Robust Adaptive Control for Delayed Fractional-Order Multi-Agent Systems with Unmodeled Cyber-Physical Attacks

Document Type : Research Paper

Authors
1 Department of Computer Science, College of Engineering and Information Technology, Onaizah Colleges, Qassim, Saudi Arabia
2 Department of Mathematics and Statistics, University of Lahore, Sargodha, Pakistan
3 Department of Biosciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, 602105, India
4 Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman 19328, Jordan
Abstract
Fractional-order multi-agent systems (MASs) with communication delays play a vital role in smart grids, autonomous vehicles, and industrial automation, but their memory-dependent dynamics increase vulnerability to cyber-physical attacks. In particular, malicious sensor and actuator attacks compromise feedback and control signals, threatening stability and consensus. Traditional control methods often require   prior knowledge of attack models, which restricts their effectiveness against unknown and time-varying disturbances. This paper proposes a robust adaptive control framework for delayed fractional-order MASs within the Caputo derivative setting. The method integrates an adaptive estimation law with Caputo-based Lyapunov analysis to counter unmodeled sensor and actuator attacks using only local and neighboring agent data. Stability conditions are rigorously established through a fractional Razumikhin approach. Simulations validate the scheme's resilience under sensor-only, actuator-only, and combined attacks, confirming its ability to maintain consensus, achieve faster convergence, and improve reliability in hostile environments.
Keywords
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Zhang, J.X., Zhang, X., Boutat, D., Liu, D.Y., Fractional-order complex systems: Advanced control, intelligent estimation and reinforcement learning image-processing algorithms, Fractal and Fractional, 9(2), 2025, 67.
[2] Wang, J., He, G., Geng, S., Zhang, S., Zhang, J., Data-driven adaptive control for uncertain nonlinear systems, Nonlinear Dynamics, 113(5), 2025, 4197-4209.
[3] Sheikh, S., Khalsa, L., Varghese, V., The impact of memory effect in the higher-order time-fractional derivative for hygrothermoelastic cylinder, Multidiscipline Modeling in Materials and Structures, 20(5), 2024, 761-783.
[4] Munoz-Pacheco, J.M., Wei, Z., Volos, C., Sambas, A., Future challenges in the fractional-order dynamical systems: from mathematics to applications, Frontiers in Applied Mathematics and Statistics, 9, 2023, 1324660.
[5] Liu, X., Sun, Y., Han, Q., Cao, K., Shen, H., Xu, J., Ji, A., A novel adaptive dynamic optimal balance control method for wheel-legged robot, Applied Mathematical Modelling, 137, 2025, 115737.
[6] Fekri, S., Athans, M., Pascoal, A., Issues, progress and new results in robust adaptive control, International Journal of Adaptive Control and Signal Processing, 20(10), 2006, 519-579.
[7] Ranjan, S., Majhi, S., Fixed-Time State Observer-Based Robust Adaptive Neural Fault-Tolerant Control for a Quadrotor Unmanned Aerial Vehicle, International Journal of Adaptive Control and Signal Processing, 39(1), 2025, 132-151.
[8] Shambhu Choudhary, S., Nath Gupta, T., Hussain, I., Intelligent solar grid integration: advancements in control strategies and power quality enhancement, International Journal of Circuit Theory and Applications, 53(6), 2025, 3462-3480.
[9] Jadidi, S., Badihi, H., Zhang, Y., Active fault-tolerant and attack-resilient control for a renewable microgrid against power-loss faults and data integrity attacks, IEEE Transactions on Cybernetics, 54(4), 2023, 2113-2128.
[10] Chen, Z., Tan, J., He, Y., Cao, Z., Decentralized observer-based event-triggered control for an interconnected fractional-order system with stochastic Cyber-attacks, AIMS Mathematics, 9(1), 2024, 1861-1876.
