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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10

To determine the number of rules

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2 3 4 5 6

-20

-30

-40

-50

-21.84

-34.39

-42.38

-43.81

-43.52

S

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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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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

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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

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1

input data

degr

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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

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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

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18

Consequent parameters generation

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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

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,,,

21

g

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now

Minimum*

Descending

21

The gradient of objective function (1)

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22

The gradient of objective function (2)

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11

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23

The gradient of objective function (3)

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24

Stop condition

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errorpercentageaverageAPE

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%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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