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

Houssem Jerbi; Obaid Alshammari; Sondess Ben Aoun; Mourad Kchaou; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis

doi:10.3390/math12010015

journal article Open Access Dec 20, 2023

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

Houssem Jerbi

Obaid Alshammari

Sondess Ben Aoun Mourad Kchaou

Theodore E. Simos Spyridon D. Mourtas

Vasilios N. Katsikis

Mathematics Vol. 12 No. 1 pp. 15 · MDPI AG

View at Publisher Save 10.3390/math12010015

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.

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References

65

[1]

Kalman "Contributions to the theory of optimal control" Bol. Soc. Mat. Mex. (1960)

[2]

Qin "Robust H∞ control of doubly fed wind generator via state-dependent Riccati equation technique" IEEE Trans. Power Syst. (2019) 10.1109/tpwrs.2018.2881687

[3]

Time-Varying Formation Tracking for Linear Multiagent Systems With Multiple Leaders

Xiwang Dong, Guoqiang Hu

IEEE Transactions on Automatic Control 2017 10.1109/tac.2017.2673411

[4]

Nonlinear Optimal Control for the Wheeled Inverted Pendulum System

G. Rigatos, K. Busawon, J. Pomares et al.

Robotica 2020 10.1017/s0263574719000456

[5]

Laub "A Schur method for solving algebraic Riccati equations" IEEE Trans. Autom. Control. (1979) 10.1109/tac.1979.1102178

[6]

Aguilar "Nonlinear H∞-control of nonsmooth time-varying systems with application to friction mechanical manipulators" Automatica (2003) 10.1016/s0005-1098(03)00148-1

[7]

Mohammadi "Convergence and sample complexity of gradient methods for the model-free linear-quadratic regulator problem" IEEE Trans. Autom. Control. (2022) 10.1109/tac.2021.3087455

[8]

Hambly "Policy gradient methods for the noisy linear quadratic regulator over a finite horizon" SIAM J. Control Optim. (2021) 10.1137/20m1382386

[9]

Kalman filtering techniques for the online model parameters and state of charge estimation of the Li-ion batteries: A comparative analysis

M. Hossain, M.E. Haque, M.T. Arif

Journal of Energy Storage 2022 10.1016/j.est.2022.104174

[10]

Hamidi, F., Jerbi, H., Alharbi, H., Leiva, V., Popescu, D., and Rajhi, W. (2023). Metaheuristic solution for stability analysis of nonlinear systems using an intelligent algorithm with potential applications. Fractal Fract., 7. 10.3390/fractalfract7010078

[11]

Kchaou, M., Regaieg, M.A., Jerbi, H., Abbassi, R., Stefanoiu, D., and Popescu, D. (2023). Admissible control for non-linear singular systems subject to time-varying delay and actuator saturation: An interval type-2 fuzzy approach. Actuators, 12. 10.3390/act12010030

[12]

Oshman "Eigenfactor solution of the matrix Riccati equation—A continuous square root algorithm" IEEE Trans. Autom. Control. (1985) 10.1109/tac.1985.1103823

[13]

Dooren "A generalized eigenvalue approach for solving Riccati equations" SIAM J. Sci. Stat. Comput. (1981) 10.1137/0902010

[14]

Zhang "Quaternions and matrices of quaternions" Linear Algebra Its Appl. (1997) 10.1016/0024-3795(95)00543-9

[15]

Rodman, L. (2014). Topics in Quaternion Linear Algebra, Princeton University Press. 10.23943/princeton/9780691161853.001.0001

[16]

Hamilton "On a new species of imaginary quantities, connected with the theory of quaternions" Proc. R. Ir. Acad. (1840)

[17]

Joldeş, M., and Muller, J.M. (2020, January 7–10). Algorithms for manipulating quaternions in floating-point arithmetic. Proceedings of the 2020 IEEE 27th Symposium on Computer Arithmetic (ARITH), Portland, OR, USA. 10.1109/arith48897.2020.00016

[18]

"Generalized commutative quaternions of the Fibonacci type" Bol. Soc. Mat. Mex. (2022) 10.1007/s40590-021-00386-4

[19]

Pavllo "Modeling human motion with quaternion-based neural networks" Int. J. Comput. Vis. (2020) 10.1007/s11263-019-01245-6

[20]

Giardino "Quaternionic quantum mechanics in real Hilbert space" J. Geom. Phys. (2020) 10.1016/j.geomphys.2020.103956

[21]

Goodyear, A.M.S., Singla, P., and Spencer, D.B. (2019, January 13–17). Analytical state transition matrix for dual-quaternions for spacecraft pose estimation. Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Maui, HI, USA.

[22]

Kansu "Quaternionic representation of electromagnetism for material media" Int. J. Geom. Methods Mod. Phys. (2019) 10.1142/s0219887819501056

[23]

Mezouar "Kinematic modeling and control of a robot arm using unit dual quaternions" Robot. Auton. Syst. (2016) 10.1016/j.robot.2015.12.005

[24]

Xiao, L., Zhang, Y., Huang, W., Jia, L., and Gao, X. (2022). A dynamic parameter noise-tolerant zeroing neural network for time-varying quaternion matrix equation with applications. IEEE Trans. Neural Netw. Learn. Syst., 1–10. 10.1109/tnnls.2022.3225309

[25]

Kovalnogov "Computing quaternion matrix pseudoinverse with zeroing neural networks" AIMS Math. (2023) 10.3934/math.20231164

[26]

Xiao, L., Cao, P., Song, W., Luo, L., and Tang, W. (2023). A fixed-time noise-tolerance ZNN model for time-variant inequality-constrained quaternion matrix least-squares problem. IEEE Trans. Neural Netw. Learn. Syst., 1–10. 10.1109/tnnls.2023.3242313

