Interpolation functions of interval type-2 fuzzy systems

Abstract

The interpolation functions of interval type-2 fuzzy systems and their universal approximation are investigated in this paper. Two types of fuzzification methods are designed to construct the antecedents and consequents of the type-2 inference rules. Then the properties of the fuzzy operator and the type-reduction algorithm are used to integrate all parts of the fuzzy system. Interpolation functions of interval type-2 fuzzy systems, which are proved to be universal approximators, are obtained based on three models, namely single input and single output, double inputs and single output, and multiple inputs and single output. The proposed approach is applied to approximate experiments of dynamic systems so as to evaluate the system performance. The system parameters are optimized by the QPSO algorithm. Experimental results for several data sets are given to show the approximation performances of the proposed interpolation functions are better than those of the interpolation function of the classical type-1 fuzzy system.

Keywords

Interval type-2 fuzzy system interval type-2 fuzzy set interpolation function universal approximation

1 Introduction

At the end of the last century, Karnik and Mendel [1 –3] proposed type-2 fuzzy logic systems (hereafter referred to as type-2 fuzzy systems) based on Zadeh’s theory of type-2 fuzzy sets [4]. Compared with the classical type-1 fuzzy system, the interval type-2 fuzzy system has a stronger ability to deal with uncertainty in the aspects of system modeling, intelligent control, self-learning robots, etc., and has been widely accepted [5 –8]. The application results show that the interval type-2 fuzzy controller has smaller tracking error and better stability and robustness than the classical type-1 fuzzy controller.

An interval type-2 fuzzy system [2] is composed of fuzzifier, inference, rules, type-reducer and defuzzifier. Among them, the type-2 fuzzy inference and type reduction require a huge amount of computation. Many studies have been devoted to simplifying the calculation of these two parts. By studying the properties of type-2 fuzzy operations [9 –12] and the truth value algebra of type-2 fuzzy sets [13 –15], the calculation of type-2 fuzzy inference has been greatly improved. As to type reduction, Karnik-Mendel (KM) algorithms [16] and its improved versions effectively transformed the interval type-2 fuzzy sets into type-1 fuzzy sets, such as enhanced KM (EKM) algorithms [17], the continuous version of KM (CKM) algorithms [18], weighted EKM (WEKM) algorithms [19]. Furthermore, Nagar-Bardini (NB) algorithms [20], weighted NB (WNB) algorithms [21] and other type reduction algorithms [22 –24] have enriched the theoretical foundations for the research and application of type-reduction algorithms.

Although the calculation of interval type-2 fuzzy systems has been greatly simplified, no concise mathematical formula has been derived to express the systems. A classical type-1 fuzzy system can be considered as an interpolation function [25]. For example, a type-1 fuzzy system with double inputs and single output is approximately a binary piecewise interpolation function. However, the interpolation formula of type-2 fuzzy system has not been derived up to now. If its interpolation functions are available, we can well analyze and have a good understanding why the interval type-2 fuzzy system is better than the interval type-1 fuzzy system in the approximation ability, stability and robustness. Therefore, it is of great significance to research the functional expressions of interval type-2 fuzzy systems. This paper will derive the interpolation functions of interval type-2 fuzzy systems by integrating all parts of interval type-2 fuzzy systems. In addition, the computation of the whole type-2 fuzzy systems is reduced and the universal approximation of these interpolation functions is ensured.

This paper is structured as follows. Section 2 introduces some related concepts and theories of type-2 fuzzy sets and systems. Section 3 provides two types of specific methods for the interval type-2 fuzzification. Sections 4 and 5 deal with the interpolation functions of interval type-2 fuzzy systems with single and multiple inputs respectively. Section 6 presents the approximation ability of the proposed interpolation functions compared with the interpolation function of type-1 fuzzy systems by using a simulation example. Section 7 draws the conclusions.

2 Preliminaries

In this section, some concepts about type-2 fuzzy sets and systems are given, and the type-reduction algorithm [16] (named KM algorithm) is introduced. After that, two types of fuzzification methods are designed to construct the antecedents and consequents of the type-2 inference rules in an interval type-2 fuzzy system.

2.1 Type-2 fuzzy sets and systems

A fuzzy set of type-2 $\tilde{A}$ on the domain X is characterized by a membership function $μ_{\tilde{A}} : X \to F ([0, 1]), x \mapsto μ_{\tilde{A}} (x)$ , where $F ([0, 1]) = {f | f : [0, 1] \to [0, 1]}$ , and $f \in F ([0, 1])$ is called fuzzy truth value. The set of type-2 fuzzy sets on the domain X is denoted by $\tilde{F} (X)$ .

Let $\tilde{A}$ , $\tilde{B} \in \tilde{F} (X)$ and ★ be a triangular norm. In this calculation, assume that ★ is continuous. Union and intersection on type-2 fuzzy sets are given as follows. For ∀x ∈ X, ∀ w ∈ [0, 1],

$\begin{matrix} \tilde{A} \cup \tilde{B} : μ_{(\tilde{A} \cup \tilde{B})} (x) (w) \\ ≜ & (μ_{\tilde{A}} (x) ⊔^{(\lor, ★)} μ_{\tilde{B}} (x)) (w) & ≜ & sup_{w = u \lor v} (μ_{\tilde{A}} (x) (u) ★ μ_{\tilde{B}} (x) (v)) \end{matrix}$ (1)

$\begin{matrix} \tilde{A} \cap \tilde{B} : μ_{(\tilde{A} \cap \tilde{B})} (x) (w) \\ ≜ & (μ_{\tilde{A}} (x) ⊓^{(★, ★)} μ_{\tilde{B}} (x)) (w) \\ ≜ & sup_{w = u ★ v} (μ_{\tilde{A}} (x) (u) ★ μ_{\tilde{B}} (x) (v)) \end{matrix}$ (2) Specially, 1 = χ_{1}, 0 = χ_{0}, where $χ_{{1}} (w) = {\begin{matrix} 1, & w = 1, 0, & w \in [0, 1] and w \neq 1, \end{matrix}$ and $χ_{{0}} (w) = {\begin{matrix} 1, & w = 0, 0, & w \in [0, 1] and w \neq 0 . \end{matrix}$ Then for $\forall f \in F ([0, 1])$ , it can be obtained that $\begin{matrix} f ⊔^{(\lor, ★)} 1 = 1, f ⊔^{(\lor, ★)} 0 = f, & & f ⊓^{(★, ★)} 1 = f, f ⊓^{(★, ★)} 0 = 0 . \end{matrix}$

The type-2 fuzzy system as shown in Fig.1 (see [2]) is very similar to the type-1 fuzzy system. Take the system with single input and single output as an example to introduce the construction process of a general type-2 fuzzy system in detail. It adopts the compositional rule of inference (CRI) algorithm in fuzzy inference, the centroid type-reduction algorithm in type-reducer and the center-of-gravity algorithm in defuzzifier.

Fig. 1

Type-2 fuzzy system

Let X and Y be the universe of the input and output variables respectively, where X = [a, b] and Y = [c, d]. Denote {(x_i, y_i)} _0⩽i⩽N a set of the input and output data, where x_i ∈ X and y_i ∈ Y. If the x_i and y_i are respectively fuzzified into the type-2 fuzzy sets ${\tilde{A}}_{i}$ and ${\tilde{B}}_{i}$ , then N fuzzy-2 inference rules are obtained:

$if x is {\tilde{A}}_{i} then y is {\tilde{B}}_{i}, i = 1, \dots, N,$ (3) where ${\tilde{A}}_{i}$ and ${\tilde{B}}_{i}$ are called the antecedent and consequent of the ith inference rules, respectively. According to the CRI algorithm, the inference relation of the ith inference rule is a type-2 fuzzy relation from X to Y. Then ${\tilde{R}}_{i} = {\tilde{A}}_{i} ⋂ {\tilde{B}}_{i}$ . The N fuzzy-2 inference rules should be joined by “or” (corresponding to set theoretical operator “⋃”). The whole type-2 fuzzy inference relation from (13) is $\tilde{R} = ⋃_{i = 1}^{N} {\tilde{R}}_{i},$ i.e.,

$\tilde{R} = ⋃_{i = 1}^{N} ({\tilde{A}}_{i} ⋂ {\tilde{B}}_{i})$ (4) For a type-2 fuzzy system, because the input x′ is a crisp quantity, it must be converted to a type-2 fuzzy set so as to use (13) as follows:

${\tilde{A}}^{*} (x) = {\begin{matrix} χ_{{1}}, & x = x^{'}, \\ {chi}_{{0}}, & x \in X and x \neq x^{'}, \end{matrix}$ (5) which is called type-2 fuzzification. ${\tilde{A}}^{*}$ is the type-2 fuzzy input set. From the expressions (1), (2), (4) and (5), the type-2 fuzzy output set ${\tilde{B}}^{*}$ can be inferred: $\begin{matrix} μ_{{\tilde{B}}^{*}} (y) & = & ⊔_{x \in X}^{(\lor, ★)} ({\tilde{A}}^{*} (x) ⊓^{(★, ★)} ({⊔^{(\lor, ★)}}_{i = 1}^{N} μ_{{\tilde{A}}_{i}} (x) & & ⊓^{(★, ★)} μ_{{\tilde{B}}_{i}} (y))) \\ = & {⊔^{(\lor, ★)}}_{i = 1}^{N} (μ_{{\tilde{A}}_{i}} (x^{'}) ⊓^{(★, ★)} μ_{{\tilde{B}}_{i}} (y)) . \end{matrix}$ We divide the output universe Y into y₁, …, y_M. By using the centroid type-reduction algorithm, the type-reduced set C, i.e., the centroid of ${\tilde{B}}^{*}$ , can be gained: $\begin{matrix} C (y) = sup_{\frac{\sum_{i = 1}^{M} y_{i} w_{i}}{\sum_{i = 1}^{M} w_{i}} = y} T_{i = 1}^{M} μ_{{\tilde{B}}^{*}} (y_{i}) (w_{i}), \end{matrix}$ where $T$ is a t-norm. Finally, the crisp output y′ obtained by using the center-of-gravity algorithm is $\begin{matrix} y^{'} = \frac{\int_{y \in Y} yC (y) dy}{\int_{y \in Y} C (y) dy} . \end{matrix}$

This calculation will take the KM algorithm [16] to give an approximate result for the type-reduced set C. Some related content is introduced below. By using the knowledge of the operation properties of interval type-2 fuzzy set, it can be known that the output ${\tilde{B}}^{*}$ is still an interval type-2 fuzzy set rewritten as $μ_{\tilde{B^{*}}} (y) = [{LMF}_{{\tilde{B}}^{*}} (y), {UMF}_{{\tilde{B}}^{*}} (y)], y \in Y$ , and the type-reduced set Y_c is a classical interval fuzzy set rewritten as Y_c = [c_l, c_r]. Discretize the universe Y into y₁, …, y_M. Then the point c_l can be obtained by the KM algorithm as follows:

Set $h_{i} = \frac{UMF (y_{i}) + LMF (y_{i})}{2}, i = 1, \dots, M$ . Compute $c_{l}^{'} = \frac{\sum_{i = 1}^{M} y_{i} h_{i}}{\sum_{i = 1}^{M} h_{i}}$ .

