Fuzzy Gain-Composite approach for robust control of chaotic fractional-order optical systems with guaranteed performance

Sepideh Nikfahm Khoubravan; Saeed Mirzajani; Aghileh Heydari; Majid Roohi; Sepideh Nikfahm Khoubravan; Saeed Mirzajani; Aghileh Heydari; Majid Roohi

doi:10.3934/math.20251191

AIMS Mathematics

2025, Volume 10, Issue 11: 27103-27128. doi: 10.3934/math.20251191

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Research article

Fuzzy Gain-Composite approach for robust control of chaotic fractional-order optical systems with guaranteed performance

1.
Department of Mathematics, Payame Noor University, Tehran, Iran
2.
Department of Basic sciences, Technical and Vocational University (TVU), Tehran, Iran
3.
Department of Mathematics, Aarhus University, 8000 Aarhus, Denmark

Received: 30 July 2025 Revised: 28 October 2025 Accepted: 04 November 2025 Published: 21 November 2025
MSC : 26A33, 37N35, 93C42

In this work, a Guaranteed-Cost Fuzzy Gain-Composite (GC-FGC) control scheme was introduced to stabilize a specialized class of chaotic fractional-order Takagi–Sugeno (T–S) fuzzy nonlinear optical systems. The controller design was grounded in fractional-order Lyapunov theory and formulated using Linear Matrix Inequality (LMI) conditions, enabling the effective handling of the complex chaotic dynamics inherent in such systems. The proposed method ensures asymptotic stability and further incorporates a dynamics-free control strategy that remains robust in the presence of system uncertainties and input saturation constraints. To decouple the control rules from the specific system dynamics, the design leverages the norm-bounded nature of the chaotic system states. Furthermore, a deep reinforcement learning framework based on the Soft Actor-Critic (SAC) algorithm was employed to fine-tune the internal coefficients of the GC-FGC controller. By optimizing a reward function through the SAC agent's neural network, an optimal policy was derived that guarantees finite-time convergence and satisfied the sliding surface reachability conditions. The effectiveness and applicability of the proposed control framework were verified through comprehensive simulations and two illustrative numerical case studies.
- T-S fuzzy model,
- fractional-order systems,
- Guaranteed-cost fuzzy gain-composite control,
- Lyapunov theory,
- nonlinear optical systems
Citation: Sepideh Nikfahm Khoubravan, Saeed Mirzajani, Aghileh Heydari, Majid Roohi. Fuzzy Gain-Composite approach for robust control of chaotic fractional-order optical systems with guaranteed performance[J]. AIMS Mathematics, 2025, 10(11): 27103-27128. doi: 10.3934/math.20251191

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Abstract

In this work, a Guaranteed-Cost Fuzzy Gain-Composite (GC-FGC) control scheme was introduced to stabilize a specialized class of chaotic fractional-order Takagi–Sugeno (T–S) fuzzy nonlinear optical systems. The controller design was grounded in fractional-order Lyapunov theory and formulated using Linear Matrix Inequality (LMI) conditions, enabling the effective handling of the complex chaotic dynamics inherent in such systems. The proposed method ensures asymptotic stability and further incorporates a dynamics-free control strategy that remains robust in the presence of system uncertainties and input saturation constraints. To decouple the control rules from the specific system dynamics, the design leverages the norm-bounded nature of the chaotic system states. Furthermore, a deep reinforcement learning framework based on the Soft Actor-Critic (SAC) algorithm was employed to fine-tune the internal coefficients of the GC-FGC controller. By optimizing a reward function through the SAC agent's neural network, an optimal policy was derived that guarantees finite-time convergence and satisfied the sliding surface reachability conditions. The effectiveness and applicability of the proposed control framework were verified through comprehensive simulations and two illustrative numerical case studies.

