Robust synchronization of Caputo–Hadamard fractional-order T–S fuzzy Lur'e systems with delay: LMI and SOS conditions

Sultan M. Alzahrani; Sultan M. Alzahrani

doi:10.3934/math.2026871

AIMS Mathematics

2026, Volume 11, Issue 7: 21486-21518. doi: 10.3934/math.2026871

Previous Article Next Article

Research article

Robust synchronization of Caputo–Hadamard fractional-order T–S fuzzy Lur'e systems with delay: LMI and SOS conditions

Sultan M. Alzahrani ^,

Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 20 May 2026 Revised: 25 June 2026 Accepted: 07 July 2026 Published: 20 July 2026
MSC : 34A08, 34D06, 90C22, 90C23, 93C42, 93C43

In this paper, the robust asymptotic synchronization conditions are obtained for uncertain nonlinear Caputo–Hadamard fractional-order Takagi–Sugeno (T–S) fuzzy Lur'e-type systems with time-varying delay. Matched fuzzy-rule weights represent the master and slave plants, and the slave system is controlled by a static output-error parallel distributed compensation controller that can utilize either the current or past output errors. Norm-bounded structures for the uncertain matrices and incremental sector bounds on the local nonlinearities are considered. Delay-free and delayed synchronization criteria are obtained through the use of a common quadratic Lyapunov function, a Caputo–Hadamard quadratic derivative inequality, the Schur complement, and a Caputo–Hadamard Halanay inequality. The constant-matrix results are further extended to polynomial T–S fuzzy Lur'e systems using a global unconstrained sum-of-squares matrix certificate. For a real-life partial-output scenario (with fewer measured outputs than state dimension), the numerical validation first solves the delayed linear matrix inequalities (LMI) as a max-margin semidefinite program using CVXPY, and then describes the polynomial extension via an explicit matrix sum-of-squares (SOS) Gram certificate and an additional simulation example. The main novelty is that, to the best of my knowledge, this is the first study to address simultaneously, within a single convex LMI/SOS framework, the Caputo–Hadamard (logarithmic-time) fractional order, an unbounded time-varying delay, matched norm-bounded parametric uncertainty, incremental sector nonlinearities, and static partial output-error feedback; unlike integer-order or Riemann–Liouville formulations, this setting cannot rely on the classical chain rule and instead requires a Caputo–Hadamard quadratic derivative inequality together with a logarithmic-time Halanay argument. Throughout, "robust synchronization" is meant in the sense of complete synchronization (asymptotic decay of the error to zero) that is guaranteed for every admissible uncertainty; a numerical study further quantifies how the fractional order $ \alpha $ shapes the synchronization transient.
Citation: Sultan M. Alzahrani. Robust synchronization of Caputo–Hadamard fractional-order T–S fuzzy Lur'e systems with delay: LMI and SOS conditions[J]. AIMS Mathematics, 2026, 11(7): 21486-21518. doi: 10.3934/math.2026871

Related Papers:

Abstract

In this paper, the robust asymptotic synchronization conditions are obtained for uncertain nonlinear Caputo–Hadamard fractional-order Takagi–Sugeno (T–S) fuzzy Lur'e-type systems with time-varying delay. Matched fuzzy-rule weights represent the master and slave plants, and the slave system is controlled by a static output-error parallel distributed compensation controller that can utilize either the current or past output errors. Norm-bounded structures for the uncertain matrices and incremental sector bounds on the local nonlinearities are considered. Delay-free and delayed synchronization criteria are obtained through the use of a common quadratic Lyapunov function, a Caputo–Hadamard quadratic derivative inequality, the Schur complement, and a Caputo–Hadamard Halanay inequality. The constant-matrix results are further extended to polynomial T–S fuzzy Lur'e systems using a global unconstrained sum-of-squares matrix certificate. For a real-life partial-output scenario (with fewer measured outputs than state dimension), the numerical validation first solves the delayed linear matrix inequalities (LMI) as a max-margin semidefinite program using CVXPY, and then describes the polynomial extension via an explicit matrix sum-of-squares (SOS) Gram certificate and an additional simulation example. The main novelty is that, to the best of my knowledge, this is the first study to address simultaneously, within a single convex LMI/SOS framework, the Caputo–Hadamard (logarithmic-time) fractional order, an unbounded time-varying delay, matched norm-bounded parametric uncertainty, incremental sector nonlinearities, and static partial output-error feedback; unlike integer-order or Riemann–Liouville formulations, this setting cannot rely on the classical chain rule and instead requires a Caputo–Hadamard quadratic derivative inequality together with a logarithmic-time Halanay argument. Throughout, "robust synchronization" is meant in the sense of complete synchronization (asymptotic decay of the error to zero) that is guaranteed for every admissible uncertainty; a numerical study further quantifies how the fractional order $ \alpha $ shapes the synchronization transient.

