确保区间2型模糊逻辑系统单调性的准则与方法

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本文主要探讨了区间2型（Interval Type-2, IT2）模糊逻辑系统（Fuzzy Logic Systems, FLS）的单调性问题。在许多系统建模和控制问题中，特别是当物理结构知识有限且训练数据点较少的情况下，定性的知识，如单调性，具有重要的价值。因为这类知识有助于确保系统的稳定性和可解释性。作者 Chengdong Li 和 Jianqiang Yi，均来自 IEEE，与 Guiqing Zhang 合作，他们在 IEEETRANSACTIONSONFUZZYSYSTEMS 杂志的2014年10月刊上发表了一篇名为《On the Monotonicity of Interval Type-2 Fuzzy Logic Systems》的研究论文。论文的核心内容是针对 IT2 FLSs设计并推导出满足单调性的充分条件，这些条件适用于规则的前件（antecedent）和后件（consequent）部分。在研究过程中，他们考察了五种常见的区间2型模糊集简化和解模糊化方法：Karnik-Mendel 方法、Du-Ying 方法、Begian-Melek-Mendel 方法、Wu-Tan 方法以及 Nie-Tan 方法。通过对比分析，论文指出，只要将IT2 FLS的规则前件和后件按照提出的单调性条件进行排列，就可以保证系统的输入和输出之间保持单调性。这意味着，即使在处理不确定性较高的信息时，通过遵循这些条件，IT2 FLS能够提供更可靠的行为预测。这篇论文的贡献在于，它不仅提供了理论框架，还为实际应用中的IT2 FLS设计提供了一个重要的指导原则，使得系统设计者能够在考虑到单调性约束的同时，更好地利用模糊逻辑的优势来处理复杂的决策问题。该研究为IT2 FLS的理论发展和工程实践提供了新的洞察，特别是在处理缺乏精确数据和特定结构知识的场景中。

1200 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014

D. Monotonicity of IT2 Fuzzy Logic Systemss Using the

Karnik–Mendel Method

The Karnik–Mendel type-reduction and COS defuzziﬁcation

method (KM method) [1], [8], [47]–[50] are the most popular

out-processing methods. The monotonicity of such IT2 FLS is

studied in this section, while the monotonicity of the IT2 FLSs

using other out-processing method is studied in Section IV.

For the increasing monotonicity, we have the following results

for all y

(x) and y

(x).

Lemma 1: y

(x) and y

(x) (n =1,...,K) are all monoton-

ically increasing w.r.t x

if we have the following.

1) For any x

≥ x

∈ X

, 1 ≤ l ≤ m ≤ N

,wehave

)μ

) ≥ μ

)μ

), where μ

means either μ



or μ



, and μ

means either μ



2) w

···i

k −1

k +1

···i

≤ w

···i

k −1

+1)i

k +1

···i

k −1

k +1

···i

≤ w

···i

k −1

+1)i

k +1

···i

for all the combinations of (i

,...,i

k−1

k+1

,...,i

where i

=1, ···,N

− 1.

Proof: See Appendix A. 

The conclusion in this lemma is very important. Most of the

proofs of the following theorems are based on the conclusion

provided by this Lemma.

For the monotonicity of the IT2 FLS using the KM method,

we have the following theorem.

Theorem 1: The IT2 FLS using the KM method is monoton-

ically increasing w.r.t x

if we have the following.

1) For any x

≥ x

∈ X

, 1 ≤ l ≤ m ≤ N

,wehave

)μ

) ≥ μ

)μ

), where μ

means either μ



or μ



, and μ

means either μ



2) w

···i

k −1

k +1

···i

≤ w

···i

k −1

+1)i

k +1

···i

k −1

k +1

···i

≤ w

···i

k −1

+1)i

k +1

···i

for all the combinations of (i

,...,i

k−1

k+1

,...,i

where i

=1, ···,N

− 1.

Proof: See Appendix B. 

E. Monotonicity of T1 Fuzzy Logic System

As discussed in Section II-B, T1 FLS can be viewed as a

special case of the IT2 FLS using the KM method. Consequently,

for the monotonicity of T1 FLS, we have the following result.

Theorem 2: The T1 FLS is monotonically increasing w.r.t x

if we have the following.

1) For any x

≥ x

∈ X

, 1 ≤ l ≤ m ≤ N

,wehave

)μ

) ≥ μ

)μ

2) w

···i

k −1

k +1

···i

≤ w

···i

k −1

+1)i

k +1

···i

for all the

combinations of (i

,...,i

k−1

k+1

,...,i

), where i

1,..., N

− 1.

Note that our result on the monotonicity of both IT2 FLS and

T1 FLS put no limitation on the shapes and types of IT2 FSs

In this study, we do not constrain the shape of the IT2 FSs, Hence, this

theorem holds for any kind of IT2 FSs, as long as this condition can be satisﬁed.

In the next section, we will show that this condition can easily be met when

Trapezoidal and Gaussian IT2 FSs are adopted.

(a) (b)

Fig. 1. Trapezoidal FSs: (a) Trapezoidal T1 FS. (b) Trapezoidal IT2 FS.

and T1 FSs. The second condition in Theorems 1 and 2 can be

easily checked. On the other hand, we need to explore how the

antecedent FSs in the rule base can meet the ﬁrst condition in

both theorems. In the next section, we will study this issue.

III. M

ONOTONICITY CONDITIONS ON THE ANTECEDENT

PARTS OF FUZZY RULES

In practical FLSs, the most frequently used FSs are the Gaus-

sian FS and the Trapezoidal FS, a special case of which is the

Triangular FS [1]–[3]. Next, we will show how the Trapezoidal

and Gaussian FSs can satisfy the ﬁrst condition in Theorems 1

and 2.

A. Monotonicity Conditions of the Trapezoidal Fuzzy Sets

Fig. 1(a) shows us a Trapezoidal T1 FS A, the MF of which

can be expressed as

(x)=

⎧

⎪

⎨

⎪

⎩

0,x≤ a or x ≥ d

x−a

b−a

,a<x≤ b

h, b < x ≤ c

x−d

c−d

, c<x<d

(11)

where 0 <h≤ 1. We denote such a Trapezoidal T1 FS as

(x)=μ

(x, a, b, c, d, h).

Note that Triangular T1 FSs are special cases of Trapezoidal

T1 FSs when b = c. Generally, in many applications, normal

Trapezoidal T1 FSs whose heights equal to 1 are adopted.

However, in this study, we consider the general case where

0 <h≤ 1.

By blurring the Trapezoidal T1 FS shown in Fig. 1(a), we can

obtain the Trapezoidal IT2 FS



A [see Fig. 1(b)], which can be

described by its lower and upper MFs μ



(x) and μ



(x) as



(x)=μ



(x, a,b,c,d,h) (12)



(x)=μ



(x, a, b, c, d, 1) (13)

where

a ≤ a, b ≤ b,c≤ c, d ≤ d. We denote such a Trape-

zoidal IT2 FS as μ



(x)=μ



(x, a, b, c, d, a,b,c,d,h).Again,

when b

= c and b = c, the Trapezoidal IT2 FS becomes a Tri-

angular IT2 FS.

Lemma 2: Consider two Trapezoidal T1 FSs μ

(x)=

(x, a

),μ

(x)=μ

(x, a

).If

≤ a

≤ b

≤ c

≤ d

(as shown in Fig. 2),

then, for any x

≥ x

∈ X,wehaveμ

)μ

) ≥

)μ

Proof: See Appendix C. 

From Lemma 2, we can make the following conclusion.

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