Finite-time boundedness of delayed Takagi-Sugeno quasi-one-sided Lipschitz systems

Omar Kahouli; Sulaiman Almohaimeed; Hamdi Gassara; Lilia El Amraoui; Mohamed Ayari; Omar Kahouli; Sulaiman Almohaimeed; Hamdi Gassara; Lilia El Amraoui; Mohamed Ayari

doi:10.3934/math.2026901

AIMS Mathematics

2026, Volume 11, Issue 7: 22258-22276. doi: 10.3934/math.2026901

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Research article

Finite-time boundedness of delayed Takagi-Sugeno quasi-one-sided Lipschitz systems

1.
Department of Electronics Engineering, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Electrical Engineering, College of Engineering, Qassim University, 52571, Saudi Arabia
3.
Laboratory of Sciences and Techniques of Automatic Control and Computer Engineering, National School of Engineering of Sfax, University of Sfax, PB 1173, 3038 Sfax, Tunisia
4.
Department of Electrical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
5.
Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Arar 91431, Saudi Arabia

Received: 06 April 2026 Revised: 14 June 2026 Accepted: 23 June 2026 Published: 24 July 2026
MSC : 93C42, 93C10, 93D05, 93B51, 34K35

This paper investigates the finite-time boundedness (FBss) of perturbed nonlinear systems with interval time-varying delay. First, the system's nonlinear dynamics are represented by a Takagi–Sugeno (TS) fuzzy model, while quasi-one-sided Lipschitz (QL) terms account for either intrinsic nonlinearities or lumped uncertainties. This formulation captures a broad class of systems, encompassing both natural system nonlinearities and aggregated perturbations. Second, to ensure the FBss of the proposed system, an observer-based controller (OC) is employed under the assumption that the premise variables are measurable, even if the QL condition itself involves unmeasurable states. Then, a single-step procedure based on linear matrix inequalities (LMIs) is proposed to design the OC gains. To this end, a Lyapunov–Krasovskii (LK) functional is employed, comprising a delay-free quadratic term, which incorporates the matrix $ \mathscr{P} $ used to verify the QL condition, along with three integral terms that capture the effects of the interval time-varying delay, including its lower and upper bounds. In addition, we employ a suitable decoupling approach that allows $ \mathscr{P} $ to be regarded as a full decision-variable matrix and enabling the LMI conditions to be derived in a single step. The effectiveness and feasibility of the proposed methods are finally verified through a simulation study.
- time delay,
- quasi-one-sided Lipschitz nonlinearity,
- finite-time boundedness,
- observer-based controller
Citation: Omar Kahouli, Sulaiman Almohaimeed, Hamdi Gassara, Lilia El Amraoui, Mohamed Ayari. Finite-time boundedness of delayed Takagi-Sugeno quasi-one-sided Lipschitz systems[J]. AIMS Mathematics, 2026, 11(7): 22258-22276. doi: 10.3934/math.2026901

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Abstract

This paper investigates the finite-time boundedness (FBss) of perturbed nonlinear systems with interval time-varying delay. First, the system's nonlinear dynamics are represented by a Takagi–Sugeno (TS) fuzzy model, while quasi-one-sided Lipschitz (QL) terms account for either intrinsic nonlinearities or lumped uncertainties. This formulation captures a broad class of systems, encompassing both natural system nonlinearities and aggregated perturbations. Second, to ensure the FBss of the proposed system, an observer-based controller (OC) is employed under the assumption that the premise variables are measurable, even if the QL condition itself involves unmeasurable states. Then, a single-step procedure based on linear matrix inequalities (LMIs) is proposed to design the OC gains. To this end, a Lyapunov–Krasovskii (LK) functional is employed, comprising a delay-free quadratic term, which incorporates the matrix $ \mathscr{P} $ used to verify the QL condition, along with three integral terms that capture the effects of the interval time-varying delay, including its lower and upper bounds. In addition, we employ a suitable decoupling approach that allows $ \mathscr{P} $ to be regarded as a full decision-variable matrix and enabling the LMI conditions to be derived in a single step. The effectiveness and feasibility of the proposed methods are finally verified through a simulation study.

