Similar to shehabi - A Classical and Fuzzy Logic Control Design and Simulation of a Longitudinal Landing Autopilot [ MSc. thises - Cairo Univ. 1995] English V.
Similar to shehabi - A Classical and Fuzzy Logic Control Design and Simulation of a Longitudinal Landing Autopilot [ MSc. thises - Cairo Univ. 1995] English V. (20)
shehabi - A Classical and Fuzzy Logic Control Design and Simulation of a Longitudinal Landing Autopilot [ MSc. thises - Cairo Univ. 1995] English V.
1. i
Cairo University
Faculty of Engineering
Aerospace Department
A Classical and Fuzzy Logic Control Design and
Simulation of a Longitudinal Landing Autopilot
Thesis Submitted To the Department of Aerospace Engineering ,
Faculty of Engineering ,
Cairo University,
In Fulfillment of The Degree of
Master of Science
By
ABDUL GHAFOOR ABDUL MAJEED AL-SHEHABI
Aleppo University
Syria
Under The Supervision of
Prof. Dr. Sayed D. Hassan
Prof. Dr. Mohammed B. Argoun Dr. Gamal M. El-Bayoumi
1995
2. ii
Acknowledgments
There are many that have helped me to write this thesis in many
deferent ways .
I wish to express my deep appreciation to Porf. Dr. Sayed D. Hassan
whose course in fuzzy control and automatic flight control systems
provided some of the basic concept on which much of the analysis in this
thesis is based .
Also I wish to thank Prof. Dr. Mohammed B. Argoun who made
many helpful suggestions and corrections and painstakingly reviewed and
corrected the final manuscript .
I would like to express my thanks for many useful comments and
suggestion provide by Dr. Gamal M. El-Bayoumi who reviewed this thesis
during the various stages of its development .
Finally , I would like to thank my Parents and my wife for supporting me
during this thesis
Abdul Ghafoor AL-Shehabi
1995
3. iii
Abstract
During the past several years, application of fuzzy set theory and
fuzzy logic in control has emerged as one of the most active and fruitful
areas of research , especially in the realm of industrial processes which lack
quantitative data regarding the input-output relation . Fuzzy logic is a
system which is closer in spirit to human logical thinking and natural
language than conventional logic .
The fuzzy logic controller based on fuzzy logic provides a means of
converting a linguistic control strategy based on expert knowledge into an
automatic control system . The main problems of static fuzzy controllers
are the derivation of the control rule base and the fixed structure of the
controller .
A detailed study of the controller structure and design parameters is
performed with applications to many systems to investigate the controller
behavior and how it is compared to conventional techniques .
As one of the aerospace control systems , an automatic landing
control system for fighter aircraft is considered in this thesis . Modeling of
the aircraft was performed , longitudinal equations of motion were used to
design an automatic landing control system . The problem is treated in this
thesis for two points of view . The first considers the system as a one-input
one-output system . The second considers the system as a two-input two-
output system .
4. iv
Finally, conventional and fuzzy control techniques are used to
design both glide slope control system and automatic flare control system .
5. v
Table of Contents
Acknowledgments i
Abstract ii
Table of Contents iv
Abbreviation & Symbols viii
CHAPTER ONE : Introduction
1.1 : Objectives of the thesis 1
1.2 : Scope of the work 6
CHAPTER TWO : Equations of Motion
2.1 Introduction 9
2.2 Translational motion 10
2.3 Angular motion 16
2.4 Orientation of the aircraft “Kinematics equations” 21
2.5 Position of the airplane ‘“Navigation equation” 23
2.6 Round earth equation 24
2.7 Flat earth equation 24
2.8 Nonlinear model of the aircraft 26
2.9 Equations of motion in wind Axes 28
2.9-1 Wind Axes force equations 29
2.9-2 Wind Axes moment equations 31
2.10 Decoupling of the nonlinear equations 34
2.11 Linearization of the equations of motion in wind Axes 36
2.11-1 Linearization of the force equations 37
2.11-2 Linearization of the moment equations 41
2.11-3 Linearization of the kinematics equations 45
2.12 Decoupled linearized equations 46
2.13 Conclusions 50
6. vi
CHAPTER THREE : Conventional Automatic Landing Control System
3.1 Introduction 51
3.2 Glide slope coupler and automatic flare geometry 53
3.3 Aircraft Dynamics 58
3.4 SISO conventional landing control systems 59
3.4-1 Pitch attitude control system 60
3.4-2 Automatic velocity control system 68
3.4-3 SISO glide slope control system 71
3.4-4 SISO Flare control system 79
3.5 MIMO automatic landing control systems 82
3.5-1 Glide slope control system 82
3.5-2 Flare control system 88
3.6 Conclusions 92
CHAPTER FOUR : Fuzzy logic control systems
4.1 Introduction 93
4.2 Linguistic variables 94
4.3 Fuzzy logic 96
4.3-1 Fuzzy sets 96
4.3-2 Definitions 96
4.3-3 Fuzzy set operation 97
4.3-4 Fuzzy relations 98
4.4 Design procedure for fuzzy logic control system 99
4.5 Knowledge Base 100
4.5-1 Data base 100
4.5-2 Rule base 104
4.6 Fuzzification 111
4.7 Design making logic 111
7. vii
4.8 Defuzzification 112
4.8-1 Center of area defuzzification 113
4.8-2 Middle of maxima defuzzification 114
4.9 Conclusions 114
CHAPTER FIVE : SISO fuzzy landing control systems
5.1 Introduction 115
5.2 SISO fuzzy controller structure 116
5.3 Pitch attitude control system 118
5.4 Glide slope fuzzy control system 123
5.5 Flare fuzzy control system 128
5.6 Velocity fuzzy control system 132
5.7 Conclusions 134
CHAPTER SIX (6) MIMO fuzzy landing control systems
6.1 Introduction 135
6.2 MIMO fuzzy control systems 135
6.3 MIMO glide slope control system 147
6.4 MIMO flare control system 151
6.5 Conclusions 155
CHAPTER SEVEN (7) : CONCLUSIONS
APPENDICES
Appendix ( A ) : Aixs coordinate system 158
Appendix ( B ) : Numerical linearizatitin equations of motion 170
Appendix ( C ) : Aircraft model 177
8. viii
Abbreviations & Symbols
ECI Earth center inertial
NED North East Down
ABC Aircraft body coordinate
P
r
The position vector of cg. of A/C in ECI frame
P
r
& Absolute velocity of A/C in ECI frame
zyx ppp ,, The components of ECI position vector
zyx ppp ,, The components of P
r
accounting the effect of Earth’s oblateness
NEDP
r
& Velocity vector in the NED geographic frame
DEN ppp ,, Are the components of position vector in NED frame
r
r
Position vector of element of mass in ABC frame
zyx ,, Axes are aligned, respectively, forward starboard and down
abcV
r
Absolute velocity of aircraft cg in ABC frame
BBB zyx ,, Axes of the aircraft-body coordinate (ABC)
BV
r
Relative velocity vector of aircraft cg. with respect to air mass
wvu ,, Are the components of BV
r
&
Bω
r
Absolute angular velocity in body axes
rqp ,, Are the body-axes angular rates of the aircraft Bω
r
SV
r
Velocity vector in Stability Axes
SSS zyx ,, Axes of the stability coordinate system
WV
r
Velocity vector in wind Axes
WWW zyx ,, Axes of the wind coordinate system
Wω
r
Angular velocity vector in wind axes
WWW rqp ,, Are the wind-axes angular rates of the aircraft
9. ix
GV
r
The velocity vector of A/C in the locally level geographic frame
GGG zyx ,, Axes of the locally coordinate system
Eω
r
Absolute angular velocity vector of Earth’s rotation in ECI frame
xω Earth’s rotation rate
pF , aF , gF Propulsion forces, aerodynamic forces and gravitational forces
BF
r
, BT
r
Net applied force and net applied torque
zyx FFF ,, Components of BF
r
NML ,, Rolling moment, pitching moment and yawing moment
BH
r
The angular momentum vector of the aircraft in (ABC) frame
zyx HHH ,, components of BH
r
WT
r
The net torque in wind axes
WWW NML ,, Components of torque in wind axes
Φ
r
Attitude vector of the aircraft
ψθϕ ,, Roll angle, pitch angle and yaw angle
α , β , γ Angle of attack, sideslip angle and Flight path angle
Γ Deviation from flight path angle
g Earth’s gravitational acceleration
g′ Earth’s gravitational acceleration for a point on the surface
og′ Earth’s gravitational acceleration at sea level and latitude 45o
321 ,, ggg The components of the gravity vector
GM Earth-mass gravitational constant
mδ An element of mass
m Mass of the aircraft
m& Change of the mass of the aircraft
T Transform matrix from ECI frame to ABC frame
10. x
αT Transform matrix from body axes to stability axes
βT Transform matrix from stability axes to wind axes
BWT Transform matrix from body axes to wind axes
GT Transform matrix from inertial frame to geographic frame
J The inertia matrix
xxJ Moment of inertia about x-axis
xyJ Cross-product of inertia
L , D , Y Lift forces, drag forces and sideforce
tF , tM Are the components of thrust
ux
rr
, state and input vector
ee ux , Equilibrium values of parameters
uxx ∇∇∇ ,, & A row vector of first partial derivative operators
l Celestial angle
aret δδδδ ,,, Thrust . Elevator, rudder and ailron angle
A , B , E Jacobean matrices
q Free stream dynamic pressure
S Wing reference area
b Wing span
c Wing mean geometric chord
FLC Fuzzy logic controller
MFLC Multi-variables fuzzy logic controller
µ Geocentric latitude or Membership function
19. 9
Chapter(2)
EQUATIONS OF MOTION
2.1 INTRODUCTION
In this chapter the equations of motion for the aircraft are derived .
The chapter proceeds as follows . In sections (2.2) and (2.3) respectively
the vector equations of translational motion and of angular motion are
derived in body axes . The attitude of the aircraft relative to the fixed frame
are described in section (2.4) and the equations that govern the orientation
of the aircraft are derived . The flight path represented by the equations of
the position of the aircraft are derived in section (2.5) . In section (2.6) all
of equations derived in previous sections are collected in matrix form
which represents the so-called round earth equations .
In the section (2.7) we consider the surface of the earth as inertial reference
frame to derive the flat earth equations . The vector equations of motion are
expanded in section (2.8) to give the nonlinear equations of motion of the
aircraft . There consist of forces, moment, kinematics and navigation
equations . Equations of motion in wind axes are derived in section (2.9-1)
for force equations and in section (2.9-2) for moment . Decoupling of the
nonlinear equations of motion in wind axes is described in section (2.10) .
All wind equations are linearized in section (2.11) using small disturbance
theory and the numerical linearization program found in Appendix ( B ) .
The separation of the linear equations in wind axes is done in section (
2.12) .
20. 10
The equations of motions are derived by applying Newton’s second
law of motion . To make this application the following assumptions are
made :
1- The mass of the aircraft remains constant during any particular
dynamics analysis . The amount of fuel consumed is neglected .
2- The aircraft is a rigid body . The distance between any points on the
aircraft do not change in flight, its motion can considered to be of six
degrees of freedom .
The form of the equations of motion depends upon the choice of axis
system . We can not apply Newton’s second law of motion directly to the
airplane, because the time rate of change of linear and angular momentum
are referred to an absolute or inertial reference frame while the forces and
velocities with which we are concerned are all related to a coordinate
system which is fixed to the airplane and moving with it . This frame is
called Aircraft Body Coordinate (ABC) frame which is centered at the
center of aircraft, it is aligned as follows : Bx -forward, By -starboard and
Bz - down . It is a non-inertial reference frame . Therefore it is necessary to
perform a transformation from the moving frame of reference to the one
that is fixed ( relative to the Earth ) .
