关于区间2型模糊逻辑系统的单调性_Begian-Melek-Mendel方法去模糊化资源-CSDN文库

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区间2型模糊逻辑系统（Interval Type-2 Fuzzy Logic Systems, IT2 FLSs）是一种在不确定性和模糊性处理方面具有独特优势的控制系统。它不仅继承了传统1型模糊逻辑系统（Type-1 FLSs）的优点，而且由于其对不确定性的更高容忍度，能够构建更加复杂、非线性的输入输出映射，进而获得更好的控制性能。在工程应用中，尤其是当需要系统模型具有单调性特性时，理解并保证这种单调性是非常重要的。单调性（Monotonicity）在数学、控制理论和工程领域中，指的是一个变量随着另一个变量变化而呈现出一致增减的特性。在模糊逻辑系统中，研究输入与输出之间的单调性意味着确保随着输入变量的变化，输出变量要么单调递增要么单调递减，这有助于系统的设计者构建更为直观的控制系统。文章《关于区间2型模糊逻辑系统的单调性》探讨了如何将单调性这一定性知识融合到IT2 FLSs中。作者首先指出，在缺乏具体物理结构知识或训练数据点较少的情况下，定性知识对于系统建模和控制问题非常有用。接着，文章为模糊规则的前提和结论部分推导出了充分条件，以保证输入和输出之间的单调性关系。在此过程中，考虑了五种类型缩减和去模糊化方法：Karnik–Mendel方法、Du–Ying方法、Begian–Melek–Mendel方法、Wu–Tan方法和Nie–Tan方法。研究结果显示，只要按照所提出的单调性条件排列IT2 FLSs的模糊规则的前提和结论部分，系统就能够表现出单调性。此外，得到的单调性条件适用于任何类型的IT2模糊集（例如梯形IT2模糊集和高斯IT2模糊集）以及1型FLSs。单调性的加入对于工程应用中单调性属性需求的IT2 FLSs的设计具有重要意义，提供了数据驱动方法、模糊逻辑系统、建模与控制、单调性、2型模糊以及类型缩减和去模糊化方法等多个领域的深入理解。文中所提出的单调性条件和应用指南将有助于设计能够更好地满足特定应用需求的IT2 FLSs。在单调性条件的研究中，作者重点研究了IT2 FLSs的结构，包括模糊规则的形成和组织，以及在不同类型的模糊集合下的单调性表达。这涉及到模糊集合理论中对不确定性的描述和处理，包括不确定性的类型（区间不确定性）、隶属度函数的形状等概念。针对不同类型和形状的模糊集，确保单调性的条件可能会有所不同，但文章表明，无论使用何种IT2模糊集，所提出的单调性条件都是有效的，这为IT2 FLSs的实践应用提供了较为通用的设计原则。文章的作者还强调了单调性在控制系统设计中的重要性，并通过研究结果说明了单调性条件的实用性和有效性。例如，在处理与物理现象相关的过程控制时，一个单调的系统模型可以确保输入和输出之间有一个明确的、可预测的因果关系。这一点对于设计具有预测能力和可靠控制行为的系统至关重要。本研究不仅为IT2 FLSs的单调性设计提供了理论基础，还提出了一些实际应用的指导，如将单调性条件应用于建模和控制系统设计的建议。这些内容对于设计能在工程应用中满足特定单调性要求的IT2 FLSs具有重要的参考价值。

