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Solving type-2 fuzzy relation equations via semi-tensor product of matrices

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Abstract

The problem of solving type-2 fuzzy relation equations is investigated. In order to apply semi-tensor product of matrices, a new matrix analysis method and tool, to solve type-2 fuzzy relation equations, a type-2 fuzzy relation is decomposed into two parts as principal sub-matrices and secondary sub-matrices; an r-ary symmetrical-valued type-2 fuzzy relation model and its corresponding symmetrical-valued type-2 fuzzy relation equation model are established. Then, two algorithms are developed for solving type-2 fuzzy relation equations, one of which gives a theoretical description for general type-2 fuzzy relation equations; the other one can find all the solutions to the symmetrical-valued ones. The results can improve designing type-2 fuzzy controllers, because it provides knowledge to search the optimal solutions or to find the reason if there is no solution. Finally some numerical examples verify the correctness of the results/algorithms.

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Authors and Affiliations

  1. College of Computer and Control Engineering, Nankai University, Tianjin, 300071, China

    Yongyi Yan, Zengqiang Chen & Zhongxin Liu

  2. Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, 300071, China

    Yongyi Yan, Zengqiang Chen & Zhongxin Liu

Authors
  1. Yongyi Yan
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  2. Zengqiang Chen
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  3. Zhongxin Liu
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Corresponding author

Correspondence to Yongyi Yan.

Additional information

This work was partially supported by the Natural Science Foundation of China (No. 61174094), the Tianjin Natural Science Foundation of China (No. 13JCYBJC17400), and the Program for New Century Excellent Talents in University of China (No. NCET-10-0506).

Yongyi YAN received the B.S. and M.S. degrees in Mathematics from Luoyang Normal University, Luoyang, China, in 2005 and 2008, respectively, and is currently pursuing the Ph.D. degree in Control Theory and Engineering at Nankai University, Tianjin, China. His current research interests are in the fields of modeling and optimization of complex systems, fuzzy control, intelligent predictive control.

Zengqiang CHEN was born in Tianjin, China in 1964. He received the B.S., M.E. and Ph.D. degrees from Nankai University, in 1987, 1990, and 1997, respectively. He is currently a professor of Control Theory and Engineering of Nankai University, and Deputy Director of Institute of Robotics and Information Automation. His current research interests are in the fields of intelligent predictive control, chaotic systems and complex dynamic network, multi-agent system control.

Zhongxin LIU received his B.E. and D.E. degrees in Nankai University, in 1997 and 2002, respectively. He is currently a professor of Control Theory and Engineering of Nankai University, Tianjin, China. His current research interests include Multi-agent systems, complex and dynamic networks, computer control and management.

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Yan, Y., Chen, Z. & Liu, Z. Solving type-2 fuzzy relation equations via semi-tensor product of matrices. Control Theory Technol. 12, 173–186 (2014). https://doi.org/10.1007/s11768-014-0137-7

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  • DOI: https://doi.org/10.1007/s11768-014-0137-7

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