Kybernetika 61 no. 2, 264-287, 2025

Stabilizability of multi-agent systems over finite fields via fully actuated system approaches

Yunsi Yang, Jun-e Feng and Lei JiaDOI: 10.14736/kyb-2025-2-0264

Abstract:

The problem of stabilizability of high-order fully actuated (HOFA) multi-agent systems over finite fields is considered in this paper. The necessary and sufficient conditions for the stabilizability of HOFA multi-agent systems are presented, which indicates the stabilizability is closely related to the interaction topology among agents. Using the full-actuation property of HOFA models, a stabilization control protocol with neighbor interaction is given for HOFA multi-agent systems. Additionally, when the multi-agent system is stabilizable, the time for the system to reach a stable state can be determined through the control protocol. Finally, the results are employed to solve the formation control problem, and some sufficient and/or necessary conditions are proposed. Numerical examples are presented to demonstrate the effectiveness of the proposed results.

Keywords:

stabilizability, multi-agent systems, finite fields, fully actuated system approach

Classification:

12E20, 93D99, 93A16

paper.pdf

References:

  1. I. Arel, C. Liu, T. Urbanik and A. G. Kohls: Reinforcement learning-based multi-agent system for network traffic signal control. IET Intell. Transp. Syst. 4 (2010), 128-135.   DOI:10.1049/iet-its.2009.0070
  2. A. Das, R. Fierro, V. Kumar, J. Ostrowski, J. Spletzer and C. Taylor: A vision-based formation control framework. IEEE Trans. Robot. Automat. 18, (2002), 5, 813-825.   DOI:10.1109/TRA.2002.803463
  3. W. Ding, G. Yan and Z. Lin: Collective motions and formations under pursuit strategies on directed acyclic graphs. Automatica 46 (2010), 1, 174-181.   DOI:10.1016/j.automatica.2009.10.025
  4. G. Duan: Fully actuated system approach for control: An overview. IEEE Trans. Cybernet. 54 (2024), 12, 7285-7306.   DOI:10.1109/TCYB.2024.3457584
  5. G. Duan: High-order system approaches: {I. Full-actuation and parametric design}. Acta Automat. Sin. 46 (2020), 7, 1333-1345.   DOI:10.16383/j.aas.c200234
  6. G. Duan: High-order system approaches: {II. Controllability and fully-actuation}. Acta Automat. Sinica 46 (2020), 8, 1571-1581.   DOI:10.16383/j.aas.c200369
  7. G. Duan: High-order fully actuated system approaches: {Part I. Models and basic procedure}. Int. J. Syst. Sci. 52 (2021), 2, 422-435.   DOI:10.1080/00207721.2020.1829167
  8. G. Duan: High-order fully actuated system approaches: {Part II. Generalized strict-feedback systems}. Int. J. Syst. Sci. 52 (2021), 3, 437-454.   DOI:10.1080/00207721.2020.1829168
  9. G. Duan: High-order fully actuated system approaches: {Part VII. Controllability, stabilizability and parametric designs}. Int. J. Syst. Sci. 52 (2021), 14, 3091-3114.   doi:10.1080/00207721.2021.1921307
  10. G. Duan: High-orderfully actuated system approaches: {Part X. Basics of discrete-time systems}. Int. J. Syst. Sci. 53 (2021), 4, 810-832.   DOI:10.1080/00207721.2021.1975848
  11. J. A. Fax: Optimal and Cooperative Control of Vehicle Formations. Ph.D. Thesis, California Institute of Technology, Pasaden 2002.   CrossRef
  12. J. A. Fax and R. M. Murray: Information flow and cooperative control of vehicle formations. IEEE Trans. Automat. Control 49 (2004), 9, 1465-1476.   DOI:10.1109/TAC.2004.834433
  13. M. Franceschelli, A. Gasparri, A. Giua and G. Ulivi: Decentralized stabilization of heterogeneous linear multi-agent systems. In: Proc. 2010 IEEE Int. Conf. Robot. Autom., 2010, pp. 3556-3561.   DOI:10.1109/ROBOT.2010.5509637
