Design of Support vector learning for Evolutionarybased fuzzy systems
Chiou JuingShian1, Liu MingTang2
Department of Electrical Engineering, Southern Taiwan University of
Technology, Tainan Hsien, Taiwan,R.O.C.
E-mail:jschiou@mail.stut.edu.tw
1
Abstract. In this study we proposed an evolutionary genetic algorithm (GA) are
discussed in which the membership function shapes and types and the fuzzy
rule set, including the number of rules inside the rule set are evolved by GA to
create a complete fuzzy controller. In addition, the genetic parameters of the
evolutionary algorithm are adapted by using fuzzy system. Moreover, this
investigation elucidates the feasibility of using SVMs to forecast genetic
evolved process and then evolutionary genetic algorithm (GA) were employed
to choose the parameters of a SVM model. Subsequently, examples of the
simulation motor and active suspension system controlled result demonstrate
that the propose method can get the faster evolution process, and the
evolutionary fuzzy system can obtain the better performance.
Keywords: FLC, genetic algorithms, optimal design, support vector machines
(SVMs)
1
Introduction
IN the design of fuzzy system, the definition of the membership functions, modeled
by fuzzy sets and the rule sets are most important considerations; them also constitute
the main difficulty [1,2]. In GA, there are two main reproduction operators, namely
crossover and mutation. The choice of the probabilities of crossover and mutation is
known to critically affect the behavior and the performance of GA, and a number of
guidelines are existing in the literature for choosing them [3]. Instead of having fixed
probabilities of crossover and mutation, a fuzzy-controlled crossover and mutation
probabilities is presented in this paper. The basic concept is based on considering the
useful knowledge and human expertise. For example, if the best fitness value of the
individual does not change during several generations, the mutation rate should be
increased to prevent the system premature convergence. When the fitness value of the
individual grows up obviously and continuously, we should emphasize the crossover
rate but the mutation rate should be reduced to keep the convergence of system. By
using fuzzy system can assist GA to adjust the evolution parameters such as crossover
rate and mutation rate.
This paper presents evolutionary genetic algorithm by using genetic algorithm and
FLC system are discussed in which the membership function shapes and types and the
fuzzy rule set including the number of rules inside it are evolved using a GAs. In
addition, crossover and mutation are two critical genetic parameters (operators).
Moreover, we use fuzzy control to dynamic adjustment the crossover rate and
mutation rate, even more using SVMs to forecast genetic evolved process and then
evolutionary genetic algorithm (GA) to enhance the efficiency of evolution.
2
CHROMOSOME ENCODING METHODS
GAs are search algorithms modeled after the mechanics of natural genetics. GAs
can be used to obtain an approximate solution for single variable or multivariable
optimal problems. While stochastic in nature, GAs perform a highly effective search
of the problem hyperspace, efficiently directing the search to promising regions. So
GAs work with a population of points rather than a single point. The analogy between
biological evolution and a binary GA, it is required to encode values as individual
chromosomes. The GA begins, like any other optimization algorithm, by defining the
optimization variables, the fitness function, and the error amount. It ends like other
optimization algorithms too, by testing for convergence.
For the object is to optimize the number of fuzzy sets, which is the problem of
membership function's distributing in fields, so fixing on membership function's
shape is considered firstly. In order to properly represent the space distributing of
fields, each membership function is determined by two values the start point x1 and
the end point x2 . Theoretically, each fuzzy variable can have many fuzzy sets with
its own membership function, but commonly used are three, five, seven, or nine fuzzy
sets for each fuzzy variable, and the membership function can be linear or nonlinear.
In this paper, we adopt six membership functions for the fuzzy set; they are left
triangle, right triangle, triangle, Gaussian, and sigmoid functions, respectively. The
membership function of each fuzzy set can be described as three parameter genes
denoting the start point x1 , the end point x2 and the function type value
respectively, the parameter gene in this paper is adopted real number coding, a total of
six types of functions are used as the membership function gene candidates shown in
Figure 1; each is represented by an integer from 0 to 6, zero is useless, one is
Triangle, two is Left_triangle, three is Right_triangle, four is Gaussian, five is
Sigmoid, six is Reverse_sigmoid. In order to have a homogeneous chromosome,
integers are chosen to represent the start point x1 and the end point x2 instead of real
values. Assume for the variable that its dynamic range is [0, 10] and that it has fuzzy
