Skip to main content
SpringerLink
Log in

Distributed multiple-bipartite consensus in networked Lagrangian systems with cooperative–competitive interactions

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In combination with the collective behavior evolution of bipartite consensus and cluster/group consensus, this paper proposes the notion of multiple-bipartite consensus in networked Lagrangian systems (NLSs), effectively integrating the above emergent collective behaviors in cooperative–competitive networks. The problems of leaderless multiple-bipartite consensus and leader-following multiple-bipartite tracking consensus are solved via deploying the designed distributed adaptive torque controllers for uncertain NLSs. Compared with the traditional bipartite consensus framework, antagonistic interactions can exist in the same subnetwork. By introducing an acyclic partition and adding the integral item in the distributed adaptive torque control protocols, the explicit expressions of the final states are eventually obtained in the leaderless case. Moreover, the leader-following case can be realized in finite time resorting to two predefined estimators embedded in the scheme. The effectiveness of two scenarios has been illustrated through numerical simulations with ten heterogeneous mechanical manipulators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data availability

The data that support the findings of this study are available on request from the corresponding author (E-mail: lihengyu@shu.edu.cn). The data are not publicly available due to their containing information that could compromise research participant privacy/consent.

References

  1. Hu, J., Wu, Y., Li, T., Ghosh, B.K.: Consensus control of general linear multiagent systems with antagonistic interactions and communication noises. IEEE Trans. Autom. Control 64(5), 2122–2127 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Wen, G., Wang, H., Yu, X., Yu, W.: Bipartite tracking consensus of linear multi-agent systems with a dynamic leader. IEEE Trans. Circuits Syst. II. Exp. Briefs 65(9), 1204–1208 (2018)

    Google Scholar 

  3. Bai, L., Shi, L., Chen, H., Shao, J., Cheng, Y.: Opinion dynamics driven under leadership in cooperation-competition social networks. IEEE 16th International Conference on Control and Automation (ICCA), (Virtual) Sapporo, Hokkaido, Japan. 9264461, pp. 141–146 (2020)

  4. Wasserman, S., Faust, K.: Social network analysis: Methods and applications. Cambridge University Press, Cambridge, U.K. (1994)

    MATH  Google Scholar 

  5. Dong, X., Tang, D., Tang, C.: Social network analysis in China’s hospital healthcare. Physica A 565, 1255461 (2021)

    Google Scholar 

  6. Hu, J., Zheng, W.: Emergent collective behaviors on coopetition networks. Phys. Lett. A 378(26–27), 1787–1796 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Bechlioulis, C.P., Demetriou, M.A., Kyriakopoulos, K.J.: A distributed control and parameter estimation protocol with prescribed performance for homogeneous lagrangian multi-agent systems. Auton. Robot 42, 1525–1541 (2018)

    Google Scholar 

  8. Li, X., Su, H., Chen, M.Z.: Flocking of networked Euler-Lagrange systems with uncertain parameters and time-delays under directed graphs. Nonlin. Dyn. 85(1), 415–424 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Abdessameud, A., Tayebi, A., Polushin, I.G.: Leader-follower synchronization of Euler-Lagrange systems with time-varying leader trajectory and constrained discrete-time communication. IEEE Trans. Autom. Control 62(5), 2539–2545 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Zheng, Y., Zhao, Q., Ma, J., Wang, L.: Second-order consensus of hybrid multi-agent systems. Syst. Control Lett. 125, 51–58 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Yuan, C.: Leader-following consensus control of general linear multi-agent systems with diverse time-varying input delays. J. Dyn. Syst. Meas. Control-Trans. ASME 140(6), 061010 (2018)