  14. Y. Guan, Z. Ji, L. Zhang and L. Wang: Decentralized stabilizability of multi-agent systems under fixed and switching topologies. Syst. Control Lett. 62 (2013), 5, 438-446.   DOI:10.1016/j.sysconle.2013.02.010
  15. Y. Guan and X. Kong: Stabilisability of discrete-time multi-agent systems under fixed and switching topologies. Int. J. Syst. Sci {mi50}, (2019), 2, 294-306.   DOI:10.1080/00207721.2018.1551975
  16. A. Jadbabaie, J. Lin and A. S. Morse: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Control 48 (2003), 6, 988-1001.   DOI:10.1109/TAC.2003.812781
  17. Z. Ji, Z. Wang and H. Lin: Controllability of multi-agent systems with time-delay in state and switching topology. Int. J. Control 83 (2010), 2, 371-386.   DOI:10.1080/00207170903171330
  18. R. E. Kalman: Contribution to the theory of optimal control. Bol. Soc. Mat. Mexicana 5 (1960), 2, 102-119.   CrossRef
  19. R. E. Kalman: Controllability of linear dynamical systems. Theory Differ. Equat. 1 (1963), 3, 189-213.   CrossRef
  20. H. Kim, H. Shim, J. Back and J. Seo: Stabilizability of a group of single integrators and its application to decentralized formation problem. In: Proc. 50th IEEE Conf. Decis. Control Euro. Control Conf., 2011, pp. 4829-4834.   DOI:10.1109/CDC.2011.6161139
  21. X. Li, M. Chen, H. Su and C. Li: Consensus networks with switching topology and time-delays over finite fields. Automatica 68 (2016), 39-43.   DOI:10.1016/j.automatica.2016.01.033
  22. Y. Li, H. Li, X. Ding and G. Zhao: Leader-follower consensus of multiagent systems with time delays over finite fields. IEEE Trans. Cybernet. 49 (2018), 8, 3203-3208.   DOI:10.1109/TCYB.2018.2839892
  23. Y. Li and H. Li: Controllability of multi-agent systems over finite fields via semi-tensor product method. In: Proc. 38th Chin. Control Conf., 2019, pp. 5606-5611.   DOI:10.23919/ChiCC.2019.8866482
  24. X. Li, H. Su and M. Chen: Consensus networks with time-delays over finite fields. Int. J. Control 89 (2016), 5, 1000-1008.   DOI:10.1080/00207179.2015.1110755
  25. A. Ligtenberg, M. Wachowicz, A. K. Bregt, A.Beulensb and D. L. Kettenis: A design and application of a multi-agent system for simulation of multi-actor spatial planning. J. Environ. Management 72 (2004), 1, 43-55.   DOI:10.1016/j.jenvman.2004.02.007
  26. G. Liu: Coordination of networked nonlinear multi-agents using a high-order fully actuated predictive control strategy. IEEE/CAA J. Autom. Sinica 9 (2022), 4, 615-623.   doi:10.1109/JAS.2022.105449
  27. T. Logenthiran, D. Srinivasan and A. M. Khambadkone: Multi-agent system for energy resource scheduling of integrated microgrids in a distributed system. Electr. Power Syst. Res. 81 (2011), 1, 138-148.   DOI:10.1016/j.epsr.2010.07.019
  28. Z. Lu, L. Zhang and L. Wang: Structural controllability of multi-agent systems with general linear dynamics over finite fields. In: Proc. 35th Chin. Control Conf., 2016, pp. 8230-8235.   DOI:10.1109/ChiCC.2016.7554667
  29. Z. Lu, L. Zhang and L. Wang: Controllability analysis of multi-agent systems with switching topology over finite fields. Sci. China Inform. Sci. 62 (2019), 12, 12201.   DOI:10.1007/s11432-017-9284-4
  30. N. Lynch: Distributed Algorithms. Elsevier, San Francisco 1996.   CrossRef
  31. M. Meng, X. Li and G. Xiao: Synchronization of networks over finite fields. Automatica 115 (2020), 108877.   DOI:10.1016/j.automatica.2020.108877
  32. G. Mullen and D. Panario: Handbook of Finite Fields. Chapman and Hall/CRC, Boca Raton 2013.   DOI:10.1201/b15006
