Analytic expression of Mamdani fuzzy system constructed by fuzzy similarity degree and its output algorithm

Abstract

Fuzzy similarity degree is a measurement of the similarity between fuzzy sets through local information, it plays an important role in the design of fuzzy system and controller. This article first proposes a new computational formula for membership functions of a consequent fuzzy set based on fuzzy similarity degree, and an analytic representation of the Mamdani fuzzy system is obtained through the Gauss fuzzification, product inference engine and center average defuzzification. Next, a specific Mamdani fuzzy system constructed by Gauss fuzzifier or singleton fuzzification be expressed through a given fuzzy similarity degree in practice. Finally, the output algorithm of the proposed fuzzy system is given by the space positioning method. The result shows that the Mamdani fuzzy system constructed by fuzzy similarity degree and Gauss fuzzification is superior to that based on singleton fuzzification in terms of approximation capability.

Keywords

Fuzzy system fuzzy similarity degree Gauss fuzzification singleton fuzzification output algorithm

1 Introduction

Fuzzification is an important method in design-ing a fuzzy system or controller. There are usually three fuzzification methods mainly include single-ton, Gauss and triangle fuzzification, where singleton fuzzification, product inference engine and center average defuzzification are commonly used to construct a fuzzy system. However, consequent fuzzy sets in the fuzzy rule bank are often ignored, reducing the approximation capacity. The main mechanism to construct the fuzzy system lies in the weighted average of the center of consequent fuzzy sets whose weight is the height of the corresponding set. Consequently, it is an indicator that cannot be neglected to know the influence of consequent fuzzy sets’ center on approximation capability, see [1 –3].

In 1992, the universal approximation of one kind of fuzzy system has been proved by Wang [4] through least square method and fuzzy basis function, and the approximation capability and reduction in the number of rules in hierarchi-cal fuzzy system have been analyzed through the hierarchy input of high-dimensional variables [5]. In 1994, the single-input single-output (SISO) fuzzy system model with the generalized fuzzifia-tion has been put forward by Ref. [6], who has also proved the approximation capability of SISO and multiple-input single-output (MISO) fuzzy systems from the perspective of infinite norm. In 1997, Yeung [7] carried out a new comparative study based on similarity and fuzzy reasoning method for the first time, later, some methods and their applications were also studied by Raha [8]. In 1997, Mouzouris and Mendel [9] studied some practical applications for non-singleton fuzzy logic systems, and gave the theoretical analysis of this fuzzy system. In 2001, a concept of piecewise linear function on multi-dimensional space was first introduced by Ref. [10]. The approximation ability of the generalized Mamdani fuzzy system to a class of integrable functions was studied in significances of p-integral norm, and then the application fields of fuzzy systems are expanded. The results represent the mainstream development and trend of the fuzzy system theory at that time. However, a complete theoretical system has not yet been developed and formed.

In 2009, Ref. [11] proposed the concept of fuzzy similarity degree based on axiomatic description and established the SISO fuzzy system model through fuzzy system degree and reasoning algor-ithm, thus testifying the approximation capability of the fuzzy system. In 2010, the notion of the norm of generalized fuzzy system had been put forward in [12]. Then, the modeling of several fuzzy systems was discussed by Ref. [13] with singleton fuzzification and implication operator, but the method was only confined to SISO fuzzy system; In 2012, Ref. [14] integrated the Mamdani and T-S fuzzy systems to establish a generalized hybrid fuzzy system, proving that the approxima-tion capability of the system has remained, and the number of rules are greatly reduced inside the system. In 2014, Ref. [15] studied the approximat-ion capability of Mamdani fuzzy system and how it is realized by introducing K-integral norm and reduction operator. Moreover, Refs. [16, 17] discu-ssed the approximation capability of a special fuzzy system to smooth function based on Bernstein polynomial expression. In 2015, in term of the Kp-integral norm Ref. [18] give a quantitative characterization and calculation of a generalized Mamdani fuzzy system using multiv-ariate piecewise linear functions. In 2017, Ref. [19] proposed a new method for mesh constructing piecewise linear functions, and the approximation process in Mamdani and T-S fuzzy system was discussed. These previous achievements provide the theoretical support for further studies on the approximation capability or error accuracy of a generalized fuzzy system.

The remainder of the paper is organized as follows: In Section 2, some basic concepts for fuzzy similarity degree, fuzzification and defuzzi-fication are briefly summarized. In Section 3, the modeling method of the Mamdani fuzzy system is proposed, the new calculation formula of the consequent fuzzy sets is given by fuzzy similarity degree, and the analytic expression of the fuzzy system are obtained by one practical example. In Section 4, a new output algorithm of Mamdani fuzzy system is put forward through space positioning method. In Section 5, a simulation example shows that the Mamdani fuzzy system constructed by a fuzzy similarity degree and the Gauss fuzzification is superior to that based on singleton fuzzification in terms of approximation capability.

2 Basic definitions

Fuzzy similarity degree can be used to measure the similarity degree between two fuzzy sets according to local information. In reality, dealing with the local information among fuzzy sets and simplifying the reasoning process is effective.

Let the $ℝ$ be a set of real numbers, $ℝ^{n}$ an n-dimensional Euclidean space, $U \subset ℝ^{n}$ a non-empty domain, F (U) a set of all fuzzy sets on U. Below is the axiomatic definition of fuzzy similarity degree.

Definition 2.1. Let a mapping FS : F (U) × F (U) → F (U). If the mapping FS satisfies the following conditions a) – c):

FS (A, B) = FS (B, A).

Ø ⊂ FS (A, B) ⊂ U, and A≠ Ø, then FS (A, A) = U.

If A ⊂ B ⊂ C, then FS (A, C) ⊂ FS (A, B) ∩FS (B, C).

Then the fuzzy mapping FS is called a fuzzy similarity degree on F (U).

