for the Degree of Doctor of Philosophy

Candidate: Ahmed Momtaz Hosny

Supervisor: Prof. Han Chao

School of Astronautics Beihang University, Beijing, China

Beijing University of Aeronautics and Astronautics Ph.D. Dissertation

i

ACKNOWLEDGMENT

All gratitude is due to ALLAH. Thanks to ALLAH for everything, who blessed me with my supervisors, and gave us the support to finish this work. I would like to express and present my thanks and deepest gratitude to my supervisor Dr. Han Chao for his continuous encouragement, valuable advice and precious help to finish this work. Great indebtedness and special thanks to General Dr. Abd Allah El Ramsisy and Col. Dr. Ahmed Abd El Aziz for the initiation of the topic and his kind supervision, advice, reviewing and valuable guidance to produce this work. I would like to acknowledge my gratitude to all members of the aircraft control department in Military Technical College for their support and help. Also I would like to express my deep thanks to all my friends especially Maj. Mohamed Fayed and Maj. Ayman El Saai. I am especially indebted to my wife for her assistance and carefree, and my son Omar. This work would not have been possible without the support of my parents and all my great family members.

Ahmed Momtaz Hosny Beijing, 2008


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ABSTRACT

This work investigates the performance on the aircraft control system during air

refuel purposes of the Linear Quadratic Regulator (LQR) control alone, and the

integration between fuzzy control and LQR. LQR is modern linear control that is

suitable for multivariable state feedback and is known to yield good performance

for linear systems or for nonlinear systems where the nonlinear aspects are

presented. The fuzzy control is known to have the ability to deal with nonlinearities

without having to use advanced mathematics. The LQR integrated fuzzy control

(LQRIFC) simultaneously makes use of the good performance of the LQR in the

region close to switching curve, and the effectiveness of the fuzzy control in region

away from switching curve. A new analysis of the fuzzy system behavior presented

helps to make possible precise integration of LQR features into the fuzzy control.

The LQRIFC is verified by simulation to suppress the uncertain instability more

effectively than the LQR alone besides minimizing the time of the mission proposed.

The aerial refueling process has been divided into 3 basic stages (initial stability

stage, tracking stage, alignment stage) to meet all the process requirements and

constraints with minimizing the whole mission time period. GA has been used to

tune the fuzzy logic controller parameters and some PID’s. The control strategy

was applied to both aerial refueling methods. And the MATLAB simulation results

show the validity and the efficiency of using the proposed control approach.


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TABLE OF CONTENTS

Acknowledgment…………………………………………………………………... i

Abstract ……………………………………………………………………………. ii

Table of contents…………………………………………………………………... iii

List of figures ……………………………………………………………………… ix

List of tables ………………………………………………………………………. xii

List of Acronyms…………………………………………………………………... xiii

List of symbols…………………………………………………………………….. xiv

Chapter (1)

Introduction to

Air Force Aerial Refueling Methods

Flying Boom versus Hose-and-Drogue

1.1 Introduction……………………………………………………………………. 1

1.2 Background……………………………………………………………………. 2

1.3 Introduction to Fuzzy Logic Controllers……………………………………. 3

1.3.1 Introduction………………………………………………………………. 3

1.3.2 A structure of fuzzy logic controller ……………………………………. 8

1.3.2.1 Preprocessing…………………………………………………….. 8

1.3.2.2 Fuzzification……………………………………………………… 9

1.3.2.3 Rule Base…………………………………………………………. 10

1.3.2.4 Inference Engine…………………………………………………. 14

1.3.2.5 Defuzzification……………………………………………………. 17

1.3.2.6 Postprocessing …………………………………………………… 18

1.4 Summary………………………………………………………………………. 19


iv

Chapter (2)

Literature Survey

Traditional and Fuzzy Logic P, PD and PID controllers

2.1 The Control Problem………………………………………………………….. 21

2.2 Controller Types………………………………………………………………. 25

2.2.1 Proportional Controller………………………………………………… 25

2.2.2 PI Controller…………………………………………………………….. 26

2.2.3 PID Controller…………………………………………………………... 26

2.2.4 Proportion Fuzzy Controller…………………………………………… 26

2.2.5 Proportional and derivative controller………………………………... 28

2.2.6 Incremental Fuzzy Controller………………………………………….. 29

2.2.7 Proportional, integral and derivative fuzzy controller……………….. 32

2.2.8 Summary…………………………………………………………………. 33

2.3 Problem Formulation…………………………………………………………. 34

2.4 Main Objective of the Thesis…………………………………………………. 34

2.5 Thesis Layout………………………………………………………………….. 35

Chapter (3)

Preliminaries

Modeling, Trimming and Linearization

3.1 Aircraft Modeling …………………………………………………………….. 36

3.1.1 Reference Frames ………………………………………………………. 36

3.1.2 General Equation of Motion …………………………………………… 38

3.1.3 Linearized Equations of Motion……………........................................... 41


v

3.2 Aircraft Trimming Using NN Algorithm…………………………………….. 42

3.2.1 Basic Architecture of Neural Network…………………………………. 42

3.2.2 Network Structure Design……………………………………………. 44

3.2.2.1 Number of Hidden Layers………………………………….. 44

3.2.2.2 Number of Hidden Units……………………………………. 45

3.2.3 Back-Propagation Technique………………………………………... 46

3.3 Optimization Algorithms Applied to F16 nonlinear model ………………... 47

3.4 Linearization Process…………………………………………………………. 49

Chapter (4)

Linear Quadratic Regulator (LQR)

4.1 Linear Quadratic Regulator with Output Feedback………………………... 52

4.2 Linear Quadratic Regulator Basis……………………………………………. 53

4.2.1 Quadratic Performance Index ………………………………………….. 53

4.2.2 Solution of the LQR problem …………………………………………… 55

4.2.3 The initial condition problem……………………………………………. 58

4.2.4 Maximum desired values of z(t) and u(t)………………………………... 64

4.3 Tracking a command………………………………………………………….. 70

4.3.1 Tracker with desired structure ………………………………………… 72

4.3.2 LQ Formulation of the Tracker Problem……………………………… 74

4.3.3 The Deviation System…………………………………………………… 75

4.3.4 Performance Index for LQ Tracker……………………………………. 76

4.3.5 Solution of the LQ Tracker Problem ………………………………….. 77

4.3.6 Design Equations for Gradient-Based Solution ………………………. 78


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Chapter (5)

Intelligent Learning of Fuzzy Logic Controller Via Genetic Algorithm

5.1 Introduction……………………………………………………………………. 84

5.2 Applying Fuzzy Inference System for Control………………………………. 84

5.2.1 Designing membership functions (Fuzzification process)……………. 85

5.2.2 Rule Derivation…………………………………………………………. 87

5.2.2.1 Structure of fuzzy rules………………………………………... 88

5.2.3 Fuzzy rule-based inference…………………………………………….. 89

5.2.4 Response pattern of a simple generic fuzzy control system………….. 92

5.3 Genetic Algorithm Process……………………………………………………. 95

5.3.1 Application of GA Functions for Tuning Fuzzy PID Controller…….. 96

5.3.1.1 Optimizing Fuzzy Output Gains ………………………………. 96

5.3.1.2 Tuning of Fuzzy Sets…………………………………………… 98

5.3.1.3 Rule Base Weight Optimization………………………………... 103

5.4 LQR Fuzzy Logic Control (LQRIFC)……………………………………….. 103

Chapter (6)

Design of Fighter Flight Formation Autopilot for aerial Refueling Racetrack

Mission Employing KC-130J Tanker with Drogue-Hose Configuration

6.1 Introduction……………………………………………………………………. 109

6.2 Receiving aircraft modeling ………………………………………………….. 110

6.3 Control Strategy……………………………………………………………….. 111

6.3.1 Controller design…………………………………………………………. 112

6.3.1.1 Formation geometry and trajectory variables………………… 112

6.3.1.2 Level plane formation definition ………………………………. 112

6.3.1.3 Vertical plane formation definition…………………………….. 114

6.3.2 Control laws………………………………………………………………. 114


vii

6.3.2.1 Lateral Distance Control ………………………………………. 114

6.3.2.2 Forward Distance Control……………………………………… 115

6.3.2.3Vertical Distance Control……………………………………….. 116

6.4 System Identification Process………………………………………………… 116

6.4.1 Estimation of Simple Process Model…………………………………… 117

6.4.2 Initial Parameter Values and Parameter Bounds…………………….. 117

6.4.3 Compensator design using system identification technique………….. 117

6.5 Alignment Stage-Formation Control Simulation Results ………………….. 117

Chapter (7)

Control Strategy and Implementation Methodology

7.1 Introduction……………………………………………………………………. 122

7.2 Aerial Refueling Stages……………………………………………………….. 122

7.2.1 Stability Augmentation System ………………………………………… 122

7.2.2 Tanker Tracking Using Command Augmentation system (CAS)……. 122

7.2.3 Alignment Terminal using Flight Formation Controller……………... 127

7.3 Fighter -Tanker Tracking Stage Optimal Trajectory Determination……... 130

7.3.1 Optimal trajectory equations…………………………………………… 130

Chapter (8)

SUMMARY AND FINAL CONCLUSION

8.1 Stability Stage Poles Verification with Flying Qualities (1ST 133 Stage) ………..

8.2 Tracking Stage Time Using Different Guidance Technique (2nd 133 Stage)...….

8.3 Alignment Stage during (Boeing KC 767) Tanker–Boom configuration

(3rd

Stage)…………………………………………………………………………... 134


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8.4 (Boeing KC 767 Tanker – Boom configuration) versus (KC-130J Tanker

with Hose – Drogue configuration)……………………………………………….

135 8.5 Final Conclusion ………………………………………………………………. 136 8.6 Recommended Future Work …………………………………………………. 136 REFERENCES…………………………………………………………………….. 137 Appendix A Appendix B


ix

List of Figures Fig. 1.1 USAF KC-10 Refueling B-52 with Flying Boom……………………... 1

Fig. 1.2 USMC KC-130 Refueling F/A-18s with Hose-and-Drogue………….. 2

Fig. 1.3 Direct control…………………………………………………………… 4

Fig. 1.4 Feedforward control…………………………………………………… 6

Fig. 1.5 Fuzzy parameter adaptive control……………………………………. 7

Fig. 1.6 Blocks of a fuzzy controller……………………………………………. 8

Fig. 1.7 Example of nonlinear scaling of an input measurement…………….. 9

Fig. 1.8 Examples of membership functions…………………………………… 14

Fig. 1.9 Graphical construction of the control signal in a fuzzy PD controller 15

Fig. 1.10 One input, one output rule base with non-singleton output sets…… 19

Fig. 2.1 A generic model for control with feedback…………………………… 22

Fig. 2.2 A classical PID controller ……………………………………………... 23

Fig. 2.3 Fuzzy proportional controller (FP)……………………………………. 27

Fig. 2.4 Fuzzy PD controller (FPD)……………………………………………. 28

Fig. 2.5 Fuzzy incremental controller (FInc)…………………………………... 31

Fig. 2.6 Fuzzy PD+I controller (FPD+I)……………………………………….. 32

Fig. 3.1 Relationship between the body-fixed reference frame FB, flight-

path reference frame FW and stability reference frame F

37

S

Fig. 3.2 Basic Neural Network Structure………………………………………. 42

Fig. 4.1 Lateral output performance when 1)0( =β ………………………….. 68

Fig. 4.2 Lateral output performance when 1)0( =ϕ ………………………….. 69

Fig. 4.3 Effect of PI weighting parameters ρ = .1, .2, .4 respectively………… 70

Fig. 4.4 plant with compensator of desired structure…………………………. 71

Fig. 4.5 Plant/ feedback structure……………………………………………… 74

Fig. 4.6 Normal acceleration controller………………………………………… 80

Fig. 4.7 Normal acceleration step response for different PI (3.2 green), (8.36

blue)

80

Fig. 4.8 Pitch rate controller……………………………………………………. 81

Fig. 4.9 Pitch-rate step response using regular LQ tracker for different PI 81


x

(4.2 blue), (3.98 green)

Fig. 4.10 Lateral –Directional controller ……………………………………… 83

Fig. 4.11 Closed loop response to command of 1=ϕr , 0=rr . Bank angle ϕ

(rad), and washout yaw rate (rad/sec)

83

Fig. 5.1 The structure of a Fuzzy Logic Control System……………………… 85

Fig. 5.2 Triangular membership function……………………………………… 86

Fig. 5.3 Membership function for error input (e)……………………………... 86

Fig. 5.4 Membership function for error rate input (ce)……………………….. 86

Fig. 5.5 Membership function for control signal output (u)…………………... 87

Fig. 5.6 Fuzzy matching process………………………………………………... 89

Fig. 5.7 Fuzzy inference process………………………………………………... 89

Fig. 5.8 Fuzzy combination process……………………………………………. 90

Fig. 5.9 Defuzzification process………………………………………………… 91

Fig. 5.10 Fuzzy logic inference system…………………………………………. 91

Fig. 5.11 Architecture of the generic fuzzy control system…………………… 92

Fig. 5.12 Step response of a simple generic fuzzy controller………………….. 93

Fig. 5.13 Nonlinear fuzzy PID controller………………………………………. 94

Fig. 5.14 Fuzzy inference system control action (PID fuzzy logic controller

output gains)

95

Fig. 5.15 The flow of GA functions and process………………………………. 95

Fig. 5.16. Nonlinear fuzzy PID controller……………………………………… 96

Fig. 5.17 Objective function J and Fitness function F ………………………… 98

Fig. 5.18 Triangular and trapezoidal membership functions………………… 98

Fig. 5.19 Bell-shaped membership functions………………………………….. 99

Fig. 5.20 Example for one-point crossover of fuzzy partitions……………….. 101

Fig. 5.21 Mutating a fuzzy partitions…………………………………………... 101

Fig. 5.22a Input variable “error”………………………………………………. 102

Fig. 5.22b Input variable “error change”……………………………………… 102

Fig. 5.22c Output variable “Kp”………………………………………………... 102

Fig. 5.22d Output variable “Ki”……………………………………………....... 102

Fig. 5.22e Output variable “Kd”……………………………………………….. 103


xi

Fig. 5.23a LQR integrated fuzzy control (LQRIFC) for pitch controller……. 104

Fig. 5.23b LQR integrated fuzzy control (LQRIFC) for lateral-directional

controller

104

Fig. 5.24 Integrated system step response in both cases with optimized fuzzy

controller and without fuzzy controller

106

Fig. 5.25 Integrated system step response in both cases with optimized fuzzy

controller and without fuzzy controller for lateral motion

106

Fig. 6.1 KC-130J tanker and F-16 receiver configuration …………………… 109

Fig. 6.2 F-16 nonlinear modeling……………………………………………….. 110

Fig. 6.3 Level plane formation geometry………………………………………. 113

Fig. 6.4 Lateral –Directional control law ……………………………………… 115

Fig. 6.5 Forward control law……………………………………………………. 115

Fig. 6.6 Vertical control law ……………………………………………………. 116

Fig. 6.7 Control strategy implementation using SIMULINK ………………... 119

Fig. 6.8 Nonlinear model output performance with compensation ………….. 120

Fig. 6.9 Root locus for lateral –directional control with and without phase

lead compensator respectively

121

Fig. 6.10 Trajectory in x-y plane for both wingman and leader……………… 121

Fig. 7.1 Tanker position transformation to aircraft wing axes…………......... 125

Fig. 7.2 Actual trajectories for tanker and receiving aircraft………………… 125

Fig. 7.3 F-16 nonlinear model performance…………………………………... 126

Fig. 7.4 Nonlinear model output performance with compensation………….. 128

Fig. 7.5 Fighter-Tanker tracking stage optimal trajectory …………………… 130

Fig. 7.6 Relative distance in X direction (ft)…………………………………… 132

Fig. 7.7 Relative distance in Y direction (ft)…………………………………… 132

Fig. 7.8 Relative distance in Z direction (ft)…………………………………… 132

Fig. 8.1 (Boeing KC 767 Tanker – Boom configuration) versus (KC-130J

Tanker with Hose – Drogue configuration)

135


xii

List of tables

Table 2.1 Quick reference to fuzzy controllers………………………………. 33

Table 2.2 Relation between fuzzy and PID gains …………………………… 34

Table 3.1 Wing level trimmed values at Mach 0.75…………………………. 49

Table 4.1 LQR with output feedback………………………………………… 59

Table 4.2 Optimal output feedback solution algorithm (LQ Tracker)…….. 61

Table 4.3 LQ Tracker with Output Feedback..……………………………… 79

Table 5.1 Control rules for a simple generic fuzzy controller………………. 93

Table 5.2 Gain Limits………………………………………………………….. 97

Table 5.3 Decoding Equation ………………………………………………… 97

Table 5.4 Linearized Matrices………………………………………………... 105

Table 5.5 Gain values…………………………………………………………. 106

Table 6.1 F-16 aircraft states ………………………………………………… 110

Table 6.2 Control channels …………………………………………………… 111

Table 6.3 Flight formation controller parameters…………………………... 117

Table 8.1 The listed values satisfy a level 1 flying quality with a category B

flight phase thus validating the design parameters for the proposed

controller

133

Table 8.2 Tracking phase time versus guidance technique (second phase) 134

Table 8.3 Relative distances in 3 directions (X, Y, Z) with upper and lower

limits according to the design constrains of the Tanker-Boom

configuration (last phase)

134


xiii

List of Acronyms

ANN Artificial Neural Network

CA Control Allocation

DEA Direct Eigenstructure Assignment

DMC Dynamic Matrix Control

EA Eigenstructure Assignment

FCS Flight Control System

FTC Fault Tolerant Control

FDI Fault Detection and Isolation

FMEA Failure Modes and Effects Analysis

GWEA Gain Weighted Eigenspace Assignment

GPC Generalized Predictive Control

ICE Innovative Control Effector

ICEM Integrated Control Effector Management

IMM Interacting Multiple Model

MIMO Multiple Input Multiple Output

MPC Model Predictive Control

PID Proportional Integral Derivative Controller

QP Quadratic Program

SISO Single Input Single Output

LQR Linear quadratic regulator

LQRIFC Linear quadratic regulator integrated fuzzy control

PI Performance index

GA Genetic Algorithms


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List of Symbols

BF

Body-fixed frame of reference

SF Stability reference frame

WF Wind reference frame

EF Earth fixed reference frame

CG Center of gravity

BV Velocity with respect to body reference frame

EV Velocity with respect to earth fixed frame of reference

Er Position of CG with respect to EF

Bω Angular velocity with respect to body reference frame

p Roll rate q Pitch rate

r Yaw rate

BI Inertia matrix

EW Wind velocity with respect to EF

α Angle of attack

β Angle of side slip

ψ Azimuth angle with respect to EF

θ Pitch angle with respect to EF

ϕ Bank angle with respect to EF

L Rolling moment

M Pitch moment

N Yaw moment

X Axial force component

Y Side force component


xv

Z Normal force component

tδ Control thrust

eδ Elevator deflection angle

alδ Aileron deflection angle

rδ Rudder deflection angle


1

Chapter (1) Introduction to

Air Force Aerial Refueling Methods Flying Boom versus Hose-and-Drogue

1.1 Introduction Air Force aerial refueling received considerable attention in the 108th and 109thCongresses.

Much attention has focused on recapitalizing the KC-135 fleet, and a proposed and ultimately

rejected lease of 100 Boeing KC-767 aircraft. This report examines the Department of Defense’s

(DOD) mix of aerial refueling methods. Currently, Air Force fixed-wing aircraft refuel with the

“flying boom.” The boom is a rigid, telescoping tube that an operator on the tanker aircraft

extends and inserts into a receptacle on the aircraft being refueled Fig 1.1. Air Force helicopters,

and all Navy and Marine Corps aircraft refuel using the “hose-and-drogue.” NATO countries and

other allies also refuel with the hose-and drogue. As its name implies, this refueling method

employs a flexible hose that trails from the tanker aircraft. A drogue (a small windsock) at the

end of the hose stabilizes it in flight, and provides a funnel for the aircraft being refueled, which

inserts a probe into the hose Fig. 1.2. All boom-equipped tankers (i.e., KC-135, KC-10), have a

single boom and can refuel one aircraft at a time with this mechanism. Many tanker aircraft that

employ

Fig. 1.1 USAF KC-10 Refueling B-52 with Flying Boom


2

the hose-and-drogue system, can simultaneously employ two such mechanisms and, refuel two

aircraft simultaneously. The boom, however, can dispense fuel faster than a hose-and-drogue.

Fig. 1.2 USMC KC-130 Refueling F/A-18s with Hose-and-Drogue

A single flying boom can transfer fuel at approximately 6,000 lbs per minute. A single hose-and-

drogue can transfer between 1,500 and 2,000 lbs of fuel per minute. Unlike bombers and other

large aircraft, however, fighter aircraft cannot accept fuel at the boom’s maximum rate. (Today’s

fighter aircraft can accept fuel at 1,000 to 3,000 lbs per minute whether from the boom or from

the hose-and-drogue.)2 Thus, the flying boom’s primary advantage over the hose-and-drogue

system is lost when refueling fighter aircraft. As decisions are made regarding the Air Force

tanker fleet, an issue that may arise for Congress is whether to examine the mix of boom or and

hose-and-drogue refuelable aircraft in the Air Force. What might be the benefits and costs of any

changes? Would DOD benefit in terms of increased combat power? If so, would this benefit

justify the cost?

1.2 Background Air Force aircraft have not always used the flying boom. All U.S. combat aircraft used the hose-

and-drogue system until the late 1950s. The Air Force’s decision to field boom-equipped tankers


3

was based on the refueling needs of long-range bombers, which required large amounts of fuel.

The Air Force’s fighter community resisted eliminating the hose-and-drogue, but was overruled

by the Strategic Air Command, which operated the tanker fleet, and during the Cold War, placed

a higher value on refueling bombers. The perceived shortcomings of using a single boom to

refuel fighter aircraft is reflected in a 1990 Air Force initiative to standardize DOD fighter

aircraft refueling on the hose-and-drogue method. As initially conceived, the initiative consisted

of three elements: (1) placing probes on all F-15 and F-16 fighters; (2) incorporating a probe in

the design of the F-22A; and (3) adding two drogue pods to at least 150 KC-135s. To provide

redundancy and flexibility, Air Force fighters would retain their boom receptacles. The 1991 war

with Iraq (Operation Desert Storm) heightened DOD concerns over a lack of uniformity in aerial

refueling methods. Navy leaders expressed frustration and dissatisfaction with the number of Air

Force aerial refueling aircraft capable of employing the hose-and-drogue. Post-conflict analyses

recommended that the Navy purchase its own fleet of land-based KC-10-sized tankers to increase

the number of hose-and-drogue aircraft and reduce its reliance on Air Force aerial refueling.

1.3 Introduction to Fuzzy Logic Controllers

1.3.1 Introduction

While it is relatively easy to design a PID controller, the inclusion of fuzzy rules creates many

extra design problems, and although many introductory textbooks explain fuzzy control, there are

few general guidelines for setting the parameters of a simple fuzzy controller. The approach here

is based on a three step design procedure which builds on PID control.

1. Start with a PID controller.

2. Insert an equivalent, linear fuzzy controller.

3. Make it gradually nonlinear.

Guidelines related to the different components of the fuzzy controller will be introduced shortly.

In the next three sections three simple realizations of fuzzy controllers are described: a table-

based controller, an input-output mapping and a Takagi-Sugeno type controller. A short section

summarizes the main design choices in a simple fuzzy controller by introducing a check list. The

terminology is based on an international standard which is underway (IEC, 1996). Fuzzy


4

controllers are used to control consumer products, such as washing machines, video cameras, and

rice cookers, as well as industrial processes, such as cement kilns, underground trains, and robots.

Fuzzy control is a control method based on fuzzy logic. Just as fuzzy logic can be described

simply as ’’computing with words rather than numbers’’ fuzzy control can be described simply as

’’control with sentences rather than equations’’. A fuzzy controller can include empirical rules,

and that is especially useful in operator controlled plants. Take for instance a typical fuzzy

controller

1. If error is Neg and change in error is Neg then output is NB

2. If error is Neg and change in error is Zero then output is NM (1)

The collection of rules is called a rule base. The rules are in the familiar if-then format, and

formally the if-side is called the condition and the then-side is called the conclusion (more often,

perhaps, the pair is called (antecedent - consequent- conclusion ). The input value ’’Neg’’ is a

linguistic term short for the word negative the output value ’’NB’’ stands for negative BIG and

’’NM’’ for negative medium . The computer is able to execute the rules and compute a control

signal depending on the measured inputs error and change in error.

The objective here is to identify and explain the various design choices for engineers. In a rule

based controller the control strategy is stored in amore or less natural language. The control

strategy is isolated in a rule base opposed to an equation based description. A rule based

controller is easy to understand and easy to maintain for a non-specialist end-user. An equivalent

controller could be implemented using conventional techniques in fact, any rule based controller

could be emulated in, say fortran it is just that it is convenient.

