Document Type : Research Articles
Authors
1 Technical and Vocational School, Technical and Vocational University, Ilam, Iran.
2 Department of Electrical and Computer Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran.
3 Faculty of Electrical Engineering, Shahid Beheshti University, Tehran, Iran.
Abstract
In this paper, an observer-based controller design for fractional-order multi-agent systems is discussed. By introducing a novel algorithm and leveraging appropriate lemmas and theoretical frameworks, we propose a stable observer and a distributed consensus protocol tailored for multi-agent systems within the Lipschitz and one-sided Lipschitz classes of nonlinear systems. Lipschitz systems have a bounded rate of change, ensuring proportional output to input differences, while one-sided Lipschitz systems relax this constraint, allowing differential growth in one direction for efficiency. The stability of the observer and the controller in achieving the consensus problem is demonstrated using the Lyapunov's second method. The proposed approach is rigorously developed, ensuring that the designed observer and controller meet the necessary stability criteria. Extensive simulation results validate the theoretical findings, showcasing the method's effectiveness and robustness in practical scenarios. Specifically, the simulations demonstrate that the proposed method achieves global Mittag-Leffler stability, with the estimated states converging to the actual states with minimal deviation. The method's advantages include its ability to handle a broader class of nonlinear systems, including those with large Lipschitz constants, and its robustness to uncertainties and nonlinearities. These simulations confirm the theoretical predictions and illustrate the practical applicability of our approach in real-world multi-agent systems, such as swarm robotics, power grids, and sensor networks.
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[2] M. Nazifi, M. Pourgholi, "Adaptive Fractional-Order Consensus Control of Cyber-Physical Power Systems in The Presence of Unbounded Perturbations." International Journal of Industrial Electronics Control and Optimization (2024).
[3] F. Aazam Manesh, M. Pourgholi, and E. Amini Boroujeni, "Fractional-Order Multi-agent Formation Using Distributed NMPC Design with Obstacles and Collision Avoidance and Connectivity Maintenance," Journal of Control, Automation and Electrical Systems, pp. 1-11, 2022.
[4] M. Sallah and C. Margeanu, "Effect of fractional parameter on neutron transport in finite disturbed reactors with quadratic scattering," 2016.
[5] A. El-Sayed, S. Rida, and A. Arafa, "On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber," International Journal of Nonlinear Science, vol. 7, no. 4, pp. 485-492, 2009.
[6] E. Balci, S. Kartal, and I. Ozturk, "Comparison of dynamical behavior between fractional order delayed and discrete conformable fractional order tumor-immune system," Mathematical Modelling of Natural Phenomena, vol. 16, p. 3, 2021.
[7] A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations. elsevier, 2006.
[8] M. Shahvali, M.-B. Naghibi-Sistani, and J. Askari, "Adaptive fault compensation control for nonlinear uncertain fractional-order systems: static and dynamic event generator approaches," Journal of the Franklin Institute, vol. 358, no. 12, pp. 6074-6100, 2021.
[9] Varga, Bence, J.K. Tar, and R. Horváth, "Fractional order inspired iterative adaptive control," Robotica, vol.42, no. 2, pp. 482-509, 2024.
[10] Chendrayan, J. Hoon Jeong, and Y. H. Joo, "Stochastic exponential stabilization and optimal control results for a class of fractional order equations," Chaos, Solitons & Fractals, vol. 185, no. 1, pp.15087, 2024.
[11] H. Kheloufi, A. Zemouche, F.Bedouhene, and M. Boutayeb, "On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties," Automatica, vol. 49, no. 12, pp. 3700-3704, 2013.
[12] A. Boroujeni and H. R. Momeni, "Observer based control of a class of nonlinear fractional order systems using LMI," International Journal of Science and Engineering Investigations, vol. 1, no. 1, pp. 48-52, 2012.
[13] M. Pourgholi and E. A. Boroujeni, "An iterative LMI-based reduced-order observer design for fractional-order chaos synchronization," Circuits, Systems, and Signal Processing, vol. 35, no. 6, pp. 1855-1870, 2016.
[14] S. Ahmad and M. Rehan, "On observer-based control of onesided Lipschitz systems," Journal of the Franklin Institute, vol. 353, no. 4, pp. 903-916, 2016.
[16] T. Kaczorek, "Reduced-order perfect nonlinear observers of fractional descriptor discrete-time nonlinear systems," International Journal of Applied Mathematics and Computer Science, vol. 27, no. 2, 2017.