References

[1]	S. Y. Nof, A. M. Weiner, G. J. Cheng, Laser and Photonic Systems: Design and Integration, Taylor & Francis, (2014).
[2]	C. He, Y. Shen, A. Forbes, Towards higher-dimensional structured light, Light Sci. Appl., 11 (2022), 205. https://doi.org/10.1038/s41377-022-00897-3 doi: 10.1038/s41377-022-00897-3
[3]	M. S. Asl, M. Javidi, Numerical evaluation of order six for fractional differential equations: stability and convergency, Bull. Belg. Math. Soc. Simon Stevin, 26 (2019), 203–221.
[4]	A. A. Alikhanov, M. S. Asl, C. Huang, Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation, Fract. Calc. Appl. Anal., 27 (2024), 102–123.
[5]	M. S. Asl, M. Javidi, An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis, J. Comput. Appl. Math., 324 (2017), 101–117.
[6]	M. Roohi, S. Mirzajani, A. R. Haghighi, A. Basse-O'Connor, Robust stabilization of fractional-order hybrid optical system using a single-input TS-fuzzy sliding mode control strategy with input nonlinearities, AIMS Math., 9 (2024), 25879–25907. https://doi.org/10.3934/math.20241264 doi: 10.3934/math.20241264
[7]	R. Shalaby, M. El-Hossainy, B. Abo-Zalam, T. A. Mahmoud, Optimal fractional-order PID controller based on fractional-order actor-critic algorithm, Neural Comput. Appl., 35 (2023), 2347–2380. https://doi.org/10.1007/s00521-022-07710-7 doi: 10.1007/s00521-022-07710-7
[8]	M. Roohi, S. Mirzajani, A. Basse-O'Connor, A no-chatter single-input finite-time PID sliding mode control technique for stabilization of a class of 4D chaotic fractional-order laser systems, Mathematics, 11 (2023), 4463.
[9]	M. T. Vu, S. H. Kim, D. H. Pham, H. L. N. N. Thanh, V. H. Pham, M. Roohi, Adaptive dynamic programming-based intelligent finite-time flexible SMC for stabilizing fractional-order four-wing chaotic systems, Mathematics, 13 (2025), 2078.
[10]	M. Roohi, S. Mirzajani, A. R. Haghighi, A. Basse-O'Connor, Robust design of two-level non-integer SMC based on deep soft actor-critic for synchronization of chaotic fractional-order memristive neural networks, Fractal Fract., 8 (2024), 548.
[11]	C. Chen, H. Wang, J. Li, Observer-based adaptive control for fractional-order strict-feedback nonlinear systems with state quantisation and input quantisation, Int. J. Syst. Sci., (2025), 1–17.
[12]	A. Jajarmi, M. Akbarian, D. Baleanu, Analysis and backstepping control of a novel 4D fractional chaotic oscillator, Math. Methods Appl. Sci., (2025), In press.
[13]	H. Zou, M. Wang, Enhanced sliding-mode control for tracking control of uncertain fractional-order nonlinear systems based on fuzzy logic systems, Appl. Sci., 15 (2025), 4686. https://doi.org/10.3390/app15134686 doi: 10.3390/app15134686
[14]	S. Tong, M. Cui, Fuzzy adaptive predefined-time decentralized fault-tolerant control for fractional-order nonlinear large-scale systems with actuator faults, IEEE Trans. Fuzzy Syst., 32 (2024), 1000–1012. https://doi.org/10.1109/TFUZZ.2023.3317014 doi: 10.1109/TFUZZ.2023.3317014
[15]	A. Sharafian, H.Y. Naeem, I. Ullah, A. Ali, L. Qiu, X. Bai, Resilience to deception attacks in consensus tracking control of incommensurate fractional-order power systems via adaptive RBF neural network, Expert Syst. Appl. 211 (2025), 127763. https://doi.org/10.1016/j.eswa.2025.127763 doi: 10.1016/j.eswa.2025.127763
[16]	S. Mirzajani, S. S. Moafimadani, M. Roohi, A new encryption algorithm utilizing DNA subsequence operations for color images, Appl. Math. 4 (2024), 1382–1403.
[17]	Z. Rasooli Berardehi, C. Zhang, M. Taheri, M. Roohi, M. H. Khooban, Implementation of T-S fuzzy approach for the synchronization and stabilization of non-integer-order complex systems with input saturation at a guaranteed cost, Trans. Inst. Meas. Control, 45 (2023), 2536–2553. https://doi.org/10.1177/01423312231155273 doi: 10.1177/01423312231155273
[18]	X. Fan, Z. Wang, A fuzzy Lyapunov function method to stability analysis of fractional-order T–S fuzzy systems, IEEE Trans. Fuzzy Syst., 30 (2022), 2769–2776. https://doi.org/10.1109/TFUZZ.2021.3078289 doi: 10.1109/TFUZZ.2021.3078289
[19]	Y. Hao, H. Liu, Z. Fang, Observer-based adaptive control for uncertain fractional-order T-S fuzzy systems with output disturbances, Int. J. Fuzzy Syst., 26 (2024), 1783–1801. https://doi.org/10.1007/s40815-024-01703-5 doi: 10.1007/s40815-024-01703-5
[20]	Y. Mu, H. Zhang, Z. Gao, J. Zhang, A fuzzy Lyapunov function approach for fault estimation of T–S fuzzy fractional-order systems based on unknown input observer, IEEE Trans. Syst., Man, Cybern. Syst., 53 (2023), 1246–1255. https://doi.org/10.1109/TSMC.2022.3196502 doi: 10.1109/TSMC.2022.3196502