[27]

Xiao "Zeroing neural networks for dynamic quaternion-valued matrix inversion" IEEE Trans. Ind. Inform. (2022) 10.1109/tii.2021.3090063

[28]

Aoun "A quaternion Sylvester equation solver through noise-resilient zeroing neural networks with application to control the SFM chaotic system" AIMS Math. (2023) 10.3934/math.20231401

[29]

Tan "New varying-parameter recursive neural networks for model-free kinematic control of redundant manipulators with limited measurements" IEEE Trans. Instrum. Meas. (2022) 10.1109/tim.2022.3216598

[30]

Abbassi, R., Jerbi, H., Kchaou, M., Simos, T.E., Mourtas, S.D., and Katsikis, V.N. (2023). Towards higher-order zeroing neural networks for calculating quaternion matrix inverse with application to robotic motion tracking. Mathematics, 11. 10.3390/math11122756

[31]

Kovalnogov "Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images" AIMS Math. (2023) 10.3934/math.2023733

[32]

Kovalnogov "A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking" AIMS Math. (2023) 10.3934/math.20231323

[33]

Zhang "Design and analysis of a general recurrent neural network model for time-varying matrix inversion" IEEE Trans. Neural Netw. (2005) 10.1109/tnn.2005.857946

[34]

Chai "A neural network for Moore-Penrose inverse of time-varying complex-valued matrices" Int. J. Comput. Intell. Syst. (2020) 10.2991/ijcis.d.200527.001

[35]

Wu "Improved recurrent neural networks for solving Moore-Penrose inverse of real-time full-rank matrix" Neurocomputing (2020) 10.1016/j.neucom.2020.08.026

[36]

Jiang, W., Lin, C.L., Katsikis, V.N., Mourtas, S.D., Stanimirović, P.S., and Simos, T.E. (2022). Zeroing neural network approaches based on direct and indirect methods for solving the Yang–Baxter-like matrix equation. Mathematics, 10. 10.3390/math10111950

[37]

Jerbi, H., Alharbi, H., Omri, M., Ladhar, L., Simos, T.E., Mourtas, S.D., and Katsikis, V.N. (2022). Towards higher-order zeroing neural network dynamics for solving time-varying algebraic Riccati equations. Mathematics, 10. 10.3390/math10234490

[38]

Katsikis "Zeroing neural network with fuzzy parameter for computing pseudoinverse of arbitrary matrix" IEEE Trans. Fuzzy Syst. (2022) 10.1109/tfuzz.2021.3115969

[39]

Alharbi, H., Jerbi, H., Kchaou, M., Abbassi, R., Simos, T.E., Mourtas, S.D., and Katsikis, V.N. (2023). Time-varying pseudoinversion based on full-rank decomposition and zeroing neural networks. Mathematics, 11. 10.3390/math11030600

[40]

Mourtas "Exploiting the Black-Litterman framework through error-correction neural networks" Neurocomputing (2022) 10.1016/j.neucom.2022.05.036

[41]

Kovalnogov, V.N., Fedorov, R.V., Generalov, D.A., Chukalin, A.V., Katsikis, V.N., Mourtas, S.D., and Simos, T.E. (2022). Portfolio insurance through error-correction neural networks. Mathematics, 10. 10.3390/math10183335

[42]

Mourtas, S.D., and Kasimis, C. (2022). Exploiting mean-variance portfolio optimization problems through zeroing neural networks. Mathematics, 10. 10.3390/math10173079

[43]

Qiao "Computing time-varying ML-weighted pseudoinverse by the Zhang neural networks" Numer. Funct. Anal. Optim. (2020) 10.1080/01630563.2020.1740887

[44]

Wang "Complex ZFs for computing time-varying complex outer inverses" Neurocomputing (2018) 10.1016/j.neucom.2017.09.034

[45]

Dai "A fuzzy adaptive zeroing neural network with superior finite-time convergence for solving time-variant linear matrix equations" Knowl.-Based Syst. (2022) 10.1016/j.knosys.2022.108405

[46]

Trenkler "Quaternions: Further contributions to a matrix oriented approach" Linear Algebra Its Appl. (2001) 10.1016/s0024-3795(00)00283-4

[47]

Gupta, A.K. (2014). Numerical Methods Using MATLAB, Apress. 10.1007/978-1-4842-0154-1

[48]

Simos "Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems" Math. Comput. Simul. (2022) 10.1016/j.matcom.2022.05.033

[49]

Carino, J., Abaunza, H., and Castillo, P. (2015, January 9–12). Quadrotor quaternion control. Proceedings of the 2015 International Conference on Unmanned Aircraft Systems (ICUAS), Denver, CO, USA. 10.1109/icuas.2015.7152367

[50]

Fresk, E., and Nikolakopoulos, G. (2015, January 8–11). Experimental evaluation of a full quaternion based attitude quadrotor controller. Proceedings of the 2015 IEEE 20th Conference on Emerging Technologies & Factory Automation (ETFA), Luxembourg. 10.1109/etfa.2015.7301555

Showing 50 of 65 references

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References

Details

Published: Dec 20, 2023
Vol/Issue: 12(1)
Pages: 15
License: View

Authors

H

Houssem Jerbi

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

O

Obaid Alshammari

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

S

Sondess Ben Aoun

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

M

Mourad Kchaou

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

T

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; Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

S

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

V

Vasilios N. Katsikis

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

Funding

Research Deanship of Ha’il University, Kingdom of Saudi Arabia Award: RG-23 072

Cite This Article

Houssem Jerbi, Obaid Alshammari, Sondess Ben Aoun, et al. (2023). Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control. Mathematics, 12(1), 15. https://doi.org/10.3390/math12010015

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

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