Find L such as $y_{L} ⩽ c_{l}^{'} ⩽ y_{L + 1}$ .

Compute $\begin{matrix} c_{l}^{″} = \frac{\sum_{i = 1}^{L} y_{i} UMF (y_{i}) + \sum_{i = L + 1}^{M} y_{i} LMF (y_{i})}{\sum_{i = 1}^{L} UMF (y_{i}) + \sum_{i = L + 1}^{M} LMF (y_{i})} . \end{matrix}$

Check if $c_{l}^{″} = c_{l}^{'}$ . If yes, stop. Let $c_{l} = c_{l}^{″}$ . If no, go to Step 5.

Set $c_{l}^{'} = c_{l}^{″}$ , go to Step 2.

The point c_r can be obtained by using a procedure similar to the one described above:

Set $h_{i} = \frac{UMF (y_{i}) + LMF (y_{i})}{2}, i = 1, \dots, M$ . Compute $c_{r}^{'} = \frac{\sum_{i = 1}^{M} y_{i} h_{i}}{\sum_{i = 1}^{M} h_{i}}$ .

Find R such as $y_{R} ⩽ c_{r}^{'} ⩽ y_{R + 1}$ .

Compute $\begin{matrix} c_{r}^{″} = \frac{\sum_{i = 1}^{R} y_{i} LMF (y_{i}) + \sum_{i = R + 1}^{M} y_{i} UMF (y_{i})}{\sum_{i = 1}^{R} LMF (y_{i}) + \sum_{i = R + 1}^{M} UMF (y_{i})} . \end{matrix}$

Check if $c_{r}^{″} = c_{r}^{'}$ . If yes, stop. Let $c_{r} = c_{r}^{″}$ . If no, go to Step 5.

Set $c_{r}^{'} = c_{r}^{″}$ , go to Step 2.

The above algorithms show that c_l and c_r have the following forms: $\begin{matrix} c_{l} = \frac{\sum_{i = 1}^{L} y_{i} UMF (y_{i}) + \sum_{i = L + 1}^{M} y_{i} LMF (y_{i})}{\sum_{i = 1}^{L} UMF (y_{i}) + \sum_{i = L + 1}^{M} LMF (y_{i})}, \\ c_{r} = \frac{\sum_{i = 1}^{R} y_{i} LMF (y_{i}) + \sum_{i = R + 1}^{M} y_{i} UMF (y_{i})}{\sum_{i = 1}^{R} LMF (y_{i}) + \sum_{i = R + 1}^{M} UMF (y_{i})}, \end{matrix}$ where L and R are switch points satisfying y_L ⩽ c_l ⩽ y_L+1, y_R ⩽ c_r ⩽ y_R+1 .

3 The methods to construct the antecedents and consequents of the inference rules of interval type-2 fuzzy systems

In this section, two methods are designed to construct the antecedents and consequents of the inference rules of interval type-2 fuzzy systems by taking the system with single input and single output as an example. Denote IOD₁={(x_i, y_i) ∣ 1 ⩽ i ⩽ N}. Assume that the data sets IOD₁ used to construct the interval type-2 fuzzy system satisfies that

$if N \to \infty then max_{1 ⩽ i ⩽ N - 1} x_{i + 1} - x_{i} \to 0 .$ (6)

Given a data set IOD₁, where a = x₁ < ⋯ < x_N = b . The antecedents ${{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}$ and the consequents ${{\tilde{B}}_{i}}_{1 ⩽ i ⩽ N}$ will be constructed from {x_i} _1⩽i⩽N and {y_i} _1⩽i⩽N, respectively. First, {x_i} _1⩽i⩽N are fuzzified into a group of triangular or parabola fuzzy sets {A_i} _1⩽i⩽N on the universe X as follows.

Triangular fuzzification: $\begin{matrix} A_{1} (x) = {\begin{matrix} \frac{x - x_{2}}{x_{1} - x_{2}}, & x_{1} ⩽ x ⩽ x_{2}, \\ 0, & x_{2} < x ⩽ x_{N}, \end{matrix} \\ A_{i} (x) = {\begin{matrix} \frac{x - x_{i - 1}}{x_{i} - x_{i - 1}}, & x_{i - 1} ⩽ x ⩽ x_{i}, \\ \frac{x - x_{i + 1}}{x_{i} - x_{i + 1}}, & x_{i} < x ⩽ x_{i + 1}, \\ i = 2, \dots, N - 1, \\ 0, & others, \end{matrix} \end{matrix}$ (7) $A_{N} (x) = {\begin{matrix} 0, & x_{1} ⩽ x ⩽ x_{N - 1}, \\ \frac{x - x_{N - 1}}{x_{N} - x_{N - 1}}, & x_{N - 1} < x ⩽ x_{N}, \end{matrix}$

Parabola fuzzification:

$\begin{matrix} A_{1} (x) = {\begin{matrix} 1 - \frac{(x - x_{1})^{2}}{(x_{2} - x_{1})^{2}}, & x_{1} ⩽ x ⩽ x_{2}, \\ 0, & x_{2} < x ⩽ x_{N}, \end{matrix} & & A_{i} (x) = {\begin{matrix} 1 - \frac{(x - x_{i})^{2}}{(x_{i - 1} - x_{i})^{2}}, & x_{i - 1} ⩽ x ⩽ x_{i}, \\ 1 - \frac{(x - x_{i})^{2}}{(x_{i + 1} - x_{i})^{2}}, & x_{i} < x ⩽ x_{i + 1}, \\ i = 2, \dots, N - 1, \\ 0, & others, \end{matrix} \\ A_{N} (x) = {\begin{matrix} 0, & x_{1} ⩽ x ⩽ x_{N - 1}, \\ 1 - \frac{(x - x_{N})^{2}}{(x_{N - 1} - x_{N})^{2}}, & x_{N - 1} < x ⩽ x_{N}, \end{matrix} \end{matrix}$ (8)

Then {A_i} _1⩽i⩽N are transformed into the type-2 interval fuzzy sets

{{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}

in the following way:

μ_{{\tilde{A}}_{i}} (x) = [δ_{i} A_{i} (x), 1 \land (γ_{i} + 1) A_{i} (x)],

(9) where the parameters δ_i, γ_i ∈ [0, 1] , i = 1, …, N. The consequents

{{\tilde{B}}_{i}}_{1 ⩽ i ⩽ N}

are fuzzified in a simple way:

Fuzzify {y_i} _1⩽i⩽N into the type-1 fuzzy set {B_i} _1⩽i⩽N with this property: $B_{i} (y_{s}) = {\begin{matrix} 1, & s = i, \\ 0, & s \in {1, \dots, N} / {i} . \end{matrix}$ (10)

Set $μ_{\tilde{B_{i}}} (y) = χ_{{B_{i} (y)}}$ , where $χ_{{B_{i} (y)}} (w) = {\begin{matrix} 1, & w = B_{i} (y), 0, & w \in [0, 1] and w \neq B_{i} (y) . \end{matrix}$

Then two methods to construct the antecedents and consequents of the inference rules in this article are showed in Table 1. For example, let N = 5, [x₁, x₂, x₃, x₄, x₅]=[-3, - 1.5, 0, 1.5, 3], δ_i = 0.7 and γ_i = 0.2, i = 1, …, 5. The figures of {A_i} _1⩽i⩽N and ${{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}$ are showed in Fig. 2 (a) and (b) (resp. Fig. 3 (a) and (b)) by using the method (▾₁)(resp. (▾₂)).

Table 1

Two methods to construct the antecedents and consequents of the inference rules

Method	A _i	$μ_{{\tilde{A}}_{i}} (x)$	B _i	$μ_{\tilde{B_{i}}} (y)$
(▾ ₁)	Triangular (7)	(9)	(10)	χ _{{B_i(y)}}
(▾ ₂)	Parabola (8)	(9)	(10)	χ _{{B_i(y)}}

Fig. 2

(a)The figure of the fuzzy set {A_i} _1⩽i⩽N; (b)The figure of the interval type-2 fuzzy set ${{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}$ .

Fig. 3

(a)The figure of the fuzzy set {A_i} _1⩽i⩽N; (b)The figure of the interval type-2 fuzzy set ${{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}$ .

4 Interval type-2 fuzzy systems with single input and single output

In this section, the interpolation function of an interval type-2 fuzzy system with single input and single output as well as its universal approximation are investigated. The data set IOD₁ used to construct the system satisfies the property (6).