    Google Scholar 

  12. Klotz, J.R., Kan, Z., Shea, J.M., Pasiliao, E.L., Dixon, W.E.: Asymptotic synchronization of a leader-follower network of uncertain Euler-Lagrange systems. IEEE Trans. Control Netw. Syst. 2(2), 174–182 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Ao, W., Huang, J., Cui, Q., Sanchez, R.V.: Finite-time leaderless consensus control of a group of Euler-Lagrangian systems with backlash nonlinearities. J. Frankl. Inst. 356(16), 9286–9301 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Chen, J., Chen, B., Zeng, Z., Jiang, P.: Event-triggered synchronization strategy for multiple neural networks with time delay. IEEE Trans. Cybern. 50(7), 3271–3280 (2020)

    Google Scholar 

  15. Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Yu, J., Yu, S.: An improved fixed-time bipartite flocking protocol for nonlinear multi-agent systems. Int. J. Control. (2020). https://doi.org/10.1080/00207179.2020.1828623

    Article  Google Scholar 

  17. Xiao, Q., Liu, H., Wang, X., Huang, Y.: A note on the fixed-time bipartite flocking for nonlinear multi-agent systems. Appl. Math. Lett. 99, 105973 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Zhu, Y., Li, S., Ma, J., Zheng, Y.: Bipartite consensus in networks of agents with antagonistic interactions and quantization. IEEE Trans. Circuits Syst. II. Exp. Briefs 65(12), 2012–2016 (2018)

  19. Liu, F., Song, Q., Wen, G., Lu, J., Cao, J.: Bipartite synchronization of Lur’e network under signed digraph. Int. J. Robust Nonlinear Control 28, 6087–6105 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Ning, B., Han, Q., Zuo, Z.: Bipartite consensus tracking for second-order multi-agent Systems: A time-varying function based preset-time approach. IEEE Trans. Autom. Control 66(6), 2739–2745 (2021)

  21. Meng, D.: Bipartite containment tracking of signed networks. Automatica 79, 282–289 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Zhang, X., Wang, X.: Bipartite consensus of multi-agent systems with reduced-order observer-based distributed control protocols. Int. J. Syst. Sci. (2021). https://doi.org/10.1080/00207721.2021.1882612

  23. Zuo, S., Lewis, F., Song, Y., Davoudi, A.: Bipartite output synchronization of heterogeneous multiagent systems on signed digraphs. Int. J. Robust Nonlinear Control 28, 4017–4031 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Zhang, H., Cai, Y., Wang, Y., Su, H.: Adaptive bipartite event-triggered output consensus of heterogeneous linear multiagent systems under fixed and switching topologies. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 4816–4830 (2020)

    MathSciNet  Google Scholar 

  25. Peng, Z., Hu, J., Shi, K., Luo, R., Huang, R., Ghosh, B.K., Huang, J.: A novel optimal bipartite consensus control scheme for unknown multi-agent systems via model-free reinforcement learning. Appl. Math. Comput. 369, 124821 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Shams, A., Rehan, M., Tufail, M.: (2021) \(\text{ H}_{\infty }\) bipartite consensus of nonlinear multi-agent systems over a directed signed graph with a leader of non-zero input. Int. J. Control (2021). https://doi.org/10.1080/00207179.2021.1888157

    Article  Google Scholar 

  27. Hu, J., Wu, Y.: Interventional bipartite consensus on coopetition networks with unknown dynamics. J. Frankl. Inst. 354(11), 4438–4456 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Zhou, Q., Wang, W., Liang, H., Basin, M.V., Wang, B.: Observer-based event-triggered fuzzy adaptive bipartite containment control of multiagent systems with input quantization. IEEE Trans. Fuzzy Syst. 29(2), 372–384 (2021)

    Google Scholar 

  29. Lui, D.G., Petrillo, A., Santini, S.: An optimal distributed PID-like control for the output containment and leader-following of heterogeneous high-order multi-agent systems. Inf. Sci. 541(1), 166–184 (2020)

    MathSciNet  Google Scholar 

  30. Lui, D.G., Petrillo, A., Santini, S.: Distributed model reference adaptive containment control of heterogeneous multi-agent systems with unknown uncertainties and directed topologies. J. Frankl. Inst. 358(1), 737–756 (2021)