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014 1197

On the Monotonicity of Interval Type-2

Fuzzy Logic Systems

Chengdong Li, Member, IEEE, Jianqiang Yi, Senior Member, IEEE, and Guiqing Zhang

Abstract—Qualitative knowledge is very useful for system mod-

eling and control problems, especially when speciﬁc physical struc-

ture knowledge is unavailable and the number of training data

points is small.This paper studies the incorporation of one common

qualitative knowledge—monotonicity into interval type-2 (IT2)

fuzzy logic systems (FLSs). Sufﬁcient conditions on the antecedent

and consequent parts of fuzzy rules are derived to guarantee the

monotonicity between inputs and outputs. We take into account ﬁve

type-reduction and defuzziﬁcation methods (the Karnik–Mendel

method, the Du–Ying method, the Begian–Melek–Mendel method,

the Wu–Tan method, and the Nie–Tan method). We show that IT2

FLSs are monotonic if the antecedent and consequents parts of

their fuzzy rules are arranged according to the proposed mono-

tonicity conditions. The derived monotonicity conditions are valid

for the IT2 FLSs using any kind of IT2 fuzzy sets (FSs) (e.g., Trape-

zoidal IT2 FSs and Gaussian IT2 FSs) and stand for type-1 FLSs

as well. Guidelines for applying the proposed conditions to model-

ing and control problems are also given. Our results will be useful

in the design of monotonic IT2 FLSs for engineering applications

when the monotonicity property is desired.

Index Terms—Data-driven method, fuzzy logic system, model-

ing and control, monotonicity, type-2 fuzzy, type-reduction and

defuzziﬁcation method.

I. INTRODUCTION

ECENTLY, type-2 (T2) fuzzy logic systems (FLSs)

[1]–[7] have attracted increasing interest, as T2 FLSs not

only have the advantages of conventional FLSs (type-1 FLSs)

but can provide the capability to model high levels of uncer-

tainties and produce more complex input–output mappings and

better results as well. To date, due to the computation com-

plexity and theoretical analysis difﬁculty, the most widely stud-

ied and applied T2 FLSs are the interval ones, where interval

type-2 (IT2) fuzzy sets (FSs)

[8]–[12] are adopted to reduce

Manuscript received February 4, 2013; revised May 12, 2013; July 14, 2013;

accepted September 17, 2013. Date of publication October 18, 2013; date of

current version October 2, 2014. This work was supported by the National Nat-

ural Science Foundation of China under Grant 61105077, Grant 61273149, and

grant 61074149, and the Excellent Young and Middle-Aged Scientist Award

Grant of Shandong Province of China (BS2012DX026).

C. Li and G. Zhang are with the School of Information and Electri-

cal Engineering, Shandong Jianzhu University, Jinan 250101, China (e-mail:

lichengdong@sdjzu.edu.cn; qqzhang@sdjzu.edu.cn).

J. Yi is with the Institute of Automation, Chinese Academy of Sciences, Bei-

jing 100190, China (e-mail: jianqiang.yi@ia.ac.cn).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TFUZZ.2013.2286416

IT2 FSs are isomorphic to interval-valued FSs [12]. Both concepts are alter-

natively used by different researchers; for example, in [13]–[17], interval-valued

FSs are utilized, while in most of the other references, IT2 FSs are adopted. As

an extension of T1 FSs, IT2 FSs (interval-valued FSs) have close relationships

with other extensions, e.g., intuitionistic FSs [18], interval-valued intuitionistic

computational complexity. IT2 FLSs have found lots of appli-

cations in many areas, especially in the modeling and control

ﬁelds [21]–[35].

For modeling, IT2 FLSs represent the input–output map-

pings of the systems to be identiﬁed and are usually constructed

through data-driven methods. When constructing IT2 FLSs us-

ing data-driven methods, we often encounter that the data points

are noisy and that the number of the data points is small. As

discussed in [36]: “in such cases, it is very important to fully

exploit the additional nonquantitative knowledge about the sys-

tem in order to obtain meaningful, interpretable models. More-

over, taking the qualitative knowledge about the system into

account renders the model-identiﬁcation process less vulner-

able to noise and inconsistencies in the data and suppresses

overﬁtting.” Monotonicity between the inputs and outputs is

one of such qualitative knowledge in many modeling problems.