Fig. 1.3 Direct control


5

To isolate the control strategy in a rule base for operator controlled systems. Fuzzy controllers are

being used in various control schemes (IEC, 1996). The most obvious one is direct control,

where the fuzzy controller is in the forward path in a feedback control system Fig. 1.3. The

process output is compared with a reference, and if there is a deviation, the controller takes action

according to the control strategy. In the figure, the arrows may be understood as hyper-arrows

containing several signals at a time for multiloop control. The sub-components in the figure will

be explained shortly. The controller is here a fuzzy controller, and it replaces a conventional

controller, say, a PID (proportional integral-derivative) controller. In feedforward control

Fig. 1.4 a measurable disturbance is being compensated. It requires a good model, but if a

mathematical model is difficult or expensive to obtain, a fuzzy model may be useful. Fig 1.4

shows a controller and the fuzzy compensator, the process and the feedback loop are omitted for

clarity. The scheme, disregarding the disturbance input, can be viewed as a collaboration of linear

and nonlinear control actions; the controller C may be a linear PID controller, while the fuzzy

controller F is a supplementary nonlinear Controller. Fuzzy rules are also used to correct tuning

parameters in parameter adaptive control schemes Fig. 1.5. If a nonlinear plant changes

operating point, it may be possible to change the parameters of the controller according to each

operating point. This is called gain scheduling since it was originally used to change process

gains. A gain scheduling controller contains a linear controller whose parameters are changed as

a function of the operating point in a preprogrammed way. It requires thorough knowledge of the

plant, but it is often a good way to compensate for nonlinearities and parameter variations. Sensor

measurements are used as scheduling variables that govern the change of the controller

parameters, often by means of a table look-up. Whether a fuzzy control design will be stable is a

somewhat open question. Stability concerns the system’s ability to converge or stay close to an

equilibrium point. A stable linear system will converge to the equilibrium asymptotically no

matter where the system state variables start from. It is relatively straight forward to check for

stability in linear systems,


6

Fig. 1.4 Feedforward control

for example by checking that all eigenvalues are in the left half of the complex plane. For

nonlinear systems, and fuzzy systems are most often nonlinear, the stability concept is more

complex. A nonlinear system is said to be asymptotically stable if, when it starts close to an

equilibrium, it will converge to it. Even if it just stays close to the equilibrium, without

converging to it, it is said to be stable (in the sense of Lyapunov). To check conditions for

stability is much more difficult with nonlinear systems, partly because the system behavior is also

influenced by the signal amplitudes apart from the frequencies. The literature is some what

theoretical and interested readers are referred to Driankov, Hellendoorn & Reinfrank (1993) or

Passino & Yurkovich (1998). They report on four methods (Lyapunov functions, Popov, circle,

and conicity), and they give several references to scientific papers. It is characteristic, however,

that the methods give rather conservative results, which translate into unrealistically small

magnitudes of the gain factors in order to guarantee stability. Another possibility is to

approximate the fuzzy controller with a linear controller, and then apply the conventional linear

analysis and design procedures on the approximation. It seems likely that the stability margins of

the nonlinear system would be close in some sense to the stability margins of the linear

approximation depending on how close the approximation is. This paper shows how to build such

a linear approximation, but the theoretical background is still unexplored. There are at least four

main sources for finding control rules (Takagi & Sugeno in Lee, 1990).

Expert experience and control engineering knowledge:. One classical example is the operator’s

handbook for a cement kiln (Holmblad & Ostergaard, 1982). The most common approach to

establishing such a collection of rules of thumb, is to question experts or operators using a


7

carefully organized questionnaire. _ based on the operator’s control action: Fuzzy IF_THEN

rules can be deduced from observations of an operator’s control actions or a log book. The rules

express input-output relationships. Based on a fuzzy model of the process. A linguistic rule base

may be viewed as an inverse model of the controlled process.

Fig. 1.5 Fuzzy parameter adaptive control

Thus the fuzzy control rules might be obtained by inverting a fuzzy model of the process. This

method is restricted to relatively low order systems, but it provides an explicit solution assuming

that fuzzy models of the open and closed loop systems are available (Braae&Rutherford in Lee,

1990). Another approach is fuzzy identification (Tong; Takagi & Sugeno; Sugeno – all in Lee,

1990; Pedrycz, 1993) or fuzzy model-based control [1], [2], [3].

Based on learning the self-organizing controller is an example of a controller that finds the rules

itself. Neural network is another possibility. There is no design procedure in fuzzy control such

as root-locus design, frequency response design, pole placement design, or stability margins,

because the rules are often nonlinear. Therefore we will settle for describing the basic

components and functions of fuzzy controllers, in order to recognize and understand the various

options in commercial software packages for fuzzy controller design.

There is much literature on fuzzy control and many commercial software tools (MIT, 1995), but

there is no agreement on the terminology, which is confusing. There are efforts, however, to

standardize the terminology, and the following makes use of a draft of a standard from the

International Electro technical Committee (IEC, 1996). Throughout, letters denoting matrices are

in bold upper case, for example A; vectors are in bold lower case, for example x; scalars are in

italics, for example n; and operations are in bold, for example min.


8

1.3.2 A structure of fuzzy logic controller

There are specific components characteristic of a fuzzy controller to support a design procedure.

In the block diagram in Fig. 1.6, the controller is between a preprocessing block and a post-

processing block. The following explains the diagram block by block.

Fig. 1.6 Blocks of a fuzzy controller

1.3.2.1 Preprocessing

The inputs are most often hard or crisp measurements from some measuring equipment, rather

than linguistic. A preprocessor, the first block in Fig. 1.6, conditions the measurements before

they enter the controller. Examples of preprocessing are:

_ Quantization in connection with sampling or rounding to integers;

_ Normalization or scaling onto a particular, standard range;

_ Filtering in order to remove noise;

_ Averaging to obtain long term or short term tendencies;

_ A combination of several measurements to obtain key indicators; and

_ Differentiation and integration or their discrete equivalences


9

A quantizer is necessary to convert the incoming values in order to find the best level in a

discrete uninverse. Assume, for instance, that the variable error has the value 4.5, but the

universe is (u=(-5,-4,-3……..3,4,5)). The quantiser rounds to 5 to fit it to the nearest level.

Quantization is a means to reduce data, but if the quantization is too coarse the controller may

oscillate around the reference or even become unstable.

Nonlinear scaling is an option Fig. 1.7. In the FL SMIDTH controller the operator is asked to

enter three typical numbers for a small, medium and large measurement respectively (Holmblad

& Østergaard, 1982). They become break-points on a curve that scales the incoming

measurements (circled in the figure). The overall effect can be interpreted as a distortion of the

primary fuzzy sets. It can be confusing with both scaling and gain factors in a controller, and it

makes tuning difficult. When the input to the controller is error, the control strategy is a static

mapping between input and control signal. A dynamic controller would have additional inputs,

for example derivatives, integrals, or previous values of measurements backwards in time. These

are created in the preprocessor thus making the controller multi-dimensional, which requires

many rules and makes it more difficult to design. The preprocessor then passes the data on to the

controller.

Fig. 1.7 Example of nonlinear scaling of an input measurement

1.3.2.2 Fuzzification

The first block inside the controller is fuzzification, which converts each piece of input data to

degrees of membership by a lookup in one or several membership functions.

The fuzzification block thus matches the input data with the conditions of the rules to determine

how well the condition of each rulematches that particular input instance. There is a degree of

membership for each linguistic term that applies to that input variable.


10

1.3.2.3 Rule Base

The rules may use several variables both in the condition and the conclusion of the rules. The

controllers can therefore be applied to both multi-input-multi-output (MIMO) problems and

single-input-single-output (SISO) problems. The typical SISO problem is to regulate a control

signal based on an error signal. The controller may actually need both the error, the change in

error, and the accumulated error as inputs, but we will call it single-loop control, because in

principle all three are formed from the error measurement. To simplify, this section assumes that

the control objective is to regulate some process output around a prescribed set point or reference.

The presentation is thus limited to single-loop control.

Rule Format: Basically a linguistic controller contains rules in the if -then format, but they can

be presented in different formats. In many systems, the rules are presented to the end-user in a

format similar to the one below,

1. If error is Neg and change in error is Neg then output is NB

2. If error is Neg and change in error is Zero then output is NM

3. If error is Neg and change in error is Pos then output is Zero

4. If error is Zero and change in error is Neg then output is NM

5. If error is Zero and change in error is Zero then output is Zero

6. If error is Zero and change in error is Pos then output is PM

7. If error is Pos and change in error is Neg then output is Zero

8. If error is Pos and change in error is Zero then output is PM

9. If error is Pos and change in error is Pos then output is PB

The names =zeros, pos, neg are labels of fuzzy sets as well as NB, NM, PB, PM

(negative big, negative medium, positive big, and positive medium respectively). The same set of

rules could be presented in a relational format, a more compact representation.


11

The top row is the heading, with the names of the variables. It is understood that the two leftmost

columns are inputs, the rightmost is the output, and each row represents a rule. This format is

perhaps better suited for an experienced user who wants to get an overview of the rule base

quickly. The relational format is certainly suited for storing in a relational database. It should be

emphasized, though, that the relational format implicitly assumes that the connective between the

inputs is always logical and_ or logical or for that matter as long as it is the same operation for all

rules_and not a mixture of connectives. Incidentally, a fuzzy rule with an or_combination of

terms can be converted into an equivalent and combination of terms using laws of logic

(DeMorgan’s laws among others). A third format is the tabular linguistic format.

This is even more compact. The input variables are laid out along the axes, and the output

variable is inside the table. In case the table has an empty cell, it is an indication of a missing

rule, and this format is useful for checking completeness. When the input variables are error and

change in error, as they are here, that format is also called a linguistic phase plane. In case there

are n > 2 input variables involved, the table grows to an n-dimensional array; rather user-un

friendly. To accommodate several outputs, a nested arrangement is conceivable. A rule with

several outputs could also be broken down into several rules with one output. Lastly, a graphical

format which shows the fuzzy membership curves is also possible Fig. 1.8. This graphical user-

interface can display the inference process better than the other formats, but takes more space on

a monitor.

Connectives In mathematics, sentences are connected with the words and, or, if_then (or

(implies), and if and (only if)), or modifications with the word not. These five are called


12

connectives. It also makes a difference how the connectives are implemented. The most

prominent is probably multiplication for fuzzy and instead of minimum. So far most of the

examples have only contained and operations, but a rule like ‘‘If error is very neg and not zero or

change in error is zero then ...’’ is also possible. The connectives and or are always defined in

pairs, for example,

There are other examples (e.g., Zimmermann, 1991) but they are more complex. Modifiers: A

linguistic modifiers, is an operation that modifies the meaning of a term. For example, in the

sentence ‘‘very close to 0’’, the word very modifies close to 0 which is a fuzzy set. A modifier is

thus an operation on a fuzzy set. The modifier very can be defined as squaring the subsequent

membership function, that is

A whole family of modifiers is generated by pa where p is any power between zero and infinity.

With ∞=p the modifier could be named exactly because it would suppress all memberships

lower than 1.0.

Universes: Elements of a fuzzy set are taken from a universe of discourse or just universe. The

universe contains all elements that can come into consideration. Before designing the

membership functions it is necessary to consider the universes for the inputs and outputs. Take

for example the rule:

If error is Neg and change in error is Pos then output is 0

Naturally, the membership functions for Neg and Pos must be defined for all possible values of

error and change in error and a standard universe may be convenient.

Another consideration is whether the input membership functions should be continuous or

discrete. A continuous membership function is defined on a continuous universe by means of


13

parameters. A discrete membership function is defined in terms of a vector with a finite number

of elements. In the latter case it is necessary to specify the range of the universe and the value at

each point. The choice between fine and coarse resolution is a trade off between accuracy, speed

and space demands. The quantiser takes time to execute, and if this time is too precious,

continuous membership functions will make the quantiser obsolete.

Membership Functions: Every element in the universe of discourse is a member of a fuzzy set to

some grade, maybe even zero. The grade of membership for all its members describes a fuzzy

set, such as |Neg. In fuzzy sets elements are assigned a grade membership, such that the

transition from membership to non-membership is gradual rather than abrupt. The set of elements

that have a non-zero membership is called the support of the fuzzy set. The function that ties a

number to each element x of the universe is called the membership function )(xµ . The designer is

inevitably faced with the question of how to build the term sets. There are two specific questions

to consider: (i) How does one determine the shape of the sets? and (ii) How many sets are

necessary and sufficient? For example, the error in the position controller uses the family of

terms Neg, Zero and pos. According to fuzzy set theory the choice of the shape and width is

subjective, but a few rules of thumb apply.

_ A term set should be sufficiently wide to allow for noise in the measurement.

_ A certain amount of overlap is desirable; otherwise the controller may run into poorly defined

states, where it does not return a well defined output

A preliminary answer to questions (i) and (ii) is that the necessary and sufficient number of sets

in a family depends on the width of the sets, and vice versa. A solution could be to ask the

process operators to enter their personal preferences for the membership curves; but operators

also find it difficult to settle on particular curves. The manual for the TILShell product

recommends the following (Hill, Horstkotte & Teichrow, 1990).

_Start with triangle set: All membership functions for a particular input or output should be

symmetrical triangles of the same width. The leftmost and the rightmost should be shouldered

ramps.

The overlap should be at least 50 %: The widths should initially be chosen so that each value of

the universe is a member of at least two sets, except possibly for elements at the extreme ends. If,

on the other hand, there is a gap between two sets no rules fire for values in the gap.

Consequently the controller function is not defined.


14

Membership functions can be flat on the top, piece-wise linear and triangle shaped, rectangular,

or ramps with horizontal shoulders. Fig. 1.8 shows some typical shapes of membership functions.

Strictly speaking, a fuzzy set A is a collection of ordered pairs:

1.3.2.4 Inference Engine

Fig. 1.9 and Fig. 1.10 are both a graphical construction of the algorithm in the core of the

controller. In Fig. 1.9, each of the nine rows refers to one rule. For example, the first row says

that if the error is negative (row 1, column 1) and the change in error is negative (row1, column

2) then the output should be negative big (row 1, column 3). The picture corresponds to the rule

base in (2). The rules reflect the strategy that the control signal should be a combination of the

reference error and the change in error, a fuzzy proportional-derivative controller. We shall refer

to that figure in the following. The instances of the error and the change in error are indicated by

the vertical lines on the first and second columns of the chart. For each rule, the inference engine

looks up the membership values in the condition of the rule.


15

Fig. 1.9 Graphical construction of the control signal in a fuzzy PD controller

Aggregation: The aggregation operation is used when calculating the degree of fulfillment or

firing strength of the condition of a rule k. A rule, say rule 1, will generate a fuzzy membership

value 1eµ coming from the error and a membership value 1ceµ coming from the change in error

measurement. The aggregation is their combination,

similarly for the other rules. Aggregation is equivalent to fuzzification, when there is only one

input to the controller. Aggregation is sometimes also called fulfillment of the rule or firing

strength.

Activation: the activation of a rule is the deduction of the conclusion, possibly reduced by its

firing strength. Thickened lines in the third column indicate the firing strength of each rule. Only

the thickened part of the singletons are activated, and min or product (*) is used as the activation

operator. It makes no difference in this case, since the output membership functions are

singletons, but in the general case of −−− andzs ,,π functions in the third column, the

multiplication scales the membership curves, thus preserving the initial shape, rather than

clipping them as the min operation does. Both methods work well in general, although the


16

multiplication results in a slightly smoother control signal. In Fig. 1.9, only rules four and five

are active.

A rule k can be weighted a priori by a weighting factor [ ]1,0∈kw , which is its degree of

confidence. In that case the firing strength is modified to

The degree of confidence is determined by the designer, or a learning program trying to adapt the

rules to some input-output relationship.

Accumulation: All activated conclusions are accumulated, using the max operation, to the final

graph on the bottom right Fig. 1.9. Alternatively, sum accumulation counts overlapping areas

more than once Fig. 1.10. Singleton output Fig. 1.9 and sum accumulation results in the simple

output

The alpha’s are the firing strengths from the n rules and s1,s2,…..sn are the output singletons.

Since this can be computed as a vector product, this type of inference is relatively fast in a matrix

oriented language.

There could actually have been several conclusion sets. An example of a one-input two- outputs

rule is ‘‘If ae is a then o1 is x and o2 is y”. The inference engine can treat two (or several)

columns on the conclusion side in parallel by applying the firing strength to both conclusion sets.

In practice, one would often implement this situation as two rules rather than one, that is, ‘‘If ae

is a then o1 is x, ‘‘If ae is a then o2 is y”.

1.3.2.5 Defuzzification

The resulting fuzzy set Fig. 1.9, bottom right; Fig. 1.10, extreme right must be converted to a

number that can be sent to the process as a control signal. This operation is called defuzzification,

and in Fig. 1.9 the x coordinate marked by a white, vertical dividing line becomes the control


17

signal. The resulting fuzzy set is thus defuzzified into a crisp control signal. There are several

defuzzification methods.

Centre of gravity (COG) The crisp output value x (black line in Fig. 1.9) is the abscissa under

the centre of gravity of the fuzzy set,

(1.1)

Here ix is a running point in a discrete universe, and )( ixµ , is its membership value in the

membership function. The expression can be interpreted as the weighted average of the elements

in the support set. For the continuous case, replace the summations by integrals. It is a much used

method although its computational complexity is relatively high. This method is also called

centroid of area.

Centre of gravity method of singletons: If the membership functions of the conclusions are

singletons Fig. 1.9, the output value is

(1.2)

Here is is the position of singleton l in the universe, and )( isµ , is equal to the firing strength

iα of rule i . This method has a relatively good computational complexity, and u is differentiable

with respect to the singletons is , which is useful in neurofuzzy systems. Bisector of area (BOA):

This method picks the abscissa of the vertical line that divides the area under the curve in two

equal halves. In the continuous case,


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(1.3)

Here x is the running point in the universe, )(xµ is its membership, min is the leftmost value of

the universe, and max is the rightmost value. Its computational complexity is relatively high, and

it can be ambiguous. For example, if the fuzzy set consists of two singletons any point between

the two would divide the area in two halves; consequently it is safer to say that in the discrete

case, BOA is not defined.

Mean of maxima (MOM) An intuitive approach is to choose the point with the strongest

possibility, i.e. maximal membership. It may happen, though, that several such points exist, and a

common practice is to take the mean of maxima (MOM). This method disregards the shape of the

fuzzy set, but the computational complexity is relatively good.

Leftmost maximum (LM), and rightmost maxima (RM): Another possibility is to choose the

leftmost maximum (LM), or the rightmost maximum (RM). In the case of a robot, for instance, it

must choose between left or right to avoid an obstacle in front of it. The defuzzifier must then

choose one or the other, not something in between. These methods are indifferent to the shape of

the fuzzy set, but the computational complexity is relatively small.

1.3.2.6 Postprocessing

Output scaling is also relevant. In case the output is defined on a standard universe this must be

scaled to engineering units for instance, volts, meters, or tons per hour. An example is the scaling

from the standard universe [-1, 1] to the physical units [-100, 100] volts. The postprocessing

block often contains an output gain that can be tuned, and sometimes also an integrator.


19

Fig. 1.10 One input, one output rule base with non-singleton output sets

1.4 Summary

In a fuzzy controller the data passes through a preprocessing block, a controller, and a

postprocessing block. Preprocessing consists of a linear or non-linear scaling as well as a

quantisation in case the membership functions are discretised (vectors); if not, the membership of

the input can just be looked up in an appropriate function. When designing the rule base, the

designer needs to consider the number of term sets, their shape, and their overlap. The rules

themselves must be determined by the designer, unless more advanced means like self-

organization or neural networks are available. There is a choice between multiplication and

minimum in the activation. There is also a choice regarding defuzzification; center of gravity is

probably most widely used. The postprocessing consists in a scaling of the output. In case the

controller is incremental, postprocessing also includes an integration. The following is a checklist

of design choices that have to be made.

_Rule base related choice. Number of inputs and outputs, rules, universes, continuous / discrete,

the number of membership functions, their overlap and width, singleton output;

_Inference engine related choices. Connectives, modifiers, activation operation, aggregation


20

operation, and accumulation operation

_ Defuzzification method. COG, COGS, BOA, MOM, LM, and RM

_ Pre-and post processing. Scaling, gain factors, quantization, and sampling time

Some of these items must always be considered, others may not play a role in the particular

design. The input-output mappings provide an intuitive insight which may not be relevant from a

theoretical viewpoint, but in practice they are well worth using. The analysis represented by plots

is limited, though, to three dimensions. Various input-output mappings can be obtained by

changing the fuzzy membership functions, and the chapter shows how to obtain a linear mapping

with only a few adjustments. The linear fuzzy controller may be used in a design procedure based

on PID control as following:

1. Tune a PID controller.

2. Replace it with a linear fuzzy controller.

3. Transfer gains.

4. Make the fuzzy controller nonlinear.

5. Fine-tune it.

It seems sensible to start the controller design with a crisp PID controller, maybe even just a P

controller, and get the system stabilized. From there it is easier to go to fuzzy control.


21

Chapter (2) Literature Survey

Traditional and Fuzzy Logic P, PD and PID controllers

2.1 The Control Problem Given a system or process whose behavior can be controlled with one or more input signals, the

problem arises of inducing a desired state or response in an optimal way. “Optimal” could refer

here to obtaining the desired system’s response in minimal time, or with a minimum number of

oscillations, or without overshooting, etc.

In this chapter we discuss the control problem from the point of view of a system with feedback.

Many feedback controlled systems were built before control theory developed in the 19th and

20th centuries. Control theory is now a fairly mathematical subject, in which processes are

described in the time or frequency domain using special transformations, most notably the

Laplace Transformation. Here we will deal with a special case of a controller, the PID controller,

which is simple but nevertheless the most popular choice for system control. If we had an

analytical model of a process valid for a broad range of environmental conditions, we could

compute the optimal input in advance, or even the optimal sequence of input signals needed to

obtain a desired behavior. Let us assume that the response of the system to an input x is provided

by the functional model f(x). Then, the required input for the response y is just f(y). Having an

“inverse model” of the system allows us to input the control variable to the system directly and

even without a feedback signal. More frequently, however, although we can have detailed

knowledge of the system, it is very difficult to compute in advance the optimal input due to

environmental disturbances and noise. We then proceed using a kind of “trial and error”

approach, testing with one input signal, and adjusting its value according to the measured error.

The corrections are not made blindly, but based in the detected past behavior of the system, as we

show below.

When a system is controlled without looking to the actual behavior or output, we have an open-

loop system (open loop because the possible feedback loop has not been established). Open loop

systems are simpler and can be adequate in some cases. A closed-loop system is one in which a

measurement of the system’s state is compared to the desired goal, and the difference can then be

used to make corrections. Fig. 2.1 shows a diagram of a system prepared for closed-loop control.


22

The controller, nowadays almost always a microprocessor, receives the desired output value as

well as the measured error of the system’s response (initialized to zero at the beginning). The

controller produces a control signal which drives the robot’s actuators. This, together with

environmental disturbances, produces the actual system’s output. The output can be monitored

using sensors possibly affected by noise. The measurement is then compared inside the controller

with the desired response, and the controller produces a corrected control signal. The controller is

activated many times and very rapidly in comparison with the process’ inertia.

Fig. 2.1 A generic model for control with feedback

The main question is therefore, what should be the structure of the controller box. A very popular

strategy for driving such closed-loop systems are PID controllers. The letter P stands here for

proportional, I for integral, and D for differential.1 The letters P, I, and D refer also to the three

gain constants used in PID controllers.

In what follows we take a very simple example, a DC motor, and discuss what happens when a

desired rotational speed is needed. We assume that the controller does not contain any analytical

model of the motor which could allow it to set the input voltage a priori according to the desired

speed. We only assume that we can make the voltage lower or higher (between −1 and 1 Volts) in

fractional steps, and that we can monitor the instantaneous rotational speed of the motor. A mass

with moment of inertia I is fixed to the motor’s rotor.

In a DC motor the relationship between the torque and the voltage is linear. In the examples

below we use the following relationship between torque D, voltage V, and angular velocity ω :


23

where a and b are constants specific to the motor. The term b! is the braking voltage produced by

the DC motor, which also behaves as a generator, when it is running at angular velocity ω . When

the DC motor reaches its maximum speed, the braking voltage is equal to the maximum voltage

and the torque is zero. The constant ‘a’ converts from units of voltage to units of torque.

Fig. 2.2 A classical PID controller

When the mass m attached to the rotor is small (i.e. I is small) the motor responds to high

voltages with a “jump-start”. When I is large, it takes much more time to accelerate the DC

motor. The PID controller should take charge of minimizing the acceleration time, while at the

same time avoiding control oscillations, which are very common in robots or any process in

which we try to reach the steady state in minimal time. In what follows we first take a look at a

purely proportional controller, then a PI controller, and finally a PID controller.

Fig. 2.2 shows a schematics of a discrete bare bones PID controller. We denote time by t. Time

advances in discrete steps. The controller receives as input not the reference signal r(t) but the

difference between the current state of the system given by s(t), and the reference signal r(t). This

we call the error e(t). The controller generates from the input e(t) a corrected control signal c(t +

1) which drives the system, produces a new output s(t + 1), a new error, which is fed back to the

controller box. It is very important to notice that in this simple feedback loop the original value of

the reference signal r(t) is not directly visible to the controller. In some variations of PID this

assumption is eliminated. The classical textbook version of the control signal generated by a PID

controller reads as follows:

(2.1)

where Kp, Kp/Ti and KpTd are the proportional, integral, and differential control gains, as we

discuss below. Sometimes the equation is written in a simpler manner

(2.2)


24

where Kp, Ki and Kd are the proportional, integral and differential gains. Equation (2.1) has a

form which allows us to associate an intuitive meaning with the constants Ti and Td, as explained

below.