2.2 TRANSLATIONAL MOTION
The sum of external forces has three components, the propulsion,
aerodynamic forces and the gravitational attraction . The propulsion system
produces reaction forces on the aircraft body, while the aerodynamic forces
are produced by the relative motion with respect to the air . These forces
21. 11
depend on the shape of the aircraft body and the orientation of the aircraft
with respect to the air flow . The aerodynamic forces are defined in terms
of dimensionless coefficients, the flight dynamics and the reference area as
follows :
sqCF ii ..= (2.2-1)
Where iC is the dimensionless coefficients
q is free-strim dynamic pressure
s is wing reference area
The Newton’s second law of motion state that the summation of all external
forces acting on the body must be equal to the time rate of change of its
linear momentum . The time rate of change is taken with respect to inertial
space . This law can be expressed by the vector equation
∑ = abc
I
vm
dt
d
F
r
.. (2.2-2)
where I subscript indicates that the derivative must be take with respect
to an inertial frame .
absV
r
is the absolute velocity of aircraft cg in ABC frame
The net force applied on the aircraft in ABC frame is
∑ ++= gp FFFF a (2.2-3)
where
Fg = T.m. g
where T is a transform matrix from Earth Centred Inertial (ECI) to ABC
gmTFF ap ..++ = abs
I
Vm
dt
d r
. (2.2-4)
)( PTVV EBabs
rrrr
×+= ω (2.2-5)
22. 12
where BV
r
is the relative velocity of aircraft cg with respect to air mass
PE
rr
×ω is the absolute velocity of surrounding air at the position of cg
of aircraft . This position indicated by the vector
r
P which is the
position vector of aircraft cg in ECI frame
Eω
r
is the absolute angular velocity of the earth
Substitute in (2.2-5) and let apB FFF +=
)]([... PTV
dt
d
mVmgmTF EB
I
abcB
rrrr
&
r
×++=+ ω (2.2-6)
where m& is the change of mass of aircraft . This term is quite significant
in the case of missiles but for aircraft it is negligible, and
therfore, will be dropped from the equation .
)]([.. PTV
dt
d
mgmTF EB
I
B
rrrr
×+=+ ω (2.2-7)
)]()([ PTVVm EBBB
r
&rrrr
& ×+×+= ωω
)]()([.
1
PTVVgTF
m
EBBBB
r
&rrrr
&
r
×+×+=+ ωω (2.2-8)
Rearranging, we get :
)()(.
1
PTVgTF
m
V EBBBB
r
&rrrrr
& ×−×−+= ωω
)()]([
1
BBEBB VPgTF
m
V
rrr
&rrr
& ×−×−+= ωω (2.2-9)
where P
r
& is the absolute velocity of aircraft in ECI frame .
Then the absolute velocity of the aircraft cg in ABC frame is calculated
from the relationship :
PTVabs
r
&
r
.= (2.2-10)
abs
T
VTP
rr
& .= (2.2-11)
23. 13
Substitute from (2.2-5) in(2.2-11)
)]([ PTVTP EB
T
rrrr
& ×+= ω
PVTP EB
T
rrrr
& ×+= ω (2.2-12)
Substitute in(2.2-9) we have
)]([).(
1
PgTVTF
m
V EEBEBBB
rrrrrrrr
& ××−+×+−= ωωωω (2.2-13)
where g is the gravitational acceleration vector . The gravitational force
acting on the mass of the aircraft is given by
2
.
r
mGM
g = (2.2-14)
where GM is the Earth mass gravitation constant equal
=1.4076431E16 ft3 3
/ sec.
The direction of the gravitational acceleration vector is aligned along the
ECI position vector i.e, pointing towards the center of the earth . Thus, the
gravitational acceleration for unit mass can be written as
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=
z
y
x
P
P
P
P
GM
g .3
(2.2-15)
The component 2
P
GM
gives the correct magnitude for g and the remaining
denominator component P serve to reduce the P
r
vector to unit length .
We note that the gravitational attraction is inversely proportional to the
square of
r
P because of the inverse square law, and with the latitude angle
of P
r
. Because of the nonspherical shape of the Earth, the magnitude of g
at the sea level varies from 32.199 2
.sec/ft at the equator to 32.257 2
.sec/ft
at the poles .
24. 14
To account for the effect of the Earth’s oblateness on g, the components
zyx PPP ,, of ECI position vector are replaced by zyx PPP ,, , respectively which
are given by :
)]sin.53.()(.5.11[ 22
2 λ−+=
P
r
JPP E
xx
)]sin.51.()(.5.11[ 22
2 λ−+=
P
r
JPP E
yy (2.2-16)
)]sin.51.()(.5.11[ 22
2 λ−+=
P
r
JPP E
zz
where 2J is the gravitational harmonic constant .
From equation (2.2-13) we find that the last term )( Pg EE
rrrr
××− ωω
contains the Earth gravitational acceleration and the centripetal acceleration
due rotating in the ECI frame with the Earth’s angular vector Eω
r
where ;
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
0
0
x
E
ω
ω
r
,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
z
y
x
P
P
P
P
r
where ; xω is the Earth rotation rate, xω =7.29E-5 rad/sec .
Since the vector P
r
lies in a plane parallel to the Earth’s Equatorial plane
and moving with the Earth
PP
x
xEE
rrrr
.
00
00
000
)(
2
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=××
ω
ωωω
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=
z
yx
P
P
0
.
2
ω (2.2-17)
25. 15
For this condition we note that )( PEE
rrr
×× ωω corresponds to the centriptal
acceleration for a circular orbit at a prependicular distance of 22
zx PP +
from the Earth’s Axis .
Hence for a point on the surface of Earth we can write
)( Pgg EE
rrrrr
××−≡′ ωω (2.2-18)
where g
v
′ is commonly called the gravity vector ,which is accurately normal
to the surface and it less magnitude than g .
Substituting from (2.2-18) into (2.3-13) we find
gTVTF
m
V BEBBB
′+×+−=
rrrrr
& .).(
1
ωω (2.2-19)
The term BEB VT
rrr
×+ ).( ωω is a tangential acceleration component resulting
from the combination of the total angular velocity and the translational
velocity where ET ω
r
. is the component of the total angular velocity which is
usually neglectable compared to the body axes angular rates of an aircraft,
thus ;
)(.
1
BBBB VgTF
m
V
rrrrr
& ×−′+= ω (2.2-20)
2.3 ANGULAR MOTION
The torques acting on the airplane are generated due to the
displacement of the aerodynamic control surfaces, or due to any reaction
control thrust and also due to any components of the engine thrust not
acting through the cg .
26. 16
Newton’s second law of motion states that the summation of the external
moments acting on the airplane equals the time rate of change of the
moment of momentum ( angular momentum ) . The time rate of change of
both linear and angular momentums is referred to an absolute or inertial
reference frame, this law can be expressed by the vector equation
I
B
B
dt
Hd
T
r
r
= (2.3-1)
where BH
r
is the angular momentum vector of rigid body
BT
r
is the net torque acting about the aircraft’s cg
B subscript indicates that the moments and moment
of momentum acting in ABC frame .
To determine the angular momentum vector consider an element of mass
δm with position vector r
r
in ABC frame, the tangential velocity can be
expressed by the cross product as follows
rv B
rrr
×= ωtan (2.3-2)
Then the translational moment resulting from this tangential velocity of the
element of mass can be expressed as
mrM B δωδ ).(
rr
×= (2.3-3)
The moment of momentum is the moment times the lever arm as in the next
vector equation
mrrH BB δωδ )]([
rrrr
××= (2.3-4)
but ∫= BB HdH
rr
when integrating the component of BH
r
δ over the entire mass
mrrH BB δω )]([
rrrr
××∫= (2.3-5)
If we express the angular velocity and the position vector as
27. 17
krjqipB
rrrr
... ++=ω
kzjyixr
rrrr
... ++=
Then
zyx
rqp
kji
r
rrr
rr
=×ω
Ssubstitute in equation (2.3-5) it becomes
mrzxqyxpzyiH δ]....)[( 22
−−+= ∫
rr
mpyxrzyqxzj δ]....)[( 22
−−++ ∫
r
(2.3-6)
mqzypzxryxk δ]....)[( 22
−−++ ∫
r
The scalar components of H
r
are
mqzympzxmryxH
mpyxmrzymqxzH
mrzxmqyxmpzyH
z
y
x
δδδ
δδδ
δδδ
]..[]..[])[(
]..[]..[])[(
]..[]..[])[(
22
22
22
∫∫∫
∫∫∫
∫∫∫
−−+=
−−+=
−−+=
but mzx δ)( 22
+∫ is defined to be the moment of enertia asbout x-axis and
doneted by xxJ , and ∫ myx δ).( is defined to be the product of inertia and
denoted by xyJ
Substituting these definitions into the angular moment we can write BH
r
as
the vector matrix product
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
=
r
q
p
JJJ
JJJ
JJJ
H
zzzyzx
yzyyyx
xzxyxx
B
r
(2.3-7)
BJ ω
r
.= (2.3-8)
where J is called the inertia matrix of the rigid body .
By differentiation the angular moment according to Newton’s law we find
28. 18
BBBB THH
rrrr
& =×+ ω (2.3-9)
Assuming that the inertia matrix constant although it changes in an abrupt
manner if the aircraft is releasing stores such dropping equipment, and
changes in a gradual manner as fuel is used up then
BBBB TJJ
rrrr
& +×−= ).(. ωωω
BBBB TJJJ
rrrr
& .)].([ 11 −−
+×−= ωωω
BBBB TJJJ
rrrr
& ... 11 −−
+×−= ωωω (2.3-10)
This is a basic vector equation for angular motion, it relates the derivatives
of the body axes angular rates p, q, r .
The inverse of the inertia matrix 1−
J can be simplified as in the next form
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=−
333231
232221
131211
1
kkk
kkk
kkk
J
where
∆
−
=
2
11
. yzzzyy JJJ
k
∆
+
=
zzxyzxyz JJJJ
k
..
12
∆
−
=
yyzxyzxy JJJJ
k
..
13
∆
−
=
2
22
. zxxxzz JJJ
k
∆
−
=
xxyzzxxy JJJJ
k
..
23
∆
−
=
2
33
. xyyyxx JJJ
k
and
222
......2.. xyzzzxyyyzxxzxyzxyzzyyxx JJJJJJJJJJJJ −−−−=∆
Usually we don’t need neither the full inertia matrix nor its inverse , Also
we note that the set of aircraft Axes can be chosen such that the cross
products of inertia are zero . This means that the inertia matrix is diagonal,
so the Axes which make the cross product of the inertial matrix is zero are
29. 19
called the principle Axes . This kind of Axes is not necessary because most
of the aircraft’s have symmetric x-z plane . For this condition every cross
product ijJ equal jiJ in magnitude but opposite in sign except cross
product xzJ which is non zero .
For symmetric aircraft’s the inertia matrix and its inverse reduce to the next
form
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
=
zzxz
yy
xzxx
JJ
J
JJ
J
0
00
0
,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
Γ
Γ
=−
xxxz
yy
xzzz
JJ
J
JJ
J
0
00
0
11
(2.3-11)
where
2
. xzzzxx JJJ −=Γ (2.3-12)
Substitute in (2.3-10) and let
[ ]T
B rqp=ω
r
and let the applied torque vector to be
[ ]T
B NMLT =
r
we find
NJLJrqJJJJqpJJJJp xzzzxzyyzzzzzzyyxxxz ...])([.][ 2
+++−−+−=Γ&
MrpJrpJJqJ xzxxzzyy +−+−= )(.)(. 22
& (2.3-13)
NJLJqpJJJJrqJJJJr xxxzxzyyxxxxzzyyxxxz ...])([.][ 2
+++−++−−=Γ&
These equations are highly coupled and non linear because the angular rate
vector occurs twice.
If we consider the body of the aircraft with two planes of symmetry, then
all cross products of inertia are zero, and the moment equations then
reduced to the gyroscopic equations .
30. 20
xx
zzyy
J
rqJJ
p
.)( −
=&
yy
xxzz
J
rpJJ
q
.)( −
=& (2.3-14)
zz
yyxx
J
qpJJ
r
.)( −
=&
2.4 ORIENTATION OF THE AIRPLANE
The attitude of the airplane can not described relative to a moving
body axis frame . It must be defined in terms of a fixed frame of reference .
