Transcript Genetic-Based Fuzzy Logic Controller for Satellites Stabilized by Reaction Wheels and Gravity Gradient Hanafy M.

Genetic-Based Fuzzy Logic Controller
for Satellites Stabilized by Reaction
Wheels and Gravity Gradient
Hanafy M. Omar
Aerospace Department
King Fahd University of Petroleum and Minerals
Saudi Arabia
Outline
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Introduction and Objective
Satellite Dynamics
Controller Design
Formulation of the Optimization
Problem
Genetic Algorithms Code
Simulation Results
Conclusion
Future Work
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Introduction
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The satellite carries on board
different equipment for remote
sensing and telemetry which needs
to be precisely pointed to the earth.
The satellite may receive an
impulsive torque from any particles
moving in the space which results in
deviation of the satellite from its
attitude.
This deviation will result in a poor
imaging and communications with
the ground stations.
An attitude control must used to
return the satellite back to its
orientation
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Introduction
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Most of the current satellite attitude control
systems use reaction because they give high
pointing accuracy.
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Unfortunately, these wheels are internal torquers
and they don not change the total momentum of
the satellite; they only transfer the momentum
from the satellite to the wheels.
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Therefore, they are needed to be desaturated by
an external torquer.
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Introduction
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In this work, we chose the gravity gradient as the
external torquer which is produced from the
gravitational field of the earth.
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This torque decreases as the altitude increases,
and increases with the increase of the moment of
inertial of the satellite.
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Hence, to increase the effectiveness of the gravity
gradient, a boom with long length is usually used
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Motivation
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The fuzzy logic control (FLC) is a rule-based
controller that is known of its robustness and ,
suitability for handling linear and non-linear models.
The core of FLC is the rules, which determine the
relation between the inputs and the output, usually
the FLC rules are obtained by mapping the
performance of a skillful operator. The generated
rules from this method is not available in satellite
operations.
Engineering experience or self organizing FLC can be
used for generating the rules but it is not necessarily
to be optimum.
The same problem is encountered also in the
determination of the distribution of membership
functions.
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Objective
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In this paper, a systematic technique is proposed to
design an optimal FLC for controlling the attitude of
a satellite stabilized by reaction wheels in the
presence of the gravity gradient.
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The FLC rules and the distribution of the
membership functions parameters are determined
by solving an optimization problem using the
Genetic Algorithms (GA) technique.
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Satellite Dynamics
The normalized equations of motion of
the satellite with a reaction wheel in
the presence of the gravity gradient
can be written as
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Equations Of Motion
Pitch


    0
  
2

q



p

 

H

   2

p

1
1
0

0     0 
0 
1 
  1 e  1Q

q
  
  dy
1  y 

H  0
1




0 

where
p
y
is the normalized liberation period
is the ratio of moment of inertia of the wheel w.r.t. the
satellite moment of inertia
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Equations Of Motion
Roll/Yaw
1
0
 0
   
1
    3n 2 a  1
1  x
 p 
 H x   3n 2 a 0
0

  
0
0
     n
 r   0
nb
0
  
 H z   0
nb
0

0 
    0
0  na
0   
p  1


0  na
0   H x  0
 

0
1
0    0
1   r  0
0 1

1  z   
 H  0
0
0
0   z  
n
0
0
0
1
0

0 ex  1
   
0   e z  0
0
1


0
0
where
n
a
is the normalized orbital speed of the satellite
I yy  I zz
I xx
,b 
I yy  I xx
I zz
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0
0
0 Qdx 
 
0 Qdz 
1

1
Pitch Motion
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Structure Of FLC Attitude
Controller
fuzzification

Attitude
Reference
+
NM NS
ZO
c1=-1
a2
c2
a3
c3
a4
c4=0
q
PM
PB
NM NS
Kq
a2
a1
c1=-1
c2
ZO
a3
c3
a5
a6
c5
c6
c7=1
PS
PM
PB
m
Membership value
NB
-
PS
K
a1
defuzzification
m
Membership value
NB
a4
c4=0
a5
c5
Fuzzy
Inference
System
NM NS
NB
ZO
PS
PM
PB
Ke
a2
a1
c1=-1
c2
a6
c6
m
Membership value
c7=1
The Fuzzy Attitude controller
(FLC)
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a3
c3
a4
c4=0
a5
c5
a6
c6
c7=1
e
Attitude
Satellite & States
RW
Dynamics
Controller Design
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Choose the number and distribution of membership functions
(MF's) for the inputs and the output variables.
m
Membership value
NB
a2
a1
c1=-1
NS
NM
c2
ZO
a3
c3
w1 
 1  c 2  c3 , c 2  c3  0
c5  c 6  1
a5
c5
1
a
w2 
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c7=1
c i 1  c i  w1 a i ,
i  1,2
c i 1  c i  w2 a i ,
i  4,5
i
1
6
a
i 4
PB
a6
c6
,
3
i 1
Or
PM
a4
c4=0
Constraints
0  c5  c 6 ,
PS
i
Controller Design
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Choosing the scaling factors
Since the ranges of the membership functions are
normalized, scaling factors are used to transform these
normalized ranges to the physical operating ranges
K 
1
 max
, Kq 
1
T d max
, K e  emax
• Generating the Fuzzy rules and the distribution of the
membership functions of the fuzzy inputs an the
fuzzy output by solving an optimization problem using
Genetic Algorithms (GA)
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Formulation of the Optimization
Problem
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Let
tf
f 
 w 
1
2

