Construction of non-convex fuzzy sets and its application

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Construction of non-convex fuzzy sets and its application Dan Hu a,b,∗, Tao Jiang a, Xianchuan Yu a a b

College of Information Science and Technology, Beijing Normal University, Beijing 100875, China Department of Radiology and BRIC, University of North Carolina at Chapel Hill, USA

a r t i c l e

i n f o

Article history: Received 17 November 2017 Revised 22 October 2018 Accepted 27 October 2018 Available online xxx Keywords: Non-convex fuzzy sets Parametric qualitative fuzzy set (PQ FS) Fuzzy logic system Fuzzy logic controller

a b s t r a c t Although non-convex fuzzy set (FS) has the high potential of great performance in data modeling and controlling, it is seldom used and discussed because the lack of linguistic explanation and normative construction way. To address this problem, we propose a method named “parametric qualitative fuzzy set (PQ FS) plus choice strategy” for the construction and linguistic explanation of non-convex FS, in which PQ FS is a collection of convex FSs with special structure, and choice strategy is an approach to choose convex FSs from PQ FS. Based on this method, a non-convex FS is obtained as the trajectory of a collection of convex FS by choosing specific convex FS under specific situation. Thus, the linguistic explanation of non-convex FS is obtained: using non-convex FSs to represent linguistic variables does not violate the routine of using convex FSs, because it shows that the linguistic variable is just represented by different convex FS at different situation. Theorems are shown to demonstrate that the “PQ FS plus choice strategy” can effectively construct a non-convex FS. Furthermore, “Why a fuzzy logic system (FLS) adopting non-convex FSs may have a higher approximation capability” is discussed by introducing a parametric qualitative FLS (PQ FLS) that is compared with a typical Mamdani FLS as a function approximator. This indicates that non-convex FSs can approximate more extrema in a given universe with smaller partition numbers or fewer rules than convex FSs. Finally, the experimental results verify that a PQ FLS designed with the proposed non-convex FS construction method can outperform traditional convex fuzzy logic controllers (FLCs). Meanwhile, using parallel computing in the model training phase of PQ FLSs can reduce the calculation time compared to single-thread mode. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Since Zadeh introduced the concept of a fuzzy set (FS) [1], it has been successfully applied in various fields [2]. In both theory and application, FSs tend to be restricted to the convex form. Few studies have focused on non-convex fuzzy systems. Mitaim and Kosko discussed the shape of membership functions (MFs) in adaptive function approximation and used the non-convex sinc function as an MF to design a fuzzy logic system [3,4]. Although experiments have revealed the importance of non-convex FS, Mitaim and Kosko reported that non-convex FSs may have no linguistic meaning. They explained that “the engineering goal of function-approximation accuracy may sometimes outweigh the linguistic or philosophical interpretations of a FS” [4]. Why the FLS adopting non-convex FSs may have a higher approximation capability has not been discussed yet. Such a situation is not beneficial for the popularization of non-convex FSs. In [5], Garibaldi

∗ Corresponding author at: College of Information Science and Technology, Beijing Normal University, Beijing 100875, China. E-mail address: [email protected] (D. Hu).

discussed non-convex FSs and classified them into elementary non-convex FSs, time-related non-convex FSs, and consequent non-convex MFs. The advantages of the non-convex FS were proved by a case study on a fuzzy expert system. In this case, Garibaldi introduced non-convex MFs for some special linguistic terms and some special uses related to human decision-making. Moreover, some other studies have reported on non-convex fuzzy data [6], fuzzy entropy of non-convex FS [7], and fuzzy variables in fuzzy structural analysis [8]. However, except the non-convex fuzzy system discussed by Mitaim and Kosko [3,4], the other studies are not related to traditional fuzzy systems and fuzzy controllers. Furthermore, although non-convex FSs have the potential to increase the performance of a fuzzy system, they are seldom used because of the lack of linguistic explanation. If only convex MFs are used in designing fuzzy systems, we may lose the opportunity to capture more complex structures in the data by producing potentially simpler or better systems. Furthermore, although non-convex FSs can be obtained through set operation between convex FSs, how to construct or choose them is still required to be discussed. Actually, non-convex FS can be regarded as the result of adaptively varying the parameters of convex FSs. Historically, the idea of adaptively

https://doi.org/10.1016/j.neucom.2018.10.111 0925-2312/© 2019 Elsevier B.V. All rights reserved.

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varying the parameters of FSs at the process of controlling has ever been introduced in [9] as stable adaptive FLS and in [10] as variable universe FLS, which fuzzy membership functions are adjusted based on specific adaptive law, Lyapunov synthesis approach in [9] and contraction-expansion factor in [10], how to measure the relationship between the two FLSs is still not discussed. Thus, to construct non-convex FSs, a new concept named parametric qualitative fuzzy set (PQ FS), which is a collection of convex FSs with a specific structure is proposed in this paper. To describe and process knowledge with variation and adaptability, the “PQ FS plus choice strategy” is utilized to construct a non-convex FS. To show the linguistic explanation of non-convex FS, a non-convex FS is explained as the trajectory FS of the chosen convex FSs under all situations. To describe the high approximation and controlling capability of non-convex FSs, the PQ FLS system and the constructednon-convex fuzzy logic controllers (PQ FLCs) are discussed in this paper. The main contributions of this study are the following. (1) The “PQ FS plus choice strategy” method is proposed to construct non-convex FSs. (2) The linguistic explanation of non-convex FS is given in this paper. (3) PQ FLS and PQ FLCs are designed to show how to apply the constructed non-convex FSs in fuzzy logic system and fuzzy logic controllers. The experimental results show the high approximation capability and high controlling performance of non-convex FSs. This paper is organized as follows. The PQ FS and trajectory FS are defined and constructed in Section 2. The theorems related to the non-convex property of “PQ FS plus choice strategy” and the algorithm for constructing non-convex FSs are presented in Section 3. The linguistic explanation of non-convex FSs is represented in Section 4. “Why the FLS adopting non-convex FSs may have a higher approximation capability ” is discussed in Section 5. PQ FLCs is designed for controlling a first order unstable nonlinear system in Section 6. The conclusions are presented in Section 7. 2. PQ FS and trajectory FSs

PQ FSs may be constructed in many ways. Three ways of constructing PQ FSs that might be useful in practice are formalized in this section. 2.1.1. Perturbation A ∈ F (X ) is a FS. PQ FSs based on A can be constructed through parameter perturbation. Suppose the membership function of A is determined by a parameter set v = {v1 , . . . , vk }. Then, A can be represented as A(x; v). A PQ FS A¨ is constructed by perturbation of the parameter subset v1 ⊆v with the complementary parameters v2 = (v1 )c are fixed on the chosen values v20 . Suppose V is the perturbation scope of v1 . Then 1 1 A¨ = {A˙ v |A˙ v = A(x; v1 , v20 ), v1 ∈ V }.

(5)

Example 1. [PQ FS constructed by perturbation] Suppose A is a Gaussian FS. The parameter set of A is v = {m, σ }. A(x ) = −

(x−m0 )2 2

G(x; m0 , σ0 ) = e 2σ0 rameter perturbation.

. PQ FSs A¨1 and A¨2 are constructed by pa-

˙m A¨1 = {A˙ m 1 |A1 = G (x; m, σ0 ), m ∈ Vm = [m0 − m1 , m0 + m2 ]},

(6)

A¨2 = {A˙ σ2 |A˙ σ2 = G(x; m0 , σ ),

(7)

σ ∈ Vσ = [σ0 − σ1 , σ0 + σ2 ]},

where m1 , m2 , σ 1 and σ 2 are constants. 2.1.2. Homotopy of LMF and UMF By the homotopy of continuous functions, a PQ FS A¨ can be constructed by the two FSs (LMF and UMF). Definition 3. [Homotopy of continuous functions] A homotopy between two continuous functions f, g: X → Y is defined as a continuous function h: X × [0, 1] → Y such that, for x ∈ X, h(x, 0 ) = f (x ) and h(x, 1 ) = g(x ). Definition 4. [PQ FS constructed by homotopy] Suppose h: X × [0, 1] → [0, 1] is a homotopy between FSs A1 and A2 that satisfies h(x, 0 ) = A1 and h(x, 1 ) = A2 (or h(x, 0 ) = A2 and h(x, 1 ) = A1 ). Then, a PQ FS A¨ can be constructed as

A¨ = {A˙ t |A˙ t (x ) = h(x, t ), t ∈ [0, 1]}.

