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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