Obviously, Definition 2.1 is presented in axiom form, which is only theoretical, but practically, it needs specific similarity degree to show its meaning. Such as, for any A, B ∈ F (U), x = (x ₁, x ₂, …, x _n) ∈ U, the membership function of a fuzzy set FS (A, B) is showed as follows: $(i) FS (A, B) (x) = {\begin{matrix} 1 - \frac{| A (x) - B (x) |}{A (x) \lor B (x)}, \\ A (x) > 0 or B (x) > 0 \\ 0, A (x) = B (x) = 0 \end{matrix}$ $(ii) FS (A, B) (x) = {\begin{matrix} \frac{2 (A (x) \land B (x))}{A (x) + B (x)}, \\ A (x) > 0 or B (x) > 0 \\ 0, A (x) = B (x) = 0 \end{matrix}$

Obviously, according to Definition 2.1, it is not difficult to verify that the mapping FS in the formulas (i) and (ii) is a fuzzy similarity degree on the domain U. However, it is noteworthy that Definition 2.1 is given in axiomatic form, it is only principled. In practical application, it is necessary to give a specific fuzzy similarity degree in order to further construct a fuzzy system. Normally, in the fuzzy rule base, a fuzzy system is mainly composed of three parts: the fuzzy inference engine, the fuzzification and defuzzification. And the fuzzy similarity degree is mainly used to describe a concrete degree of similarity between two fuzzy sets. Therefore, it is not hard to imagine utilizing the fuzzy similarity degree FS to design some inference rules, and then a generalized fuzzy system may be established with the inference calculation and consequent fuzzy sets. Of course, the proposed new method for constructing a fuzzy systems is also of great significance.

Definition 2.2. The Gauss fuzzification is a process of mapping any real-valued point x ^* = $(x_{1}^{*}$ , $x_{2}^{*}, \dots, x_{n}^{*})$ $\in U \subset ℝ^{n}$ into fuzzy set A′ on U, the membership function of A′ is $\begin{matrix} A^{'} (x) = exp (- {(\frac{x_{1} - x_{1}^{*}}{σ_{1}})}^{2}) \\ * exp (- {(\frac{x_{2} - x_{2}^{*}}{σ_{2}})}^{2}) * \\ \dots * exp (- {(\frac{x_{n} - x_{n}^{*}}{σ_{n}})}^{2}) . \end{matrix}$

In the above expression, the arbitrary point x = (x ₁, x ₂, …, x _n) ∈ U, the parameter σ _i > 0, i = 1, 2, …, n, and operation “*” is a t- norm, which is usually used as the algebra product or minimal operator. The Fig. 1 below shows the geometric interpretation of a Gauss fuzzification in two-dimensional space.

Fig. 1

Geometric representation of Gauss fuzzification when n = 2.

Note 1. Fuzzification is a process of turning a clear input variable into a fuzzy set for easier fuzzy reasoning calculation. It equates to a fuzzy transformation Δ : U → F (U). Actually, a common point of the three kinds of fuzzification (single point, Gauss and triangular) is that they satisfy the equation A′ (x ^*) =1. The single-point fuzzification is simple but easily lose information, the Gauss fuzzification is complicated, but it can describe more complete information.

Definition 2.3. A singleton fuzzification is a process of mapping any real-valued point x ^* = $(x_{1}^{*}, x_{2}^{*}$ , $\dots, x_{n}^{*}) \in U \subset ℝ^{n}$ into a singleton fuzzy set A′ on U, the membership function of A′ is $A^{'} (x) = {\begin{matrix} 1, x = x^{*} \\ 0, x \neq x^{*} \end{matrix} .$

Definition 2.4. Let $U \subset ℝ$ , a fuzzy set A ∈ F (U). Then $sup_{x \in U} A (x)$ is the height of A. Especially, when $sup_{x \in U} A (x) = 1$ , A is a standard fuzzy set; if all the points where A (x) reaches the maximum value have limited mean value, the mean value is the center of A.

Definition 2.5. Let {A ₁, A ₂, …, A _N} be a fuzzy set family on U, with the following definitions:

For any x ∈ U, there is i ₀ ∈ {1, 2, …, N} which make A _i
₀ (x) >0, then it is called that {A ₁, A ₂, …, A _N} is complete on U, i.e., the completeness means that U must be entirely covered by the given set family without any inter space.

For arbitrary x ∈ Ker (A _i), j = 1, 2, …, N, and satisfy the condition A _i (x) =0(i ≠ j), then it is called that {A ₁, A ₂, …, A _N} is consistent on U, i.e., the consistency means that the membership functions of two adjacent fuzzy sets must intersect, but the intersection is not beyond bounds within their own domain.

Definition 2.6. Suppose B′ is the integration of output fuzzy set {B ¹, B ², …, B ^M} corresponding to M, IF-THEN rules in the fuzzy rule library. If the center ${\bar{y}}^{l}$ of M fuzzy sets is weighted by the height w _l, then $y^{*} = \sum_{l = 1}^{M} {\bar{y}}^{l} w_{l} / - \sum_{l = 1}^{M} w_{l}$ is the center average defuzzification of B′, where the weight is equal to the height w _l of the corresponding fuzzy sets.

Defuzzification is a process of transforming B′ in $V \subset ℝ$ into y ^* ∈ V. The center average defuzzification is intuitive and reasonable, although the calculation amount is so large to conduct; the maximum defuzzification is easier in calculation, but even a tiny change in B′ is likely to result in huge change in y ^*, so the cen-ter average defuzzification is the most practical method at present.

3 The construction of Mamdani fuzzy system

A fuzzy system is mainly composed of fuzzy inference engine,fuzzification and defuzzification. Fuzzy similarity degree describes the similarity degree between fuzzy sets. In this section, a fuzzy similarity degree FS was used to build the rules of inference synthesis, and the membership function of output fuzzy sets will be calculated according to independent inference synthesis to set up a specific Mamdani fuzzy system.

First, a generalized modus ponens is selected:

IF x is A, THEN y is B

IF x is A′

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––,

Conclusion y is B′

where the membership function of the output fuzzy set B′ on $V \subset ℝ$ is $B^{'} (y) = sup_{x \in U} t (A^{'} (x), (A \to B) (x, y)),$ where A and A′ are the input fuzzy sets on U, B and B′ are the output fuzzy set on V, t is a t-norm, and “→” means some kind of implication. The fuzzy rule base is made up by the IF-THEN rules as follows: $R_{u}^{l} : IF x_{1} is A_{1}^{l}, x_{2} is A_{2}^{l}, \dots, x_{n} is A_{n}^{l},$ $THEN y is B^{l},$ where l = 1, 2, …, M, $A_{i}^{l}$ is the input fuzzy set along the i coordinate axis on $U_{i} \subset ℝ$ , whereas B ^l is the output fuzzy set of l rule on U _i, and in the input variable x = (x ₁, x ₂, …, x _n) ∈U ₁ × U ₂ ×… × U _n = U ⊂ $ℝ^{n}, y \in V \subset ℝ$ .