2.1. PQ FS and its construction To discuss PQ FS, firstly, we must restate the accepted definitions of convex and non-convex fuzzy sets. Definition 1. [Convex and non-convex FSs] [1] Fuzzy set, A is said to be convex if and only if

A(λx1 + (1 − λ )x2 ) ≥ min{A(x1 ), A(x2 )}

(1)

where x1 , x2 ∈ X, λ ∈ [0, 1]. A is said to be non-convex if it is not convex.

(8)

Example 2. [PQ FS constructed by linear homotopy] Suppose Al , Au ∈ F (X ), and ∀x ∈ X, Au (x) ≥ Al (x). Then, linear homotopy h: X × [0, 1] → [0, 1] is defined as

h(x, t ) = (1 − t )Al + tAu . Then, a PQ FS A¨ constructed by lows:

(9) Al ,

Au ,

and h is shown as fol-

A¨3 = {A˙ t3 |A˙ t3 (x ) = (1 − t )Al (x ) + tAu (x ), t ∈ [0, 1]}. Clearly, μ ¯ A¨ = 3

Au ,

μA¨ = 3

(10)

Al .

Definition 2. [PQ FS] Let PF (X ) be all possible collections of FSs defined on the domain X, that is, PF (X ) = {A ⊆ F (X )|F (X ) = {A|A : X → [0, 1]}}.∀A¨ ∈ PF (X ), A¨ is called a parametric qualitative fuzzy set (PQ FS), if all elements in A¨ can be parameterized by a vector v, that is,

2.1.3. Transformation Transformation of FSs can also be used to construct PQ FSs.

A¨ = {A˙ v |A˙ v ∈ F (X ), v ∈ V }.

A¨ = {A˙ t |A˙ t = A( ft (x )), t ∈ T }.

A˙ v is called the parametric embedded FS of PQ FS A¨ . is the image of x under A¨ and defined as

(2)

∀x ∈ X, A¨ (x )

A¨ (x ) = {A˙ v (x )|A˙ v ∈ A¨ }.

(3)

μA¨ and μ¯ A¨ are the upper and lower membership functions(UMF and LMF, respectively) of A¨ and defined as

μA¨ (x ) = min{A˙ v (x )|A˙ v ∈ A¨ }, μ¯ A¨ (x ) = max{A˙ v (x )|A˙ v ∈ A¨ }.

(4)

Definition 5. [PQ FS constructed by transformation] ∀A ∈ F (X ). ft : X → X is a transformation from X to X, t ∈ T. A PQ FS A¨ can be constructed as:

(11)

Example 3. [PQ FS constructed by flexible transformation] {A} is the given fuzzy set. Suppose T = [0, 1], ft : X → X is a flexible transformation: ft (x ) = x/t. Then, a PQ FS A¨ 4 can be constructed as A¨ 4 = {A˙ t4 |A˙ t4 (x ) = A(x/t ), t ∈ (0, 1]}. That is, a PQ FS is a collection of FSs with a specific structure, which is determined by the approach used for construction.

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2.2. Choice strategy and trajectory FSs The parametric embedded FSs in PQ FSs are restricted to the convex form. Therefore, the collection of convex FSs in a PQ FS indicates the variation of representing a single concept with FS under different situations (such as different person, environment, and time). Each parametric embedded convex FS represents a specific representation. Then, a choice strategy should be designed that determines the corresponding parametric embedded convex FSs for each situation p ∈ P, where P is the situation set that comprises all concerned situations. Since the choice strategy is a correspondence relationship from P to parameter set V, the parameter functions mapping from V to P can be considered as the most simple and convenient choice strategy. Suppose the parameter set of PQ FS A¨ is V. A¨ = {A˙v |v ∈ V }. The parameter function η: P → V can be designed with different properties, such as monotonicity, periodicity, symmetry and boundedness. Some families of functions that might be useful in practice are shown as follows (with examples) : (1) Periodic:

η ( p) = sin(ω p + α ), ω ∈ R, α ∈ R;

Fig. 1. Trajectory FS of A¨5 determined by parameter function η1 .

(12)

(2) Monotonic:

η ( p) = 1 − α e−kp , α > 0, k > 0; 2

 η ( p) =  +

| p| P 

α ,

α > 0,  > 0;

(13) (14)

(3) Differential time-series such as the Mackey–Glass equation given by

d η ( p) 0.2η ( p − τ ) = − 0.1η ( p ), τ > 0; dp 1 + η10 ( p − τ )

(15) Fig. 2. Trajectory FS of A¨6 determined by parameter function η2 .

(4) Piecewise function:

η ( p) =

⎧ v1 , ⎪ ⎪ ⎪ ⎪v2 , ⎨ ···

vk , ⎪ ⎪ ⎪ ⎪ ⎩· · · vn ,

p ∈ P1 p ∈ P2 p ∈ Pk

Parameter function η1 : P1 → [4, 6] is

(16)

p ∈ Pn

m = η1 ( p) = 5 + sin p ∈ [4, 6], p ∈ P1 = 2X. Parameter function η2 : P2 → [0, 1] is

t = η2 ( p ) =

where vk ∈ V (k = 1, 2, . . . , n ) and P1 ∪ P2 ∪ · · · ∪ Pn = P . As the domain of η, situation set P is always chosen as time, input domain X, or other domains of context factors. Definition 6. [Trajectory FSs] Suppose A¨ is a PQ FS, A¨ = {A˙v |v ∈ V }, X is the domain of A˙ v , P is the situation set, and η: P → V is the parameter function (choice strategy). A˙ η ( p) is an FS defined by X × P and named as trajectory FS of A¨ (under η). In particular, when the situation set P is considered as kX = {kx|x ∈ X, k ∈ R}, A˙ η (kx ) (x ) is a trajectory FS of A¨ defined in X. In this paper, only the trajectory FS defined in X are considered. Example 4. A, Al , Au ∈ F (X ), A(x ) = G(x; 5, 1.5 ), and Al (x ) = G(x; 5, 1 ), Au (x ) = G(x; 5, 3 ). A¨5 is constructed by A and the perturbation of the mean. A¨6 is constructed by Al , Au , and the linear homotopy. A¨7 is constructed by A and the flexible transformation.

(20)

1 ∈ [0, 1], p ∈ P2 = 0.3X. 1 + e−p

(21)

Parameter function η3 : P3 → [0, 1] is t = η3 ( p) = 1 − 0.9e−p ∈ [0, 1], p ∈ P3 = X. The red curves in Figs. 1–3 represent the trajectory FSs of PQ FSs in Example 4, which are determined by the PQ FSs and the corresponding chosen parameter functions. The black curves in Figs. 1–3 represent some parametric embedded FSs of A¨5 , A¨6 , and A¨7 , respectively. This approach of generating trajectory FSs is called the “PQ FSs plus choice strategy.” For each value p ∈ P, only one parametric embedded FS has an intersection with the trajectory, which is the single parametric embedded FS (opinion) chosen for the current situation (such as input, time, and context). Thus, a trajectory FS actually records all the choices for the entire universe. Parameter functions with different properties and PQ FSs with different structures will lead to different trajectory FSs. 2

3. Construction of non-convex FSs

˙m A¨5 = {A˙ m 5 |A5 = G (x; m, 1.5 ), m ∈ [4, 6]},

(17)

A¨6 = {A˙ t6 |A˙ t6 (x ) = (1 − t )Al (x ) + tAu (x ), t ∈ [0, 1]},

(18)

A¨7 = {A˙ t7 |A˙ t7 (x ) = G(x/t; 5, 2 ), t ∈ (0, 1]}.