References

[1]	Z. Zhang, J. Dong, A novel $H_\infty$ control for T-S fuzzy systems with transform matching membership functions, Fuzzy Set. Syst., 467 (2023), 108582. https://doi.org/10.1016/j.fss.2023.108582 doi: 10.1016/j.fss.2023.108582
[2]	O. Kahouli, T. Maatoug, F. Delmotte, M. A. Hammami, N. B. Ali, M. Alshammari, Analysis and design for a class of fuzzy control nonlinear systems with state observer under disturbances, AIMS Math., 10 (2025), 9595–9613. https://doi.org/10.3934/math.2025442 doi: 10.3934/math.2025442
[3]	A. Alessandri, A. Cioncolini, D. Padovani, Design of observer-based controllers for Lipschitz nonlinear systems by using LMIs, In: 2024 IEEE 63rd Conference on Decision and Control (CDC), MiCo, Milan, Italy, 2024. https://doi.org/10.1109/CDC56724.2024.10886406
[4]	G. Hu, Observers for one-sided Lipschitz nonlinear systems, IMA J. Math. Control Inform., 23 (2006), 395–401. https://doi.org/10.1093/imamci/dni068 doi: 10.1093/imamci/dni068
[5]	S. Ahmad, M. Rehan, On observer-based control of one-sided Lipschitz systems, J. Franklin I., 353 (2016), 903–916. https://doi.org/10.1016/j.jfranklin.2016.01.010 doi: 10.1016/j.jfranklin.2016.01.010
[6]	B. El Haiek, H. El Aiss, A. Hmamed, A. El Hajjaji, E. H. Tissir, New approach to robust observer-based control of one-sided Lipschitz non-linear systems, IET Control Theory A., 13 (2018), 333–342. https://doi.org/10.1049/iet-cta.2018.5389 doi: 10.1049/iet-cta.2018.5389
[7]	Y. Mu, H. Zhang, Y. Yan, Z. Wu, A design framework of nonlinear $ H_\infty $ PD observer for one-sided Lipschitz singular systems with disturbances, IEEE T. Circuits-II, 69 (2022), 3304–3308. https://doi.org/10.1109/TCSII.2022.3166677 doi: 10.1109/TCSII.2022.3166677
[8]	G. D. Hu, A note on observer for one-sided Lipschitz non-linear systems, IMA J. Math. Control Inform., 25 (2008), 297–303. https://doi.org/10.1093/imamci/dnm024 doi: 10.1093/imamci/dnm024
[9]	M. Yadegar, M. Davoodi, Observer-based tracking controller design for quasi-one-sided Lipschitz nonlinear systems, Optim. Control Appl. Met., 39 (2018), 1638–1647. https://doi.org/10.1002/oca.2432 doi: 10.1002/oca.2432
[10]	W. Q. Dong, G. D. Hu, Y. H. Cong, Separation principle for discrete-time quasi-one-sided Lipschitz nonlinear systems, IET Control Theory A., 15 (2021), 136–147. https://doi.org/10.1049/cth2.12043 doi: 10.1049/cth2.12043
[11]	N. N. Krasovskii, On the application of the second method of Lyapunov for equations with time delays, J. Appl. Math. Mech., 20 (1956), 315–327.
[12]	Y. Y. Cao, P. M. Frank, Stability analysis and synthesis of nonlinear time delay systems via linear Takagi-Sugeno fuzzy models, Fuzzy Set. Syst., 124 (2001) 213–229. https://doi.org/10.1016/S0165-0114(00)00120-2 doi: 10.1016/S0165-0114(00)00120-2
[13]	R. Datta, R. Saravanakumar, R. Dey, B. Bhattacharya, Further results on stability analysis of Takagi–Sugeno fuzzy time-delay systems via improved Lyapunov–Krasovskii functional, AIMS Math., 7 (2022), 16464–16481. https://doi.org/10.3934/math.2022901 doi: 10.3934/math.2022901