Taking the identiﬁcation of the water heating system [37] for

example, the temperature of water will change with respect to

the heat power monotonically. Therefore, the identiﬁed fuzzy

model (type-2 or type-1) for the water heating system should be

monotonic between the heat power and the temperature.

For control applications, IT2 FLSs are utilized to realize con-

trol laws to reduce control errors. In many cases, the control

signal (output of IT2 FLS) should be monotonic with respect to

the error and/or the change of error (inputs of IT2 FLS). One

typical example is the control of a liquid level in a tank. An

appropriate fuzzy controller (type-2 or type-1) for this system

needs to open the valve larger as the liquid level deviates more

from the required level. Another example is the temperature

control of the refrigerator. The more the real temperature in the

refrigerator deviates from the setpoint, the lager the control ac-

tion is needed to be generated by IT2 FLS to increase the motor

speed in the compressor.

From the previous discussion, we can see that it would be very

helpful to ﬁnd the conditions under which the FLSs can give

monotonic input–output mappings. There are several meaning-

ful papers on the monotonicity of type-1 (T1) FLSs [36]–[42].

Broekhoven et al. [36], [38] have studied the monotonicity is-

sue on the Mamdani–Assilian models under the mean of max-

ima defuzziﬁcation and the center-of-gravity defuzziﬁcation.

In [37], [39], and [40], sufﬁcient parameter conditions are given

to ensure a monotonic input–output mapping of the TSK T1

FLS. Kouikoglou et al. [41] have discussed how to ensure the

monotonicity of the hierarchical sum-product T1 FLSs. Seki

et al. [42] have derived the monotonicity conditions of the sin-

gle input rule modules (SIRMs) connected T1 FLSs.

FSs [18], and L-FSs [19]. Deschrijver and Kerre [20] have made a comprehen-

sive study on the relationships among such popular extensions of T1 FSs.

See http://www.ieee.org/publications

standards/publications/rights/index.html for more information.

1198 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014

TABLE I

XISTING RESULTS FOR MONOTONICITY OF IT2 FLSS

However, for the monotonicity of IT2 FLSs, there have been

only a few studies up until now because of the complexity of

the input–output mappings of IT2 FLSs. Only for some special

IT2 FLSs, e.g., SIRMs-connected IT2 FLS [43] which is es-

sentially linear combination of the single input IT2 FLSs, the

Begian–Melek–Mendel method-based IT2 FLS [44], and the

single-input IT2 FLS [45], [46], the parameter conditions for

the monotonicity have been given. Related results are shown in

Table I. From this table, we can observe that the existing results

have the following limitations.

1) Only two kinds of type-reduction and defuzziﬁcation

methods have been considered. For IT2 FLSs, different

type-reduction and defuzziﬁcation methods can give dif-

ferent input–output mappings. The popular type-reduction

and defuzziﬁcation methods include the Karnik–Mendel

(KM) method [1], [8], [47]–[50], the Du–Ying (DY)

method [51], the Begian–Melek–Mendel (BMM) method

[52], the Wu–Tan (WT) method [53], the Nie–Tan (NT)

method [54], [55], and the uncertainty bound (UB) method

[56]. There exist no monotonicity results for the other

type-reduction and defuzziﬁcation methods.

2) The most widely used type-reduction and defuzziﬁcation

method is the KM method. However, for this method, up

until now, the results are only for single input IT2 FLSs.

3) There exist strict limitations on the membership functions

(MFs) including both the kinds of IT2 FSs and the differ-

entiability of the MFs of IT2 FSs.

Therefore, to be more practical, more work needs to be done

to guarantee the monotonicity of multiinput IT2 FLSs with dif-

ferent kinds of MFs and out-processing methods. In this study,

we present a uniﬁed framework for the monotonicity of IT2

FLSs. We derive the parameter conditions under which the IT2

FLSs can give monotonic input–output mappings. The derived

conditions are composed of two parts: the conditions on the an-

tecedent IT2 FSs and the conditions on the consequent weights.