[21]	Y. Hao, Z. Fang, H. Liu, Stabilization of delayed fractional‐order T‐S fuzzy systems with input saturations and system uncertainties, Asian J. Control, 26 (2024), 246–264.
[22]	X. Fan, Z. Wang, Event‐triggered integral sliding mode control for fractional order T‐S fuzzy systems via a fuzzy error function, Int. J. Robust Nonlinear Control, 31 (2021), 2491–2508.
[23]	P. Mahmoudabadi, M. Tavakoli-Kakhki, Tracking control with disturbance rejection of nonlinear fractional order fuzzy systems: Modified repetitive control approach, Chaos Soliton. Fract., 150 (2021), 111142. https://doi.org/10.1016/j.chaos.2021.111142 doi: 10.1016/j.chaos.2021.111142
[24]	R. Kavikumar, R. Sakthivel, O. M. Kwon, P. Selvaraj, Robust tracking control design for fractional-order interval type-2 fuzzy systems, Nonlinear Dyn., 107 (2022), 3611–3628. https://doi.org/10.1007/s11071-021-07163-y doi: 10.1007/s11071-021-07163-y
[25]	R. Elavarasi, G. Nagamani, Region bounded observer design for T-S fuzzy systems with nonlinear perturbations, Commun. Nonlinear Sci. Numer. Simul., 151 (2025), 108998. https://doi.org/10.1016/j.cnsns.2025.108998 doi: 10.1016/j.cnsns.2025.108998
[26]	Y. Hao, Z. Fang, H. Liu, Adaptive T-S fuzzy synchronization for uncertain fractional-order chaotic systems with input saturation and disturbance, Inf. Sci., 666 (2024), 120423. https://doi.org/10.1016/j.ins.2024.120423 doi: 10.1016/j.ins.2024.120423
[27]	K. Liu, W. Tang, Y. Sheng, Z. Zeng, N. R. Pal, Event-triggered finite-time stabilization of delayed T–S fuzzy systems on time scales, IEEE Trans. Cybern., (2025), 1–10. https://doi.org/10.1109/TCYB.2025.3573795 doi: 10.1109/TCYB.2025.3573795
[28]	J. Sun, Y. Yan, S. Yu, Adaptive fuzzy control for T-S fuzzy fractional order nonautonomous systems based on Q-learning, IEEE Trans. Fuzzy Syst., 32 (2024), 388–397. https://doi.org/10.1109/TFUZZ.2023.3298812 doi: 10.1109/TFUZZ.2023.3298812
[29]	G. Narayanan, S. Lee, S. Ahn, Optimal fractional fuzzy sliding-mode control for fractional-order fuzzy systems based on actor–critic reinforcement learning scheme, J. Franklin Inst., 362 (2025), 107804. https://doi.org/10.1016/j.jfranklin.2025.107804 doi: 10.1016/j.jfranklin.2025.107804
[30]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press (1998), Vol. 198.
[31]	T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072. https://doi.org/10.1016/j.aml.2019.106072 doi: 10.1016/j.aml.2019.106072
[32]	T. Zhang, Y. Li, Exponential Euler scheme of multi-delay Caputo–Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
[33]	T. Zhang, Y. Li, Global exponential stability of discrete-time almost automorphic Caputo–Fabrizio BAM fuzzy neural networks via exponential Euler technique, Knowl.-Based Syst., 246 (2022), 108675. https://doi.org/10.1016/j.knosys.2022.108675 doi: 10.1016/j.knosys.2022.108675
[34]	C. Li, W. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777–784. https://doi.org/10.1016/j.amc.2006.08.163 doi: 10.1016/j.amc.2006.08.163
[35]	Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965–1969.
[36]	S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM, (1994).
[37]	M. Haklidir, H. Temeltaş, Guided soft actor critic: A guided deep reinforcement learning approach for partially observable Markov decision processes, IEEE Access, 9 (2021), 159672–159683.
[38]	A. A. Alikhanov, P. Yadav, V. K. Singh, M. S. Asl, A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation, Comput. Appl. Math., 44 (2025), 115. https://doi.org/10.1007/s40314-024-03077-8 doi: 10.1007/s40314-024-03077-8
[39]	A. A. Alikhanov, M. S. Asl, D. Li, A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application, Comput. Math. Appl., 175 (2024), 330–344. https://doi.org/10.1016/j.camwa.2024.10.016 doi: 10.1016/j.camwa.2024.10.016
[40]	M. S. Abdelouahab, N. E. Hamri, J. Wang, Hopf bifurcation and chaos in fractional-order modified hybrid optical system, Nonlinear Dyn., 69 (2012), 275–284.
[41]	H. Natiq, M. R. M. Said, N. M. Al-Saidi, A. Kilicman, Dynamics and complexity of a new 4D chaotic laser system, Entropy, 21 (2019), 34.
[42]	F. Yang, J. Mou, C. Ma, Y. Cao, Dynamic analysis of an improper fractional-order laser chaotic system and its image encryption application, Opt. Lasers Eng., 129 (2020), 106031. https://doi.org/10.1016/j.optlaseng.2020.106031 doi: 10.1016/j.optlaseng.2020.106031
[43]	H. Liu, S. Li, H. Wang, Y. Sun, Adaptive fuzzy control for a class of unknown fractional-order neural networks subject to input nonlinearities and dead-zones, Inf. Sci., 454–455 (2018), 30–45. https://doi.org/10.1016/j.ins.2018.04.069 doi: 10.1016/j.ins.2018.04.069

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