(19)

3.1. Construction of non-convex FS through “PQ FS plus choice strategy” η ( 2x ) η ( 0.3x ) η (x ) In Example 4, A˙5 1 (x ), A˙6 2 (x ) and A˙7 3 (x ) are all η1 (2x ) η3 (x ) trajectory FSs defined on X. A˙5 (x ) and A˙7 (x ) are non-

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(iii). If on [c, d]⊆[a, b], Al monotonically decreases and the inequality

η ( x1 ) Au ( x2 ) − Al ( x1 ) > η ( x2 ) Au ( x1 ) − Al ( x1 )

(24)

holds for x1 < x2 , x1 , x2 ∈ [c, d], then A˙ η (x ) (x ) monotonically decreases on [c, d]. (iv). If η(x) monotonically decreases on [γ 1 , γ 2 ] < [β 1 , β 2 ], then A˙ η (x ) (x ) monotonically decreases on [γ 1 , γ 2 ]. (v). If η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[a, b], and ∃x1 < x2 ∈ [ψ 1 , ψ 2 ],

η ( x1 ) < Fig. 3. Trajectory FS of A¨7 determined by parameter function η3 .

convex, indicating that the “PQ FS plus choice strategy” can be considered an approach of constructing non-convex FSs. However, the trajectory FS could also be convex, such as η ( 0.3x ) A˙6 2 (x ) in Example 4. Thus, the “PQ FS plus choice strategy” for constructing non-convex FSs comprises three steps: Step 1: Construct a PQ FS based on a convex fuzzy set and a specific construction method. Step 2: Determine a choice strategy. Step 3: Demonstrate the non-convexity of the trajectory FS. Whether the trajectory FS is non-convex can be discussed by the specific construction approach of PQ FS and the properties of the choice function. Several methods exist for constructing PQ FSs and designing the choice functions. In this paper, we only discuss the trajectory FSs of PQ FSs constructed by linear homotopy. Theorem 1. Suppose Al and Au are continuous convex FSs defined on [a, b], Au (b) = Al (b) = 0. Al < Au . Al and Au are all not constant. Then, ∃[α 1 , α 2 ]⊆[a, b], [β 1 , β 2 ]⊆[a, b], Al and Au monotonically increase on [α 1 , α 2 ]. Al and Au monotonically decrease on [β 1 , β 2 ]. Proof. Suppose Au monotonically increases on [a1 , b1 ]⊆[a, b]. Because Al is convex and not constant, ∃c ∈ [a, b], Al (c) is the maximum of Al (x). Thus, ∃[a2 , b2 ]⊆[a1 , c], Al monotonically increases on [a2 , b2 ]. Let [α1 , α2 ] = [a2 , b2 ] ∩ [a1 , b1 ] ⊂ [a, b], Al and Au monotonically increase on [α 1 , α 2 ]. The existence of [β 1 , β 2 ] can be proved similarly.  Theorem 2. A PQ FS A¨ is constructed by linear homotopy and continuous convex FSs Al , and Au defined on [a, b]. Al < Au . A¨ = {A˙ t |A˙ t (x ) = (1 − t )Al + tAu , t ∈ [0, 1]}. Situation set P = [a, b]. Parameter function η: [a, b] → [0, 1] is continuous. Al and Au monotonically increase on [α 1 , α 2 ]. Al and Au monotonically decrease on [β 1 , β 2 ]. (i). If η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[a, b], and ∃x1 < x2 ∈ [γ 1 , γ 2 ],

A˙ η (x1 ) (x ) − Al (x ) η ( x1 ) > u 2 l 1 , A ( x1 ) − A ( x1 )

(22)

then ∃[ 1 ,  2 ]⊆[γ 1 , γ 2 ], A˙ η (x ) (x ) monotonically decreases on [ 1 ,  2 ]. (ii). If η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[a, b], ∃x1 < x2 ∈ [γ 1 , γ 2 ], Au (x2 ) − Al (x2 ) < Au (x1 ) − Al (x1 ), and

η ( x1 ) >

(

Au

Al ( x1 ) − Al ( x2 ) , (x2 ) − Al (x2 )) − (Au (x1 ) − Al (x1 ))

(23)

then ∃[ 1 ,  2 ]⊆[γ 1 , γ 2 ], A˙ η (x ) (x ) monotonically decreases on [ 1 ,  2 ].

A˙ η (x1 ) (x2 ) − Al (x1 ) , Au ( x1 ) − Al ( x1 )

(25)

then ∃[ 1 , 2 ]⊆[ψ 1 , ψ 2 ], A˙ η (x ) (x ) monotonically increases on [ 1 ,

2 ]. (vi). If η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[a, b], ∃x1 < x2 ∈ [ψ 1 , ψ 2 ], Au (x2 ) − Al (x2 ) > Au (x1 ) − Al (x1 ), and

η ( x1 ) >

(

Au

Al ( x1 ) − Al ( x2 ) , (x2 ) − Al (x2 )) − (Au (x1 ) − Al (x1 ))

(26)

then ∃[ 1 , 2 ]⊆[ψ 1 , ψ 2 ], A˙ η (x ) (x ) monotonically increases on [ 1 ,

2 ]. (vii). If on [e, f]⊆[a, b], Al monotonically increases and the inequality

η ( x1 ) Au ( x2 ) − Al ( x1 ) < η ( x2 ) Au ( x1 ) − Al ( x1 )

(27)

holds for x1 < x2 , x1 , x2 ∈ [e, f], then A˙ η (x ) (x ) monotonically increases on [e, f]. (viii). If η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[α 1 , α 2 ], then A˙ η (x ) (x ) monotonically increases on [ψ 1 , ψ 2 ]. Proof. (i). Because A˙ η (x ) (x ) = η (x )(Au (x ) − Al (x )) + Al (x ),

η (x2 )(Au (x2 ) − Al (x2 )) = A˙ η(x2 ) (x2 ) − Al (x2 ).

(28)

By contrast,

η (x1 )(Au (x1 ) − Al (x1 )) > A˙ η(x1 ) (x2 ) − Al (x1 ).

(29)

Thus, ∃[ 1 ,  2 ]⊆[x1 , x2 ]⊆[γ 1 , γ 2 ], on [ 1 ,  2 ],

[η (x )(Au (x ) − Al (x ))] < [A˙ η (x ) (x2 )] − [Al (x )] .

(30)

Then

[A˙ η (x ) (x )] < [A˙ η (x ) (x2 )] = η (x )(Au (x2 ) − Al (x2 )) < 0.

(31)

(ii). For x1 < x2 ∈ [γ 1 , γ 2 ],

A˙ η (x1 ) (x2 ) − Al (x1 ) − η ( x1 ) Au ( x1 ) − Al ( x1 ) 1 = u · ( η ( x 1 )[ ( A u ( x 2 ) A ( x1 ) − Al ( x1 ) − Al (x2 )) − (Au (x1 ) − Al (x1 ))] + (Al (x2 ) − Al (x1 ))). Au ( x2 ) − Al ( x2 ) < Au ( x1 ) − Al ( x1 ) and Al (x1 )−Al (x2 ) A˙ η (x1 ) (x2 )−Al (x1 ) , we have (Au (x2 )−Al (x2 ))−(Au (x1 )−Al (x1 )) Au (x1 )−Al (x1 )

When

(32)

η ( x1 ) > − η ( x1 ) < 0.

Then, based on (i), the conclusion in (ii) holds. (iii). ∀x1 < x2 ∈ [c, d], Al (x1 ) − Al (x2 ) < 0 because Al monotonically decreases on [c, d].

A˙ η (x2 ) (x2 ) − A˙ η (x1 ) (x1 ) = η (x2 )(Au (x2 ) − Al (x2 )) − η (x1 )(Au (x1 ) − Al (x1 )) + Al (x2 ) − Al (x1 ) η (x )

When η (x1 ) > 2

Au ( x

2

)−Al (x

1)

Au (x1 )−Al (x1 )

(33)

,

A˙ η (x2 ) (x2 ) − A˙ η (x1 ) (x1 ) < (Al (x1 ) − Al (x2 ))(η (x2 ) − 1 ) < 0.

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Fig. 4. The graphical representation of Theorem 3. (b)–(f) are respectively corresponding to (i)–(v) of Theorem 3. The PQ FS in figure (i) is chosen randomly and the locations of cl and cu will not affect the conclusion. In the subfigures, “” and “” are used to represent the monotonicity of A˙ η (x ) (x ) in the corresponding subinterval, where “”(“”) means A˙ η (x ) (x ) monotonically increases (decreases) in a subinterval bounded by the dotted lines.

(iv). Because η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[β 1 , β 2 ], η (x), [Au (x)] and [Al (x)] are all less than zero. Then,

[A˙ η (x ) (x )] = η (x )(Au (x ) − Al (x )) + η (x )[Au (x )] + (1 − η (x ))[Al (x )] < 0.