sets. If the fuzzy membership functions are uniformly distributed over the range with
half-way overlap as shown in Fig. 1, then the center point ci (i  1,...,7) of the ith
membership function is located at
ci  x i1 
xi 2  xi1
2
(1)
Fig. 1. Six uniformly distributed membership functions.
In this paper, the fuzzy numbers 1~7 on the fuzzy control rule table represent the
linguistic values NL, NM, NS, ZE, PS, PM, and PL, respectively. The total length of
the chromosome representing the system is (3×7)+(7×7)=70 and the system can be
represented as
S1 S 2 S 3 ...........S 21S 22 S 23 S 24 ...............S 68 S 69 S 70 where S1 S 2 S 3 ...........S 21
represents (start point, end point, type), S 22 to
3
3.1
S 70 represent fuzzy rule.
EVOLUTIONARY GENETIC ALGORITHM
ADAPTIVE GENETIC ALGORITHM
Crossover and mutation are two critical operators; they facilitate an efficient search
and guide the search into new regions. They can be varied during the running, usually
starting out by running the GA with a relatively higher value for crossover and lower
value for mutation, then tapering off the crossover value and increasing the mutation
rate toward the end of the evolution. In order to make GA can explore the potential
solutions in the searching field as soon as possible, the mutation rate and crossover
rate should be duly adjusted during the evolution. Although there is no a rule for how
to adjust these evolutionary parameters, but by using human experience can help us to
achieve this purpose. For example, when the best fitness did not increasing after a few
generations of the evolution, it means that the system could be stuck at a local
minimum, so the system should probably concentrate on exploiting rather than
exploring; in another word, the crossover rate should be decreased and the mutation
rate should be increased. According to this kind of knowledge, we use fuzzy system
to adjust the crossover and mutation rates. The best fitness (BF), the number of
generations for unchanged best fitness (UN), and the variance of fitness (VF) are the
input variables, and mutation rate (MR) and crossover rate (CR) are the output
variables of the fuzzy system. The examples of the fuzzy control rules based on the
linguistic description of the fuzzy implications are stated as
If UN is high and VF is medium, then MR is high, and CR is low.
For simplicity, each variable has three fuzzy sets: Low, Medium, and High. BF
variable 0~4, VF variable 0~5, UN variable 0~10, MR variable 0.075~0.1, CR
variable 0.65~0.9, so the fuzzy rule of CR and MR as shown in Table 1.
Table 1 The fuzzy table of CR and MR
(MR,CR)
BF
VF
L
M
H
L
M
H
(L,H)
(H,H)
(H,H)
(H,L)
(H,L)
(L,L)
(L,H)
(M,M)
(M,M)
(H,M)
(H,M)
(L,M)
(L,H)
(L,H)
(L,H)
(H,H)
(M,H)
(L,H)
H
M
UN
L
For example, assume input [UN, BF, VF] = [5, 2, 2] for UN-VF, UN-VF inference,
respectively.
UN-BF inference:(M.M)
If UN is 5 and BF is 2 then MR is 0.0875 and CR is 0775.
UN-VF inference:(H,M)
If UN is 5 and VF is 2 then MR is 0.0966 and CR is 0775.
Finally, max MR, CR  [0.0966,0775] .
After the fuzzy inference, we can obtain the dynamic crossover and mutation rates
which are used in the evolution. The flowchart of adaptive genetic algorithm is shown
in Figure 2.
Start
Generate an Initial
Population
Decode
chromosomes
(BF,UN,VF)
Optimize
Fuzzy system
Generate the Next
Generation or Stop
Yes
No
FLC
Reproduction
Support Vector
Learning
(CR)
Crossover
e
R +
_
k1
(MR)
EVOLUTIONARY
GENETIC ALGORITHM
y
FLC
Mutation
d
dt
k3
u
Plant
k2
FUZZY SYSTEM
Fig. 2. Support vector learning for Evolutionary-based fuzzy systems
3.2
SUPPORT VECTOR MACHINES INTEGRATED GA
GA can be applied in high dimensional, multi-modal, and complex problems.
Hence, multi-modal to search the following question, namely calculate the increase of
quantity. Therefore, we present a genetic fuzzy feature transformation method for
support vector machines to do more accurate data (every gene variation) classification.
Given data are first transformed into a high feature space by a genetic evolution, and
then SVMs are used to map data into a higher feature space and then construct the
hyperplane to make a final decision. Through frame of probability reference, each
gene rate of change high to show system is it not convergence and is it is it evolve to
continue to need to have. If the rate of change of gene equals zero one does not evolve
again and asymptotically stable system.
Support vector machines (SVMs) are a classification technique of machine learning
based on statistical learning theory [10,12]. Considering a classification problem with
two classes, SVMs are to construct an optimal hyper-plane that maximizes the margin
between two classes. According to Vapnik statistical learning theory [10,11], the
maximum of margin implies the extraordinary generalization capability and good
performances of SVM classifiers [5,6,7]. So far, SVMs have already been
successfully applied to many real fields. This paper method is to collect the best
chromosome within every generation, and then the larger variation of gene to record
in the chromosome, and determine the learning sample of SVM. The learning sample
function is defined as follow:
(2)
 S m _ 1 _ 1 , S m _ 1 _ 2 , S m _ 1 _ 3 ,..S m _ 1 _ 70 
S