Since fuzzy rules stand for a fuzzy relation, let $R_{u}^{l} = A_{1}^{l} \times A_{2}^{l}$ $\times \dots \times A_{n}^{l}$ →B ^l, $A^{l} = A_{1}^{l}$ $\times A_{2}^{l} \times \dots \times A_{n}^{l}$ , where A ^l is a fuzzy relation on U ₁ × U ₂ × … × U _n, its membership function can be represented as $\begin{matrix} A^{l} (x) = (A_{1}^{l} \times A_{2}^{l} \times \dots \times A_{n}^{l}) (x_{1}, x_{2}, \dots, x_{n}) \\ = A_{1}^{l} (x_{1}) * A_{2}^{l} (x_{2}) * \dots * A_{n}^{l} (x_{n}) . \end{matrix}$

Particularly, when the operation * is used as the algebra product, the membership function of the fuzzy relation $R_{u}^{l}$ can be defined as $R_{u}^{l} (x, y) = (\prod_{i = 1}^{n} A_{i}^{l} (x_{i})) B^{l} (y), l = 1, 2, \dots, M .$

Hence, the common fuzzy inference model can be described as:

Hypothesis A ¹ → B ¹, A ² → B ², …, A ^M → B ^M

Known A′

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– .

Result B′

Here, $A^{l} = A_{1}^{l} \times A_{2}^{l} \times \dots \times A_{n}^{l}$ , $A_{i}^{l}$ , $A_{i}^{l} \in F (U_{i})$ ; B ^l, B′ ∈ F (V), l = 1, 2, …, M.

According to every rule in the fuzzy rule base of independent inference, a consequent fuzzy set can be determined while the output of the whole fuzzy inference engine is the combination or integration of M-independent fuzzy sets. Using the generalized modus ponens, the membership function of a consequent fuzzy set $B_{l}^{'}$ of fuzzy rule $R_{u}^{l}$ can be shown as $B_{l}^{'} (y) = sup_{x \in U} t (A^{'} (x), R_{u}^{l} (x, y)) .$ (1)

If fuzzy similarity degree FS is used to replace t-norm in Formula (1) and an independent inference of “fuzzy combination” and Mamdani product are used, the membership function of the output fuzzy set B′ is $B^{'} (y) = {max}_{l = 1}^{M} (sup_{x \in U} FS (A^{'}, A^{l}) (x) B^{l} (y)),$ (2) where A′ and $A^{l} = A_{1}^{l} \times A_{2}^{l} \times \dots \times A_{n}^{l}$ are the fuzzy sets on U, $A_{i}^{l}$ is the input fuzzy set on $U_{i} \subset ℝ$ , x = (x ₁, x ₂, …, x _n) ∈ U are the input variables, and B ^l is the output fuzzy set of l rule on $V \subset ℝ$ . Usually, we choose B ^l as the standard fuzzy set, and ${\bar{y}}^{l}$ is the center of B ^l and $B^{l} ({\bar{y}}^{l}) = 1$ .

Since FS is presented according to Definition 2.1. In practice, only specific fuzzy similarity degree and fuzzification contribute to the production of analytic expression of Mamdani fuzzy set, the Gauss fuzzification was used to map a real-valued point $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) \in U$ into a fuzzy set A′ on U.

In Formula (2), if let $C^{l} (y) = sup_{x \in U} FS (A^{'}, A^{l}) (x) B^{l} (y),$ then C ^l is a fuzzy set on V, the expression $sup_{x \in U} FS (A^{'}, A^{l}) (x)$ is equivalent to a compression constant. It affects only the height of the fuzzy set C ^l but does not affect the center of C ^l, so the center of C ^l equals to ${\bar{y}}^{l}$ .

For input x = (x ₁, x ₂, …, x _n) ∈ U, the height w _l of output fuzzy set C ^l can be taken as $w_{l} = C^{l} ({\bar{y}}^{l}) = sup_{x \in U} FS (A^{'}, A^{l}) (x) .$

Without considering which fuzzification to be selected, according to product inference engine and center average fuzzification, the abstract expression of a specific Mamdani fuzzy system by integrating w _l into Definition 2.6 will be obtained as follows: $y^{*} = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} {sup}_{x \in U} FS (A^{'}, A^{l}) (x)}{\sum_{l = 1}^{M} {sup}_{x \in U} FS (A^{'}, A^{l}) (x)} .$ (3)

Especially, for any real valued point $x^{*} = (x_{1}^{*}, x_{2}^{*}$ , $\dots, x_{n}^{*}) \in U$ , there will be $sup_{x \in U} FS (A^{'}, A^{l}) (x) =$ $(sup_{x \in U - {x^{*}}} FS (A^{'}, A^{l}) (x)) \lor sup_{x \in {x^{*}}} FS (A^{'}, A^{l}) (x) =$ $(sup_{x \in U - {x^{*}}} FS (A^{'}, A^{l}) (x)) \lor FS (A^{'}, A^{l}) (x^{*}),$ where $sup_{x \in U - {x^{*}}} FS (A^{'}, A^{l}) (x)$ is a constant related to l rather than x = (x ₁, x ₂, …, x _n) ∈ U, A′ is an input fuzzy set on U, and $A^{l} = A_{1}^{l} \times A_{2}^{l}$ $\times \dots \times A_{n}^{l}$ is a fuzzy relation, its membership function is $A^{l} (x) = \prod_{i = 1}^{n} A_{i}^{l} (x_{i})$ .

For arbitrariness based on real-valued point x ^*, if let x ^* = x and y ^* = F (x), Formula (3) will be generalized as follows:

$F (x) =$

$\begin{matrix} \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} ((sup_{x^{'} \in U - {x}} FS (A^{'}, A^{l}) (x^{'})) \lor FS (A^{'}, A^{l}) (x))}{\sum_{l = 1}^{M} ((sup_{x^{'} \in U - {x}} FS (A^{'}, A^{l}) (x^{'})) \lor FS (A^{'}, A^{l}) (x))}, \end{matrix}$ (4) where the center ${\bar{y}}^{l}$ of the output fuzzy set B ^l can be expressed as ${\bar{y}}^{l} = {\bar{y}}_{i_{1} i_{2} \dots i_{n}} = g (x_{i_{1} i_{2} \dots i_{n}}^{*})$ , here, g (x) is the approximating function for the given data pair, and $x_{i_{1} i_{2} \dots i_{n}}^{*} = (x_{1}^{i_{1}}, x_{2}^{i_{2}}, \dots, x_{n}^{i_{n}})$ is any vertex coordinate in subdivision domain.