(34) 

Finally, (vi–viii) of this theorem can be proved similarly to (i–iv). Theorem 3. Suppose Al (a ) = Al (b) = Au (a ) = Au (b) = 0. Al and Au are all not constant functions. When one of the following conditions is satisfied, the trajectory FS A˙ η (x ) (x ) must be non-convex. (i). η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[α 1 , α 2 ], ∃x1 , x2 ∈ [γ 1 , γ 2 ], x1 < x2 , η (x1 ) >

A˙ η (x1 ) (x2 )−Al (x1 ) ; Au (x1 )−Al (x1 )

η(x) monotonically in-

A˙ η (x1 ) (x2 )−Al (x1 ) ; Au (x1 )−Al (x1 )

η(x) monotonically de-

creases on [ψ 1 , ψ 2 ]⊆[α 1 , α 2 ], γ 2 ≤ ψ 1 . (ii). η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[β 1 , β 2 ], ∃x1 , x2 ∈ [ψ 1 , ψ 2 ], x1 < x2 , η (x1 ) <

creases on [γ 1 , γ 2 ]⊆[β 1 , β 2 ], γ 2 ≤ ψ 1 . (iii). η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[α 1 , α 2 ], ∃x1 , x2 ∈ [γ 1 , γ 2 ], x1 < x2 , η (x1 ) >

A˙ η (x1 ) (x2 )−Al (x1 ) ; Au (x1 )−Al (x1 )

η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[β 1 , β 2 ], ∃x1 , x2 ∈ [ψ 1 , ψ 2 ], x3 < x4 , η (x3 ) < A˙ η (x3 ) (x4 )−Al (x3 ) . Au (x3 )−Al (x3 )

(iv). η(x) monotonically decreases on [γ 1 , γ 2 ]⊆[α 1 , α 2 ], ∃x1 , x2 ∈ [γ 1 , γ 2 ], Au (x2 ) − Al (x2 ) < Au (x1 ) − Al (x1 ) and η (x1 ) > Al (x1 )−Al (x2 ) ; ∃[α 3 , α 4 ]⊆(γ 2 , b), Al monotonically in(Au (x2 )−Al (x2 ))−(Au (x1 )−Al (x1 )) η (x ) Au (x )−Al (x ) creases on [α 3 , α 4 ] and η (x3 ) < u 4 l 3 holds for x3 < x4 ∈ [α 3 , A (x3 )−A (x3 ) 3

α 4 ].

(v). η(x) monotonically increases on [ψ 1 , ψ 2 ]⊆[β 1 , β 2 ], ∃x1 , x2 ∈ [ψ 1 , ψ 2 ], Au (x2 ) − Al (x2 ) > Au (x1 ) − Al (x1 ) and η (x1 ) > Al (x1 )−Al (x2 ) ; ∃[β 3 , β 4 ]⊆(a, ψ 1 ), Al monotonically de(Au (x2 )−Al (x2 ))−(Au (x1 )−Al (x1 )) η (x ) Au (x )−Al (x ) creases on [β 3 , β 4 ] and η (x3 ) > u 4 l 3 holds for x3 < x4 ∈ [β 3 , A (x3 )−A (x3 ) 4

β 4 ].

Al ( a )

Al ( b )

Au ( a )

Au ( b )

A˙ η (a ) (a ) =

Proof. Because = = = = 0, A˙ η (b) (b) = 0. Thus, ∃ 1 ,  2 > 0, A˙ η (x ) (x ) monotonically increases on (a, a + 1 ) and monotonically decreases on (b − 2 , b). Then, based on Theorem 2, the monotonicity of A˙ η (x ) (x ) in the specific interval can be determined. The graphical representations

η (x )

Fig. 5. Comparison between η2 (x1 ) and 2 2

Au (x2 )−Al (x1 ) Au (x1 )−Al (x1 )

when x1 < x2 .

of Theorem 4 are shown in Fig. 4. Fig. 4(b)–(f) corresponding to (i-v) of this theorem, respectively. Under each situation, the trajectory FS A˙ η (x ) (x ) has at least two maximums on [a, b]. Thus, Theorem 3 holds.  Therefore, based on Theorem 3 and the “PQ FS plus choice strategy”, non-convex FSs can be constructed using suitable parameter functions. Example 4 (continued) A¨ 6 is constructed by linear homotopy. A¨6 = {A˙ t6 |A˙ t6 (x ) = (1 − t )G(x; 5, 1 ) + tG(x; 5, 3 ), t ∈ [0, 1]}. X = [−4, 14]. t is determined by P2 = 0.3X and parameter function η ( 0.3x ) η2 : P2 → [0, 1], η2 ( p) = 1−p . As shown in Fig. 2, A˙6 2 (x ) 1+e

is convex. It’s convexity will be explained by Theorem 2. To be consistent with Theorem 2, let P2 = X and η2 ( p) = 1+e1−0.3 p . Then,

η (x ) η ( 0.3x ) A˙6 2 (x ) is the same as A˙6 2 (x ). η2 (x ) monotonically increases on [−4, 14]. On [−4, 5], because η2 (x ), Al (G(x; 5, 1)) and Au (G(x; 5,

η (x ) 3)) monotonically increase, A˙6 2 (x ) monotonically increases based

η (x ) on Theorem 2(viii). On [5,12], based on Theorem 2(iii), A˙6 2 (x ) monotonically decreases because ∀x1 < x2 ,

η2 (x1 ) Au (x2 ) − Al (x1 ) > η2 (x2 ) Au (x1 ) − Al (x1 )

(35)

holds. Fig. 5 is the representation of inequality (35): to avoid large values and for ease of interpretation, only x1 < x2 ∈ [6, 14] are shown.

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Algorithm 1 constructing given non-convex FSs by homotopy. Input: A non-convex FS A(x ) Output: A PQ FS A¨ and the homotopy function η (x ) 1: 2:

Al and Au are chosen to satisfy the constraint that Al  A  Au . A PQ FS A¨ is obtained by Al , Au and homotopy,

A¨ = {A˙ t |A˙ t = (1 − g(t ))Al + g(t )Au , t ∈ [0, 1]},

(36 )

where g is an invertible function satisfies that g(0 ) = 0, g(1 ) = 1. 3: Construct the parameter function η : X → [0, 1] as

η (x ) = g−1 ((A(x ) − Al (x ))/(Au (x ) − Al (x ))), 4:

(37 )

where g−1 is the inverse function of g. The non-convex FS A is constructed by A¨ and η,A(x ) = A˙ η (x ) (x ).

Fig. 7. Parameter function η and some parametric embedded FSs A˙ t of A¨ .

3.3. Constructing any given non-convex FSs by using the piecewise function If the given non-convex FS has the piecewise property, piecewise function is the optimal parameter function for the construction of it. Suppose the membership function of a non-convex FS, A, in domain X is a piecewise function shown in Eq. (41)

A (x ) =

3.2. Constructing given non-convex FSs by homotopy

Example 5. Mitaim and Kosko once took the sinc function as the MF to design a fuzzy system [4]. In this example, we constructed the sinc function by using the “PQ FS plus parameter strategy”. To ensure that the membership degree of the fuzzy set is positive, sinc function is modified as fuzzy set A: sin x x

+ 0.217 , x ∈ [−20, 20]. 1.217

A is shown as the red curve in Fig. 6. Then, A can be constructed by {A¨ , η}, where A¨ is a PQ FS, A¨ = {A˙ t |A˙ t = (1 − t )Al + tAu , t ∈ [0, 1]},

Au ( x ) =

⎪ ⎪ ⎩

1−

 A (x ) = l

7.72

1

0.71(x−0.5 ) , 7.72

0, 1 + (1/3.51 )x, 1 − (1/3.51 )x,

x ∈ [−20, −7.72 ) ∪ (7.72, 20] x ∈ [−7.72, −0.5 ) x ∈ [−0.5, 0.5 ) x ∈ [0.5, 7.72] x ∈ [−20, 3.51 ) ∪ (3.51, 20] x ∈ [−3.51, 0 ) x ∈ [0, 3.51]

η (x ) = (A(x ) − Al (x ))/(Au (x ) − Al (x )).

x ∈ [ak−1 , ak )

(41)

x ∈ [an−1 , an )

A¨ = {A˙ t = At (x )|t = 1, 2, . . . , n}.

Homotopy is the simplest method to construct a non-convex FS based on the “PQ FS plus choice strategy”. Suppose A is the given non-convex FS; then, the algorithm for constructing given nonconvex FSs by homotopy is shown as follows:

⎧ 0.29, ⎪ ⎪ ⎨1 + 0.71(x+0.5) ,

···

Ak ( x ), ⎪ ⎪ ⎪ ⎪ ⎩ ··· An ( x ),

x ∈ [a0 , a1 ) x ∈ [a1 , a2 )

where [a0 , a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [an−1 , an ) = X and Ak (x) is the kth part of the piecewise function. PQ FS A¨ can be obtained by the n parts of A,

Fig. 6. Given non-convex FS A and the bounds of PQ FS A¨ in example 5.

A = (sinc + 0.217 )/1.217 =

⎧ A1 ( x ), ⎪ ⎪ ⎪ ⎪ ⎨A2 (x ),

(38)

Constructing the parameter function η: X → n as

η (x ) =

⎧ 1, ⎪ ⎪ 2, ⎪ ⎪ ⎨

··· ⎪ k, ⎪ ⎪ ⎪ ⎩· · · n,

The parameter function η and some parametric embedded FSs A˙ t of A¨ are shown in Fig. 7.

x ∈ [ak−1 , ak )

(43)

x ∈ [an−1 , an )

Example 6. Garibaldi and Musikasuwan considered the desirability (drinkability) of a glass (cup) of milk according to the temperature of the milk is a continuous non-convex FS [5]. The non-convex fuzzy set, drinkability, can be constructed by using the “PQ FS plus parameter strategy”. As shown in Fig. 8, A is the curve and can be divided into three convex fuzzy sets: A1 , A2 and A3 . The expression of A is given as Eq. (44). Then, A can be constructed by {A¨ , η}, where A¨ is a PQ FS, A¨ = {A˙ t = At (x )|t = 1, 2, 3} and η(x) is expressed as in Eq. (48).



