Fixed-Time Consensus of Fractional-Order Single Integrator Multi-Agent Systems
Subject Areas : electrical and computer engineering
Hossein
Zamani
1
(Tarbiat Modares University)
وحيد
جوهري مجد
2
(Tarbiat Modares University)
Khosro
Khandani
3
(Arak University)
Keywords: Fractional-order integrators, multi-agent systems, consensus, fixed-time convergence,
Abstract :
The problem of consensus in fractional order single-integrator multi-agent systems has been studied in this paper. The effect of memory is considered using the Riemann-Liouville fractional derivative in the dynamics of the agents. In order to achieve convergence among the agents, a fractional order control protocol based on the error signal between neighboring agents is proposed. Using Lyapunov's stability theorem, a Lyapunov function is introduced that shows that the agents converge over a specified settling time and the upper bound of the settling time is obtained. The merit of the proposed bound for the settling time is that it is independent of the initial conditions. Finally, some simulations are provided to confirm the introduced method.
[1] S. Westerlund, Dead Matter Has Memory, Kalmar, Sweden: Causal Consulting, 2002.
[2] S. K. Samko and B. Ross, "Integration and differentoiation to a variable fractional order," Integral Transforms Special Func., vol. 1, no. 4, pp. 277-300, Apr. 2007.
[3] B. Ross and S. K. Samko, "Fractional integration operator of variable order in the Holder space Hu(x)," Int. J. Math. Math. Sci., vol. 18, no. 4, pp. 777-788, Jan. 1995.
[4] I. Podlubny, "Fractional-order systems and PIλDµ controllers," IEEE Trans. on Automatic Control, vol. 44, no. 1, pp. 208-214, Jan. 1999.
[5] C. A. Monje, B. M. Vinagre, V. Feliu, and Y. Q. Chen, "On auto-tuning of fractional order PIλDµ controllers," in Proc. of Fractional Derivatives and Applications, vol. 2, 6 pp., Porto, Portugal, 19-21 Jul. 2006.
[6] C. Ma and Y. Hori, "Fractional order control: theory and applications in motion control," IEEE Industrial Electronics Magazine, vol. 1, no. 4, pp. 6-16, Winter 2007.
[7] H. Chao, Y. Luo, L. Di, and Y. Q. Chen, "Roll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle," Control Engineering Practice, vol. 18, no. 7, pp. 761-772, Jul. 2010.
[8] R. Melicio, V. M. F. Mendes, and J. P. S. Catalao, "Fractional-order control and simulation of wind energy systems with PMSG/full-power converter topology," Energy Conversion and Management, vol. 51, no. 6, pp. 1250-1258, Jun. 2010.
[9] M. Zamani, M. Karimi-Ghartemani, N. Sadati, and M. Parniani, "Design of a fractional order PID controller for an AVR using particle swarm optimization," Control Engineering Practice, vol. 17, no. 12, pp. 1380-1387, Dec. 2009.
[10] M. S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, and M. Siami, "Some applications of fractional calculus in suppression of chaotic oscillations," IEEE Trans. on Industrial Electronics, vol. 55, no. 11, pp. 4094-4101, Nov. 2008.
[11] F. L. Lewis, H. Zhang, K. Hengster-Movric, and A. Das, Cooperative Control of Multi-Agent Systems: Optimal and Adaptive Design Approaches, Springer Science and Business Media, 2013.
[12] R. Olfati-Saber and R. Murray, "Consensus problems in networks of agents with switching topology and time-delays," IEEE Trans. on Automatic Control, vol. 49, no. 9, pp. 1520-1533, Sept. 2004.
[13] W. Ren and R. Beard, "Consensus seeking in multiagent systems under dynamically changing interaction topologies," IEEE Trans. on Automatic Control, vol. 50, no. 5, pp. 655-661, May 2005.
[14] F. Chen, Z. Chen, L. Xiang, Z. Liu, and Z. Yuan, "Reaching a consensus via pinning control," Automatica, vol. 45, no. 5, pp. 1215-1220, May 2009.
[15] Z. Meng, Z. Lin, and W. Ren, "Robust cooperative tracking for multiple nonidentical second-order nonlinear systems," Automatica, vol. 49, no. 8, pp. 2363-2372, Aug. 2013.
[16] S. Ferik, A. Qureshi, and F. Lewis, "Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems," Automatica, vol. 50, no. 3, pp. 798-808, Mar. 2014.
[17] F. Xiao, L. Wang, J. Chen, and Y. Gao, "Finite-time formation control for multi-agent systems," Automatica, vol. 45, no. 11, pp. 2605-2611, Nov. 2009.
[18] S. Li, H. Du, and X. Lin, "Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics," Automatica, vol. 47, no. 8, pp. 1706-1712, Aug. 2011.
