# Abstract:

In this paper, a novel consensus algorithm is presented to handle with the leader-following consensus problem for lower-triangular nonlinear MASs (multi-agent systems) with unknown controller and measurement sensitivities under a given undirected topology. As distinguished from the existing results, the proposed consensus algorithm can tolerate to a relative wide range of controller and measurement sensitivities. We present some important matrix inequalities, especially a class of matrix inequalities with multiplicative noises. Based on these results and a dual-domination gain method, the output consensus error with unknown measurement noises can be used to construct the compensator for each follower directly. Then, a new distributed output feedback control is designed to enable the MASs to reach consensus in the presence of large controller perturbations. In view of a Lyapunov function, sufficient conditions are presented to guarantee that the states of the leader and followers can achieve consensus asymptotically. In the end, the proposed consensus algorithm is tested and verified by an illustrative example.

# Keywords:

output feedback, consensus, lower-triangular, nonlinear multi-agent systems, measurement noises, controller sensitivity

93C10

# References:

1. M. S. Ali, R. Agalya, V. Shekher and Y. H. Joo: Non-fragile sampled data control for stabilization of non-linear multi-agent system with additive time varying delays, Markovian jump and uncertain parameters. Nonlinear Analysis: Hybrid Systems 36 (2020), 100830.   DOI:10.1016/j.nahs.2019.100830
2. A. Amini, A. Azarbahram and M. Sojoodi: ${H_\infty }$ Consensus of nonlinear multi-agent systems using dynamic output feedback controller: an LMI approach. Nonlinear Dynamics 85 (2016), 3, 1865-1886.   DOI:10.1007/s11071-016-2801-6
3. A. Bidram, A. Davoudi, F. L. Lewis and J. M. Guerrero: Distributed cooperative secondary control of microgrids using feedback linearization. IEEE Trans. Power Systems 28 (2013), 3, 3462-3470.   DOI:10.1109/tpwrs.2013.2247071
4. H. Cai, G. Hu, F. L. Lewis and A. Davoudi: A distributed feedforward approach to cooperative control of AC microgrids. IEEE Trans. Power Systems 31 (2016), 5, 4057-4067.   DOI:10.1109/tpwrs.2015.2507199
5. M. Chen, B. Jiang and W. W. Guo: Fault-tolerant control for a class of non-linear systems with dead-zone. Int. J. Systems Science 47 (2016), 7, 1689-1699.   DOI:10.1080/00207721.2014.945984
6. C. C. Chen, C. Qian, Z. Y. Sun and Y. W. Liang: Global output feedback stabilization of a class of nonlinear systems with unknown measurements sensitivity. IEEE Trans. Automat. Control 63 (2018), 7, 2212-2217.   DOI:10.1109/tac.2017.2759274
7. G. Chen and Y. D. Song: Robust fault-tolerant cooperative control of multi-agent systems: A constructive design method. J. Franklin Inst. 352 (2015), 10, 4045-4066.   DOI:10.1016/j.jfranklin.2015.05.031
8. J. Chen, W. Zhang, Y. Y. Cao and H. Chu: Observer-based consensus control against actuator faults for linear parameter-varying multiagent systems. IEEE Trans. Systems Man Cybernet.: Systems 47 (2017), 7, 1336-1347.   DOI:10.1109/tsmc.2016.2587300
9. C. Deng, D. Zhang and G. Feng: Resilient practical cooperative output regulation for MASs with unknown switching exosystem dynamics under DoS attacks. Automatica 139 (2022), 110172.   DOI:10.1016/j.automatica.2022.110172
10. H. Du and R. Jia: Synchronization of a class of nonlinear multi-agent systems with sampled-data information. Nonlinear Dynamics 82 (2015), 3, 1483-1492.   DOI:10.1007/s11071-015-2255-2
11. 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