To describe the attitude of the aircraft with respect to inertial space, it is
necessary to specify the attitude of one axis system with respect to the
other . The orientations of the airplane can be described by three
consecutive rotations, whose order is important . The angular rotations
produce what is called the Eular angles .
The orientations of the body frame with respect to the fixed frame can be
determined as follows .
Imagine the airplane to be positioned so that the body axis system is
parallel to the fixed frame . When we consider the NED frame as a
reference frame then the Eular angles ϕ , θ and ψ specify the orientation
of the aircraft axis system with respect to the reference frame NED, and the
orientation of the body frame with respect to NED frame can be determined
in the following manner .
Starting from reference frame
1- Rotate about the Gz Axis ,nose right (positive “yaw” ψ )
31. 21
G
GG
GG
zz'
.cosy.sinxy'
.siny.cosxx'
=
+−=
+=
ψψ
ψψ
2- Rotate about y’ Axis, nose up (positive “pitch” θ )
θθ
θθ
cos'.sin'.''
'''
sin'.cos'.''
zxz
yy
zxx
+=
=
−=
3-rotate about x’’ Axis, right wing down (positive “roll”ϕ)
ϕϕ
ϕϕ
cos'.'sin'.'
sin'.'cos'.'
''
zyz
zyy
xx
B
B
B
+−=
+=
=
In matrix form
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
=
G
G
G
B
B
B
z
y
x
.
100
0cossin
0sincos
.
cos0sin
010
sin0cos
.
cossin0
sincos0
001
z
y
x
ψψ
ψψ
θθ
θθ
ϕϕ
ϕϕ (2.4-1)
The relation between Angular velocities in the body frame p, q and r, and
the Eulare rates ϕ& , θ& and ψ& can be determined from Figure (2.4-1)
XG
ZG
YG
x1
y1
z1
y2
z2
x2 XB
ZB
YB
32. 22
Figure( 2.4-1) Sketch of fixed aircraft Axes
ϕθϕθψ
ϕθψϕθ
θψϕ
sin.cos.cos.
sin.cos.cos.
sin.
&&
&&
&&
−=
+=
+=
r
q
p
(2.4-2)
In matrix form
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
ψ
θ
ϕ
ϕθϕ
ϕθϕ
θ
&
&
&
.
cos.cossin0
sin.coscos0
sin01
r
q
p
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
r
q
p
.
cos
cos
cos
sin0
sincos0
tan.costan.sin1
θ
ϕ
θ
ϕ
ϕϕ
θϕθϕ
ψ
θ
ϕ
&
&
&
Bf ω
rrr
& ).(Φ=Φ (2.4-4)
θ
ϕ
θ
ϕ
ψ
ϕϕθ
θϕθϕϕ
cos
cos
cos
sin
.
sin.cos.
tan.cos.tan.sin.
rq
rq
rqp
+=
−=
++=
&
&
&
By integrating the above equations, we can determine Eular angles .
2.5 POSITION OF THE AIRPLANE ( flight path )
33. 23
We wish to have the coordinates of the flight path relative to the
fixed frame, in the ECI frame the position of the aircraft cg will be denoted
by the inertial position vector )(tP
r
and the rotation matrix that takes vector
from ECI frame to ABC frame is T(t) which is an orthogonal matrix
).(
.
pTvv
vTp
EBabs
abs
T
rrrr
rr
&
×+=
=
ω
)].(.[ pTvTp EB
T rrrr
& ×+= ω
pvTp EB
T rrrr
& ×+= ω. (2.5-1)
2.6 ROUND EARTH EQUATION
The state variables of the aircraft module will be the three
components of each of inertial position vector P
r
, the relative velocity
vector Bv
r
, the angular velocity vector Bω
r
, and the Eular angles rates, a
complete state model consist of forces, moments, position and attitude
equations, these equations will be written in matrix form which a vector-
cross product operation ×Bω
r
( ) and ×Eω
r
( ) replaced by the matrix BΩ
and EΩ respectively as the next form
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
Φ⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Φ
Ω−
Ω+Ω−Ω−
Ω
=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Φ
−−
0
.
0
0)(00
0..00
00).(..
00
11
2
B
B
B
B
B
EBE
T
E
B
B
TJ
m
F
v
p
f
JJ
TTgT
T
v
p
r
r
r
r
rr
&
r
&
r
&
r
&
ωω
( 2.6-1)
Where
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=Ω
00
00
000
x
xE
ω
ω
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
=Ω
0
0
0
PQ
PR
QR
B
xω is the earth rotation rate .
34. 24
We note that these equations are non linear because the coefficient matrix
contains T, and BΩ which are function of the state variables, also the
gravitational acceleration is a function of the inertial position vector
according to (2.2-18) .
2.7 FLAT EARTH EQUATIONS
For these equation we consider NED on the surface of the earth as
inertial reference frame although it is accelerating and rotating we will
neglect the acceleration associated with the Earth rate compared with that
can be produced by the maneuvering of the aircraft . Also most of vehicles
are limited to altitude below 100000 ft which is considered small distance
compared with the Earth radius which is equal to about 12 million feet, the
magnitude of the inertial position vector
r
P will be constant and equal to the
Earth radius . Therefore the vector g
r
and centripetal acceleration
][ pEE
rrr
××ωω in (2.2-13) can be replaced by the g
r
′, which it’s magnitude at
the sea level and latitude 450
is og′ =32.17 ft/sec2
. The position is then
tracked by resolving Bv
r
vector into the NED frame and integrating these
components to obtain three NED position states .
The term BEB vT
rrr
×+ ).( ωω is a tangential acceleration component resulting
from the combination of the total angular velocity and the translational
velocity . Where ET ω
r
× is the component of the total angular velocity which
is usually neglectable compared to the body axes angular rates of an aircraft
and its cross product with Bv
r
is also negligible for normal speeds then
the flat earth equations obtained from (1.3-6) can be rewritten simply as
)(.
1
BBoBB VgTF
m
V
rrrr
& ×−′+= ω
35. 25
BBBB TJJJ
rrr
& .... 11 −−
−Ω−= ωω (2.7-1)
Bg ω
rr
& ).(Φ=Φ
B
T
vTP
rr
& .=
Then the state vector is
],,,[ TTT
B
T
B
T
Pvx
rrrrr
Φ= ω (2.7-2)
2.8 NON LINEAR MODEL OF THE AIRCRAFT
By expanding the vector of flat Earth equations of motion, and using
the aerodynamic knowledge, we will describe how aerodynamic forces and
moments can be incorporated into these equations
The elements of the state vector consists of :
The components of the velocity vector Bv
r
The components of the angular rate vector Bω
r
The components of the Eular Angles Φ
The components of the position vector P
r
Then the state vector are
],,,,,,,,,,,[ DEN
T
ppprqpwvux ψθϕ=
r
(2.8-1)
where hpD = in NED frame .
We will use the constants iI in the nonlinear equations of motion, which
defined by
2
2
1
.
)(
xzzzxx
xzzzyyzz
JJJ
JJJJ
I
−
−
= 22
.
)(
xzzzxx
yyxxzzxz
JJJ
JJJJ
I
−
−+
=
23
. xzzzxx
zz
JJJ
J
I
−
= 24
. xzzzxx
xz
JJJ
J
I
−
=
yy
xxzz
J
JJ
I
−
=5
yy
xz
J
J
I =6
36. 26
yyJ
I
1
7 = 2
2
8
.
)(
xzzzxx
xxyyxxxz
JJJ
JJJJ
I
−
−−
=
29
. xzzzxx
xx
JJJ
J
I
−
= (2.8-2)
Γ=− 2
. xzzzxx JJJ (2.8-3)
When the matrix-vector products are multiplied out, then the standard set of
body Axes state equations are
m
Fz
gvpuqw
m
Fy
gwpurv
m
Fx
gwqvru
o
o
o
+′+−=
+′++−=
+′−−=
θϕ
θϕ
θ
cos.cos...
cos.sin...
sin...
&
&
&
(2.8-4)
NILIqrIpIr
MIrpIrpIq
NILIqpIrIp
..)...(
.)(..
..)...(
9428
7
22
65
4321
++−=
+−−=
+++=
&
&
&
(2.8-5)
θ
ϕϕ
ψ
ϕϕθ
ϕϕθϕ
cos
cos.sin.
sin.cos.
)cos.sin.(tan
rq
rq
rqp
+
=
−=
++=
&
&
&
(2.8-6)
θϕθϕθ
ψθϕψϕψθϕψϕψθ
ψθϕψϕψθϕψϕψθ
cos.cos.cos.sin.sin.
)sin.sin.coscos.sin()sin.sin.sincos.(cossin.cos.
)sin.sin.cossin.(sin)cos.sin.sinsin.cos(cos.cos.
wvuh
wvup
wvup
E
N
−−=
+−+++=
+++−+=
&
&
&
We know that the aerodynamic forces and moment depend on the
aerodynamic angles and true airspeed . Therefore we have to calculate
them.
37. 27
It is usually convenient to replace the state variable u, v, w by vt, α, β
where u, v and w are calculate from
W
T
B vTv
rr
.= (2.8-10)
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
0
0.
vt
T
w
v
u
T
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
βα
β
βα
cos.sin.
sin.
cos.cos.
vt
vt
vt
and vt, α, β are calculated from
u
w
=αtan (2.8-11)
vt
v
=βsin (2.8-12)
2
1
222
)( wvuvt ++= (2.8-13)
Then we can derive the following expression for new state derivatives
vt
wwvvuu
tv
&&&
&
... ++
= (2.8-14)
β
β
cos.
..
2
vt
tvvvtv &&& −
= (2.8-15)
22
..
wu
uwwu
+
−
=
&&
&α (2.8-16)
The new state vector is
],,,,,,,,,,,[ hpprqpvtx EN
T
ψθϕβα=
r
(2.8-17)
2.9 EQUATIONS OF MOTION IN WIND AXES
The wind axis equations of motion are derived from the body axis
equations of motion .
38. 28
The wind axes are the natural axes for the aerodynamic forces and change
their orientation relative to the aircraft body during flight .
Conversion of body axes equations to wind axes equations is to transform
the vector matrix equation using the body to wind rotation matrix BWT .
Using this approach we start from flat earth equation, pre multiply through
by BWT and make use of the property R
T
BWBW TT Ω=&. where RΩ is a cross
product matrix containing the relative an Eular rates of wind axes with
respect to the body axes, for simplicity the engine thrust vector will be
assumed to be parallel to the body x-axis, the body axes thrust force
component will be denote by Ft and the thrust moment by Mt
We will use the subscript w to denote wind-axes quantities
2.9-1 WIND AXES FORCE EQUATIONS
We know the vector force equation in the body axes are given by
vector equation
BBBB F
m
gTvv
rrrr
& .
1
. ++×= ω
gTvvF
m
BBBB ..
1
+×+=
rrr
&
r
ω (2.9-1)
Multiply both side by BWT we find
gTTvTvTF
m
T BWBBBWBBWBBW ..)(..
1
. −×+=
rrr
&
r
ω (2.9-2)
We can write
W
T
BWB vTv
rr
.=
Then
39. 29
BBWBBWBBBW vTTvT
rrr
..)( ×=× ωω
WWBBBW vvT
rrrr
×=× ωω )( (2.9-3)
also
)(.. BBWBBW v
dt
d
TvT
rr
& =
).(.. W
T
BWBWBBW vT
dt
d
TvT
rr
& =
)...(. W
T
BWW
T
BWBWBBW vTvTTvT
r
&
r&
r
& +=
WW
T
BWBWBBW vvTTvT
r
&
r&
r
& += ... (2.9-4)
Substitute from(2.9-3) and (2.9-4) into (2.9-2) we get
gTTvvvF
m
BWWWWRWW ....
1
−×+Ω+=
rrrr
&
r
ω
gTTvF
m
vv BWWWWWRW ...
1
. +×−=Ω+
rrrrr
& ω (2.9-5)
where
T
BWBWR TT &.=Ω
T T TBW = β α.
