Type-2 Fuzzy Neural Networks

CHAPTER 4

Type-2 Fuzzy Neural Networks Contents 4.1 Type-1 Takagi-Sugeno-Kang Model 4.2 Other Takagi-Sugeno-Kang Models 4.2.1 Model I 4.2.2 Model II 4.2.2.1 Interval Type-2 TSK FLS 4.2.2.2 Numerical Example of the Interval Type-2 TSK FLS 4.2.3 Model III 4.3 Conclusion References

37 38 38 39 40 41 42 43 43

Abstract The two most common artificial intelligence techniques, FLSs and ANNs, can be used in the same structure simultaneously, namely as “fuzzy neural networks.” The advantages of ANNs such as learning capability from input-output data, generalization capability, and robustness and the advantages of fuzzy logic theory such as using expert knowledge are harmonized in FNNs. In this chapter, type-1 and type-2 TSK fuzzy logic models are introduced. Instead of using fuzzy sets in the consequent part (as in Mamdani models), the TSK model uses a function of the input variables. The order of the function determines the order of the model, e.g., zeroth-order TSK model, first-order TSK model, etc.

Keywords Type-1 fuzzy neural networks, Type-2 fuzzy neural networks, TSK models, Artificial intelligence, Fuzzy logic, Neural networks

4.1 TYPE-1 TAKAGI-SUGENO-KANG MODEL A type-1 TSK model can be described by fuzzy If-Then rules. For instance, in a first-order type-1 TSK model, the rule base is as follows: IF x1 is Aj1 and x2 is Aj2 and …and xn is Ajn n  wij xi + bj THEN uj =

(4.1)

i=1 Fuzzy Neural Networks for Real Time Control Applications http://dx.doi.org/10.1016/B978-0-12-802687-8.00004-9

Copyright © 2016 Elsevier Inc. All rights reserved.

37

38

Fuzzy Neural Networks for Real Time Control Applications

where x1 , x2 , . . . ,xn are the input variables, uj ’s are the output variables, and Aji ’s are type-1 fuzzy sets for the jth rule and the ith input. The parameters in the consequent part of the rules are wij and bj (i = 1, . . . , n, j = 1, . . . , M). The final output of the system can be written as: M

j=1 fj uj

u = M

(4.2)

j=1 fj

where fj is given by: fj (x) = μAj1 (x1 ) ∗ · · · ∗ μAjn (xn )

(4.3)

in which ∗ represents the t-norm, which is the prod operator in this book.

4.2 OTHER TAKAGI-SUGENO-KANG MODELS Other TSK models (shown in Table 4.1) can be classified into three groups [1]: 1. Model I: Antecedents are type-2 fuzzy sets, and consequents are type-1 fuzzy sets (A2-C1) 2. Model II: Antecedents are type-2 fuzzy sets, and consequents are crisp numbers (A2-C0) 3. Model III: Antecedents are type-1 fuzzy sets, and consequents are type-1 fuzzy sets (A1-C1)

4.2.1 Model I Type-2 Model I can be described by fuzzy If-Then rules in which the antecedent part is type-2 fuzzy sets. In the consequent part, the structure is similar to that of a type-1 TSK fuzzy system, however, the parameters are type-1 fuzzy sets rather than crisp numbers. They are therefore named as “Type-2 TSK Model I” systems. In Model I, the rule base is as follows: Table 4.1 Classification of other TSK models Other TSK FLSs Model I Model II

Antecedent Consequent

Model III

Type-2 fuzzy sets Type-2 fuzzy sets Type-1 fuzzy sets Type-1 fuzzy sets Crisp numbers Type-1 fuzzy sets

39

Type-2 Fuzzy Neural Networks

˜ j1 and x2 is A ˜ j2 and …and xn is A ˜ jn IF x1 is A n  Wij xi + Bj THEN Uj =

(4.4)

i=1

where x1 , x2 , . . . ,xn are the input variables, Uj ’s are the output variables, and ˜ ji ’s are type-2 fuzzy sets for the jth rule and the ith input. The parameters in A the consequent part of the rules are Wij and Bj (i = 1, . . . , n, j = 1, . . . , M), which are type-1 fuzzy sets. The final output of the first-order type-2 TSK Model I is as follows [1]:  U(U1 , . . . , UM , F1 , . . . , FM ) =