  33. R. Olfati-Saber and R. M. Murray: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Automat. Control 49 (2004), 9, 1520-1533.   DOI:10.1109/TAC.2004.834113
  34. F. Pasqualetti, D. Borra and F. Bullo: Consensus networks over finite fields. Automatica 50 (2014), 2, 349-358.   DOI:10.1016/j.automatica.2013.11.011
  35. J. Reger: Linear systems over finite fields - modeling, analysis, and synthesis. Automatisierungstechnik 53 (2005), 1, 45.   DOI:10.1524/auto.53.1.45.56703
  36. H. Ren, Z. Cheng, J. Qin and R. Lu: Deception attacks on event-triggered distributed consensus estimation for nonlinear systems. Automatica 154 (2023), 111100.   DOI:10.1016/j.automatica.2023.111100
  37. H. Ren, R. Liu, Z. Cheng, H. Ma and H. Li: Data-driven event-triggered control for nonlinear multi-agent systems with uniform quantization. IEEE Trans. Circuits Syst. II: Express Br. 71 (2023), 2, 712-716.   DOI:10.1109/TCSII.2023.3305946
  38. S. Shreyas and H. Christoforos: Structural controllability and observability of linear systems over finite fields with applications to multi-agent systems. IEEE Trans. Automat. Control 58 (2013), 1, 60-73.   DOI:10.1109/TAC.2012.2204155
  39. H. Su, M. Chen, J. Lam and Z. Lin: Semi-global leader-following consensus of linear multi-agent systems with input saturation via low gain feedback. IEEE Trans. Circuits Syst. I: Regul. Pap. 60 (2013), 7, 1881-1889.   DOI:10.1109/TCSI.2012.2226490
  40. Y. Sun, Z. Ji, Y. Liu and C. Lin: On stabilizability of multi-agent systems. Automatica {mi144} (2022), 110491.   DOI:10.1016/j.automatica.2022.110491
  41. T. Tay, I. Mareels and J. Moore: High performance control. Springer Science Business Media, New York 1998.   DOI:10.1007/978-1-4612-1786-2
  42. H. G. Tanner: On the controllability of nearest neighbor interconnections. In: Proc. 43rd IEEE Conf. Decis. Control 3 (2005), pp. 2467-2472.   DOI:10.1109/CDC.2004.1428782
  43. R. A. H. Toledo: Linear finite dynamical systems. Commun. Algebra 33 (2005), 9, 2977-2989.   DOI:10.1081/AGB-200066211
  44. L. Xiang, J. Zhu, F. Chen and G. Chen: Controllability of weighted and directed networks with nonidentical node dynamics. Math. Probl. Engrg. (2013).   DOI:10.1155/2013/405034
  45. X. Xu and Y. Hong: Leader-following consensus of multi-agent systems over finite fields. In: Proc. 53rd IEEE Conf. Decis. Control, 2014, pp. 2999-3004.   DOI:10.1109/CDC.2014.7039850
  46. Y. Yang, J. Feng and L. Jia: Recent advances of finite-field networks. Math. Model. Control 3 (2023), 3, 244-255.   DOI:10.3934/mmc.2023021
  47. Y. Yang, J. Feng and L. Jia: Stabilisation of multi-agent systems over finite fields based on high-order fully actuated system approaches. Int. J. Syst. Sci. 55 (2024), 12, 2478-2493.   DOI:10.1080/00207721.2024.2307951
  48. D. Zhang, G. Liu and L. Cao: Coordinated control of high-order fully actuated multiagent systems and its application: A predictive control strategy. IEEE/ASME Trans. Mechatronics 27 (2022), 6, 4362-4372.   DOI:10.1109/TMECH.2022.3156587
  49. D. Zhang, G. Liu and L. Cao: Proportional integral predictive control of high-order fully actuated networked multiagent systems with communication delays. IEEE Trans. Syst. Man Cybern.: Syst. 53 (2022), 2, 801-812.   DOI:10.1109/TSMC.2022.3188504
  50. D. Zhang, G. Liu and L. Cao: Constrained cooperative control for high-order fully actuated multiagent systems with application to air-bearing spacecraft simulators. IEEE/ASME Trans. Mechatron. 28 (2023), 3, 1570-1581.   DOI:10.1109/TMECH.2022.3223927