12. Z. Guo, H. Xue and Y. Pan: Neural networks-based adaptive tracking control of multi-agent systems with output-constrained and unknown hysteresis. Neurocomputing 458 (2021), 24-32.   DOI:10.1016/j.neucom.2021.05.079
13. C. C. Hua, K. Li and X. P. Guan: Semi-global/global output consensus for nonlinear multiagent systems with time delays. Automatica 103 (2019), 480-489.   CrossRef
14. C. C. Hua, X. You and X. P. Guan: Leader-following consensus for a class of high-order nonlinear multi-agent systems. Automatica 73 (2016), 138-144.   DOI:10.1016/j.automatica.2016.06.025
15. B. Kaviarasan, O. M. Kwon, M. J. Park and R. Sakthivel: Stochastic faulty estimator-based non-fragile tracking controller for multi-agent systems with communication delay. Appl. Math. Comput. 392 (2021), 125704.   DOI:10.1016/j.amc.2020.125704
16. B. Kaviarasan, R. Sakthivel, Y. Li, D. Zhao and Y. Ren: Non-fragile control protocol for finite-time consensus of stochastic multi-agent systems with input time-varying delay. Int. J. Machine Learning Cybernet. 11 (2020), 2, 325-337.   DOI:10.1007/s13042-019-00976-9
17. K. H. Khalil: Nonlinear Systems. (Third edition.) Prentice-Hall, Upper Saddle River, NJ 2002.   CrossRef
18. M. S. Koo and H. L. Choi: Output feedback regulation of a class of lower triangular nonlinear systems with arbitrary unknown measurement sensitivity. Int. J. Control Automat. Systems 18 (2020), 9, 2186-2194.   DOI:10.1007/s12555-019-0721-1
19. M. Krstic and H. Deng: Stabilization of Uncertain Nonlinear Systems. Springer, New York 1998.   CrossRef
20. H. Lei and W. Lin: Universal adaptive control of nonlinear systems with unknown growth rate by output feedback. Automatica 42 (2006), 10, 1783-1789.   DOI:10.1016/j.automatica.2006.05.006
21. H. Lei and W. Lin: Adaptive regulation of uncertain nonlinear systems by output feedback: A universal control approach. Systems Control Lett. 56 (2007), 7-8, 529-537.   DOI:10.1016/j.sysconle.2007.03.002
22. K. Li, C. C. Hua, X. You and C. K. Ahn: Leader-following asynchronous consensus for multiagent systems with unknown control and output directions. Automatica 132 (2021), 109832.   DOI:10.1016/j.automatica.2021.109832
23. K. Li, C .C. Hua, X. You and X. P. Guan: Output feedback-based consensus control for nonlinear time delay multiagent systems. Automatica 111 (2020), 108669.   DOI:10.1016/j.automatica.2019.108669
24. K. Li, C. C. Hua, X. You and X. P. Guan: Distributed output-feedback consensus control of multiagent systems with unknown output measurement sensitivity. IEEE Trans. Automat. Control 66 (2021), 7, 3303-3310.   DOI:10.1109/tac.2020.3017711
25. H. Li, Q. Liu, G. Feng and X. Zhang: Leader-follower consensus of nonlinear time-delay multiagent systems: A time-varying gain approach. Automatica 126 (2021), 109444.   DOI:10.1016/j.automatica.2020.109444
26. C. Li, S. Tong and W. Wang: Fuzzy adaptive high-gain-based observer backstepping control for SISO nonlinear systems. Inform. Sci. 181 (2011), 11, 2405-2421.   DOI:10.1016/j.ins.2011.01.040
27. W. Li, X. Yao and M. Krstic: Adaptive-gain observer-based stabilization of stochastic strict-feedback systems with sensor uncertainty. Automatica 120 (2020), 109112.   DOI:10.1016/j.automatica.2020.109112
28. C. Liu, X. Yue and Z. Yang: Are nonfragile controllers always better than fragile controllers in attitude control performance of post-capture flexible spacecraft?. Aerospace Sci. Technol. 118 (2021), 107053.   DOI:10.1016/j.ast.2021.107053
29. H. Ma, Z. Wang, D. Wang, D. Liu, P. Yan and Q. Wei: Neural-network-based distributed adaptive robust control for a class of nonlinear multiagent systems with time delays and external noises. IEEE Trans. Systems Man Cybernet.: Systems 46 (2016), 6, 750-758.   DOI:10.1109/tsmc.2015.2470635
