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Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation

Author

Listed:
  • Bolin Liao

    (College of Computer Science and Engineering, Jishou University, Jishou 416000, China)

  • Cheng Hua

    (College of Computer Science and Engineering, Jishou University, Jishou 416000, China)

  • Xinwei Cao

    (School of Business, Jiangnan University, Wuxi 214122, China)

  • Vasilios N. Katsikis

    (Department of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559 Athens, Greece)

  • Shuai Li

    (School of Engineering, Swansea University, Swansea SA2 8PP, UK)

Abstract

Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order 10 − 5 when solving CTDLE under complex linear noises, which is much lower than order 10 − 1 of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to 2 ∥ A ∥ F / ζ 3 quickly and stably, while the residual error of the NTZNN model is divergent.

Suggested Citation

  • Bolin Liao & Cheng Hua & Xinwei Cao & Vasilios N. Katsikis & Shuai Li, 2022. "Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation," Mathematics, MDPI, vol. 10(15), pages 1-17, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2817-:d:883297
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    References listed on IDEAS

    as
    1. Yongjun He & Bolin Liao & Lin Xiao & Luyang Han & Xiao Xiao, 2021. "Double Accelerated Convergence ZNN with Noise-Suppression for Handling Dynamic Matrix Inversion," Mathematics, MDPI, vol. 10(1), pages 1-21, December.
    2. Wendong Jiang & Chia-Liang Lin & Vasilios N. Katsikis & Spyridon D. Mourtas & Predrag S. Stanimirović & Theodore E. Simos, 2022. "Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation," Mathematics, MDPI, vol. 10(11), pages 1-13, June.
    3. Yi, Chenfu & Zhang, Yunong & Guo, Dongsheng, 2013. "A new type of recurrent neural networks for real-time solution of Lyapunov equation with time-varying coefficient matrices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 92(C), pages 40-52.
    4. Guo, Dongsheng & Zhang, Yunong, 2015. "ZNN for solving online time-varying linear matrix–vector inequality via equality conversion," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 327-338.
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