[14]	Z. Z. Liu, Y. He, L. Jin, W. H. Chen, Stability analysis of delayed Takagi–Sugeno fuzzy systems via a membership-dependent reciprocally convex inequality, J. Franklin I., 361 (2024), 106776. https://doi.org/10.1016/j.jfranklin.2024.106776 doi: 10.1016/j.jfranklin.2024.106776
[15]	S. Idrissi, Y. El Fezazi, N. El Fezazi, E. H. Tissir, Further improvements on robust stability analysis for Takagi-Sugeno fuzzy systems with time-varying delay, Int. J. Autom. Control, 17 (2023), 487–505. https://doi.org/10.1504/IJAAC.2023.133976 doi: 10.1504/IJAAC.2023.133976
[16]	L. Etienne, L. Hetel, D. Efimov, Observer analysis and synthesis for perturbed Lipschitz systems under noisy time-varying measurements, Automatica, 106 (2019), 406–410. https://doi.org/10.1016/j.automatica.2019.04.003 doi: 10.1016/j.automatica.2019.04.003
[17]	C. M. Nguyen, P. N. Pathirana, H. Trinh, Reduced-order observer design for one-sided Lipschitz time-delay systems subject to unknown inputs, IET Control Theory A., 10 (2016), 1097–1105. https://doi.org/10.1049/iet-cta.2015.1173 doi: 10.1049/iet-cta.2015.1173
[18]	W. Dong, Y. Zhao, Y. Cong, Reduced-order observer-based controller design for quasi-one-sided Lipschitz nonlinear systems with time delay, Int. J. Robust Nonlin., 31 (2021), 817–831. https://doi.org/10.1002/rnc.5312 doi: 10.1002/rnc.5312
[19]	A. H. Esmail, I. Ghous, Z. Duan, M. H. Jaffery, S. Li, Observer-based control for time-delayed quasi-one-sided Lipschitz nonlinear systems under input saturation, Fuzzy Set. Syst., 361 (2024), 107326. https://doi.org/10.1016/j.jfranklin.2024.107326 doi: 10.1016/j.jfranklin.2024.107326
[20]	A. Aloui, O. Kahouli, M. Ayari, H. Gassara, L. El Amraoui, Observer-based local stabilization of state-delayed quasi-one-sided Lipschitz systems with actuator saturation, Mathematics, 13 (2025), 22. https://doi.org/10.3390/math13223610 doi: 10.3390/math13223610
[21]	P. Dorato, Short time stability in linear time-varying systems, Polytechnic Institute of Brooklyn, Microwave Research Institute, 1961.
[22]	F. Amato, M. Ariola, C. T. Abdallah, P. Dorato, Finite-time control for uncertain linear systems with disturbance inputs, In: Proceedings of the American Control Conference, 1999, 1776–1780. https://doi.org/10.1109/ACC.1999.786148
[23]	F. Amato, M. Ariola, Finite-time control of discrete-time linear systems, IEEE T. Automat. Contr., 50 (2005), 724–729. https://doi.org/10.1109/TAC.2005.847042 doi: 10.1109/TAC.2005.847042
[24]	F. Amato, G. Carannante, G. De Tommasi, A. Pironti, Input–output finite-time stability of linear systems: Necessary and sufficient conditions, IEEE T. Automat. Contr., 57 (2012), 3076–3081. https://doi.org/10.1109/TAC.2012.2199151 doi: 10.1109/TAC.2012.2199151
[25]	F. Amato, R. Ambrosino, M. Ariola, G. De Tommasi, A. Pironti, On the finite-time boundedness of linear systems, Automatica, 107 (2019), 454–466. https://doi.org/10.1016/j.automatica.2019.06.002 doi: 10.1016/j.automatica.2019.06.002