Next, for a given fuzzy similarity degree we utilize the Gauss fuzzification and singleton fuzzification methods through an example to calculate the analytic expression of the specific Mamdani fuzzy system as follows:

Example 1. Assume that a fuzzy similarity degree FS is selected according to Formula (ii), for any real-valued point $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) \in U$ . The membership function of output fuzzy set B′ and the analytic expression of Mamdani fuzzy system can be worked out according to the Gauss fuzzification (or singleton fuzzification), product inference engine, center average defuzzification.

In fact, according to the preceding Formula (ii), the fuzzy similarity degree FS is shown as $FS (A, B) (x) = \frac{2 (A (x) \land B (x))}{A (x) + B (x)},$ where A (x) >0 or B (x) >0.

Application of the Gauss fuzzification. For arbitrary x = (x ₁, x ₂, …, x _n) ∈ U, according to the condition A′ (x ^*) =1 and Formula (2), the membership function of the output fuzzy set B′ is $B^{'} (y) = {max}_{l = 1}^{M} (sup_{x \in U} (\frac{2 (A^{'} (x) \land A^{l} (x)}{A^{'} (x) + A^{l} (x)}) B^{l} (y))$ $= {\begin{matrix} {max}_{l = 1}^{M} sup_{x \in U} (\frac{2 (\prod_{i = 1}^{n} G (x_{i}) \land \prod_{i = 1}^{n} A_{i}^{l} (x_{i}))}{\prod_{i = 1}^{n} G (x_{i}) + \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}) B^{l} (y), \\ x \neq x^{*} \\ {max}_{l = 1}^{M} sup_{x \in U} (\frac{2 \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}{1 + \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}) B^{l} (y), x = x^{*} \end{matrix}$ $= {\begin{matrix} {max}_{l = 1}^{M} β_{l} B^{l} (y), x \neq x^{*} \\ {max}_{l = 1}^{M} (\frac{2 \prod_{i = 1}^{n} A_{i}^{l} (x_{i}^{*})}{1 + \prod_{i = 1}^{n} A_{i}^{l} (x_{i}^{*})}) B^{l} (y), x = x^{*} \end{matrix},$ where the Gauss function G (x _i) is shown as $G (x_{i}) = exp (- ((x_{i} - x_{i}^{*}) / - σ_{i})^{2}), G (x_{i}^{*}) = 1 .$

Here, the constant $\begin{matrix} β_{l} & = sup_{x \in U} (\frac{2 (\prod_{i = 1}^{n} G (x_{i}) \land \prod_{i = 1}^{n} A_{i}^{l} (x_{i}))}{\prod_{i = 1}^{n} G (x_{i}) + \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}) \\ \geq 0, l = 1, 2, \dots, M . \end{matrix}$

Similarly, we can also obtain that the height ω _l of output fuzzy set C ^l is $ω_{l} = {\begin{matrix} β_{l}, x \neq x^{*} \\ \frac{2 \prod_{i = 1}^{n} A_{i}^{l} (x_{i}^{*})}{1 + \prod_{i = 1}^{n} A_{i}^{l} (x_{i}^{*})}, x = x^{*} . \end{matrix}$

According to Formula (4), for any x = (x ₁, x ₂, …, x _n) ∈ U, the Mamdani fuzzy system F ₁ (x) based on Gauss fuzzification can be expressed as $F_{1} (x) = F_{1} (x_{1}, x_{2}, \dots, x_{n})$ $= \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} (β_{l} \lor \frac{2 \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}{1 + \prod_{i = 1}^{n} A_{i}^{l} (x_{i})})}{\sum_{l = 1}^{M} (β_{l} \lor \frac{2 \prod_{i = 1}^{n} A_{i}^{l} (x_{i})}{1 + \prod_{i = 1}^{n} A_{i}^{l} (x_{i})})} .$ (5)

Application of single-point fuzzification. For arbitrary x = (x ₁, x ₂, …, x _n) ∈ U, according to A′ (x ^*) =1 and Formula (2), the membership function of the output fuzzy set B′ is

B^{'} (y) = {max}_{l = 1}^{M} (sup_{x \in U} (\frac{2 (A^{'} (x) \land A^{l} (x)}{A^{'} (x) + A^{l} (x)}) B^{l} (y))

= {max}_{l = 1}^{M} sup_{x \in U} {\begin{matrix} (\frac{2 A^{l} (x^{*})}{1 + A^{l} (x^{*})} B^{l} (y)), x = x^{*} \\ 0, x \neq x^{*} \end{matrix}

= {max}_{l = 1}^{M} (\frac{2 A^{l} (x^{*})}{1 + A^{l} (x^{*})} B^{l} (y)) .

In this moment, there is $β_{l} = sup_{x \in U - {x^{*}}} (\frac{2 (A^{'} (x) \land A^{l} (x)}{A^{'} (x) + A^{l} (x)}) = 0 .$

In addition, the height of C ^l is $w_{l} = A^{l} (x^{*}) = \prod_{i = 1}^{n} \frac{2 A^{l} (x_{i}^{*})}{1 + A^{l} (x_{i}^{*})} .$

According to Formula (4), the Mamdani fuzzy system F ₂ (x) can be expressed as $F_{2} (x) = F_{2} (x_{1}, x_{2}, \dots, x_{n})$ $= \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} (\prod_{i = 1}^{n} \frac{2 A_{i}^{l} (x_{i})}{1 + A_{i}^{l} (x_{i})})}{\sum_{l = 1}^{M} (\prod_{i = 1}^{n} \frac{2 A_{i}^{l} (x_{i})}{1 + A_{i}^{l} (x_{i})})} .$ (6)

It is not hard to see from Example 1, the fuzzy system (6) is calculated with the application of singleton fuzzification are much easier than the fuzzy systems (5) worked out with application of the Gauss fuzzification, but the approximation capability of the former two is rather weaker than that of the latter two mainly because in singleton fuzzification, only x ^* is changed into singleton fuzzy set, leading to serious loss of useful information. That is why in Formulas (5) the output value F _i (x ₁, x ₂, …, x _n) of the Mamdani fuzzy system can be calculated by the antecedent fuzzy sets ${A_{i}^{l}}$ , i = 1, 2, …, n; l = 1, 2, …, M.

