Caputo Derivative-Based FTC Design and Sensor Fault Estimation for Lipschitz Fractional Nonlinear Systems

Ali Omar M. Alsharif; Ausama H. Ahmed; Bashir Ghariba; Omar Ibrahim Marey; Mahmoud Y. Khamaira; Ahmed. M. A Zargoun

doi:10.65419/albahit.v5i3.152

المؤلفون

Ali Omar M. Alsharif Department of Electrical & Computer Engineering, Faculty of Engineering, Elmergib University, Alkhums, Libya المؤلف
Ausama H. Ahmed Department of Electrical & Computer Engineering, Faculty of Engineering, Elmergib University, Alkhums, Libya المؤلف
Bashir Ghariba Department of Electrical & Computer Engineering, Faculty of Engineering, Elmergib University, Alkhums, Libya المؤلف
Omar Ibrahim Marey Department of Electrical & Computer Engineering, Faculty of Engineering, Elmergib University, Alkhums, Libya المؤلف
Mahmoud Y. Khamaira Department of Electrical & Computer Engineering, Faculty of Engineering, Elmergib University, Alkhums, Libya المؤلف
Ahmed. M. A Zargoun Libyan authority for scientific research, Tripoli , Libya المؤلف

DOI:

https://doi.org/10.65419/albahit.v5i3.152

الكلمات المفتاحية:

مشتق كابوتو الكسري، ، التحكم المتسامح مع الأخطاء، تقدير خطأ الاستشعار، ، المراقب التكيفي، الانظمة الكسرية والغير الخطية، متباينة المصفوفة الخطية

الملخص

لا يزال تقدير خطأ المستشعر والتحكم المتسامح مع الخطأ من المشكلات المهمة لضمان استقرار الأنظمة غير الخطية ذات الترتيب الكسري في وجود أخطاء المستشعر. للتعامل مع هذه المشكلة، تظهر هذه الورقة نهج مشتق كابوتو الكسري لبناء استراتيجية موحدة لتقدير خطأ الاستشعار في وقت واحد والتحكم المتسامح مع الخطأ. حيث أخد في الاعتبار شرط متباينة المصفوفة الخطية linear matrix inequalities لدراسة الاستقرار على أساس نظرية ليابونوف Lyapunov theory. يتم تطبيق النهج المقترح على نظام فضاء الحالة من الدرجة الثانية مع خطأ استشعار متحيز وذلك لإثبات أدائه. كما أثبتت نتائج المحاكاة أن المراقب التكيفي يمكنه تقدير حجم الخطأ بدقة ولديه قدرة تتبع قوية مع أخطاء تقدير أولية كبيرة. علاوة على ذلك، فإن وحدة التحكم المتسامحة مع الأخطاء قادرة على ايصال النظام لحالة الاستقرار بنجاح مع عدم وجود خطأ في حالة الاستقرار وبدون أي تجاوز519..

المراجع

[1] C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications. in Advances in Industrial Control. London: Springer, 2010.

[2] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, 1998.

[3] N. Mojahed, H. Fatoorehchi, and S. Nazari, “Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey,” arXiv.org.

[4] “Linear Models of Dissipation whose Q is almost Frequency Independent—II | Geophysical Journal International | Oxford Academic.”

[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, 2006.

[6] R. L. Bagley and R. A. Calico, “Fractional order state equations for the control of viscoelasticallydamped structures,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 2, pp. 304–311, 1991,

[7] I. Podlubny, “Fractional-order systems and PIαDβcontrollers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999.

[8] O. Krejcar and H. Namazi, “Review of the applications of different fractional models in engineering,” Applied Mathematical Modelling, vol. 153, p. 116623, May 2026.

[9] N. Verma and A. Jain, “Implementation of class topper optimized fractional order proportional-integral-derivative controller for cart-inverted pendulum system,” Majlesi Journal of Electrical Engineering, vol. 19, no. 1, pp. 1–10, 2025.

[10] M. Sánchez-Rivero, M. A. Duarte-Mermoud, L. Bárzaga-Martell, M. E. Orchard, and G. Ceballos-Benavides, “Fractional Gradient-Based Model Reference Adaptive Control Applied on an Inverted Pendulum-Cart System,” Fractal and Fractional, vol. 9, no. 8, p. 485, Aug. 2025,

[11] J. Lunze, “From fault diagnosis to reconfigurable control: A unified concept,” in 2016 3rd Conference on Control and Fault-Tolerant Systems (SysTol), Sep. 2016, pp. 413–421.

[12] Z. Zhao, L. Zhu, J. Li, and D. Du, “Actuator and sensor fault detection for a nonlinear fractional-order system with time-varying delay,” Chaos, Solitons & Fractals, vol. 199, p. 116647, 2025.

[13] G. Narayanan, J. J. Hoon, and J. Y. Hoon, “Resilient sampled-data control for fractional-order PMVG-based WTS with actuator saturation and probabilistic faults using fuzzy Lyapunov function method,” Information Sciences, vol. 686, p. 121294, 2025.

[14] S. M. Alzahrani and H. Rguigui, “Algebraic linear matrix inequality design of robust sensor fault estimators for ς$$ varsigma $$‐tempered fractional‐order systems with monotone nonlinearities,” Asian Journal of Control, p. asjc.70000, Oct. 2025.

[15] Y. Yan, H. Zhang, Y. Mu, and J. Sun, “Fault-tolerant fuzzy-resilient control for fractional-order stochastic underactuated system with unmodeled dynamics and actuator saturation,” IEEE Transactions on Cybernetics, vol. 54, no. 2, pp. 988–998, 2023.

[16] M. Elouni, H. Hamdi, B. Rabaoui, N. B. Braiek, and M. Rodrigues, “Robust sensor faults observer design for continuous-time switched systems using descriptor approach,” in 2023 20th International Multi-Conference on Systems, Signals & Devices (SSD), IEEE, 2023, pp. 825–831.

[17] I. Ahmed, M. Rehan, and N. Iqbal, “A novel exponential approach for dynamic event-triggered leaderless consensus of nonlinear multi-agent systems over directed graphs,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 69, no. 3, pp. 1782–1786, 2021.

[18] O. Naifar, “Practical observer design for nonlinear systems using caputo fractional derivative with respect to another function,” in 2025 IEEE 22nd International Multi-Conference on Systems, Signals & Devices (SSD), IEEE, 2025, pp. 411–418.

[19] A. Jmal, O. Naifar, A. B. Makhlouf, and N. Derbel, “Fault Tolerant Control for Linear Fractional-Order Systems with Sensor Faults,” in 2018 15th International Multi-Conference on Systems, Signals & Devices (SSD), Mar. 2018, pp. 105–110.

[20] Y. Li, Y. Chen, and I. Podlubny, “Mittag–Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969,

[21] J. Xu, R. Wang, J. Mao, and Z. Xiang, “New results on predefined-time consensus for nonlinear multi-agent systems under Lipschitz condition: A dynamic-gains-based method,” Journal of the Franklin Institute, vol. 362, no. 13, p. 107894.

[22] A. O. M. Alsharif, A. Jmal, O. Naifar, A. H. Ahmed, H. Gassara, and A. B. Makhlouf, “Sensor Fault Estimation and Ftc Design for Fractional Linear Systems Using the Conformable Derivative,” in 2025 IEEE 22nd International Multi-Conference on Systems, Signals & Devices (SSD), IEEE, 2025, pp. 404–410. Accessed: Jun. 06, 2026.

[23] M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, and R. Castro-Linares, “Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1–3, pp. 650–659, 2015.