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Interval Type-2 Fuzzy Set-Theoretic Control Design for Uncertain Dynamical Systems

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Abstract

Fuzzy set plays an important role in handling vagueness for controlling uncertain dynamical systems. However, conventional type-1 fuzzy set (T1FS) requires precisely defined membership function, which is usually unavailable in practical control applications. This study pioneers the use of IT2FS for the control design of uncertain dynamical systems to relax this limitation of conventional T1FS-based control design. Concretely, the considered (possibly fast) time-variant uncertainty is bounded with the bound lying within IT2FS (hence interval type-2 fuzzy dynamical system, IT2FDS). A robust control design approach is proposed without invoking any IF-THEN fuzzy rules, providing a two-layer performance. The lower layer renders uniform boundedness and uniform ultimate boundedness for the system by the Lyapunov analysis, ensuring the bottom line. The upper layer improves the fuzzy-based control performance by optimal gain design oriented by a two-player zero-sum game, taking advantage of the interval description of membership function. It is shown that the equilibrium of the game which contains the optimal gain always exists. A semi-analytical solution strategy for the equilibrium is given. This is the first endeavor in exploring IT2FS-based control design without using any IF-THEN rules. It is shown in the demonstrative examples that the proposed IT2FS-based approach can achieve better control performance than the conventional T1FS-based approach.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 52105096, 52175087), in part by the GuangDong Basic and Applied Basic Research Foundations, China (Grant No. 2021A1515011752), in part by the the Fundamental Research Funds for the Central Universities (No. 531110810651), in part by the Natural Science Foundation of Hunan Province of China (Grant No. 2022JJ40084), and in part by the Scientific Research Project of the Education Department of Hunan Province of China (No. 21B0029).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Yifan Liu, Hui Yin, Baizhan Xia & Dejie Yu

  2. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Ye-Hwa Chen

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  1. Yifan Liu
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Correspondence to Hui Yin.

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Liu, Y., Yin, H., Xia, B. et al. Interval Type-2 Fuzzy Set-Theoretic Control Design for Uncertain Dynamical Systems. Int. J. Fuzzy Syst. 26, 1069–1087 (2024). https://doi.org/10.1007/s40815-023-01654-3

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