30. W. Ni and D. Cheng: Leader-following consensus of multi-agent systems under fixed and switching topologies. Systems Control Lett. 59 (2010), 3-4, 209-217.   DOI:10.1016/j.sysconle.2010.01.006
31. S. K. Pandey, S. R. Mohanty and N. Kishor: A literature survey on load-frequency control for conventional and distribution generation power systems. Renewable Sustainable Energy Rev. 25 (2013), 318-334.   DOI:10.1016/j.rser.2013.04.029
32. C. Qian and W. Lin: Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control 47 (2002), 10, 1710-1715.   DOI:10.1109/tac.2002.803542
33. W. Ren and N. Sorensen: Distributed coordination architecture for multi-robot formation control. Robotics Autonomous Syst. 56 (2008), 4, 324-333.   DOI:10.1016/j.robot.2007.08.005
34. G. Rigatos, P. Siano and N. Zervos: Sensorless control of distributed power generators with the derivative-free nonlinear kalman filter. IEEE Trans. Industr. Electronics 61 (2014), 11, 6369-6382.   DOI:10.1109/tie.2014.2300069
35. R. Sakthivel, A. Parivallal, N. H. Tuan and S. Manickavalli: Nonfragile control design for consensus of semi-Markov jumping multiagent systems with disturbances. Int. J. Adaptive Control Signal Process. 35 (2021), 6, 1039-1061.   DOI:10.1002/acs.3245
36. Z. Y. Sun, J. W. Xing and Q. Meng: Output feedback regulation of time-delay nonlinear systems with unknown continuous output function and unknown growth rate. Nonlinear Dynamics 100 (2020), 2, 1309-1325.   DOI:10.1007/s11071-020-05552-3
37. X. H. Wang and H. B. Ji: Leader-follower consensus for a class of nonlinear multi-agent systems. Int. J. Control Automat. Systems 10 (2012), 1, 27-35.   DOI:10.1007/s12555-012-0104-3
38. W. Wang, C. Wen and J. Huang: Distributed adaptive asymptotically consensus tracking control of nonlinear multi-agent systems with unknown parameters and uncertain disturbances. Automatica 77 (2017), 133-142.   DOI:10.1016/j.automatica.2016.11.019
39. R. Xu, X. Wang and Y. Zhou: Observer-based event-triggered adaptive containment control for multiagent systems with prescribed performance. Nonlinear Dynamics 107 (2022), 3, 2345-2362.   DOI:10.1007/s11071-021-07064-0
40. B. Yang and W. Lin: Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback. IEEE Trans. Automat. Control 49 (2004), 7, 1069-1080.   DOI:10.1109/tac.2004.831186
41. G. Yu and Y. Shen: Relative-output-based consensus for nonlinear multi-agent systems with unknown measurement sensitivities. Neurocomputing 382 (2020), 21-31.   DOI:10.1016/j.neucom.2019.11.082
42. Y. Yuan, Z. Wang, P. Zhang and H. Dong: Nonfragile near-optimal control of stochastic time-varying multiagent systems with control- and state- dependent noises. IEEE Trans. Cybernet. 49 (2019), 7, 2605-2617.   DOI:10.1109/tcyb.2018.2829713
43. C. Zhang, L. Chang and X. Zhang: Leader-follower consensus of upper-triangular nonlinear multi-agent systems. IEEE/CAA J. Automat. Sinica 1 (2014), 2, 210-217.   DOI:10.1109/jas.2014.7004552
44. D. Zhang, C. Deng and G. Feng: Resilient cooperative output regulation for nonlinear multi-agent systems under DoS attacks. IEEE Trans. Automat. Control (2022), 1-8.   DOI:10.1109/tac.2022.3184388
45. J. Zhang, D. Qi and G. Zhao: A new game model for distributed optimization problems with directed communication topologies. Neurocomputing 148 (2015), 278-287.   DOI:10.1016/j.neucom.2013.11.054
46. J. Zhang, J. Song, J. Li, F. Han and H. Zhang: Observer-based non-fragile ${H_\infty }$-consensus control for multi-agent systems under deception attacks. Int. J. Systems Sci. 52 (2021), 6, 1223-1236.   DOI:10.1080/00207721.2021.1884917
47. T. Zhou, Q. Liu and W. Wang: Non-fragile disturbance observer-based containment control of multi-agent systems over switching topologies. IEEE Access 9 (2021), 123430-123437.   DOI:10.1109/access.2021.3109984