The formulas describe a continuous PID controller in which the process is running continuously

and the feedback loop is so fast as to seem continuous (for example, using an analog circuit). The

integral of the current instantaneous error e(t) divided by Ti is a measure of how far off the

system has been from the desired response in the past, and the derivative of the error is a measure

of how fast the error is changing. The derivative of the error, multiplied by a time step Td, can be

used as a predictor of the future error. A PID controller, therefore, adjusts the control signal

according to the current error, a measure of the error in the past, and the predicted error in the

near future.

(2.3)

If we take Ti = t and Td = 1, then the current error is e(t), the I-term is the average error in the

past, and the D-term is the predicted error in the next time step. They all enter in the computation.

Their sum is multiplied by the constant Kp and is used as the control signal. Variations in the

values of Ti and Td provide different weighted averages of past, current, and future error.

According to surveys of the kind of control methods most popular in industry, PID controllers

account for the bulk of all devices used in process control [3]. They are popular because they are

simple, analyzable, and because there is already a rich literature which deals with them.

2.2 Controller Types

2.2.1 Proportional Controller A P-controller (proportional controller) runs in a closed-loop, which in the case of robots can be

monitored up to several thousand times per second. During the control loop there is a difference

at time t between the desired system state r(t) (the reference signal), and the current system state


25

s(t). Therefore, the error computed in the controller at time t is e(t) = r(t) − s(t). Let us denote by

c(t) the control signal applied at time t. In the discrete case, the control signal for the next control

iteration can be computed as

(2.4)

where Kp is a proportionality constant which transforms from state to control units, and which

determines how large the control signal should be. In the case of the DC motor described above,

we can experiment with several control signals. Starting from c(0) = 0, for a desired fixed

angular velocity r(·) = 0.5, for given motor parameters and a fixed mass with moment of inertia I,

we obtain the results shown in Fig. 2.2. The graphic shows how the rotational velocity increases

until the system comes close to the desired rotational speed. The motor does not reach the desired

velocity of 0.5 because should the error (e(t) fall to zero, then the control variable c(t + 1) (in this

case the voltage) would also fall to zero. The system reaches the steady state when a certain error

e(t) is such that the computed control variable c(t + 1) produces the error e(t + 1) = e(t). The error

stabilizes at a non-zero level which produces a constant control signal. In the example shown in

Fig. 2.2 the controller produces at the beginning voltages higher than 1 Volt. In such case the

control signal has become saturated and we clip it at 1 Volt. This introduces a further

complication in the control process, a nonlinearity which in this case is not so important, but

which can afflict the control of other kind of systems. Since the motor does not reach 0.5 as

angular velocity, we can try to force it to reach that speed, but only at the expense of increasing

excessively the proportional gain. The rotational speed would oscillate then around an

equilibrium value which is still different from 0.5.

2.2.2 PI Controller The solution to the problem of the system settling on a steady state s(t) which is different from

the desired state r(t) is to add another term to the computation of the control signal. An integral

controller adds up the error during the successive control cycles (as shown in Equation 1). This

integrated error can then be used to adjust the control variable c(t). If, for example, the error has


26

been positive for many past cycles, the correction to the control variable should be increased

more drastically. The control expression used in a PI controller is of the form

(2.5)

where Ti is the integration time constant, and ∑t

tie )( is the discrete integral of the past errors

until time t. Adjusting Ti and for a given Kp, we obtain different weights for the integral of the

past error. The factors Ti/(1+Ti) and 1/(1+Ti) are the relative weights of the P-term and the

I-term before the are multiplied by the constant Kp. 2.2.3 PID Controller The effect of the integral term is to eliminate the systematic offset from the desired response r(t).

However, the introduction of the integral term can lead to undesirable oscillations and makes the

controller act “harder”. This problem can be ameliorated using a differential term in the

computation of the control signal. For the differential term we use the discrete derivative of the

error signal, that is, the difference e(t) − e(t − 1). If the derivative is large, that is, if the error is

changing rapidly, we can decide to make the control signal change smaller by an amount

KpTd(e(t) − e(t − 1)). The control signal at (t + 1) becomes then

(2.6)

2.2.4 Proportion Fuzzy Controller

Input to a fuzzy proportional (FP) controller is error, and the output is the control signal, cf. the

block diagram in Fig. 2.3. This is the simplest fuzzy controller there is. It is relevant for state- or

output-feedback in a state space controller.Compared to crisp proportional control the fuzzy P

controller has two gains GE and GU in stead of just one. instead of just one. As a convention,

signals are written in lower case before gains and upper case after gains, for instance eGEE *= .

The gains are mainly for tuning the response, but since there are two gains, they can also be used

for scaling the input signal onto the input universe to exploit it better.


27

Fig. 2.3 Fuzzy proportional controller (FP)

The controller output is the control signal nU a nonlinear function of ne ,

(2.7)

The function f is the fuzzy input-output map of the fuzzy controller. Using the linear

approximation nn eGEeGEf *)*( = then

(2.8)

Compared with the conventional proportion gain, the product of the gain factors is equivalent to

the proportional gain,i.e.,

(2.9)

The accuracy of the approximation depends mostly on the membership functions and the rules.

The approximation is best, however, if we choose the same universe on both input and output

side, for example [-100, 100]. The rule base

1. If H is Pos then x is 100

2. If H is Neg then x is -100

with Pos and Neg triangular as defined previously, is equivalent to a P-controller. Given a target

Ns, for example from the Ziegler-Nichols rules, equation (2.9) helps to choose the gains. The

equation has one degree of freedom, since the fuzzy P controller has one more gain factor than

the crisp P controller. This is used to exploit the full range of the input universe. If for example

the maximal reference step is 1, whereby the maximal error hq is 1, and the universe for E is [-

100, 100], then fix GE at 100. Since GE is now fixed, GU is determined by equation (2.9).

Because of the process dynamics it will take some time before a change in the control signal is


28

noticeable in the process output, and the proportional controller will be more or less late in

correcting for an error.

2.2.5 Proportional and derivative fuzzy controller Derivative action helps to predict the error and the proportional-derivative controller uses the

derivative action to improve closed-loop stability. The basic structure of a PD controller is

(2.10)

The control signal is thus proportional to an estimate of the error dT seconds ahead, where the

estimate is obtained by linear extrapolation. For 0=dT the control is purely proportional, and

when dT is gradually increased, it will dampen oscillations. If dT becomes too large the system

becomes overdamped and it will start to oscillate again. Input to the fuzzy proportional-derivative

(FPD) controller is the error and the derivative of the error Fig. 2.4. In fuzzy control the latter

term is usually called change in error.

(2.11)

Fig. 2.4 Fuzzy PD controller (FPD)

This is a discrete approximation to the differential quotient using a backward difference. Other

approximations are possible, as in crisp PD controllers. Notice that this definition deviates from

the straight difference 1−−= nnn eece used in the early fuzzy controllers. The controller output is

a nonlinear function of error and change in error.


29

(2.12)

Again the function i is the input-output map of the fuzzy controller, only this time it is a surface.

Using the linear approximation nn ceGCEeGE ** + , then

(2.13)

By comparison, the gains in (2.10) and (2.13) are related in the following way,

(2.14)

(2.15)

The approximation corresponds to replacing the fuzzy input-output surface with a plane. The

approximation is best if we choose the output universe to be the sum of the input universes.

Assume, for example, that the input universes are both [-100, 100] and we choose output

singletons on [-200, 200], then the input-output map will be the plane CEEu += . Therefore, by

that choice, we can exploit (2.14) and (2.15). The fuzzy PD controller may be applied when

proportional control is inadequate. The derivative term reduces overshoot, but it may be sensitive

to noise as well as an abrupt change of the reference causing a derivative kick The usual counter-

measures may overcome these problems: in the former case insert a filter, and in the latter use the

derivative of the process output ny instead of the error.

2.2.6 Incremental Fuzzy Controller

If there is a sustained error in steady state, integral action is necessary. The integral action will

increase the control signal if there is a small positive error, no matter how small the error is; the

integral action will decrease it if the error is negative. A controller with integral action will

always return to zero in steady state. It is possible to obtain a fuzzy PI controller using error and

change in error as inputs to the rule base. Experience shows, however, that it is rather difficult to

write rules for the integral action. Problems with integrator wind up also have to be dealt with.

Windup occurs when the actuator has limits, such as maximum speed for a motor or maximum

opening of a valve. When the actuator saturates, the control action stays constant, but the error


30

will continue to be integrated, the integrator winds up. The integral term may become very large

and it will then take a long time to wind it down when the error changes sign. Large overshoots

may be the consequence. There are methods to avoid it (Åström & Hägglund, 1995). It is often a

better solution to configure the controller as an incremental controller. An incremental controller

adds a change in control signal u∆ to the current control signal

(2.16)

using (2.2) with 0=dT . It is natural to use an incremental controller when for example a stepper

motor is the actuator. The controller output is an increment to the control signal, and the motor

itself performs an integration. It is an advantage that the controller output is driven directly from

an integrator, then it is easy to deal with windup and noise. A disadvantage is that it cannot

include D-action well. The fuzzy incremental (FInc) controller in Fig. 2.5 is almost the same

configuration as the FPD controller except for the integrator on the output The output from the

rule base is therefore called change in output and the gain in the output has changed name

accordingly to GCU. The control signal nU is the sum of all previous increments,

(2.17)

Notice again that this definition deviates from the early fuzzy controllers, where nU =

Fig. 2.5 Fuzzy incremental controller (FInc)


31

∑ )*( icuGCU the difference is the sampling period sT . The linear approximation to this controller is

(2.18)

By comparing (2.16) and (2.18) it is clear that the gains are related in the following way,

(2.19) Notice that the proportional gain now depends on GCE. The gain on the integral action is

determined by the ratio between the two input gains, and it is the inverse of the derivative gain in

FPD control. It is as if GE and GCE have changed roles. The controller is really a PI controller,

compare (2.2) and (2.18). The usual problem of Integrator windup can be overcome by simply

limiting the integrator. 2.2.7 Proportional, integral and derivative fuzzy controller It is straight forward to envision a fuzzy PID controller, with three input terms: error, integral

error and derivative error. A rule base with three inputs, however, easily becomes rather big and,

as mentioned earlier, rules concerning the integral action are troublesome. Therefore it is

common to separate the integral action as in the fuzzy PD+I (FPD+I) controller in Fig. 2.6. The

integral error is computed as,

(2.20)


32

Fig. 2.6 Fuzzy PD+I controller (FPD+I)

The controller is thus a function of the three inputs

(2.21)

Its linear approximation is

(2.22)

In the last line we have assumed a nonzero GE. Comparing (2.2) and (2.22) the gains are related

in the following way,

(2.23)

This controller provides all the benefits of PID control, but also the disadvantages regarding

derivative kick and integrator windup


33

2.2.8 Summary Advantages and disadvantages of all four fuzzy controllers are listed in Table 2.1. The fuzzy P

controller may be used in state space models or for practicing. To improve the settling time and

reduce overshoot, the fuzzy PD is the choice. If there is a problem with a steady state error, a

fuzzy incremental controller or a fuzzy PD+I is the choice. The relationships between the PID

gains and the fuzzy gains are summarized in Table 2.2. To emphasize, the controllers with i

replaced by a summation are linear approximations to the corresponding fuzzy configurations;

the relations hold for the approximations only. They are valid when FH and FX are true

difference quotients instead of just differences. If these are implemented as differences anyway,

( 1−−=∆= nnnn eeece and 1−−=∆= nnnn uuucu ) the sample period must be taken into account

in the equations, and the table modified accordingly. Also, for fixed universe controllers, the

output universe must be the sum of the input universes. With input universes [-100, 100], for

instance, the output universe of an FPD controller should be [-200, 200].

Table 2.1 Quick reference to fuzzy controllers

Table 2.2 Relation between fuzzy and PID gains


34

2.3 Problem Formulation In order to design air refueling autopilot that containing multiple stages, there are some basic

features that should be satisfied such that the fighter-tanker configuration constrains, flying

qualities standard and the minimum possible mission time that allow the tanker to refuel multiple

aircrafts within small time. On the other hand it is required to design a suitable controller that

meets the mentioned criteria together with a safety margin. consequently the aerial refueling

process should be divided into three basic stages that minimize the total time period with a

control discipline to switch between controllers in such a way without losing system stability.

These three basic stages can be defined (initial stability stage, tracking stage and alignment

stage). Each one of these stages has an important role during the predefined mission. Initial

stability stage is to recover the aircraft from any past maneuvers, tracking stage has a significant

impact to minimize the whole mission time and the alignment stage is to secure the aircraft

docking under the tanker satisfying the safety constrains of the tanker.

2.4 Main Objective of the Thesis The main objective of this thesis is to design and implement an integrated control system that

combines both advantages of an optimal controller and fuzzy logic controller. Genetic algorithms

will be used to optimize the fuzzy logic parameters such that the output gains, the membership

functions locations and rule base firing strength. In mean while it is required to apply the three

stages control strategy to construct the overall aircraft autopilot. Providing a complete mission

simulation describing the three basic stages.

2.5 Thesis Layout In the beginning of this thesis in chapter 1 some brief introductory is presented to explain the

aerial refueling aspects and the main concept of the fuzzy logic controller. Chapter 2 illustrates

many types of traditional and smart (fuzzy logic) controllers. Chapter 3 presents the basic

preliminaries needed to construct aircraft model and the basic principles to compute the

linearized state space from the F16 nonlinear model. An optimal controller type (LQR) will be

discussed in Chapter 4 with an iterative algorithm to compute the output gain matrix for both


35

longitude and lateral motions. Chapter 5 will discuss the intelligent learning of fuzzy logic

controller integrated with LQR controller. In this part GA will be used to optimize the fuzzy logic

parameters that results in the minimum output performance index. GA code will be provided with

different number of samples and different number of generations with different initializations to

cope with the different types fuzzy logic parameters. After testing the integrated LQR Fuzzy

logic controller via SIMULINK / MATLAB for longitudinal and lateral motions separately, the

complete integrated control system will be applied and simulated for fighter-tanker configuration.

Applying the whole control strategy for the mission proposed with multiple stages, consequently

evaluating the system performance and verifying with the required constrains as mentioned

before in problem formulation in the previous section. Chapter 6 will discuss the last stage of the

aerial refueling controller (Alignment stage) and all design constrains that should be verified for

safe and secured docking. Chapter 7 illustrates the complete methodology for designing an aerial

refueling autopilot. Chapter 8 introduces the final conclusion of the thesis with the future work

expected.


36

Chapter 3

Preliminaries

Modeling, Trimming and Linearization

3.1 Aircraft Modeling The first successful study of control and dynamics of an aircraft is credited to Bryan and

Lanchester (1908). The equations of motion have changed little in the last century, and the

original formulations are still used. The following sections give an overview of the various

reference frames used to describe an aircraft’s state, provide an overview of the general nonlinear

and linear equations of motion for an aircraft and review the various ways that failures can be

modeled.

3.1.1 Reference Frames There are a variety of reference frames used to describe aircraft movement and orientation. Some

are suited to navigation (FE) and some to control (FB, FS, FW

). The relevant frames are described

in the following list.

Body-fixed reference frame FB: this is a right-hand orthogonal reference system which has its

origin, OB, at the centre of gravity of the aircraft. The XBOBZB plane coincides with the aircraft’s

plane of symmetry if it is symmetric, or it is located in a plane, approximating what would be the

plane of symmetry if it is not. The XB axis is directed towards the nose of the aircraft, the YB axis

points to the right wing, and the ZB

axis points towards the bottom of the aircraft.

Stability reference frame FS: this is a special body-fixed reference frame, used in the study of

small deviations from the nominal flight condition. The reference frames FB and FS differ in the

orientation of their respective X axis. The XS axis is chosen parallel to the projection of the true

airspeed vector V on the OBXBZB plane (if the aircraft is symmetric this is the plane of

symmetry), or parallel to V itself in case of a symmetrical nominal flight condition. The YS axis

coincides with the YB axis.


37

Flight path or wind reference frame FW: this reference frame, also called the wind reference

frame, has its origin in the C.G. of the aircraft. The Xw axis is aligned with the velocity vector of

the aircraft and ZW axis coincides with the ZS

axis.

Earth fixed reference frame FE : this reference frame, also called the topodetic reference frame, is

a right-handed orthogonal system which is considered to be fixed in space. Its origin can be

placed at an arbitrary position, but will be chosen to coincide with the aircraft’s centre of gravity

at the start of a flight test maneuver. The ZE axis points downwards, parallel to the local direction

of gravity. The XE axis is directed to the north, the YE

axis to the east.

The relationships between the various frames of reference are shown in Fig. 3.1.

Fig. 3.1 Relationship between the body-fixed reference frame FB, flight- path reference

frame FW and stability reference frame FS


38

3.1.2 General Equation of Motion Our goal in this section is to describe the position and orientation of the aircraft in appropriate

reference frames, as well as the dynamics and relationships between the various frames.

The velocity of the aircraft is defined as follows:

[ ]TB wvuV = : Velocity with respect to FB

[ ]TEEEE wvuV = : Velocity with respect to F

[ ]zyxE WWWW =

E

: Velocity with respect to F

W

The velocity VB

EBEBWB WLVV +=

does not include the effects of wind. If wind is to be explicitly included a

superscript W will be used. , where 1−= EBBE LL is the coordinate transform from

FE to FB. LEB

is defined as:

−−++−

=θϕθϕθ

ψϕψθϕψϕψθϕψθψϕψθϕψϕψθϕψθ

coscoscossinsincossinsinsincoscoscossinsinsinsincossinsincossincossincoscossinsincoscos

EBL (3.1)

Where ϕψθ ,, describes the orientation of the aircraft as defined below.

Two important relationships between the various reference frames can now be defined. As shown

in Fig. 3.1 the angle of attack,α , is the angle between the body-fixed axis and the stability axis.

The sideslip angle β is the angle between the wind axis and the stability axis. These values can

be computed as follows:

uw1tan −=α (3.2)

222

1sinwvu

vB++

= − (3.3)


39

The position of the aircraft is most often used for navigation, and hence its dynamics are given in

the earth-frame as follows:

WBEB

WE

E

E

E

E VLVzyx

r ==

=•

•

•

• (3.4)

Note that ZE points into the ground and is therefore negative for positive height. As a result, the

altitude or height is often used instead of ZE EZh −=:

The orientation of the airplane is defined relative to the earth axes and is given by the Euler

angles ),,( ϕθψ . The Euler angles define the rotations from the earth-fixed reference to the body-

fixed axis. The ordering of the rotations is important and is done as follows:

1. Rotate ψ about OEZ

2. Rotate

E

θ about OEYE

3. Rotate

ϕ about OEZ

E

Euler angles are not unique, and so in order to avoid ambiguities, the following ranges are

defined:

πψπ ≤≤− or πψ 20 ≤≤

22πθπ

≤≤−

πϕπ ≤≤− or πϕ 20 ≤≤

The angular rotation rates are defined relative to the body-fixed inertial frame FB

[ ]TB rqp=ω

as

, where p is the roll rate, q is the pitch rate, and r is the yaw rate. The angular

rates are related to the rate of change of the Euler angles by the following coordinates transforms:


40

−=

•

•

•

ψθϕ

θϕϕθϕϕ

coscossin0cossincos0

001

rqp

(3.5)

−=

•

•

•

rqp

θϕθϕϕϕ

θϕθϕ

ψθϕ

seccossecsin0sincos0

tancostansin1 (3.6)

Note that when perturbations are small, such that ),,( ψθϕ may be treated as small angles,

equations 3.5 and 3.6 may be approximated by:

=

•

•

•

ψθϕ

rqp

(3.7)

It is ready now to compute the dynamics equations of the aircraft. The dynamics are built from

basic Newtonian dynamics of a rigid body, as given by the general force and moment equations:

)( VtVmF ×Ω+

∂∂

= (3.8)

)()(Ω×Ω+

∂Ω∂

= It

IM (3.9)

Where V is linear velocity, Ω is angular velocity and I is the moment of inertia. Equation (3.8)

and (3.9) written in terms of the variables defined in this section is:

)(sin WWW rvqwummgX −+=−•

θ (3.10)

)(sincos WWW pwruvmmgY −+=+•

ϕθ (3.11)

)(coscos WWW qupvwmmgZ −+=+•

ϕθ (3.12)

qrIIrpqIpqrLrqIpIL zyxyzxyzx )()()()( 22 −−−−+−−−= ••• (3.13)

rpIIpqrIqrpIprIqIM xzyzxyzxy )()()()( 22 −−−−+−−−= ••• (3.14)

pqIIqrpIrpqIqpIrIN yxzxyzxyz )()()()( 22 −−−−+−−−= ••• (3.15)


41

Where

−−−−−−

=

zzyzx

yzYyx

xzxyx

B

IIIIIIIII

I is the inertia, [ ]ZYX and [ ]NML are the forces and

moments acting on the aircraft respectively. These forces and moments are functions of the

control surfaces, thrust and drag. They can be written as functions of the six linear and angular

velocities ),,,,,( rqpwvu and the actuator positions, which is usually given by the aileron,

elevator, rudder and throttle positions. In some cases it is easier to use the true velocity BVV = ,

sideslip angle β and angle of attack α rather than the body-fixed velocities ),,( wvu .equations

3.10 to 3.12 can be replaced by:

)sinsinsincoscos(1 βαββα ZYXm

V ++=• (3.16)

βααααβ

α tan)sincos()cossin(1cos1 rpqZX

mV+−+

+−=• (3.17)

ααβαββαβ cossin)sinsincossincos(11 rpZYXmV

−+

−+−=• (3.18)

Equations 3.10 to 3.18 can be combined together to form the full nonlinear dynamic of the

airplane. Other commonly used variables, which can be computed directly from the aircraft’s

state, are the azimuth or heading angle, χ , and the flight path angle, γ , these are defined as:

ψβχ += (3.19)

αθγ −==•

− )(sin 1

Vh (3.20)

3.1.3 Linearized Equations of Motion The equations of motion developed in the previous section are nonlinear. If it is assumed that the

motion of the airplane consists of small deviations from a reference condition of steady state

flight, then a linearized model around the trim condition can be obtained.

Steady flight can be defined, for example, as one of the following:

Steady wings-level flight: 0,,, =••• ψθϕϕ

Steady turning flight: 0, =•• θϕ , •ψ =turn rate


42

Steady pull-up: 0,, =•• ψϕϕ , •θ =pull-up rate

Steady roll: 0, =•• ψθ , •ϕ =roll rate

Where 0,,,,, =•••••• βαVrqp and all control surface inputs are zero.

3.2 Aircraft Trimming Using NN Algorithm

3.2.1 Basic Architecture of Neural Network

A layered feed-forward network consists of a certain number of layers, and each layer contains a

certain number of units. There is an input layer, an output layer, and one or more hidden layers

between the input and the output layer. Each unit receives its inputs directly from the previous

layer (except for input units) and sends its output directly to units in the next layer (except for

output units). Unlike the Recurrent network, which contains feedback information, there are no

connections from any of the units to the inputs of the previous layers nor to other units in the

same layer, nor to units more than one layer ahead. Every unit only acts as an input to the

immediate next layer. Obviously, this class of networks is easier to analyze theoretically than

other general topologies because their outputs can be represented with explicit functions of the

inputs and the weights[6], [7], [8], [9].

Fig. 3.2 Basic Neural Network Structure

An example of a layered network with one hidden layer is shown in Error! Reference source not

found. In this network there are l inputs, m hidden units, and n output units. The output of the jth

hidden unit is obtained by first forming a weighted linear combination of the l input values, then

adding a bias,


43

where

(3.14)

is the weight from input i to hidden unit j in the first layer and is the bias for

hidden unit j. If we are considering the bias term as being weights from an extra input eqn (3.14)

can be rewritten to the form of,

The activation of hidden unit j then can be obtained by transforming the linear sum using an

activation function

(3.15)

The outputs of the network can be obtained by transforming the activation of the hidden units

using a second layer of processing units. For each output unit k, first we get the linear

combination of the output of the hidden units,

(3.16)

Again we can absorb the bias and rewrite the above equation to,

(3.17)

Then applying the activation function

(3.18)

to (3.18) we can get the kth output

Combining eqn (3.16), (3.17), (3.18) and (3.19) we get the complete representation of the

network as

(3.19)

(3.20)


44

The simple network is a network with one hidden layer. We can extend it to have two or more

hidden layers easily as long as we make the above transformation further. One thing we need to

note is that the input units are very special units. They are hypothetical units that produce outputs

equal to their supposed inputs. No processing is done by these input units.

3.2.2 Network Structure Design

Though theoretically there exists a network that can simulate a problem to any accuracy, there is

no easy way to find it. To define an exact network architecture such as how many hidden layers

should be used, how many units should there be within a hidden layer for a certain problem is

always a painful job. Here we will give a brief discussion about the issues to be considered when

we design a network.

3.2.2.1 Number of Hidden Layers

Because networks with two hidden layers can represent functions with any kind of shapes, there

is no theoretical reason to use networks with more than two hidden layers. It has also been

determined that for the vast majority of practical problems, there is no reason to use more than

one hidden layer. Problems that require two hidden layers are only rarely encountered in practice.