[26]	N. El Akchioui, N. El Fezazi, A. Frih, M. Taoussi, R. Farkous, E. H. Tissir, Finite time stability of linear time varying delay systems using free matrix based integral inequalities, Results Control Opti., 10 (2023), 100200. https://doi.org/10.1016/j.rico.2023.100200 doi: 10.1016/j.rico.2023.100200
[27]	N. El Fezazi, M. Fahim, S. Idrissi, R. Farkous, T. Alvarez, E. H. Tissir, New delay-dependent finite-time stabilization of Takagi–Sugeno fuzzy approaches for delayed nonlinear systems, Asian J. Control, 27 (2024), 1119–1134. https://doi.org/10.1002/asjc.3509 doi: 10.1002/asjc.3509
[28]	Y. Ruan, T. Huang, K. Zhou, Finite-time control for Takagi–Sugeno fuzzy systems with time-varying delay, Int. J. Control Automat. Syst., 18 (2020), 1353–1366. https://doi.org/10.1007/s12555-019-0069-x doi: 10.1007/s12555-019-0069-x
[29]	H. Gholami, T. Binazadeh, Sliding-mode observer design and finite-time control of one-sided Lipschitz nonlinear systems with time-delay, Soft Comput., 23 (2019), 6429–6440. https://doi.org/10.1007/s00500-018-3297-4 doi: 10.1007/s00500-018-3297-4
[30]	W. Wang, X. Qi, S. Zhong, F. Liu, The finite-time boundedness and control analysis of switched nonlinear time-varying delay systems based on auxiliary matrices, J. Franklin I., 360 (2023), 14290–14308. https://doi.org/10.1016/j.jfranklin.2023.10.025 doi: 10.1016/j.jfranklin.2023.10.025
[31]	J. P. Arango, L. Etienne, E. Duviella, K. Langueh, P. Segovia, V. Puig, An observer-based tracking controller for one sided Lipschitz–quadratically inner bounded perturbed nonlinear systems, Int. J. Syst. Sci., 2026. https://doi.org/10.1080/00207721.2025.2601783
[32]	G. D. Hu, W. Dong, Y. Cong, Separation principle for quasi-one-sided Lipschitz nonlinear systems with time delay, Int. J. Robust Nonlin., 30 (2020), 2430–2442. https://doi.org/10.1002/rnc.4881 doi: 10.1002/rnc.4881
[33]	X. H. Chang, L. Zhang, J. H. Park, Robust static output feedback $H_\infty$ control for uncertain fuzzy systems, Fuzzy Set. Syst., 273 (2015), 87–104. https://doi.org/10.1016/j.fss.2014.10.023 doi: 10.1016/j.fss.2014.10.023
[34]	M. S. Shanmugam, M. S. Ali, G. Narayanan, M. Rhaima, Finite-time boundedness of switched time-varying delay systems with actuator saturation: Applications in water pollution control, IEEE Access, 12 (2023), 11700–11710. https://doi.org/10.1109/ACCESS.2023.3347613 doi: 10.1109/ACCESS.2023.3347613
[35]	O. Kahouli, L. El Amraoui, M. Ayari, H. Gassara, A. El Hajjaji, Exponential stabilization of quasi-one-sided Lipschitz systems with time delay, AIMS Math., 10 (2025). http://dx.doi.org/10.3934/math.20251173 doi: 10.3934/math.20251173
[36]	B. El Haiek, H. El Aiss, A. Hmamed, A. El Hajjaji, E. H. Tissir, New approach to robust observer-based control of one-sided Lipschitz non-linear systems, IET Control Theory A., 13 (2019), 333–342. https://doi.org/10.1049/iet-cta.2018.5389 doi: 10.1049/iet-cta.2018.5389

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