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A new fuzzy system based on rectangular pyramid.

Jiang M, Yuan X, Li H, Wang J - ScientificWorldJournal (2015)

Bottom Line: It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain.Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives.Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China.

ABSTRACT
A new fuzzy system is proposed in this paper. The novelty of the proposed system is mainly in the compound of the antecedents, which is based on the proposed rectangular pyramid membership function instead of t-norm. It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain. Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives. Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation.

No MeSH data available.


(a) The original and simulation surfaces of f1. (b) The approximation error surface of f1. (c) The original and simulation surfaces of f2. (d) The approximation error surface of f2. (e) The original and simulation surfaces of f3. (f) The approximation error surface of f3. (g) The original and simulation surfaces of f4. (h) The approximation error surface of f4.
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fig5: (a) The original and simulation surfaces of f1. (b) The approximation error surface of f1. (c) The original and simulation surfaces of f2. (d) The approximation error surface of f2. (e) The original and simulation surfaces of f3. (f) The approximation error surface of f3. (g) The original and simulation surfaces of f4. (h) The approximation error surface of f4.

Mentions: For convenience, some notations are stated as follows: System I represents rectangular pyramid fuzzy system and System II represents the conventional fuzzy system (mentioned in Section 2) using Gaussian membership functions. Let the distances between the peak points of the two fuzzy systems be 0.2, and the distances between the sample points are chosen as 0.01. The membership functions of rule antecedents of RPFS are given in Figure 4. The original and simulation surfaces and the approximation error surfaces are shown in Figure 5. The max approximation errors and the standard deviations of System I and System II are presented in Tables 1 and 2, respectively.


A new fuzzy system based on rectangular pyramid.

Jiang M, Yuan X, Li H, Wang J - ScientificWorldJournal (2015)

(a) The original and simulation surfaces of f1. (b) The approximation error surface of f1. (c) The original and simulation surfaces of f2. (d) The approximation error surface of f2. (e) The original and simulation surfaces of f3. (f) The approximation error surface of f3. (g) The original and simulation surfaces of f4. (h) The approximation error surface of f4.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4383317&req=5

fig5: (a) The original and simulation surfaces of f1. (b) The approximation error surface of f1. (c) The original and simulation surfaces of f2. (d) The approximation error surface of f2. (e) The original and simulation surfaces of f3. (f) The approximation error surface of f3. (g) The original and simulation surfaces of f4. (h) The approximation error surface of f4.
Mentions: For convenience, some notations are stated as follows: System I represents rectangular pyramid fuzzy system and System II represents the conventional fuzzy system (mentioned in Section 2) using Gaussian membership functions. Let the distances between the peak points of the two fuzzy systems be 0.2, and the distances between the sample points are chosen as 0.01. The membership functions of rule antecedents of RPFS are given in Figure 4. The original and simulation surfaces and the approximation error surfaces are shown in Figure 5. The max approximation errors and the standard deviations of System I and System II are presented in Tables 1 and 2, respectively.

Bottom Line: It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain.Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives.Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China.

ABSTRACT
A new fuzzy system is proposed in this paper. The novelty of the proposed system is mainly in the compound of the antecedents, which is based on the proposed rectangular pyramid membership function instead of t-norm. It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain. Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives. Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation.

No MeSH data available.