Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control
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) me...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/1/15 |
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| author | Houssem Jerbi Obaid Alshammari Sondess Ben Aoun Mourad Kchaou Theodore E. Simos Spyridon D. Mourtas Vasilios N. Katsikis |
| author_facet | Houssem Jerbi Obaid Alshammari Sondess Ben Aoun Mourad Kchaou Theodore E. Simos Spyridon D. Mourtas Vasilios N. Katsikis |
| author_sort | Houssem Jerbi |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T15:02:08Z |
| format | Article |
| id | doaj.art-26dd91a8f0584465af52158160236e2f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T15:02:08Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-26dd91a8f0584465af52158160236e2f2024-01-10T15:03:18ZengMDPI AGMathematics2227-73902023-12-011211510.3390/math12010015Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor ControlHoussem Jerbi0Obaid Alshammari1Sondess Ben Aoun2Mourad Kchaou3Theodore E. Simos4Spyridon D. Mourtas5Vasilios N. Katsikis6Department of Industrial Engineering, College of Engineering, University of Hail, Háil 81481, Saudi ArabiaDepartment of Electrical Engineering, College of Engineering, University of Hail, Háil 81481, Saudi ArabiaDepartment of Computer Engineering, College of Computer Science and Engineering, University of Hail, Háil 81451, Saudi ArabiaDepartment of Electrical Engineering, College of Engineering, University of Hail, Háil 81481, Saudi ArabiaLaboratory of Interdisciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, RussiaDepartment of Economics, Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, 10559 Athens, GreeceDepartment of Economics, Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, 10559 Athens, GreeceThe 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.https://www.mdpi.com/2227-7390/12/1/15zeroing neural networkquaternionalgebraic Riccati equationquadrotor control |
| spellingShingle | Houssem Jerbi Obaid Alshammari Sondess Ben Aoun Mourad Kchaou Theodore E. Simos Spyridon D. Mourtas Vasilios N. Katsikis Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control Mathematics zeroing neural network quaternion algebraic Riccati equation quadrotor control |
| title | Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control |
| title_full | Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control |
| title_fullStr | Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control |
| title_full_unstemmed | Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control |
| title_short | Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control |
| title_sort | hermitian solutions of the quaternion algebraic riccati equations through zeroing neural networks with application to quadrotor control |
| topic | zeroing neural network quaternion algebraic Riccati equation quadrotor control |
| url | https://www.mdpi.com/2227-7390/12/1/15 |
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