A1 ( x ), A2 (x ) = 0.12, A3 ( x ),



(40)

x ∈ [a0 , a1 ) x ∈ [a1 , a2 )

Thus, the non-convex FS A is constructed by A¨ and η, A(x ) = A˙ η (x ) (x ).

A (x ) =

(39)

(42)

A˙ 1 (x ) =

A1 ( x ), 0.12,

A˙ 2 (x ) = 0.12,

A˙ 3 (x ) =

0.12, A3 ( x ),

x ∈ [0, 22 ) x ∈ [22, 44 ) x ∈ [44, 60]

x ∈ [0, 22 ) x ∈ [22, 60 ) x ∈ [0, 60] x ∈ [0, 44 ) x ∈ [44, 60]

(44)

(45) (46)

(47)

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7

Suppose F¨1i and G¨ i are PQ FSs, i i,vi F¨1i = {F˙1 1 |vi1 ∈ V1i }, G¨ i = {G˙ i,w |wi ∈ W i }.

(50)

Then, R is the parametric qualitative rule set (PQ rule set). According to the parametric embedded FSs of PQ FSs, rule Rl can be represented by a collection of parametric embedded rules. Each i i parametric embedded rule R˙ i,(v1 ,w ) corresponds with the choices i

i,v i of F˙1 1 and G˙ i,w . i

i i i i,v R˙ i,(v1 ,w ) : IF x1 is F˙1 1 , THEN y is G˙ i,w ,

(51)

i i R¨i = {R˙ i,(v1 ,w ) |vi1 ∈ V1i , wi ∈ W i }.

(52)

{R i , i

For the rule set R = = 1, . . . , M}, a parametric embedded rule set R˙ v of R is obtained by choosing one para

i i metric embedded rule from each R¨i . ∀v ∈ M i=1 V1 × W , v = 1 1 k k M M i i ( v1 , w ; . . . ; v1 , w ; . . . ; v1 , w ), v|i = ( v1 , w ),

Fig. 8. Drinkability of milk according to its temperature.

 η (x ) =

1, 2, 3,

x ∈ [0, 22 ) x ∈ [22, 44 ) x ∈ [44, 60]

i i R˙ v = {R˙ i,(v1 ,w ) |i = 1, . . . , M, (vi1 , wi ) = v|i }.

(48) ¨ = R

By constructing and representing non-convex FSs through the “PQ FS plus parameter function”, the essence and linguistic explanation of the non-convex FSs can be clarified. A non-convex FS is the trajectory of a collection of convex FSs under choices. Although it is better to express words in natural language by convex functions or regions [11] at a cognitive level, the words are restricted to the same person at the same time or situation. The higher-order uncertainty existing in “words differ in meaning with people and time” is negligible. When collections of convex FSs are used to express the words, non-convex FSs may be obtained by choosing the convex FS according to specific situation. The final trajectories are non-convex FSs does not violate the routine of representing linguistic variables by convex FSs, because convex FSs are the actual objects chosen to represent the linguistic variables under specific situations (such as time, and input). Thus, non-convex FSs are reasonable and have their own linguistic interpretation. 5. Why the fuzzy logic system adopting non-convex FSs may have higher approximation capability Although Mitaim and Kosko [3,4] demonstrated that adopting non-convex FSs, such as the sinc function, in fuzzy logic system (FLS) can result in a higher capability of function approximation through experiments, no studies have yet investigated this because of a lack of knowledge about non-convex FSs. In this paper, based on the proposed construction method “PQ FS plus parameter function” of non-convex FSs, “Why the FLS adopting non-convex FSs may have a higher approximation capability” could be discussed using parametric qualitative FLSs.

M 

V1i × W i .

(54)

Each R˙ v is a fuzzy rule set. The FLS that takes R˙ v as its rule set is defined as S˙ v and is called the parametric embedded FLS of S. For an input x, an output yv will be obtained according to S˙ v . If S˙ v is an FLS with a singleton fuzzifier, product inference and centroid defuzzifier, it can be described as an interpolation function f˙ v : U ⊆ R → R, R is a set of real numbers,

M

i,vi1

i

y¯ i,w · F˙1

i=1

f˙ v (x ) =

M

i ˙ i,v1 i=1 F1

(x )

(x )

,

(55)

i i where y¯ i,w is the point in the output space R at which G˙ i,w (y ) achieves its maximum value. Consider a multiple-input single-output system S with M rules in its rule base R = {Ri , i = 1, . . . , M}. The ith rule Ri is in the form:

Ri : IF x1 is F¨1i , and . . . and xρ is F¨ρi , THEN y is G¨ i , i = 1, . . . , M. Suppose F¨ki , k = 1, . . . , ρ , and G¨ i are PQ FSs, i

i i,v F¨ki = {F˙k k |vik ∈ Vki }, G¨ i = {G˙ i,w |wi ∈ W i }.



¨ = R

R˙ |v ∈ v

M 



i=1

ρ 

(56)



Vki

×W

i

,

(57)

k=1

M M v = (v11 , . . . , v1ρ , w1 ; v21 , . . . , v2ρ , w2 ; . . . ; vM 1 , . . . , vρ , w ).

(58)

R˙ v = {R˙ i,(v1 ,...,vρ ,w ) |i = 1, . . . , M, (vi1 , . . . , viρ , wi ) = v|i }.

(59)

i

i

i

i

i i i i,v R˙ i,(v1 ,...,vρ ,w ) : IF x1 is F˙1 1 , and . . . and i

i i,v xρ is F˙ρ ρ , THEN y is G˙ i,w .

5.1. Parametric qualitative FLSs An FLS that uses PQ FSs in the rule antecedents or consequences is called a parametric qualitative FLS (PQ FLS). Consider a single-input single-output system S that considers M rules as its rule base R = {Ri , i = 1, . . . , M}. The ith rule Ri is in the form:1

1

R˙ v |v ∈

(53)



i=1

4. Linguistic explanation of non-convex FSs

Ri : IF x1 is F¨1i , THEN y is G¨ i , i = 1, . . . , M.



(49)

Although it is unnecessary to use the subscript 1 on x for a single-antecedent rule, by doing so we will make the multiple-antecedent case easier to understand.

R˙ v

(60) R˙ v

Each is a fuzzy rule set. The FLS with as its rule set is defined as S˙ v and called the parametric embedded FLS of S. For an input x, an output yv will be obtained according to S˙ v . If S˙ v is a FLS with singleton fuzzifier, product inference and centroid defuzzifier, it can be described as an interpolation function f˙ : U ⊆ Rρ → R, x = (x1 , x2 , . . . , xρ ) ∈ U,

ρ i,vi i y¯ i,w · k=1 F˙k k (xk ) ,

M ρ i,vik ˙ F ( x ) k i=1

M f˙ v (x ) =

i=1

(61)

k=1 k

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i i where y¯ i,w is the point in the output space R at which G˙ i,w (y ) v ˙ achieves its maximum value. f is called as an embedded interpolation function.

5.2. Comparison of PQ FLS and a typical Mamdani FLS as a function approximator In general, compared with other approximators, the FLS approximators offer a unique ability of utilizing not only numerical data but also linguistically-expressed human knowledge. The shapes of fuzzy MFs affect how well an FLS approximates a given function or controls a given object [3]. Although fuzzy MFs can have any shape, MFs are usually restricted to convex forms. What will be improved if the convex FSs are replaced by trajectory FSs in typical Mamdani FLSs? As universal approximators, the approximated functions by typical Mamdani FLSs can only have one extremum in a single partition subarea. If the function to be approximated has many extrema, high accuracy on approximation implies a large number of fuzzy partitions and rules. Thus, the FLS becomes high complicated. For a general SISO FLS, the input universe [a, b] is divided into N( > 1) subintervals:

a = C0 < C1 < C2 < · · · < CN−1 < CN = b.

