Caputo Derivative-Based FTC Design and Sensor Fault Estimation for Lipschitz Fractional Nonlinear Systems
DOI:
https://doi.org/10.65419/albahit.v5i3.152Keywords:
Caputo fractional derivative (CFD), fault-tolerant control (FTC), sensor fault estimation, adaptive observer, fractional nonlinear systems, linear matrix inequalities (LMIs)Abstract
Sensor fault estimation and active fault-tolerant control (FTC) are still important problems for guaranteeing the fractional-order nonlinear systems' (FONS) stability under sensor faults. To deal with this problem, this work uses the Caputo fractional derivative framework to construct a unified strategy for simultaneous sensor fault estimation and active (FTC). Sufficient conditions obtained are in the form of linear matrix inequalities (LMIs) from a stability study based on Lyapunov theory. The proposed approach is applied to a two-state system with a biased sensor fault to demonstrate its performance. The simulation results indicate that the adaptive observer can accurately estimate the fault magnitude and has robust tracking ability with large initial estimation errors. Moreover, the fault tolerant controller is capable of stabilizing the system successfully with zero steady state error and without any overshoot.
References
[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.