Theorem 4.1. Assume that the antecedents ${{\tilde{A}}_{i}}_{1 ⩽ i ⩽ N}$ and the consequents ${{\tilde{B}}_{i}}_{1 ⩽ i ⩽ N}$ of the inference rules (13) are constructed by the method (▾ ₁) or (▾ ₂), each input x is fuzzified by (5), the union and intersection operators in the CRI algorithm are respectively taken by ⊔^(∨,★) and ⊓^(★,★), and centroid type-reduction and center-of-gravity algorithm are applied to get the crisp output. Then there exists a group of base functions such that the interval type-2 fuzzy system with single input and single output is approximately a unary piecewise interpolation function F (x): $\begin{matrix} F (x) & = & \frac{1}{2} (\frac{1 \land (γ_{i} + 1) A_{i} (x)}{1 \land (γ_{i} + 1) A_{i} (x) + δ_{i + 1} A_{i + 1} (x)} \\ + \frac{δ_{i} A_{i} (x)}{δ_{i} A_{i} (x) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}) y_{i} \\ + \frac{1}{2} (\frac{δ_{i + 1} A_{i + 1} (x)}{1 \land (γ_{i} + 1) A_{i} (x) + δ_{i + 1} A_{i + 1} (x)} \\ + \frac{1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}{δ_{i} A_{i} (x) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}) y_{i + 1}, \end{matrix}$ (11) where x ∈ [x_i, x_i+1] , i = 1, …, N - 1, and δ_s, γ_s ∈ [0, 1] , s = 1, …, N.

Proof. For the ith inference rule, there is $\begin{matrix} μ_{{\tilde{A}}_{i} (x)} ⊓^{(★, ★)} μ_{{\tilde{B}}_{i} (y)} = & [(δ_{i} A_{i} (x)) ★ B_{i} (y), \\ (1 \land (γ_{i} + 1) A_{i} (x)) ★ B_{i} (y)] . \end{matrix}$ Then the whole type-2 fuzzy inference relation is $\begin{matrix} μ_{\tilde{R} (x, y)} & = {⊔^{(\lor, ★)}}_{i = 1}^{N} (μ_{{\tilde{A}}_{i} (x)} ⊓^{(★, ★)} μ_{{\tilde{B}}_{i} (y)}) \\ = [⋁_{i = 1}^{N} ((δ_{i} A_{i} (x)) ★ B_{i} (y)), \\ ⋁_{i = 1}^{N} ((1 \land (γ_{i} + 1) A_{i} (x)) ★ B_{i} (y))] . \end{matrix}$ Since each input x′ is fuzzified by (5), the output set ${\tilde{B}}^{*}$ of the interval type-2 fuzzy is $\begin{matrix} μ_{{\tilde{B}}^{*} (y)} & = & μ_{\tilde{R} (x^{'}, y)} \\ = & [⋁_{i = 1}^{N} ((δ_{i} A_{i} (x^{'})) ★ B_{i} (y)), \\ ⋁_{i = 1}^{N} ((1 \land (γ_{i} + 1) A_{i} (x^{'})) ★ B_{i} (y))] . \end{matrix}$ Let $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y) & = ⋁_{i = 1}^{N} ((1 \land (γ_{i} + 1) A_{i} (x^{'})) ★ B_{i} (y)), \\ {LMF}_{{\tilde{B}}^{*}} (y) & = ⋁_{i = 1}^{N} ((δ_{i} A_{i} (x^{'})) ★ B_{i} (y)) . \end{matrix}$ Next, we’re going to find the type-reducted set [CL, CR] according to ${UMF}_{{\tilde{B}}^{*}}$ and ${LMF}_{{\tilde{B}}^{*}}$ . Just discretize Y into y₁, …, y_N. Then permutate k_l = ψ (l) so that $y_{k_{1}} ⩽ y_{k_{2}} ⩽ \dots ⩽ y_{k_{N}} .$ From to the KM algorithm, there are $\begin{matrix} CL = \\ \frac{\sum_{j = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) y_{k_{j}} + \sum_{j = L + 1}^{N} {LMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) y_{k_{j}}}{\sum_{j = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) + \sum_{j = L + 1}^{N} {LMF}_{{\tilde{B}}^{*}} (y_{k_{j}})}, \\ CR = \\ \frac{\sum_{j = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) y_{k_{j}} + \sum_{j = R + 1}^{N} {UMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) y_{k_{j}}}{\sum_{j = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{k_{j}}) + \sum_{j = R + 1}^{N} {UMF}_{{\tilde{B}}^{*}} (y_{k_{j}})}, \end{matrix}$ where k_L and k_R are the indexes of the transition point and $\begin{matrix} y_{k_{L}} ⩽ CL ⩽ y_{k_{L} + 1}, \\ y_{k_{R}} ⩽ CR ⩽ y_{k_{R + 1}} . \end{matrix}$ When x′ ∈ [x_i, x_i+1], there is $A_{j} (x^{'}) = 0, j \in {1, \dots, N} / {i, i + 1} .$ Together with the properties of B_i (y), we have $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y_{s}) & = {\begin{matrix} 1 \land (γ_{s} + 1) A_{s} (x^{'}), s = i, i + 1, \\ 0, s \in {1, \dots, N} / {i, i + 1} . \end{matrix} \\ {LMF}_{{\tilde{B}}^{*}} (y_{s}) & = {\begin{matrix} δ_{s} A_{s} (x^{'}), s = i, i + 1, \\ 0, s \in {1, \dots, N} / {i, i + 1} . \end{matrix} \end{matrix}$ Assume that y_i ⩽ y_i+1. Since CL is the left endpoint of the type-reduced set, it can be obtained according to the KM algorithm that $\begin{matrix} CL = & \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} & \land \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} & \land \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} . \end{matrix}$ Then we have $\begin{matrix} \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ geqslant \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \end{matrix}$ and $\begin{matrix} \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} ⩾ \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} . \end{matrix}$ }} Therefore, $\begin{matrix} CL & = & \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ = & \frac{(1 \land (γ_{i} + 1) A_{i} (x^{'})) y_{i} + δ_{i + 1} A_{i + 1} (x^{'}) y_{i + 1}}{1 \land (γ_{i} + 1) A_{i} (x^{'}) + δ_{i + 1} A_{i + 1} (x^{'})} . \end{matrix}$ Since CR is the right endpoint of the type-reduced set, it can be obtained according to KM algorithm that $\begin{matrix} CR & = & \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ ⋁ \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ ⋁ \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} . \end{matrix}$ Then we also have $\begin{matrix} \frac{{UMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{UMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ leqslant \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \end{matrix}$ and $\begin{matrix} \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {LMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ leqslant \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} . \end{matrix}$ Therefore, $\begin{matrix} CR & = & \frac{{LMF}_{{\tilde{B}}^{*}} (y_{i}) y_{i} + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1}) y_{i + 1}}{{LMF}_{{\tilde{B}}^{*}} (y_{i}) + {UMF}_{{\tilde{B}}^{*}} (y_{i + 1})} \\ = & \frac{δ_{i} A_{i} (x^{'}) y_{i} + (1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})) y_{i + 1}}{δ_{i} A_{i} (x^{'}) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})} . \end{matrix}$ When y_i ⩾ y_i+1, the indexes of CL and CR are swapped. From the above, it can be known that when x′ ∈ [x_i, x_i+1], the crisp output of interval type-2 fuzzy system is $\begin{matrix} y^{'} & \approx & F (x^{'}) = \frac{1}{2} (CL + CR) \\ = & \frac{1}{2} (\frac{(1 \land (γ_{i} + 1) A_{i} (x^{'})) y_{i} + δ_{i + 1} A_{i + 1} (x^{'}) y_{i + 1}}{1 \land (γ_{i} + 1) A_{i} (x^{'}) + δ_{i + 1} A_{i + 1} (x^{'})} \\ + \frac{δ_{i} A_{i} (x^{'}) y_{i} + (1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})) y_{i + 1}}{δ_{i} A_{i} (x^{'}) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})}) \\ = & \frac{1}{2} (\frac{1 \land (γ_{i} + 1) A_{i} (x^{'})}{1 \land (γ_{i} + 1) A_{i} (x^{'}) + δ_{i + 1} A_{i + 1} (x^{'})} \\ + \frac{δ_{i} A_{i} (x^{'})}{δ_{i} A_{i} (x^{'}) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})}) y_{i} \\ + \frac{1}{2} (\frac{δ_{i + 1} A_{i + 1} (x^{'})}{1 \land (γ_{i} + 1) A_{i} (x^{'}) + δ_{i + 1} A_{i + 1} (x^{'})} \\ + \frac{1 \land ((γ_{i + 1} + 1) A_{i + 1} (x^{'}))}{δ_{i} A_{i} (x^{'}) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x^{'})}) y_{i + 1} . \end{matrix}$

Moreover, because the values of $U M F_{{\tilde{B}}^{*}}$ and $L M F_{{\tilde{B}}^{*}}$ are related to ${x^{'}}_{1}, {x^{'}}_{2}$ , we denote

$U_{s t} ({x^{'}}_{1}, {x^{'}}_{2}) = (1 \land (γ_{1 s} + 1) A_{1 s} (x_{1^{'}})) ⋆ (1 \land (γ_{2 t} + 1) A_{2 t} (x_{2^{'}})), L_{s t} ({x^{'}}_{1}, {x^{'}}_{2}) = (δ_{1 s} A_{1 s} (x_{1^{'}})) ⋆ (δ_{2 t} A_{2 t} (x_{2^{'}})), (s, t) \in I_{i j} .$

When $({x^{'}}_{1}, {x^{'}}_{2}) \in [x_{1 i}, x_{1, i + 1}] \times [x_{2 j}, x_{2, j + 1}]$ , there is

$F ({x^{'}}_{1}, {x^{'}}_{2}) = \frac{1}{2} (C L + C R) = \frac{1}{2} ((\frac{U_{ψ_{(i j)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(i j)}} + \sum_{(s, t) \in I_{i j} / {(i, j)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}}}{U_{ψ_{(i j)}} ({x^{'}}_{1}, {x^{'}}_{2}) + \sum_{(s, t) \in I_{i j} / {(i, j)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2})} \land \frac{\sum_{(s, t) \in {(i, j), (i + 1, j)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}} + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}}}{\sum_{(s, t) \in {(i, j), (i + 1, j)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2})} \land \frac{\sum_{(s, t) \in I_{i j} / {(i + 1, j + 1)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}} + L_{ψ_{(i + 1, j + 1)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in I_{i j} / {(i + 1, j + 1)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) + L_{ψ_{(i + 1, j + 1)}} ({x^{'}}_{1}, {x^{'}}_{2})}) + (\frac{L_{ψ_{(i j)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(i j)}} + \sum_{(s, t) \in I_{i j} / {(i, j)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}}}{L_{ψ_{(i j)}} ({x^{'}}_{1}, {x^{'}}_{2}) + \sum_{(s, t) \in I_{i j} / {(i, j)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2})} \lor \frac{\sum_{(s, t) \in {(i, j), (i + 1, j)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}} + \sum_{s, t) \in {(i, j + 1), (i + 1, j + 1)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}}}{\sum_{(s, t) \in {(i, j), (i + 1, j)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) + \sum_{s, t) \in {(i, j + 1), (i + 1, j + 1)}} U_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2})} \lor \frac{\sum_{(s, t) \in I_{i j} / {(i + 1, j + 1)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(s t)}} + U_{ψ_{(i + 1, j + 1)}} ({x^{'}}_{1}, {x^{'}}_{2}) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in I_{i j} / {(i + 1, j + 1)}} L_{ψ_{(s t)}} ({x^{'}}_{1}, {x^{'}}_{2}) + U_{ψ_{(i + 1, j + 1)}} ({x^{'}}_{1}, {x^{'}}_{2})})) .$

From the arbitrariness of x′, the expression of (14) is obtained. Next the interpolation property of (14) will be proved.