Then
T
R TTTTTT )...(. αβαβαβ
&& +=Ω
)...(. TTTT
R TTTTTT βαβααβ
&& +=Ω
TTT
R TTTTTT βααβββ .... && +=Ω
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
=Ω
0sin.cos.
sin.0
cos.0
βαβα
βαβ
βαβ
&&
&&
&&
R (2.9-6)
We know
41. 31
BBWBBWBBBW JTTJT ωωωω
rrrr
...)..( +=×
W
T
BWBWWBBBW TJTJT ωωωω
rrrr
...)..( +=× (2.9-11)
Also
).(.... W
T
BWBWBBW T
dt
d
JTJT ωω
rr
& =
)...(... W
T
BWW
T
BWBWBBW TTJTJT ωωω
r
&
r&
r
& +=
)........ W
T
BWBWW
T
BWBWBBW TJTTJTJT ωωω
r
&
r&
r
& += (2.9-14)
Substitute from (2.9-13) and (2.9-14) in (2.9-10) we find
=wT
r
W
T
BWBWW
T
BWBW TJTTJT ωω
r
&
r& ...... + W
T
BWBWW TJT ωω
rr
...++ (2.915)
Multiply the first term by BW
T
BW TT .
=wT
r
W
T
BWBWW
T
BWBW
T
BWBW TJTTTTJT ωω
r
&
r& ........ + W
T
BWBWW TJT ωω
rr
...++
Then we have
WWWWWWRWW JJJT ωωωω
rrr
&
rr
.... ×++Ω= (2.9-16)
WWWWWWWRW TJJJ
rrrr
& .).(. 11 −−
+×+Ω−= ωωωω (2.9-17)
where
T
BWBWW TJTJ ..= (2.9-18)
TT
W TTJTTJ βααβ ....=
where J is the inertial matrix and for symmetric aircraft, it is equal to
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
=
zzxz
yy
xzxx
JJ
J
JJ
J
0
00
0
(2.9-19)
let
T
S TJTJ αα ..= (2.9-20)
43. 33
Decoupling means that the equations of motion are separated into
two independent sets . One describes the longitudinal motion, and the other
set describes the lateral motion .
We note that the wind Axes force equations of motion will be considerably
simplified if either or both of the bank angle ϕ and the side slip angle β
are Zero .
If 0=ϕ and 0≠β then we have level skidding flight
If 0≠ϕ and 0=β then we have coordinated turning flight
If 0=ϕ and 0=β then we have non-sideslipping flight
We will consider the last case, because it leads to decoupling of flat Earth
equations of motion .
First we will consider that 0=ϕ then the g-component equation becomes
)sin(.cos.01 αθβ −′−= gg
)sin(.sin.02 αθβ −′= gg (2.10-1)
)cos(.03 αθ −′= gg
Also if we consider that 0=β , then the force equation are reduced to
γαα
β
γα
cos....sin...
....
sin..cos..
0
0
gmqvtmLFvtm
rvtmYvtm
gmDFvtm
WT
W
T
′++−−=
−=
′+−=
&
& (2.10-2)
where γ ⏐ αθβϕ −=== 0
For 0=β and qqw =
44. 34
Then the third equation can be written as
γαα cos..sin... 0gmLFvtm T
′−+=& (2.10-3)
From the above equations we note that the longitudinal equations have
become independent of the lateral equations .
Also from kinematics equations (2.8-6 ) If 0=ϕ . We have
q=θ& (2.10-4)
Also for the moment equations (2.8-5 ) and if 0≡≡ rp , we find that
the pitching moment equation is not coupled to the rolling and yawing
moment equations
MqJ yy =&. (2.10-5)
2.11 LINEARIZATION OF THE EQUATIONS OF MOTION IN
WIND AXES
45. 35
Study of nonlinear equations show that there are six simultaneous
non linear equations of motion to completely describe the behavior of a
rigid aircraft . In this form the solution can be obtained only by the use of
analog or digital computer or by manual numerical integration .
We can write the non linear equations of motion in wind axes in the form
0),,( =uxxfi
rrr
& I=0 ......n (2.11-1)
It can be obtained from wind axes equation by moving all nonzero term to
the right hand side of the equations .
If we consider the state vector as
],,,,,,,,[ ψθϕβα rqpvtx
T
i =
r
(2.11-2)
And the control vector as
],,,[ aretuT
δδδδ=
r
(2.11-3)
By applying small perturbations from steady state conditions x
r
, u
r
, and
then expanding the nonlinear state equations 0),,( =uxxfi
rrr
& using Tylor
series around the equilibrium point ),( ee ux and keeping only first order
terms, we find that the perturbations in states, state’s derivatives and
control vector must satisfy
0...... =∇+∇+∇ ufxfxf iuixix δδδ&& (2.11-4)
Where
],,.........,,[.
321 n
iiii
ix
x
f
x
f
x
f
x
f
f
∂
∂
∂
∂
∂
∂
∂
∂
=∇ (2.11-5)
These equations can be written in simplest linear state variable form as
uBxAxE
rrr
& ... += (2.12-6)
Where x and u are the perturbations from equilibrium values of the states
and control vector and the coefficient matercies are defined as
46. 36
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
nx
x
x
f
f
f
E
.
.
.
2
1
&
&
&
M
,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
nx
x
x
f
f
f
A
.
.
.
2
1
M
,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
nu
u
u
f
f
f
B
.
.
.
2
1
M
(2.11-7)
These matrices are called Jacobean matrices and must be calculated at the
equilibrium point .
We will evaluate the Jacobean matrices for steady ,level flight condition
with the additional constraint of no sideslip β=0 therefore the equilibrium
steady state condition are
0,,,, ≡rqpϕβ
all derivatives≡ 0 (2.12-8)
Another Jacobean matrices in different flight conditions can be obtained by
numerical linearization, see Appendix (B) .
2.11-1 LINEARIZATION OF THE FORCE EQUATIONS
Applying the steady state conditions on the force equations in wind
axes become
γαα
β
γα
cos....sin...
....
sin..cos..
0
0
gmqvtmLFvtm
rvtmYvtm
gmDFvtm
WT
W
T
′++−−=
−=
′+−=
&
& (2.11-9)
Writing every equation as is the next form
0),,( =uxxf
rrr
&
We find that the first equation for example can be written as
γα sin..cos..),,(1 gmDFtvmuxxf T +−+−= &
rrr
&
By consider small perturbations we have
47. 37
0111
=++ u
u
f
x
x
f
x
x
f
δ
∂
∂
δ
∂
∂
δ
∂
∂
&
&
(2.11-10)
Where
x
x
gm
x
x
D
x
x
F
x
x
tvm
x
x
f T
&
&
&
&
&
&
&
&
&
&
&
δ
∂
γ∂
δ
∂
∂
δ
∂
α∂
δ
∂
∂
δ
∂
∂ )sin..()()cos.().(1
+−+−=
x
x
gm
x
x
D
x
x
F
x
x
tvm
x
x
f T
δ
∂
γ∂
δ
∂
∂
δ
∂
α∂
δ
∂
∂
δ
∂
∂ )sin..()()cos.().(1
+−+−=
&
u
u
gm
u
u
D
u
u
F
u
u
tvm
u
u
f T
δ
∂
γ∂
δ
∂
∂
δ
∂
α∂
δ
∂
∂
δ
∂
∂ )sin..()()cos.().(1
+−+−=
&
Then
x
x
gmx
x
D
x
x
Fx
x
F
x
x
tv
mx
x
f
T
T &
&
&
&
&
&
&
&
&
&
&
&
&
δ
∂
γ∂
γδ
∂
∂
δ
∂
α∂
αδ
∂
∂
αδ
∂
∂
δ
∂
∂ )(
cos..
)()(
sin.
)(
cos
)(1
−−−+−=
x
x
gmx
x
D
x
x
Fx
x
F
x
x
tv
mx
x
f
T
T δ
∂
γ∂
γδ
∂
∂
δ
∂
α∂
αδ
∂
∂
αδ
∂
∂
δ
∂
∂ )(
cos..
)()(
sin.
)(
cos
)(1
−−−+−=
&
x
x
gmu
u
D
u
u
Fu
u
F
u
u
tv
mu
u
f
T
T δ
∂
γ∂
γδ
∂
∂
δ
∂
α∂
αδ
∂
∂
αδ
∂
∂
δ
∂
∂ )(
cos..
)()(
sin.
)(
cos
)(1
−−−+−=
&
Similarly , we find that the partial derivatives with respect to x
r
&
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇+∇
∇−∇
∇+∇
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
−
=
Lvtm
Yvtm
Dtvm
f
f
f
xx
xx
xx
xxx
x
x
e
&&
&&
&&
&
&
&
&
&
&
α
β
...
..
..
3
2
1
(2.11-11)
Also we find (2-12.12)
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇−∇′+∇+∇−∇−∇−
∇+∇′+∇−∇+∇−
∇−∇′−∇−∇+∇−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
).sin..(sin.....sin.cos.
).cos..(sin.....cos.
)(cos...cos.sin.
3
2
1
θγαγααα
ϕθβγβα
αθγααα
xxowxxxx
xxowxxx
xxoxxx
xxx
x
x
gmqvtmLFtFt
gmrvtmYþFt
gmDFtþFt
f
f
f
e
Where
αθγ xxx ∇−∇=∇
let
αsin.FtL −=
αcos.FtD −=
50. 40
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
T
T
T
A
A
A
B
B
B
B
B
B
B
B
B
B
N
M
L
N
M
L
N
M
L
T
r
TA BBB TTT
rrr
+= (2.11-18)
In wind Axes then we have
TA WWW TTT
rrr
+= ⇒
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
w
w
w
W
N
M
L
T
r
(2.11-19)
where
β
β
cos.
sin.
MtMM
MtLL
A
A
ww
ww
+=
+=
(2.11-20)
We shall consider only the variation of thrust with speed and the derivative
vt
Mt
∂
∂
is broken down into two components vM and TvM .
From the vector moment equation of motion
WWWWWWWRW TJJJ
rrrr
& .).(. 11 −−
+×+Ω−= ωωωω
Writing this equation as form 0),,( =uxxf
rrr
&
Then
=),,( uxxf
rrr
& WWWWWwWRW TJJJ
rrrr
& .).(. 11 −−
+×+Ω−− ωωωω (2.11-21)
From the first and second terms which are
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
w
w
w
w
r
q
p
&
&
&
r
&ω ,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
+−
−−
=Ω
βαβα
βαβ
βαβ
ω
sin..cos..
sin...
cos...
.
ww
ww
ww
wR
qp
rp
rq
&&
&&
&&
r
(2.11-22)
We note that there is no contibution to the linear equations for the steady
state nonturning flight conditions
0=== www rqp
51. 41
Only the first term contributes an identity block to the E matrix in p,q and r
columns
The next term ).(1
WWWW JJ ωω
r
×−
can also be discarded
The last term when substituting the steady state condition 0=β equals
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
Γ
=−
w
w
w
xxxz
yy
xzzz
WW
N
M
L
JJ
J
JJ
TJ
0
0
1
0
0
1
.1
r
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
+
Γ
=
wxxwxz
yy
w
wxzwzz
NJLJ
J
M
NJLJ
..
..
1
(2.11-23)
Substitute in (2.12-21) we find
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
+
Γ
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
wxxwxz
yy
w
wxzwzz
w
w
w
NJLJ
J
M
NJLJ
r
q
p
f
f
f
..
..
1
9
8
7
&
&
&
(2.11-24)
Then
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Γ
∇+∇
∇
Γ
∇+∇
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
wxxxwxxz
yy
wx
wxxzwxzz
x
x
x
NJLJ
J
M
NJLJ
f
f
f
....
.
....
09
8
7
β
(2.11-25)
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Γ
∇+∇
∇
Γ
∇+∇
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
wUxxwUxz
yy
wU
wUxzwUzz
U
U
U
NJLJ
J
M
NJLJ
f
f
f
....
.
....