m _ 2 _ 1 , S m _ 2 _ 2 , S m _ 2 _ 3 ,..S m _ 2 _ 70 
Gm  
,





 S m _ l _ 1 , S m _ l _ 2 , S m _ l _ 3 ,..S m _ l _ 70 
S LearningSample  G m ( S m _ 1 ) - G m-1 ( S m _ 1 )
if S LearningSample  0 then Z m _ i  1 else Z m _ i  -1.
where Gm is the generation of evolution, S m _ l is the chromosome, S is the gene.
Because the gene to larger variation represents chromosome can be optimized again,
and then the opposite fixed of gene position form has already had good evolution that
needn't be optimized again. It is in (2) that the chromosome best chromosome after
classifying is represented. We can regard S m _ 1 as the learning sample and S m _ 2 as
and train the dataset. Given a training dataset:


X m  S m _ 2 _ i , Z m _ i  S m _ 2 _ i  R n , Z m _ i   1,1, i  1,  n
(3)
The optimal hyper-plane of X is defined as f ( S m _ 2 )  0 , where
f ( S m _ 2 )  sign w  S m _ 2 _ i  b ,
(4)
In SVM literature, w is often referred to as the weight vector; b is called the bias. We
choose the separating hyperplane w  S m _ 2 _ i  b  0 that is furthest away from the
data points S m _ 2 _ i that is, that has maximal margin. Suppose S m _ 2 _ 1 and S m _ 2 _ 2 with
Z m _ 1  1 and Z m _ 2  1 are positive and negative points closest to the
hyperplane. For maximal separation, the hyperplane should be as far away as possible
from each of them:
 w  S m _ 2 _ 1  b  1

w  S m _ 2 _ 2  b  1
 w  S m _ 2 _ 1  S m _ 2 _ 2   2

w
2
 S m _ 2 _ 1  S m _ 2 _ 2  
w
w
Maximising the margin is equivalent to maximizing 2 w , it turns out that the optimal
separating hyperplane solving can be found as the solution to the equivalent
optimization problem:
(5)
1 2
w
w ,b
2
subject to Z m _ i w  S m _ 2 _ i  b   1,
min
Then, constructing the optimal hyperplane is therefore a convex quadratic problem.
This is the crucial property that allows generalization to the non-linear case. The
Lagrange function differentiating it with regard to w, b and  i  0 are introduced for
each of the constraints (4) to get the following Lagrangian :
Lw, b,   
(6)
1 2 n
w    i Z m _ i w  S m _ 2 _ i  b   1
2
i 1
Thus it possible to equivalently solve the dual optimization problem of maximizing
(6), such that the gradient of L with respect to w and b vanishes, and requiring that
 i  0 , This is ,
(7)
n



L
w
,
b
,


0
,

Z m _ i i  0

 b
i 1

n
 Lw, b,   0,  w   Z m _ i i S m _ 2 _ i
 w
i 1
By substituting (7) into (6), the dual form of the optimization problem is derived.
n
M ( )    i 
Maximum
i 1
n
Subject
to
Z
j 1
m_ j
1 n
 Z m _ i Z m _ j  i j S m _ 2 _ i  S m _ 2 _ j ,
2 i , j 1
 j  0;
Q   i  0,
i  1,2,, n,
To do this, we use the idea of SVM for crisp nonlinear regression. The basic idea is
that a nonlinear regression function is achieved by simply preprocessing input
 : Rn  F
patterns S m _ 2 _ i by a map
into some feature space F and then
applying the standard ridge regression learning algorithm. Hence it suffices to know