Let x′ = x_i. We have A_i+1 (x_i) =0. Together with (14), there is $\begin{matrix} F (x_{i}) = \frac{1}{2} (1 + 1) y_{i} = y_{i} . \end{matrix}$ Therefore, the expression of (14) has interpolation property. From the above, the conclusion is established.

Theorem 4.2. Let f ∈ C [a, b]. Assume that IOD₁ about the function f (x) satisfies the condition y_i = f (x_i) , i = 1, …, N. Then the interval type-2 fuzzy system with single input and single output (14) is a universal approximator.

Proof. It needs to prove that for arbitrary ɛ > 0, there exists a positive integer N ∈ Z⁺ such that ∥F (x) - f (x) ∥ _∞ < ɛ, if N > N₁. Let $\begin{matrix} w_{i} (x) & = & \frac{1}{2} (\frac{1 \land (γ_{i} + 1) A_{i} (x)}{1 \land (γ_{i} + 1) A_{i} (x) + δ_{i + 1} A_{i + 1} (x)} & & + \frac{δ_{i} A_{i} (x)}{δ_{i} A_{i} (x) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}), \\ w_{i + 1} (x) & = & \frac{1}{2} (\frac{δ_{i + 1} A_{i + 1} (x)}{1 \land (γ_{i} + 1) A_{i} (x) + δ_{i + 1} A_{i + 1} (x)} & & + \frac{1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}{δ_{i} A_{i} (x) + 1 \land (γ_{i + 1} + 1) A_{i + 1} (x)}) . \end{matrix}$ Then $F (x) = w_{i} (x) y_{i} + w_{i + 1} (x) y_{i + 1}$ and $w_{i} (x) + w_{i + 1} (x) = 1 .$ Thereby, $\begin{matrix} | F (x) - f (x) | & = & | w_{i} (x) y_{i} + w_{i + 1} (x) y_{i + 1} - f (x) | \\ = & | w_{i} (x) y_{i} + w_{i + 1} (x) y_{i + 1} \\ - (w_{i} (x) f (x) + w_{i + 1} (x) f (x)) | \\ = & | w_{i} (x) f (x_{i}) + w_{i + 1} (x) f (x_{i + 1}) \\ - (w_{i} (x) f (x) + w_{i + 1} (x) f (x)) | \\ = & | w_{i} (x) (f (x_{i}) - f (x)) \\ + w_{i + 1} (x) (f (x_{i + 1}) - f (x)) | \\ ⩽ & w_{i} (x) | f (x_{i}) - f (x) | \\ + w_{i + 1} (x) | f (x_{i + 1}) - f (x) | . \end{matrix}$ Since f ∈ C [a, b], there exist ξ_i, η_i ∈ [x_i, x_i+1] such that $\begin{matrix} f (x_{i}) - f (x) & = f^{'} (ξ_{i}) (x_{i} - x), \\ f (x_{i + 1}) - f (x) & = f^{'} (η_{i}) (x_{i + 1} - x) \end{matrix}$ by differential medial value theorem. Hence, $\begin{matrix} | F (x) - f (x) | & ⩽ & w_{i} (x) | f^{'} (ξ_{i}) (x_{i} - x) | \\ + w_{i + 1} (x) | f^{'} (η_{i}) (x_{i + 1} - x) | \\ ⩽ & w_{i} (x) | f^{'} (ξ_{i}) | | x_{i} - x_{i + 1} | \\ + w_{i + 1} (x) | f^{'} (η_{i}) | | x_{i + 1} - x_{i} | \\ ⩽ & w_{i} (x) ∥ f^{'} ∥_{\infty} | x_{i} - x_{i + 1} | \\ + w_{i + 1} (x) ∥ f^{'} ∥_{\infty} | x_{i} - x_{i + 1} | \\ = & ∥ f^{'} ∥_{\infty} | x_{i} - x_{i + 1} |, \end{matrix}$ Define $h = max_{1 ⩽ i ⩽ N - 1} x_{i + 1} - x_{i}$ . Then for every x ∈ [a, b], we have $\begin{matrix} ∥ F (x) - f (x) ∥_{\infty} & ⩽ & ∥ f^{'} ∥_{\infty} h . \end{matrix}$ Due to the property of the data set IOD₁, there exists a N₁ ∈ Z⁺ such that if N > N₁ then $h < \frac{ɛ}{∥ f^{'} ∥_{\infty}}$ for every N ∈ Z⁺. Therefore, $\begin{matrix} ∥ F (x) - f (x) ∥_{\infty} < ɛ . \end{matrix}$ From the above, the conclusion is established.

5 Interval type-2 fuzzy systems with multiple inputs and single output

This section deals with the interpolation functions of interval type-2 fuzzy systems with multiple inputs and single output. The case of double inputs and single output is analyzed firstly.

5.1 The interpolation function of interval type-2 fuzzy systems with double inputs and single output

The data set of constructing interval type-2 fuzzy system with double inputs and single output is given as IOD₂ = {(x_1i, x_2j, y_ij) , 1 ⩽ i ⩽ p, 1 ⩽ j ⩽ q}. Its elements both satisfy the property (6) and the following ordering relationship

$\begin{matrix} a_{1} = x_{11} < x_{12} < \dots < x_{1 p} = b_{1}, \\ a_{2} = x_{21} < x_{22} < \dots < x_{2 q} = b_{2}, \end{matrix}$ (12) and $\begin{matrix} y_{ψ_{(11)}} ⩽ \dots ⩽ y_{ψ_{(p 1)}} ⩽ y_{ψ_{(12)}} ⩽ \\ \dots ⩽ y_{ψ_{(p 2)}} ⩽ y_{ψ_{(13)}} ⩽ \dots ⩽ y_{ψ_{(pq)}}, \end{matrix}$ where ψ is a permutation function. Similar to the case of single input and single output, we also fuzzify x_1i (resp. x_2j, y_ij) into ${\tilde{A}}_{1 i}$ (resp. ${\tilde{A}}_{2 j}$ , ${\tilde{B}}_{ij}$ ) on the universe X₁ (resp. X₂, Y) by the method (▾ ₁) or (▾ ₂). Then the type-2 fuzzy inference rules are

$if x_{1} is {\tilde{A}}_{1 i} and x_{2} is {\tilde{A}}_{2 j} then y is {\tilde{B}}_{ij},$ (13) where i = 1, …, p, j = 1, …, q . The interpolation function of interval type-2 fuzzy systems with double inputs and single output will be given as follows.

Theorem 5.1. Assume that the antecedents ${{\tilde{A}}_{1 i}}_{1 ⩽ i ⩽ p}$ and ${{\tilde{A}}_{2 j}}_{1 ⩽ j ⩽ q}$ and the consequents ${{\tilde{B}}_{ij}}_{1 ⩽ i ⩽ p, 1 ⩽ j ⩽ q}$ of the inference rules (13) are constructed by the method (▾ ₁) or (▾ ₂) and the other conditions are the same as Theorem 4. Then there exists a group of base functions such that the interval type-2 fuzzy system with double inputs and single output is approximately a binary piecewise interpolation function F (x₁, x₂):

$\begin{matrix} F (x_{1}, x_{2}) = & \frac{1}{2} ((\frac{U_{ψ_{(ij)}} (x_{1}, x_{2}) y_{ψ_{(ij)}} + \sum_{(s, t) \in I_{ij} / {(i, j)}} L_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}}}{U_{ψ_{(ij)}} (x_{1}, x_{2}) + \sum_{(s, t) \in I_{ij} / {(i, j)}} L_{ψ_{(st)}} (x_{1}, x_{2})} & ⋀ \frac{\sum_{(s, t) \in {(i, j), (i + 1, j)}} U_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}} + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} L_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}}}{\sum_{(s, t) \in {(i, j), (i + 1, j)}} U_{ψ_{(st)}} (x_{1}, x_{2}) + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} L_{ψ_{(st)}} (x_{1}, x_{2})} \\ ⋀ \frac{\sum_{(s, t) \in I_{ij} / {(i + 1, j + 1)}} U_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}} + L_{ψ_{(i + 1, j + 1)}} (x_{1}, x_{2}) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in I_{ij} / {(i + 1, j + 1)}} U_{ψ_{(st)}} (x_{1}, x_{2}) + L_{ψ_{(i + 1, j + 1)}} (x_{1}, x_{2})}) \\ + (\frac{L_{ψ_{(ij)}} (x_{1}, x_{2}) y_{ψ_{(ij)}} + \sum_{(s, t) \in I_{ij} / {(i, j)}} U_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}}}{L_{ψ_{(ij)}} (x_{1}, x_{2}) + \sum_{(s, t) \in I_{ij} / {(i, j)}} U_{ψ_{(st)}} (x_{1}, x_{2})} & ⋁ \frac{\sum_{(s, t) \in {(i, j), (i + 1, j)}} L_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}} + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} U_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}}}{\sum_{(s, t) \in {(i, j), (i + 1, j)}} L_{ψ_{(st)}} (x_{1}, x_{2}) + \sum_{(s, t) \in {(i, j + 1), (i + 1, j + 1)}} U_{ψ_{(st)}} (x_{1}, x_{2})} \\ ⋁ \frac{\sum_{(s, t) \in I_{ij} / {(i + 1, j + 1)}} L_{ψ_{(st)}} (x_{1}, x_{2}) y_{ψ_{(st)}} + U_{ψ_{(i + 1, j + 1)}} (x_{1}, x_{2}) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in I_{ij} / {(i + 1, j + 1)}} L_{ψ_{(st)}} (x_{1}, x_{2}) + U_{ψ_{(i + 1, j + 1)}} (x_{1}, x_{2})})), \end{matrix}$ (14)