Even for problems requiring more than one hidden layer theoretically, most of the time, using

one hidden layer performs much better than using two hidden layers in practice. Training often

slows dramatically when more hidden layers are used. There are several reasons why we should

use as few layers as possible in practice:

1) Most training algorithms for feed-forward network are gradient-based. The additional

layer through which errors must be back propagated makes the gradient very unstable. The

success of any gradient-directed optimization algorithm is dependent on the degree to which the

gradient remains unchanged as the parameters vary.

2) The number of local minima increases dramatically with more hidden layers. Most of the

gradient-based optimization algorithms can only find local minima, thus they miss the global

minima. Even though the training algorithm can find the global minima, there is a higher

probability that after many time-consuming iterations, we will find ourselves stuck in a local

minimum and have to escape or start over.


45

Of course, it is possible that for a certain problem, using more hidden layers of just a few units is

better than using fewer hidden layers requiring too many units, especially for networks that need

to learn a function with discontinuities. In general, it is strongly recommended that one hidden

layer be the first choice for any practical feed-forward network design. If using a single hidden

layer with a large number of hidden units does not perform well, then it may be worth trying a

second hidden layer with fewer processing units.

3.2.2.2 Number of Hidden Units

Another important issue in designing a network is how many units to place in each layer. Using

too few units can fail to detect the signals fully in a complicated data set, leading to underfitting.

Using too many units will increase the training time, perhaps so much that it becomes impossible

to train it adequately in a reasonable period of time. A large number of hidden units might cause

overfitting, in which case the network has so much information processing capacity, that the

limited amount of information contained in the training set is not enough to train the network.

The best number of hidden units depends on many factors – the numbers of input and output

units, the number of training cases, the amount of noise in the targets, the complexity of the error

function, the network architecture, and the training algorithm.

There are lots of “rules of thumb” for selecting the number of units in the hidden layers [10]-[15]:

• •

• •

- between the input layer size and output layer size

• •

- two third of the input layer size plus the output layer size

• •

- less than twice the input layer size

Those rules can only be taken as a rough reference when selecting a hidden layer size. They do

not reflect the facts well because they only consider the factor of the input layer size and output

layer size but ignore other important factors that we previously mentioned.

- squared input layer size multiplied by output layer size

In most situations, there is no easy way to determine the optimal number of hidden units without

training using different numbers of hidden units and estimating the generalization error of each.


46

The best approach to find the optimal number of hidden units is trial and error. In practice, we

can use either the forward selection or backward selection to determine the hidden layer size.

Forward selection starts with choosing an appropriate criterion for evaluating the performance of

the network. Then we select a small number of hidden units, like two if it is difficult to guess

how small it is; train and test the network; record its performance. Next we slightly increase the

number of hidden units; train and test until the error is acceptably small, or no significant

improvement is noted, whichever comes first. Backward selection, which is in contrast with

forward selection, starts with a large number of hidden units, and then decreases the number

gradually. This process is time-consuming, but it works well.

3.2.3 Back-Propagation Technique

Back-propagation is the most commonly used method for training multi-layer feed-forward

networks. It can be applied to any feed-forward network with differentiable activation functions.

This technique was popularized by Rumelhart, Hinton and Williams.

For most networks, the learning process is based on a suitable error function which is then

minimized with respect to the weights and bias. If a network has differential activation functions,

then the activations of the output units become differentiable functions of input variables, the

weights and bias. If we also define a differentiable error function of the network outputs such as

the sum-of-square error function, then the error function itself is a differentiable function of the

weights. Therefore, we can evaluate the derivative of the error with respect to weights, and these

derivatives can then be used to find the weights that minimize the error function, by either using

the popular gradient descent or other optimization methods. The algorithm for evaluating the

derivative of the error function is known as back-propagation, because it propagates the errors

backward through the network.

3.3 Optimization Algorithms Applied to F16 nonlinear model

Though the gradient descent optimization method used in the standard back-propagation learning

algorithm is widely used and proven very successful in many applications such as aircraft

trimming process. The following algorithm flowchart was done using MATLAB code to


47

calculate the trimming values of the aircraft F-16 at 0.75 mach. Table 3.1 shows the trimming

values at different altitudes after using 101 random samples through the NN algorithm.

The algorithm procedure in brief:

1- Initializing random number generator to produces bounded elevator angle and throttle

positions values required for neural network model.

2- Applying all these values directly to the F16 nonlinear model.

3- Computing the integral square error in the model simulation for both altitude and mach

number.

4- Checking the ISE (integral square error) for convergence therefore selecting the suitable

values of elevator angle and throttle position for trimming.

5- Collecting the previous selected values to apply to the neural network model in shape of

epochs.

6- Applying the back propagation algorithm for a feed forward neural network model thus

producing the trimming values for the required conditions.

7- Comparing the output index with the reference index for verification.


48

Random Number Initialization

Random No. Generator (mean-variance)

Nonlinear Aircraft model

ISE (Alt) ISE ( M)

ISE Converge

Yes No

Input Pool (M, H, ele, thr)

Simulation Results

Elevator Throttle

Trimmed Conditions

Modify

Repeat

200 counts

Multilayer Feed Forward NN (Inverse Model ) B/P Training Algorithm

Generating Trimming data for several operating conditions

Model Verification for evaluating NN fitness according to reference index

Not verified

Increasing NN training epochs

Optimized trimming data for steady level flight at operating conditions

M Alt


49

Table 3.1 Wing level trimmed values at Mach 0.75

3.4 Linearization Process Often have a nonlinear set of dynamics given by:

),( uxfx =• (3.21)

Where x is once gain of the state vector, u is the vector of inputs, and ),( ••f is a nonlinear vector function that describes the dynamics. Typically assume that the system is operating about some nominal state solution )(0 tx (possibly requires a nominal input )(0 tu ).

- Then write the actual state as )(tx = )()(0 txtx δ+ and the actual inputs as )()()( 0 tututu δ+=

- The δ is included to denote the fact that we expect the variations about the nominal to be “small”

- Then develop the linearized equations by using Taylor series expansion of ),( ••f about )(0 tx and )(0 tu

- Recalling the vector equation ),(0 uxfx = , each equation of which ),(0 uxfx ii =

can be expanded as ),()( 000 uuxxfxxdtd

iii δδδ ++=+

uufx

xfuxf ii

i δδ00

00 ),(∂∂

+∂∂

+≈ (3.22)

Where

∂∂

∂∂

=∂∂

n

iii

xf

xf

xf

.......1

And

0• means that we should evaluate the function at the nominal values of 0x and 0u .


50

The meaning of “small” deviations now clear the variations in xδ and uδ must be small enough that we can ignore the higher order terms in the Tayolr expansion of ),( uxf . Since

),( 000

uxfdt

dxi

i = we thus have that

uuf

xxf

xdtd ii

i δδδ00

)(∂∂

+∂∂

≈ (2.23)

Combining for all n state equations, gives (note that we also set ≈→= ) that

u

uf

ufuf

x

xf

xfxf

xdtd

nn

δδδ

∂∂

⋅⋅

∂∂∂∂

+

∂∂

⋅⋅

∂∂∂∂

=

0

0

2

0

1

0

0

2

0

1

(2.24)

utBxtA δδ )()( +=

Where

021

2

2

22

1

2

1

1

1

....

1)(

∂∂

∂∂

∂∂

⋅⋅

∂∂

⋅⋅⋅⋅∂∂

∂∂

∂∂

⋅⋅⋅⋅∂∂

∂∂

≡

n

nnn

n

n

xf

xf

xf

xf

xf

xf

xf

xf

xf

tA

And

021

2

2

2

1

2

1

2

1

1

1

....

)(

∂∂

∂∂

∂∂

⋅⋅

∂∂

⋅⋅⋅⋅∂∂

∂∂

∂∂

⋅⋅⋅⋅∂∂

∂∂

≡

m

nnn

m

m

uf

uf

uf

uf

uf

uf

uf

uf

uf

tB

Similarly, if the nonlinear measurement equation is ),( uxgy = , can show that, if


51

u

ug

ugug

x

xg

xgxg

y

pp

δδδ

∂

∂⋅⋅

∂∂∂∂

+

∂

∂⋅⋅

∂∂∂∂

=

0

0

2

0

1

0

0

2

0

1

(2.25)

utDxtC δδ )()( += Typically think of these nominal conditions 0x and 0u as “set points” or “operating points” for

nonlinear system. The equations

utBxtAxdtd δδδ )()( +=

utDxtCy δδδ )()( +=

Then give us a linearized model of the system dynamic behavior about these operating / set

points.

Note that if 0x and 0u then the partial fractions in the expressions of A-D are all constants → LTI

linearized model.

One particularly important set of operating points are the equilibrium points of the system.

Defined as the states & control input combinations for which

0),( 00 ≡=• uxfx (2.26) Provides n algebraic equations to find the equilibrium points. Usually in practice we drop the δ as they are rather than cumbersome, and (abusing notation) we

write the state equations as:

)()()()()( tutBtxtAtx +=• (2.27)

)()()()()( tutDtxtCty += (2.28) Which is of the same form as the previous linear models. Consequently, by applying the previous procedure through MATLAB code at desired operating

(Trimmed) point results in F-16 State space matrices for both longitudinal and lateral motions as

shown in Appendix B.


52

Chapter (4)

Linear Quadratic Regulator (LQR)

4.1 Linear Quadratic Regulator with Output Feedback In this section it will be shown how to use modern techniques to design stability augmentation

systems (SAS) and autopilots. This is accomplished by regulating certain states of the aircraft to

zero while obtaining desirable closed-loop response characteristics. It involves the problem by

placing the closed-loop poles at desirable locations.

Using classical control theory, it was forced to take a one-loop-at-a-time approach to designing

multivariable SAS and autopilots. In this section a performance criteria will be selected to reflect

the concern with closed- loop stability and good time responses and then derive matrix equations

that may be solved for all the control gains simultaneously. These matrix equations are solved

using digital computer programs. This approach thus closes all the loops simultaneously and

results in a simplified design strategy for MIMO systems or SISO systems with multiple

feedback loops.

Once the performance criterion has been selected, the control gains are explicitly computed by

matrix design equations, and closed-loop stability will generally be guaranteed. This means that

the engineering judgment in modern control enters in the selection of the performance criterion.

Different criteria will result in different closed-loop time responses and robustness properties.

Assuming that the plant is given by linear time-invariant state-variable model it is possible to

describe the aircraft model as shown in the following equations.

CxyBuAxx

=+=•

(4- 1, 2)

With nRtx ∈)( the state, mRtx ∈)( the control input, and pRty ∈)( the measured output. The

controls will be output feedbacks of the form


53

Kyu −= , (4- 3)

Where K is an pm × matrix of constant feedback coefficients to be determined by the design

procedure. Since the regulator problem only involves stabilizing the aircraft and inducing good

closed-loop time responses, )(tu will be taken as a pure feedback with no auxiliary input, as it

will be shown in next section output feedback permits to design aircraft controllers of any desired

structure. This is the main advantage and the reason for preferring it over full state feedback.

In the regulator problem, the main goal is obtaining good time responses as well as in the stability

of the closed-loop system. Therefore, the performance criterion will be selected in the time

domain as shown in next section.

4.1 Linear Quadratic Regulator Basis

4.2.1 Quadratic Performance Index The objective of state regulation for the aircraft is to drive any initial condition error to zero, thus

guaranteeing stability. This maybe achieved by selecting the control input )(tu to minimize a

quadratic cost or performance index (PI) of the type

dtRuuQxxJ TT )(21

0

+= ∫∞

(4- 4)

Where Q and R are symmetric positive semi-definite weighting matrices. Positive semi-

definiteness of a square matrix M (denoted )0≥M is equivalent to all its eigen values being

nonnegative, and also to the requirement that the quadratic form MxxT be nonnegative for all

vectors x. Therefore, the definiteness assumptions on Q and R guarantee that J is nonnegative

and lead to a sensible minimization problem. This quadratic PI is a vector version of the integral-

squared PI of the sort used in classical control [D’Azzo and Houpis, 1988].

To understand the motivation for the choice eqn. 4, consider the following. If the square-root

M of a positive semi definite matrix M is defined by

MMMT

= , (4- 5)


54

Then, it can be written as

dtuRxQJ )(21

0

22

∫∞

+= , (4- 6)

With w the Euclidean norm of a vector w (i.e., www T=2 ). If we are able to select the control

input u(t) so that J takes on a minimum finite value, certainly the integrand must become zero

for large time. This means that both the linear combination )(txQ of the states and the linear

combination )(tuR of the controls must go to zero. In different designs we may select Q and R

for different performance requirements, corresponding to specified functions of the state and

input. In particular, if Q and R are chosen nonsingular, the entire state vector )(tx and all the

controls )(tu will go to zero with time if J has a finite value.

Since a bounded value for J will guarantee that )(txQ and )(tuR go to zero with time, this

formulation for the PI is appropriate for the regulator problem, as any initial condition errors will

be driven to zero.

If the state vector )(tx consists of a capacitor voltages )(tv and inductor currents )(ti , then

2x will contain terms like )(2 tv and )(2 ti . Similarly, if velocity )(ts is a state component,

2x will contain terms like )(2 ts . Therefore, the minimization of the PI is a generalized

minimum energy problem. We are concerned with minimizing the energy in the states without

using too much control energy.

The relative magnitudes of Q and R may be selected to trade of requirements on the smallness of

the state against requirements on the smallness of the input. For instance, a larger control

weighting matrix R will make it necessary for )(tu to be smaller to ensure that )(tuR is near

zero. We say that a larger R penalizes the control more, so that they will be smaller in norm

relative to state vector. On the other hand, to make )(tx go to zero more quickly with time we

may select a larger Q .


55

As a final remark on the PI, we shall see that the position of the closed loop poles depend on the

choices for the weighting matrices Q and R . That is, Q and R may be chosen to yield good time

responses in the closed loop system.

Let us now derive matrix design equations that maybe used to solve for the control gain K that

minimizes the PI.

4.2.2 Solution of the LQR problem The LQR problem with output feedback is the following. Given the linear system equations eqn.

1,2, find the feedback coefficient matrix K in the control input eqn. 3 that minimizes the value of

the quadratic PI eqn. 4.

By substituting eqn. 3 intro eqn. 1 the closed loop system equations are found to be:

xAxBKCAx c=−=• )( (4-7)

The PI may be expressed in terms of K as

∫∞

+=0

)(21 xdtRKCKCQxJ TTT (4- 8)

The design problem is now to select the gain K so that J is minimized. This dynamical

optimization problem may be converted into an equivalent static one that is easier to solve as

follows. Suppose that we can find a constant, symmetric, positive semi-definite matrix P so that

xRKCKCQxPxxdtd TTTT )()( +−= (4- 9)

Then J may be written as

)()(21)0()0(

21 lim tPxtxPxxJ T

t

T

∞→

−= (4- 10)

Assuming that the closed loop system is asymptotically stable so that )(tx vanishes with time,

this becomes

)0()0(21 PxxJ T= (4- 11)


56

If P satisfies eqn. 9, we may use eqn. 7 to see that

•• +==+− PxxPxxPxxdtdxRKCKCQx TTTTTT )()( (4- 12)

xPAPAx CTc

T )( +=

Since this must hold for all initial conditions, and hence for all state trajectories )(tx , we may

write

0=+++≡ QRKCKCPAPAg TTc

Tc (4- 13)

If K and Q are given and P is to be solved for, this is called a Lyapunov equation.

In summery for any fixed feedback matrix K if there exists a constant symmetric, positive semi-

definite matrix P that satisfies eqn. 13, and if the closed loop system is stable, the cost J is given

in terms if P by eqn. 11. This is an important result in that the nn × auxiliary matrix P is

independent of the state. Given a feedback matrix K , P may be computed from the Lyapunov

equation eqn. 13. Then only the initial condition )0(x is required to compute the closed loop cost

under the influence of the feedback control Kyu −= .

It is now necessary to use this result to compute the gain K that minimizes the PI. By using the

trace identity

)()( BAtrABtr = (4- 14)

For any compatibly dimensioned matrices A and B (with the trace of a matrix the sum of its

diagonal elements), we may write eqn. 11 as

)(21 PXtrJ = (4- 15)

Where the nn × symmetric matrix X is defined by

)0()0( TxxX ≡ (4-16)


57

It is now clear that the problem of selecting K to minimize eqn. 8 subject to the dynamical

constraint eqn. 7 on the states is equivalent to the algebraic problem of selecting K to minimize

eqn. 15 subject to the constraint eqn. 13 on the auxiliary matrix P .

To solve this modified problem, we use the Lagrange multiplier approach [Lewis. 1986] to

modify the problem yet again. Thus adjoin the constraint to the PI by defining the Hamiltonian

)()( gStrPXtr += (4-17)

With S a symmetric nn × matrix of Lagrange multipliers which still needs to be determined.

Then our constraint optimization problem is equivalent to the simpler problem of minimizing

eqn. 17 without constraints. To accomplish this we need only set the partial derivatives of

with respect to all the independent variables P , S and K equal to zero. Using the facts that for

any compatibly dimensioned matrices A, B and C and any scalar y,

TT CABABCtr

=∂

∂ )( (4-18)

And

T

T By

By

∂∂

=∂∂ (4-19)

The necessary conditions for the solution of the LQR problem with output feedback are given by:

QRKCKCPAPAgS

TTc

Tc +++==

∂∂

=0 (4-20)

XSASAP

Tcc ++=

∂∂

=0 (4-21)

TTT PSCBRKCSCK

−=∂∂

=

210 (4-22)


58

The first two of these are Lyapunov equations and the third equation for the gain K . If R is

positive definite (i.e., all eigenvalues greater than zero, which implies nonsingularity; denoted

R>0) and TCSC is nonsingular, then eqn. 22 may be solved for K to obtain

11 )( −−= TTT CSCPSCBRK (4-23)

To obtain the output feedback gain K minimizing the PI eqn. 4, we need to solve the three

coupled equations eqn. 20, 21, 23. This situation is quite strange, for to find K we must

determine along the way the values of two auxiliary and apparently unnecessary nn × matrices,

P and S . These auxiliary quantities may, however, not be as unnecessary as it appears, for note

that the optimal cost may be determined directly from P and the initial state by using eqn. 11.

4.2.3 The initial condition problem Unfortunately, the dependence of X in eqn. 16 on the initial state )0(x is undesirable, since it

makes the optimal gain dependent on the initial state through eqn. 21. in many applications

)0(x may not be known. This dependence is typical of output feedback design. Meanwhile, it is

usual [Levine and Athans, 1970] to sidestep this problem by minimizing not the PI but its

expected value, that is, )(JE . Then eqn. 11 and 16 are replaced by

),(21))0()0((

21)( PXtrPxxEJE T == (4-24)

Where the symmetric nn × matrix

))0()0(( TxxEX ≡ (4-25)

is the initial autocorrelation of the state. It is usual to assume that nothing is known of

)0(x except that it is uniformly distributed on the surface described by X . That is, we assume that

the actual initial state is unknown, but that it is nonzero with a certain expected Euclidean norm.


59

For instance, if the initial are assumed to be uniformly distributed on the unit sphere, then IX = ,

the identity. This is a sensible assumption for the regulator problem, where we are trying to drive

arbitrary nonzero initial states to zero. The design equations for the LQR with output feedback

are collected in Table 4.1 for convenient reference. We shall now discuss their solution for K .

Table 4.1 LQR with output feedback

System Model

CxyBuAxx

=+=•

Control

Kyu −= Performance Index

dtRuuQxxJ TT )(21

0

+= ∫∞

With 0≥Q , 0≥R Optimal gain design equations

QRKCKCPAPA TTc

Tc +++=0 (4-26)

XSASA T

cc ++=0 (4-27)

11 )( −−= TTT CSCPSCBRK (4-29) Where

BKCAAc −= , ))0()0(( TxxEX = Optimal cost

)(21 PXtrJ =

Determining the optimal feedback gain


60

The impotence of this modern LQ approach to controls design is that the matrix equations in

Table 1 are used to solve for all the pm × elements of K at once. This corresponds to closing all

the feedback loops simultaneously. Moreover, as long as certain reasonable conditions on the

plant and PI weighting matrices hold, the closed loop is generally guaranteed to be stable. In view

of the trial and error successive loop closure approach used in stabilizing multivariable systems

using classical approaches, this is quite important. The equations for P , S and K are coupled

nonlinear matrix equations in three unknowns. It is important to discuss some aspects of their

solution for the optimal feedback gain matrix K .

Conditions for convergence of the LQ solution algorithm

1. There exists a gain K such that cA is stable. If this is true, we call the system output

stabilizable.

2. The output matrix C has full row rank p .

3. Control weighing matrix R is positive definite matrix. This means that all the control

inputs should be weighted in the PI.

4. Q is positive semi definite and ),( AQ is detectable. That is, the observability matrix

polynomial

−

−≡

QAsI

sO )( (4-30)

Has full rank n for all values of the complex variable s not contained in the left half plane.

If these conditions hold, the algorithm finds an output-feedback gain that stabilizes the plant and

minimizes the PI. The delectability condition means that any unstable system modes must be

observable in the PI. Then if the PI is bounded, which it is if the optimization algorithm is

successful, signals associated with the unstable modes must go to zero as t becomes large; that is,

they are stabilized in the closed-loop system.


61

Initial stabilizing gain since all three algorithms for solving the matrix equations in

Table 4.2 for K are iterative in nature, a basic issue for all of them is the selection of an initial

stabilizing output-feedback gain K0, that is to start the algorithms, it is necessary to provide a K0

such that (A-B K0

C) is stable.

Table 4.2 Optimal output feedback solution algorithm (LQR)

1. Initialize:

Set k=0

Determine a gain K0 so that A-B K0

2. k

C is asymptotically stable th

Set

iteration:

CBkAAk 0−=

Solve for kP and kS in

QCRKKCAPPA kT

kT

kkkTk +++=0

XASSA Tkkkk ++=0

Set )(21 XPtrJ kk =

Evaluate the gain update direction

kTTT kCSCPSCBRK −=∆ −− 11 )(

Update the gain by

kkk kk ∆+=+ α1

Where α is chosen so that

CBkAA kk 1+−= is asymptotically stable

kkk JXPtrJ ≤= ++ )(21

11

If 1+kJ and kJ are close enough to each other, go to step 3, otherwise set 1+= kk and go to

step 2.

3. Terminate:


62

Set 1,1 ++ == kk JJkk

Stop

Iterative design: Software that solves for the optimal output-feedback gain K is described in

appendix A. Given good software, design using LQ approach is straightforward. A design

procedure would involve selecting the design parameters Q and R, determining the optimal gain

K, and simulating the closed loop response and frequency-domain characteristics. If the results

are not suitable, different matrices Q and R are chosen and the design is repeated. The designed

software makes a design iteration take only few minutes. This approach introduces the notion of

tuning the design parameters Q and R for good performance.

Selection of the PI weighting matrices Once the PI weighting matrices Q and R have been selected, the determination of the optimal

feedback gain K is a formal procedure relying on the solution of the nonlinear coupled matrix

equations. Therefore, the engineering judgment in the modern LQ design appears in the selection

of Q and R. there are some guidelines for this which we shall now discuss.

Observability in the choice of Q

For stabilizing solutions to the output-feedback problem, it is necessary for ),( AQ to be

detectable. The detestability condition basically means that Q should be chosen so that all

unstable states are weighted in the PI. Then, if J is bounded so that )(txQ vanishes for large t,

the open-loop unstable states will be forced to zero through the action of the control. This means

exactly that the unstable poles must have been stabilized by the feedback control gain.

A stronger condition than detectability is observability, which amounts to the full rank of O(s) for

all values of s. observability is easier to check than detectability since it is equivalent to the full

rank n of the observability matrix :


63

−1

.nAQ

AQQ

(4-31)

Which is a constant matrix and so easier to deal with than O(s). in fact, O has full rank n if and

only if the observability gramian OOT is nonsingular. Since the gramian is an nn × matrix, its

determinant is easily examined using available software (singular value decomposition /condition

number ). The observability of ),( AQ means basically that all states are weighted in the PI.

From a numerical point of view, if ),( AQ is observable, a positive definite solution P to

equation 26 results; otherwise, P may be singular. Since P helps determine K through equation

28, it is found that if P is singular, it may result in some zero-gain elements in K. that is, if

),( AQ is not observable, the LQ algorithm can refuse to close some feedback loops.

This observability condition amounts to a restriction on the selection of Q, and is a drawback of

modern control.

The structure of Q The choice of Q can be confronted more easily by considering the performance objectives of the

LQR. Suppose that a performance output

Hxz = (4-32)

Is required to be small in the closed-loop system. For instance, in F-16 aircraft lateral regulator it

is desired for the sideslip angle, yaw rate roll angle, and roll rate to be small. See output

responses Fig. 4.1. Therefore, we might select [ ]Tprz ϕβ= . Once )(tz has been chosen,

the performance output matrix H may be formally written down. The signal )(tz may be made

small by LQR design by selecting the PI

∫∞

+=0

)(21 dtRuuzzJ TT (4-33)


64

Which amounts to using the PI in Table 1 with HHQ T= , so that Q may be computed from H.

that is, by weighting performance outputs in the PI, Q is directly given.

