| Title:
|
$\beta $-exponential stability of time-delay systems based on sliding mode control (English) |
| Author:
|
Athmouni, Nassim |
| Author:
|
Brahmia, Nejib |
| Author:
|
Fajraoui, Tarek |
| Author:
|
Mabrouk, Fehmi |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2026 |
| Pages:
|
99-114 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper investigates sliding mode control for one-sided Lipschitz non-linear systems with time-delays and uncertainties. A suitable integral sliding surface is introduced, explicitly accounting for delay terms and distinguishing itself from existing approaches. To guarantee the $\beta$-exponential stability, a new sufficient condition is derived in the form of a linear matrix inequality. Furthermore, an appropriate sliding mode control law is developed to enforce finite-time convergence of the system states to the sliding surface and guarantee their persistence on it. (English) |
| Keyword:
|
time-delay systems |
| Keyword:
|
$\beta $-exponentially stable |
| Keyword:
|
one-sided Lipschitz |
| Keyword:
|
sliding mode control |
| Keyword:
|
linear matrix inequality |
| MSC:
|
93D05 |
| MSC:
|
93D15 |
| MSC:
|
93D23 |
| DOI:
|
10.14736/kyb-2026-1-0099 |
| . |
| Date available:
|
2026-03-03T22:30:43Z |
| Last updated:
|
2026-03-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153537 |
| . |
| Reference:
|
[1] Abbaszadeh, M., Marquez, H. J.: Nonlinear observer design for one-sided Lipschitz systems..In: Proc. 2010 American Control Conference, IEEE 2010, pp. 5284-5289. |
| Reference:
|
[2] Echi, N.: Observer design and practical stability of nonlinear systems under unknown time-delay..Asian J. Control 23 (2021), 685-696. |
| Reference:
|
[3] Echi, N., Mabrouk, F.: Observer based control for practical stabilization of one-sided Lipschitz nonlinear systems..Rocky Mt. J. Math. 54 (2024), 109-120. |
| Reference:
|
[4] Ekramian, M.: Static output feedback problem for Lipschitz nonlinear systems..J. Franklin Inst. 357 (2020), 1457-1472. |
| Reference:
|
[5] Ghanes, M., Leon, J. De, Barbot, J.: Observer design for nonlinear systems under unknown time-varying delays..IEEE Trans. Automat. Control 58 (2013), 1529-1534. |
| Reference:
|
[6] Gu, K., Kharitonov, V. L., Chen, J.: Stability of Time-Delay Systems..Birkhäuser, 2003. |
| Reference:
|
[7] Hale, J. K., Lunel, S. M. V.: Introduction to Functional Differential Equations..Springer, New York 1993. Zbl 0787.34002 |
| Reference:
|
[8] Huang, L., Lin, X., Zhong, B., Xu, D.: Robust control for one-sided Lipschitz non-linear systems with time-varying delays and uncertainties..IET Control Theory Appl. 14 (2020), 2116-2126. |
| Reference:
|
[9] Jia, X., Chen, X., Xu, S., Zhang, B., Zhang, Z.: Adaptive output feedback control of nonlinear time-delay systems with application to chemical reactor systems..IEEE Trans. Ind. Electron. 64 (2017), 4792-4799. |
| Reference:
|
[10] Kang, W.: Approximate linearization of nonlinear control systems..Systems Control Lett. 23 (1994), 43-52. |
| Reference:
|
[11] Khalil, H. K.: Nonlinear Syst..3rd ed. Prentice-Hall, Upper Saddle River, NJ, USA 2001. |
| Reference:
|
[12] Kang, Z., Lin, X., Shen, X., Liu, Z., Gao, Y., Liu, J.: Event-Triggered Generalized Super-Twisting Sliding Mode Control for Position Tracking of PMSMs..IEEE Trans. Ind. Informat. (2025). |
| Reference:
|
[13] Lili, C., Ying, Z., Xian, Z.: Guaranteed cost control for uncertain genetic regulatory networks with interval time-varying delays..Neurocomputing 131 (2014), 105-112. |
| Reference:
|
[14] al., Z. Liu et: Fixed-time sliding mode control for DC/DC buck converters with mismatched uncertainties..IEEE Trans. Circuits Syst. I 70 (2023), 472-480. |
| Reference:
|
[15] Lin, X., Wu, C., Yao, W., Liu, Z., Shen, X., Xu, R., Sun, G., Liu, J.: Observer-based fixed-time control for permanent-magnet synchronous motors with parameter uncertainties..IEEE Trans. Power Electron. 38 (2023), 4335-4344. |
| Reference:
|
[16] Lin, X., Xu, R., Yao, W., Gao, Y., Sun, G., Liu, J., Peretti, L., Wu, L.: Observer-based prescribed performance speed control for PMSMs: A data-driven RBF neural network approach..IEEE Trans. Ind. Informat. 20 (2024), 7502-7512. |
| Reference:
|
[17] Lin, X., Liu, J., Liu, Z., Gao, Y., Peretti, L., Wu, L.: Model-Free Current Predictive Control for PMSMs With Ultra-Local Model Employing Fixed-Time Observer and Extremum-Seeking Method..IEEE Trans. Power Electron. 40 (2025), 10682-10693. |
| Reference:
|
[18] Mathiyalagan, K., Ragul, R.: Observer-based finite-time dissipativity for parabolic systems with time-varying delays..Appl. Math. Comput. 413 (2022), 126605. |
| Reference:
|
[19] Muroya, Y., Kuniya, T., Wang, J. L.: Stability analysis of a delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure..J. Math. Anal. Appl. 425 (2015), 415-439. |
| Reference:
|
[20] Niu, Y., Ho, W. C., Lam, J.: Robust integral sliding mode control for uncertain stochastic systems with time-varying delay..Automatica 41 (2005), 873-880. |
| Reference:
|
[21] Onyeka, A. E., Yan, X.-G., Mu, J.: Sliding mode control of time-delay systems with delayed nonlinear uncertainties..IFAC-PapersOnLine 50 (2017), 2696-2701. |
| Reference:
|
[22] Saad, W., Sellami, A., Garcia, G.: Robust integral sliding mode-$H_\infty$ control of one-sided Lipschitz non-linear systems..IET Control Theory Appl. 2 (2018), 2357-2367. |
| Reference:
|
[23] Shen, X., Liu, J., Liu, Z., Gao, Y., Vazquez, J. I. L. S., Wu, L., Franquelo, L. G.: Sliding mode control of neutral-point-clamped power converters with gain adaptation..IEEE Trans. Power Electron. 1 (2024), 1-23. |
| Reference:
|
[24] Sipahi, R., Niculescu, S.-I., Abdallah, C. T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay..IEEE Control Syst. 31 (2011), 38-65. |
| Reference:
|
[25] Utkin, V.: Sliding Modes in Control and Optimization..Springer-Verlag, Berlin 1992. Zbl 0748.93044 |
| Reference:
|
[26] Wang, Y., Xie, L., Souza, C. E. D.: Robust control of a class of uncertain nonlinear system..Syst. Control Lett. 19 (1992), 139-149. |
| Reference:
|
[27] Zhang, Z., Guo, Y., Zhu, S., Liu, J., Gong, D.: Adaptive integral sliding-mode finite-time control with integrated extended state observer for uncertain nonlinear systems..Information Sciences 667 (2024), 120456. |
| . |