09
8
7
β
(2.11-26)
We will assume that all derivatives equal to zero except the derivatives
shown in table (2.11-2) will be considered to be nonzero .
52. 42
Table (2.11-2) The Moments Dimentional Derivatives (wind Axes)
______________________________________________________________________
Roll Pitch Yaw
----------------------------------------------------------------------------------------------------
∂β
∂
β
Aw
xx
L
J
L
1
=
vt
M
J
M Aw
yy
v
∂
∂1
=
∂β
∂
β
Aw
zz
N
J
N
1
=
w
w
xx
p
p
L
J
L A
∂
∂1
=
∂α
∂
α
Aw
yy
M
J
M
1
=
w
w
zz
p
p
N
J
N A
∂
∂1
=
w
w
xx
r
r
L
J
L A
∂
∂1
=
α∂
∂
α
&
&
Aw
yy
M
J
M
1
=
w
w
zz
r
r
N
J
N A
∂
∂1
=
a
L
J
L Aw
xx
a
∂δ
∂
δ
1
=
w
w
yy
q
q
M
J
M A
∂
∂1
=
a
N
J
N Aw
zz
a
∂δ
∂
δ
1
=
r
L
J
L Aw
xx
r
∂δ
∂
δ
1
=
e
M
J
M Aw
yy
e
∂δ
∂
δ
1
=
r
N
J
N Aw
zz
r
∂δ
∂
δ
1
=
t
M
J
M Aw
yy
t
∂δ
∂
δ
1
=
∂β
∂
β
Tw
zz
T
N
J
N
1
=
vt
M
J
M T
V
w
yy
T
∂
∂1
=
∂α
∂
α
Tw
yy
T
M
J
M
1
=
----------------------------------------------------------------------------------------
Then
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
−
=
100000000
01000000
001000000
9
8
7
α&
&
&
&
M
f
f
f
exxx
x
x
(2.11-27)
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+++
+
++++
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
NrLrNpLpNL
MqMMtMv
LrNrLpNpMtNL
f
f
f
v
xxx
x
x
e
..0..0000..0
000000
..0..0000...0
9
8
7
µσµσβµβσ
α
µσµσνβσβµ
53. 43
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
++
++
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
=
= rLrNaLaN
eMtM
rNrLaNaL
f
f
f
e
e
uu
xxU
U
U
δσδµδσδµ
δδ
δσδµδσδµ
....00
00
....00
9
8
7
(2.11-29)
2.11-3 LINEARIZATION OF THE KINEMATICS EQUATIONS
We find that the kinematics equations in body Axes are given by the
next vector equation
Bg ω
rrr
& ).(Φ=Φ
For the wind Axes ,we can write
wBWB T ωω
rr
.1−
=
w
TT
B TT ωω βα
rr
..=
Substitute in kinematics body Axes
w
TT
TTg ωβα
rrr
& ..).(Φ=Φ (2.11-30)
Writing this equation as form 0),,( =uxxf
rrr
&
Then
=),,( uxxf
rrr
& w
TT
TTg ωβα
rrr
& ..).(Φ+Φ− (2.11-31)
We note that there is no aerodynamic force or moment involved in these
equations, therefore it is easy to see that the contribution to E matrix is
given by
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
−
000100000
000010000
000001000
6
5
4
f
f
f
x
x
x
&
&
&
(2.11-32)
Also we note that equation (2.12-31) is linear in wrand, ww qp , so all partial
derivatives of the coefficient marix elements will be eliminated when we
set
54. 44
0r= w == ww qp
Hence to evaluate the coefficient marix under steady state condition 0=β
Then the transformation T
Tβ reduces to identity matrix and equation(2.11-
31)
is given by
=),,( uxxf
rrr
& w
T
Tg ωα
rrr
& .).(Φ+Φ− (2.11-33)
where
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
+−+
=Φ
θ
ϕα
θ
ϕ
θ
ϕα
ϕαϕαϕ
αϕϕαϕθαϕθα
α
cos
cos.cos
cos
sin
cos
cos.sin
sin.coscossin.sin
cos.cos.tansinsin.tansin.cos.tancos
).( T
Tg
Then
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∇
∇
∇
e
e
e
e
e
e
e
e
x
x
x
f
f
f
θ
α
θ
α
θ
γ
θ
γ
cos
cos
0
cos
sin
000000
010000000
cos
sin
0
cos
cos
000000
6
5
4
( 2.11-35)
The partial derivatives of the kinematic variables with respect to the control
vector are all zero .
2.12 DECOUPLED LINEARIZED EQUATIONS
The motion of the aircraft in free flight can be extremely complicated
. the airplane has three translational motion, vertical, horizontal and
transverse and has three rational motion pitch ,yaw and roll and numerous
elastic degree of freedom .
55. 45
However we can make some simplifying assumptions which will reduce
the complicity of the problem . First we shall assume that the aircraft’s
motion consist of small deviations from its equilibrium flight condition .
Second we shall assume that the motion of airplane can be analyzed by
decoupling, which means that the equations of motion are separated into
two independent sets, one set describes the longitudinal motion pitching
and translational in the x-z plane and the other set describe the lateral -
directional motion rolling, sideslipping and yawing of the airplane .
The thrust moment will be retained in the longitudinal equations, for
simplicity the effect will be drooped from lateral equations, the decoupling
also occurs in the linear small perturbation equations and the reduced order
of the two sets greatly simplifies the study of the dynamic model of an
aircraft.
2.12-1 LONGITUDINAL EQUATIONS OF MOTION
The longitudinal states and control are respectively
[ ]qvtxT
θα=
r
[ ]etuT
δδ=
r
The longitudinal equations are obtained from the first and last row of (2.11-
15 ), (2.11-16 ) and (2.11-17 ) divided by m, and the middle rows of (2.11-
32), (2.11-35), and the middle rows of (2.11-27), (2.11-28), (2.11-29) .
Thus the longitudinal coefficient matrices are given by
57. 47
The lateral states and control are
[ ]ww
T
rpx ϕβ=
r
[ ]rauT
δδ=
r
Where the state Ψ has been deposed
The state equation are obtained from the second rows of (2.11-15), ( 2.11-
16) and (2.11-17), the first rows of (2.11-32) and (2.11-35), and the first
and the second rows of ( 2.11-27), (2.11-28) and (2.11-29) .
Thus the lateral coefficient matrices are given by
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1000
0100
0010
000vt
E
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
ra
ra
ra
NN
LL
YY
B
δδ
δδ
δδ
00
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡ −′
=
rp
rp
e
e
e
e
rpe
NNN
LLL
vtYYgY
A
0
0
cos
sin
cos
cos
00
cos.0
β
β
β
θ
γ
θ
γ
θ
2.13 CONCLUSIONS
In this chapter we have derived the equations of motion of an an
aircraft in different sets of coordinates, also we have described how the
aerodynamic forces and moments acting on an aircraft and we have related
these forces and moments to the equations of motion of a rigid aircraft .
Steady-state flight conditions have been defined . It has been shown that
the equations of motion can be linearized around a steady-state condition
and that they can be separated into two decoupled sets . One of these sets
59. 49
chapter(3)
AUTOMATIC LANDING SYSTEM
USING CONVENTIONAL CONTROL METHODS
3.1 INTRODUCTION
Automatic landing system is used to reduce the pilot’s workload, and
to guide the aircraft to a safe landing in poor visibility and in all weather
operations . This system will guide the aircraft down a predetermined glide
slope, and then at a preselected altitude will reduce the rate of descent and
cause the aircraft to flare out and touch down with an acceptably low rate
of descent .
The design of glide slope coupler, an automatic flare control system
and altitude hold control system have one thing in common, that is, the
control of flight path angle γ . The automatic control of flight path angle
without control of air speed either manual or automatic is impossible .
Hence the automatic control of trajectory requires simultaneous control of
engine thrust and pitch attitude . This is because using control elevator
only to attempt to gain altitude will result in a loss of speed and an eventual
stall .
In landing condition the aircraft operates on the back side of the
power curve as shown in Figure ( 3.1-1 ) . This region is to the left of the
minimum of the power required . We note that opening the throttle
produces an increase in altitude, while it decreases speeds . The speed is
60. 50
then controlled by the elevator . This means that the throttle is used to
control altitude ( increasing the power causes a gain in altitude ), while the
elevator is used to control airspeed (down elevator causes a gain in speed )
[9] .
Figure (3.1-1) power curve
Some researchers solve the automatic landing problem as single-input
single-output system [10], in this case, the automatic landing system is
assumed to consist of three SISO systems, which are glide slope coupler,
automatic flare control and velocity control system . Others solve the
problem as a multi-input multi-output system [9] . In this case, the
automatic landing system consists of two MIMO systems, namely ; glide
slope coupler, and automatic flare control system . The design of all of
these systems is performed using conventional control technology .
In this chapter we will design automatic landing control systems
using conventional control methods . The chapter proceeds as follow . The
flight path of the aircraft included both of glide slope and flare geometry is
61. 51
described in section (3.2) . In section (3.3) Jacobean matrices are derived
using numerical linearization . The details of the numerical linearization
algorithm are given in appendix ( B ) . Section (3.4) presents the design of
SISO conventional control systems for pitch attitude, flight speed, glide
slope and flare control for automatic landing system . In both sections
(3.5) and (3.6) MIMO control systems are designed using pole placement
technique for both glide slope coupler and automatic flare control system
.
3.2 GLIDE SLOPE COUPLER AND AUTOMATIC FLARE
GEOMETRY
Glide slope geometry :
Glide slope coupler is the first stage of an automatic landing system
. In this stage the control system uses a radio beam directed upward from
the ground at 2 50
. to 3 50
. . Equipment onboard the airplane measure the
angle deviation Γ from the beam and compute the perpendicular
displacement of the aircraft from the glide path . Denoting the
perpendicular displacement as d we have
Γ= sin.Rd (3.2-1)
Where
Γ is the difference between the actual and the desired glide path
angle.
R is the radial distance of the aircraft from the glide slope transmitter
d is the deviation of the aircraft from the glide path . It is the normal
distance of the aircraft above or below the desired glide path .
62. 52
The geometry of glide slope, for which the reference trajectory has angle
0
5.2−=rγ as shown in Figure (3.2-1) .
d
R vt
station
cg of A/C
2.5 Ground
C.L. of
glide slope
o
Γ
γr
γ
Figure (3.2-1) Glide Slope Geometry
If the aircraft is below the centerline of glide slope, then d is considered
negative, as is γ , when the velocity vector is below the horizon, that is, the
aircraft is descanting [10].
The component of tv perpendicular to the glide slope center line is d& and is
given by
)sin( rtvd γγ −=& (3.2-2)
where 0
5.2−=rγ
)5.2sin( 0
+= γtvd&
for small angle 0
5.2<γ
)5.2( += γtvd& (3.2-3)
d& is positive if 0
5.2+γ positive, and as d initially was negative the aircraft
is approaching the glide slope from below .
Integrating equation (3.2-3) we find
63. 53
)5.2(
3.57
0
+= γ
s
v
d t
(3.2-4)
The automatic landing equipment does not measure the perpendicular
distance to the glide slope center line, but measure the angular error and the
range R and calculate d . Thus for a given value of d the angular error
increases as the aircraft nears the station, which has the effect of increasing
the system gain as the range to the station decreases as shown in Figure
(3.2-1) . From this Figure, we have
R
d
=Γtan (3.2-5)
or for small angles
R
d3.57
=Γ (3.2-6)
where Γ is given in degrees .
The glide slope intercept at about 1500 ft altitude with an airspeed about
250 ft / sec. . Then an automatic control system will be engaged and the
aircraft will descend with :
1- constant flight path angle 0
5.2−=rγ
2- constant airspeed vt = 250ft / sec.
3- pitch attitude between −50
and 50
4- rate of descend 10 ft/sec. or greater
Flare Geometry :
As the aircraft gets very close to the runway threshold, the glide path
control system disengages and the flare maneuver is executed . This means
that the flare is the final phase of landing of the aircraft . This maneuver
64. 54
occurs at an altitude between 20 and 70 ft above the end of the runway .