where (x₁, x₂) ∈ [x_1i, x_1,i+1] × [x_2j, x_2,j+1] , I_ij = {(i, j) , (i + 1, j) , (i, j + 1) , (i + 1, j + 1)} , i = 1, …, p - 1, j = 1, …, q - 1, and $\begin{matrix} U_{st} (x_{1}, x_{2}) = (1 \land (γ_{1 s} + 1) A_{1 s} (x_{1})) ★ \\ (1 \land (γ_{2 t} + 1) A_{2 t} (x_{2})), & L_{st} (x_{1}, x_{2}) = (δ_{1 s} A_{1 s} (x_{1})) ★ (δ_{2 t} A_{2 t} (x_{2})), & γ_{1 s}, γ_{2 t}, δ_{1 s}, δ_{2 t} \in (0, 1], \\ s = 1, \dots, p, t = 1, \dots, q . \end{matrix}$

Proof. For the ijth inference rule, there is $\begin{matrix} μ_{{\tilde{A}}_{1 i} (x_{1})} ⊓^{(★, ★)} μ_{{\tilde{A}}_{2 j} (x_{2})} ⊓^{(★, ★)} μ_{{\tilde{B}}_{ij} (y)} \\ = & [(δ_{1 i} A_{1 i} (x_{1})) ★ (δ_{2 j} A_{2 j} (x_{2})) ★ B_{ij} (y), \\ (1 \land (γ_{1 i} + 1) A_{1 i} (x_{1})) ★ (1 \land (γ_{2 j} + 1) A_{2 j} (x_{2})) \\ ★ B_{ij} (y)] . \end{matrix}$ Then the whole type-2 fuzzy inference relation is $\begin{matrix} μ_{\tilde{R} (x_{1}, x_{2}, y)} & = & {⊔^{(\lor, ★)}}_{i = 1}^{p} {⊔^{(\lor, ★)}}_{j = 1}^{q} (μ_{{\tilde{A}}_{1 i} (x_{1})} \\ ⊓^{(★, ★)} μ_{{\tilde{A}}_{2 j} (x_{2})} ⊓^{(★, ★)} μ_{{\tilde{B}}_{ij} (y)}) \\ = & [⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (δ_{1 i} A_{1 i} (x_{1})) \\ ★ (δ_{2 j} A_{2 j} (x_{2})) ★ B_{ij} (y), \\ ⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (1 \land (γ_{1 i} + 1) A_{1 i} (x_{1})) \\ ★ (1 \land (γ_{2 j} + 1) A_{2 j} (x_{2})) ★ B_{ij} (y)] . \end{matrix}$ Since each input $(x_{1}^{'}, x_{2}^{'})$ is fuzzified by (5), the interval type-2 fuzzy output set ${\tilde{B}}^{*}$ is $\begin{matrix} μ_{{\tilde{B}}^{*} (y)} & = & μ_{\tilde{R} (x_{1}^{'}, x_{2}^{'}, y)} \\ = & [⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (δ_{1 i} A_{1 i} (x_{1}^{'})) ★ (δ_{2 j} A_{2 j} (x_{2}^{'})) ★ B_{ij} (y), \\ ⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (1 \land (γ_{1 i} + 1) A_{1 i} (x_{1}^{'})) \\ ★ (1 \land (γ_{2 j} + 1) A_{2 j} (x_{2}^{'})) ★ B_{ij} (y)] . \end{matrix}$ Let $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y) & = & ⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (1 \land (γ_{1 i} + 1) A_{1 i} (x_{1}^{'})) \\ ★ (1 \land (γ_{2 j} + 1) A_{2 j} (x_{2}^{'})) ★ B_{ij} (y), \end{matrix}$ (15) $\begin{matrix} label chapt 1 : e 02 {LMF}_{{\tilde{B}}^{*}} (y) & = & ⋁_{i = 1}^{p} ⋁_{j = 1}^{q} (δ_{1 i} A_{1 i} (x_{1}^{'})) \\ ★ (δ_{2 j} A_{2 j} (x_{2}^{'})) ★ B_{ij} (y) . \end{matrix}$ (16) Next, we’re going to find the type-reduced set [CL, CR] according to ${UMF}_{{\tilde{B}}^{*}}$ and ${LMF}_{{\tilde{B}}^{*}}$ . We discretize Y into y_ij, i = 1, …, p, j = 1, …, q. From the properties of B_ij (y), it can be obtained respectively from (15) and (16) that $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y_{ij}) = & (1 \land (γ_{1 i} + 1) A_{1 i} (x_{1}^{'})) \\ ★ (1 \land (γ_{2 j} + 1) A_{2 j} (x_{2}^{'})), {LMF}_{{\tilde{B}}^{*}} (y_{ij}) = & (δ_{1 i} A_{1 i} (x_{1}^{'})) ★ (δ_{2 j} A_{2 j} (x_{2}^{'})) . \end{matrix}$ Let (y_{k
₁}, …, y_{k
_pq}) = (y_ψ(1), …, y_ψ(pq)). Then $\begin{matrix} c = & y_{k_{1}} ⩽ \dots ⩽ y_{k_{p}} ⩽ y_{k_{p + 1}} ⩽ \dots ⩽ y_{k_{2 p}} ⩽ \\ \dots ⩽ y_{k_{(q - 1) p + 1}} ⩽ \dots ⩽ y_{k_{pq}} = d . \end{matrix}$ By the KM algorithm, there are $\begin{matrix} CL = \\ \frac{\sum_{l = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) y_{k_{l}} + \sum_{l = L + 1}^{pq} {LMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) y_{k_{l}}}{\sum_{l = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) + \sum_{l = L + 1}^{pq} {LMF}_{{\tilde{B}}^{*}} (y_{k_{l}})}, \end{matrix}$ $\begin{matrix} CR = \\ \frac{\sum_{l = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) y_{k_{l}} + \sum_{l = R + 1}^{pq} {UMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) y_{k_{l}}}{\sum_{l = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{k_{l}}) + \sum_{l = R + 1}^{pq} {UMF}_{{\tilde{B}}^{*}} (y_{k_{l}})}, \end{matrix}$ where k_L, k_R are the indexes of the transition point. Denote $\begin{matrix} I = {(s, t) ∣ 1 ⩽ s ⩽ p, 1 ⩽ t ⩽ q}, & I_{ij} = {(i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)}, \\ i \in {1, \dots, p - 1}, j \in {1, \dots, q - 1} . \end{matrix}$ When $(x_{1}^{'}, x_{2}^{'}) \in [x_{1 i}, x_{1, i + 1}] \times [x_{2 j}, x_{2, j + 1}]$ , there are $\begin{matrix} A_{1 s} (x_{1}^{'}) = 0, s \in {1, \dots, p} / {i, i + 1}, \\ A_{2 t} (x_{2}^{'}) = 0, t \in {1, \dots, q} / {j, j + 1} . \end{matrix}$ Thereby, ${UMF}_{{\tilde{B}}^{*}} (y_{st}) = 0, {LMF}_{{\tilde{B}}^{*}} (y_{st}) = 0,$ (17) $(s, t) \in I / I_{ij} .$ Permutate y_ij, y_i+1,j, y_i,j+1, y_i+1,j+1 such that $y_{ψ_{(ij)}} ⩽ y_{ψ_{(i + 1, j)}} ⩽ y_{ψ_{(i, j + 1)}} ⩽ y_{ψ_{(i + 1, j + 1)}} .$ Denote $\begin{matrix} L_{C} (st) = {LMF}_{{\tilde{B}}^{*}} (y_{ψ_{(st)}}), \\ U_{C} (st) = {UMF}_{{\tilde{B}}^{*}} (y_{ψ_{(st)}}), \\ V_{1} = I_{ij} / {(i, j)}, \\ V_{2} = {(i, j), (i + 1, j)}, \\ V_{3} = {(i, j + 1), (i + 1, j + 1)}, \\ V_{4} = I_{ij} / {(i + 1, j + 1)} . \end{matrix}$ Since CL is the left endpoint of type-reduced set, it can be gained according to the KM algorithm and (17) that $\begin{matrix} CL = \frac{\sum_{(s, t) \in I_{ij}} L_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} L_{C} (st)} \\ ⋀ \frac{U_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} L_{C} (st) y_{ψ_{(st)}}}{U_{C} (ij) + \sum_{(s, t) \in V_{1}} L_{C} (st)} \\ ⋀ \frac{\sum_{(s, t) \in V_{2}} U_{C} (st) y_{ψ_{(st)}} + \sum_{(s, t) \in V_{3}} L_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in V_{2}} U_{C} (st) + \sum_{(s, t) \in V_{3}} L_{C} (st)} & & ⋀ \frac{\sum_{(s, t) \in V_{4}} U_{C} (st) y_{ψ_{(st)}} + L_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} U_{C} (st) + L_{C} (i + 1, j + 1)} \\ ⋀ \frac{\sum_{(s, t) \in I_{ij}} U_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} U_{C} (st)} . \end{matrix}$ Then we have $\begin{matrix} \frac{\sum_{(s, t) \in I_{ij}} L_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} L_{C} (st)} ⩾ \\ \frac{U_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} L_{C} (st) y_{ψ_{(st)}}}{U_{C} (ij) + \sum_{(s, t) \in V_{1}} L_{C} (st)} \end{matrix}$ and $\begin{matrix} \frac{\sum_{(s, t) \in I_{ij}} U_{C} (ij) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} U_{C} (ij)} ⩾ \\ \frac{\sum_{(s, t) \in V_{4}} U_{C} (ij) y_{ψ_{(st)}} + L_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} U_{C} (ij) + L_{C} (i + 1, j + 1)} . \end{matrix}$ Therefore, $\begin{matrix} CL = \frac{U_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} L_{C} (st) y_{ψ_{(st)}}}{U_{C} (ij) + \sum_{(s, t) \in V_{1}} L_{C} (st)} \\ ⋀ \frac{\sum_{(s, t) \in V_{2}} U_{C} (st) y_{ψ_{(st)}} + \sum_{(s, t) \in V_{3}} L_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in V_{2}} U_{C} (st) + \sum_{(s, t) \in V_{3}} L_{C} (st)} & & ⋀ \frac{\sum_{(s, t) \in V_{4}} U_{C} (st) y_{ψ_{(st)}} + L_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} U_{C} (st) + L_{C} (i + 1, j + 1)} . \end{matrix}$ Since CR is the right endpoint of type-reduced set, it can be obtained in a similar way according to the KM algorithm and (17) that $\begin{matrix} CR = \frac{\sum_{(s, t) \in I_{ij}} U_{C} (st) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} U_{C} (st))} \\ ⋁ \frac{L_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} U_{C} (st)) y_{ψ_{(st)}}}{L_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} U_{C} (st)) y_{ψ_{(st)}}} & & ⋁ \frac{\sum_{(s, t) \in V_{2}} L_{C} (st)) y_{ψ_{(st)}} + \sum_{(s, t) \in V_{3}} U_{C} (st)) y_{ψ_{(st)}}}{\sum_{(s, t) \in V_{2}} L_{C} (st)) + \sum_{(s, t) \in V_{3}} U_{C} (st))} & & ⋁ \frac{\sum_{(s, t) \in V_{4}} L_{C} (st)) y_{ψ_{(st)}} + U_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} L_{C} (st)) + U_{C} (i + 1, j + 1)} \\ ⋁ \frac{\sum_{(s, t) \in I_{ij}} L_{C} (st)) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} L_{C} (st))} . \end{matrix}$ Then we also have $\begin{matrix} \frac{\sum_{(s, t) \in I_{ij}} U_{C} (st)) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} U_{C} (st))} ⩽ \\ \frac{L_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} U_{C} (st)) y_{ψ_{(st)}}}{L_{C} (ij) + \sum_{(s, t) \in V_{1}} U_{C} (st))} \end{matrix}$ and $\begin{matrix} \frac{\sum_{(s, t) \in I_{ij}} L_{C} (st)) y_{ψ_{(st)}}}{\sum_{(s, t) \in I_{ij}} L_{C} (st))} ⩽ \\ \frac{\sum_{(s, t) \in V_{4}} L_{C} (st)) y_{ψ_{(st)}} + U_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} L_{C} (st)) + U_{C} (i + 1, j + 1)} . \end{matrix}$ Therefore, $\begin{matrix} CR = \frac{L_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} U_{C} (st)) y_{ψ_{(st)}}}{L_{C} (ij) y_{ψ_{(ij)}} + \sum_{(s, t) \in V_{1}} U_{C} (st)) y_{ψ_{(st)}}} & & ⋁ \frac{\sum_{(s, t) \in V_{2}} L_{C} (st)) y_{ψ_{(st)}} + \sum_{(s, t) \in V_{3}} U_{C} (st)) y_{ψ_{(st)}}}{\sum_{(s, t) \in V_{2}} L_{C} (st)) + \sum_{(s, t) \in V_{3}} U_{C} (st))} & & ⋁ \frac{\sum_{(s, t) \in V_{4}} L_{C} (st)) y_{ψ_{(st)}} + U_{C} (i + 1, j + 1) y_{ψ_{(i + 1, j + 1)}}}{\sum_{(s, t) \in V_{4}} L_{C} (st)) + U_{C} (i + 1, j + 1)} . \end{matrix}$ From the arbitrariness of $x_{1}^{'}, x_{2}^{'}$ , the expression of (14) is obtained. Next the interpolation property of (14) will be proved.