 w2e dt  w3t f
2
0
The optimization problem can be formulated as
zR
min f ( z)
Subjected to
 q0
at
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t  tf
m
Penalty Function
Po  k1  (t f )  k 2 q (t f )
 Po
P( z )  
 K Po
0   (t f )   max 4
 (t f )   max 4
The augmented optimization function will be
F  f P
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Genetic Algorithms
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Genetic algorithms (GA) are stochastic search
algorithms based on the mechanics of natural selection
and survival-of-the-fittest.
GAs operate on a population of potential solutions
applying the principle of survival of the fittest to
produce better and better approximations to a solution.
At each generation, a new set of approximations is
created by the process of selecting individuals according
to their level of fitness in the problem domain then
apply some operations on them that borrowed from
natural genetics (i.e. Crossover and Mutation).
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Advantages of GA
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GA optimize a performance index based on
input/output relationships only.
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Therefore, derivative information is not needed
in the execution of the algorithm and hence
many pitfalls that gradient search methods suffer
can be overcome especially for dynamic systems.
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Structure of the GA individual
Re al










MF for 
MF for q
MF for e
Rules

 

 

 


z  {r1 , r2 ,...r49 a1 , a2 ,...a6 , a1 , a2 ,...a6 , a1 , a2 ,...a6 , t f }
Integer
m
Membership value
NB
NS
NM
ZO
The encoding system for the FLC output
PS
PM
PB
MF
a2
a1
c1=-1
c2
a3
c3
a4
c4=0
a5
c5
Code
a6
c6
c7=1
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NB NM NS ZO PS PM PB
1
2
3
4
5
6
7
Genetic Algorithm
Start
The values used for the GA
parameters are:
• Population size: 50
(randomly generated)
• Crossover rate: 0.7
• Mutation Rate: 0.01
Generate randomly 50 individual
for the initial generation
Evaluate the Objective function for
multiple initial conditions
Rank the individuals and use
Method of Roulette Wheel to
select some of them
Apply the genetic operations
(crossover and mutation) on the
selected individuals to obtain the
new generation
Evaluate the Objective function for
the new generation
Check if
the objective function does not
improve over the last thirty
generations
yes
end
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Initial Conditions
• If a single initial condition is used to determine the
objective function, GA can produce a controller that
works well around this operating condition while it
may fail elsewhere.
• To be able to find a controller with a satisfactory
performance which operates over the entire range
of the input spaces, we choose multiple initial
condition which are a combination of (max, qmax, 0)
• In this case, the total value of the objective
function is the sum of the objective functions from
all the initial conditions
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Initial Conditions
q
qmax
qmax
2
qmax
4

 qmax
4
max
max
4
2
 qmax
2
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max
Stability Analysis
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To make sure that the designed controller is stable
over the whole operating conditions, another large
set of initial conditions, which is different from that
one used in the training process, is tested.
It is found that the designed controller was able to
damp all the disturbances
Therefore, the satellite system is stable with the
designed controller in a broad range of operating
conditions.
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Proposed Distributed Controller
for Roll/Yaw Motion
ref  0 +
-
K
FLC
+
1  
Kp
+
FLC
e
+

Ke
p
 +
Satellite
Dynamics
 ref  0
FLC 
K
+
 +
+
Ke
-
Kr
FLC
1 +
-
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+
e

r
Objective and Penalty Functions
for Roll/Yaw Motion
Objective Function
 w 
tf
f 
1
2

 w2  w3ex  w4ez dt  w5t f
2
2
2
0
Penalty Function
Po  k1  (t f )  k 2 q (t f )  k3  (t f )  k4 r (t f )
 Po
P( z )  
 K Po
0   (t
)   max 4 & 0   (t f )   max 4 
( (t f )  max 4) & ( (t f )   max 4)
f
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Simulation Results
The following data are used for in the simulations
Satellite Dynamics
a  b 1
Pitch Motion
n 
5
wp  n 3
I xx  I zz
 1.68n
I yy
wi  w2  w3  1; k1  k 2  K  1000
 max  25 o
Roll/Yaw Motion

pi
rad , q max  Td max  1.0, e y max  1.3
180
max  max  25o , p max  r max  1.0, ex max  ex max  1.3
wi  1, ki  1000, i  1 : 4, K  1000
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Pitch Motion
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Evolution of the
Best Objective Function
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FLC Parameters
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Final Time and Fuzzy Rules
q
e y

NB
NB
NM
NS
ZO
PS
PM
PB
NB
ZO
PS
NB
ZO
ZO
PM
NM
NB
PB
NM
NS
ZO
PB
NM
NS
NM
NS
PB
ZO
NM
NS
PB
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ZO
NM
NM
NS
ZO
PS
PM
PM
PS
NB
PS
PM
ZO
NB
PS
PM
PM
PM
NB
ZO
PS
PM
NB
PB
PB
NM
ZO
ZO
PB
NS
ZO
PB
Time History
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Roll/Yaw Motion
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Evolution of the
Best Objective Function
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Final Time and Linking Parameters
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Membership Functions
FLC
FLC 
FLC
FLC
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Fuzzy Rules
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Time History
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Conclusion
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A GA code was developed to design an optimal FLC,
which includes the generation of the controller rules
and the distribution of its membership functions.
Without any prior knowledge of the rules or the
distribution of membership function, the developed
code was able to generate a fuzzy controller with a
satisfactory performance.
With the use of multiple initial conditions in
determining the objective function, the designed
controller ha an acceptable performance in a broad
range of the satellite operating conditions.
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Suggestions and Future Work

GA requires heavy computations to able to get the
large number of the FLC parameters. The microgenetic algorithm may be a good candidate to
overcome this problem.
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Another difficulty in the optimization is the choice of
the optimization function and its parameters, which
still depends on the engineering experience. This
difficulty may be resolved using the multi-objectives
technique.
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Questions?
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