(62)

Then, N + 1 continuous FSs are defined on [a, b] to fuzzify the input variable x. Each input FS is denoted by Ai (0 ≤ i ≤ N). Among them, Ai (1  i  (N − 1 )) has a MF Ai (x) whose value is nonzero only on [Ci−1 , Ci+1 ]. If this FLS is a typical Mamdani FLS, then all Ai (0 ≤ i ≤ N) are convex. In particular, Ai (x )(1  i  N − 1 ) equals 0 at x = Ci−1 , increases monotonically on [Ci−1 , Ci−1 + αi−1 ], 0 < αi−1  Ci − Ci−1 , and reaches 1 at x = Ci−1 + αi−1 . Ai (x) is 1 on [Ci−1 + αi−1 , Ci+1 − βi+1 ], 0 < βi−1  Ci+1 − Ci , and decreases monotonically and becomes 0 at x = Ci+1 . More details can be found in [12]. Suppose there are M( > 1) fuzzy rules in the form

ri : If x is Ai , Then y is Bi , i = 1, 2, . . . , M.

(63)

Bi is a convex FS for the output variable y, whose membership value is 1 only at y = yi and is < 1 elsewhere. After being defuzzified by the centroid defuzzifier, the output of the SISO FLS is

M

Ai ( x )yi y = F (x ) = i=1 . M i=1 Ai (x )

(64)

Theorem 4. If only two fuzzy rules ri and ri+1 are assigned to each subinterval [Ci , Ci+1 ] , the function defined in Eq. (64) either monotonically increases or decreases on [Ci , Ci+1 ](0  i < N ) [12]. On the basis of Theorem 4, if a typical Mamdani FLS tries to approximate a continuous function f with K extrema, [a, b] must be divided into at least K + 1 subintervals. Thus, when f has several extrema, the number of subintervals will be huge, which will lead to high computational complexity of FLS. In a SISO PQ FLS S, suppose there are M( > 1) fuzzy rules in the form

ri : If x is A¨ i , then y is Bi , i = 1, 2, . . . , M,

(65)

where A¨ i are PQ FSs. The interpolation function of the PQ FLS is

M

i=1

F ( x ) = M

η (x ) A˙ i i (x )yi

i=1

η (x ) A˙ i i (x )

,

(66)

η (x ) where ηi is the parameter function corresponding with A¨ i , A˙ i i (x ) ¨ is the trajectory FS of Ai .

Theorem 5. Assume yi < yi+1 . Suppose PQ FS A¨ i is constructed by convex FSs {ALi , AUi } and linear homotopy, ALi < AUi , A¨ i = {A˙ ti |A˙ ti =

(1 − t )ALi + tAUi , t ∈ [0, 1]}. ηi : [a, b] → [0, 1] is the continuous parameter function. If only two fuzzy rules ri and ri+1 are assigned to each subinterval [Ci , Ci+1 ](i = 0, 1, . . . , N − 1 ), ∀0 < K ∈ Z+ , the interpolation function F(x) of PQ FLS S can have K extrema in [Ci , Ci+1 ] by choosing an appropriate parameter function ηi and ηi+1 . Proof. When x ∈ [Ci , Ci+1 ], only the FSs Ati and Ati+1 may have nonzero values. Hence only ri and ri+1 are activated. The output of the PQ FLS is given by

F (x ) =

ηi+1 (x ) η (x ) A˙ i i (x )yi + A˙ i+1 (x )yi+1 ηi+1 (x ) η (x ) A˙ i i (x ) + A˙ i+1 (x )

= yi+1 + ϕ (x )(yi − yi+1 ), (67)

where

ϕ (x ) =

˙ ηi (x )

Ai

η (x ) A˙ i i (x )

i+1 (x ) (x ) + A˙ ηi+1 (x )

=

1 . ηi+1 (x ) η (x ) ˙ 1 + Ai+1 (x )/A˙ i i (x )

(68) 

As x increases from Ci to Ci+1 , ALi (x ) and AUi (x ) monotonically decreases from 1 to 0, while ALi+1 (x ) and AUi+1 (x ) monotonically increases from 0 to 1. If F(x) is supposed to be maximum in interval [σ1 , σ2 ] ⊆ [Ci , Ci+1 ], the parameter functions ηi and ηi+1 should be designed as: ηi+1 monotonically decreases on [σ 1 , x0 ] and monotonically increases on [x0 , σ 2 ](σ 1 < x0 < σ 2 ); ηi monotonically increases on [σ 1 , x0 ] and monotonically decreases on [x0 , σ 2 ]. In interval [x0 , σ 2 ], ηi (x ) < 0, (ALi ) (x )  0 and (AUi ) (x ) < 0; ηi +1 (x ) > 0, (ALi+1 ) (x )  0 and (AUi+1 ) (x ) > 0. Then, based η (x ) ) (x ) < 0

on Theorem 3(iv) and Theorem 3(viii), (A˙ i i i (x ) (A˙ ηi+1 ) (x ) > 0 hold. Thus, in [x0 , ˙ ηi+1 (x ) ˙ ηi (x )

and

σ 2 ], F(x) monotonically decreases

(x )/Ai (x ) monotonically increases. In interval [σ 1 , x0 ], ηi (x ) > 0, (ALi ) (x )  0 and (AUi ) (x ) < 0. (A˙ ηi i (x) ) (x ) > 0 does not hold necessarily. However, if ∃x∗ ∈ [σ 1 ,

because Ai+1

x0 ],

ηi ( x ∗ ) <

η ( x∗ ) A˙ i i (x0 ) − ALi (x∗ )

AUi (x∗ ) − ALi (x∗ )

,

(69)

based on Theorem 3(v), there is an interval (α i , β i )⊆[σ 1 , x0 ] such η (x ) that A˙ i i (x ) monotonically increases in (α i , β i ). If ∀x ∈ [σ 1 , x0 ],

ηi ( x ) 

η (x ) A˙ i i (x0 ) − ALi (x )

AUi (x ) − ALi (x )

,

(70)

and a new parameter function can be constructed as ηinew ,

ηinew (x ) =



ηi ( x ) , ηi ( θ i ( x − x 0 ) + x 0 ) ,

x ∈ [ x 0 , σ2 ] , x ∈ [ σ1 +(θθi −1)x0 , x0 ]

(71)

i

σ1 +(θi −1 )x0 < x0 holds. Because ηi (x ) > 0, θi σ + ( θ −1 ) x 0 ηinew (x ) < ηi (x ) on [ 1 θi , x0 ]. θ i is chosen to make the i σ +(θ −1 )x0 value of ηinew (x ) is sufficiently small. Then, ∃x∗ ∈ [ 1 θi , x0 ], i σ1 +(θi −1 )x0 Eq. (69) holds, and there is an interval (αi , βi ) ⊆ [ , x0 ] θi ηinew (x ) ηinew (x )

where θ i > 1. σ1 <

such that (A˙ i ) (x ) > 0. A˙ i monotonically increases in this interval. For the same reason, we can construct ηinew similar to ηinew +1 σ + (θ

−1 )x

0 and there is an interval (αi+1 , βi+1 ) ⊆ [ 1 θi+1 , x0 ], such i+1 new ηi+1 (x ) that A˙ i+1 (x ) monotonically decreases in (αi+1 , βi+1 ). It is easy to choose θ i and θi+1 , such that (αi+1 , βi+1 ) ∩ (αi , βi ) = ∅. Then, in (αi+1 , βi+1 ) ∩ (αi , βi ), F(x) monotonically increases beηinew (x ) ηnew (x ) +1 cause A˙ i+1 (x )/A˙ i i (x ) monotonically decreases.

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Fig. 9. PQ FSs and parametric functions in Example 7.