In general, the modeling methods of the above mentioned fuzzy systems are mainly shown in Formula (3) or (4), where the abstract fuzzy similarity degree FS plays a crucial role. In addition, how to calculate the consequent fuzzy set $B_{l}^{'}$ and output fuzzy set B′ are also two key indicators. However, the usual modeling of the Mamdani fuzzy system or T-S fuzzy system can be constructed only through several steps: the fuzzy inference engine, fuzzification and defuzzification. Intuitively, their analytical expressions are only related to the subdivision number and input antecedent. see [18, 19], and their approximation performance is not very good, see the simulation example below. In this paper, from another point of view a new tool FS called “fuzzy similarity degree “ is introduced to construct a generalized Mamdani fuzzy system or T-S fuzzy system. This is mainly because the fuzzy similarity degree FS is abstractly given and has more generality. So once the mapping FS is given, the analytical expression of the corresponding fuzzy system will be determined. Of course, there may be many ways to construct a fuzzy system waiting for us to continue to explore, but whether a fuzzy system has good performance is a common concern.

4 Output algorithm

In the above section, the analytic expression of Mamdani fuzzy system, namely, F ₁ (x) and F ₂ (x) were presented according to fuzzy similarity degree. However, how to work out the system’s output value corresponding to x = (x ₁, x ₂, …, x _n) ∈ U has not been discussed. Hence, a output algorithm is to be designed with subdivision input space U to work out accordingly the output value of the Mamdani fuzzy system.

Assuming U = U ₁ × U ₂ × … × U _n, where U _i = [a _i, b _i], i = 1, 2, …, n, the procedures to design the output algorithm of Mamdani fuzzy system in Formulas (3) and (4) are as follows:

Step 1. For any fixed $m \in ℕ$ , choosing m - 1 equidistant subdivision points are to be inserted equidistantly in the closed interval [a _i, b _i] corresponding to the subdivision points $t_{i}^{j}$ : $a_{i} = t_{i}^{0} < t_{i}^{1} < t_{i}^{2} < \dots < t_{i}^{m - 1} < t_{i}^{m} = b_{i},$ where i = 1, 2, …, n. The [a _i, b _i] is divided into m smaller closed intervals $[t_{i}^{j - 1}, t_{i}^{j}]$ , and a family of fuzzy set is defined with $t_{i}^{j}$ as a peak point, where a triangular or trapezoid fuzzy number is commonly selected to serve as the output, as is shown in Fig. 2.

Step 2. Arbitrarily determine the real valued points $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) \in Δ_{ij}$ within the small space region Δ _ij which x = (x ₁, x ₂, …, x _n) ∈ U belongs to, where x ^* can be picked randomly but is supposed to be close to x. Each σ _i is a given parameter, i = 1, 2, …, n.

Step 3. The total number of non-zero fuzzy sets corresponding to every component x _i in Δ _ij is located. The product of the number of nonzero fuzzy sets on each x _i-axis is the total number of actual rules which determines the number of “sum expression” of the system’s output value.

Step 4. According to x = (x ₁, x ₂, …, x _n) ∈ Δ _ij in determining the vertex coordinates (the black dots in Fig. 2 in Δ _ij. If the vertex coordinates may be expressed abstractly as $(t_{1}^{j_{1}}, t_{2}^{j_{2}}, \dots, t_{n}^{j_{n}})$ , and let g be the to-be approximating function, then the center ${\bar{y}}^{j_{1} j_{2} \dots j_{n}}$ of the output fuzzy set B ^{j₁j₂…j_n} is selected as ${\bar{y}}^{j_{1} j_{2} \dots j_{n}} = g (t_{1}^{j_{1}}, t_{2}^{j_{2}}, \dots, t_{n}^{j_{n}}) .$

Fig. 2

Fuzzy subdivision figure on [0, 1] × [0, 1] when n = 2 and m ₀ = 5.

Step 5. Calculating the supremum of fuzzy similarity $sup_{x \in U - {x^{*}}} FS (A^{'}, A^{l}) (x)$ based on a given fuzzy similarity degree and the Gauss fuzzification formula, where A′ is expressed as Definition 2.2, the membership function of A ^l is $A^{l} (x) = \prod_{i = 1}^{n} A_{i}^{l} (x_{i})$ , for arbitrarily x = (x ₁, $x_{2}, \dots, x_{n}) \in U \subset ℝ^{n}$ .

Step 6. For an input variable x = (x ₁, x ₂, …, x _n) ∈U, integrate all the parameter set in the above steps into Formulas (3) and (4) to work out the output value of this system.

Note 2. The sign $\sum_{l = 1}^{M} (\cdot)$ can be decomposed as $\sum_{j_{1} = 1}^{N_{1}} \sum_{j_{2} = 1}^{N_{2}} \dots \sum_{j_{n} = 1}^{N_{n}} (\cdot)$ and the total number of fuzzy rules M = N ₁ N ₂ … N _n. Fuzzy similarity degree FS is used to replace t-norm in the Mamdani fuzzy system established in this section, which means that the membership function of B′ on V must be re-inferred. Besides, the selection of the center ${\bar{y}}^{j_{1} j_{2} \dots j_{n}}$ is quite important.

Lemma 1. [22] Let g be a continuous differentiable function in compact set U = [a ₁, b ₁]× [a ₂, b ₂] $\times \dots \times [a_{n}, b_{n}] \subset ℝ^{n}$ . Then, for any ɛ > 0, there is a dissection number $m_{0} \in ℕ$ , when m > m ₀, the Mamdani fuzzy system (4) will satisfy ||F - g||_∞ < ɛ, where $m_{0} = [\frac{b - a}{ɛ} \sum_{j = 1}^{n} | | \frac{\partial g}{\partial x_{j}} | |_{\infty}],$ $b = max_{1 \leq i \leq n} b_{i}$ and $a = min_{1 \leq i \leq n} a_{i}$ , and || · ||_∞is an infinite norm.

5 Simulation example

The output value of a certain Mamdani fuzzy system can be worked out by the above output algorithm. For simplicity, only the actual output value of some sample points in Mamdani fuzzy system are calculated on a two dimensional Euclidean space, and its approximation effect will be compared with a singleton fuzzification and Gauss fuzzification, respectively.

Example 2. Assume that the to be approximating function is $g (x, y) = \frac{1}{2} sin (x + y)$ .

