Article Contents
Article Contents

# Bounded consensus of double-integrator stochastic multi-agent systems

• * Corresponding author: Jinrong Wang
• In the framework of fixed topology and stochastic switching topologies, we study the mean-square bounded consensus(MSBC) of double-integrator stochastic multi-agent systems(SMASs) including additive system noises and communication noises. Combining algebra, graph theory and random analysis, we obtain several equivalent conditions for double-integrator SMASs to reach MSBC. In addition, the simulation examples also verify the correctness of the theoretical results.

Mathematics Subject Classification: 60H10, 93D50.

 Citation:

• Figure 1.  Topology $\mathcal{G}$

Figure 2.  The state trajectories

Figure 3.  Network topology $\{\mathcal{G}_{1}, \mathcal{G}_{2}, \mathcal{G}_{3}\}$

Figure 4.  The topological transformation diagram

Figure 5.  The state trajectories

Table 1.  The position of each agent

 Agent 0 10 20 30 40 50 1 8 98.8028 126.813 154.609 179.828 203.348 2 2 80.8029 121.441 153.733 180.202 204.014 3 5 90.3149 124.265 154.215 180.111 203.65 4 4 87.0743 123.181 153.327 179.376 203.095 5 10 78.4122 120.934 153.445 180.121 203.67 6 6 93.0754 125.01 153.793 179.501 202.966

Table 2.  The velocity of each agent

 Agent 0 10 20 30 40 50 1 80 2.4374 2.8897 2.8699 2.2523 2.6406 2 20 4.5277 3.5997 3.0241 2.2809 2.5409 3 50 3.3306 3.2226 2.9195 2.2264 2.5746 4 40 3.6902 3.3191 3.0047 2.2885 2.571 5 10 4.8084 3.6863 2.9844 2.2629 2.5924 6 60 2.9621 3.0973 2.8919 2.2493 2.6171

Table 3.  The position of each agent

 Agent 0 20 40 60 80 100 120 1 1 28.5681 50.5773 72.6362 94.757 116.347 138.372 2 2 29.2191 50.7588 73.2699 95.6504 115.763 138.011 3 3 33.0884 49.7954 73.5687 96.0563 115.574 137.825 4 4 29.7302 50.489 73.1723 95.2234 115.863 138.231 5 5 34.3277 48.9584 73.0675 94.9233 116.168 138.358