Particularly, we consider ﬁve kinds of type-reduction and de-

fuzziﬁcation methods which cover most of practical IT2 FLSs.

We also show how to use the proposed results to design appropri-

ate monotonic IT2 FLSs for modeling and control applications.

To the best of the authors’ knowledge, this is the most com-

prehensive study on the monotonicity of IT2 FLSs. The main

novelties of this paper are listed as follows.

1) The presented results are valid for all kinds of IT2 FSs

including the Trapezoidal IT2 FSs, the Gaussian IT2 FSs,

Triangular IT2 FSs, and even the general IT2 FSs.

2) The presented conditions are useful for the ﬁve widely

used type-reduction and defuzziﬁcation methods which

cover most of IT2 FLSs in practice.

3) The derived conditions are for multiinput IT2 FLSs, no

matter which type-reduction and defuzziﬁcation method

is adopted.

4) Except the proposed antecedent and consequent condi-

tions, there exist no other limitations on the MFs of IT2

FSs. Hence, the monotonicity conditions can be more eas-

ily satisﬁed.

5) The presented results are also valid for T1 FLSs and gen-

eral TSK FLSs and more loose than some existing results

for T1 FLSs [37], [39], [42].

The rest of this paper is organized as follows. Section II

studies the monotonicity of T1 FLS and the IT2 FLS using

the KM method. Section III derives the monotonicity condi-

tions on the antecedent T1 FSs and IT2 FSs. Section IV studies

the monotonicity of the IT2 FLSs using the DY method, the

BMM method, the WT method, the NT method, and the other

out-processing methods. Section V extends the monotonicity

conditions to general TSK IT2 FLSs, presents guidelines for

applying the derived conditions to modeling and control prob-

lems and summarizes other fundamental properties of IT2 FLSs.

Finally, conclusions are drawn in Section VI. The proofs of all

lemmas and theorems are given in the Appendix.

II. M

ONOTONICITY OF FUZZY LOGIC SYSTEMS

In this study, we consider the general multiinput single-

output FLS, whose input variables are supposed to be x=

,...,x

) ∈ X

× X

×···×X

. However, our results can

be readily extended to multiinput multioutput FLS, for the lat-

ter can be decomposed into several multiinput single-output

FLSs [1], [8].

First, the structures of T1 FLSs and IT2 FLSs are introduced

brieﬂy.

A. T1 Fuzzy Logic Systems

By assigning the jth input variable N

T1 FSs, we can obtain



j=1

fuzzy rules, each of which has the following form:

Rule(i

...i

):Ifx

is A

, ..., x

is A

, Then

(x) is w

...i

, where i

=1, 2,...,N

···i

s are the crisp

consequent weights, A

s are antecedent T1 FSs for the input

variables. This rule base can be seen as the simplest TSK model

and the Mamdani model with height defuzziﬁcation, where

···i

represents the point with the maximum membership

degree of the consequent T1 FS of Rule(i

...i

) [57]. In fact,

it represents the most frequently used T1 FLSs in engineering

problems [57], [58].

LI et al.: ON THE MONOTONICITY OF INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS 1199

Once a crisp input x=(x

,...,x

) is applied to the T1

FLS, through the singleton fuzziﬁer, the ﬁring strength of Rule

...i

) can be calculated by the product operation as follows:

...i

(x) =



j=1

). (1)

Then, the output of the T1 FLS is computed as

(x) =



···



···i

(x)w

···i



···



···i

(x)



···



···i



j=1

)



···





j=1

)

. (2)

B. IT2 FLS

By blurring the T1 FSs and crisp weights, we can obtain the

following type-2 fuzzy rule base:

Rule(i

...i

):Ifx



, ..., x



, Then,

(x) is [w

...i

, w

···i

], where i

=1, 2,...,N



sarethe

antecedent IT2 FSs for the input variables, and [w

···i

, w

···i

are the interval consequent weights. This rule base can be seen

as the Mamdani model with KM method where [w

···i

, w

···i

]

can be viewed as the centroid of the consequent IT2 FS [57].