For arbitrary $(x, y) \in [- 1, 1] \times [- 1, 1] \subset ℝ^{2}$ , themembership function of a given fuzzy similarity degree FS is $FS (A, B) (x, y) = {\begin{matrix} 1 - \frac{| A (x, y) - B (x, y) |}{A (x, y) \lor B (x, y)}, \\ A (x, y) > 0 or B (x, y) > 0 \\ 0, A (x, y) = B (x, y) = 0 \end{matrix}$

In fact, by the above method of the Example 1 it is easy to obtain that the specific Mamdani fuzzy system based on the Gauss fuzzification and singleton fuzzification. In fact, for a particular Mamdani fuzzy system F ₃ based on the Gauss fuzzification can expressed as $F_{3} (x, y) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} (α_{l} (y) \lor A_{1}^{l} (x) A_{2}^{l} (y))}{\sum_{l = 1}^{M} (α_{l} (y) \lor A_{1}^{l} (x) A_{2}^{l} (y))},$ (7) where $F_{3} (x, y) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} (α_{l} (y) \lor A_{1}^{l} (x) A_{2}^{l} (y))}{\sum_{l = 1}^{M} (α_{l} (y) \lor A_{1}^{l} (x) A_{2}^{l} (y))},$ In addition, a specific Mamdani fuzzy system F ₄ based on the singleton fuzzification can expressed as $F_{4} (x, y) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} A_{1}^{l} (x) A_{2}^{l} (y)}{\sum_{l = 1}^{M} A_{1}^{l} (x) A_{2}^{l} (y)} .$ (8) Let a given approximation accuracy ɛ = 0.4 > 0, adjusting parameter σ _i ≡ 1, and selected the real value point $x^{*} = (0, \frac{4}{5}) \in [- 1, 1] \times [- 1, 1]$ . Below, we will analyze the approximation capability of the Mamdani fuzzy system F ₂ to a given continuous function g in the sense of the Gauss fuzzification.

According to the six steps in output location algorithm, the output value of F ₁ (x, y) can be worked out. Apparently, g (x, y) is continuous and differentiable, and with $\frac{\partial g}{\partial x} = \frac{\partial g}{\partial y} = \frac{1}{2} cos (x + y),$ there is ${∥ \frac{\partial g}{\partial x} ∥}_{\infty} = {∥ \frac{\partial g}{\partial y} ∥}_{\infty} = sup_{(x, y) \in [- 2, 2]^{2}} | \frac{1}{2} cos (x + y) | = \frac{1}{2} .$ In accordance with Lemma 1, the subdivision number m ₀ (or M) can be determined by the formula: $M = m_{0} = \frac{b - a}{ɛ} \sum_{j = 1}^{2} {∥ \frac{\partial g}{\partial x_{j}} ∥}_{\infty} = \frac{2}{0.4} (\frac{1}{2} + \frac{1}{2}) = 5 .$ For simplicity, only the first quadrant [0, 1] × [0, 1] is divided at the same interval. Thus, the equal subdivision points (center point) along the x-axis are $0 < \frac{1}{5} < \frac{2}{5} < \frac{3}{5} < \frac{4}{5} < 1$ , and the smaller squares after the division is recorded as Δ _ij (i, j = 1, 2, …, 5), as shown in Fig. 2.

Obviously, the square region [0, 1] × [0, 1] is divided into 25 identical smaller ones whose side length is $\frac{1}{5}$ . Of course, some triangular fuzzy numbers $A_{1}^{j} (j = 0, 1, \dots, 5)$ can be defined by all the equal points of [0, 1] as the peak points (see Fig. 2), where the fuzzy numbers $A_{1}^{- 5}$ and $A_{1}^{5}$ are not presented. Particularly, the membership functions of $A_{1}^{0}$ , $A_{1}^{- 5}$ , and $A_{1}^{5}$ are as follows, respectively: $A_{1}^{0} (x) = {\begin{matrix} 1 + 5 x, - \frac{1}{5} \leq x < 0 \\ 1 - 5 x, 0 \leq x < \frac{1}{5} \\ 0, otherwise, \end{matrix}$ $A_{1}^{- 5} (x) = {\begin{matrix} - 5 x - 4, - 1 \leq x < \frac{4}{5} \\ 0, otherwise, \end{matrix}$ $A_{1}^{5} (x) = {\begin{matrix} 5 x - 4, \frac{4}{5} \leq x < 1 \\ 0, otherwise . \end{matrix}$ If the membership function of $A_{1}^{0}$ is moved in parallel to the left and right $\frac{1}{5}$ unit length on [-1, 1], i.e., $A_{1}^{j} (x) = A (x \pm \frac{j}{5})$ , j = 1, 2, 3, 4. Then a family of triangular fuzzy numbers ${A_{1}^{j}}$ can be obtained along [-1, 1] the x-axis.

Similarly, let $A_{2}^{j} (y) = A_{1}^{j} (y) (j = 0, \pm 1, \dots$ , ±5), the other family of triangular antecedent fuzzy numbers ${A_{2}^{j}}$ can be obtained along [-1, 1] the y-axis. If the center ${\bar{y}}^{i_{1} i_{2}}$ of the output fuzzy set is selected as ${\bar{y}}^{i_{1} i_{2}} = g (\frac{i_{1}}{5}, \frac{i_{2}}{5}), i_{1}, i_{2} = 0, \pm 1, \pm 2, \dots, \pm 5 .$ According to the analytic expression of the given function g, all center values ${\bar{y}}^{i_{1} i_{2}}$ can be determined. In addition, the analytic expressions of each Mamdani fuzzy system F ₃ (x, y) and F ₄ (x, y) can be obtained by substituting ${\bar{y}}^{i_{1} i_{2}} = g (\frac{i_{1}}{5}, \frac{i_{2}}{5})$ in Formulas (7) and (8), respectively.

For example, the analytic expression and space surface of Formula (8) can be shown as $F_{4} (x, y) =$ $\frac{\sum_{i_{1} = 0}^{5} \sum_{i_{2} = 0}^{5} (A_{1}^{i_{1}} (x) A_{2}^{i_{2}} (y)) g (\frac{i_{1}}{5}, \frac{i_{2}}{5})}{\sum_{i_{1} = 0}^{5} \sum_{i_{2} = 0}^{5} A_{1}^{i_{1}} (x) A_{2}^{i_{2}} (y)} .$ (9) Comparison between approximation function g and the space surface of Formula (9) is shown in Figs. 3, 4. For the sake of intuition, we draw their images on [-3, 3] × [-3, 3] as

Fig. 3

Space surface graph of the given function. g(x,y) on [-3, 3] × [-3, 3].

Fig. 4

Space surface graph of Mamdani fuzzy system. F ₄ (x, y) on [-3, 3] × [-3, 3].