Without loss of generality, let $(x_{1}^{'}, x_{2}^{'}) = (x_{1 i}, x_{2 j})$ and y_{ψ
_(ij)} = y_ij. There are $\begin{matrix} U_{ψ_{(ij)}} (x_{1 i}, x_{2 j}) = & U_{ij} (x_{1 i}, x_{2 j}) = 1, Ł_{ψ_{(ij)}} (x_{1 i}, x_{2 j}) = & L_{ij} (x_{1 i}, x_{2 j}) = δ_{1 i} ★ δ_{2 j} . U_{st} (x_{1 i}, x_{2 j}) = & L_{st} (x_{1 i}, x_{2 j}) = 0, \\ (s, t) \in I_{ij} / {(i, j)} . \end{matrix}$ Together with (14), there is $\begin{matrix} F (x_{1 i}, x_{2 j}) & = & \frac{1}{2} (y_{ij} \land y_{ij} \land y_{ij} + y_{ij} \lor y_{ij} \lor y_{ij}) \\ = & y_{ij} . \end{matrix}$ Therefore, the expression of (14) has interpolation property. From the above, the conclusion is established.

5.2 The interpolation functions of interval type-2 fuzzy systems with multiple inputs and single output

Since the interpolation function of the interval type-2 fuzzy system contains permutation, for the sake of simplicity, the construction data set of the interval type-2 fuzzy system with multiple inputs and single output are required as $\begin{matrix} {IOD}_{n} = & {(x_{1 j}, \dots, x_{nj}, y_{j}) \in X_{1} \times \dots \times X_{n} \times Y, \\ 1 ⩽ j ⩽ m}, \end{matrix}$ which both satisfies the property (6) and the following ordering relationship

$x_{i 1} ⩽ x_{i 2} ⩽ \dots ⩽ x_{im}, i = 1, \dots, n,$ (18) and $\begin{matrix} y_{ψ (1)} ⩽ y_{ψ (2)} ⩽ \dots ⩽ y_{ψ (m)}, \end{matrix}$ where ψ is a permutation function. We fuzzify x_ij and y_j into interval type-2 fuzzy sets ${\tilde{A}}_{ij}$ and ${\tilde{B}}_{j}$ , respectively. The type-2 fuzzy inference rules are obtained in the following:

$if x_{1} is {\tilde{A}}_{1 j} and \dots and x_{n} is {\tilde{A}}_{nj} then y is {\tilde{B}}_{j},$ (19) where j = 1, 2, …, m. Notice that when x_ij = x_i,j+1, there is ${\tilde{A}}_{ij} = {\tilde{A}}_{i, j + 1}$ . Denote x=(x₁, x₂, ⋯, x_n).

Theorem 5.2. Assume that the antecedents ${{\tilde{A}}_{ij}}_{1 ⩽ i ⩽ n,}$ _{1 ⩽j ⩽m} and the consequents ${{\tilde{B}}_{j}}_{1 ⩽ j ⩽ m}$ of the inference rules (19) are constructed by the method (▾ ₁) or (▾ ₂), and the other conditions are the same as Theorem 4. Then there exists a group of base functions such that the interval type-2 fuzzy system with multiple inputs and single output is approximately a multivariate piecewise interpolation function F (x):

