Finite-time stabilization of stochastic systems with varying parameters

Wajdi Kallel; Noura Allugmani; Wajdi Kallel; Noura Allugmani

doi:10.3934/math.2023903

AIMS Mathematics

2023, Volume 8, Issue 8: 17687-17701. doi: 10.3934/math.2023903

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Research article

Finite-time stabilization of stochastic systems with varying parameters

Wajdi Kallel ^{1
,
,},
Noura Allugmani ²

1.
Mathematics Department, Faculty of applied sciences, Umm Al-Qura University, Saudi Arabia
2.
Mathematics Department, College of Sciences and Arts at Alkamil, University of Jeddah, Jeddah, Saudi Arabia

Received: 13 March 2023 Revised: 08 May 2023 Accepted: 12 May 2023 Published: 24 May 2023
MSC : 93B52, 93C10, 93D40, 93E15

This research deals with the stabilization of the stochastic nonlinear systems. In order to achieve the asymptotic stability in probability with respect to unknown bounded disturbances, a control Lyapunov function is applied to present a modified Sontag's homogeneous controller. The obtained results reveal that the presented control achieves the desirable robust asymptotic stability in probability. The finite-time stability in probability for stochastic nonlinear systems is also discussed in this manuscript. Simulation examples are provided to demonstrate the effectiveness of the controllers.
- homogeneous system,
- Lyapunov function,
- finite time stability,
- stochastic systems,
- stabilization
Citation: Wajdi Kallel, Noura Allugmani. Finite-time stabilization of stochastic systems with varying parameters[J]. AIMS Mathematics, 2023, 8(8): 17687-17701. doi: 10.3934/math.2023903

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Abstract

This research deals with the stabilization of the stochastic nonlinear systems. In order to achieve the asymptotic stability in probability with respect to unknown bounded disturbances, a control Lyapunov function is applied to present a modified Sontag's homogeneous controller. The obtained results reveal that the presented control achieves the desirable robust asymptotic stability in probability. The finite-time stability in probability for stochastic nonlinear systems is also discussed in this manuscript. Simulation examples are provided to demonstrate the effectiveness of the controllers.