[11] Khan, A., Niazi, A.U.K., Abbasi, W., Awan, F., Khan, M.M.A., Imtiaz, F., Cyber secure consensus of fractional order multi-agent systems with distributed delays: Defense strategy against denial-of-service attacks, Ain Shams Engineering Journal, 15, 2024, 102609.
[12] Hu, T., Zhang, X., Shi, K., Secure intermittent impulsive consensus control for fractional-order multiagent systems under denial-of-service and deception attacks, Information Sciences, 678, 2024, 120949.
[13] Wu, J., Peng, C., Yang, H., Wang, Y.L., Recent advances in event-triggered security control of networked systems: A survey, International Journal of Systems Science, 53(12), 2022, 2624-2643.
[14] Dimitrov, Y., Georgiev, S., Todorov, V., Approximation of Caputo fractional derivative and numerical solutions of fractional differential equations, Fractal and Fractional, 7(10), 2023, 750.
[15] Strässer, R., Berberich, J., Allgöwer, F., Robust data-driven control for nonlinear systems using the Koopman operator, IFAC-Papers OnLine, 56(2), 2023, 2257–2262.
[16] Yang, Y., Zhang, H., Chen, J., Adaptive consensus for multi-agent systems with nonlinear dynamics and switching topologies, Automatica, 49(7), 2013, 2107–2115.
[17] Chen, G., Yang, Y., Fixed-Time Stability of Time-Varying Hybrid Systems with Time-Delay, Circuits, Systems, and Signal Processing, 43(5), 2024, 2758-2781.
[18] Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R., Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications in Nonlinear Science and Numerical Simulation, 22(1–3), 2015, 650–659.
[19] Zhao, L., Chen, X., Yu, J., Shi, P., Output feedback-based neural adaptive finite-time containment control of non-strict feedback nonlinear multi-agent systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 69(2), 2021, 847–858.
[20] Cong, D., Tuan, H.T., Asymptotic stability criteria for fractional differential equations with delay via fractional Lyapunov functions, Communications in Nonlinear Science and Numerical Simulation, 100, 2019, 224–232.
[21] Liu, S., Yang, R., Li, X., Xiao, J., Global attractiveness and consensus for Riemann–Liouville’s nonlinear fractional systems with mixed time-delays, Chaos, Solitons and Fractals, 143, 2021, 110577.
[22] Zheng, X., Ma, H., Zhou, Q., Li, H., Neural-based prescribed-time consensus control for multiagent systems via dynamic memory event-triggered mechanism, Science China Technological Sciences, 68(3), 2025, 1-11.
[23] Yang, H., Li, S., Yang, L., Ding, Z., Leader-following consensus of fractional-order uncertain multiagent systems with time delays, Neural Processing Letters, 54(6), 2022, 4829-4849.
[24] Wang, C., Ji, H., Leader-following consensus of multi-agent systems under directed communication topology via distributed adaptive nonlinear protocol, Systems and Control Letters, 70, 2014, 23–29.
[25] Lian, B., Koru, A.T., Xue, W., Lewis, F.L., Davoudi, A., Distributed Dynamic Clustering and Consensus in Multiagent Systems, IEEE Transactions on Automatic Control, 69(9), 2024, 6474-6481.
[26] Hattaf, K., On the stability and numerical scheme of fractional differential equations with application to biology, Computation, 10(6), 2022, 97.
[27] Shao, K., Shao, J., He, C., Hu, R., Class function-based adaptive disturbance observer for uncertain nonlinear systems, International Journal of Systems Science, 56(4), 2025, 841-849.
[28] Zhu, B., Liang, H., Niu, B., Wang, H., Zhao, N., Zhao, X., Observer-based reinforcement learning for optimal fault-tolerant consensus control of nonlinear multi-agent systems via a dynamic event-triggered mechanism, Information Sciences, 689, 2025, 121350.
[29] Li, Z., Xia, Y., Fu, M., Liu, H., Resilient consensus control of fractional-order multi-agent systems under sensor and actuator attacks, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(12), 2019, 2503–2513.