$\begin{matrix} F (x) & = & \frac{1}{2} (\frac{\sum_{j = 1}^{L} U_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = L + 1}^{m} L_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{L} U_{ψ (j)} (x) + \sum_{j = L + 1}^{m} L_{ψ (j)} (x)} \\ + \frac{\sum_{j = 1}^{R} L_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = R + 1}^{m} U_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{R} L_{ψ (j)} (x) + \sum_{j = R + 1}^{m} U_{ψ (j)} (x)}), \end{matrix}$ (20)where $\begin{matrix} U_{j} (x) = T_{i = 1}^{m} (1 \land (γ_{ij} + 1) A_{ij} (x_{i})), & L_{j} (x) = T_{i = 1}^{m} (δ_{ij} A_{ij} (x_{i})), & δ_{ij}, γ_{ij} \in (0, 1], i = 1, \dots, n, j = 1, \dots, m, & CL = min_{1 ⩽ h ⩽ m} { \\ \frac{\sum_{j = 1}^{h} U_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h + 1}^{m} L_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h} U_{ψ (j)} (x) + \sum_{j = h + 1}^{m} L_{ψ (j)} (x)}}, & CR = max_{1 ⩽ h ⩽ m} { \\ \frac{\sum_{j = 1}^{h} L_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h + 1}^{m} U_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h} L_{ψ (j)} (x) + \sum_{j = h + 1}^{m} U_{ψ (j)} (x)}}, \\ label L & L = min {h_{0} : \\ \frac{\sum_{j = 1}^{h_{0}} U_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h_{0} + 1}^{m} L_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h_{0}} U_{ψ (j)} (x) + \sum_{j = h_{0} + 1}^{m} L_{ψ (j)} (x)} = CL} \end{matrix}$ (21) $\begin{matrix} label R & R = max {h_{0} : \\ \frac{\sum_{j = 1}^{h_{0}} L_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h_{0} + 1}^{m} U_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h_{0}} L_{ψ (j)} (x) + \sum_{j = h_{0} + 1}^{m} U_{ψ (j)} (x)} = CR} \end{matrix}$ (22) Proof. For each inference rule, there is $\begin{matrix} ({⊓^{(★, ★)}}_{i = 1}^{n} μ_{{\tilde{A}}_{ij} (x_{i})}) ⊓^{(★, ★)} μ_{{\tilde{B}}_{j} (y)} \\ = [(T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}))) ★ B_{j} (y), \\ (T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}))) ★ B_{j} (y)] . \end{matrix}$ Then the whole type-2 fuzzy inference relation is $\begin{matrix} μ_{\tilde{R} (x, y)} \\ = & {⊔^{(\lor, ★)}}_{j = 1}^{m} (({⊓^{(★, ★)}}_{i = 1}^{n} μ_{{\tilde{A}}_{ij} (x_{i})}) ⊓^{(★, ★)} μ_{{\tilde{B}}_{j} (y)}) \\ = & [⋁_{j = 1}^{m} ((T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}))) ★ B_{j} (y)), \\ ⋁_{j = 1}^{m} ((T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}))) ★ B_{j} (y))] . \end{matrix}$ Since that each input x’=(x’₁, …, x’_n) is fuzzified by (5), the interval type-2 fuzzy output set ${\tilde{B}}^{*}$ is $\begin{matrix} μ_{{\tilde{B}}^{*} (y)} = μ_{\tilde{R} (x^{'}, y)} \\ = & [⋁_{j = 1}^{m} ((T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}^{'}))) ★ B_{j} (y)), \\ ⋁_{j = 1}^{m} ((T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}^{'}))) ★ B_{j} (y))] . \end{matrix}$ Let $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y) & = & ⋁_{j = 1}^{m} ((T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}^{'}))) \\ ★ B_{j} (y)), {LMF}_{{\tilde{B}}^{*}} (y) & = & ⋁_{j = 1}^{m} ((T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}^{'}))) ★ B_{j} (y)) . \end{matrix}$ Next, we’re going to find the type-reduced set [CL, CR] according to ${UMF}_{{\tilde{B}}^{*}}$ and ${LMF}_{{\tilde{B}}^{*}}$ . Just discretize Y into y_j, j = 1, …, m. Then permutate {y_j} _1⩽j⩽m such that $\begin{matrix} y_{ψ (1)} ⩽ y_{ψ (2)} ⩽ \dots ⩽ y_{ψ (m)} . \end{matrix}$ By the KM algorithm, there are $\begin{matrix} CL = \\ \frac{\sum_{j = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) y_{ψ (j)} + \sum_{j = L + 1}^{m} {LMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) y_{ψ (j)}}{\sum_{j = 1}^{L} {UMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) + \sum_{l = L + 1}^{m} {LMF}_{{\tilde{B}}^{*}} (y_{ψ (j)})}, \\ label h 2_{M} ISO & CR = \\ \frac{\sum_{l = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) y_{ψ (j)} + \sum_{j = R + 1}^{m} {UMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) y_{ψ (j)}}{\sum_{j = 1}^{R} {LMF}_{{\tilde{B}}^{*}} (y_{ψ (j)}) + \sum_{j = R + 1}^{m} {UMF}_{{\tilde{B}}^{*}} (y_{ψ (j)})}, \end{matrix}$ where L, R are the indexes of the transition point. By the properties of B_ij (y), there are $\begin{matrix} {UMF}_{{\tilde{B}}^{*}} (y_{j}) = & T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}^{'})), \\ {LMF}_{{\tilde{B}}^{*}} (y_{j}) = & T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}^{'})) . \end{matrix}$ Since the values of ${UMF}_{{\tilde{B}}^{*}} (y_{j})$ and ${LMF}_{{\tilde{B}}^{*}} (y_{j})$ are related to x’, we denote $\begin{matrix} U_{j} (x^{'}) = & T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{i}^{'})), Ł_{j} (x^{'}) = & T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{i}^{'})) . \end{matrix}$ Then the left endpoints CL and the right endpoints CR of type-reduced set can be gained $\begin{matrix} CL = & min_{1 ⩽ h ⩽ m} { \\ \frac{\sum_{j = 1}^{h} U_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h + 1}^{m} L_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h} U_{ψ (j)} (x) + \sum_{j = h + 1}^{m} L_{ψ (j)} (x)}} \\ = & \frac{\sum_{j = 1}^{L} U_{ψ (j)} (x^{'}) y_{ψ (j)} + \sum_{j = L + 1}^{m} L_{ψ (j)} (x^{'}) y_{ψ (j)}}{\sum_{j = 1}^{L} U_{ψ (j)} (x^{'}) + \sum_{j = L + 1}^{m} L_{ψ (j)} (x^{'})}, \\ CR = & max_{1 ⩽ h ⩽ m} { \\ \frac{\sum_{j = 1}^{h} L_{ψ (j)} (x) y_{ψ (j)} + \sum_{j = h + 1}^{m} U_{ψ (j)} (x) y_{ψ (j)}}{\sum_{j = 1}^{h} L_{ψ (j)} (x) + \sum_{j = h + 1}^{m} U_{ψ (j)} (x)}} \\ = & \frac{\sum_{j = 1}^{R} L_{ψ (j)} (x^{'}) y_{ψ (j)} + \sum_{j = R + 1}^{m} U_{ψ (j)} (x^{'}) y_{ψ (j)}}{\sum_{j = 1}^{R} L_{ψ (j)} (x^{'}) + \sum_{j = R + 1}^{m} U_{ψ (j)} (x^{'})}, \end{matrix}$ where L and R are defined by (21) and (22), respectively. From the arbitrariness of x’, the expression of (20) is obtained. Next its interpolation property will be proved.

Without loss of generality, let x’=(x_1j, …,x_nj) and ψ (j) = j. According to the feature of fuzzy set A_ij, we know $\begin{matrix} U_{j} (x_{1 j}, \dots, x_{nj}) & = & T_{i = 1}^{n} (1 \land (γ_{ij} + 1) A_{ij} (x_{ij}^{'})) \\ = & 1, Ł_{j} (x_{1 j}, \dots, x_{nj}) & = & T_{i = 1}^{n} (δ_{ij} A_{ij} (x_{ij}^{'})) = T_{i = 1}^{n} δ_{ij} . \end{matrix}$ Next we will calculate the value of U_s (x_1j, …, x_nj) and L_s (x_1j, …, x_nj) where s ∈ {1, …, m}/{j}. Because (x_1j, …, x_nj) ≠ (x_1s, …, x_ns), i.e., there exists i₀ ∈ {1, …, n} such that x_{i
₀
j} ≠ x_{i
₀
s}, we have A_{i
₀
s} (x_{i
₀
j}) =0. Then $\begin{matrix} 1 \land (γ_{is} + 1) A_{i_{0} s} (x_{i_{0} j}) = δ_{i_{0} s} A_{i_{0} s} (x_{i_{0} j}) = 0 . \end{matrix}$ Furthermore, $\begin{matrix} U_{s} (x_{1 j}, \dots, x_{nj}) & = & T_{i = 1}^{n} (1 \land (γ_{is} + 1) A_{is} (x_{ij})) \\ = & 0, Ł_{s} (x_{1 j}, \dots, x_{nj}) & = & T_{i = 1}^{n} (δ_{is} A_{is} (x_{ij})) = 0 . \end{matrix}$ Substitute the above values into the function (20), we can obtain that $\begin{matrix} F (x_{1 j}, \dots, x_{nj}) = \frac{1}{2} (y_{j} + y_{j}) = y_{j} . \end{matrix}$ Therefore, the expression of (20) has interpolation property. From the above, the conclusion is established.

5.3 Universal approximation of interval type-2 fuzzy systems with multiple inputs and single output

Theorem 5.3. Let f ∈ C ([a₁, b₁] × ⋯ × [a_n, b_n] × [c, d]). Assume that IOD_n about the function f (x) satisfies the condition y_j = f (x_1j, …, x_nj) , j = 1, …, m. Then the interval type-2 fuzzy system with multiple inputs and single output (20) is a universal approximator.

Proof. It needs to prove that for arbitrary ɛ > 0, there exists a positive integer N₀ ∈ Z⁺ such that ∥F ( x ) - f ( x ) ∥ _∞ < ɛ if N > N₀. Without loss of generality, assume y₁ ⩽ y₂ ⩽ ⋯ ⩽ y_m. Denote $\begin{matrix} ξ_{1 j} (x) & = & {\begin{matrix} \frac{U_{j} (x)}{\sum_{j = 1}^{L} U_{j} (x) + \sum_{j = L + 1}^{m} L_{j} (x)}, & 1 ⩽ j ⩽ L, \\ frac L_{j} (x) \sum_{j = 1}^{L} U_{j} (x) + \sum_{j = L + 1}^{m} L_{j} (x), & L + 1 ⩽ j ⩽ m, \end{matrix} \\ {xi}_{2 j} (x) & = & {\begin{matrix} \frac{L_{j} (x)}{\sum_{j = 1}^{R} L_{j} (x) + \sum_{j = R + 1}^{m} U_{j} (x)}, & 1 ⩽ j ⩽ R, \\ frac L_{j} (x) \sum_{j = 1}^{R} L_{j} (x) + \sum_{j = R + 1}^{m} U_{j} (x), & R + 1 ⩽ j ⩽ m . \end{matrix} \end{matrix}$ Then $\begin{matrix} F (x) = \frac{1}{2} \sum_{j = 1}^{m} (ξ_{1 j} (x) + ξ_{2 j} (x)) y_{j} . \end{matrix}$ and