Apparently, the sample point $(x, y) = (\frac{1}{20}, \frac{4}{5})$ ∈Δ ₁₄ is taken as an example to calculate the output value of Formula (9): $F_{4} (\frac{1}{20}, \frac{4}{5})$ . In fact, the sample point $(\frac{1}{20}, \frac{4}{5})$ falls on the border of square region Δ ₁₄; $x = \frac{1}{20}$ is located between $A_{1}^{0}$ and $A_{1}^{1}$ on the x-axis, and $y = \frac{4}{5}$ is located on $A_{2}^{4}$ along the y-axis, and there is $\begin{matrix} A_{1}^{0} (\frac{1}{20}) & = 1 - 5 \times \frac{1}{20} = \frac{3}{4}, A_{1}^{1} (\frac{1}{20}) \\ = 5 \times \frac{1}{20} = \frac{1}{4}, A_{2}^{4} (\frac{4}{5}) = 1 . \end{matrix}$ The centers of their corresponding output fuzzy sets are ${\bar{y}}^{04} = g (0, \frac{4}{5}) = \frac{1}{2} sin \frac{4}{5} = 0.35867,$ ${\bar{y}}^{14} = g (\frac{1}{5}, \frac{4}{5}) = \frac{1}{2} sin (\frac{1}{5} + \frac{4}{5}) = 0.42074 .$ With σ _i ≡ 1 and $x^{*} = (0, \frac{4}{5})$ , substituting $(\frac{1}{20}, \frac{4}{5})$ in Formula (9) to obtain $F_{4} (\frac{1}{20}, \frac{4}{5}) =$ $\frac{(1 - \frac{e^{\frac{- 1}{400}} - \frac{3}{4} \times 1}{e^{\frac{- 1}{400}} \lor \frac{3}{4} \times 1}) \times {\bar{y}}^{04} + (1 - \frac{e^{\frac{- 1}{400}} e^{\frac{- 1}{25}} - \frac{1}{4} \times 1}{e^{\frac{- 1}{400}} e^{\frac{- 1}{25}} \lor \frac{1}{4} \times 1}) \times {\bar{y}}^{14}}{2 - (\frac{e^{\frac{- 1}{400}} - \frac{3}{4} \times 1}{e^{\frac{- 1}{400}} \lor \frac{3}{4} \times 1} + \frac{e^{\frac{- 1}{400}} e^{\frac{- 1}{25}} - \frac{1}{4} \times 1}{e^{\frac{- 1}{400}} e^{\frac{- 1}{25}} \lor \frac{1}{4} \times 1})}$ $\approx \frac{0.75188 \times 0.35867 + 0 . 29638 \times 0 . 42074}{2 - 0.98727}$ $\approx 0.38396 .$ Actually, the exact value of g (x, y) at sample point $(\frac{1}{20}, \frac{4}{5})$ is $g (\frac{1}{20}, \frac{4}{5}) = \frac{1}{2} sin \frac{17}{20} \approx 0.37564$ .

If the other five sample points on [0, 1] × [0, 1] are randomly selected as $(\frac{1}{20}, \frac{1}{20}), (\frac{1}{2}, \frac{1}{2}), (\frac{2}{5}, \frac{2}{5}), (\frac{9}{20}, \frac{13}{20}), (\frac{7}{20}, \frac{19}{20}) .$ Analogously, according to the output location algorithm given in the front, the output values and errors of Mamdani fuzzy systems F ₃ and F ₄ can be calculated, as shown in Table 1.

Table 1

Comparison of the output values and errors of Mamdani fuzzy systems F ₃ and F ₄

Number	Sample points	Real-value	g (x, y)	F₃ (x, y)	F₄ (x, y)	\|F₃ (.) - g (.) \|\|	\|F₄ (.) - g (.) \|
	(x, y)	points $(x_{1}^{}, x_{2}^{})$
1	$(\frac{1}{20}, \frac{1}{20})$	(0, 0)	0.04992	0.04942	0.04942	0.00050	0.00050
2	$(\frac{1}{20}, \frac{4}{5})$	$(0, \frac{4}{5})$	0.37564	0.38396	0.36218	0.00832	0.01346
3	$(\frac{1}{2}, \frac{1}{2})$	$(\frac{1}{2}, \frac{3}{5})$	0.42074	0.41654	0.41654	0.00420	0.00420
4	$(\frac{2}{5}, \frac{2}{5})$	$(\frac{2}{5}, \frac{2}{5})$	0.35868	0.35868	0.35868	0.00000	0.00000
5	$(\frac{9}{20}, \frac{13}{20})$	$(\frac{2}{5}, \frac{3}{5})$	0.44560	0.44343	0.44262	0.00217	0.00298
6	$(\frac{7}{20}, \frac{19}{20})$	$(\frac{2}{5}, 1)$	0.48178	0.49043	0.46254	0.00865	0.01924

Similar method, the output values and errors of the Mamdani fuzzy systems F ₁ and F ₂ can also be calculated, as shown in Table 2.

Table 2

Comparison of the output values and errors of Mamdani fuzzy systems F ₁ and F ₂

Number	Sample points	Real-value	g (x, y)	F₁ (x, y)	F₂ (x, y)	\|F₁ (.) - g (.) \|	\|F₂ (.) - g (.) \|
	(x, y)	points $(x_{1}^{}, x_{2}^{})$
1	$(\frac{1}{20}, \frac{1}{20})$	(0, 0)	0.04992	0.05761	0.06281	0.00769	0.01289
2	$(\frac{1}{20}, \frac{4}{5})$	$(0, \frac{4}{5})$	0.37564	0.37413	0.37842	0.00151	0.00278
3	$(\frac{1}{2}, \frac{1}{2})$	$(\frac{1}{2}, \frac{3}{5})$	0.42074	0.41654	0.41654	0.00420	0.00420
4	$(\frac{2}{5}, \frac{2}{5})$	$(\frac{2}{5}, \frac{2}{5})$	0.35868	0.35868	0.35868	0.00000	0.00000
5	$(\frac{9}{20}, \frac{13}{20})$	$(\frac{2}{5}, \frac{3}{5})$	0.44560	0.44618	0.45982	0.00058	0.01422
6	$(\frac{7}{20}, \frac{19}{20})$	$(\frac{2}{5}, 1)$	0.48178	0.48954	0.44598	0.00776	0.03580

Tables 1, 2 clearly show that on the premise of given fuzzy similarity degrees (i) and (ii), the approximation accuracy of F ₁ and F ₃ are superior to that of F ₂ and F ₄. The output value of the other six sample points are displayed in Figs. 5, 6.