The automatic landing system must start to reduce the rate of descent of the
aircraft . To achieve the correct pitch attitude for landing and begin to
reduce the airspeed . The flare maneuver is required to decrease the vertical
rate of descent to a level consistent with the ability of landing to dissipate
the energy of the impact at landing .
The trajectory shown in Figure (3.2-2) represents the path of the
aircraft wheels as the landing is carried out . During this flare maneuver,
the flight path angle of the aircraft has to change from -2.50
to the
positive value recommended for touch down . In other words, during the
flare maneuver the control system must control of the high of the center of
gravity and it’s rate of change such that the resulting trajectory corresponds
as nearly possible to the idealized exponential path during the flare .
C.L of
Glide slope
ho
2.5
Flare path
Glide slope transmitter
Touchdown
2500 ft.
o
x
γr =
Figure (3.2-2) Automatic flare geometry
In the final phase of landing, the aircraft is commanded to fly at an
exponential path from the initial flare until touch down, thus
τ
t
o ehh
−
= . (3.2-7)
where h is the high above the runway
65. 55
oh is the high at the start of flare
τ is the flare time constant
Differentiating Equation (2.2-7) yields
τ
τ
t
o
e
h
h
−
−= .& (3.2-8)
So that the commanded altitude should obey the differential equation
r
h
h +−=
τ
& (3.2-9)
τ and oh are chosen for desired flare characteristics, oh can be determined
from equation (3.3-25) if τ is known, which at the initial of flare t=0 is
τ
o
o
h
h −=& (3.2-10)
The distance from ho to the point of touchdown depends on the value of ho ,
the flare entry height, and the approach speed of the aircraft, tv . Usually
the point of touchdown, which is aimed for, is 2500 ft from the runway
threshold which is the nominal location of the glide path transmitter.
Assuming that the airspeed does not change significantly through the flare
trajectory then the rate of descent at the beginning of the flare is.
.sec/91.10
3.57
5.2
250sin ftvh to −=−≈= γ& (3.2-11)
From Figure (3.2-2) the value of distance from oh to the glide slope
transmitter location is given by
τ
)5.2tan(
t
o
v
h
x == (3.2-12)
xxh o
o 0435.0)5.2tan( == (3.2-13)
From equations (3.2-10) and (3.2-13) we find
x0435.091.10 =− τ
66. 56
x
91.10
0435.0
=τ
The desired value of τ can be obtained by specifying the distance to the
touch down point from the glide slope transmitter . If it is desired to touch
down 2500 ft beyond the glide slope transmitter and if we assume that h
vanishes in τ4 second then
25004 += xvt τ (3.2-14)
25004 += ττ vtvt (3.2-15)
25003 =τtv (3.2-16)
so that .sec333.3=τ
then
ftho 38.36333.3*91.10 ==
hence the ideal flare maneuver is assumed to take
r
h
h c
c +−=
333.3
&
rhh cc +−= 3.0& (3.2-17)
3.3 AIRCRAFT DYNAMICS
In our problem we will design an automatic landing control system
for longitudinal dynamics of fighter aircraft described in Appendix (C) [9] .
The longitudinal dynamics of the aircraft were linearized about the
equilibrium values of the next parameters
cg=0.2 .sec/250 ftvt = 0
5.0=α .sec/3.0 radq −= 0
5.2−=θ 40=po
51.0=tδ fth 750= other parameters are zero’s
67. 57
The model can be obtained by linearizing the equations of motion about
equilibrium values or using directly the linearization program, given in
Appendix ( B ) .
vt α q θ po
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−−−
−−−−−−
−−−
=
9999.00000
00100
00815.0302.14090.2
5293.13734.6909.0535.04837.5
372.0125.32121.0159.373830.6
E
EEE
E
A
δt δe
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
−−
=
094.64
00
2553.40
3040.10
3577.10
E
E
E
B (3.3-1)
3.4 SISO CONVENTIONAL LANDING CONTROL SYSTEMS
To guide the aircraft down toward the runway, the aircraft must be
guided laterally and vertically . The localizer beam is used to position the
aircraft on a trajectory so that the aircraft will intercept the center line of
runway . To be able to land the aircraft without any visual aid to the
runway, it requires an Automatic landing system that can intercept the
localizer and glide path signals, then guide the aircraft down the glide path
to some preselected altitude at which the aircraft’s descent rate is reduced
and the aircraft executes a flare maneuver so that it touches down with an
acceptable sink rate . In a completely automatic landing, the autopilot
guides the aircraft all the way down to touchdown and flare .
68. 58
As the aircraft descends along the glide path, its pitch attitude and
speed must be controlled . This can be accomplished by means of pitch
displacement and speed control autopilot . So the autoland system must
comprise a number of automatic control systems which include a localizer
and glide slope coupler, attitude and airspeed control, and an automatic
flare control system .
3.4-1 PITCH ATTITUDE CONTROL SYSTEM
This autopilot is normally used only when the aircraft is in wings
level flight . The controlled variable is θ, ( αγθ += ), and the sensor is an
attitude reference gyro ( which provides an error signal proportional to the
deviation from a preset orientation in inertial space ) . The controller does
not hold the flight path angle, γ , constant because the angle of attack
changes with flight conditions .
Thus, if the thrust is increased, the angle of attack will tend to
decrease and the aircraft will climb, and as aircraft weight decreases ( fuel
is burned ), angle of attack will decrease, also causing a gradual climb, a
preset climb will gradually level out as decreasing air density causes α to
increase .
The pitch attitude hold autopilot is not very important in its own
right .Therefore, in the case of the jet aircraft it is desired to increase the
damping of short period oscillation, this can be accomplished by adding an
inner feedback loop utilizing a rate gyro . However, the same feedback
69. 59
configuration is used in inner loops in other autopilots, such as altitude hold
and automatic landing . Figure (3.4-2) is a blogdigram of such a system
-
+ ue θc eθ δe
θ
AIRCRAFT
DYNAMICS
ELEVATOR
SERVO
AMPLIFIER
RATE
GYRO
VERTICAL
GYRO
θ
.
Figure (3.4-2) Displacement autopilot with pitch rate feedback for damping
Figure (3.4-3) show the block diagram for the aircraft when pitch rate is
incorporated into the design . For this problem we have two parameters to
select, namely Kq and Kθ . The root locus method can be used to pick both
parameters . The procedure is essentially a trial and error one . First, the
root locus plot is determined for the inner loop . The gyro gain is selected
and then the outer loop root locus plot is constructed . Several iterations
may be required until the desired overall system performance is achieved .
K -10
S+10
1
S
K
q q
-
++
-
ue θc
q
eθ δe
δeθ
θ
Figure (3.4-3) Blockdiagram of the attitude hold with pitch rate feedback
The design of pitch attitude control system can be performed using short
70. 60
period approximation of the aircraft dynamic described in section (3.3), we
get a truncated 3rd model of the aircraft which includes only α, q and θ as
state variables .
[ ]eE
E
q
E
q δ
θ
α
θ
α
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−−
−−
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
0
255.4
304.1
010
0815.0302.1
373.6909.0535.0
&
&
&
The transfer function for this system is given by
)0711.16723.0)(005.0(
)505.0(s0455.0
iss
s
e
q
±++
+−
=
δ
(3.4-1)
We note that the elevator angle to pitch rate transfer function has a dc gain
of zero (because of the zero at the origin), indicating that a constant
elevator deflection will not sustain a steady pitch rate . If the pole s=-0.005
is canceled with the zero at the origin, a short period approximation for the
transfer function can be obtained as
612.13445.1
)505.0(0455.0
2
++
+−
=
ss
s
e
q
δ
(3.4-2)
This transfer function has a finite dc gain and shows that constant elevator
angle tends to produce constant pitch rate over an interval of time that is
short compared to the phugoid period .
We note that the root locus gain is negative, the sign of the elevator servo
tranfer function
eu
eδ
is made negative so that the forward transfer function
is positive . This is done so that a positive θr causes a positive change in θ ,
which will be helpfull for analyzing glide slope control system, so we
choose
10
10
+
−
=
su
e
e
δ
(3.4-3)
71. 61
Thus we have
)612.1s3445.12)(10(
)505.0(k455.0 q
+++
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ss
s
ole
u
q
(3.4-4)
The inner loop of pitch attitude control system is shown in Figure (3.4-4)
K
q
q
-
+
ue
Rq
-0.0455(s+o.505)
s^2+1.3445S+1.612
qδ e
S+10
-10
Figure(3.4-4)Blockdiagram of the inner loop of pitch attitude control
system
The root locus plot for inner loop as the rate gyro sensitivity is increased
for zero is shown in Figure (3.4-5)
-15
-10
-5
0
5
10
15
20
ImagAxis
Figure (3.4-5) root locus for inner loop
72. 62
Selection of the rate gyro gain
As seen from the root locus for inner loop, we note that there is a range of
rate gyro sensitivities . We have to find the value of rate gyro sensitivity
that gives the best overall system performance after closing the outer loop .
Indeed there is no simple rule to aid us in selecting the final sensitivity, but
the final choice usually results by trial and error . We have to find the
value which gives us a rapid response with smaller overshoot . After
choosing more than one value for qK and from root locus we note that the
higher gain of the inner loop requires higher gains for the outer loop . The
final choose of qk and θk is depended on the simulation program of the
aircraft which we have considered the approximate values of both qk and
θk whose give us the minimum error between θ and cθ .
For the closed loop indicated in Figure (3.4-4), we chose 75=qK
The rate gyro sensitivity was selected so that the complex poles would lie
on the branches approaching the 900
asymptote with damping ratio ζ ≥
0.823
we chose a high damping ratio because the outer loop will tend to decrease
the damping of the complex roots .
The resulting closed loop transfer function of the inner loop is given by :
37.33125.4934.11
)505.0(455.0
23
+++
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
sss
s
r
q
clq
(3.4-5)
Figure(3.4-6) shows the transient response of the aircraft with pitch rate
feedback for 75=qk , for step input of qr .
73. 63
Figure (3.4-6) Transient response of the aircraft for step input
Using the closed loop poles from the root locus shown in Figure (3.4-5)
,the block diagram for outer loop can be drawn as the seen in Figure (3.4-
7).
K 1
S
0.455(S+0.505)
S^3+11.34S^2+49.05S+33.37
q
-
+ Rq
ecθ
θ
θ
θ
Figure (3.4-7) Block diagram for the outer loop of the pitch control
system
74. 64
The root locus for outer loop is shown in Figure (3.4-8)
where
s37.33s125.4934.11
)505.0(455.0
234
+++
+
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ss
sK
e ol
θ
θ
θ
(3.4-6)
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
ImagAxis
Figure (3.4-8) root locus for outer loop
From root locus we chose 130=θk
After choosing 75=qK and 130=θK simulation using of the complete
three degree of freedom longitudinal equations of motion was done, for
which
9.29s52.92s125.49s34.11
)505.0(15.59
234
++++
+
=
s
s
cθ
θ
(3.4-7)
The response of the aircraft alone for a step input with no velocity control
is illustrated in Figure (3.4-9)
75. 65
Figure (3.4-9) Transient response of pitch attitude control system
with pitch rate feedback .
3.4-2 AUTOMATIC VELOCITY CONTROL SYSTEM
The automatic velocity control system is the same as that have been
used in both glide slope and flare control systems . The block diagram of
velocity control system that could be used to maintain a constant speed
along the flight path is shown in Figure (3.4-10) . The difference in flight
speed is used to produce a proportional displacement of engine throttle so
that the speed difference is reduced .
76. 66
If we consider vt as the change in airspeed, then the input to the
aircraft dynamics is the change in throttle from that required for straight
and level flight .
10s+1
1+ s A/C
vtrv=0 0.999 0.2
S+0.2
64.94Kc
S+0.999
-
+
powerDtutev
yv
Figure (3.4-10) Block diagram for velocity control system
The automatic velocity control system consists of proportional plus
integral (PI) network in order to make the system Type 1, a compensetor
that adds a Zero at s = −0 9999. .