References

[1]	J. Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, J. Mathématiques Pures Appl., 8 (1892), 101–186.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[3]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[4]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142. https://doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
[5]	B. B. He, H. C. Zhou, C. H. Kou, Stability analysis of Hadamard and Caputo–Hadamard fractional nonlinear systems without and with delay, Fract. Calc. Appl. Anal., 25 (2022), 2420–2445. https://doi.org/10.1007/s13540-022-00106-3 doi: 10.1007/s13540-022-00106-3
[6]	B. B. He, H. C. Zhou, Caputo–Hadamard fractional Halanay inequality, Appl. Math. Lett., 125 (2022), 107723. https://doi.org/10.1016/j.aml.2021.107723 doi: 10.1016/j.aml.2021.107723
[7]	H. Gassara, M. Tlija, L. Mchiri, A. Ben Makhlouf, Finite-time stability analysis for delayed fuzzy Hadamard fractional-order systems, Fractal Fract., 9 (2025), 63. https://doi.org/10.3390/fractalfract9020063 doi: 10.3390/fractalfract9020063
[8]	O. Naifar, A. B. Makhlouf, L. Mchiri, M. Rhaima, Finite time stability for Hadamard fractional-order systems, Ain Shams Eng. J., 16 (2025), 103263. https://doi.org/10.1016/j.asej.2025.103263 doi: 10.1016/j.asej.2025.103263
[9]	L. Ma, W. Zhang, Finite-time stability for Caputo–Hadamard type fractional differential systems without and with proportional delays, Chaos, 36 (2026), 013131. https://doi.org/10.1063/5.0311074 doi: 10.1063/5.0311074
[10]	S. M. A. Abusalim, A. Ben Makhlouf, R. Fakhfakh, Stochastic stability analysis for neutral systems with Hadamard fractional derivatives, Axioms, 15 (2026), 263. https://doi.org/10.3390/axioms15040263 doi: 10.3390/axioms15040263
[11]	R. Fakhfakh, S. M. Boulaaras, G. Alomani, R. Alzahrani, A. Ben Makhlouf, Averaging principle for stochastic delay differential equations with Hadamard fractional integral, J. Nonlinear Math. Phys., 33 (2026), 27. https://doi.org/10.1007/s44198-026-00398-6 doi: 10.1007/s44198-026-00398-6
[12]	K. Abuasbeh, M. Arab, Data-driven variable-order fractional control for grid resilience: A hybrid Caputo–Hadamard framework validated with US power system data, AIMS Math., 11 (2026), 5492–5531. https://doi.org/10.3934/math.2026227 doi: 10.3934/math.2026227
[13]	S. Momani, I. H. Jebril, I. M. Batiha, L. S. Calucag, A. Biswas, Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems, AIMS Math., 11 (2026), 12204–12232. https://doi.org/10.3934/math.2026501 doi: 10.3934/math.2026501
[14]	Y. Alruwaily, R. Alzahrani, F. Alshahrani, B. Meftah, R. Fakhfakh, Parametric inequalities for $s$-convex stochastic processes via Caputo fractional derivatives, Axioms, 15 (2026), 147. https://doi.org/10.3390/axioms15020147 doi: 10.3390/axioms15020147
[15]	M. E. Yalçın, J. A. K. Suykens, J. Vandewalle, Master–slave synchronization of Lur'e systems with time-delay, Int. J. Bifurcation Chaos, 11 (2001), 1707–1722. https://doi.org/10.1142/S021812740100295X doi: 10.1142/S021812740100295X
[16]	J. Cao, H. X. Li, D. W. C. Ho, Synchronization criteria of Lur'e systems with time-delay feedback control, Chaos Soliton Fract., 23 (2005), 1285–1298. https://doi.org/10.1016/j.chaos.2004.06.025 doi: 10.1016/j.chaos.2004.06.025
[17]	Y. Wu, L. Xiong, G. Zhai, T. Wu, Improved synchronization analysis for delayed Lur'e systems using improved technique, Int. J. Control Autom. Syst., 19 (2021), 1480–1490. https://doi.org/10.1007/s12555-020-0111-8 doi: 10.1007/s12555-020-0111-8
[18]	Y. Li, Y. Q. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965–1969. https://doi.org/10.1016/j.automatica.2009.04.003 doi: 10.1016/j.automatica.2009.04.003
[19]	M. S. Bouridah, T. Bouden, M. E. Yalçın, Chaos synchronization of fractional-order Lur'e systems, Int. J. Bifurcation Chaos, 30 (2020), 2050206. https://doi.org/10.1142/S0218127420502065 doi: 10.1142/S0218127420502065
[20]	A. Halanay, Differential equations: Stability, oscillations, time lags, New York: Academic Press, 1966.
