A New Type of Fuzzy Membership Function Designed for Interval Type-2 Fuzzy Neural Network
-
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
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.-
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
-
-
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 $(0 < a < 1)$ $\overline{\mu}$ $(1-|\frac{x-m}{\sigma}|^{a_1})^{\frac{1}{a_1}}$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|^{\frac{1}{a}}$ $\underline{\mu}$ $(1-|\frac{x-m}{\sigma}|^{a_2})^{\frac{1}{a_2}}$ $1-|\frac{x-m}{\sigma}|$ $1-|\frac{x-m}{\sigma}|^{a}$ $1-|\frac{x-m}{\sigma}|^{\frac{1}{a}}$ $1-|\frac{x-m}{\sigma}|^{a}$ 
下载: 导出CSV
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 $ $0$
下载: 导出CSV
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}}) $ $-(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
下载: 导出CSV
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}})\ $ $-\frac{1}{a_{ij}^2}(\frac{x_i-m_{ij}}{\sigma_{ij}})^\frac{1}{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$ $\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $-(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})$ $(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
下载: 导出CSV
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}}) $ $\frac{\partial\underline{\mu}_{ij}}{\partial a_{ij}}$ $(\frac{m_{ij}-x_i}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{m_{ij}-x_i}{\sigma_{ij}})$ $-(\frac{x_i-m_{ij}}{\sigma_{ij}})^{a_{ij}}{\rm ln} (\frac{x_i-m_{ij}}{\sigma_{ij}})$
下载: 导出CSV
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 $m_{ij}=0, ~~a_{ij}=1, ~~d_{ij}=1$
下载: 导出CSV
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 $4.68\times 10^{-3}~~~~$ $5.18\times 10^{-2}~~~~$
下载: 导出CSV
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 $2.16\times 10^{-2}~~~~$ $3.27\times 10^{-2}~~~~$
下载: 导出CSV
-
[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. 期刊类型引用(12)
1. 周林娜，王宵，丛香怡，陈正升，杨春雨. 协作机械臂碰撞环境下的安全控制. 控制理论与应用. 2024(02): 292-302 . 
百度学术2. 陆玮铭，胡陟，王泽华，倪双涛. 基于费兹定律变导纳控制的手术机器人人机交互. 传感器与微系统. 2023(02): 17-19+23 . 百度学术3. 乐宇倚，郭帅. 基于变导纳控制的上肢康复机器人柔顺控制方法研究. 工业控制计算机. 2021(10): 9-11+14 . 百度学术4. 周伟杰，韩亚丽，朱松青，周一鸣，李沈炎. 基于阻抗控制的下肢康复外骨骼随动控制. 科学技术与工程. 2020(05): 1934-1939 . 百度学术5. 周一鸣，韩亚丽，吴枫. 膝关节康复机械腿的摆动控制研究. 机电工程. 2020(05): 565-571 . 百度学术6. 丛明，马鸿江，刘冬，张佳琦. 康复外骨骼柔性髋关节设计与人机交互控制. 华中科技大学学报(自然科学版). 2020(10): 38-43 . 百度学术7. 付兴，徐海波，李月，陈磊，王黎光. 基于导纳控制的喷涂机器人直接示教方法研究. 现代制造工程. 2020(12): 49-54+82 . 百度学术8. 林安迪，干旻峰，葛涵，唐宇存，徐海东，匡绍龙，黄立新，孙立宁. 基于模糊模型参考学习控制的手术机器人人机交互. 机器人. 2019(04): 543-550 . 百度学术9. 张燕，王建宙，李威，王婕，陈玲玲，杨鹏. 基于数据驱动的膝关节外骨骼控制. 浙江大学学报(工学版). 2019(10): 2024-2033 . 百度学术10. 唐宇存，张建法，武帅，孙凤龙，匡绍龙，孙立宁. 基于虚拟夹具的手术机器人导纳控制安全策略. 机器人. 2019(06): 842-848 . 百度学术11. 李剑锋，张子康，张雷雨，董明杰，左世平，张凯. 并联踝康复机器人的系统搭建与运动控制策略. 中南大学学报(自然科学版). 2019(11): 2753-2762 . 百度学术12. 王斌锐，靳明涛，沈国阳，金英连，陈迪剑. 气动肌肉肘关节的滑模内环导纳控制设计. 兵工学报. 2018(06): 1233-1238 . 百度学术其他类型引用(18)
-
图(8) / 表(8)
计量
- 文章访问数: 2641
- HTML全文浏览量: 411
- PDF下载量: 479
- 被引次数: 30