4.2.4 Maximum desired values of z(t) and u(t) A convenient guideline for selecting Q and R is given in [Bryson and Ho, 1975]. Suppose that the

performance output in equation 32 has been defined so that H is given. Consider the PI

∫∞

+=0

~)(

21 dtRuuzQzJ TT (4-34)

Then, in Table 4.1 we have HQHQ T~

= . To select ~Q and R , one must proceed as follows,

using the maximum allowable deviations in )(tz and )(tu . Define maximum allowable deviation

in component )(tzi of )(tz as iMz and the maximum allowable deviation in component )(tui of

the control input )(tu as iMu . Then ~Q and R may be selected as

~Q =diagqi R, =diagri

,

with

2

1

iMi z

q = , 2

1

iMi r

r = (4- 35)

The rationale for this choice is easy to understand. For instance, as the allowed limits iMz on

)(tzi decrease, the weighting in PI placed on )(tzi increase, which requires smaller excursions in

)(tzi in the closed-loop system.

In our approach we shall select the design parameters Q and R in the table and then use the

design equations there to close all the feedback loops simultaneously by computing. The

objective is to design a closed-loop controller to provide for the function of SAS as well as the

closure of the roll-attitude loop. This objective involves the design of two feedback channels with

multiple loops, but it is straightforward to deal with using modern control techniques. The

simplicity of MIMO design using LQR will be evident.


65

We used the F-16 linearized lateral dynamics at the nominal flight condition ( 502=TV ft/s, 300

psf dynamic pressure, CG at 0.35 c ) retaining the lateral states sideslip β , bank angle ϕ , roll rate

p , and yaw rate r ,. Additional states aδ and rδ are introduced by the aileron and rudder

actuators.

aa us 2.20

2.20+

=δ , rr us 2.20

2.20+

=δ (4- 36)

A washout filter

rs

srw 1+= (4-37)

is used, with r the yaw rate and wr the wash-out yaw rate. The washout filter state is denoted

wx . Thus the entire state vector is:

[ ]Twra xrpx δδϕβ= (4-38)

The full-state-variable model of the aircraft plus actuators, washout filter, and control dynamics is of the form

BuAxx +=• (4-39) where

−−

−=

102958.57000002.2000000002.2000000000

ra

ra

ra

ra

rrrrrprr

ppprpppp

rp

rp

BBAAAABBAAAABBAAAABBAAAA

A δδϕβ

δδϕβ

βδϕδϕϕϕϕϕβ

βδβδβββϕββ


66

=

002.200

02.2000000000

B,

=

r

a

uu

u ,

=

ϕβpr

y

w

thus Cxy = where

−

=

000002958.5700000002958.5700002958.57001002958.57000

C

For the computation of the feedback gain K, it is necessary to select PI weighting matrices Q and

R in Table 4.1. Then the code described in appendix A is used to compute the optimal gain K.

Our philosophy for selecting Q and R as follows.

First, let us discuss the choice of Q. It is desired to obtain good stability of the dutch roll mode,

so that 2β and 2r should be weighted in the PI by factors of drq . To obtain stability of the roll

mode, which in closed-loop will consist primarily of p and ϕ , we may weight 2p and 2ϕ in the

PI by factors of rq . We don’t care about aδ and rδ , so it is not necessary to weight them in the

PI; the control weighting matrix R will prevent unreasonably large control inputs. Thus so far we

have

0,0,0,,,, drrrdr qqqqdiagQ = (4-40) We don’t care directly about wx ; however, it is necessary to weight it in the PI. This is because

omitting it would cause problems with observability condition. A square root of Q in 40 is


67

]0,0,0,,,,[ drrrdr qqqqQ = (4-41) Consequently, the observability matrix in equation 31 has a right-hand column of zero; hence the

system is unobservable. This may be noted in simpler fashion by examining the A matrix , where

the seventh state wx is seen to have no influence on the states that are weighted in equation 40.

to correct this potential problem, we choose

1,0,0,,,, drrrdr qqqqdiagQ = (4-42) As far as the R matrix goes, it is generally satisfactory to select it as

IR ρ= (4-43) With I the identity matrix and ρ a scalar design parameter. After few trials, we obtained a good result using 1.0=ρ , drq =50, 100=rq for this selection the optimal feedback gain was

−−−

−−−=

26.044.021.019.135.011.044.056.0

K (4-44)

The resulting closed loop poles were as

83.013.3 j±− - Dutch roll mode ),( βr 11.082.0 j±− Roll mode ),( ϕp

18.1747.11 j±− -15.02 To verify the design a simulation was performed. The initial state was selected as [ ]Tx 0000001)0( = ; that is, we choose 1)0( =β . The following figures show the output performance of the states


68

Fig. 4.1 Lateral output performance when 1)0( =β

To verify the design a simulation was performed. The initial state was selected as

[ ]Tx 0000010)0( = ; that is, we choose 1)0( =ϕ . The following figures show the output performance of the output states


69

Fig. 4.2 Lateral output performance when 1)0( =ϕ

Effect of PI weighting parameters ρ = .1, .2, .4 respectively

Fig. 4.3 Effect of PI weighting parameters ρ = .1, .2, .4 respectively


70

4.3 Tracking a command In an aircraft control we are often interested not in regulating the state near zero, which discussed

in the preceding section, but in the following a nonzero reference command signal. In this part we

are interested in designing a control system for optimal step-response shaping. This reference-

input tracking or servo design problem is important in the design of command augmentation

systems (CAS), where the reference command may be, for instance, desired pitch rate or normal

acceleration. It should be mentioned that the optimal linear quadratic (LQ) tracker of modern

control is not a causal system [Lewis, 1986]. It depends on solving an “adjoint” system of

differential equations backward in time, and so is impossible to implement. A suboptimal

“steady-state” tracker using full state-variable feedback is available, but it offers no convenient

structure for the control system in terms of desired dynamics such as PI control, washout filters,

and so on. Thus there have been problems with using it in aircraft control. Modified versions of

LQ tracker have been presented in [Davison and Ferguson, 1981] and [Gangsaas et al., 1986].

There, controllers of desired structure can be designed since the approaches are output-feedback

based. The optimal gains are determined numerically to minimize a PI with, possibly, some

constraints. It is possible to design a tracker by the first designing a regulator using, for instance

Table 4.1. Then some feedforward terms are added to guarantee perfect tracking [Kwakernaak

and Sivan, 1972]. The problem with this technique is that the resulting tracker has no convenient

structure and often requires derivatives of the reference command input. Moreover, servosystems

designed using this approach depends on knowing the dc gain exactly. If the dc gain is not known

exactly, the performance deteriorates.

Here we discuss an approach to the design of tracking control systems which is more useful in

aircraft control applications [Stevens et al., 1991]. This approach will allow us to design a servo

control system that has any structure desired. This approach will include a unitary –gain outer

loop that feeds the performance output back and subtracts it from the reference command, thus

defining a tracking error ( )te which should be kept small see Fig. 4.4.


71

Fig. 4.4 plant with compensator of desired structure

It can also include compensator dynamics such as a washout filter and an integral controller. The

control gains are chosen to minimize a quadratic performance index (PI). We are able to give

explicit design equations for the control gains see Table 4.3, solved using the software described

in appendix A.

A problem with the tracker developed in this section is the need to select the design parameters Q

and R in the PI in Table 4.3. There are some intuitive techniques available for choosing these

parameters, however, in the preceding section we shall show how modified PIs may be used to

make the selection of Q and R almost transparent, yielding tracker design techniques that are very

convenient for use in aircraft control systems design. We shall show, in fact, that the key to

achieving required performance using modern design strategies is in selecting an appropriate PI.

4.3.1 Tracker with desired structure In aircraft controls design there is a wealth of experience and knowledge that dictates in many

situations what sort of compensator dynamics yield good performance from the point of view of

both the controls engineer and the pilot. For instance, a washout circuit maybe required, or it may


72

be necessary to augment some feedforward channels with integrators to obtain a steady state error

of exactly zero.

The control system structures used in classical aircraft design also give good robustness

properties. That is, they perform well even if there are disturbances or uncertainties in the system.

Thus the multivariable approach developed here usually affords this robustness.

Our approach to tracker design allows controller dynamics of any desired structure and then

determines the control gains that minimize a quadratic PI over that structure. Before discussing

the tracker design, we shall derive the compensator state equations first as follows.

Consider the situation in Fig. 4.4 where the plant is described by.

CxyBuAxx

=+=•

(4-45)

With state )(tx , control input )(tu , and )(ty the measured output available for feedback

purposes. In addition,

Hxz = (4-46)

Is a performance output, which must track the given reference input )(tr . The performance output

)(tz is not generally equal to )(ty .

It is important to realize that for perfect tracking it is necessary to have as many control inputs in

vector )(tu as there are command signals to track in )(tr .

The dynamic compensator has the form

GeFww +=• (4-47)

JeDwv += (4-48)

With state )(tw , output )(tv , and input equal to the tracking error

)()()( tztrte −= (4-49)

F, G, D, and J are known matrices chosen to include the desired structure in the compensator.

The allowed form for the plant control input is


73

LvKyu −−= (4-50)

Where the constant gain matrices K and L are to be chosen in the controls design step to result in

satisfactory tracking of )(tr . This formulation allows for both feedback and feedforward

compensator dynamics.

These dynamics maybe written in augmented form as

rG

uB

wx

FGHA

wx

dtd

+

+

−

=

00

0 (4-51)

rJw

xDJH

Cvy

+

−

=

00 (4-52)

[ ]

=

wx

Hz 0 (4-53)

[ ]

−=

vy

LKu (4-54)

Note that in terms of the augmented plant/ compensator state description, the admissible controls

are represented as a constant output feedback [K L]. in the augmented description, all matrices

are known except the gains K and L, which need to be selected to yield acceptable closed-loop

performance.

4.3.2 LQ Formulation of the Tracker Problem By redefining the state, the output, and the matrix variables to streamline the notation, we see that

the augmented equations as follows contain the dynamics of both the aircraft and the

compensator.

GrBuAxx ++=• (4-55)

FrCxy += (4-56)

Hxz = (4-57)

In this description, let us take the state nRtx ∈)( , control input mRtu ∈)( , reference input qRtr ∈)( ,


74

Performance output qRtz ∈)( , and measured output pRty ∈)( . The admissible controls are then

proportional output feedbacks of the form

KFrKCxKyu −−=−= (4-58)

With constant gain K to be determined. This situation corresponds to the block diagram in

Fig. 4.5.

Fig. 4.5 Plant/ feedback structure

Since K is an pm × matrix, we intend to close all the feedback loops simultaneously by

computing K.

Using these equations the closed-loop system is found to be

rBKFGxBKCAx )()( −+−=• (4-59)

rBxAx cc +≡• (4-60)

Our formulation differs sharply from the traditional formulations of the optimal tracker problem.

Note that the previous equations include both feedback and feedforward terms, so that both the

closed-loop poles and compensator zeros maybe affected by varying the gain K. thus we should

expect better success in shaping the step response than by placing only the poles.


75

Let us now formulate an optimal control problem for selecting the control gain K to guarantee

tracking of )(tr . Then we shall derive the design equations in Table 4.3, which are used to

determine the optimal K. these equations are solved using a code in appendix A.

4.3.3 The Deviation System Denote steady-state values by overbars and deviations from the steady-state values by tildes.

Then the state, output, and control deviations are given by

xtxtx −= )()(~ (4-61)

(4-62)

xHztztz ~)()(~ =−= (4-63)

)(~)()()(~00 txKCKFrxKCKFrKCxututu −=−−−−−=−= (4-64)

Or

yKu ~~ −= (4-65)

The tracking error )()()( tztrte −= is given by

etete −= )(~)( (4-66)

With the error deviation given by

xHxHrHxretete ~)()()()(~00 −=−−−=−= (4-67)

Or

ze ~~ −=

Since in any acceptable design the closed-loop plant will be asymptotically stable, cA is

nonsingular. According to equation 60, at steady state

00 rBxA cc += (4-68)

So that the steady-state state response is

01 rBAx cc

−−= (4-69)

And the steady the steady-state error is

01

0 )( rBHAIxHre cc−+=−= (4-70)

To understand this expression, note that the closed-loop transfer function from 0r to z is

The steady-state behavior may be investigated by considering the dc value of )(sH (i.e., s=0);

this is just cc BHA 1−− , the term appearing in equation 70.

xCytyty ~)()(~ =−=


76

Using above equations the closed loop dynamics of the state deviation are seen to be

xAx c~~ = (4-71)

xCy ~~ = (4-72)

exHz ~~~ −== (4-73)

And the control input to the deviation system equation 71 is now equation 65. Thus the step-

response shaping problem has been converted to a regulator problem for the deviation system.

uBxAx ~~~ += (4-74) 4.3.4 Performance Index for LQ Tracker To make the tracking error )(te small, we propose to attack two equivalent problems: the problem

of regulating the error deviation )(~)(~ tzte −= to zero, and the problem of making small steady-

state error e .

To make small both the error deviation and the steady-state error, we propose selecting K to

minimize the performance index (PI)

∫∞

++=0 2

1)~~~~(21 eVedtuRueeJ TTT (4-75)

With R>0, V>0, design parameters.

We can generally select rIR = and vIV = , with r and v scalars. This simplifies the design since

now only a few parameters must be tuned during the interactive design process.

According equation 67, xHHxee TTT ~~~~ = according to Table 4.1, therefore, it follows that the

matrix Q there is equal to HH T , where H is known. That is weighting the error deviation in the

PI has already shown us how to select the design parameter Q.

4.3.5 Solution of the LQ Tracker Problem It is now necessary to solve for optimal feedback gain K that minimizes the PI. The design

equations needed are now derived. They appear in Table 4.3.


77

By using equation 71 and the same procedure used in previous section in designing LQR. The

optimal cost is found to satisfy

eVexPxJ T

21)0(~)0(~

21

+= (4-76)

With P > 0 the solution to

RKCKCQPAPAg TTc

Tc +++≡=0 (4-77)

With HHQ T= and e given by equation 70

In our discussion of the linear quadratic regulator we assumed that the initial conditions were

uniformly distributed on a surface with known characteristics. While this is satisfactory for the

regulator problem, it is an unsatisfactory assumption for the tracker problem. In the latter

situation the system the system starts at rest and must achieve a given final state that is dependent

on the reference input. To find the correct value of )0(~x , we note that since the plant starts at rest

[i.e., 0)0( =x ], according to equation 61

xx −=)0(~ , (4-78)

So that the optimal cost becomes

eVePXtreVexPxJ TTT

21)(

21

21

21

+=+= (4-79)

With P given by equation 77, e given by equation 70, and T

cTc

Tcc

T ABrrBAxxX −−== 001 (4-80)

With Tc

Tc AA )( 1−− = .

The optimal solution to the unit-step tracking problem with initially at rest, maybe may now be

determined by minimizing J over the gains K, subject to constraint according to equation 77,


78

equation 70 and equation 80. This algebraic optimization problem can be solved by the code

presented in appendix A.

4.3.6 Design Equations for Gradient-Based Solution As an alternative solution procedure, one may use gradient-based techniques which is faster than

non-gradient-based approaches.

To find the gradient of the PI with respect to the gains, define the Hamiltonian

eVegStrPXtr T

21)()( ++= (4-81)

With S a lagrange multiplier. Now, using the basic matrix calculus identities,

11

1−−

−

∂∂

−=∂

∂ YxYY

xY (4-82)

xVUV

xU

xUV

∂∂

+∂∂

=∂

∂ (4-83)

∂

∂∂∂

=∂∂

xz

zytr

xy T

. (4-84)

To find K by a gradient minimization algorithm, it is necessary to provide the algorithm with values of J and K

J∂

∂ for a given K. the value of J is given by the expression in Table 4.3 for the

optimal cost. To find KJ

∂∂ given K, solve the following algorithm.

Table 4.3 LQ Tracker with Output Feedback System Model

HxzFrCxy

GrBuAxx

=+=

++=

Control

Kyu −= Performance index


79

eVedtuRuxQxJ TTT

21)~~~~(

21

0

++= ∫∞

, with HHQ T=

Optimal Output Feedback Gain

TTTc

TTTTc

TTTT

Tcc

TTc

Tc

yVrHAByxVHHPABPSCBRKCSCK

XSASAP

RKCKCQPAPAS

0)(210

0

0

−− −++−=∂∂

=

++=∂∂

=

+++=∂∂

=

With r a unit step of magnitude 0r and

01 rBAx cc

−−=

0FrxCy += T

cTc

Tcc

T ABrrBAxxX −−== 001

vIVrIR == , Where

BKFGBBKCAA

c

c

−=−= ,

Optimal cost

eVePXtrJ21)(

21

+=

Fig. 4.6 shows the block diagram representation for the normal acceleration controller. Fig. 4.7

shows the step response of the normal acceleration. In the same manner pitch rate controller can

be represented in Fig. 4.8 with pitch rate performance as in Fig. 4.9, appendix B2.

Fig. 4.6 Normal acceleration controller


80

Fig. 4.7 Normal acceleration step response for different PI (3.2 green), (8.36 blue)

[ ]T

Fepitch qx εαδα= , [ ]TFpitch qy εα=

GrBuAxx ++=•

FrCxy +=

HxZ =

Kyu −=

−−

−=

0002958.570010000.10002.200000~~~00~~~

eqqqq

eq

BAABAA

Aδα

αδααα

,

=

00

2.2000

B ,

=

10000

G ,

=

000

F

=

100000002958.57002958.57000

C , [ ]0002958.570=H , [ ]yKKKKyu lqα−=−=


81

Fig. 4.8 Pitch rate controller

Fig. 4.9 Pitch-rate step response using regular LQ tracker for different PI (4.2 blue), (3.98

green)

[ ]Twralateral xrpx εδδφβ= , [ ]Trlateral epey φε=

GrBuAxx ++=•

FrCxy +=

HxZ =

Kyu −=


82

−−

−−

=

0000001001002958.57000002.20000000002.20000000~~~~~~00~~~~~~00~~~~~~00~~~~~~

rrarrrprrr

rpaprppppp

rarp

rarp


A δδϕβ

δδϕβ

ϕδϕδϕϕϕϕβϕ

βδβδββϕβββ

,

=

0000

2.20002.2000000000

B

,

=

0100000000000000

G

Hxxr

zw

=

−

=

=

01002958.5700000000010ϕ

cr FrCx

epe

y +=

∈

=

ϕ

−

−=

000000100000010000002958.5700010000000

C

=

01001000

F

The control input [ ]Tra uuu = may be expressed as Kyu −=

Constrained Feedback Gain Matrix

In order to get certain elements ijk of K very small or equal to zero, gain-element weight

ijg is chosen large to make the ijk of the feedback matrix K very small or equal to zero

as shown in equation 85.

∑∑+=i j

ijij kgPXtrJ 2)(21 (4-85)

with

=

0000

2

431

kkkk

K

Fig. 4.10 shows Lateral-Directional controller implying two inputs, Figure 11 shows output

performance according to bank step input.


83

Fig. 4.10 Lateral –Directional controller

Fig. 4.11 Closed loop response to command of 1=ϕr , 0=rr . Bank angle ϕ (rad), and washout

yaw rate (rad/sec)


84

Chapter (5)

Intelligent Learning of Fuzzy Logic Controller Via Genetic

Algorithm

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 C78C] 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 this 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


85

the fuzzification process is the measured or estimated variable that appears in the antecedent part

of the if_then rule. This input 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 [18, 19].

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. Fig. 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.


86

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). Fig. 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)


87

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 [19, 20, 21].

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 [22, 23, 24].

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 off human operator.

3-Fuzzy model of the plant [25, 26].

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 [27, 28].

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. The first rule base is

the standard control fuzzy rule base. The second level contains a fuzzy rule base consisting of


88

meta-rules, which attempt to assess the performance of the close 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.

5.2.2.1 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 is nonfuzzy numeric value or symbolic value.

(b) Fuzzy consequent: IF….THEN y is A

Where A is a fuzzy set.


89

5.2.3 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. Fig. 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. Fig. 5.7 shows

the clipping method for fuzzy inference for the first rule at matching degree =0.3.

Fig. 5.7 Fuzzy inference process

(3) Combination: Combine the conclusion inferred by all fuzzy rules into a final conclusion.

Fig. 5.8 shows the combining fuzzy conclusions inferred by the clipping method.


90

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 purpose such as MOA (Maximum of area) and COA (Center of

area). The centroid (or COA) defuzzification method calculates he 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 i

i

A ii

( )y yy

( )y

µ

µ

×∑=

∑ (5.1)

where Aµ is the membership of area A.

Fig. 5.9 Defuzzification process


91

In this work these steps were applied to the inputs of the fuzzy logic controller (system error,

change in error) the control action is then generated at the defuzzification process. Fig. 5.10

represents 49 rule-based inference of Mamdani type fuzzy logic controller and final

defuzzification step.

Fig. 5.10 Fuzzy logic inference system

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

e

(t), defined respectively as

n (t)=ne

ce

e(t) (5.2)

n (t)=nce (e(t)-e(t-T)) (5.3)


92

where ne and nce

δare the corresponding normalization factors, into changes in the control action

un

If e

(t) via rules of the form

n (t) is P and cen δ(t) is N then un

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

(t) is Z.

Fig. 5.11 Architecture of the generic fuzzy control system

In reference to Fig. 5.11 the computed value of (δ u) is subsequently denormalized and

integrated (added to the past value of (u) to form the present value of (u)) as follows:

u nu(t) u(t T) (t)de uδ δ= − + (5.4)

where

uedδ is the corresponding denormalization factor.


93

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 49 rule-based Mamdani type fuzzy PID controller Fig. 5.13 was established

and simulated through MATLAB according to the following rule base table then generating the

final control action as shown in Fig. 5.14. 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 as it will be discussed later on.


94

Rule error error rate p i d Firing Strength 1 NB NB PB NB PS 0.89736 2 NB NM PB NB NS 0.89736 3 NB NS PM NM NB 0.89736 4 NB Z PM NM NB 0.89736 5 NB PS PS NS NB 0.89736 6 NB PM Z Z NM 0.89736 7 NB PB Z Z PS 0.89736 8 NM NB PB NB PS 0.85494 9 NM NM PB NB NS 0.85494

10 NM NS PM NM NB 0.85494 11 NM Z PS NS NM 0.85494 12 NM PS PS NS NM 0.85494 13 NM PM Z Z NS 0.85494 14 NM PB NS Z Z 0.85494 15 NS NB PM NB Z 0.65068 16 NS NM PM NM NS 0.65068 17 NS NS PM NS NM 0.65068 18 NS Z PS NS NM 0.65068 19 NS PS Z Z NS 0.65068 20 NS PM NS PS NS 0.65068 21 NS PB NS PS Z 0.65068 22 Z NB PM NM Z 0.67087 23 Z NM PM NM NS 0.67087 24 Z NS PS NS NS 0.67087 25 Z Z Z Z NS 0.67087 26 Z PS NS PS NS 0.67087 27 Z PM NM PM NS 0.67087 28 Z PB NM PM Z 0.67087 29 PS NB PS NM Z 0.98392 30 PS NM PS NS Z 0.98392 31 PS NS Z Z Z 0.98392 32 PS Z NS PS Z 0.98392 33 PS PS NS PS Z 0.98392 34 PS PM NM PM Z 0.98392 35 PS PB NM PB Z 0.98392 36 PM NB PS Z PB 0.66334 37 PM NM Z Z PS 0.66334 38 PM NS NS PS PS 0.66334 39 PM Z NM PS PS 0.66334 40 PM PS NM PM PS 0.66334 41 PM PM NM PB PS 0.66334 42 PM PB NB PB PB 0.66334 43 PB NB Z Z PB 0.95689 44 PB NM Z Z PM 0.95689 45 PB NS NM PS PM 0.95689 46 PB Z NM PM PM 0.95689 47 PB PS NM PM PS 0.95689 48 PB PM NB PB PS 0.95689 49 PB PB NB PB PB 0.95689


95

Fig. 5.13 Nonlinear fuzzy PID controller

Fig. 5.14 Fuzzy inference system control action (PID fuzzy logic controller output gains)

5.3 Genetic Algorithm Process GA uses a direct analogy of natural behavior Fig. 5.15, They work with a population of

individuals, each reprinting a possible solution to a given problem. Each individual is assigned a

fitness score according to how good its solution to the problem is. The highly fit individuals are

given opportunities to reproduce, by cross breeding with other individuals in the population. This

produces new individuals as offspring, who share some features taken from each parent. The least

fit members of the population are less likely to get selected for reproduction, and will eventually

die out. Adaptive GA’s whose parameters, such as the population, size, cross over probability and

mutation probability are varied during the GA is running thus converging to the optimum

solution faster.


96

Fig. 5.15 The flow of GA functions and process

5.3.1 Application of GA Functions for Tuning Fuzzy PID Controller Parameters As shown in the following figure Fig. 5.16, the fuzzy PID controller consists of output

denormalized gains, input-output membership functions and rule bases, [40], [41]. The main

objective of the pre-explained GA procedure is to optimize (minimize) the output performance

index (maximize fitness function) of the whole integrated control system used in the pitch control

motion of F-16 aircraft. Application of GA MATLAB code combined with SIMULINK control

model is implemented to minimize the performance index as explained later [43].

Fig. 5.16 Nonlinear fuzzy PID controller with membership functions

Using GA to minimize the value of the output performance index:


97

∫∞

=0

2)_( dtsimpitcheJ (5.5)

Where e(pitch_sim) is the pitch angle error to workspace

The previous index can be obtained from the nonlinear simulation of the pitch control model

consequently GA can use it as the fitness function.