[21]	J. Li, H. L. Li, Y. Wang, H. Zhou, J. Cao, Synchronization of Caputo–Hadamard fractional-order fuzzy neural networks with time delays and uncertain parameters, J. Appl. Math. Comput., 72 (2026), 56. https://doi.org/10.1007/s12190-025-02722-7 doi: 10.1007/s12190-025-02722-7
[22]	D. T. Hong, N. T. Thanh, The synchronization criteria for uncertain nonlinear Caputo–Hadamard fractional-order systems with time-delay output feedback control, Rend. Circ. Mat. Palermo, Ⅱ. Ser., 74 (2025), 52. https://doi.org/10.1007/s12215-024-01164-7 doi: 10.1007/s12215-024-01164-7
[23]	Z. Zhang, H. L. Li, P. Liu, Y. Chen, J. Cao, Complete synchronization of Caputo–Hadamard fractional-order fuzzy competitive neural networks with time-varying delays, Nonlinear Dyn., 114 (2026), 650. https://doi.org/10.1007/s11071-026-12530-8 doi: 10.1007/s11071-026-12530-8
[24]	Z. Zhang, H. L. Li, D. D. Li, L. Zhang, J. Cao, Synchronization of Caputo–Hadamard fractional-order competitive neural networks with discrete and distributed delays, Math. Methods Appl. Sci., 2026, 1–19. https://doi.org/10.1002/mma.70745
[25]	B. Zhang, J. Zhuang, H. Liu, J. Cao, Y. Xia, Master–slave synchronization of a class of fractional-order Takagi–Sugeno fuzzy neural networks, Adv. Differ. Equ., 2018 (2018), 473. https://doi.org/10.1186/s13662-018-1918-y doi: 10.1186/s13662-018-1918-y
[26]	J. Ran, Y. Zhou, H. Pu, Global stability and synchronization of stochastic discrete-time variable-order fractional-order delayed quaternion-valued neural networks, Math. Comput. Simul., 226 (2024), 413–437. https://doi.org/10.1016/j.matcom.2024.07.017 doi: 10.1016/j.matcom.2024.07.017
[27]	T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., SMC-15 (1985), 116–132. https://doi.org/10.1109/TSMC.1985.6313399 doi: 10.1109/TSMC.1985.6313399
[28]	K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets Syst., 45 (1992), 135–156. https://doi.org/10.1016/0165-0114(92)90113-I doi: 10.1016/0165-0114(92)90113-I
[29]	H. O. Wang, K. Tanaka, M. F. Griffin, An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Trans. Fuzzy Syst., 4 (1996), 14–23. https://doi.org/10.1109/91.481841 doi: 10.1109/91.481841
[30]	X. J. Ma, Z. Q. Sun, Y. Y. He, Analysis and design of fuzzy controller and fuzzy observer, IEEE Trans. Fuzzy Syst., 6 (1998), 41–51. https://doi.org/10.1109/91.660807 doi: 10.1109/91.660807
[31]	K. Tanaka, T. Ikeda, H. O. Wang, Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs, IEEE Trans. Fuzzy Syst., 6 (1998), 250–265. https://doi.org/10.1109/91.669023 doi: 10.1109/91.669023
[32]	H. J. Lee, J. B. Park, G. Chen, Robust fuzzy control of nonlinear systems with parametric uncertainties, IEEE Trans. Fuzzy Syst., 9 (2001), 369–379. https://doi.org/10.1109/91.919258 doi: 10.1109/91.919258
[33]	K. Tanaka, H. O. Wang, Fuzzy control systems design and analysis: A linear matrix inequality approach, New York: John Wiley & Sons, 2001. https://doi.org/10.1002/0471224596
[34]	G. Feng, A survey on analysis and design of model-based fuzzy control systems, IEEE Trans. Fuzzy Syst., 14 (2006), 676–697. https://doi.org/10.1109/TFUZZ.2006.883415 doi: 10.1109/TFUZZ.2006.883415
[35]	S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, Philadelphia: SIAM, 1994. https://doi.org/10.1137/1.9781611970777
[36]	J. G. VanAntwerp, R. D. Braatz, A tutorial on linear and bilinear matrix inequalities, J. Process Control, 10 (2000), 363–385. https://doi.org/10.1016/S0959-1524(99)00056-6 doi: 10.1016/S0959-1524(99)00056-6
[37]	P. A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph.D. thesis, California Institute of Technology, Pasadena, 2000.
[38]	S. Prajna, A. Papachristodoulou, P. Seiler, P. A. Parrilo, New developments in sum of squares optimization and SOSTOOLS, In: Proceedings of the 2004 American control conference, 6 (2004), 5606–5611. https://doi.org/10.23919/ACC.2004.1384747
[39]	H. Gassara, O. Naifar, M. Chaabane, A. Ben Makhlouf, H. Arfaoui, M. Aldandani, Observer-based control for nonlinear Hadamard fractional-order systems via SOS approach, Asian J. Control, 27 (2025), 912–920. https://doi.org/10.1002/asjc.3497 doi: 10.1002/asjc.3497

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)