References

[1]	C. Ma, H. Shi, P. Nie, J. Wu, Finite-time stochastic stability analysis of permanent magnet synchronous motors with noise perturbation, Entropy, 24 (2022), 791. https://doi.org/10.3390/e24060791 doi: 10.3390/e24060791
[2]	J. Yin, S. Khoo, Z. Man, Finite-time stability theorems of homogeneous stochastic nonlinear systems, Syst. Control Lett., 100 (2017), 6–13. https://doi.org/10.1016/j.sysconle.2016.11.012 doi: 10.1016/j.sysconle.2016.11.012
[3]	S. P. Bhat, D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751–766. https://doi.org/10.1137/S0363012997321358 doi: 10.1137/S0363012997321358
[4]	E. Moulay, W. Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl., 323 (2006), 1430–1443. https://doi.org/10.1016/j.jmaa.2005.11.046 doi: 10.1016/j.jmaa.2005.11.046
[5]	J. Yin, D. Ding, Z. Liu, S. Khoo, Some properties of finite-time stable stochastic nonlinear systems, Appl. Math. Comput., 259 (2015), 686–697. https://doi.org/10.1016/j.amc.2015.02.088 doi: 10.1016/j.amc.2015.02.088
[6]	X. Chen, F. Zhao, Y. Liu, H. Liu, T. Huang, J. Qiu, Reduced-order observer-based preassigned finite-time control of nonlinear systems and its applications, IEEE Trans. Syst. Man Cy. Syst., in press. https://doi.org/10.1109/TSMC.2023.3241365
[7]	S. Wang, J. Mei, D. Xia, Z. Yang, J. Hu, Finite-time optimal feedback control mechanism for knowledge transmission in complex networks via model predictive control, Chaos Soliton. Fract., 164 (2022), 112724. https://doi.org/10.1016/j.chaos.2022.112724 doi: 10.1016/j.chaos.2022.112724
[8]	E. Moulay, Stability and stabilization of homogeneous systems depending on a parameter, IEEE Trans. Automat. Contr., 54 (2009), 1382–1385. https://doi.org/10.1109/TAC.2009.2015560 doi: 10.1109/TAC.2009.2015560
[9]	W. Zhang, H. Su, X. Cai, H. Guo, A control Lyapunov function approach to stabilization of affine nonlinear systems with bounded uncertain parameters, Circuits Syst. Signal Process., 34 (2015), 341–352. https://doi.org/10.1007/s00034-014-9848-8 doi: 10.1007/s00034-014-9848-8
[10]	P. Florchinger, On the stabilization of homogeneous control stochastic systems, Proceedings of 32nd IEEE Conference on Decision and Control, San Antonio, TX, USA, 1993,855–856. https://doi.org/10.1109/CDC.1993.325028
[11]	C. Boulanger, Stabilization of nonlinear stochastic systems using control Lyapunov function, Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA, 1997,557–558. https://doi.org/10.1109/CDC.1997.650688
[12]	L. P. Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247–320. https://doi.org/10.1007/BF02392419 doi: 10.1007/BF02392419
[13]	E. Moulay, Stabilization via homogeneous feedback controls, Automatica, 44 (2008), 2981–2984. https://doi.org/10.1016/j.automatica.2008.05.003 doi: 10.1016/j.automatica.2008.05.003
[14]	L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett., 19 (1992), 467–473. https://doi.org/10.1016/0167-6911(92)90078-7 doi: 10.1016/0167-6911(92)90078-7
[15]	K. Hoshino, Y. Nishimura, Y. Yamashita, Convergence rates of stochastic homogeneous systems, Syst. Control Lett., 124 (2019), 33–39. https://doi.org/10.1016/j.sysconle.2018.11.013 doi: 10.1016/j.sysconle.2018.11.013
[16]	R. Khasminskii, Stochastic stability of differential equations, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-23280-0
[17]	J. Yin, S. Khoo, Z. Man, X. Yu, Finite-time stability and instability of stochastic nonlinear systems, Automatica, 47 (2011), 2671–2677. https://doi.org/10.1016/j.automatica.2011.08.050 doi: 10.1016/j.automatica.2011.08.050
[18]	Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163–1173. https://doi.org/10.1016/0362-546X(83)90049-4 doi: 10.1016/0362-546X(83)90049-4
[19]	J. Huang, L. Yu, S. Xia, Stabilization and finite time stabilization of nonlinear differential inclusions based on control Lyapunov function, Circuits Syst. Signal Process., 33 (2014), 2319–2331. https://doi.org/10.1007/s00034-014-9741-5 doi: 10.1007/s00034-014-9741-5
[20]	M. Krstić, P. V. Kokotović, Control Lyapunov functions for adaptive nonlinear stabilization, Syst. Control Lett., 26 (1995), 17–23. https://doi.org/10.1016/0167-6911(94)00107-7 doi: 10.1016/0167-6911(94)00107-7
[21]	P. Florchinger, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), 1430–1443. https://doi.org/10.1137/S0363012993252309 doi: 10.1137/S0363012993252309
[22]	X. Chen, H. Liu, G. Wen, Y. Liu, J. Cao, J. Qiu, Adaptive neural preassigned-time control for macro-micro composite positioning stage with displacement constraints, IEEE Trans. Ind. Inform., in press. https://doi.org/10.1109/TII.2023.3254602
[23]	C. Wang, X. Chen, J. Cao, J. Qiu, Y. Liu, Y. Luo, Neural network-based distributed adaptive pre-assigned finite-time consensus of multiple TCP/AQM networks, IEEE Trans. Circuits Syst. I, 68 (2021), 387–395. https://doi.org/10.1109/TCSI.2020.3031663 doi: 10.1109/TCSI.2020.3031663
[24]	A. Baklouti, L. Mifdal, S. Dellagi, A. Chelbi, An optimal preventive maintenance policy for a solar photovoltaic system, Sustainability, 12 (2020), 4266. https://doi.org/10.3390/su12104266 doi: 10.3390/su12104266
[25]	A. Baklouti, L. Mifdal, S. Dellagi, A. Chelbi, Selling or leasing used vehicles considering their energetic type, the potential demand for leasing, and the expected maintenance costs, Energy Rep., 8 (2022), 1125–1135. https://doi.org/10.1016/j.egyr.2022.07.074 doi: 10.1016/j.egyr.2022.07.074
[26]	A. Baklouti, Quadratic Hom-Lie triple systems, J. Geom. Phys., 121 (2017), 166–175. https://doi.org/10.1016/j.geomphys.2017.06.013 doi: 10.1016/j.geomphys.2017.06.013
[27]	A. Baklouti, S. Hidri, Tools to specify semi-simple Jordan triple systems, Differ. Geom. Appl., 83 (2022), 101900. https://doi.org/10.1016/j.difgeo.2022.101900 doi: 10.1016/j.difgeo.2022.101900
[28]	A. Baklouti, M. Mabrouk, Essential numerical ranges of operators in semi-Hilbertian spaces, Ann. Funct. Anal., 13 (2022), 16. https://doi.org/10.1007/s43034-021-00161-6 doi: 10.1007/s43034-021-00161-6

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