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模糊模型規則庫自動建立之演算法 An improved approach to automatically build fuzzy model rules. 王乃堅 ( Nai-Jian Wang) 台灣科技大學電機系 中華民國九十年十月二十日 地點:政大經濟系. Outline. Motivations The concept of system identification The improved algorithm Simulations and Discussions Conclusions and Future Works. Motivation. - PowerPoint PPT Presentation
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模糊模型規則庫自動建立之演算法 An improved approach to
automatically build fuzzy model rules
王乃堅 (Nai-Jian Wang)台灣科技大學電機系
中華民國九十年十月二十日地點:政大經濟系
2
Outline
Motivations
The concept of system identification
The improved algorithm
Simulations and Discussions
Conclusions and Future Works
3
Motivation
Only I/O data
Model construction
I/O relation
Modification
4
The concept of system identification
Structure Identification I
a: Input candidatesb: Input variables
Structure Identification II
a: Number of rulesb: Partition of input
spaceParameter Identification
5
Takagi and Sugeno’s model
1 2 3 4 5 6 7 8 9
1
3
6
9
Small Big
x
y
2R
1R
isxIfR :1
isxIfR :2
36.0, xythen
62.0, xythen
1 4 7
1 4 7
6
Sugeno and Yasukawa’s model
1 2 3 4 5 6 7 8 9
1
3
6
9
Small Big
x
y
2R1R
isxIfR :2 then,
1 4 7
isxIfR :3 then,
1 4 7
isxIfR :1 then,
1 4 7 5 6 9
5 6 9
5 6 9
3R
Medium
7
Fuzzy modeling開始
決定線性系統數目
建立初步的模糊系統參數
模糊系統參數的最佳化調整
是否滿足停止條件
結束
是
否線性系統數目 加 1
8
To decide the number of rules
開始
設定初始分類數目 為 2
執行FCM演算法
S(c)值是否達到最小
結束
是
否分類數目 1加
9
Fuzzy C-means clustering
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1X
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1S
2S
10
To determine the number of rules
n
kiik
c
i
mik xvvxcS
1
22
1
2 3 4 5 6
-20
-30
-40
-50
-21.84
-34.39
-42.38
-43.81
-43.52
S
c
11
Coarse fuzzy modeling
Fuzzy C-Regression Model (FCRM)
Premise parameters generation
Consequent parameters generation
12
Fuzzy C-Regression Model (1)
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ystep
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istep1
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13
Fuzzy C-Regression Model (2)
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14
Premise parameters generation (1)
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0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
the order of data set
degr
ee
15
Premise parameters generation (2)
0 5 10 15 20 25 30 35 40 45 501
1.5
2
2.5
3
3.5
4
4.5
5
the order of data set
inpu
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ta
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
input data
degr
ee
16
Premise parameters generation (3)
1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
input data
degr
ee
1 1.5 2 2.5 3 3.5 4 4.5 50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
input data
degr
ee
17
Premise parameters generation (4)
2
2
121 exp,
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18
Consequent parameters generation
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xaxaxaay
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11
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19
Fine tuning開始
取得目前的前件部參數、後件部參數、
step size
建立新的一組參數
是否滿足停止條件
結束
是
否規則庫step size調整
將目標函數各別對前件部和後件部參數取
偏微分
gθθ nownext
20
The steepest decent method
g nownext θθ
T
n
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,,,
21
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dd
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Descending
21
The gradient of objective function (1)
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22
The gradient of objective function (2)
2
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1
2
1
2
11
exp2
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ikk
ik
ikk
ik
ik
ik
ik
i
p
px
p
px
p
p
A
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22
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ik
ikk
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ik
ik
ik
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23
The gradient of objective function (3)
ik
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1ˆ
24
Stop condition
n
j
jyjyn
indexeperformancPI
1
2ˆ1
P
i iT
iOiT
P
errorpercentageaverageAPE
1
%1001
thresholdPI
25
Example 1 (1)
5,1,1 21
25.12
21 xxxxy
Rule
3.095 3.2013.518 -0.249 -0.265
1.477 1.5112.751 2.406
6.504 -0.672 -0.4692.072 2.1562.828 2.437
4.842 -0.381 -0.4211.831 1.8392.667 2.805
4.136 -0.387 -0.3571.026 1.3692.897 2.544
5.052 -0.559 -0.2432.005 1.924
iA1iA2
iC0iC1
iC2
1R
2R
3R
4R
5R
26
Example 1 (2)
Rule
3.806 2.8425.165 -1.094 -0.224
0.957 1.4712.767 1.853
4.741 -1.117 -1.0721.080 0.6572.023 2.682
3.671 -0.572 -0.8840.590 1.3232.973 3.221
3.447 -0.317 -0.5510.951 1.1202.894 2.363
8.415 -0.376 -0.7851.984 2.230
iA1iA2
iC0iC1
iC2
1R
2R
3R
4R
5R
The optimal parameters
27
Example 1 (3)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
0.2
0.4
0.6
0.8
1
1.2
1.4
the number of iteration
the
num
ber
of p
erfo
rman
ce in
dex
Method Kim Ourrule
number3 5
PI 0.0197 0.006691run time 23674sec 1281secdata size 50 50
28
Example 2 (1)
xy
yxyxcz
)sin()sin(),(sin
Rule-1.550 2.9542.155 3.851-2.202 -0.0036.802 9.5403.642 -0.4665.567 8.214
2.008 0.307 -0.252
-0.065 -0.011 0.004
0.007 0.005 -0.002
iA1iA2
iC0iC1
iC2
1R
2R
3R
Rule-1.479 3.3493.485 5.370-1.879 -0.1976.859 9.3573.334 -0.5695.677 8.049
0.854 0.099 -0.087
0.018 0.001 0.00001
-0.078 0.009 -0.00036
iA1iA2
iC0iC1
iC2
1R
2R
3R
29
Example 2 (2)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
the number of iteration
the
num
ber
of p
erfo
rman
ce in
dex
Method Jang Ourrule
number16 3
data size 121 100paramet
ernumber
72 21
PI - 0.019934APE estimated 0.01% 0.001%
30
Example 3 (1)
25.115.01 zyxoutput
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010
-8
10-6
10-4
10-2
100
102
the number of iteration
the
num
ber
of p
erfo
rmac
e in
dex
31
Example 3 (2)
ModelTraining
errorChecking
error
Thenumberof rule
Thenumber ofparameter
Trainingdata size
Checkingdata size
ANFIS[1] 0.04% 1.07% 8 50 216 125GMDH
model[34] 4.70% 5.70% - - 20 20
Sugeno andKang[14] -
11.50% 2.10% 3 22 20 20
Sugeno andKang[14] -
20.59% 3.40% 4 32 20 20
Ourmethod 0.0023% 1.51% 6 54 20 20
32
Conclusions and Future Works
架構精簡,彈性大易於在電腦上實現 不錯的運算效率和較佳的近似結果 有較佳的能力去描述未知系統改進 FCM方法不足之處 以其他的最佳化方法取代最陡坡降法
33
Least-squares estimator
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θ
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AθyAθyeeθaθ
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TT
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02
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