$\sum_{j = 1}^{m} ξ_{1 j} (x) = \sum_{j = 1}^{m} ξ_{2 j} (x) = 1 .$ (23) Therefore, $\begin{matrix} | F (x) - f (x) | \\ = | \frac{1}{2} \sum_{j = 1}^{m} (ξ_{1 j} (x) + ξ_{2 j} (x)) y_{j} - f (x) | \\ = | \frac{1}{2} \sum_{j = 1}^{m} (ξ_{1 j} (x) + ξ_{2 j} (x)) f (x_{1 j}, \dots, x_{nj}) \\ - \frac{1}{2} \sum_{j = 1}^{m} (ξ_{1 j} (x) + ξ_{2 j} (x)) f (x) | & & ⩽ \frac{1}{2} \sum_{j = 1}^{m} (ξ_{1 j} (x) + ξ_{2 j} (x)) | f (x_{1 j}, \dots, x_{nj}) - f (x) | \end{matrix}$ Denote $h_{i} = max_{1 ⩽ j ⩽ m - 1} x_{i, j + 1} - x_{ij}, i = 1, \dots, n$ , $h = max_{1 ⩽ i ⩽ n} h_{i}$ . By Taylor function, f ( x ) can be expanded in the point (x_1j, …, x_nj), j = 1, …, m as follows: $\begin{matrix} f (x) = & f (x_{1 j}, \dots, x_{nj}) \\ + (x_{1} - x_{1 j}) \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{1}} \\ + \dots + (x_{n} - x_{nj}) \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{n}} \\ + O (h^{2}) . \end{matrix}$ It is known that for every ɛ₁ > 0, there exists δ₁ > 0 such that $\frac{O (h^{2})}{h^{2}} < ɛ_{1}$ if h < δ₁. Furthermore, because of the property of IOD_n, there exists N₁ ∈ Z⁺ such that h < min {δ₁, 1} if N > N₁. Together with the two aspects, there exists N₁ ∈ Z⁺ such that O (h²) < ɛ₁h² < ɛ₁h if N > N₁. Then it can be gained that $\begin{matrix} | f (x) - f (x_{1 j}, \dots, x_{nj}) | \\ = & | (x_{1} - x_{1 j}) \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{1}} + \dots \\ + (x_{n} - x_{nj}) \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{n}} + O (h^{2}) | \\ < & | \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{1}} | h + \dots \\ + | \frac{\partial f (x_{1 j}, \dots, x_{nj})}{\partial x_{n}} | h + ɛ_{1} h \\ < & {| | \frac{\partial f (x)}{\partial x_{1}} | |}_{\infty} h + \dots + {| | \frac{\partial f (x)}{\partial x_{n}} | |}_{\infty} h + ɛ_{1} h \\ = & ({| | \frac{\partial f (x)}{\partial x_{1}} | |}_{\infty} + {| | \frac{\partial f (x)}{\partial x_{n}} | |}_{\infty} + ɛ_{1}) h . \end{matrix}$ According to the above inequality and the expression of (23), there is $\begin{matrix} ∥ F (x) - f (x) ∥_{\infty} < & ({| | \frac{\partial f (x)}{\partial x_{1}} | |}_{\infty} + \dots + \\ {| | \frac{\partial f (x)}{\partial x_{n}} | |}_{\infty} + ɛ_{1}) h . \end{matrix}$ Due to the property of IOD_n, there exists N₂ ∈ Z⁺ such that $\begin{matrix} h < \frac{ɛ}{{| | \frac{\partial f (x)}{\partial x_{1}} | |}_{\infty} + \dots + {| | \frac{\partial f (x)}{\partial x_{n}} | |}_{\infty} + ɛ_{1}} \end{matrix}$ if N > N₂. Let N₀ = max {N₁, N₂}. Then ∥F ( x ) - f ( x ) ∥ _∞ < ɛ when N > N₀. Other situations can be proved similarly. From the above, the conclusion is established.

6 Simulation example

This section will take the interpolation function of an interval type-2 fuzzy system (14) to approximate a dynamic system with double inputs and single output, then compare the experiment results of three classes of interpolation functions of interval type-2 fuzzy systems, denoted by IT2-t-min-FS, IT2-t-pro-FS and IT2-p-pro-FS, and a class of interpolation function of type-1 fuzzy system in [25], denoted by T1-t-pro-FS, respectively. The methods of fuzzification and fuzzy operators adopted in each functions are shown in Table 3.

Select Van der Pol equation as the real model:

${\begin{matrix} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = μ (1 - x_{1}^{2}) x_{2} - x_{1}, \end{matrix}$ (24) where μ = 1. Four classes of interpolation functions of fuzzy systems will be established according to Table 3, and their solutions will be compared with the real solution of the model (24) under the same initial value condition. Let T = 20s and the initial value (x₁ (0) , x₂ (0)) = (2, 0). A total of 2001 points will be sampled uniformly as the experimental data and their root mean square error (RMSE) will be used as the evaluation index. For x₁ (t_s) and x₂ (t_s), their errors between the true value and the approximate value are set as ɛ_1s and ɛ_2s, respectively. Calculate the following root mean square errors: $\begin{matrix} {RMSE}_{x_{1}} = \sqrt{\frac{\sum_{s = 1}^{2001} ɛ_{1 s}^{2}}{2001}}, {RMSE}_{x_{2}} = \sqrt{\frac{\sum_{s = 1}^{2001} ɛ_{2 s}^{2}}{2001}}, \\ {RMSE}_{\sqrt{x_{1}^{2} + x_{2}^{2}}} = \sqrt{\frac{\sum_{s = 1}^{2001} {ɛ_{1 s}}^{2} + {ɛ_{2 s}}^{2}}{2001}} . \end{matrix}$

Let p = 6, q = 7. Then the quantum-behaved particle swarm optimization (QPSO) algorithm [26] is applied to optimize the construction parameters of the interval type-2 fuzzy system δ₁₁—δ_1p, δ₂₁—δ_2q, γ₁₁—γ_1p and γ₂₁—γ_2q, and ${RMSE}_{\sqrt{x_{1}^{2} + x_{2}^{2}}}$ is taken as the optimization target. In the QPSO algorithm, the contraction-expansion coefficient increases linearly from 0.5 to 1, the number of particles is 20, and the number of iterations is 500. Therefore, the parameter values of three classes of interpolation functions of interval type-2 fuzzy systems are shown in Table 2.

Table 2

The structure information of 4 classes of interpolation functions of fuzzy systems

fuzzy system	the method of fuzzification	triangular norm ★
T1-t-pro-FS	(refchapt1 : eq3) and (refqujian_B)	product ·
IT2-t-min-FS	(▾ ₁)	minimum ∧
IT2-t-pro-FS	(▾ ₁)	product ·
IT2-p-pro-FS	(▾ ₂)	product ·

Table 3

The parameters values of 3 classes of interpolation functions of interval type-2 fuzzy systems

Fuzzy system	δ ₁₁	δ ₁₂	δ ₁₃	δ ₁₄	δ ₁₅	δ ₁₆
IT2-t-min-FS	0.2337	0.415	0.9747	0.9936	0.9022	0.1782
IT 2-t-pro-FS	0.5083	0.351	0.9943	0.9926	0.6608	0.0913
IT2-p-pro-FS	0.0093	0.7889	0.9987	0.9979	0.6537	0.1994
Fuzzy system	δ ₂₁	δ ₂₂	δ ₂₃	δ ₂₄	δ ₂₅	δ ₂₆	δ ₂₇
IT2-t-min-FS	0.9543	0.9933	0.4046	0.9849	0.2285	0.9844	0.9684
IT2-t-pro-FS	0.9873	0.98	0.1485	0.9954	0.8459	1	0.9374
IT2-p-pro-FS	0.9886	0.9557	0.4733	0.9752	0.2067	0.9999	0.9997
hline Fuzzy system	γ ₁₁	γ ₁₂	γ ₁₃	γ ₁₄	γ ₁₅	γ ₁₆
IT2-t-min-FS	0.0152	0.0066	0.9998	0.9402	0.0063	0.0473
IT2-t-pro-FS	0.0438	0.0014	0.958	0.9089	0.0165	0.1552
IT2-p-pro-FS	0.9644	0.0069	0.8804	0.9889	0.0023	0.0054
Fuzzy system	γ ₂₁	γ ₂₂	γ ₂₃	γ ₂₄	γ ₂₅	γ ₂₆	γ ₂₇
IT2-t-min-FS	0.8631	0.0026	0.0073	0.3608	0.2175	0.0624	0.651
IT2-t-pro-FS	0.9297	0.9896	0.0071	0.3654	0.1489	0.9783	0.955
IT 2-p-pro-FS	0.4895	0.8839	0.0231	0.9766	0.0062	0.0107	0.9099

Three kinds of RMSEs for the four classes of interpolation functions are showed in Table 4. It can be seen that every kinds of RMSEs of IT2-t-min-FS (or IT2-t-pro-FS, IT2-p-pro-FS) are smaller than those of T1-t-pro-FS. When the white noise with mean value 0 and variance 0.1 is added to the sampled data set, the results remain the same (see Table 5). Therefore, simulation results show that the approximation performances of IT2-t-min-FS, IT2-t-pro-FS and IT2-p-pro-FS are better than those of T1-t-pro-FS.

Table 4

RMSEs of 4 classes of interpolation functions of fuzzy systems

Fuzzy system	RMSE _{x ₁}	RMSE _{x ₂}	${RMSE}_{\sqrt{x_{1}^{2} + x_{2}^{2}}}$
T1-t-pro-FS	0.2519	0.32	0.593
IT 2-t-min-FS	0.1353	0.2318	0.4623
IT 2-t-pro-FS	0.1979	0.2902	0.5217
IT 2-p-pro-FS	0.1312	0.2082	0.4524

Table 5

RMSEs of 4 classes of interpolation functions of fuzzy systems with noise

Fuzzy system	RMSE _{x ₁}	RMSE _{x ₂}	${RMSE}_{\sqrt{x_{1}^{2} + x_{2}^{2}}}$
T1-t-pro-FS	0.274	0.3326	0.6064
IT 2-t-min-FS	0.0982	0.1775	0.4152
IT 2-t-pro-FS	0.1992	0.3015	0.5185
IT2-p-pro-FS	0.1367	0.1702	0.4304

7 Conclusions

Under certain conditions, an interval type-2 fuzzy system can be transformed into a piecewise interpolation function with the property of universal approximation. We can just use the interpolation function instead of link-by-link design in the systems, and continue to research other undetected properties of type-2 fuzzy systems by studying their interpolation functions. The parameters of every interpolation functions of type-2 fuzzy systems play an important role in system modeling. In our simulation, the QPSO algorithm is used to optimize the construction parameters of interval type-2 fuzzy systems. By adjusting antecedent factors for different models, a better approximation effect can be achieved.

It can be seen that the functional expressions of interval type-2 fuzzy systems are very similar in forms with type-1 fuzzy systems. Li [25] pointed out that type-1 fuzzy control algorithm is equivalent to the finite element method of mathematical physics, and it is a kind of direct method or numerical method of control theory. The conclusion of this paper provides a useful supplement for the numerical form of high type fuzzy control algorithm and enriches the theoretical research of fuzzy control.

Footnotes

Acknowledgment

This paper is funded by the National Natural Science Foundation of China (No. 12071325).

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