 ···

u1



 ··· uM

M μFj (fj ) τj=1

f1

 M

fM

M τj=1 μUj (uj )

j=1 fj uj

M

j=1 fj

(4.5)

where M is the number of rules fired, uj ∈ Uj , fj ∈ Fj , and τ and  indicate the t-norm. Fj is the firing strength which is defined as: Fj = μA˜ j1 (x1 )  μA˜ j2 (x2 )  · · ·  μA˜ jn (xn )

(4.6)

where  shows the meet operation. Although the calculation of (4.5) is difficult, some general concepts are explained in Ref. [2]. When interval type-2 sets are used in the antecedent part and type-1 sets are used in the consequent part, it is shown in Ref. [1] that the output of an interval T2FLS is: f (x) =

ul + ur 2

(4.7)

where ur and ul are the maximum and minimum values of u, respectively. The reader is encouraged to refer [1] and [2] for further information about the calculation process of ur and ul . A broad survey exists in literature in which number of different type reducers for Model I are compared [4].

4.2.2 Model II Model II can be regarded as a special case of Model I where the antecedents are type-2 fuzzy sets and the consequents are polynomials. A type-2 TSK

40

Fuzzy Neural Networks for Real Time Control Applications

Model II can be described by fuzzy If-Then rules. For instance, in a firstorder type-2 TSK Model II, the rule base is as follows [1]: ˜ j1 and x2 is A ˜ j2 and …and xn is A ˜ jn IF x1 is A n  THEN uj = wij xi + bj

(4.8)

i=1

where x1 , x2 , . . . ,xn are the input variables, uj ’s are the output variables, ˜ ji ’s are type-2 fuzzy sets for the jth rule and the ith input. The parameters in A the consequent part of the rules are wij and bj (i = 1, . . . , n, j = 1, . . . , M). The final output of the model is as follows [1]:  U(F1 , . . . , FM ) =

 M

 ··· f1

fM

j=1 fj uj

M τj=1 μFj (fj )

M

j=1 fj

(4.9)

where M is the number of rules fired, fj ∈ Fj , and τ indicates the t-norm. Note that (4.9) is a special case of (4.5), because each Uj in (4.5) is converted into a crisp value here. The firing strength is the same as (4.6). 4.2.2.1 Interval Type-2 TSK FLS In the structure of the interval type-2 TSK FLS, (4.9) is given as follows [3]:  YTSK/A2−C0 =

 M

 ···

1

f 1 ∈[ f 1 ,f ]

f M ∈[f M ,f

M

]

1

j=1 fj uj

M

j=1 fj

(4.10)

where f and f j are given by: j

f (x) = μA (x1 ) ∗ · · · ∗ μA (xn ) j

j1

jn

(4.11)

f j (x) = μAj1 (x1 ) ∗ · · · ∗ μAjn (xn ) in which ∗ represents the t-norm, which is the prod operator in this book. The output of the fuzzy system in closed form is approximated by [3]: M YTSK/A2−C0 = M

j=1 f j uj

j=1 f j

+

M

j=1 f j

M

j=1 f j uj M j=1 f j + j=1 f j

+ M

(4.12)

Type-2 Fuzzy Neural Networks

41

4.2.2.2 Numerical Example of the Interval Type-2 TSK FLS In order to be able to give a clear explanation about the inference of this type FLS, a numerical example is given: Let’s assume a T2FLS (A2-CO) with two inputs and two type-2 fuzzy MFs for each. While the antecedent type-2 fuzzy MFs are given in Fig. 4.1, the consequent part of the rules are given as follows: u1 = 4x1 + x2 and u2 = 2x1 + 3x2 . 1 ~

~

A11

A12

0.8

m(x1)

0.6

0.4

0.2

0

0

2

4 x1* = 5.25

6 x1

8

10

12

10

12

1 ~

~

A21

A22

0.8

m(x2)

0.6

0.4

0.2

0

0

2

4

6 x2

8 x2* =6.5

Figure 4.1 Two rules each having two type-2 triangular fuzzy MFs.