    MathSciNet  MATH  Google Scholar 

  31. Zhang, X., Liu, X., Feng, Z.: Distributed containment control of singular heterogeneous multi-agent systems. J. Frankl. Inst. 357(3), 1378–1399 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Haghshenas, H., Badamchizadeh, M.A., Baradarannia, M.: Containment control of heterogeneous linear multi-agent systems. Automatica 54, 210–216 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Yang, Q., Lyu, L., Li, X., Chen, C., Lewis, F.L.: Adaptive distributed synchronization of heterogeneous multi-Agent systems over directed graphs with time-Varying edge weights. J. Frankl. Inst. 358(4), 2434–2452 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Qin, J., Yu, C.: Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition. Automatica 49(9), 2898–2905 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Morǎrescu, I.C., Martin, S., Girard, A., Muller-Gueudin, A.: Coordination in networks of linear impulsive agents. IEEE Trans. Autom. Control 61(9), 2402–2415 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Ji, L., Yu, X., Li, C.: Group consensus for heterogeneous multiagent systems in the competition networks with input time delays. IEEE Trans. Syst. Man Cybern. 50(11), 4655–4663 (2020)

    Google Scholar 

  37. Wang, Y., Ma, Z., Chen, G.: Avoiding congestion in cluster consensus of the second-order nonlinear multiagent systems. IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3490–3498 (2018)

    MathSciNet  Google Scholar 

  38. Mirzaei, M., Atrianfar, H., Mehdipour, N., Abdollahi, F.: Asynchronous consensus of continuous-time Lagrangian systems with switching topology and non-uniform time delay. Robot. Auton. Syst. 83, 106–114 (2016)

    Google Scholar 

  39. Yao, X., Ding, H., Ge, M., Park, J.H.: Event-triggered synchronization control of networked Euler-Lagrange systems without requiring relative velocity information. Inform. Sci. 508, 183–199 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Dong, Y., Chen, J.: Adaptive control for rendezvous problem of networked uncertain Euler-Lagrange systems. IEEE Trans. Cybern. 49(6), 2190–2199 (2019)

    MathSciNet  Google Scholar 

  41. Ghapania, S., Mei, J., Ren, W., Song, Y.: Fully distributed flocking with a moving leader for Lagrange networks with parametric uncertainties. Automatica 67, 67–76 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Zhan, J., Li, X.: Cluster consensus in networks of agents with weighted cooperative-competitive interactions. IEEE Trans. Circuits Syst. II. Exp. Briefs 65(2), 241–245 (2018)

    MathSciNet  Google Scholar 

  43. Hu, H., Wen, G., Yu, W., Cao, J., Huang, T.: Finite-time coordination behavior of multiple Euler-Lagrange systems in cooperation-competition networks. IEEE Trans. Cybern. 49(8), 2967–2979 (2019)

    Google Scholar 

  44. Liu, J., Li, H., Luo, J.: Bipartite consensus in networked Euler-Lagrange systems with uncertain parameters under a cooperation-competition network topology. IEEE Control Syst. Lett. 3(3), 494–498 (2019)

    MathSciNet  Google Scholar 

  45. Wu, Y., Ge, M., Ding, T., Chen, C., Ling, G.: Task-space bipartite tracking of networked robotic systems via hierarchical finite-time control. Nonlin. Dyn. 100(4), 3469–3483 (2020)

    Google Scholar 

  46. Hu, H., Wen, G., Yu, W., Xuan, Q., Chen, G.: Swarming behavior of multiple Euler-Lagrange systems with cooperation-competition interactions: An auxiliary system approach. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5726–5737 (2018)

    MathSciNet  Google Scholar 

  47. Zhao, L., Wang, R., Li, W., Wu, Q.: Cluster oscillatory synchronization of networked Lagrangian systems with the distributed adaptive observers. Nonlin. Dyn. 93, 1219–1230 (2018)