When w

···i

= w

···i

, this IT2 FLS can be viewed as the

simpliﬁed TSK model [57]. Again, this rule base represents the

most widely used IT2 FLSs in engineering applications, e.g.,

modeling problems [21] and control applications [30], [51],

[58].

Once a crisp input x=(x

,...,x

) is applied to the IT2

FLS, through the singleton fuzziﬁer and the type-2 inference

process, the interval ﬁring strength of Rule (i

,...,i

) can

be calculated by the product operation as follows:

···i

(x) = [f

···i

(x), f

···i

(x)] (3)

where

···i

(x) =



j=1



) (4)

···i

(x) =



j=1



) (5)

in which μ



and μ



denote the lower and upper MFs of the

IT2 FS



To generate crisp output, the output processing including

type-reduction and defuzziﬁcation is needed. There exist sev-

eral different type-reduction and defuzziﬁcation methods. The

IT2 FLSs with different type-reduction and defuzziﬁcation

methods have different input–output mappings. In this section,

we only discuss the most widely used Karnik–Mendel type-

reduction and Center-Of-Sets (COS) defuzziﬁcation method

(KM method) [1], [47]–[50], while other output processing

methods will be given in Section IV. The KM type-reducer

which uses the Karnik–Mendel algorithms to realize the type-

reduction are reported as the COS type-reducer in some early

literatures on IT2 FLSs [8].

Let

(x) =



···



···i

(x)w

···i



···



···i

(x)

(6)

(x) =



···



···i

(x)w

···i



···



···i

(x)

(7)

where f

···i

(x) means f

···i

(x) or f

···i

(x). Consequently,

we have a total of 2



j =1

different combinations for both

(x) and y

(x), i.e., n =1,...,K where K =2



j =1

By the KM type-reducer, the left and right end points of the

interval output of the IT2 FLS are computed as

(x) = min

n=1

(x) (8)

(x) = max

n=1

(x). (9)

The Karnik–Mendel algorithm [1], [8] or its enhanced ones

[47]–[49] can be used to ﬁnd y

(x) and y

(x) effectively. This

is why, we usually call this COS type-reducer as the KM type-

reducer. As the expressions in (8) and (9) are very convenient for

our theoretical analysis, the details of Karnik–Mendel algorithm

are omitted here. For more details, see [1], [8], and [47]–[49].

If the KM method is adopted, the crisp output of the IT2 FLS

is calculated as

(x) =

(x) + y

(x)]. (10)

When all sources of uncertainty disappear, IT2 FSs



s be-

come T1FSs A

s, and interval weights [w

···i

, w

···i

]s be-

come crisp weights w

···i

s. Simultaneously, y

(x) and y

(x)

all turn to the same function y

(x), which is the input–output

mapping of a T1 FLS. Hence, the T1 FLS can be viewed as a

special case of the IT2 FLS using the KM method.

C. Deﬁnition of Monotonicity

To present monotonicity conditions for FLSs, let us give the

deﬁnition of monotonicity ﬁrst.

Deﬁnition 1 [37], [39]: An FLS is said to be monotonically

increasing with respect to (w.r.t) the kth input variable

,if∀x

≤ x

∈ X

implies that y

,...,x

, ..., x

) ≤

,...,x

) for all the combinations of (x

,...,

k−1

k+1

,...,x

). And, an FLS is said to be monotonically

decreasing w.r.t the kth input variable x

,if∀x

≤ x

∈ X

implies that y

,...,x

) ≥ y

,...,x

,..., x

)

for all combinations of (x

,..., x

k−1

k+1

,...,x

In this study, we just take the increasing monotonicity into

account. Similar results can be obtained for decreasing mono-

tonicity.

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