42

Fuzzy Neural Networks for Real Time Control Applications

The mathematical form of triangular MFs are as follows:  μ(x) ˜ =

1− 0

|x−c| d

if c − d < x < c + d else

(4.13)

where c 11 = c 21 = c 11 = c 21 = 4, c 12 = c 22 = c 12 = c 22 = 8, and d11 = d21 = d12 = d22 = 3, d11 = d21 = d12 = d22 = 4. The input 1 (x∗1 ) and the input 2 (x∗2 ) are selected as 5.25 and 6.5, respectively. The firing strengths are as follows: f 1 = 0.6875 ∗ 0.6250 = 0.4297 f = 0.5833 ∗ 0.5000 = 0.2917

(4.14)

1

f 2 = 0.3125 ∗ 0.3750 = 0.1172 f = 0.0833 ∗ 0.1667 = 0.0139 2

u1 = 4x1 + x2 = 4 ∗ 5.25 + 6.5 = 27.50 u2 = 2x1 + 3x2 = 2 ∗ 5.25 + 3 ∗ 6.5 = 30.00 0.4297 ∗ 27.50 + 0.0139 ∗ 30.00 = 27.5783 0.4297 + 0.0139 0.2917 ∗ 27.50 + 0.1172 ∗ 30.00 ur = = 28.2166 0.2917 + 0.1172 ul =

(4.15)

(4.16)

The final output is calculated as follows: u∗ =

ul + ur = 27.8974 2

(4.17)

4.2.3 Model III The TSK Model III can be described by fuzzy If-Then rules. For instance, in a first-order type-2 TSK Model III the rule base is as follows [1]: IF x1 is Aj1 and x2 is Aj2 and …and xn is Ajn n  Wij xi + Bj THEN Uj = i=1

(4.18)

Type-2 Fuzzy Neural Networks

43

where x1 , x2 , . . . ,xn are the input variables, Uj ’s are the output variables, and Aji ’s are type-1 fuzzy sets for jth rule and the ith input. The parameters in the consequent part of the rules are Wij and Bj (i = 1, . . . , n, j = 1, . . . , M). Note that both the consequent parameters and the outputs of the rules above are type-1 fuzzy sets. Also, Ajk ’s are type-1 fuzzy sets (k = 1, . . . , n). The final output of the model is as follows:  U(U1 , . . . , UM ) =

··· u1

M

 uM

M τj=1 μUj (uj )

j=1 fj uj

M

j=1 fj

(4.19)

where M is the number of rules fired, uj ∈ Uj . fj is the firing strength, which is defined as: fj = μAj1 (x1 ) ∗ μAj2 (x2 ) ∗ · · · ∗ μAjn (xn )

(4.20)

4.3 CONCLUSION The advantages of ANNs such as learning capability from input-output data, generalization capability, and robustness and the advantages of fuzzy logic theory such as using expert knowledge are harmonized in FNNs. Inspired by the conventional FNNs, T2FNNs have been designed in which the MFs are type-2. These systems are stronger to deal with uncertainties in the rule base of the system compared to their type-1 counterparts.

REFERENCES [1] Q. Liang, Fading channel equalization and video traffic classification using nonlinear signal processing techniques, University of Southern California, USA, 2000. [2] N. Karnik, J. Mendel, An introduction to type-2 fuzzy logic systems 1998, October 1998, URL http://sipi.usc.edu/mendel/report. [3] M. Begian, W. Melek, J. Mendel, Parametric design of stable type-2 TSK fuzzy systems, in: Fuzzy Information Processing Society, 2008. NAFIPS 2008. Annual Meeting of the North American, 2008, pp. 1-6. [4] Wu. Dongrui, Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: overview and comparisons. Fuzzy Systems, IEEE Transactions on 21, no. 1 (2013) 80-99.