    MATH  Google Scholar 

  48. Zhao, L., Li, W., Wei, F., Wang, J., Bo, H.: Coordinated motion of Lagrangian systems with auxiliary oscillators under cooperative and cooperative-competitive interactions. Nonlin. Dyn. 100, 2415–2426 (2020)

  49. Miao, Z., Yu, J., Ji, J., Zhou, J.: Multi-objective region reaching control for a swarm of robots. Automatica 103, 81–87 (2019)

  50. Liu, J., Ji, J., Zhou, J., Xiang, L., Zhao, L.: Adaptive group consensus in uncertain networked Euler-Lagrange systems under directed topology. Nonlin. Dyn. 82(3), 1145–1157 (2015)

  51. Wu, W., Zhou, W., Chen, T.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I. Regul. Pap. 56(4), 829–839 (2009)

  52. Liu, J., Li, H., Ji, J., Luo, J.: Group-bipartite consensus in the networks with cooperative-competitive interactions. IEEE Trans. Circuits. Syst. II. Exp. Briefs 67(12), 3292–3296 (2020)

    Google Scholar 

  53. Cao, Y., Ren, W., Meng, Z.: Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking. Syst. Control Lett. 59(9), 522–529 (2010)

    MathSciNet  MATH  Google Scholar 

  54. Mei, J., Ren, W., Ma, G.: Distributed containment control for Lagrangian networks with parametric uncertainties under a directed graph. Automatica 48, 653–659 (2012)

    MathSciNet  MATH  Google Scholar 

  55. Sakthivel, R., Parivallal, A., Kaviarasan, B., Lee, H., Lim, Y.: Finite-time consensus of Markov jumping multi-agent systems with time-varying actuator faults and input saturation. ISA Trans. 83, 89–99 (2018)

    Google Scholar 

  56. Kaviarasan, B., Sakthivel, R., Li, Y., Zhao, D., Ren, Y.: Non-fragile control protocol for finite-time consensus of stochastic multi-agent systems with input time-varying delay. Int. J. Mach. Learn. Cybern. 11(2), 325–337 (2020)

    Google Scholar 

  57. Cheah, C.C., Liu, C., Slotine, J.J.E.: Adaptive tracking control for robots with unknown kinematic and dynamic properties. Int. J. Robot. Res. 25(3), 283–296 (2006)

    Google Scholar 

  58. Ge, M., Guan, Z., Yang, C., Chen, C., Zheng, D., Chi, M.: Task-space coordinated tracking of multiple heterogeneous manipulators via controller-estimator approaches. J. Frankl. Inst. 353(15), 3722–3738 (2016)

    MathSciNet  MATH  Google Scholar 

  59. Ning, B., Han, Q., Zuo, Z.: Practical fixed-time consensus for integrator-type multi-agent systems: A time base generator approach. Automatica 105, 406–414 (2019)

Download references

Acknowledgements

This study was funded by the National Natural Science Foundation of China (Grant Numbers 62073209, 61625304, and 61703181) and the Natural Science Foundation of Shandong Province (Grant Number ZR2020KA005).

Author information

Authors and Affiliations

  1. School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, 200072, China

    Tiehui Zhang, Hengyu Li, Jun Liu, Shaorong Xie & Jun Luo

  2. College of Elementary Education, Jining University, Qufu, 273155, Shandong Province, China

    Tiehui Zhang

  3. Department of Mathematics, Jining University, Qufu, 273155, Shandong Province, China

    Jun Liu & Daowei Lu

Authors
  1. Tiehui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hengyu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daowei Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shaorong Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jun Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hengyu Li.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, T., Li, H., Liu, J. et al. Distributed multiple-bipartite consensus in networked Lagrangian systems with cooperative–competitive interactions. Nonlinear Dyn 106, 2229–2244 (2021). https://doi.org/10.1007/s11071-021-06674-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06674-y

Keywords

Search

Navigation