and use K S m _ 2 _ i , S m _ 2 _  j   ( S m _ 2 _ i ),  ( S m _ 2 _ j ) instead of
 ()
explicitly.
Hence, we obtain the following dual optimization problem:
n
M ( )    i 
Maximum
i 1
n
Subject
to
Z
j 1
m_ j
1 n
 Z m _ i Z m _ j i j K S m _ 2 _ i  S m _ 2 _ j ,
2 i , j 1
 j  0;
Q   i  0,
i  1,2,, n,
(8)
where Q is a positive constant and K S m _ 2 _ i  S m _ 2 _ j  is conventionally called a
kernel function satisfying the Mercer theorem [10]. The kernel function that this
paper uses is Gaussian function. The constant b is given by
n


b  Z m _ i   S m _ 2 _ i  Z m _ j j S m _ 2 _ j 
j 1


Substituting (7) for
(9)
w in (4), we have
f ( S m _ 2 )   Z m _ i i K S m _ 2 _ i  S m   b
n
(10)
i 1
We can know separability of two subsets through checking whether the following
inequalities
Z m _ i wS m _ 2 _ i  b   1;
i  1,2,, n
(11)
A procedure to compute maximum margin for two subsets is described below [12].
Step 1. Solving the quadratic programming (8).
Step 2. Determining the separating hyper-plane (10) according to (9).
Step 3. Checking the separability between two subsets according to inequalities
(11).
Step 4. Let the margin be 0 if the two subsets are not separable.
Step 5. Computing the maximum margin according to 1 /( w w) for the separable
case where the vector w is determined by (7).
4
SIMULATION
The active suspension control system of an automobile is currently of great interest,
both academically and in the automobile industry worldwide. In 1999, Yi-Pin Kuo [9]
uses GA-based fuzzy PI/PD controller is proposed for an automotive active
suspension system (AASS). By using the merit of GA’s, the optimal decision-making
rules for both types of controllers are constructed. The real-time simulation results
demonstrate that the fusion of GA’s and fuzzy controller for an AASS can provide
passengers much more ride comfort.
In this study we proposed an intelligent fuzzy logic-based genetic algorithms by
using genetic algorithm and FLC system are discussed in which the membership
function shapes and types and the fuzzy rule set including the number of rules inside
it are evolved using a GA’s. In addition, crossover and mutation are two critical
genetic parameters (operators). We use fuzzy control to dynamic adjustment the
crossover rate and mutation rate to enhance evolution velocity, and consider the
comfortable that the passenger takes and automobile controlling at the same time. In
order to evolve with higher speed, the system simulation uses the following support
vector learning for Evolutionary-based fuzzy systems shown as Fig. 2. The input
variables of the fuzzy controller are error e and error’s derivative e , and the output
variable is control u . We simulation example uses the active suspension control
system shown as Fig. 3. It consists of a quarter-car model, road profile model, and an
active suspension controller. Without further explanation, it is assumed that the tire
does not leave the ground and that z s and zu are measured from the static
equilibrium position. In addition, the velocity of sprung mass
zs and relative
z s  zu assumed to be measurable.
velocity between unsprung and sprung mass are
The dynamic equations are calculated as following:
where
ms z  ks ( zu  zs )  bs ( zu  zs )  f a
(12)
mu zu  k s ( zu  zs )  bs ( zu  zs )  f a  kt ( zr  zu )
(13)
k s is constant of the spring, k t is spring constant of the tire, bs is
damping, and zr is road surface interference. The active suspension control system
is evaluated by the following aspects: passenger ride quality, suspension deflection,
and road holding ability, as mentioned in the previous section
from sensors: position and velocity
Active
Suspension
Controller
Force
ms
zs
Sprung
fa
ks
u
bs
mu
zu
Unsprung
kt
zr
Road terrain
Fig. 3. The active suspension system of a quarter-car
To emulate the road terrain, we introduce the well-known step-like and
pseudorandom road profiles in designing the controller. Mathematics model of road
surface simulation as following
S ( m) 
(14)
CP sin( wct s i )
w
( c ) 2.5
2V
where is the wave number (cycle/m); wc  7.7 (rad/s), the sprung mass natural
frequency; V (m/s) is the vehicle speed; t s  0.01 s , sampling time of simulation;