[19] H. Li, X. Liao, and G. Chen, "Leader-following finite-time consensus in second-order multi-agent networks with nonlinear dynamics," International J. of Control, Automation and Systems, vol. 11, pp. 422-426, 2013.
[20] M. Shi and S. Hu, "Leader-following consensus for a class of fractional-order nonlinear multi-agent systems under fixed and switching topologies," in Proc. 36th Chinese Control Conf., CCC'17, pp. 11351-11356, Dalian, China, 26-28 Jul. 2017.
[21] H. Liu, L. Cheng, M., Tan, and Z.-G. Hou, "Exponential finite-time consensus of fractional-order multiagent systems," IEEE Trans. on Systems, Man, and Cybernetics, vol. 50, no. 4, pp. 1549-1558, Apr. 2020.
[22] H. Pan, Z. Liu, and C. Na, "Finite-time output leader-following consensus of fractional-order linear multi-agent systems," in Proc. 38th Chinese Control Conf., CCC'19, pp. 958-963, 27-30 Jul. 2019.
[23] A. Polyakov, "Nonlinear feedback design for fixed-time stabilization of linear control systems," IEEE Trans. on Automatic Control, vol. 57, no. 8, pp. 2106-2110, Aug. 2012.
[24] A. Polyakov, "Fixed-time stabilization of linear systems via sliding mode control," in Proc. 12th Int. Workshop on Variable Structure Systems, 6 pp., Mumbai, India, 12-14 Jan. 2012.
[25] M. Defoort, A. Polyakov, G. Demesure, M. Djemai, and K. Veluvolu, "Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics," IET Control Theory and Applications, vol. 9, no. 4, pp. 2165-2170, Sept. 2015.
[26] Z. Zuo, B. Tian, M. Defoort, and Z. Ding, "Fixed-time consensus tracking for multi-agent systems with high-order integrator dynamics," IEEE Trans. on Automatic Control, vol. 63, no. 2, pp. 563-570, Feb. 2017.
[27] H. Du, G. Wen, D. Wu, Y. Cheng, and J. Lü, "Distributed fixed-time consensus for nonlinear heterogeneous multi-agent systems," Automatica, vol. 113, Article ID: 108986, Mar. 2020.
[28] L. Wang and J. Dong, "Event-based distributed adaptive fuzzy consensus for nonlinear fractional-order multiagent systems," IEEE Trans. on Systems, Man, and Cybernetics: Systems, vol. 52, no. 9, pp. 5901-5912, Sept. 2022.
[29] C. Wang, H. Tnunay, Z. Zuo, B. Lennox, and Z. Ding, H. Liu, L. Cheng, M. Tan, and Z. G. Hou, "Fixed-Time formation control of multirobot systems: design and experiments," IEEE Trans. on Industrial Electronics, vol. 66, pp. 6292-6301, Aug. 2019.
[30] H. Zamani, V. J. Majd, and K. Khandani, "Formation tracking control of fractional-order multi-agent systems with fixed-time convergence," Proc. of the Institution of Mechanical Engineers, Part I: J. of Systems and Control Engineering, vol. 236, no. 9, pp. 1618-1629, Jul. 2022.
[31] E. Semsar-Kazerooni and K. Khorasani, "Switching control of a modified leader-follower team of agents under the leader and network topological changes," IET Control Theory and Applications, vol. 5, no. 12, pp. 1369-1377, Aug. 2011.
[32] K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, Academic Press, Inc. 1974.
[33] S. P. Bhat and D. S. Bernstein, "Finite-time stability of continuous autonomous systems," SIAM J. on Control and Optimization, vol. 38, no. 3, pp. 751-766, 2000.
[34] N. Junkang, L. Ling, L. Chongxin, H. Xiaoyu, and L. Shilei, "Further improvement of fixed-time protocol for average consensus of multi-agent systems," IFAC-PapersOnLine, vol. 50, no. 1, pp. 2523-2529, Jul. 2017.
[35] Y. Li, Y. Q. Chen, and I. Podlubny, "Mittag-Leffler stability of fractional order nonlinear dynamic systems," Automatica, vol. 45, no. 8, pp. 1965-1969, Aug 2009.
[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," Commun. Nonlinear Sci., vol. 22, no. 1-3, pp. 650-659, May. 2015.
[37] F. L. Sun, J. C. Chen, Z. H. Guan, D. Li, and T. Li, "Leader-following finite-time consensus for multi-agent sys-tems with jointly-reachable leader," Nonlinear Anal.: Real World Appl., vol. 13, no. 5, pp. 2271-2284, Oct. 2012.
[38] G. Hardy, J. Littlewood, and G. Polya, Inequalities, London: Cambridge Univ. Press, 1951.
[39] Z. Yaghoubi, "Robust cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller," Mathematics and Computers in Simulation, vol. 172, pp. 15-32, Jun. 2020.