Fig. 5

A column contrast diagram of the output values of F ₁ and F ₂ at six sample points.

Fig. 6

A column contrast diagram of the output values of F ₃ and F ₄ at six sample points.

In addition, from Figs. 5, 6, the approximation of F _i (x, y) at the border point or vertex is better, especially at the vertexes where there F _i (x, y) = g (x, y) (i = 1, 2, 3, 4) is unchanged. Such as $\begin{matrix} F_{1} (\frac{2}{5}, \frac{2}{5}) = F_{2} (\frac{2}{5}, \frac{2}{5}) = F_{3} (\frac{2}{5}, \frac{2}{5}) = \\ F_{4} (\frac{2}{5}, \frac{2}{5}) = g (\frac{2}{5}, \frac{2}{5}) = 0.35868 . \end{matrix}$ In summary, the application of the singleton fuzzification often makes the fuzzy system lose some information. This phenomenon leads that the approximation capabilities of F ₂ and F ₄ are weaker. The fuzzy systems F ₁ and F ₃ based on the Gauss fuzzification have higher approximation accuracy, but the calculations within the system are quite complicated. Therefore, in practice, the singleton fuzzification and Gauss fuzzification should be properly selected according to the given accuracy. In addition, we can also apply the t-test method in statistical inference to verify the approximation effect of the fuzzy system.

6 Conclusions

Fuzzy similarity degree is used to describe the approximating degree between two fuzzy sets through local information, so it is widely applied in the modeling of fuzzy system and design of fuzzy controller. In this paper, the formula of the membership function of consequent fuzzy set was put forward according to fuzzy similarity degree and Gauss fuzzification; the analytic expressions of Mamdani fuzzy system through the Gauss fuzzification (or singleton fuzzification), product inference engine, center average defuzzification were established, namely, F ₁ and F ₂, followed by the location algorithm. Results show that with a given fuzzy similarity degree, approximation capability of the fuzzy system F ₁ and F ₃ is superior to that of F ₂ and F ₄. Especially, the approximation capability is much better at the border of subdivision domain or vertex. However, improving the approximation accuracy of the Mamdani fuzzy system and finding a more generalized fuzzy similarity degree or fuzzificati-ion formula should be considered.

Footnotes

Acknowledgments

This work has been supported by National Natural Science Foundation of China (Grant No. 61374009, 61463019).

References

Buckley

J.J.

, Sugeno type controllers are universal controllers, Fuzzy Sets and Systems 53(3) (1993), 299–303.

Kosko

, Fuzzy systems as universal approximators, IEEE Transactions on Computers 43(11) (1994), 1329–1333.

Combs

W.E.

and Andrews

J.E.

, Combinatorial rule explosion eliminated by a fuzzy rule configuration, IEEE Transactions on Fuzzy Systems 6(1) (1998), 1–11.

Wang

L.X.

and Mendel

, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transanctions Neural Networks 3(5) (1992), 807–814.

Wang

L.X.

, Universal approximation by hierarchical fuzzy systems, Fuzzy Set and Systems 93(1) (1998), 223–230.

Zeng

X.J.

and Singh

M.G.

, Approximation theory of fuzzy system-SISO case, IEEE Transaction on Fuzzy Systems 2(2) (1994), 162–176.

Yeung

D.S.

and Tsang

E.C.C.

, A comparative study on similarity-based fuzzy reasoning methods, IEEE Transaction on Systems, Man and Cybernetics-B 27 (1997), 216–227.

Raha

, Pal

and Ray

, Similarity-based approximate reasoning: methodology and application, IEEE Transaction on Systems, Man and Cybernetics-A 32(4) (2002), 541–547.

Mouzouris

G.C.

and Mendel

J.M.

, Non-singleton fuzzy logic systems: Theory and applications, IEEE Transaction on Fuzzy Systems 5 (1997), 56–71.

10.

Liu

P.Y.

and Li

H.X.

, Analyses for Lp-norm approximation capability of generalized Mamdani fuzzy system, Information Sciences 138(2) (2001), 195–210.

11.

Wang

D.G.

, Meng

Y.P.

, Song

W.Y.

and Li

H.X.

, Analysis of fuzzy similarity-based reasoning method and approximation properties of corresponding fuzzy systems, Chinese Journal of Engineering Mathematics 26(3) (2009), 423–430.

12.

H.X.

, Yuan

X.H.

and Wang

J.Y.

, The normal numbers of the fuzzy systems and their classes, Science China. Information Science 53(11) (2010), 2215–2229.

13.

Yuan

X.H.

, Li

H.X.

and Sun

K.B.

, Fuzzy systems and their approximation capability based on parameter singleton fuzzifier methods, Acta Electronica Sinica 39(10) (2011), 2372–2377.

14.

Wang

G.J.

and Duan

C.X.

, Generalized hierarchical hybrid fuzzy system and its universai approximation, Control Theory and Application 29(5) (2012), 673–680.

15.

Wang

G.J.

, Li

X.P.

and Sui

X.L.

, Universal approximation and its realize process of generalized Mamdani fuzzy system in K-integral norms, Acta Automatica Sinica 40(1) (2014), 143–148.

16.

Wang

D.G.

, Song

W.Y.

and Li

H.X.

, Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives, Neurocomputing 149 (2015), 265–274.

17.

Wang

D.G.

, Chen

C.L.P.

, Song

W.Y.

and Li

H.X.

, Error compensated marginal linearization method for modeling a fuzzy system, IEEE Transactions on Fuzzy Systems 23(1) (2015), 215–222.

18.

Tao

Y.J.

, Wang

H.Z.

and Wang

G.J.

, Approximation ability and its realization of the generalized Mamdan fuzzy system in the sense of Kp-integral norm, Acta Electronica Sinica 43(11) (2015), 2284–2291.

19.

Wang

G.J.

and Li

X.P.

, Mesh construction of PLF and its approximation process in Mamdani fuzzy system, Journal of Intelligent & Fuzzy Systems 32(6) (2017), 4213–4225.

20.

Wang

L.X.

, A Course in Fuzzy Systems and Control (Chinese Version). Beijing: Tsinghua University Press, 2003.

21.

Wang

G.J.

, Polygonal Fuzzy Neural Network and Fuzzy System Approximation, Beijing: Science Press, 2017.

22.

Gao

and Wang

G.J.

, Generalized Mamdani fuzzy system based on fuzzy similarity degree and its approximation, Fuzzy Systems and Mathematics 32(1) (2018), 137–143.