[30] Khan, A., Niazi, A.U.K., Abbasi, W., Awan, F., Khan, A., Fractional-order nonlinear multiagent systems: A resilience-based approach to consensus analysis with distributed and input delays, Fractal and Fractional, 7(4), 2023, 322.
[31] Qin, Z., Wu, R., Lu, Y., Stability analysis of fractional-order systems with the Riemann–Liouville derivative, Systems Science and Control Engineering, 2(1), 2014, 727–731.
[32] Sun, Y., Shi, Y., Adaptive self-triggered control-based cooperative output regulation of heterogeneous multi-agent systems under sensor and actuator attack, Journal of Cleaner Production, 402, 2024, 139432.
[33] Azizi, A., Naderi Soorki, M., Vedadi Moghaddam, T., Soleimanizadeh, A., A new fractional-order adaptive sliding-mode approach for fast finite-time control of human knee joint orthosis with unknown dynamic, Mathematics, 11(21), 2023, 4511.
[34] Azizi, A., A case study on designing a sliding mode controller to stabilize the stochastic effect of noise on mechanical structures: residential buildings equipped with ATMD, Complexity, 2020(1), 2020, 9321928.
[35] Azizi, A., Mobki, H., Ouakad, H.M., Speily, O.R.B., Applied mechatronics: on mitigating disturbance effects in MEMS resonators using robust nonsingular terminal sliding mode controllers, Machines, 10(1), 2022, 34.
[36] Mobki, H., Jalilirad, M., Vatankhah Moradi, M., Azizi, A., Multi input versus single input sliding mode for closed-loop control of capacitive micro structures, SN Applied Sciences, 1(7), 2019, 676.
[37] Mobki, H., Sabegh, A.M., Azizi, A., Ouakad, H.M., On the implementation of adaptive sliding mode robust controller in the stabilization of electrically actuated micro-tunable capacitor, Microsystem Technologies, 26(12), 2020, 3903-3916.
[38] Ganesan, T., Yousaf, Z., Bhatti, M.Z., Algebraic and Spectral Analysis of a Novel Hermitian Spin Basis, Symmetry, 17(3), 2025, 450.
[39] Zhu, Z., Zhao, H., Xian, Y., Sun, H., Chen, Y. H., Ma, J., Cooperative game-theoretic optimization of adaptive robust constraint-following control for fuzzy mechanical systems under inequality constraints, Nonlinear Dynamics, 113(5), 2025, 4703-4726.
[40] Parvareh, A., Naderi Soorki, M., Azizi, A., The robust adaptive control of leader–follower formation in mobile robots with dynamic obstacle avoidance, Mathematics, 11(20), 2023, 4267.
[41] Mohammadzaheri, M., Al-Humairi, A., Vakili-Nezhaad, G., Azizi, A., Jones, S., Hamlin, C., ..., Soltani, P., Digital Model-Based Motor Servo Control Considering a Sampling-Induced Time Delay, In 12th International Conference on Control, Dynamic Systems, and Robotics, 2025.
[42] Mobki, H., Sedighi, M.H., Azizi, A., Eskandari, M.M., Designing an efficient observer for the non-linear Lipschitz system to troubleshoot and detect secondary faults considering linearizing the dynamic error, Facta Universitatis, Series: Mechanical Engineering, 20(3), 2022, 677-691.
[43] Koochakzadeh, A., Naderi Soorki, M., Azizi, A., Mohammadsharifi, K., Riazat, M., Delay-dependent stability region for the distributed coordination of delayed fractional-order multi-agent systems, Mathematics, 11(5), 2023, 1267.
[44] Riazat, M., Azizi, A., Naderi Soorki, M., Koochakzadeh, A., Robust consensus in a class of fractional-order multi-agent systems with interval uncertainties using the existence condition of Hermitianmatrices, Axioms, 12(1), 2023, 65.

Articles in Press, Corrected Proof
Available Online from 22 March 2026