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Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control

Author

Listed:
  • Houssem Jerbi

    (Department of Industrial Engineering, College of Engineering, University of Hail, Háil 81481, Saudi Arabia)

  • Obaid Alshammari

    (Department of Electrical Engineering, College of Engineering, University of Hail, Háil 81481, Saudi Arabia)

  • Sondess Ben Aoun

    (Department of Computer Engineering, College of Computer Science and Engineering, University of Hail, Háil 81451, Saudi Arabia)

  • Mourad Kchaou

    (Department of Electrical Engineering, College of Engineering, University of Hail, Háil 81481, Saudi Arabia)

  • Theodore E. Simos

    (Laboratory of Interdisciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
    Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref 32093, Kuwait
    Data Recovery Key Laboratory of Sichun Province, Neijing Normal University, Neijiang 641100, China)

  • Spyridon D. Mourtas

    (Department of Economics, Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, 10559 Athens, Greece
    Laboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, 660041 Krasnoyarsk, Russia)

  • Vasilios N. Katsikis

    (Department of Economics, Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, 10559 Athens, Greece)

Abstract

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

Suggested Citation

  • Houssem Jerbi & Obaid Alshammari & Sondess Ben Aoun & Mourad Kchaou & Theodore E. Simos & Spyridon D. Mourtas & Vasilios N. Katsikis, 2023. "Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control," Mathematics, MDPI, vol. 12(1), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:15-:d:1304270
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    References listed on IDEAS

    as
    1. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Andrey V. Chukalin & Vasilios N. Katsikis & Spyridon D. Mourtas & Theodore E. Simos, 2022. "Portfolio Insurance through Error-Correction Neural Networks," Mathematics, MDPI, vol. 10(18), pages 1-14, September.
    2. Ben Hambly & Renyuan Xu & Huining Yang, 2020. "Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon," Papers 2011.10300, arXiv.org, revised Jun 2021.
    3. Spyridon D. Mourtas & Chrysostomos Kasimis, 2022. "Exploiting Mean-Variance Portfolio Optimization Problems through Zeroing Neural Networks," Mathematics, MDPI, vol. 10(17), pages 1-20, August.
    4. Rabeh Abbassi & Houssem Jerbi & Mourad Kchaou & Theodore E. Simos & Spyridon D. Mourtas & Vasilios N. Katsikis, 2023. "Towards Higher-Order Zeroing Neural Networks for Calculating Quaternion Matrix Inverse with Application to Robotic Motion Tracking," Mathematics, MDPI, vol. 11(12), pages 1-21, June.
    5. Hadeel Alharbi & Houssem Jerbi & Mourad Kchaou & Rabeh Abbassi & Theodore E. Simos & Spyridon D. Mourtas & Vasilios N. Katsikis, 2023. "Time-Varying Pseudoinversion Based on Full-Rank Decomposition and Zeroing Neural Networks," Mathematics, MDPI, vol. 11(3), pages 1-14, January.
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