A New Type of Fuzzy Membership Function Designed for Interval Type-2 Fuzzy Neural Network
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摘要: 对于区间二型模糊神经网络（IT2FNN），论文给出了一种新型的模糊隶属函数（FMF）设计方法.通过所设计的模糊隶属函数，可以衍生出三种区间二型模糊隶属函数（IT2FMF）.每种区间二型模糊隶属函数都具有不同的不确定域.论文将三种衍生模糊隶属函数应用于简化区间二型模糊神经网络辨识两个非线性系统.通过仿真，将衍生区间二型模糊隶属函数的辨识性能与高斯和椭圆型模糊隶属函数进行了对比.仿真结果表明，通过调节简化区间二型模糊神经网络的参数，本文所设计的区间二型模糊隶属函数比高斯和椭圆型模糊隶属函数具有更好的辨识性能.
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关键词:
- 模糊隶属函数(FMF) /
- 区间二型模糊神经网络(IT2FNN) /
- 非线性系统 /
- 系统辨识
Abstract: A new type of fuzzy membership function (FMF) is proposed for interval type-2 fuzzy neural network (IT2FNN) in this paper. Three types of interval type-2 FMF (IT2FMF) can be derived from the proposed type of FMF. And each type of IT2FMF has different shape of footprint of uncertainty (FOU). The derived IT2FMFs are applied to a simplified T2FNN to identify two nonlinear systems. The identification performance of the derived IT2FMFs are compared with Gaussian and ellipsoidal type of IT2FMFs through simulation. Simulation results certify that the derived IT2FMFs can achieve better identification performance than Gaussian and ellipsoidal type of IT2FMFs with elaborately tuning of the parameters for the simplified IT2FNN.-
Key words:
- Fuzzy membership function (FMF) /
- interval type-2 fuzzy neural network (IT2FNN) /
- nonlinear system /
- system identification
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Fig. 3 The structure of the simplified IT2FNN with two input, three rules and one final output.
Table Ⅰ The MFs of the Ellipsoidal and Derived IT2FMFs
FMF Ellipsoidal EL-type LE-type EE-type $(a>1)$ EE-type $(0 < a < 1)$ $\overline{\mu}$ $(1-|\frac{x-m}{\sigma}|^{a_1})^{\frac{1}{a_1}}$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|^{\frac{1}{a}}$ $\underline{\mu}$ $(1-|\frac{x-m}{\sigma}|^{a_2})^{\frac{1}{a_2}}$ $1-|\frac{x-m}{\sigma}|$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|^{\frac{1}{a}}$ $1-|\frac{x-m}{\sigma}|^{a}$ 
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Table Ⅱ $\frac{\partial \overline{\mu}_{ij}}{\partial X}$ and $\frac{\partial\underline{\mu}_{ij}}{\partial X}$ of EL-type IT2FMF
$~\frac{\partial \mu}{\partial X}$ $ m_{ij}-\sigma_{ij}<x_i \leq m_{ij}$ $m_{ij}<x_i\leq m_{ij}+\sigma_{ij}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{(a_{ij}-1)}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{(a_{ij}-1)}$ $~\frac{\partial\underline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{1}{\sigma_{ij}}$ $\frac{1}{\sigma_{ij}}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial \sigma_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}$ $~\frac{\partial\underline{\mu}_{ij}}{\partial\sigma_{ij}}$ $\frac{m_{ij}-x_i}{\sigma_{ij}^2}$ $\frac{x_i-m_{ij}}{\sigma_{ij}^2}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial a_{ij}}$ $-(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln}(\frac{m_{ij}-x_i}{\sigma_{ij}}) $ $-(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln}(\frac{x_i-m_{ij}}{\sigma_{ij}})$ $~\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $0 $ $0$
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Table Ⅲ $\frac{\partial \overline{\mu}_{ij}}{\partial X}$ and $\frac{\partial\underline{\mu}_{ij}}{\partial X}$ of LE-type IT2FMF