P  3.14  10 6 , representation of a poorer-than-average
i  1 is the iteration step;
(European) quality principal road; and C  10 , coefficient of S (m) which amplifies
the disturbance to present the worse than normal road profile[9] . In order to consider
the comfortableness that the passenger takes and automobile controlling at the same
time, the two input variables to the fitness are defined as
es1  ( R  zs (0)) 2 , es2  ( R  zs (1)) 2 ,..., esn  ( R  zs (n)) 2
(15)
eu1  ( R  zu (0)) 2 , eu2  ( R  zu (1)) 2 ,..., eun  ( R  zu (n)) 2
(16)
where R=0 is reference target,
n is calculate the number of times,
n
F   es i  [ w  (eui )] is error amount, the weight is w  10 . This page is control
i 0
system minimum value of F by way of evolutionary fuzzy system. For more
objective observation simulation result, the control method that we adopt the optimal
linear feedback control law to study with this plan is compared, its control model is as
follows:
J (u ) 
where
Q
is symmetric and positive definite,
corresponding to the control
defined as
x1  z s  z u
where
(17)
1  T
( X QX  u T Ru )dt
2 0
R
is a weighting factor
u , and X is the state vector and its entries are
x 2  z s
x3  z u  z r
x 4  z u
x1 , x2 , x3 and x4 are the suspension deflection , sprung mass velocity, tire
deflection, and tire velocity, respectively. For the quarter-car suspension system
discussed in this section, the typical parameters for the suspension model are selected
as
mu  30kg, ms  250kg, k s  15000 N m
k t  150000 N m , bs  1000 N .s m
passive:Dashed line(--) , SVM for Evolutionary GA fuzzy:Dotted line(:) , Optimal:Dash-dotline(-.)
Suspension Deflection(m)
0.05
0
Tire-Ground Contact (m)
Suspension Acceleration(m/sec 2)
-0.05
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
10
5
0
-5
0.02
0.01
0
-0.01
-0.02
Fig. 4. Time response to a rough road (passive: Dashed line; SVM for Evolutionary GA fuzzy:
Dotted line; Optimal: Dash-dotline). (a) Suspension deflection. (b) Sprung mass acceleration.
(c) The beating in the distance of the tire and ground.
In this example, SVM for evolutionary Active suspension system was superior to
passive suspension system and the optimal linear feedback control law of Active
suspension system, the number value of different performance is arranged in table 2.
Evolve the membership function appearing to see from Fig. 6. The fuzzy rules gene
evolving finishing finally is:
[6 5 3 7 5 1 5 1 7 3 2 1 2 6 2 7 6 5 7 7 7 2 5 6 7 6 4 7 3 2 2 1 1 1 2 2 7 3 6 2 7
6 6 6 3 4 1 1 3]
Table 1.
Active suspension system (SVM for Evolutionary GA fuzzy）gene code in
the fuzzy table.
e
y
e
N
P
P
P
Z
N
N
N
Table 2.
L
M
S
E
S
M
L
N
6
6
3
4
1
1
3
M
2
7
3
6
2
7
6
N
S
Z
3
2
2
1
1
1
2
E
P
2
5
6
7
6
4
7
S
P
2
7
6
5
7
7
7
M
P
1
7
3
2
1
2
6
L
6
5
3
7
5
1
5
The maximum overtake measure of the road surface response
Suspension
deflection
(improvement %)
Sprung mass
acceleration
(improvement %)
The beating in the
distance of the tire
and
ground
(improvement %)
L
Passive
Active (Opimal state
feedback control)
SVM for
Evolutionary GA fuzzy
0.0498m
0.0433m （ 13.05%
0.0375m
（0%）
）
（23.1%）
4.847m/sec2
4.3563m/sec2
（0%）
10.12%）
56.2%）
0.0134m
0.0125m
0.0112m （ 16.42%
（0%）
（6.72%）
）
（
Active
2.124m/sec2
（
Fig. 5. Fitness function of evolutionary Active suspension system.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fig. 6. Membership function received after evolving.
3
CONCLUSION
In the study of evolutionary-based fuzzy system have been discussed in which the
membership function shapes and types and the fuzzy rule set, including the number of
rules inside the rule set, are evolved using a GA. In addition, the crossover rate and
mutation rate of GA are dynamic adjusted by fuzzy system. Moreover, this
investigation elucidates the feasibility of using SVMs to forecast genetic evolved
process and then evolutionary GA were employed to choose the parameters of a SVM
model. The GA can help the fuzzy logic system to reach the optimization by using
GA’s powerful searching ability, moreover, utilize the advantage of fuzzy logic can
assist GA to increase the efficiency of evolution. The computer simulations
demonstrate that the effectiveness for our proposed scheme.
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