[17] A. Jmal, O. Naifar, A. Ben Makhlouf, N. Derbel, and M. A. Hammami, "On observer design for nonlinear Caputo fractional‐order systems," Asian journal of control, vol. 20, no. 4, pp. 1533-1540, 2018.
[18] M. Karkhaneh and M. Pourgholi, "Adaptive observer design for one-Sided Lipschitz class of nonlinear systems," The Modares Journal of Electrical Engineering, vol. 11, no. 4, pp. 45-51, 2012.
[19] Y. Bai, A. Marino, and J. Wang, "Observer-based distributed fault detection and isolation for second-order multi-agent systems using relative information," Journal of the Franklin Institute, vol. 358, no. 7, pp. 3779-3802, 2021.
[20] X. Xue, X. Yue, and J. Yuan, "Connectivity Preservation and Collision Avoidance Control for Spacecraft Formation Flying with Bounded Actuation," in Advances in Guidance, Navigation and Control: Springer, 2022, pp. 3685-3697.
[21] C. Thompson and P. Reverdy, "Multi-Agent Recurrent Rendezvous Using Drive-Based Motivation," arXiv preprint arXiv:2109.13408, 2021.
[22] Y. Xu, T. Li, and S. Tong, "Event-triggered adaptive fuzzy bipartite consensus control of multiple autonomous underwater vehicles," IET Control Theory & Applications, vol. 14, no. 20, pp. 3632-3642, 2020.
[23] M. Rodriguez, "Intelligent Intersection Management through Gradient-Based Multi-Agent Coordination of Traffic Lights and Vehicles," University of Maryland, College Park, 2021.
[24] H. Bai and J. T. Wen, "Cooperative load transport: A formation-control perspective," IEEE Transactions on Robotics, vol. 26, no. 4, pp. 742-750, 2010.
[25] W. Ren and R. W. Beard, Distributed consensus in multivehicle cooperative control (no. 2). Springer, 2008.
[26] Z. Li, Z. Duan, and G. Chen, "Dynamic consensus of linear multi-agent systems," IET Control Theory & Applications, vol. 5, no. 1, pp. 19-28, 2011.
[27] H. Zhao, S. Xu, and D. Yuan, "An LMI approach to consensus in second-order multi-agent systems," International Journal of Control, Automation and Systems, vol. 9, no. 6, pp. 1111-1115, 2011.
[28] W. Yu, G. Chen, M. Cao, and J. Kurths, "Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 40, no. 3, pp. 881-891, 2009.
[29] M. Rehan, A. Jameel, and C. K. Ahn, "Distributed consensus control of one-sided Lipschitz nonlinear multiagent systems," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 8, pp. 1297-1308, 2017.
[30] M. Doostinia, M. T. Beheshti, S. A. Alavi, and J. M. Guerrero, "Distributed control strategy for DC microgrids based on average consensus and fractional-order local controllers," IET Smart Grid, 2021.
[31] L. Chen, X. Li, Y. Chen, R. Wu, A. M. Lopes, and S. Ge, "Leader-follower non-fragile consensus of delayed fractional-order nonlinear multi-agent systems," Applied Mathematics and Computation, vol. 414, p. 126688, 2022.
[32] R. Olfati-Saber, "Flocking for multi-agent dynamic systems: Algorithms and theory," IEEE Transactions on automatic control, vol. 51, no. 3, pp. 401-420, 2006.
[33] W. Ren and R. W. Beard, "Consensus seeking in multiagent systems under dynamically changing interaction topologies," IEEE Transactions on automatic control, vol. 50, no. 5, pp. 655-661, 2005.
[34] I. Podlubny, "Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications", Elsevier, vol 198, 1998.
[35] B. Duan and Z. Zhang, "A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input," Journal of Scientific Computing, vol. 87, no. 3, pp. 1-20, 2021.
[36] 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.
[37] M. N. Soorki and M. S. Tavazoei, "Adaptive robust control of fractional-order swarm systems in the presence of model uncertainties and external disturbances," IET control theory & applications, vol. 12, no. 7, pp. 961-969, 2018.
[38] Y. Li, Y. Chen, and I. Podlubny, "Mittag–Leffler stability of fractional order nonlinear dynamic systems," Automatica, vol. 45, no. 8, pp. 1965-1969, 2009.
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