$~\frac{\partial \mu}{\partial X}$ $ m_{ij}-\sigma_{ij} < x_i \leq m_{ij}$ $m_{ij} < x_i\leq m_{ij}+\sigma_{ij}$ $\frac{\partial\overline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{1}{\sigma_{ij}}$ $\frac{1}{\sigma_{ij}}$ $~\frac{\partial\underline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{(a_{ij}-1)}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{(a_{ij}-1)}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial \sigma_{ij}}$ $\frac{m_{ij}-x_i}{\sigma_{ij}^2}$ $\frac{x_i-m_{ij}}{\sigma_{ij}^2}$ $~\frac{\partial\underline{\mu}_{ij}}{\partial \sigma_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial a_{ij}}$ $0 $ $0$ $~\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $-(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}}) $ $-(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
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Table Ⅳ $\frac{\partial \overline{\mu}_{ij}}{\partial X}$ and $\frac{\partial\underline{\mu}_{ij}}{\partial X}$ of EE-type IT2FMF When $0 < a < 1$
$~\frac{\partial \mu}{\partial X}$ $ m_{ij}-\sigma_{ij} < x_i \leq m_{ij}$ $m_{ij} < x_i\leq m_{ij}+\sigma_{ij}$ $\frac{\partial\overline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{1}{a_{ij}\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{\frac{(1-a_{ij})}{a_{ij}}}$ $\frac{1}{a_{ij}\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{\frac{(1-a_{ij})}{a_{ij}}}$ $\frac{\partial\underline{\mu}_{ij}}{\partial m_{ij}}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{(a_{ij}-1)}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{(a_{ij}-1)}$ $\frac{\partial\overline{\mu}_{ij}}{\partial \sigma_{ij}}$ $\frac{1}{a_{ij}\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{\frac{1}{a_{ij}}}$ $-\frac{1}{a_{ij}\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{\frac{1}{a_{ij}}}$ $\frac{\partial\underline{\mu}_{ij}}{\partial \sigma_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial a_{ij}}$ $\frac{1}{a_{ij}^2}(\frac{m_{ij}-x_i}{\sigma_{ij}})^\frac{1}{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})\ $ $-\frac{1}{a_{ij}^2}(\frac{x_i-m_{ij}}{\sigma_{ij}})^\frac{1}{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$ $\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $-(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})$ $(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
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Table Ⅴ $\frac{\partial \overline{\mu}_{ij}}{\partial X}$ and $\frac{\partial\underline{\mu}_{ij}}{\partial X}$ of EE-type IT2FMF When $a>1$
$\frac{\partial \mu}{\partial X}$ $ m_{ij}-\sigma_{ij} < x_i \leq m_{ij}$ $m_{ij} < x_i\leq m_{ij}+\sigma_{ij}$ $~\frac{\partial\overline{\mu}_{ij}}{\partial m_{ij}}$ $\frac{1}{a_{ij}\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{\frac{(1-a_{ij})}{a_{ij}}}$ $-\frac{1}{a_{ij}\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{\frac{(1-a_{ij})}{a_{ij}}}$ $\frac{\partial\underline{\mu}_{ij}}{\partial m_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{(a_{ij}-1)}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{(a_{ij}-1)}$ $\frac{\partial\overline{\mu}_{ij}}{\partial \sigma_{ij}}$ $-\frac{1}{a_{ij}\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{\frac{1}{a_{ij}}}$ $\frac{1}{a_{ij}\sigma_{ij}}(\frac{x_i-m_{ij}} {\sigma_{ij}})^{\frac{1}{a_{ij}}}$ $\frac{\partial\underline{\mu}_{ij}}{\partial \sigma_{ij}}$ $-\frac{a_{ij}}{\sigma_{ij}}(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}$ $\frac{a_{ij}}{\sigma_{ij}}(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}$ $\frac{\partial\overline{\mu}_{ij}}{\partial a_{ij}}$ $-\frac{1}{a_{ij}^2}(\frac{m_{ij}-x_i}{\sigma_{ij}})^\frac{1}{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})\ $ $\frac{1}{a_{ij}^2}(\frac{x_i-m_{ij}}{\sigma_{ij}})^\frac{1}{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}}) $ $\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})$ $-(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
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Table Ⅵ The Initial Antecedent Parameters for Different Type of IT2FMFs
FMF Antecedent parameters Gaussian-type $m_{ij1}=0, ~~m_{ij2}=0, ~~d_{ij}=1$ Ellipsoidal-type $m_{ij}=0, ~~a_{ij1}=2, a_{ij2}=0.5, ~~d_{ij}=1$ EL-type $m_{ij}=0, ~~a_{ij}=2, ~~d_{ij}=1$ LE-type $m_{ij}=0, ~~a_{ij}=0.5, ~~d_{ij}=1$ EE-type $m_{ij}=0, ~~a_{ij}=1, ~~d_{ij}=1$
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Table Ⅶ The Comparison of the IAEs for Example 1
FMF Without disturbance With disturbance Gaussian-type $6.74\times 10^{-3}~~~~$ $8.65\times 10^{-2}~~~~$ Ellipsoidal-type $6.82\times 10^{-3}~~~~$ $6.3\times 10^{-2}~~~~$ EL-type $6.62\times 10^{-3}~~~~$ $6.83\times 10^{-2}~~~~$ LE-type $6.47\times 10^{-3}~~~~$ $6.12\times 10^{-2}~~~~$ EE-type $4.68\times 10^{-3}~~~~$ $5.18\times 10^{-2}~~~~$
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Table Ⅷ The Comparison of the IAEs for Example 2
FMF Without disturbance With disturbance Gaussian-type $2.25\times 10^{-2}~~~~$ $4.72\times 10^{-2}~~~~$ Ellipsoidal-type $2.68\times 10^{-2}~~~~$ $3.63\times 10^{-2}~~~~$ EL-type $2.2\times 10^{-2}~~~~$ $3.43\times 10^{-2}~~~~$ LE-type $2.21\times 10^{-2}~~~~$ $3.51\times 10^{-2}~~~~$ EE-type $2.16\times 10^{-2}~~~~$ $3.27\times 10^{-2}~~~~$
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[1] L. A. Zadeh, "The concept of a linguistic variable and its application to approximate reasoning-I, " Inform. Sci. , vol. 8, no. 3, pp. 199-249, 1975. doi: 10.1007%2F978-1-4684-2106-4_1 [2] N. N. Karnik, J. M. Mendel, and Q. L. Liang, "Type-2 fuzzy logic systems, " IEEE Trans. Fuzzy Syst. , vol. 7, no. 6, pp. 643 -658, Dec. 1999. http://dl.acm.org/citation.cfm?id=2234815 [3] Q. L. Liang and J. M. Mendel, "Interval type-2 fuzzy logic systems: Theory and design, " IEEE Trans. Fuzzy Syst. , vol. 8, no. 5, pp. 535-550, Oct. 2000. http://dl.acm.org/citation.cfm?id=2234876 [4] J. M. Mendel, R. I. John, and F. L. Liu, "Interval type-2 fuzzy logic systems made simple, " IEEE Trans. Fuzzy Syst. , vol. 14, no. 6, pp. 808-821, Dec. 2006. http://ieeexplore.ieee.org/document/4016089/ [5] R. H. Abiyev and O. Kaynak, "Type 2 fuzzy neural structure for identification and control of time-varying plants, " IEEE Trans. Ind. Electron. , vol. 57, no. 12, pp. 4147-4159, Dec. 2010. [6] C. T. Lin, N. R. Pal, S. L. Wu, Y. T. Liu, and Y. Y. Lin, "An interval type-2 neural fuzzy system for online system identification and feature elimination, " IEEE Trans. Neural