5.3.1.1 Optimizing Fuzzy Output Gains

1- Using binary system coding

Coding：using 10-bit binary genes to express DIP KKK ,, . Table 5.2 shows the upper and lower

gain limits. For example (KP) from bit 1 (0)0000000000 , to bit 10 is (1023)1111111111 . Then

string DIP KKK ,, to 30-bit binary cluster. 0000100010 1 110111000 0000110111:x express

a gene，the former 10-bit express KP, the second portion express KI and the third one express

the KD

Decoding：Cut one string of 30-bit binary string to three 10-bit binary string，then converts

them to decimal system values: y

.

1, y2 and y3.

Table 5.3 shows the decoding equations.

Table 5.2 Gain Limits

Denormalized Gain

Minimum limit

Maximum limit

K 1 P 10 K -3 D 3 K -3 I 3

Table 5.3 Decoding Equation

Denormalized gains Decoding equation K

11023

9 1 +×yP

K3

10236 2 −×

yD

K3

10236 3 −×

yI

2- Evaluation of fitness function


98

Fitness function is the main criterion of the GA algorithm, as it represents how much the system

is optimum and stable. The following equation describes the relation between the fitness function

and the performance index. 1))_((),,( −= simpitchJKKKf DIP (5.6)

3- Design operators

Proportion selection operator，single point crossover operator，basic bit mutation operator.

4- Parameters of GA

Population size is 02=M , generation 03=G , crossover probability 0.60c =P ，mutation

probability 0.10m =P 。

Adopting the above steps, after 100 steps iteration, we get the best individuals When KP=7.190,

KD =2.958 and KI= -2.870 the output performance index has the minimum value that is 1.023

shown in Fig. 5.17.

Fig. 5.17 Objective function J and Fitness function F

5.3.1.2 Tuning of Fuzzy Sets It is presented a universal algorithm for solving a very general class of optimization problems

applied to fuzzy sets. From studying how such a GA can be applied to the optimization of fuzzy

sets. An appropriate coding, genetic operators (in case that the standard variants are not

sufficient), and a fitness measure is presented [43], [44].

1- Coding fuzzy subsets of an interval


99

A reasonable resolution for encoding the membership degrees is n = 8. Such an 8-bit coding is

used in several software systems, too. For most problems, however, simpler representations of

fuzzy sets are sufficient. Many real-world applications use triangular and trapezoidal

membership functions Fig. 5.18.

Not really surprising, a triangular fuzzy set can be encoded as

Fig. 5.18 Triangular and trapezoidal membership functions

where δ is an upper boundary for the size of the offsets, for example δ = (b − a)/2. The same

can be done analogously for trapezoidal fuzzy sets:

In specific control applications, where the smoothness of the control surface plays an important

role, fuzzy sets of higher differentiability must be used. The most prominent representative is the

bell-shaped fuzzy set whose membership function is given by a Gaussian bell function:

2

2

2)(

)( urx

ex−

−=µ (5.7)

The “bell-shaped analogue” to trapezoidal fuzzy sets are so-called radial basis functions:

qq

rxrx

ififex u

qrx

≤≥

−−

=−−

−

1)(

2

2

2

)(

µ (5.8)


100

Fig. 5.19 shows a typical bell-shaped membership function. Again the coding method is

straightforward, i.e.

Fig. 5.19 Bell-shaped membership functions

where ε is a lower limit for the spread u. Analogously for radial basis functions:

The final step is simple and obvious, In order to define a coding of the whole configuration, i.e.

of all fuzzy sets involved, it is sufficient to put coding of all relevant fuzzy sets into one large

string. This type of membership functions were used in describing the fuzzy PID controller

denormalized gains ( DIP KKK ,, ) as it has better representation of system uncertainty.

2- Coding whole fuzzy partitions

In this work a simple example of an increasing sequence of trapezoidal-triangular fuzzy sets has

been presented. Such a “fuzzy partition” is uniquely determined by an increasing sequence of 2N

points, where N is the number of linguistic values we consider. The mathematical formulation is

the following:

],[],[

0

1)( 21

10

12

21 xx

xxxx

otherwiseifif

xxxxx ∈

∈

−−

=µ (5.9)


101

12

],[],[],[

0

1)(122

1222

2232

122

2

3222

32

−≤≤∈

∈∈

−−

−−

=−

−−

−−

−

−−

−

Niforxxxxxxxxx

otherwiseififif

xxxx

xxxx

xii

ii

ii

ii

i

ii

i

iµ (5.10)

22

22323222

32],[

01)( −

−−−−

−

≥∈

−

−

= N

NNNN

N

N xxx

xx

otherwiseififxx

xx

xµ (5.12)

Fig. 5.20 shows a typical example of cross over process with N = 3 that was used to

represent (error, error change) in the proposed fuzzy PID controller. It is not wise to encode

the values xi as they are, since this requires constraints for ensuring that xi are non-

decreasing. A good alternative is to encode the offsets:

Fig. 5.20 Example for one-point crossover of fuzzy partitions

3- Genetic operators

Since this GA can only deal with binary representations of fuzzy sets and partitions, all the GA

operators that mentioned before are also applicable here. However, that the offset encoding of

fuzzy partitions is highly epistemic. More specifically, if the first bit encoding x1 is changed, the

whole partition is shifted. If this results in bad convergence, the crossover operator should be


102

modified. A suggestion can be found in [43]. Fig. 5.20 shows an example what happens if two

fuzzy partitions are crossed with normal one-point crossover. Fig. 5.21 shows the same for

bitwise mutation.

Fig. 5.21 Mutating a fuzzy partitions

4- Rescaling process (Decoding of all fuzzy partitions)

Using the following simple equation to normalize the membership variables (error, error change)

by:

1020/)10202( −×= yx

Therefore the optimum (normalized) membership functions can be obtained for both (error, error

change). The following figures (Fig. 5.22a, Fig. 5.22b, Fig. 5.22c, Fig. 5.22d, Fig. 5.22e)

illustrate the optimum input-output membership functions that obtained from the adaptive genetic

algorithm for 40 samples and 20 generations at some operating point of (250 ft/s speed and 1000

ft altitude).

Fig. 5.22a Input variable “error”

Fig. 5.22b Input variable “error change”


103

Fig. 5.22c Output variable “Kp”

Fig. 5.22d Output variable “Ki”

Fig. 5.22e Output variable “Kd”

5.3.1.3 Rule Base Weight Optimization The genetic algorithm has been applied to 100 samples of input-output simulation data obtained

from fuzzy PID controller. The rules obtained by the algorithm are shown below. The values in

parentheses after each rule represent the firing strength of the corresponding rule.

1.if (e is NB) and (ec is NB) then (kp is PB)(Ki is NB)(Kd is PS)(.5811)

: : : : : : : : : :

49. if (e is PB) and (ec is PB) then (kp is NB)(Ki is PB)(Kd is PB)(.3121


104

5.4 LQR Fuzzy Logic Control (LQRIFC) The LQR integrated fuzzy control utilizes both advantages from the LQR controller and fuzzy

logic controller as LQR controller can easily satisfies the flying qualities and pilot rating

requirements and fuzzy control can cope with the nonlinearity of the system introducing a smart

way to modify the output gains according to the actual performance blending the dynamic

response that generating better performance than using LQR alone. Fig. 5.23a and Fig. 5.23b

describes the LQR integrated fuzzy control (LQRIFC) for pitch controller.

Fig. 5.23a LQR integrated fuzzy control (LQRIFC) for pitch controller

Fig. 5.23b LQR integrated fuzzy control (LQRIFC) for lateral-directional controller


105

System dynamics is described by the following state space through LQR design algorithm at

several trimmed points for pitch control; Table 5.4 shows the linearized matrices of F-16 model

at different operating points (different altitudes). The LQR closed loop system poles satisfy the

flying qualities specifications [47] such as the damping ratio and the natural frequencies. The

tuned fuzzy logic controller copes with the LQR output error thus enhancing the net output

system performance. Membership functions could be sliding back and forth so that it suppress

low frequencies disturbance on the other hand the low pass filter introduced will cope with high

frequencies noise that produce a robust and optimal control system, as the integrated control

system utilizes both advantages of the LQR and FLC techniques (optimal and robust design).

Fig. 5.24 and Fig 5.25 show the difference between both cases with the optimized fuzzy

controller and without fuzzy controller for longitudinal and lateral motions as the additional FLC

enhances the performance index (PI) more than 8.5%. Table 5.5 shows the different optimized

LQRIFC gains at different altitude. The following linear equations describe the dynamics of the

longitudinal and lateral motions of the aircraft at velocity 500 (ft/s) appendix B5.

Table 5.4 Linearized Matrices for longitudinal and lateral motion

Alt (ft) ααA~ qAα

~ eBαδ~ αqA~ qqA~ eqB δ

~

4000 -0.8972 0.9502 -0.0019 -3.6241 -1.2147 -0.2650 4500 -0.8325 0.9533 -0.0017 -3.3307 -1.1265 -0.2438 5000 -0.7716 0.9562 -0.0017 -3.0570 -1.0433 -0.2235 Alt (ft) ββA~ βϕA~ pAβ

~ rAβ~ aBβδ

~ rBβδ~

4000 -0.3220 0.0640 0.0364 -0.9917 0.0003 0.0008 4500 -0.3221 0.0642 0.0364 -0.9920 0.0003 0.0008 5000 -0.3221 0.0642 0.0364 -0.9925 0.0003 0.0008 Alt (ft) ϕβA~ ϕϕA~ pAϕ

~ rAϕ~ aBϕδ

~ rBϕδ~

4000 0 0 1 0.0037 0 0 4500 0 0 1 0.0039 0 0 5000 0 0 1 0.0040 0 0 Alt (ft) βpA~ ϕpA~ ppA~ prA~ apB δ

~ rpB δ~

4000 -30.6492 0 -3.6784 0.6646 -0.7333 0.1315 4500 -29.5264 0 -3.6555 0.6621 -0.7192 0.1301 5000 -27.0021 0 -3.6321 0.6511 Alt (ft) βrA~ ϕrA~ rpA~ rrA~ arB δ

~ rrB δ~


106

4000 8.5395 0 -0.0254 -0.4764 -0.0319 -0.0620 4500 8.9422 0 -0.0254 -0.4722 -0.0307 -0.0520 5000 9.3323 0 -0.0254 -0.4612 -0.0303 -0.0488

Table 5.5 Gain values for longitudinal and lateral controllers

Fig. 5.24 Integrated system step response in both cases with optimized fuzzy controller and

without fuzzy controller for longitudinal motion

Alt (ft) 4000 4500 5000

αK 0.2086 0.2104 0.2127

qK -0.7971 -0.8157 -0.9091

iK 4.6106 4.6518 5.0274

1K 15.0421 16.4435 17.2216 2K 0.1822 0.1936 0.2111 3K -5.3483 -5.4541 -5.6172 4K 22.5222 24.5569 26.3897 pK 7.190 8.8725 8.1053

DK 2.958 2.754 2.333

IK -2.870 -2.650 -2.230


107

Fig. 5.25 Integrated system step response in both cases with optimized fuzzy controller and

without fuzzy controller for lateral motion

The states and output of the plant plus the compensator for both pitch and lateral motions are

[ ]TFepitch qx εαδα= , [ ]T

Fpitch qy εα=

[ ]Twralateral xrpx εδδφβ= , [ ]Trlateral epey φε=

GrBuAxx ++=•

FrCxy +=

HxZ =

Kyu −=

−−

−=

0002958.570010000.10002.200000~~~00~~~

eqqqq

eq

BAABAA

Aδα

αδααα

,

=

00

2.2000

B ,

=

10000

G ,

=

000

F

=

100000002958.57002958.57000

C , [ ]0002958.570=H , [ ]lq KKKKyu α−=−=


108

−−

−−

=

0000001001002958.57000002.20000000002.20000000~~~~~~00~~~~~~00~~~~~~00~~~~~~

rrarrrprrr

rpaprppppp

rarp

rarp


A δδϕβ

δδϕβ

ϕδϕδϕϕϕϕβϕ

βδβδββϕβββ

,

=

0000

2.20002.2000000000

B

,

=

0100000000000000

G

Hxxr

zw

=

−

=

=

01002958.5700000000010ϕ

cr FrCx

epe

y +=

∈

=

ϕ

−

−=

000000100000010000002958.5700010000000

C

=

01001000

F

The control input [ ]Tra uuu = may be expressed as Kyu −=

with

=

0000

2

431

kkkk

K


109

Chapter (6) Design of Fighter Flight Formation Autopilot for aerial Refueling

Racetrack Mission Employing KC-130J Tanker with Drogue-Hose Configuration

6.1 Introduction Autonomous formation flight is currently an important research area in the aerospace community.

The aerodynamic benefits of formation and, in particular, close formation flight, have been well

documented (Ref. [48], [49]). In earlier efforts [50], a leader-wingman formation flight control

problem was investigated and a PID-type of formation controller was developed. Ref. [51]

describes the application of an “extreme seeking” algorithm to the formation control problem. In

Ref. [52] a formation flight control scheme was proposed based on the concept of Formation

Geometry Center, also known as Formation Virtual Leader. Some of the initial experimental

results of formation flight were reported in Ref [53]. However, in all the previous efforts, the

formation control problem is considered with the aircraft flying at straight level flight conditions

and/or under mild maneuvering. In this work the formation control strategy will be implemented

in the case study of aerial refueling racetrack mission. KC-130J tanker and F-16 receiver

configuration Fig. 6.1 was used as a case study in this paper.

Fig. 6.1 KC-130J tanker and F-16 receiver configuration


110

6.2 Receiving aircraft modeling The F-16 nonlinear model [45], [46] as shown in Fig. 6.2 has been constructed using

SIMULINK/ MATLAB. The plant requires four controls, eighteen states, leading edge flap

deflection and a model flag as inputs as shown in Table 6.1 and Table 6.2. Starting with

calculating the trimming points then performing the linearization algorithms, therefore the

aircraft state space will be available to calculate the optimal gain matrix that is required for the

formation controller design.

Fig. 6.2 F-16 nonlinear modeling

Table 6.1 F-16 aircraft states

Table 6.2 Control channels


111

6.3 Control Strategy The formation control problem can be basically classified as a Dynamic 3-D Target-Tracking

problem, where the objective is to track a certain point (desired position) dynamically specified

by the leader. The main difference between conventional ‘trajectory-following’ flight and

formation flight is that in the first case the trajectory is typically pre-defined and stored within the

on-board computer while in the second case the trajectory to be followed is ‘produced’ on-line by

the leader aircraft flown under specific trajectory; thus, the trajectory information has to be

obtained in real time from some of the relevant states of the leader aircraft (position, velocity,

etc.). Ideally, to achieve desirable trajectory tracking performance, the formation flight control

strategy should be based on full state tracking strategy. This concept can be concisely expressed

as: Wingman’s control inputs = Leader’s control inputs + State error feedback where the control inputs include deflections for the throttle, elevator, aileron and rudder, while

state error feedback consists of internal state variable errors and trajectory state variable errors

between leader and wingman. Particularly, internal state variable errors are angular rate errors

and Euler angle errors (pitch and bank angles); trajectory state variable error are instead given by

projected 3-D position and velocity errors (i.e., forward distance, lateral distance and vertical

distance, and their time derivatives, as defined in next section).This approach is based on the fact

that, if the wingman flies at the same position of the leader, a perfect position tracking could be

achieved under any reasonable maneuvering the leader aircraft might execute, since the leader

and wingman aircraft are sharing very similar dynamics (assuming same type of aircraft). In

reality, extra compensation might be needed to account for the trajectory variable difference

between the leader and the ideal wingman. This is because the desired wingman position is

shifted with respect to the leader’s position. Since both the leader’s state and input vectors are


112

needed to calculate the wingman input, a high communication bandwidth between the leader and

wingman is required. Among conventional formation control schemes, the simplest scheme in

terms of the minimum amount of information from leader is based upon an existing autopilot

(functioning as an “inner loop” controller) with an additional “formation-autopilot” added on to

an “outer loop” controller. This outer loop controller uses only trajectory measurements from the

leader available from GPS. Unfortunately, this simple formation control scheme has shown

desirable performance only if the leader is flying at level straight and/or performing mild

maneuvers. A reasonable tradeoff between the simplest and the most complete schemes

introduced above is given by the use of Euler angles error feedback along with trajectory error

feedback by the wingman. The control strategy discussed in this paper is based on this approach.

6.3.1 Controller design Since formation control is a 3-D tracking problem, the control task can be decomposed into three

sub-tasks: vertical distance (height) control, lateral distance control, and forward distance control.

On the other hand, since the dynamics of the aircraft attitude (angular movement) is much faster

than the trajectory dynamics (translational movement), the whole dynamics exhibit a typical two-

time-scale feature. Therefore, the design of the control system can be decomposed into two

separate phases, that is, the inner loop and the outer loop design. The function of the inner loop

controller is to maintain and/or track the desired pitch/bank angle command; the outer loop

controller – which is based on the designed inner loop controller and uses the desired pitch/bank

angle command as its output – tries to maintain and/or track the desired formation flight.

6.3.1.1 Formation geometry and trajectory variables As described above, formation flight control problem can be decomposed as a level plane and a

vertical plane dynamic trajectory-tracking problem.

6.3.1.2 Level plane formation definition Fig. 6.3 shows the level plane formation geometry. All the trajectory measurements, i.e., leader/

wingman position and velocity, are defined with respect to a pre-defined Earth-Fixed Reference

x-o-y plane and are measured by the on-board GPS’s. The pre-defined formation geometric

parameters are the forward clearance, fc, and the lateral clearance, lc. The formation trajectory


113

variables in level plane – the forward distance, f, and lateral distance, l, can be calculated from

the trajectory measurements and formation geometric parameters as:

Fig. 6.3 Level plane formation geometry

( ) ( )

cLxy

FLLxFLLy lV

yyVxxVl −

−−−= (6.1)

( )c

Lxy

FLLxFLLy fV

xxVyyVf −

−+−=

)( (6.2)

Where 22LyLxLxy VVV += is the projection of the leader’s velocity onto X-Y plane. Accordingly,

the relative forward speed and relative lateral speed of the wingman are defined as the time

derivatives of the forward distance and lateral distance respectively and are needed for formation

control purposes which can be calculated as:

•• Ω++−

= LcLxy

FxLyFyLx ffV

VVVVl )( (6.3)

•• Ω+−+

−= LcLxy

FyLyFxLxLxy ll

VVVVV

Vf )( (6.4)

There are basically two methods to obtain the angular velocity (around the vertical axis) •Ω L . One

method is to first calculate & LΩ from the GPS measurement ),( LyLx VV , then apply conventional

numerical derivative techniques to estimate the time derivatives of LΩ ; within this approach

particular caution should be exercised due to the sensitivity of the numerical derivative with

respect to measurement noise. A second approach consists in using additional measurements

from the leader aircraft, that is using the following kinematical relation (assuming 0=•Lβ ).


114

LLLLLLL rq Θ+=Ψ≅Ω •• cos/)cossin( φφ

This second approach requires not only the vertical gyro (to measure bank angle Lφ and pitch

angle LΘ ) but also the angular rate gyros (to measure pitch rate q and yaw rate)) on the leader

aircraft. In this study the first approach was used with the definitions provided above. The level

plane formation control problem can be sub-divided into a lateral distance control problem and a

forward distance control problem.

6.3.1.3 Vertical plane formation definition At nominal conditions, the leader and the wingman aircraft are separated by a vertical clearance

h. the vertical distance, zδ , can then be calculated by:

hzzz wL −−=δ (6.5)

While its time derivative is given by:

wzLz VVz −=•δ (6.6)

6.3.2 Control laws

6.3.2.1 Lateral Distance Control The objective of the lateral distance control is to minimize the lateral distance l. the basic

physical principle of the lateral distance control can be expressed by the following action-

consequence logic: cedisLateralspeedlateralbankanglerollrateaileron tan__ →→→→

In addition, the function of the rudder is to augment the lateral-directional stability (by increasing

the Dutch Roll damping ). Therefore the lateral formation control law consists of an inner loop

controller – controlling the bank angle and augmenting the lateral-directional stability and an

outer loop controller maintaining the predefined flight formation with respect to the leader. The

control law, represented in Fig. 6.4, can be expressed by:

Inner loop control law:


115

Fig. 6.4 Lateral –Directional control law

)( gWWpAW KpK φφδ φ −+= (6.7)

WrRW rK=δ (6.8) Outer loop control law:

lKlK lldotLg ++= •φφ (6.9) 6.3.2.2 Forward Distance Control The objective of the forward distance control is to minimize the forward distance f . This task

can only be accomplished through the involvement of the throttle control channel. In fact, by

increasing/decreasing the throttle, the thrust of the engine and there for the speed of the aircraft is

increased/decreased; this, in turn, allows to control the forward distance between leader and

wingman. The forward distance control law, represented in Fig. 6.5, is given by:

Fig. 6.5 Forward control law


116

fKfK ffdotTT ++= •0δδ (6.10)

6.3.2.3Vertical Distance Control The objective of the vertical distance control is to minimize the vertical distance. This task is

accomplished through the use of the elevator control channel. The vertical distance control law is

similar to the conventional altitude-hold autopilot with the only difference being that the altitude

reference may vary according to the leader’s altitude. Similar to the lateral distance controller, the

vertical distance control scheme can be designed using an inner loop control scheme - which is

basically a pitch angle controller - and an outer loop controller which provide an altitude control-

capabilities. This control law is presented in Fig. 6.6.

Fig. 6.6 Vertical control law

Inner loop control law: )(0 gWWqEE KqK θθδδ θ −++= (6.11)

Outer loop control law: zKzK zzdotLg δδθθ ++= • (6.12)

6.4 System Identification Process In order to evaluate each branch in the complete control system introducing reasonable damping

ratio, system identification process is needed to evaluate the final response considering the

coupling between the different SISO branches and the kinematics nonlinearities.


117

6.4.1 Estimation of Simple Process Model The System Identification allows to estimate simple continuous-time process models

characterizing the static gain, dominating time constants, and possible time delays (dead time).

They are variants of the transfer function model structure.

6.4.2 Initial Parameter Values and Parameter Bounds If no prior knowledge is available about the parameters, a startup routine is invoked to come up

with initial parameter estimates. These are further iterated upon to give the best possible model fit

to the data. Actually in our process there is no initial guess is provided and an automatic process

is invoked to estimate the initial values. If no qualified guess is available, this is usually a better

alternative than entering an unstable value. However, if the estimation process gives parameter

values that seem unreasonable, it might be worthwhile to try out various initial guesses and upper

and/or lower limits of the parameters.

6.4.3 Compensator design using system identification technique In this work it is introduced some structure forward control model as shown in Fig. 6.5 that was

used to generate the linear model in order to choose the optimum PID controller gains and the

compensator time constant so that achieving the desired damping ratio through root locus tool.

Additional pole compensator structure will be as following:

cc Kps

G ×+

=1

1 (6.13)

The final design parameters are given by Table 6.3.

Table 6.3 Flight formation controller parameters

K Kp KΦ Kr Ki Kl KL KV Kq KӨ Kzdot Pz 1

.431 .29 .5 .425 .331 6.9 .98 .333 .123 .5 .8 1.18

6.5 Alignment Stage-Formation Control Simulation Results The formation control law developed in the previous section was based on a linear aircraft model.

Such a controller, if properly designed, can be guaranteed to perform nominally only when the

system is operating around the design point where the linear model was derived from. However,

non-linear dynamic effects, particularly those associated with the kinematics, cannot be ignored


118

when the aircraft is undergoing a significant trajectory maneuver, i.e., flying at a large bank

angle. Thus, it is necessary that the designed controller be validated through simulation using a

nonlinear model so that any major non-linearities in terms of trajectory dynamics, such as non-

linear reference transformation and kinematical non-linearity, can be accounted for. The non-

linear aircraft model of dynamics can be described by the following 6 DOF equations plus

kinematical equations. By applying the scenario of the system identification technique it is

possible to reach the desired performance with a reasonable damping ratio. Fig. 6.7 shows the

control system implementation using SIMULINK model appendix B8. Fig. 6.8 shows the

nonlinear output response and the relative distances in all directions that satisfy the predefined

boundaries according to the KC-130J specifications. Fig. 6.9 shows the effect of adding

compensation on the damping ratio explained by root locus (SISO) tool for lateral control branch

through MATLAB tool box. It is clear that the damping ratio has reached a reasonable value

(ζ=.35) .Fig. 6.10 shows the predefined trajectory of the tanker and the receiving aircraft

describing the plane relative distance for the whole mission. It is clear from the Fig. 6.5 that the

maximum peak in the relative distances between the receiver and the tanker usually become

larger in the transient points of switching the racetrack from and to steady state flight after and

before the steady horizontal turn. Fig. 6.8 shows also that the maximum relative distances

between the tanker and the receiving aircraft may reach over 50 (ft) in some places of the

racetrack trajectory, therefore it is recommended to increase the radius of the tanker horizontal

turn over 25000 (ft) with bank angle limit of 30 (deg) in case of performing racetrack mission as

in Fig. 6.10.