Finally, because F(x) monotonically increases in (αi+1 , βi+1 ) ∩ (αi , βi ) ⊆ [σ1 , x0 ] and monotonically decreases in [x0 , σ 2 ], F(x) achieves at least one maximum in [σ 1 , σ 2 ]. In addition, if F(x) is supposed to be minimum in the interval [σ1 , σ2 ] ⊆ [Ci , Ci+1 ], the proof is similar. Since the extremum can be achieved in any specific interval by constructing parameter functions, F(x) could have as many as extrema in the given universe. Thus, F can obviously achieve K extrema in [Ci , Ci+1 ]. Thus, PQ FLSs have strong capability to approximate complex functions without increasing the number of partitions or rules. We can potentially produce simpler or better systems by using PQ FLSs. In PQ FLS, different requirements of extrema and accuracy can be satisfied by constructing the choice strategy. When trajectory FSs are approximated, the monotonicity in the subintervals is also approximated, ensuring a high accuracy of function approximation. Thus, PQ FLSs are considerably more economical and efficient than the FLSs designed by convex MFs. More strict proofs related to the approximation properties of PQ FLSs will be discussed in our further work. PQ FLSs have such capability in approximation because PQ FLS can result in complicated non-convex FLSs by changing the choice strategy. Therefore, the high accuracy of PQ FLSs can be explained by the high accuracy of non-convex FLSs. Example 7. How the trajectory FSs reach more extra by changing parameter function: Suppose Ci = 2, Ci+1 = 5. ALi (x ) = 3 − x, AUi (x ) = 5/3 − x/3, ALi+1 (x ) = x − 4, and AUi+1 (x ) = x/3 − 2/3. A¨ i = {A˙ ti |(1 − t )A˙ Li + t A˙ Li , t ∈ [0, 1]}, A¨ i+1 = {A˙ ti+1 |(1 − t )A˙ Li+1 + t A˙ Li+1 , t ∈ [0, 1]}. A¨ i and A¨ i+1 are shown in Fig. 9(a). The peak points of Bi and Bi+1 are yi = 1 and yi+1 = 2, respectively. The corresponding parameter functions for A¨ i and Ai¨+1 are ηi = 1/2(sin θi x + 1 ) and ηi+1 = | sin(θi+1 x )|, respectively. Let θi = 2, θi+1 = 3; the trajecηi+1 η tory FSs A˙ i i and A˙ i+1 are shown in Fig. 10(a). When different parameter functions are chosen, different output function F(x) will

9

Fig. 10. Trajectory FSs and PQ input-output functions with different parameter functions.

be obtained. The PQ input-output functions with θi = 2, θi+1 = 3 and θi = 7, θi+1 = 9 are shown in Fig. 10(b). In a single partition interval (Ci , Ci+1 ), the PQ input-output function can have three maxima(θi = 2, θi+1 = 3) or six maxima(θi = 7, θi+1 = 9) by changing parameter functions. Furthermore, we can obtain as many extrema as many as required by choosing appropriate θ i and θi+1 . This discussion clarifies that the higher approximation capability of PQ FLS is due to the freedom of choice strategy. When the choice strategy changes, the trajectory FSs demonstrate flexible non-convexity. By increasing the non-convexity of the trajectory FSs, the FLS can approximate the function with as many extrema as required in the given universe. Thus, the FLS adopting non-convex FSs may have a higher approximation capability because non-convex FSs can approximate more extrema in the given universe with a smaller partition number or less rules than convex FSs. 6. Experiments The validity of PQ FLS is further validated by controlling a first order unstable nonlinear system. Three performance indices, integral of absolute error(IAE), integral of the squared error(ISE) and integral of the time-weighted absolute error(ITAE), were employed as quantitative measures to compare the control performances of PQ FLCs and optimized convex FLCs. The convex FLCs were designed by convex FSs with tunable parameters. The number of the tunable parameters in PQ FLCs and convex FLCs were the same in each comparison. All experiments were run in the same environment (Matlab R2017a, Windows 10 professional 1709, 3.4 GHz Intel Core i7-6700 processor (8 cores), 16GB 2400 MHz DDR4 Memory). 6.1. Controlling first order plant with time delay Here we compare PQ FLC with a convex FLC by controlling a first order plus dead-time plant:

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D. Hu, T. Jiang and X. Yu / Neurocomputing xxx (xxxx) xxx Table 1 Parameters of the five plants. Parameter\Plant

I

II

III

IV

V

K

1 10 2.5

1 5 2.5

1 20 2.5

0.5 10 2.5

2 10 2.5

τ L

Table 2 IAE of the five FLCs. FLC\Plant

I

II

III

IV

V

Sum

Convex Trans-Exp Homo-Exp Trans-Sin Homo-Sin

7.887 9.100 5.325 5.536 4.129

8.224 6.687 7.47 4.582 4.309

11.69 14.40 13.19 7.787 6.096

9.511 14.03 7.395 5.661 5.315

8.957 7.720 8.861 5.297 4.800

46.269 51.937 42.241 28.863 24.649

Table 3 ISE of the five FLCs. FLC\Plant

I

II

III

IV

V

Sum

Convex Trans-Exp Homo-Exp Trans-Sin Homo-Sin

4.749 5.464 3.527 3.916 3.230

4.475 3.776 4.113 2.899 3.207

6.842 7.770 7.246 5.478 4.303

5.547 7.932 4.241 4.085 3.842

5.177 4.615 4.986 3.764 3.545

26.790 29.557 24.113 20.142 18.127

Fig. 11. Step response with K = 1 and τ = 10.

Table 4 ITAE of the five FLCs. FLC\Plant

I

II

III

IV

V

Sum

Convex Trans-Exp Homo-Exp Trans-Sin Homo-Sin

58.44 77.80 33.08 39.02 22.27

73.4 50.06 64.42 39.22 17.63

130.5 227.1 185.9 50.52 32.28

92.02 207.0 63.76 26.05 27.77

89.78 58.20 82.8 37.98 20.14

444.14 620.16 429.96 192.79 120.09

G (s ) =

Y (s ) = U (s )

K

τs + 1

e−Ls .

(72)

Details about the plant and the rule base of the FLCs can be found in [13,14]. Since an equivalent convex FS is adopted in [13] to handle the type-reduction of type-2 fuzzy sets, we call the FLC discussed in [13] as a convex FLC. To evaluate the feasibility of PQ FLC, a comparative study using the following five FLCs were performed: -Convex: convex FLC; -Homo-Sin: A PQ FLC using homotopy to construct PQ FSs and using sine function as choice mechanism; -Trans-Sin: A PQ FLC using transformation to construct PQ FSs and using sine function as choice mechanism; -Homo-Exp: A PQ FLC using linear homotopy to construct PQ FSs and using exponential function as choice mechanism; -Trans-Exp: A PQ FLC using translation to construct PQ FSs and using exponential function as choice mechanism. The way to construct PQ FSs by translation was just like:

A¨ T rans = {A˙ tT rans |A˙ tT rans (x ) = μ ¯ A (x + t ), t ∈ [0, 1]}.

(73)

Choice mechanisms were taken as sine :

t = |sin(ax + b)|, a, b ∈ (−∞, +∞ )

(74)

and exponential function

t = 1 − a · e−bx , a ∈ (0, 1 ), b > 0, 2

(75)

where x is the input of FLSs, a and b are parameters tuned by GA to optimize IAE. The performances of these five FLCs were compared by using the 5 plants given in Table 1 as testbeds. Tables 2–4 show the performance indices for the five FLCs in the comparative study. The

step response in Fig. 11 indicated that Homo-Sin and Trans-Sin get a better performance than other FLCs with short rise time and high steady state accuracy. From the overall performances, Homo-Sin is the best of the five FLCs, Trans-Sin is the second. The improvement obtained by PQ FLCs is significant. PQ FLCs outperform the other FLCs by low overshoot, short rise time and high steady state accuracy. Besides, the construction of PQ FSs and choice mechanism may affect the performance of PQ FLCs. In this experiment, it looks that homotopy is a good way to construct PQ FSs because the average performance of Homo-Sin and Homo-Exp was better than Trans-Sin and Trans-Exp. Furthermore, if we compare Homo-Sin and TransSin with Homo-Exp and Trans-Exp, it looks that sine function is better than exponential function to be the choice mechanism. Finally, choice mechanism is more crucial than PQ FSs construction method in the performance of PQ FLCs because the difference is bigger when the choice mechanism is changed. All these results depends only on this exact experiment, the combination of choice mechanism and PQ FSs construction method will be explored in our further work. 6.2. Using parallel genetic algorithm to train models To get the best controlling capability of FLCs, some parameters need to be optimized to get the best solution of the FCLs model. Genetic algorithms (GA) belong to the larger class of evolutionary algorithms (EA), which are commonly used to generate highquality solutions to optimization and search problems [15]. The GA has been used for PQ FLCs systems to find the best solution for the choosen model. But due to the slow convergence speed and high computational complexity problem, its use for PQ FLCs has been limited. Inspired by some parallel algorithms [16–21] for the data modle and controlling, parallel computing could reduce the calculation time that GA takes to show its results. Fine-grained parallel genetic algorithms were adopted in our experiments to fit the PQ FLCs systems and find the global optimum. Parallel computation has overhead to transfer data to the workers and to transfer results back. In Matlab R2017a, each parallel worker can be allocated only a single core by default, which can degrade performance. The parameters for experimental models were set as follows: The parallel pools was configured to allocate two cores per worker, and then the number of parallel tasks was four (8 cores in total). The parameters number as the size of ‘vector’ in FLCs was set as 8. Some other options like “generations” was set to 200, “stall generations” was set to 100.