Netw. Learn. Syst. , vol. 26, no. 7, pp. 1442-1455, Jul. 2015. [7] B. I. Choi and F. C. H. Rhee, "Interval type-2 fuzzy membership function generation methods for pattern recognition, " Inform. Sci. , vol. 179, no. 13, pp. 2102-2122, Jun. 2009. [8] M. A. Khanesar, E. Kayacan, M. Teshnehlab, and O. Kaynak, "Extended Kalman filter based learning algorithm for type-2 fuzzy logic systems and its experimental evaluation, " IEEE Trans. Ind. Electron. , vol. 59, no. 11, pp. 4443-4455, Nov. 2012. http://ieeexplore.ieee.org/document/5764520/ [9] Y. Y. Lin, J. Y. Chang, and C. T. Lin, "Identification and prediction of dynamic systems using an interactively recurrent self-evolving fuzzy neural network, " IEEE Trans. Neural Netw. Learn. Syst. , vol. 24, no. 2, pp. 310-321, Feb. 2013. [10] Y. Y. Lin, J. Y. Chang, N. R. Pal, and C. T. Lin, "A mutually recurrent interval type-2 neural fuzzy system (MRIT2NFS) with self-evolving structure and parameters, " IEEE Trans. Fuzzy Syst. , vol. 21, no. 3, pp. 492-509, Jun. 2013. [11] C. H. Wang, C. S. Cheng, and T. T. Lee, "Dynamical optimal training for interval type-2 fuzzy neural network (T2FNN), " IEEE Trans. Syst. Man Cybernet. B, vol. 34, no. 3, pp. 1462-1477, Jun. 2004. http://europepmc.org/abstract/MED/15484917 [12] J. R. Castro, O. Castillo, P. Melin, and A. Rodriguez-Díaz, "A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks, " Inform. Sci. , vol. 179, no. 13, pp. 2175-2193, Jun. 2009. [13] C. F. Juang and C. Y. Chen, "Data-driven interval type-2 neural fuzzy system with high learning accuracy and improved model interpretability, " IEEE Trans. Cybernet. , vol. 43, no. 6, pp. 1781-1795, Dec. 2013. [14] C. F. Juang and Y. W. Tsao, "A self-evolving interval type-2 fuzzy neural network with online structure and parameter learning, " IEEE Trans. Fuzzy Syst. , vol. 16, no. 6, pp. 1411-1424, Dec. 2008. http://ieeexplore.ieee.org/document/4529083 [15] C. S. Chen, "TSK-type self-organizing recurrent-neural-fuzzy control of linear microstepping motor drives, " IEEE Trans. Power Electron. , vol. 25, no. 9, pp. 2253-2265, Sep. 2010. [16] C. S. Chen, "Supervisory interval type-2 TSK neural fuzzy network control for linear microstepping motor drives with uncertainty observer, " IEEE Trans. Power Electron. , vol. 26, no. 7, pp. 2049-2064, Jul. 2011. [17] Y. Y. Lin, J. Y. Chang, and C. T. Lin, "A TSK-type-based self-evolving compensatory interval type-2 fuzzy neural network (TSCIT2FNN) and its applications, " IEEE Trans. Ind. Electron. , vol. 61, no. 1, pp. 447-459, Jan. 2014. http://ieeexplore.ieee.org/document/6469210/ [18] X. P. Xie, H. J. Ma, Y. Zhao, D. W. Ding, and Y. C. Wang, "Control synthesis of discrete-time T-S fuzzy systems based on a novel Non-PDC control scheme, " IEEE Trans. Fuzzy Syst. , vol. 21, no. 1, pp. 147-157, Feb. 2013. [19] Y. Y. Lin, S. H. Liao, J. Y. Chang, and C. T. Lin, "Simplified interval type-2 fuzzy neural networks, " IEEE Trans. Neural Netw. Learn. Syst. , vol. 25, no. 5, pp. 959-969, May2014. [20] C. F. Juang, R. B. Huang, and Y. Y. Lin, "A recurrent self-evolving interval type-2 fuzzy neural network for dynamic system processing, " IEEE Trans. Fuzzy Syst. , vol. 17, no. 5, pp. 1092-1105, Oct. 2009. -
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