Using long period approximation with α = 0, we get :
[ ]t
po
q
vt
EE
E
op
q
tv
δ
θθ
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−−−−
−−−
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
94.64
0
0
0
9999.0000
0010
5293.10815.0409.2
37219.0125.2312146.03830.6
&
&
&
&
We will use velocity and change of velocity in feedback path, then the
aircraft transfer function is
0067.00123.08273.08217.1
6885.191577.24
234
2
++++
+
=
ssss
ss
t
vt
δ
(3.4-9)
Also from the model of the aircraft, we have
tpoop δ94.649999.0 +=&
77. 67
)9999.0(
94.64
+
=
st
po
δ
(3.4-10)
Since
po
t
t
vt
po
vt δ
δ
.=
For the servo
2.0
2.0
+
=
su
t
t
δ
(3.4-11)
s
s
e
u
v
t 9999.0+
=
Then the open loop transfer function for the automatic velocity control
system is
)0067.0s123.08287.08217.1)(2.0(
)6885.191577.24)(9999.0)(110(2.0
234
+++++
+++
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ssss
sssKc
r
y
olv
v
(3.4-12)
-1.5
-1
-0.5
0
0.5
1
1.5
2
ImagAxis
Figure (3.4-11) root locus for open loop for varying kc
78. 68
From root locus we find that the system is always open loop stable .
Increasing Kc results an increase overshoot and decrease settling time . By
trial and error we find that a value of Kc=0.08 is suitable .
Figure (3.4-12) transient response of change of vt for SISO
3.4-3 SISO GLIDE SLOPE CONTROL SYSTEM
79. 69
To maintain the aircraft along the glide path, one must make Γ equal
to zero . The block diagram of an autopilot that will keep the aircraft on the
glide path is shown in Figure(3.4-13) . The transfer functions for d and Γ
are obtained form the geometry of the glide path .
From equation (3.2-1) we note that the sensitivity of d to flight path
change will depend on the range R, so an automatic control system can be
designed for some nominal value of R .
A/c and
autopilot
coupler
R
57.3vt
57.3S
+
e
2.5
o
-
γ
θ
Γc
Γ Γ
=0
θ θc γ
Figure (3.4-13)Block diagram of glide slope coupler
The glide slope coupler contains
- Elevator Servo
eu
eδ
;
10
10
+
−
=
su
e
e
δ
- Transfer function
θ
γ
;
This transfer function is required to convert θ output from the
aircraft transfer function to γ , and can be derived from the relation
αθγ −= .
θ
α
θ
γ
−=1 (3.4-13)
80. 70
but
θ
δ
δ
α
θ
α e
e
.=
From the aircraft dynamic described in section (3.3) and using short period
approximation we get
0088.0s6195.1s35.1
0003.0s0422.0s0010.0
23
2
+++
−−−
=
seδ
α
(3.4-14)
0088.0s6195.1s35.1
023.0s0455.0
23
+++
−−
=
seδ
θ
(3.4-15)
Then
)505.0(
)92.3)(692.6(0219.0
+
+−−
=
s
ss
θ
γ
(3.4-16)
- Coupler ;
The coupler consists of a proportional plus integral circuit to obtain
zero steady state error and good transient response to make the system
Type 1 system . It also consists of the lead compensator pole / zero ratio of
10 to obtain tighter control of pitch attitude .
))(
1.0
1(10
1
1
ps
zs
s
Sc
e
c
+
+
+=
Γ
θ
(3.4-17)
The zero of the lead network is selected so that it cancels the closed loop
pole at s = −0 4. .
The factor Sc is included to provide a steady state gain of 1 for lead
circuit, and its value equals Sc = 10, and using the transfer functions in
(3.4-7), (3.4-16), (3.4-17), then the forward transfer function for a glide
slope control system can be obtained .
)016.156)(937.4)(4.0(
)505.0(15.59
.
)505.0(
)392.3)(692.6(0219.0
.
)4(
)4.0)(1.0(
.
.
250
.10)( 2
++++
+
+
+−−
+
++
=
Γ
Γ ssss
s
s
ss
ss
ss
Rs
Sc
r
OL
)0516.156)(937.4)(4(.
)392.3)(692.6)(1.0(3238
)( 22
++++
+−+
−=
Γ
Γ ssssRs
sssSc
r
OL
(3.4-18)
81. 71
ٍٍ3238Sc(S+.1)(S-6.692)(S+3.392)
S^2*R(S+4)(s+4.937)(
+
_
e
-
S^2+6S+15)
Γc Γ Γ
Figure (3.4-14)Block diagram of automatic glide slope control system
For which vt = 250ft / sec., the root locus for the glide slope control system
is shown in figure (3.4-15) .
-10 -5 0 5 10
-10
-5
0
5
10
Real Axis
ImagAxis
Figure (3.4-15) root locus for automatic glide slope control system
For root locus shown in Figure(3.4-15) we find that the range to the
station R, provide the varying gain . This mean that the dynamics of the
system change as the aircraft approaches the station . Then the root locus
can be used to determine the minimum range for stable operation, for this
can we note from root locus that the value of kt where the root locus
crosses the imaginary axis is kt = 29.16 =
R
cS3653
, which gives a value of
82. 72
R= 111.058 Sc
A value of Sc= 10 gives the minimum rang of R=1110 .
The transient response of automatic glide slope for step input is
illustrated in Figure (3.4-16)
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (secs)
Amplitude
Figure (3-4.15) Transient response of glide slope for step input
Actually the minimum value of R is the value of the range R at the end of
flight path of glide slope where automatic flare control system will engage .
If we suppose that the automatic flare control system engages at altittude
ft.36=oh , then the value of .825min ftR = , and the range of values of
]1110,825[=R makes the system unstable . Therefore we have to find the
value of Sc for .825min ftR = which the system is cretiecal stable .
83. 73
From root locus shown in figure(3.4-15) we choose 115=
Sc
R
then
174.7
115
825
==Sc ,the next figures show the time response of glide slope only
without velocity controller
Figure (3-4.16) Transient response of Γand d
Figure (3-4.17) Deviation from trim velocity
84. 74
Figure (3.4-18) Altittude of the aircraft
The time responses of the glide slope controller with velocity controller for
Kc= 0.002 , Sc= 7.174 and the minimum value of R= 825 ft. are shown
in Figures (3.4-19),(3.4-20),(3.4-21) and (3.4-22)
Figure (3.4-19) Deviation from trim velocity
86. 76
Figure (3.4-22) Altittude of the aircraft
3.4-4 SISO FLARE CONTROL SYSTEM
The block diagram for automatic flare control system is shown in
Figure (3.4-23) . Note that because pitch attitude control system is used,
changing θ results in a change in flight path angle, and consequently, a
change in height . Because the heights involved are very low, and accurate
measurement of height is necessary for this control system, a low range
altimeter is used . The control law used can be simply
hkcc
&.−=θ (3.4-33)
while, to ensure accuracy, it is usual to add an integral to the proportional
term so
]..[ 21 ∫+−= dthkchkcc
&&θ (3.4-35)
hkchkcc .. 21 −−= &θ (3.4-36)
87. 77
The addition of the integral term, and the need to remove by filtering, any
noise from the height signal obtained from the radio altimeter, tends to
destabilize the closed loop system . Consequently, it is customary to
include a phase advance network with the feedback term to improve the
stability, i.e.
h
ps
zs
skc
kc
kcc
&))(
1
.1(
1
2
1
+
+
+−=θ
then a coupler is
))(
1.0
1(10
1
1
ps
zs
s
Sc
h
c
+
+
+=
&
θ
(3.4-36)
It is the same as that for the glide slope control system . To obtain a higher
sensitivity of coupler additional lead must be added to the coupler, thus
preventing the aircraft from flying into the run way too soon,
))()(
1.0
1(10
2
2
1
1
ps
zs
ps
zs
s
Sccoupler
+
+
+
+
+=
A/CCOUPLER
& h
θ
h h&θ 1
s
1
τ
--
&hr=0 hr θceh
&
Figure (3.4-23)Block diagram of conventional flare control system
The second zero of the coupler is selected so that it cancels the closed pole
at s = - 4.937, thus the coupler transfer function is
)
49.37s
4.937s
)(
4s
0.4s
)(
s
0.1
(1Sc10coupler
+
+
+
+
+= (3.4-37)
88. 78
The outer loop simply supplies the rate of descent command rh& . The
transfer function for the aircraft and autopilot is the same as the one used
for the glide slope analysis as given in the equation
9.2952.92125.4934.11
)505.0(15.59
234
++++
+
=
ssss
s
cθ
θ
(3.4-38)
The transfer function
θ
h&
can be obtained from
θ
γ
γsintvh =&
for small γ we can write γvth =& , thus
θ
γ
θ 3.57
tvh
=
&
)505.0(
)92.3)(692.6(0955.0
+
+−−
=
s
ss
(3.4-39)
The open loop transfer function from (3.3-37), (3.3-38) and(3.3-39) is
given by
sssssss
sssssSc
e
h
olh
652.5904626.198669.14668744.495382.85171.64
482.45919.57495.131132.38137.2)(15.59)(0955.0(.10
234567
2345
++++++
−−−−+−
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
&
&
The root locus for the inner loop of automatic flare control system is shown
in Figure (3.4-24)
89. 79
-100 -50 0 50 100
-100
-80
-60
-40
-20
0
20
40
60
80
100
Real Axis
ImagAxis
Figure (3.4-24) root locus for inner loop of conventional flare control
If we choose from root locus K= 209.6
Then 209.6 = 56.48825 Sc ⇒ Sc = 3.710
Figure ( 3.4-) show the step response of conventional flare control system
0.2
0.4
0.6
0.8
1
1.2
Amplitude
Figure(3.3-19) transient response of conventional control system
90. 80
chapter (4)
FUZZY CONTROL SYSTEMS
4.1 INTRODUCTION
In designing a conventional control system the initial step is to obtain
a mathematical model for the plant, this model represents the formulation
of prior information into an analytic structure . Many real world systems
have unknown parameters or highly complex and nonlinear characteristics,
and thus do not fend themselves to this type of analysis . For this type of
systems fuzzy logic provides an attractive alternative . Since its
introduction by Zadeh [22] in 1973 . Many applications [1-8],[11] and [12]
of fuzzy control have been successfully performed since Zadeh introduced
the concept of fuzzy sets . They have been used to control ill defined or
complex linear and nonlinear systems .
A fuzzy logic control system can be considered as a control expert
system which simulates the human thinking in the interpretation of real
world data using fuzzy set labels, and performs an approximate reasoning
using the compositional rule of inference .
The main advantage of using fuzzy control approach is that no
mathematical model is needed . If the mathematical model is difficult to
obtain then a fuzzy model which consist of a set of linguistic fuzzy process
then can be employed . In general the fuzzy dynamic system modeling is
91. 81
based on the idea of determining a set of input-output relations
corresponding to the plant.
Conventional control theories can be divided into two classes , that is
; classical control theory whose objective is to minimize the control error ,
and modern control theory whose objective is to obtain a dynamic control
which optimize an integral type performance index . Both the two classes
of control theories are different in design concept and structure . Control
actions of most of fuzzy control system is a fuzzified PD , PI or PID
control action used in classical control theory .
This chapter proceed as follows . In Section (4.2) linguistic variables are
defined and in Section (4.3) definitions related to fuzzy sets as well as
fuzzy relations are presented . In Section (4.4) the design procedure for a
fuzzy logic control system is described . In Section (4.5) the components of
knowledge base, namely ; data base and rule base are explained . Section
(4.6) describes the fuzzification method . Decision making logic is
described in Section (4.7) and in Section (4.8) two defuzzification methods
are described (4.8) .
4.2 LINGUISTIC VARIABLES
A linguistic variables means a variables whose values are word or
sentence in a natural or artificial language , they are defined by the
quintuple
),,),(,( SRUuTu
where
92. 82
u is the symbolic name of a linguistic variable such as speed , heat
hight , etc.
T(u) is a group of linguistic values that describe or name the variable u
and denote a symbol for a particular property of u .