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D. Hu, T. Jiang and X. Yu / Neurocomputing xxx (xxxx) xxx Table 5 The training time of the five FLCs in Plant I. FLC\Runing mode

Four-threads(s)

Single-thread(s)

Convex Trans-Exp Homo-Exp Trans-Sin Homo-Sin

216.5 668.0 497.2 338.4 284.3

341.4 1690.7 1214.6 839.2 682.7

Table 6 The training time of the five FLCs in Plant III. FLC\Runing mode

Four-threads(s)

Single-thread(s)

Convex Trans-Exp Homo-Exp Trans-Sin Homo-Sin

224.6 628.8 512.5 386.0 293.6

373.9 1703.5 1323.7 924.7 741.3

The performances of these five FLCs using the parallel computing and single-threaded mode were given in Tables 5 and 6. From the overall performances in plant I and III, Convex FLC is the fastest of the five FLCs, and Homo-Sin is the second. Using the parallel computing in GA got a huge improvement. Four-threads parallel genetic algorithms reduced the calculation time compared to single-thread mode. The training time of single-thread mode was about two to three times that of four-threads. 7. Conclusion In this paper, the construction of non-convex FSs is discussed. While a PQ FS is defined to represent the variation in understanding the concept or words by using convex FS, a non-convex FS is constructed using the “PQ FS plus choice strategy”, which represents the trajectory of choosing different convex FSs under different situations. The algorithm for constructing given non-convex FSs by homotopy is presented. Linguistic explanation of non-convex FSs is obtained, and that non-convex FS can be used to manage higher-order uncertainty is demonstrated. PQ FLSs are proposed; a PQ FLS has the higher approximation power than does the typical Mamdani T1 FLS. Furthermore, it is not necessary to ensure that each trajectory FS constructed by the “PQ FS plus choice strategy” is non-convex. Non-convexity is not the essence of the description of complicated concepts or linguistic variables but the result of “using PQ FSs to describe higher-order uncertainty and using the choice function to determine the convex set for each situation”. Whether a trajectory FS is convex or non-convex only depends on the method of construction of the PQ FS and the corresponding choice function. Then, “Why the FLS adopting non-convex FSs may have a higher approximation capability” is answered by comparing the PQ FLS and the typical Mamdani FLS as a function approximator, which shows that non-convex FSs can approximate more extrema in the given universe with a smaller partition number or lesser rules than convex FSs, resulting in a higher approximation capability with fixed inference rules. Finally, The first order unstable nonlinear systems have been designed to verify the performances of PQ FLCs, the experimental results show the higher control performances of PQ FLCs compared to convex T1 FLCs. Thus, the “PQ FS plus choice strategy” can be used to build and explain non-convex fuzzy systems. Because the basic form of PQ FS is convex and the choice strategy may be diversified, this type of

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non-convex fuzzy systems have higher interpretability and flexibility. This will be discussed in our future work. Conflict of Interest None. Acknowledgment This research is supported in part by National Natural Science Foundation of China (Grant No.11471045, 41672323) and Major Scientific Research Projects of Universities in Guangdong (2016KTSCX167). References [1] L.A. Zadeh, Fuzzy sets, Information and control 8 (3) (1965) 338–353. [2] D. Dubois, H. Prade, Towards fuzzy differential calculus part 3: differentiation, Fuzzy sets and systems 8 (3) (1982) 225–233. [3] S. Mitaim, B. Kosko, What is the best shape for a fuzzy set in function approximation? in: Proceedings of the Fifth IEEE International Conference on Fuzzy Systems, 2, IEEE, 1996, pp. 1237–1243. [4] S. Mitaim, B. Kosko, The shape of fuzzy sets in adaptive function approximation, IEEE Trans. Fuzzy Syst. 9 (4) (2001) 637–656. [5] J.M. Garibaldi, S. Musikasuwan, T. Ozen, R.I. John, A case study to illustrate the use of non-convex membership functions for linguistic terms, in: Proceedings of the IEEE International Conference on Fuzzy Systems, 3, IEEE, 2004, pp. 1403–1408. [6] A. Calcagnì, L. Lombardi, E. Pascali, Non-convex fuzzy data and fuzzy statistics: a first descriptive approach to data analysis, Soft Comput. 18 (8) (2014) 1575–1588. [7] S. Lee, S. Kim, N.-Y. Jang, Design of fuzzy entropy for non convex membership function, in: Proceedings of the International Conference on Intelligent Computing, Springer, 2008, pp. 55–60. [8] U. Reuter, Application of non-convex fuzzy variables to fuzzy structural analysis, in: Soft Methods for Handling Varizability and Imprecision, Springer, 2008, pp. 369–375. [9] L.-X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Syst. 1 (2) (1993) 146–155. [10] H.-X. Li, Z.-H. Miao, E. Lee, Variable universe stable adaptive fuzzy control of a nonlinear system, Comput. Math. Appl. 44 (5-6) (2002) 799–815. [11] P. Gärdenfors, Conceptual Spaces: The Geometry of Thought, MIT press, 2004. [12] H. Ying, G. Chen, Necessary conditions for some typical fuzzy systems as universal approximators, Automatica 33 (7) (1997) 1333–1338. [13] D. Wu, W.W. Tan, Computationally efficient type-reduction strategies for a type-2 fuzzy logic controller, in: Proceedings of the 14th IEEE International Conference on Fuzzy Systems. FUZZ’05, IEEE, 2005, pp. 353–358. [14] W.-W. Tan, D. Wu, Design of type-reduction strategies for type-2 fuzzy logic systems using genetic algorithms, in: Advances in Evolutionary Computing for System Design, Springer, 2007, pp. 169–187. [15] L. Davis, Handbook of Genetic Algorithms, CUMINCAD, 1991. [16] X. Zhou, K. Li, Y. Zhou, K. Li, Adaptive processing for distributed skyline queries over uncertain data, IEEE Trans. Knowl. Data Eng. 28 (2) (2016) 371–384. [17] X. Zhou, K. Li, G. Xiao, Y. Zhou, K. Li, Top k favorite probabilistic products queries, IEEE Trans. Knowl. Data Eng. 28 (10) (2016) 2808–2821. [18] J. Chen, K. Li, Z. Tang, K. Bilal, S. Yu, C. Weng, K. Li, A parallel random forest algorithm for big data in a spark cloud computing environment, IEEE Trans. Parallel Distrib. Syst. 28 (4) (2017) 919–933. [19] C. Liu, K. Li, C. Xu, K. Li, Strategy configurations of multiple users competition for cloud service reservation, IEEE Trans. Parallel Distrib. Syst. 27 (2) (2016) 508–520. [20] C.X. Chubo Liu Kenli Li, K. Li, Strategy configurations of multiple users competition for cloud service reservation, IEEE Trans. Parallel Distrib. Syst. 27 (2) (2016) 508–520. [21] K.L. Guoqing Xiao Kenli Li, X. Zhou, Efficient top-(k,l) range query processing for uncertain data based on multicore architectures, Distrib. Parallel Datab. 33 (3) (2015) 381–413. Dan Hu received the B.Sc. degree and M.Sc. degree in mathematics from Sichuan Normal University, Chengdu, China, in 1999 and 2002, respectively. And the Ph.D. degree in applied mathematics from Beijing Normal University, Beijing, China. She is currently an Associate Professor at college of Information Science and Technology, Beijing Normal University, working in the field of data mining. Her main area of interest is fuzzy sets and systems, artificial intelligence, and image processing.

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D. Hu, T. Jiang and X. Yu / Neurocomputing xxx (xxxx) xxx Tao Jiang received the B.Sc. degree in computer science from University of Electronic Science and Technology of China in 2016, Chengdu, China. He is currently a graduate student at college of Information Science and Technology, Beijing Normal University, working in the field of data mining. His main area of interest is fuzzy sets and systems, artificial intelligence, and image processing.

Xianchuan Yu received the B.Sc. degree in geo-sciences, the M.Sc. and Ph.D. degree in Mathematical Geology (Computer Application Tech.) from Jilin University, Changchun, China, in 1989, 1992, and 1995, respectively. He is currently a professor and an academic leader of intelligent information processing Center and College of Information Science and Technology of Beijing Normal University, Beijing, China. He is a vice director of China Mathematical Geology and geological information processing professional committee since 2012 and a vice chairman of Branch China National Committee of International Mathematical Geosciences Society (IAMG) since 2009. He is the author or coauthor of more than 120 academic papers on intelligent spatial information processing and its application, including three book (chapters). His current research interests include modeling uncertainty in geosciences, blind source separation, remote image processing, mineral resources appraisement and astronomical data analysis.

Please cite this article as: D. Hu, T. Jiang and X. Yu, Construction of non-convex fuzzy sets and its application, Neurocomputing, https: //doi.org/10.1016/j.neucom.2018.10.111