119

Fig. 6.7 Flight formation controller using SIMULINK


120

Fig. 6.8 Nonlinear model output performance with compensation


121

Fig. 6.9 Root locus for lateral –directional control with and without phase lead compensator

respectively

Fig. 6.10 Trajectory in x-y plane for both wingman and leader


122

Chapter (7) Control Strategy and Implementation Methodology

7.1 Introduction In order to design a proper and efficient aerial refueling autopilot, it should verify the basic flying

quality requirements, moreover minimizing the total mission time and verifying the Tanker-

Fighter configuration design and safety constraints such as the upper and lower limits of relative

distances and Euler angles error between the tanker and the fighter. To satisfy all the basic

requirements mentioned above a 3-stage controller has been proposed in this work starting from

the wing-level steady state flight until the final alignment in the terminal stage appendix B9. The

3-stage autopilot will be classified as following:

1- Stability Augmentation System (SAS)

2- Tanker Tracking using Command Augmentation system (CAS)

3- Alignment Terminal using Flight Formation Controller

7.2 Aerial Refueling Stages

7.2.1 Stability Augmentation System In this stage a Linear Quadratic Regulator (LQR) will be introduced in this stage, the regulator

was mentioned in chapter 4 as an optimal stabilizer for both the longitudinal and lateral motions.

In this stage this stabilizer will recover the fighter aircraft from any past maneuvers and

stabilizing all the flight parameters before applying the next stage controller. In other words it is a

transition stage before the pilot commanded stage to pilot un-commanded stage.

7.2.2 Tanker Tracking Using Command Augmentation system (CAS) The second phase is the tracking phase controlling the receiving aircraft (fighter aircraft) from the

end of the first stage to the 1000 (ft) behind the tanker at the desired lower point which is straight

and lower from the tanker with 25 (ft) at this phase it is clear that the receiving aircraft speed

profile maintained at almost constant and high speed as long as the receiving aircraft is away

from the tanker then decelerating with different acceleration rates till starting the next phase

which allow to minimize the total mission time appendix B6, B7.


123

The following equations explain the transformation procedure starting from the tanker position

with respect to fixed frame of reference to wing coordinate of the receiving aircraft with

initializing small perturbed step (0.0001) to avoid the singularities during simulation calculation.

X = Tanker X component relative to the fixed frame of reference

Y = Tanker Y component relative to the fixed frame of reference

Z = Tanker Z component relative to the fixed frame of reference

XF

Y

= Fighter X component relative to the fixed frame of reference

F

Z

= Fighter Y component relative to the fixed frame of reference

F

X

= Fighter Z component relative to the fixed frame of reference

BO

Y

= Tanker X component relative to aircraft body axes at the fixed frame of reference

BO

Z

= Tanker Y component relative to aircraft body axes at the fixed frame of reference

BO

X

= Tanker Z component relative to aircraft body axes at the fixed frame of reference

B

Y

= Tanker X component relative to aircraft body axes

B

Z

= Tanker Y component relative to aircraft body axes

B

θ

= Tanker Z component relative to aircraft body axes

, Ф, ψ = Rotation around X,Y, Z axes

A = cos θ cos ψ

B = (sin Ф sin θ cos ψ – cos Ф sin ψ)

C = (cos Ф sin θ cos ψ+ sin Ф sin ψ)

D = cos θ sin ψ

E = (sin Ф sin θ sin ψ + cos Ф cos ψ)

F = (cos Ф sin θ sin ψ – sin Ф cos ψ)

G = – sin θ

M = sin Ф cos θ

N = cos Ф cos θ


124

)])(())([()])(())([(

BDEANACGMABGCDFABDEAZAXGMABGXDYAZ BO −−−−−

−−−−−= (7.1)

)()]()[(

BDEACDFAZXDYAY BO

BO −−−−

= (7.2)

ACZBYX

X BOBOBO

)( −−= (7.3)

)])(())([()])(())([(

BDEANACGMABGCDFABDEAAZGXMABGDXAYZ FFFF

SHIFT −−−−−−−−−−

= (7.4)

)()]()[(

BDEACDFAZDXAYY SHIFTFF

SHIFT −−−−

= (7.5)

ACZBYX

X SHIFTSHIFTFSHIFT

)( −−= (7.6)

XB = XBO – X shift

Y

(7.7)

B = YBO – Y shift

Z

(7.8)

B = ZBO – Z shift

(7.9)

Fig. 7.3 shows all the control efforts of the control surfaces and the aircraft engine power during

the whole mission starting from the start point as in Fig. 7.2 appendix B3. until to the point of

contact to the tanker which is the most essential interval in the mission. The alignment terminal

control secures the position of the receiving aircraft beneath the tanker of about 30 (ft) and the

steady flight speed is equal the same as the tanker flight speed. The figures show also the relative

speed and distances between the receiving aircraft and the tanker.


125

Fig. 7.1 Tanker position transformation to aircraft wing axes

Fig. 7.2 Actual trajectories for tanker and receiving aircraft


126

Fig. 7.3 F-16 nonlinear model performance (Tracking stage)


127

7.2.3 Alignment Terminal using Flight Formation Controller The formation control law developed in the previous chapter was based on a linear aircraft

model. Such a controller, if properly designed, can be guaranteed to perform nominally only

when the system is operating around the design point where the linear model was derived from.

However, non-linear dynamic effects, particularly those associated with the kinematics, cannot be

ignored when the aircraft is undergoing a significant trajectory maneuver, i.e., flying at a large

bank angle. Thus, it is necessary that the designed controller be validated through simulation

using a nonlinear model so that any major non-linearities in terms of trajectory dynamics, such as

non-linear reference transformation and kinematical non-linearity, can be accounted for. The non-

linear aircraft model of dynamics can be described by the following 6 DOF equations plus

kinematical equations. By applying the scenario of the system identification technique it is

possible to reach the desired performance with a reasonable damping ratio. Fig. 7.4 shows the

nonlinear output response and the relative distances in all directions that satisfy the predefined

boundaries according to the KC-130J specifications. It is clear from the Fig. 7.4 that the

maximum peak in the relative distances between the receiver and the tanker usually become

larger in the transient points of switching the racetrack from and to steady state flight after and

before the steady horizontal turn. Fig. 7.4 shows also that the maximum relative distances

between the tanker and the receiving aircraft may reach over 50 (ft) in some places of the

racetrack trajectory, therefore it is recommended to increase the radius of the tanker horizontal

turn over 25000 (ft) with bank angle limit of 30 (deg) in case of performing racetrack mission.


128

Fig. 7.4 Nonlinear model output performance with compensation

Finally the overall implementation methodology and control strategy for the three basic

mentioned stages (phases) can by described clearly through the following flowchart.


129

Operating point Alt. Mach

Trimming Algorithm

Linearization Algorithm

State Space Models

LQR Algorithm (SAS)

Gain Matrix Flying Qualities Verification

Integration with (FLC type PID)

Lateral-Directional motion

GA using different types of PI For both longitudinal and lateral motions step response

PID parameters Membership Functions

Rule Base Strength

Tracking Controller during Tanker tracking phase (second stage)

Design 3 stage transformation to map the tanker position to the

receiving A/C wind axes

Optimal Guidance Law algorithm

Alignment stage controller to align the receiving A/C with the tanker C/L

Total mission time and the final relative errors

Verification 1-Design constrains 2-Flying qualities 3-Time minimization compared with diff. x

Longitudinal motion

Final states

Initial states

State Space Models plus desired compensator structure

LQ Algorithm (CAS)

Gain Matrix

Zero states Initial states

Aerial refueling Procedure


130

7.3 Fighter -Tanker Tracking Stage Optimal Trajectory Determination

Fig. 7.5 Fighter-Tanker tracking stage optimal trajectory

7.3.1 Optimal trajectory equations

Ryyxxtv RTfRTffR −−+−=⋅ 2/12)0(

2)0( ])()[( (7.10)

)]/()[(tan 1RfTfRfTff xxyy −−= −θ (7.11)

RRR vx ψcos⋅=• (7.12)

RRR vy ψsin⋅=• (7.13)

Since fR ψψ =

fRfRf vtx ψcos⋅= (7.14)


131

fRfRf vty ψsin⋅= (7.15)

Where ))),((( Rvvfavgv TRR −=

Since the tanker’s motion could be detected easily

∫+=tf

TTTTF dtvxx0

0 cosψ (7.16)

∫+=tf

TTTTF dtvyy0

0 sinψ (7.17)

Where tf = time at the end of tracking stage By using numerical method such as quadratic convergent Newton raphson method, the optimal

trajectory equation can be resolved, therefore values tf and fψ could be evaluated.

//])()[( 2/12)0(

2)0( RRRTfRTff vRvyyxxt −−+−= (7.18)

Therefore the value of tf and fψ can be obtained, Then the acceleration command equation will be as following:

•= RROPG Vu ψ (7.19)

The following Figures Fig. 7.6, Fig. 7.7, and Fig. 7.8 show the relative distances in X, Y and Z

directions after applying the previous optimal guidance law. °= 9.1requiredψ , °= 5.0requiredθ with

more than 35 % reduction in mission time


132

Fig. 7.6 Relative distance in X direction (ft)

Fig. 7.7 Relative distance in Y direction (ft)

Fig. 7.8 Relative distance in Z direction (ft)


133

Chapter (8)

SUMMARY AND FINAL CONCLUSION

8.1 Stability Stage Poles Verification with Flying Qualities (1ST

The first stage (stability phase) is need to recover the fighter aircraft from any past maneuvers

using the stability system augmentation to stabilize the aircraft to perform the next stages such as

(tracking and alignment stages). It is required in this transition phase to satisfy all the closed loop

poles with standard flying qualities for securing better and acceptable performance of the aircraft.

The following table shows the short period poles in different altitudes which satisfy the required

flying qualities as mentioned. In the same manner it is possible to satisfy the other poles with the

associated standard values as described in Table 8.1.

Stage)

Table 8.1 The listed values satisfy a level 1 flying quality with a category B flight phase thus

validating the design parameters for the proposed controller

Altitude Eigen values spω spξ )//(2 αω nsp 20000 -6.23 ± 11.3i 12.8621 0.4841 2.7572 22000 -6.15 ± 10.6i 12.2414 0.5026 2.4975 24000 -6.20 ± 10.7i 12.4004 0.5000 2.5628 26000 -6.19 ± 10.5i 12.2004 0.5073 2.4808 28000 -6.18 ± 10.3i 11.9991 0.5149 2.3996 30000 -6.17 ± 10.1i 11.8036 0.5226 2.3220 32000 -6.16 ± 9.85i 11.6181 0.5303 2.2496 34000 -6.15 ± 9.65i 11.4424 0.5378 2.1821

8.2 Tracking Stage Time Using Different Guidance Technique(2nd

The second stage is very important stage in designing aerial refueling autopilot as it affects the

total time consumed for the whole mission. There are different types of guidance laws could be

used in this phase such as (Line Of Sight (LOS), and Optimal Guidance Law (OGL)) both

guidance laws were designed and implemented in this work. Consequently it is easy to compare

between both of them as shown in Table 8.2 that results in using the second guidance law

Stage)


134

(Optimal Guidance Law) is much better more than the first one as it has less time and lower

energy consumed compared with the first guidance law.

Table 8.2 Tracking phase time versus guidance technique (second phase)

Tracking phase guidance

technique

Phase time Total energy consumed

Line Of Sight (LOS) 495 (sec) High

Optimal Guidance Law 365 (sec) Low

8.3 Alignment Stage during (Boeing KC 767) Tanker–Boom configuration

(3rd

The last stage is also very important stage as it should satisfy some important constrains to secure

safety and security during last approaching to perform the boom contact with the receiving

aircraft. Table 8.3 shows the minimum and maximum errors during alignment stage with the

design constrains that calculated according to the Tanker-Boom details. It is easy to figure out

that the Simulation errors satisfy the permitted errors with acceptable tolerances. Note that the

actual engagement starts after about 50 seconds from starting the alignment phase.

Stage)

Table 8.3 Relative distances in 3 directions (X, Y, Z) with upper and lower limits according

to the design constrains of the Tanker-Boom configuration (last phase)

Relative Distances Min ASE Max

Forward distance (ft) 87 88-97 100 Lateral distance(ft) 0 .5-3 6 Vertical distance(ft) 11 22 25

8.4 (Boeing KC 767 Tanker – Boom configuration) versus (KC-130J Tanker

with Hose – Drogue configuration) The following figures illustrate some of the simulation output results such that the relative error

in the three basic directions for both mentioned configurations that show the difference in relative


135

error specially in the forward and lateral directions at the turn points. On the other hand it is

possible to use the second configuration during performing both racetrack and alignment stage

missions while it is recommended only to use the first configuration in normal alignment stage

only. As the first configuration requires some tight constrains to be satisfied as shown in Fig. 8.1

while the second configuration has larger constrains that fit with the Hose-Drogue configuration.

Fig. 8.1 (Boeing KC 767 Tanker – Boom configuration) versus (KC-130J Tanker with Hose

– Drogue configuration)

8.5 Final Conclusion It is clear from the present work that there is some basic requirements should be satisfied in

order to establish efficient aerial refueling autopilots such as (1- flying qualities requirements, 2-

safety and secured approaching in the final docking, 3- minimization the whole mission time


136

period) which implies that a multistage autopilot is needed for this purpose. Therefore LQR

technique was used to satisfy the flying qualities, providing control integration with fuzzy logic

controller that enhance the output performance index. Applying an optimal trajectory during the

tracking stage had a great effect in minimizing the total mission time. Using flight formation

principle in last stage (alignment stage) had satisfied the required tanker geometrical constrains

mentioned before. Thus a multistage autopilot could be realized in such described sequence

verifying the safety and security with minimizing the mission time interval that permit several

aircrafts to be refueled in such an optimum time period. In this way it is possible to avoid the any

mishaps from the pilots during manual process.

8.6 Recommended Future Work It is recommended to apply the final multistage autopilot for a several aircrafts or a complete

squadron to optimize the air refueling process time for a number of aircrafts so that decrease the

time required for the tanker aircraft to fly especially in the operations time. Consequently

reducing the time needed for the fighter aircrafts to be refueled thus decreasing the time to be

exposed to any aggressors. This will require also different guidance laws according to the aircraft

position with central controller installed in the tanker aircraft.


137

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141

Appendix A

Matlab Codes

A1. Lateral Stability Regulator LQR A2. LQ Tracker for Normal Acceleration (G) Control A3. LQ Tracker for Pitch Motion Control A4. Normal Acceleration CAS Tracker Weighted Time A5. Genetic Algorithms For (f1, f2, f3) Optimization (Membership Function Optimization) A6. Fuzzy Logic Controller (called by A5) A7. Genetic Algorithms For ( PK , iK , dK ) Optimization (Fuzzy Logic Controller Output Gain Optimization) A8. Newton Raphson Method for Calculating the time required for the second stage (using Optimal Guidance Law) (XY Direction) A9. Newton Raphson Method for Calculating the time required for the second stage (using Optimal Guidance Law) (XZ Direction) A10. Fuzzy Logic Controller rules with optimum firing strength


142

%A1. Lateral stability Regulator A=[-.3220 .0640 .0364 -.9917 .0003 .0008 0;0 0 1 .0037 0 0 0;-30.6492 0 -3.6784 .6646 -.7333 0.1315 0;8.5396 0 -.0254 -.4764 -0.0319 -0.0620 0;0 0 0 0 -20.2 0 0;0 0 0 0 0 -20.2 0;0 0 0 57.2958 0 0 -1] B=[0 0; 0 0;0 0;0 0;20.2 0;0 20.2;0 0] C=[0 0 0 57.2958 0 0 -1; 0 0 57.2958 0 0 0 0;57.2958 0 0 0 0 0 0;0 57.2958 0 0 0 0 0] Q=[50 0 0 0 0 0 0;0 100 0 0 0 0 0;0 0 100 0 0 0 0;0 0 0 50 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 1] R=.1*eye(2) K0=[0 0 0.11 -.35;-1 -.21 -.44 .26] X=eye(7) t1=K0*C t2=B*t1 A0=A-t2 t3=C'*K0'*R*K0*C+Q P0=lyap(A0',t3) S0=lyap(A0,X) t4=(C*S0*C') t5= inv(t4) K1=inv(R)*B'*P0*S0*C'*t5 dK=K1-K0 K2=K1+1*dK A1=A-B*K2*C t3=C'*K2'*R*K2*C+Q P1=lyap(A1',t3) S1=lyap(A1,X) t4=(C*S1*C') t5= inv(t4) K2=inv(R)*B'*P1*S1*C'*t5 e1=eig(A1) dK=K2-K1 K3=K2-.1*dK A2=A-B*K3*C t3=C'*K3'*R*K3*C+Q P2=lyap(A2',t3) S2=lyap(A2,X) t4=(C*S2*C') t5= inv(t4) K3=inv(R)*B'*P2*S2*C'*t5 e2=eig(A2) dK=K3-K2 K4=K3-.5*dK A3=A-B*K4*C t3=C'*K4'*R*K4*C+Q P3=lyap(A3',t3) S3=lyap(A3,X) t4=(C*S3*C') t5= inv(t4) K4=inv(R)*B'*P3*S3*C'*t5 e3=eig(A3)


143

dK=K4-K3 K5=K4-.2*dK A4=A-B*K5*C t3=C'*K5'*R*K5*C+Q P4=lyap(A4',t3) S4=lyap(A4,X) t4=(C*S4*C') t5= inv(t4) K5=inv(R)*B'*P4*S4*C'*t5 e4=eig(A4) dK=K5-K4 K6=K5-.5*dK A5=A-B*K6*C t3=C'*K6'*R*K6*C+Q P5=lyap(A5',t3) S5=lyap(A5,X) t4=(C*S5*C') t5= inv(t4) K6=inv(R)*B'*P5*S5*C'*t5 e5=eig(A5) dK=K6-K5 K7=K6-1*dK A6=A-B*K7*C t3=C'*K7'*R*K7*C+Q P6=lyap(A6',t3) S6=lyap(A6,X) t4=(C*S6*C') t5= inv(t4) K7=inv(R)*B'*P6*S6*C'*t5 e6=eig(A6) dK=K7-K6 K8=K7-1*dK A7=A-B*K8*C t3=C'*K8'*R*K8*C+Q P7=lyap(A7',t3) S7=lyap(A7,X) t4=(C*S7*C') t5= inv(t4) K8=inv(R)*B'*P7*S7*C'*t5 e7=eig(A7) dK=K8-K7 K9=K8-1.2*dK A8=A-B*K9*C t3=C'*K9'*R*K9*C+Q P8=lyap(A8',t3) S8=lyap(A8,X) t4=(C*S8*C')


144

t5= inv(t4) K9=inv(R)*B'*P8*S8*C'*t5 e8=eig(A8) dK=K9-K8 K10=K9-2*dK A9=A-B*K10*C t3=C'*K10'*R*K10*C+Q P9=lyap(A9',t3) S9=lyap(A9,X) t4=(C*S9*C') t5= inv(t4) K10=inv(R)*B'*P9*S9*C'*t5 e9=eig(A9) [v,d]=eig(A9) e=eig(A0) J1=.5*trace(P1*X) J2=.5*trace(P2*X) J3=.5*trace(P3*X) J4=.5*trace(P4*X) J5=.5*trace(P5*X) J6=.5*trace(P6*X) J7=.5*trace(P7*X) J8=.5*trace(P8*X) J9=.5*trace(P9*X) B1=[0 0;0 0;0 0;0 0;0 0;0 0;0 0] C1=[0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0;0 0 0 0 0 0 0] D=[0 0;0 0;0 0;0 0] sys0=ss(A1,B1,C,D) sys1=ss(A5,B1,C,D) sys2=ss(A4,B1,C,D) damp(sys) initial(sys2,sys1,sys0,[1 0 0 0 0 0 0]);


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%A2. LQ Tracker for Normal G control A=[-1.01887 .90506 -.00215 0 0;-2.4977 -1.3859 -.17555 0 0;0 0 -20.0 0 0;10 0 0 -10 0;-16.26 -.9788 .04852 0 0] B=[0; 0; 20.2; 0; 0] G=[0; 0; 0; 0; 1] C=[0 0 0 57.2958 0;0 57.2958 0 0 0;-16.26 -.9788 .04852 0 0;0 0 0 0 1] F=[0; 0; 1; 0] H=[16.26 .9788 -.04852 0 0] Q=[264 16 1 0 0;16 60 0 0 0;1 0 0 0 0;0 0 0 0 0;0 0 0 0 100] R=.01*eye(1) V=502*eye(1) K=[0 -.53 10 5] Ac0=A-B*K*C Bc0=G-B*K*F X=eye(5) t1=Q+C'*K'*R*K*C P0=lyap(Ac0',t1) S0=lyap(Ac0,X) r0=1 x=-inv(Ac0)*Bc0*r0 y=C*x+F*r0 X=inv(Ac0)*Bc0*r0*r0'*Bc0'*(inv(Ac0))' t2=B'*P0*S0*C'-B'*(inv(Ac0))'*(P0+H'*V*H)*x*y'+B'*(inv(Ac0))'*H'*V*r0*y' t3=inv(C*S0*C') K1=inv(R)*t2*t3 ei0=eig(Ac0) e0=(eye(1)+H*inv(Ac0)*Bc0)*r0 dK=K1-K K1=K1-1*dK Ac1=A-B*K1*C Bc1=G-B*K1*F X=eye(5) t1=Q+C'*K1'*R*K1*C P1=lyap(Ac1',t1) S1=lyap(Ac1,X) r0=1 x=-inv(Ac1)*Bc1*r0 y=C*x+F*r0 X=inv(Ac1)*Bc1*r0*r0'*Bc1'*(inv(Ac1))' t2=B'*P1*S1*C'-B'*(inv(Ac1))'*(P1+H'*V*H)*x*y'+B'*(inv(Ac1))'*H'*V*r0*y' t3=inv(C*S1*C') K2=inv(R)*t2*t3 dK=K2-K1 ei1=eig(Ac1) e1=(eye(1)+H*inv(Ac1)*Bc1)*r0 K2=K2-.5*dK Ac2=A-B*K2*C Bc2=G-B*K2*F X=eye(5) t1=Q+C'*K2'*R*K2*C P2=lyap(Ac2',t1) S2=lyap(Ac2,X)


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r0=1 x=-inv(Ac2)*Bc2*r0 y=C*x+F*r0 X=inv(Ac2)*Bc2*r0*r0'*Bc2'*(inv(Ac2))' t2=B'*P2*S2*C'-B'*(inv(Ac2))'*(P2+H'*V*H)*x*y'+B'*(inv(Ac2))'*H'*V*r0*y' t3=inv(C*S2*C') K3=inv(R)*t2*t3 dK=K3-K2 ei2=eig(Ac2) e2=(eye(1)+H*inv(Ac2)*Bc2)*r0 K3=K3-.1*dK Ac3=A-B*K3*C Bc3=G-B*K3*F X=eye(5) t1=Q+C'*K3'*R*K3*C P3=lyap(Ac3',t1) S3=lyap(Ac3,X) r0=1 x=-inv(Ac3)*Bc3*r0 y=C*x+F*r0 X=inv(Ac3)*Bc3*r0*r0'*Bc3'*(inv(Ac3))' t2=B'*P3*S3*C'-B'*(inv(Ac3))'*(P3+H'*V*H)*x*y'+B'*(inv(Ac3))'*H'*V*r0*y' t3=inv(C*S3*C') K4=inv(R)*t2*t3 dK=K4-K3 ei3=eig(Ac3) e3=(eye(1)+H*inv(Ac3)*Bc3)*r0 j0=.5*trace(P0*X)+.5*e0'*V*e0 j1=.5*trace(P1*X)+.5*e1'*V*e1 j2=.5*trace(P2*X)+.5*e2'*V*e2 j3=.5*trace(P3*X)+.5*e3'*V*e3 D=[0] sys=ss(Ac0,Bc0,H,D) sys0=ss(Ac3,Bc3,H,D) sys1=ss(Ac2,Bc2,H,D) figure (1) step(sys) figure (2) step(sys0,sys)


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%A3. LQ Tracker for Pich Motion Control A=[-1.01887 .90506 -.00215 0 0;.82225 -1.07741 -.17555 0 0;0 0 -20.0 0 0;10 0 0 -10 0;0 -57.2958 0 0 0] B=[0; 0; 20.2; 0; 0] G=[0; 0; 0; 0; 1] C=[0 0 0 57.2958 0;0 57.2958 0 0 0;0 0 0 0 1] F=[0; 0; 0] H=[0 57.2958 0 0 0] Q=[264 16 1 0 0;16 60 0 0 0;1 0 0 0 0;0 0 0 0 0;0 0 0 0 100] R=1.5*eye(1) V=502*eye(1) K=[0 -.5 2] Ac0=A-B*K*C Bc0=G-B*K*F X=eye(5) t1=Q+C'*K'*R*K*C P0=lyap(Ac0',t1) S0=lyap(Ac0,X) r0=1 x=-inv(Ac0)*Bc0*r0 y=C*x+F*r0 X=inv(Ac0)*Bc0*r0*r0'*Bc0'*(inv(Ac0))' t2=B'*P0*S0*C'-B'*(inv(Ac0))'*(P0+H'*V*H)*x*y'+B'*(inv(Ac0))'*H'*V*r0*y' t3=inv(C*S0*C') K1=inv(R)*t2*t3 ei0=eig(Ac0) e0=(eye(1)+H*inv(Ac0)*Bc0)*r0 dK=K1-K K1=K1-1*dK Ac1=A-B*K1*C Bc1=G-B*K1*F X=eye(5) t1=Q+C'*K1'*R*K1*C P1=lyap(Ac1',t1) S1=lyap(Ac1,X) r0=1 x=-inv(Ac1)*Bc1*r0 y=C*x+F*r0 X=inv(Ac1)*Bc1*r0*r0'*Bc1'*(inv(Ac1))' t2=B'*P1*S1*C'-B'*(inv(Ac1))'*(P1+H'*V*H)*x*y'+B'*(inv(Ac1))'*H'*V*r0*y' t3=inv(C*S1*C') K2=inv(R)*t2*t3 dK=K2-K1 ei1=eig(Ac1) e1=(eye(1)+H*inv(Ac1)*Bc1)*r0 K2=K2-.5*dK Ac2=A-B*K2*C Bc2=G-B*K2*F X=eye(5) t1=Q+C'*K2'*R*K2*C P2=lyap(Ac2',t1) S2=lyap(Ac2,X)