Table 4.  The velocity of each agent

 Agent 0 20 40 60 80 100 120 1 6 1.1 1.1 1.1 1.1 1.1 1.1 2 5 0.9616 1.1125 1.1042 1.1115 1.1478 1.0758 3 2 0.432 1.1781 1.1128 1.0867 1.1135 1.11 4 4 0.7428 1.1314 1.1051 1.1116 1.186 1.0625 5 0 0.2022 1.1915 1.1197 1.0935 1.1285 1.0718
•  [1] J. W. Brewer, Kronecker products and matrix calculus in system theory. Special issue on the mathematical foundations of system theory, IEEE Trans. Circuits and Systems, 25 (1978), 772-781.  doi: 10.1109/TCS.1978.1084534. [2] X. K. Cao, M. Fečkan, D. Shen and J. Wang, Iterative learning control for multi-agent systems with impulsive consensus tracking, Nonlinear Anal. Model. Control, 26 (2021), 130-150.  doi: 10.15388/namc.2021.26.20981. [3] B. B. Chang, X. W. Mu, Z. Yang and J. Y. Fang, Event-based secure consensus of muti-agent systems under asynchronous DoS attacks, Applied Mathematics and Computation, 401 (2021), 126120, 11 pp. doi: 10.1016/j.amc.2021.126120. [4] G. Chen, L. Y. Wang, C. Chen and G. Yin, Critical connectivity and fastest convergence rates of distributed consensus with switching topologies and additive noises, IEEE Trans. Automat. Control, 62 (2017), 6152-6167.  doi: 10.1109/TAC.2017.2696824. [5] Z. Y. Chen, Y. Liu, W. He, H. Qiao and H. Ji, Adaptive-neural-network-based trajectory tracking control for a nonholonomic wheeled mobile robot with velocity constraints, IEEE Transactions on Industrial Electronics, 68 (2021), 5057-5067.  doi: 10.1109/TIE.2020.2989711. [6] O. L. V. Costa and M. D. Fragoso, Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl., 179 (1993), 154-178.  doi: 10.1006/jmaa.1993.1341. [7] D. R. Ding, Z. D. Wang, Q. L. Han and G. L. Wei, Security control for discrete-time stochastic nonlinear systems subject to deception attacks, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 779-789.  doi: 10.1109/TSMC.2016.2616544. [8] L. Ding, S. Li, Y.-J. Liu, H. Gao, C. Chao and Z. Deng, Adaptive neural network-based tracking control for full-state constrained wheeled mobile robotic system, IEEE Transactions on Systems Man Cybernetics: Systems, 47 (2017), 2410-2419.  doi: 10.1109/TSMC.2017.2677472. [9] T. Dong and Y. L. Gong, Leader-following secure consensus for second-order multi-agent systems with nonlinear dynamics and event-triggered control strategy under DoS attack, Neurocomputing, 416 (2020), 95-102.  doi: 10.1016/j.neucom.2019.01.113. [10] X. Q. Feng, Y. C. Yang and D. X. Wei, Adaptive fully distributed consensus for a class of heterogeneous nonlinear multi-agent systems, Neurocomputing, 428 (2021), 12-18.  doi: 10.1016/j.neucom.2020.11.043. [11] M. D. Fragoso and O. L. V. Costa, A unified approach for stochastic and mean square stability of continuous-time linear systems with Markovian jumping parameters and additive disturbances, SIAM J. Control Optim., 44 (2005), 1165-1191.  doi: 10.1137/S0363012903434753. [12] R. Hou, L. Z. Cui, X. H. Bu and J. Q. Yang, Distributed formation control for multiple non-holonomic wheeled mobile robots with velocity constraint by using improved data-driven iterative learning, Applied Mathematics and Computation, 395 (2021), 125829, 15 pp. doi: 10.1016/j.amc.2020.125829. [13] N. Huang, Z. S. Duan and Y. Zhao, Consensus of multi-agent systems via delayed and intermittent communications, IET Control Theory and Applications, 9 (2015), 62-73.  doi: 10.1049/iet-cta.2014.0729. [14] B.-Y. Kim and H.-S. Ahn, Distributed coordination and control for a freeway traffic network using consensus algorithms, IEEE Systems Journal, 10 (2016), 162-168.  doi: 10.1109/JSYST.2014.2318054. [15] M. L. Li and F. Q. Deng, Necessary and sufficient conditions for consensus of continuous-time multiagent systems with Markovian switching topologies and communication noises, IEEE Transactions on Cybernetics, 50 (2020), 3264-3270.  doi: 10.1109/TCYB.2019.2919740. [16] S. Li, Z. Li, J. Li, T. Fernando, H. Ho-Ching Iu, Q. Wang and X. Liu, Application of event-triggered cubature Kalman filter for remote nonlinear state estimation in wireless sensor network, IEEE Transactions on Industrial Electronics, 68 (2021), 5133-5145.  doi: 10.1109/TIE.2020.2987279. [17] S. E. Li, Z. Wang, Y. Zheng, D. Yang and K. You, Stability of general linear dynamic multi-agent systems under switching topologies with positive real eigenvalues, Engineering, 6 (2020), 688-694.  