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Adaptive Controller Design for a Class of Uncertain Fractional-Order Nonlinear Systems: An Adaptive Fuzzy Approach

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Abstract

An adaptive fuzzy controller is designed for uncertain fractional-order nonlinear systems in this paper. Fuzzy logic system is utilized to estimate an unknown nonlinear function. By using fractional-order Lyapunov stability criterion, an adaptive fuzzy controller, which is valid for both the Caputo fractional-order systems and the Riemann–Liouville fractional-order systems, is constructed. Fuzzy system parameters are updated by fractional-order differential equations online, and the boundedness of the parameters can be guaranteed. In stability analysis, quadratic Lyapunov functions are utilized, and some basic results which are useful in the stability analysis of fractional-order nonlinear systems are proposed. Finally, simulation results are presented to confirm our results.

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Notes

  1. \(N=N_1\times N_2\times \cdots \times N_n\) rules defined as follows are also need to approximate the unknown function \(\hat{f}(x(t))\) (\(j\in J\)): Rule \(j\ \ (j=(F_1,F_2,\ldots , F_n))\): if x(t) is \(\left( F_1(t),F_2(t),\ldots , F_n(t)\right)\), then \(\hat{f}(x(t))\) is \(B^{j}(t)\).

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Acknowledgements

The authors are indebted to the anonymous reviewers’ valuable comments, which improved the presentation and quality of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11401243, 61403157), the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (Grant Nos. KJ2015A256, KJ2016A665 and KJ2016A666), the Foundation for Distinguished Young Talents in Higher Education of Anhui Province of China (Grant No. GXYQZD2016257), the Special Fund for Scientific and Technological Bases and Talents of Guangxi (Grant No. 2016AD05050) and the Special Fund for Bagui Scholars of Guangxi.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Heng Liu & Shenggang Li

  2. Department of Applied Mathematics, Huainan Normal University, Huainan, 232038, China

    Heng Liu & Guanjun Li

  3. School of Science, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, 530006, China

    Hongxing Wang

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  1. Heng Liu
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Correspondence to Heng Liu.

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Liu, H., Li, S., Li, G. et al. Adaptive Controller Design for a Class of Uncertain Fractional-Order Nonlinear Systems: An Adaptive Fuzzy Approach. Int. J. Fuzzy Syst. 20, 366–379 (2018). https://doi.org/10.1007/s40815-017-0371-5

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