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r0=1 x=-inv(Ac2)*Bc2*r0 y=C*x+F*r0 X=inv(Ac2)*Bc2*r0*r0'*Bc2'*(inv(Ac2))' t2=B'*P2*S2*C'-B'*(inv(Ac2))'*(P2+H'*V*H)*x*y'+B'*(inv(Ac2))'*H'*V*r0*y' t3=inv(C*S2*C') K3=inv(R)*t2*t3 dK=K3-K2 ei2=eig(Ac2) e2=(eye(1)+H*inv(Ac2)*Bc2)*r0 K3=K3-.1*dK Ac3=A-B*K3*C Bc3=G-B*K3*F X=eye(5) t1=Q+C'*K3'*R*K3*C P3=lyap(Ac3',t1) S3=lyap(Ac3,X) r0=1 x=-inv(Ac3)*Bc3*r0 y=C*x+F*r0 X=inv(Ac3)*Bc3*r0*r0'*Bc3'*(inv(Ac3))' t2=B'*P3*S3*C'-B'*(inv(Ac3))'*(P3+H'*V*H)*x*y'+B'*(inv(Ac3))'*H'*V*r0*y' t3=inv(C*S3*C') K4=inv(R)*t2*t3 dK=K4-K3 ei3=eig(Ac3) e3=(eye(1)+H*inv(Ac3)*Bc3)*r0 j0=.5*trace(P0*X)+.5*e0'*V*e0 j1=.5*trace(P1*X)+.5*e1'*V*e1 j2=.5*trace(P2*X)+.5*e2'*V*e2 j3=.5*trace(P3*X)+.5*e3'*V*e3 D=[0] sys=ss(Ac0,Bc0,H,D) sys0=ss(Ac3,Bc3,H,D) sys1=ss(Ac2,Bc2,H,D) figure (1) step(sys) figure (2) step(sys0,sys1)


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%A4. Normal Acceleration CAS TRacker Weighted time close all clear all %in this file the matrices A B C D are from the book Page 387 global Ao;global Bo;global Co;global Do;global Fo;global Go;global Ho; Ao=[-1.01887 0.90506 -0.00215 0 0;0.82225 -1.07741 -0.17555 0 0; 0 0 -20.2 0 0;10 0 0 -10 0; -16.26 -0.9788 0.04852 0 0 ]; Bo=[0 ; 0;20.2 ;0 ; 0]; Co=[0 0 0 57.2958 0; 0 57.2958 0 0 0;-16.26 -0.9788 0.04852 0 0 ; 0 0 0 0 1]; Fo=[0;0;1;0]; Do=Fo; Go=[0;0;0;0;1]; Ho=[16.26 0.9788 -0.04852 0 0 ]; %initial gain matrix K0=Init_Gain(Ao,Bo,Co,-2,2); %K0=[-0.847 -0.452 1.647 8.602]; %assumed Q used for output feedback design %Q=[264 16 1 0 0;16 60 0 0 0; 1 0 0 0 0;0 0 0 0 0;0 0 0 0 100]; %Uk(1,:)=[0.006 -0.152 1.17 0.996]; %0.03651433971552 from the book %Uk(2,:)=[-1.629 -1.316 18.56 77.6]; %0.03651433971552 from the book %Uk(1,:)=[-0.847 -0.452 1.647 8.602]; %0.03651433971552 from the book Uk(1,:)=[0 0 1 0]; % open loop response, when a unity gain is applied in the proportional feedforward path options = optimset('MaxFunEvals',10000000,'TolFun',1e-8,'Display','final','MaxIter',100000,'GradObj','on'); [Uk(2,:),Jmin]=fminunc(@lqrytracktime,K0,options); Uk Ukn=size(Uk,1); %creating closed loop state space for tracker for(i=1:1:Ukn) Ac=Ao-Bo*Uk(i,:)*Co; Bc=Go-Bo*Uk(i,:)*Fo; Cc=Ho; %so that the output from the new closed loop tracker is the performance output z = Hx Dc=0; sys=ss(Ac,Bc,Cc,Dc,'statename','Alpha' 'q' 'de' 'AlphaF' 'epsilon' ,'inputname','reference I/P' ,'outputname','Perf. O/P z (norm acc)'); ro=1; t=[0:0.01:10]'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x0=[0 0 0 0 0]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u=[0*t+ro];


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[z(:,i),tout,x]=lsim(sys,u,t,x0); end figure('Name','perf. O/P (Norm Acc nz)') plot(tout,z(:,:));grid on; % color order: blue,green,red,cyan function [Ko]=Init_Gain(A,B,C,a,b) Sflag=0; while(Sflag<size(A,1)) Sflag=0; Kv= a + (b-a) * rand(size(B,2),size(C,1)); %size= #ofinputs*#of outputs Ac=A-B*Kv*C; Ev=eig(Ac); for(i=1:1:length(Ev)) if (Ev(i,1)<0) Sflag=Sflag+1;end end end Ko= Kv ; function [f,g]=lqrytracktime(K) %Ao,Bo,Co,Do are the state space matrices for the open loop system, %Q is weighting matrices, r and v are multipliers for the R and V matrices %rho used in PI for weighting steady state error, r0 required reference input %K initial gain matrix, H performance output matrix z=Hx global Ao;global Bo;global Co;global Do;global Fo;global Go;global Ho; r=.005; kn=2; %t^kn as kn increases the response is relaxed more(slower response) ro=1; R=eye(1)*r;%assumed R used for output feedback LQR design Q=0; %'Optimizing, Please wait...' Ac=Ao-Bo*K*Co; Bc=Go-Bo*K*Do; xbar=-1*inv(Ac)*Bc*ro; ybar=Co*xbar + Do*ro; X=xbar*xbar'; Pki=Ho'*Ho; Pk(:,:,1)=lyap(Ac',Pki); for(i=2:1:kn) Pk(:,:,i)=lyap(Ac',Pk(:,:,i-1)); end Q=0;


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Pk(:,:,kn+1)=lyap(Ac',(Co'*K'*R*K*Co + Q + Pk(:,:,kn)*factorial(kn))); Sk(:,:,kn+1)=lyap(Ac,X); Sk(:,:,kn)=lyap(Ac,Sk(:,:,kn+1)*factorial(kn)); for(i=kn-1:-1:1) Sk(:,:,i)=lyap(Ac,Sk(:,:,i+1)); end %Sk*Pk PkSk=0; for(i=1:1:kn+1) PkSk=PkSk+Pk(:,:,i)*Sk(:,:,i); end %Performance Index f=0.5*trace(Pk(:,:,kn+1)*X); %Gradient dJ/dK g=R*K*Co*Sk(:,:,kn+1)*Co' - Bo'*PkSk*Co' + Bo'*((inv(Ac))')*Pk(:,:,kn+1)*xbar*ybar';


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%A5. Genetic Algorithm for function f(f1,f2,f3) optimization (membership function optimization) %Genetic Algorithm for function f(x1,x2,x3) optimization %Parameters Size=5 G=2 CodeL=10; umax1=1.2; umin1=.9; umax2=1.2; umin2=.9; umax3=1.7; umin3=1.3; E=round(rand(Size,3*CodeL)); %Initial Code %Main Program for k=1:1:G time(k)=k; for s=1:1:Size m=E(s,:); y1=0;y2=0;y3=0; %Uncoding m1=m(1:1:CodeL); for i=1:1:CodeL y1=y1+m1(i)*2^(i-1); end x1=(umax1-umin1)*y1/1023+umin1; m2=m(CodeL+1:1:2*CodeL); for i=1:1:CodeL y2=y2+m2(i)*2^(i-1); end x2=(umax2-umin2)*y2/1023+umin2; m3=m(2*CodeL+1:1:3*CodeL); for i=1:1:CodeL y3=y3+m3(i)*2^(i-1); end x3=(umax3-umin3)*y3/1023+umin3; f1=x1; f2=x2; f3=x3; fuzzy_controller_genetic sim ('F16block_pitch_contr_fuzzyPD' , [0 5]); h(s)=PI(5001) F(s)=1/h(s) end Ji=1./(F+1e-10); %****** Step 1 : Evaluate BestJ ******


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BestJ(k)=min(Ji); fi=F; %Fitness Function [Oderfi,Indexfi]=sort(fi); %Arranging fi small to bigger Bestfi=Oderfi(Size); %Let Bestfi=max(fi) BestS=E(Indexfi(Size),:); %Let BestS=E(m), m is the Indexfi belong to max(fi) bfi(k)=Bestfi; %****** Step 2 : Select and Reproduct Operation****** fi_sum=sum(fi); fi_Size=(Oderfi/fi_sum)*Size; fi_S=floor(fi_Size); %Selecting Bigger fi value kk=1; for i=1:1:Size for j=1:1:fi_S(i) %Select and Reproduce TempE(kk,:)=E(Indexfi(i),:); kk=kk+1; %kk is used to reproduce end end %************ Step 3 : Crossover Operation ************ pc=0.60; n=ceil(30*rand); for i=1:2:(Size-1) temp=rand; if pc>temp %Crossover Condition for j=n:1:30 TempE(i,j)=E(i+1,j); TempE(i+1,j)=E(i,j); end end end TempE(Size,:)=BestS; E=TempE; %************ Step 4: Mutation Operation ************** %pm=0.001; %pm=0.001-[1:1:Size]*(0.001)/Size; %Bigger fi, smaller Pm %pm=0.0; %No mutation pm=0.1; %Big mutation for i=1:1:Size for j=1:1:3*CodeL temp=rand; if pm>temp %Mutation Condition if TempE(i,j)==0 TempE(i,j)=1; else TempE(i,j)=0; end end end end


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%Guarantee TempPop(30,:) is the code belong to the best individual(max(fi)) TempE(Size,:)=BestS; E=TempE; end Max_Value=Bestfi BestS x1 x2 x3 figure(1); plot(time,BestJ); xlabel('Times');ylabel('Best J'); figure(2); plot(time,bfi); xlabel('times');ylabel('Best F');


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%A6. Fuzzy Controller Design a=newfis('fuzzf'); a=addvar(a,'input','e',[-3*f1,3*f1]); %Parameter e a=addmf(a,'input',1,'NB','zmf',[-3*f1,-1*f1]); a=addmf(a,'input',1,'NM','trimf',[-3*f1,-2*f1,0]); a=addmf(a,'input',1,'NS','trimf',[-3*f1,-1*f1,1*f1]); a=addmf(a,'input',1,'Z','trimf',[-2*f1,0,2*f1]); a=addmf(a,'input',1,'PS','trimf',[-1*f1,1*f1,3*f1]); a=addmf(a,'input',1,'PM','trimf',[0,2*f1,3*f1]); a=addmf(a,'input',1,'PB','smf',[1*f1,3*f1]); a=addvar(a,'input','ec',[-3*f2,3*f2]); %Parameter ec a=addmf(a,'input',2,'NB','zmf',[-3*f2,-1*f2]); a=addmf(a,'input',2,'NM','trimf',[-3*f2,-2*f2,0]); a=addmf(a,'input',2,'NS','trimf',[-3*f2,-1*f2,1*f2]); a=addmf(a,'input',2,'Z','trimf',[-2*f2,0,2*f2]); a=addmf(a,'input',2,'PS','trimf',[-1*f2,1*f2,3*f2]); a=addmf(a,'input',2,'PM','trimf',[0,2*f2,3*f2]); a=addmf(a,'input',2,'PB','smf',[1*f2,3*f2]); a=addvar(a,'output','u',[-3*f3,3*f3]); %Parameter u a=addmf(a,'output',1,'NB','zmf',[-3*f3,-1*f3]); a=addmf(a,'output',1,'NM','trimf',[-3*f3,-2*f3,0]); a=addmf(a,'output',1,'NS','trimf',[-3*f3,-1*f3,1*f3]); a=addmf(a,'output',1,'Z','trimf',[-2*f3,0,2*f3]); a=addmf(a,'output',1,'PS','trimf',[-1*f3,1*f3,3*f3]); a=addmf(a,'output',1,'PM','trimf',[0,2*f3,3*f3]); a=addmf(a,'output',1,'PB','smf',[1*f3,3*f3]); rulelist=[1 1 1 1 1; %Edit rule base 1 2 1 1 1; 1 3 2 1 1; 1 4 2 1 1; 1 5 3 1 1; 1 6 3 1 1; 1 7 4 1 1; 2 1 1 1 1; 2 2 2 1 1; 2 3 2 1 1; 2 4 3 1 1; 2 5 3 1 1; 2 6 4 1 1; 2 7 5 1 1; 3 1 2 1 1; 3 2 2 1 1; 3 3 3 1 1; 3 4 3 1 1; 3 5 4 1 1; 3 6 5 1 1; 3 7 5 1 1;


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4 1 2 1 1; 4 2 3 1 1; 4 3 3 1 1; 4 4 4 1 1; 4 5 5 1 1; 4 6 5 1 1; 4 7 6 1 1; 5 1 3 1 1; 5 2 3 1 1; 5 3 4 1 1; 5 4 5 1 1; 5 5 5 1 1; 5 6 6 1 1; 5 7 6 1 1; 6 1 3 1 1; 6 2 4 1 1; 6 3 5 1 1; 6 4 5 1 1; 6 5 6 1 1; 6 6 6 1 1; 6 7 7 1 1; 7 1 4 1 1; 7 2 5 1 1; 7 3 5 1 1; 7 4 6 1 1; 7 5 6 1 1; 7 6 7 1 1; 7 7 7 1 1]; a=addrule(a,rulelist); %showrule(a) % Show fuzzy rule base a1=setfis(a,'DefuzzMethod','mom'); % Defuzzy writefis(a1,'fuzzf'); % save to fuzzy file "fuzz.fis" which can be % simulated with fuzzy tool a2=readfis('fuzzf'); disp('-------------------------------------------------------'); disp(' fuzzy controller table:e=[-3,+3],ec=[-3,+3] '); disp('-------------------------------------------------------'); Ulist=zeros(7,7); for i=1:7 for j=1:7 e(i)=-4+i; ec(j)=-4+j; Ulist(i,j)=evalfis([e(i),ec(j)],a2); end end Ulist=ceil(Ulist)


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figure(1); plotfis(a2); figure(2); plotmf(a,'input',1); figure(3); plotmf(a,'input',2); figure(4); plotmf(a,'output',1);


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%A7. Genetic Algorithms For f( PK , iK , dK ) Optimization (Fuzzy Logic

Controller Output Gain Optimization) %Genetic Algorithm for function f(x1,x2,x3) optimization %Parameters Size=20 G=10 CodeL=10; umax1=8; umin1=2; umax2=3; umin2=-3; umax3=3; umin3=-3; E=round(rand(Size,3*CodeL)); %Initial Code %Main Program for k=1:1:G time(k)=k; for s=1:1:Size m=E(s,:); y1=0;y2=0;y3=0; %Uncoding m1=m(1:1:CodeL); for i=1:1:CodeL y1=y1+m1(i)*2^(i-1); end x1=(umax1-umin1)*y1/1023+umin1; m2=m(CodeL+1:1:2*CodeL); for i=1:1:CodeL y2=y2+m2(i)*2^(i-1); end x2=(umax2-umin2)*y2/1023+umin2; m3=m(2*CodeL+1:1:3*CodeL); for i=1:1:CodeL y3=y3+m3(i)*2^(i-1); end x3=(umax3-umin3)*y3/1023+umin3; kp=x1; ki=x2; kd=x3; sim ('F16block_pitch_contr_fuzzynewPID' , [0 5]); h(s)=PI(5001) F(s)=1/h(s) end Ji=1./(F+1e-10); %****** Step 1 : Evaluate BestJ ******


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BestJ(k)=min(Ji); fi=F; %Fitness Function [Oderfi,Indexfi]=sort(fi); %Arranging fi small to bigger Bestfi=Oderfi(Size); %Let Bestfi=max(fi) BestS=E(Indexfi(Size),:); %Let BestS=E(m), m is the Indexfi belong to max(fi) bfi(k)=Bestfi; %****** Step 2 : Select and Reproduct Operation****** fi_sum=sum(fi); fi_Size=(Oderfi/fi_sum)*Size; fi_S=floor(fi_Size); %Selecting Bigger fi value kk=1; for i=1:1:Size for j=1:1:fi_S(i) %Select and Reproduce TempE(kk,:)=E(Indexfi(i),:); kk=kk+1; %kk is used to reproduce end end %************ Step 3 : Crossover Operation ************ pc=0.60; n=ceil(30*rand); for i=1:2:(Size-1) temp=rand; if pc>temp %Crossover Condition for j=n:1:30 TempE(i,j)=E(i+1,j); TempE(i+1,j)=E(i,j); end end end TempE(Size,:)=BestS; E=TempE; %************ Step 4: Mutation Operation ************** %pm=0.001; %pm=0.001-[1:1:Size]*(0.001)/Size; %Bigger fi, smaller Pm %pm=0.0; %No mutation pm=0.1; %Big mutation for i=1:1:Size for j=1:1:3*CodeL temp=rand; if pm>temp %Mutation Condition if TempE(i,j)==0 TempE(i,j)=1; else TempE(i,j)=0; end end end end


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%Guarantee TempPop(30,:) is the code belong to the best individual(max(fi)) TempE(Size,:)=BestS; E=TempE; end Max_Value=Bestfi BestS x1 x2 x3 figure(1); plot(time,BestJ); xlabel('Times');ylabel('Best J'); figure(2); plot(time,bfi); xlabel('times');ylabel('Best F');


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%A8. Newton Raphson Method for Calculating the time required for the second stage (using Optimal Guidance Law) (XY Direction) %Newton Raphson Method (Optimal Trajectory XY) x0 = input(' Input the initial guess value (time required) x0 '); epsilon = input(' Input the tolerance epsilon '); k=0 x(1)=x0 e=1 while abs(e) > epsilon k=k+1 f=(((20000+450*x(k))^2+6000*6000)^(0.5))/502-1000/502-x(k) f1=((((20000+450*x(k))^2+6000*6000)^(-0.5))*(20000+450*x(k))*450/502) -1 x(k+1)=x(k)-f/f1 e=x(k+1)-x(k) end fprintf(' The approximating value of the root (tf) is %8.5f\n',x(k+1)); fprintf(' The number of iteration is %d \n',k);


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%A9. Newton Raphson Method for Calculating the time required for the second stage (using Optimal Guidance Law) (XZ Direction) %Newton Raphson Method (Optimal Trajectory XZ) x0 = input(' Input the initial guess value (time required) x0 '); epsilon = input(' Input the tolerance epsilon '); k=0 x(1)=x0 e=1 while abs(e) > epsilon k=k+1 f=(((20000+450*x(k))^2+6000*6000)^(0.5))/502-1000/502-x(k) f1=((((20000+450*x(k))^2+6000*6000)^(-0.5))*(20000+450*x(k))*450/502) -1 x(k+1)=x(k)-f/f1 e=x(k+1)-x(k) end fprintf(' The approximating value of the root (tf) is %8.5f\n',x(k+1)); fprintf(' The number of iteration is %d \n',k);


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Appendix B

B1. The overall implementation methodology (Flow chart) B2. LQ pitch and lateral-directional controller integration (Simulink model) B3. Proposed transformation for mapping the tanker position to fighter aircraft body axes (Simulink model) B4. Proposed transformation subsystem (stage1, stage2) tanker to fighter aircraft body axes at NED (Simulink model) B5. LQR Fuzzy logic control integration for pitch motion control (Simulink model) called by GA A5 and A7 B6. Integrated system for the tracking stage controller utilizing LQR Fuzzy logic controller integration and proposed transformation with fuzzy logic speed controller B7. LQRIFL controllers for pitch and lateral aircraft motions B8. Alignment stage controller (terminal phase) using the principle of the flight formation control B9. Complete system description for the aerial refueling autopilot (3 phases design)


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Publications 1-Ahmed Momtaz, Galal Hassan “Development of a neurofuzzy control

system for the guidance of air to air missiles”, 5th Cairo university

conference in mechanical design and production, Cairo, 2004.

2-Ahmed Momtaz, Han Chao “ LQR integrated with fuzzy logic control for

aerial refueling autopilot” CADDM Journal, Beijing, 2007.

3-Ahmed Momtaz, Han Chao “Design of fighter flight formation for aerial

refueling racetrack mission with drogue-hose configuration” CADDM

Journal, Beijing, 2007.

4-Ahmed Momtaz, Han Chao “Intelligent learning of fuzzy logic controller

via adaptive genetic algorithm applied to aircraft longitudinal motion”

ASAT-12, Military Technical College, Cairo, 2007

5-Ahmed Momtaz, Han Chao “Development of fuzzy logic LQR control

integration for aerial refueling autopilot” ASAT-12, Military Technical college,

Cairo, 2007


i

Operating point Alt. Mach

Trimming Algorithm

Linearization Algorithm

State Space Models

LQR Algorithm (SAS)

Gain Matrix Flying Qualities Verification

Integration with (FLC type PID)

Lateral-Directional motion

GA using different types of PI For both longitudinal and lateral motions step response

PID parameters Membership Functions

Rule Base Strength

Tracking Controller during Tanker tracking phase (second stage)

Design 3 stage transformation to map the tanker position to the

receiving A/C wind axes

Optimal Guidance Law algorithm

Alignment stage controller to align the receiving A/C with the tanker C/L

Total mission time and the final relative errors

Verification 1-Design constrains 2-Flying qualities 3-Time minimization compared with diff. x

Longitudinal motion

Final states

Initial states

State Space Models plus desired compensator structure

LQ Algorithm (CAS)

Gain Matrix

Zero states Initial states

Aerial refueling Procedure


ii

Rule error error rate p i d Firing Strength 1 NB NB PB NB PS 0.89736 2 NB NM PB NB NS 0.89736 3 NB NS PM NM NB 0.89736 4 NB Z PM NM NB 0.89736 5 NB PS PS NS NB 0.89736 6 NB PM Z Z NM 0.89736 7 NB PB Z Z PS 0.89736 8 NM NB PB NB PS 0.85494 9 NM NM PB NB NS 0.85494

10 NM NS PM NM NB 0.85494 11 NM Z PS NS NM 0.85494 12 NM PS PS NS NM 0.85494 13 NM PM Z Z NS 0.85494 14 NM PB NS Z Z 0.85494 15 NS NB PM NB Z 0.65068 16 NS NM PM NM NS 0.65068 17 NS NS PM NS NM 0.65068 18 NS Z PS NS NM 0.65068 19 NS PS Z Z NS 0.65068 20 NS PM NS PS NS 0.65068 21 NS PB NS PS Z 0.65068 22 Z NB PM NM Z 0.67087 23 Z NM PM NM NS 0.67087 24 Z NS PS NS NS 0.67087 25 Z Z Z Z NS 0.67087 26 Z PS NS PS NS 0.67087 27 Z PM NM PM NS 0.67087 28 Z PB NM PM Z 0.67087 29 PS NB PS NM Z 0.98392 30 PS NM PS NS Z 0.98392 31 PS NS Z Z Z 0.98392 32 PS Z NS PS Z 0.98392 33 PS PS NS PS Z 0.98392 34 PS PM NM PM Z 0.98392 35 PS PB NM PB Z 0.98392 36 PM NB PS Z PB 0.66334 37 PM NM Z Z PS 0.66334 38 PM NS NS PS PS 0.66334 39 PM Z NM PS PS 0.66334 40 PM PS NM PM PS 0.66334 41 PM PM NM PB PS 0.66334 42 PM PB NB PB PB 0.66334 43 PB NB Z Z PB 0.95689 44 PB NM Z Z PM 0.95689 45 PB NS NM PS PM 0.95689 46 PB Z NM PM PM 0.95689 47 PB PS NM PM PS 0.95689 48 PB PM NB PB PS 0.95689 49 PB PB NB PB PB 0.95689


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v


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x

Tanker position to receiving aircraft Wind axes Trans. Tracking stage

KC-135 Tanker

Relative distances

calculations Alignment stage

F-16 Receiving Aircraft

Controllers switch

Pitch controller Alignment stage

Lat. controller Alignment stage

Speed controller Alignment stage

Pitch controller Tracking stage

Lat. controller Tracking stage

Speed controller Tracking stage

SAS long. motion

SAS Lat. motion

SAS speed controller

Optimal Trajectory

Design Constraint

Reference Command

=0

1

2

3

Velocity vector and

GPS Positions

GPS Positions

Euler angles

Relative rotations with respect to fighter

wind axes

Relative distances with respect to

Tanker body axes

Output states feedback

GPS Positions

B9


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