U is the universe of discourse i.e. it is the range of values which a
linguistic variables may take U=[0 , 10 , 20 , ..........., 100] .
R is the actual physical domain over which the linguistic variable
u takes its quantitative (crisp ) values .
S is a semantic function which gives a meaning of a linguistic value
in
terms of the quantitative elements of R .
Each crisp input into a fuzzy system can have multiple label assigned to it .
The greater the number of labels assigned to describe an input variable ,
the higher the resolution of the resultant fuzzy control system, resulting in a
smoother control response . But a large number of labels requires an added
computation a time and may lead to anunstable fuzzy system . Commonaly,
a number of labels ranging between 3 and 9 is used . The number of lables
is useually taken to be an odd number . Because the control surface fuzzy
sets on each side of zero or normal action set should be balanced and
symmetric . For example if u is the speed of an aircraft, then
T(speed)={very slow , slow, moderate , fast ,very fast}
AIRCRAFT SPEED
VERY SLOW SLOW MODERATE FAST VERY FASTFUZZY LABLE
CRISP INPUT
I/O PROGRAM
Figure (4.2-1) linguistic labels of linguistic variable
93. 83
4.3 FUZZY LOGIC
4.3-1 FUZZY SETS
Fuzzy sets are sets containing elements which have varying degree of
membership in the set . A fuzzy set A in universe of discourse U is
characterized by the membership function )(uAµ . It is established to give
numerical meaning to each label . Each membership function identifies the
range of input values that corresponds to a label . A membership function
takes the values in the interval [0,1] . Thus , a fuzzy set A in U may be
represented as a set of order pairs of a generic element u and its grade of
membership function Aµ as
}
)(
,{ U
u
u
uA A ∈=
µ
(4.3-1)
where )(uAµ is called the membership function of u in U .
Note that the membership function )(uAµ denotes the degree to which u
belongs to A and is normally limited to values between 0 and 1 . In fuzzy
set theory the set A is defined by function Aµ which maps the elements of
A to real numbered value between Zero and one
]1,0[: →AAµ 1)(; =uAµ , means u is that totally in A
0)(; =uAµ , mean u is not in A (4.3-2)
1)(0; << uAµ , mean u is partially in A
4.3-2 DEFINITIONS
1- Support :The support of a fuzzy set is a ‘crisp’ set of all points Uu ∈
such that 0)( >uAµ
94. 84
2- Cross point : The cross point uC
in U such that 5.0)( =CuAµ
3- Fuzzy singleton : A fuzzy singleton is a fuzzy set whose support is a
single point in universe of discourse U , if A is a fuzzy singleton whose
support is the point u1 then
1
)( 1
u
u
A Aµ
= , a nonfuzzy singleton is simply
written as
1
1u .
4- Height of fuzzy set hgt(A) : The height of fuzzy set hgt(A) is given by
superemum of membership function over all Uu ∈
)()(
sup
uAhgt A
Uu
µ
∈
= (4.3-3)
4.3-3 FUZZY SET OPERATIONS
1- The AND operator : The interaction of two fuzzy sets
let A and B be two fuzzy sets with membership functions )(uAµ and
)(uBµ respectively . The membership function of the interaction (AND) ,
)( BAC I= is defined by )}(),(min{)( uuu BAC µµµ = , Uu ∈
),min( BABAC =∧= (4.3-4)
2- The OR operator : The union of two fuzzy sets
let A and B be two fuzzy sets with membership functions )(uAµ and
)(uBµ respectively , the membership function of the union (OR) ,
)( BAC U= is defined )}(),(max{)( uuu BAC µµµ = , Uu ∈
),max( BABAC =∨= (4.3-5)
3- The NOT operator : The complement of fuzzy set
let A be a fuzzy set with membership functions )(uAµ , the membership
function of the complement of A is defined by
95. 85
)(1)( uu AA
µµ −= , Uu ∈ (4.3-6)
4- Cartesian product : if nAA ,,1 L are fuzzy sets in nUU ,,1 L with
membership functions AnA µµ ,,1 L respectively . Then the Cartesian product
of nAA ,,1 L is a fuzzy set in the product space nUU ××L1 and is defined by
=×× nAA L1 ∫××
××
21
),,/(),,( 21211
UU
AnA uuuu
L
LLL µµ (4.3-7)
where the membership function of the product can be defined as either
)()(),,( 1111 nAnAnAnA uuuu µµµµ ∧∧=×× LLL
or )(.).(),,( 1111 nAnAnAnA uuuu µµµµ LLL =×× (4.3-8)
where “ . ” denote the ordinary algebraic product .
5- Algebraic product :The algebraic product of fuzzy sets A and B is
defined as ∫=
U
BA
u
u
BA
)(
. .µ
(4.3-9)
where
)().()(. uuu BABA µµµ = (4.3-10)
4.3-4 FUZZY RELATIONS
If U is the Cartesian product of n universe of discourse nUU ,,1 L that
is nUUU ××= L1 then an n-Array fuzzy relation R in U is a fuzzy subset of
U and it is defined as
∫=
U
nnR uuuuR ),,/(),,( 11 LLµ (4.3-11)
where ),,( 1 nR uu Lµ is the membership function of R .
4.4 DESIGN PROCEDURE FOR A FUZZY LOGIC CONTROL
SYSTEM
96. 86
A basic fuzzy logic control system is shown in Figure (4.4-1)which
has four main components : the knowledge base, fuzzification,
defuzzification and decision making logic .
FUZZIFICATION D.M.L DEFUZZIFICATION PLANT
OUTPUT
REFERENCE
SENSOR
KNOWLEDGE BASE
Figure( 4.4-1) Basic FLC system
1- A knowledge base ; which functions like digital control theorems . The
knowledge base contains a knowledge about application domain and the
control goals to meet . It consists of two main components, the data base
and a linguistic(fuzzy) rule base .
(i)- The data base: provides the necessary definitions used to define
linguistic control rules and fuzzy data manipulation in fuzzy logic control .
(ii)-The rule base: characterizes the control goals and control policy of the
domain experts by means of a set of linguistic control rules .
2- A fuzzification interface ; which is some what like an A/D converter in
digital control , a Fuzzification interface performs the following functions :
(i) - Measurement of the values of input variables
(ii) - Scale mapping , which transfers the range of values of input
variables into corresponding universe of discourse .
(iii) - Fuzzification ,which convert input data into suitable linguistic
values that may be viewed as labels of fuzzy sets
97. 87
3- A decision making logic ; Which is like a digital controller ,it has the
capability of simulating human decision making based on fuzzy concept ,it
has infer fuzzy control action employing fuzzy implication and the rules of
inference in fuzzy logic .
4- A defuzzification interface ; which like a D/A converter in digital
control , a defuzzification interface performs the following function :
(i)- Scale mapping , which converts the range of value of outputvariables
into corresponding universes of discourse .
(ii)-Defuzzification ,which yields a nonfuzzy control action from an
inferred fuzzy control action
4.5 KNOWLEDGE BASE
Consist of two components data base and rule base
4.5-1 DATA BASE
Provides informationfor the proper operation of fuzzification
inference and defuzzification , this information consists of the input and
output membership functions and other information vital to the process of
FLC . The construction of the knowlege base in FLC is application
dependent.
The following aspects should be addressed
* Discretization and normalization of the universe of discourse;
Discretization of universe of discourse is frequently refereed to as
quantization . A universe is discretized or quantized into certain number of
segments . Each segment is labeled as a generic element and forms a
98. 88
discrete universe . A fuzzy set is defined by assigning the grade of
membership value of each generic element of the new discrete universe .
A lookup table , which defines the output of a controller for all possible
combination input signals , can be implemented by off-line processing in
order to decrease the running time of the controller . The normalization of a
universe requires a discretization of the universe of discourse into finite
number of segments , Each segment is then mapped into a suitable
normalized universe .
* Fuzzy partition of input and output space ; A linguistic variable is
associated with a term set . Each term in the term set is defined on the same
universe of discourse . A fuzzy partition determines how many terms
should exist in the term set . The primary fuzzy sets ( linguistic terms )
usually have meaning such as NB (negative big ), NM (negative medium ),
NS (negative small ), ZE (Zero), PS (positive small), PM (positive
medium), and PB (positive big) .
* Completeness ; it is the ability of a fuzzy control algorithm to infer a
proper control action . For every state of the process . The completeness of
a FLC relate to its data base , rule base or both .
* Membership function of a primary fuzzy set ; Two methods can be used
to define fuzzy sets , depending on whether the universe of discourse is
discrete or continuous .
(i)-The numerical case : The grade of membership function of a set is
represented as a vector of number whose dimension depends on the degree
of discretization .
(ii)-The continuous case : The membership of a fuzzy set is expressed in a
functional form , such as a bell- shaped function .
99. 89
Since all information contained in a fuzzy set is described by its
membership function , it is useful to develop a lexicon of terms to describe
various special features of this function .
The core of a membership function for some fuzzy set A is defined as that
region of the universe that is characterized by complete and full
membership in the set A . That is , the core is comprised of those elements
of universe U where 1)( =uAµ
The support of a membership function for some fuzzy set A is defined as
that region of the universe that is characterized by non-zero membership in
the set A . that is , the support is comprised of those elements of the
universe U where 0)( ≠uAµ
The boundaries of a membership function for some fuzzy set A are defined
as that region of the universe that contains elements that have a non-zero
membership ,but not complete membership . That is , the boundaries are
comprised of those elements of the universe U where 1)(0 << uAµ . These
elements of the universe are those with some degree of fuzziness or only
partial membership in fuzzy set A . Figure (4-5.1) illustrates the regions in
the universe
support
boundary
1
U
membership
Figure(4.5-1) Membership function of fuzzy set
100. 90
A normal fuzzy set is one whose membership function has at least one
element in the universe , whsoe membership is unity . some member ship
function are illustrate in figure (4.5-2 )
GaysianTrapezoidalTriangular
1
Quadratic
Figure (4.5-2) deferent shapes of membership function
* Choice of scaling factors : The concept of universe of discourse
(domain) requires a scale transformation which maps the physical values of
the process state variable into normalized domain . This is called input
normalization . Also output denormalization maps the normalized values of
the control output variables into their respective physical domain . Such
scale transformations are required both for discrete and continuos domains .
The scaling factors which describe the particular input normalization and
output denormalization play a role similar to that of the gain coefficients in
a conventional controller . The scaling factors are important with respect to
controller performance and stability i.e. they are the source of possible
instabilities ,oscillation problems and deteriorated damping effects .
There are basically two major approaches for the determination of the
scaling factors heuristic and formal (analytic) .
(i)-Heuristic : This approach has trial and error nature , the iterations are
continued until the desired performance values are obtained . By changing
the scaling factors for each one of e and ∆e . We are infect changing the
101. 91
weights given to this particular process state variable . This means if the
process is slower than the desired one , then we need to increase the effect
of error in the process , hence the scaling factor of error GE is increased .
Similarly if the overshoot or amplitude of oscillation is higher than desired
one , then we need to increase the effect of change of error , hence the
scaling factor of changing of error GDE is increased .
(ii)-Formal : This approach aims at establishing an analytic relationship
between the values of the scaling factors and the closed loop behavior of
the controlled process .
4.5-2 RULE BASE
The rule base is the collection of linguistic statements relating the
input signals to the desired output . The fuzzy rule base is characterized by
construction of a set of linguistic statements based on expert knowledge .
The expert knowledge is usually in the form of IF-THEN rules .
The following aspects are related to the construction of the rule base :
* Choice of process state variables and control output variables of fuzzy
control rules : Fuzzy control rules are more conveniently formulated in
linguistic rather than numerical terms . The proper choice of process state
variables and control variables is essential to the characterization of the
operation of a fuzzy system . In particular , the choice of linguistic
variables and their membership functions has a strong influence on the
linguistic structure of FLC . If one has made the choice of designing P, PI ,
PD or PID , this already implies the choice of process state and control
output variables . The process state variables representing the contents of
the rule antecedent (if-part of a rule)are selected among