doi: 10.1016/j.eng.2020.05.006. [18] X. D. Li, D. W. C. Ho and J. D. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024. [19] X. D. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2021), 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336. [20] X. D. Li, D. X. Peng and J. D. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558. [21] X. D. Li, S. J. Song and J. H. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Transactions on Automatic Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271. [22] X. D. Li, X. Y. Yang and S. J. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031. [23] X. D. Li, X. Y. Yang and J. D. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981. [24] X. D. Li, H. T. Zhu and S. J. Song, Input-to-state stability of nonlinear systems using observer-based event-triggered impulsive control, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51 (2021), 6892-6900.  doi: 10.1109/TSMC.2020.2964172. [25] Z. Y. Li and Y. G. Liu, Consensus of vehicle platoon with the hidden layer against stochastic noise, Proceedings of the 38th Chinese Control Conference, (2019). doi: 10.23919/ChiCC.2019.8865730. [26] C. Y. Liu, X. Q. Wu, X. X. Wan and J. H. Lu, Time-varying output formation tracking of heterogeneous linear multi-agent systems with dynamical controllers, Neurocomputing, 441 (2021), 36-43.  doi: 10.1016/j.neucom.2021.01.113. [27] J. Liu, Q. Shao and C. C. Hua, Consensus-based cubature information filtering for sensor networks with incomplete measurements, Neurocomputing, 364 (2019), 49-62.  doi: 10.1016/j.neucom.2019.07.030. [28] D. Luo, J. Wang and D. Shen, Consensus tracking problem for linear fractional multi-agent systems with initial state error, Nonlinear Anal. Model. Control, 25 (2020), 766-785.  doi: 10.15388/namc.2020.25.18128. [29] D. Luo, J. Wang and D. Shen, $PD^{\alpha}$-type distributed learning control for nonlinear fractional-order multi-agent systems, Math. Methods Appl. Sci., 42 (2019), 4543-4553.  doi: 10.1002/mma.5677. [30] D. Luo, J. Wang and D. Shen, Learning formation control for fractional-order multiagent systems, Math. Methods Appl. Sci., 41 (2018), 5003-5014.  doi: 10.1002/mma.4948. [31] X. Mao, Stochastic Differential Equations and Application, Second edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [32] S. X. Miao and H. S. Su, Second-order consensus of multiagent systems with matrix-weighted network, Neurocomputing, 433 (2021), 1-9.  doi: 10.1016/j.neucom.2020.12.056. [33] N. K. Mu, X. F. Liao and T. W. Huang, Consensus of second-order multi-agent systems with random sampling via event-triggered control, J. Franklin Inst., 353 (2016), 1423-1435.  doi: 10.1016/j.jfranklin.2016.01.014. [34] H. Neudecker, Some theorems on matrix differentiation with special reference to Kronecker matrix products, Journal of the American Statistical Association, 327 (2012), 953-963.  doi: 10.1080/01621459.1969.10501027. [35] H. Q. Pei and Q. Lai, Consensus of second-order multiagent systems with directed signed networks and communication delays, Complexity, 2020 (2020), 1712643, 10 pp. doi: 10.1155/2020/1712643. [36] W. Z. Qiu and J. Wang, Iterative learning control for multi-agent systems with noninstantaneous impulsive consensus tracking, Internat. J. Robust Nonlinear Control, 31 (2021), 6507-6524.  doi: 10.1002/rnc.5627. [37] W. Ren and E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, Internat. J. Robust Nonlinear Control, 17 (2007), 1002-1033.  doi: 10.1002/rnc.1147. [38] M. Sader, Z. Q. Chen, Z. X. Liu and C. Deng, Distributed robust fault-tolerant consensus control for a class of nonlinear multi-agent systems with intermittent communications, Applied Mathematics and Computation, 403 (2021), 126166, 15 pp. doi: 10.1016/j.amc.2021.126166. [39] Y. Shang, C. L. Liu and K. C. Cao, Event-triggered consensus control of nonlinear multi-agent systems based on first-order hold, International Journal of Control Automation and Systems, 19 (2021), 1461-1469.  doi: 10.1007/s12555-020-0145-y. [40] Y. C. Si and J. Wang, Relative controllability of multi-agent systems with pairwise different delays in states, Nonlinear Anal. Model. Control, 27 (2022), 289-307.  doi: 10.15388/namc.2022.27.25333. [41] F. L. Sun, C. Y. Lei and J. Kurths, Consensus of heterogeneous discrete-time multi-agent systems with noise over Markov switching topologies, Internat. J. Robust Nonlinear Control, 31 (2021), 1530-1541.  doi: 10.1002/rnc.5360. [42] H. Sun, Y. G. Liu and F. Z. Li, Distributed optimal consensus of second-order multi-agent systems, Science China-Information Sciences, 64 (2021), 209201, 3 pp. doi: 10.1007/s11432-018-9879-3. [43] S. S. Tian, Y. X. Li, Y. L. Kang and J. N. Xia, Multi-robot path planning in wireless sensor networks based on jump mechanism PSO and safety gap obstacle avoidance, Future Generation Computer Systems-The International Journal of Escience, 118 (2021), 37-47.  doi: 10.1016/j.future.2020.12.012. [44] C. Wang, C. L. Liu and S. Liu, Robust fixed-time connectivity-preserving consensus for second-order multi-agent systems with external disturbances, IET Control Theory and Applications, 14 (2020), 2674-2681.  doi: 10.1049/iet-cta.2019.1487. [45] C. Y. Wang, H. Tnunay, Z. Zuo, B. Lennox and Z. Ding, Fixed-time formation control of multirobot systems: Design and experiments, IEEE Transactions on Industrial Electronics, 66 (2019), 6292-6301.  doi: 10.1109/TIE.2018.2870409. [46] M. Wang and T. Zhang, Leader-following formation control of second-order nonlinear systems with time-varying communication delay, International Journal of Control Automation and Systems, 19 (2021), 1729-1739.  doi: 10.1007/s12555-019-0759-0. [47] Z. M. Wang, H. F. Sun, H. S. Zhang and X. Y. Liu, Bounded consensus control for stochastic multi-agent systems with additive noises, Neurocomputing, 408 (2020), 72-79.  doi: 10.1016/j.neucom.2019.11.027. [48] T. D. Wei, X. Xie and X. D. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002. [49] C. K. Wong, C. Cai and B. G. Heydecker, Adaptive traffic signal control using approximate dynamic programming, Transportation Research Part C: Emerging Technologies, 17 (2009), 456-474.  doi: 10.1016/j.trc.2009.04.005. [50] S. T. Yang and B. Yang, A semi-decentralized feudal multi-agent learned-goal algorithm for multi-intersection traffic signal control, Knowledge-Based Systems, 213 (2021), 106708, 15 pp. doi: 10.1016/j.knosys.2020.106708. [51] Y. Yang, Y. F. Li, D. Yue, Y. C. Tian and X. H. Ding, Distributed secure consensus control with event-triggering for multiagent systems under DoS attacks, IEEE Transactions on Cybernetics, 51 (2021), 2916-2928.  doi: 10.1109/TCYB.2020.2979342. [52] K. Y. You, Z. K. Li and L. H. Xie, Consensus condition for linear multi-agent systems over randomly switching topologies, Automatica, 49 (2013), 3125-3132.  doi: 10.1016/j.automatica.2013.07.024. [53] S. Yuan, C. P. Yu and J. Sun, Adaptive event-triggered consensus control of linear multi-agent systems with cyber attacks, Neurocomputing, 442 (2021), 1-9.  doi: 10.1016/j.neucom.2021.02.040. [54] H. Zhang, X. Zhou, H. Yan and J. Sun, Adaptive consensus-based distributed target tracking with dynamic cluster in sensor networks, IEEE Transactions on Cybernetics, 49 (2019), 1580-1591.  doi: 10.1109/TCYB.2018.2805717. [55] J. Zhang, H. G. Zhang, Y. L. Cai and W. H. Li, Containment control of general linear multi-agent systems by event-triggered control mechanisms, Neurocomputing, 433 (2021), 263-274.  doi: 10.1016/j.neucom.2020.11.008. [56] Y. Zhang and Y. P. Tian, Consentability and protocol design of multi agent systems with stochastic switching topology, Automatica J. IFAC, 45 (2009), 1195-1201.  doi: 10.1016/j.automatica.2008.11.005. [57] Y. Y. Zhang, R. F. Li and X. M. Huo, Stochastic consensus of discrete-time second-order multi-agent systems with measurement noises and time delays, Journal of the Franklin Institute, 355 (2018), 2791-2807.  doi: 10.1016/j.jfranklin.2018.01.015. [58] X. F. Zong, T. Li and J. F. Zhang, Consensus conditions of continuous-time multi-agent systems with additive and multiplicative measurement noises, SIAM Journal on Control and Optimization, 56 (2018), 19-52.  doi: 10.1137/15M1019775. [59] X. F. Zong, T. Li and J. F. Zhang, Consensus of nonlinear multi-agent systems with multiplicative noises and time-varying delays, IEEE Conference on Decision and Control, (2018), 5415-5420.  doi: 10.1109/CDC.2018.8618969.

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