American Institute of Mathematical Sciences

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April  2021, 14(4): 1535-1551. doi: 10.3934/dcdss.2020396

Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances

Ke Yang 1, , Wencheng Zou 1, , Zhengrong Xiang 1,, and  Ronghao Wang 2,

1. 

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
2. 

National Defense Engineering College, Army Engineering University, Nanjing 210007, China

* Corresponding author: Zhengrong Xiang

Received  October 2019 Revised  April 2020 Published  April 2021 Early access  June 2020

Fund Project: This work was supported by the National Natural Science Foundation of China number 61873128 and 61603414

In this paper, the distributed consensus problem is investigated for a class of higher-order nonlinear multi-agent systems with unmatched disturbances. By the back-stepping technique, a new distributed protocol is designed to solve the consensus problem for multi-agent systems without using the information of the Laplacian matrix and Lipschitz constants. It is proved that the practical consensus of multi-agent systems with unmatched disturbances can be achieved by the proposed protocol. Finally, the validity of the proposed scheme is verified by a simulation.

Keywords: Fully distributed consensus, higher-order nonlinear systems, multi-agent systems, unmatched disturbances, back-stepping.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396
References:
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H. B. Du, Y. G. He and Y. Y. Cheng, Finite-time synchronization of a class of second-order nonlinear multi-agent systems using output feedback control, IEEE Transactions on Circuits and Systems I: Regular Papers, 61 (2014), 1778-1788.  doi: 10.1109/TCSI.2013.2295012.  Google Scholar

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H. B. Du, G. H. Wen, G. R. Chen, J. D. Cao and F. E. Alsaadi, A distributed finite-time consensus algorithm for higher-order leaderless and leader-following multiagent systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1625-1634.  doi: 10.1109/TSMC.2017.2651899.  Google Scholar

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M.-M. Duan, C.-L. Liu and F. Liu, Event-triggered consensus seeking of heterogeneous first-order agents with input delay, IEEE Access, 5 (2017), 5215-5223.  doi: 10.1109/ACCESS.2017.2696026.  Google Scholar

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K. Fathian, T. H. Summers and N. R. Gans, Robust distributed formation control of agents with higher-order dynamics, IEEE Control Systems Letters, 2 (2018), 495-500.  doi: 10.1109/LCSYS.2018.2841941.  Google Scholar

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G. X. Gu, L. Marinovici and F. L. Lewis, Consensusability of discrete-time dynamic multiagent systems, IEEE Transactions on Automatic Control, 57 (2012), 2085-2089.  doi: 10.1109/TAC.2011.2179431.  Google Scholar

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G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, 1952.  Google Scholar

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B. Hou, F. C. Sun, H. B. Li and G. B. Liu, Consensus of second-order multi-agent systems with time-varying delays and antagonistic interactions, Tsinghua Science and Technology, 20 (2015), 205-211.  doi: 10.1109/TST.2015.7085634.  Google Scholar

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H. H. Ji, H. T. Zhang, Z. Y. Ye, H. Zhang, B. W. Xu and G. R. Chen, Stochastic consensus control of second-order nonlinear multiagent systems with external disturbances, IEEE Transactions on Control of Network Systems, 5 (2018), 1585-1596.  doi: 10.1109/TCNS.2017.2736959.  Google Scholar

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Z. K. Li, Z. S. Duan and F. L. Lewis, Distributed robust consensus control of multi-agent systems with heterogeneous matching uncertainties, Automatica J. IFAC, 50 (2014), 883-889.  doi: 10.1016/j.automatica.2013.12.008.  Google Scholar

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H. Q. Li, X. F. Liao, T. W. Huang, W. Zhu and Y. B. Liu, Second-order global consensus in multiagent networks with random directional link failure, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 565-575.  doi: 10.1109/TNNLS.2014.2320274.  Google Scholar

[13]

G. P. Li, X. Y. Wang and S. H. Li, Finite-time output consensus of higher-order multiagent systems with mismatched disturbances and unknown state elements, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 2751-2581.  doi: 10.1109/TSMC.2017.2759095.  Google Scholar

[14]

G. P. Li, X. Y. Wang and S. H. Li, Distributed composite output consensus protocols of higher-order multi-agent systems subject to mismatched disturbances, IET Control Theory and Applications, 11 (2017), 1162-1172.  doi: 10.1049/iet-cta.2016.0814.  Google Scholar

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[16]

C.-L. Liu, L. Shan, Y.-Y. Chen and Y. Zhang, Average-consensus filter of first-order multi-agent systems with disturbances, IEEE Transactions on Circuits and Systems II: Express Briefs, 65 (2018), 1763-1767.  doi: 10.1109/TCSII.2017.2762723.  Google Scholar

[17]

C. Q. Ma, T. Li and J. F. Zhang, Consensus control for leader-following multi-agent systems with measurement noise, Journal of Systems Science and Complexity, 23 (2010), 35-49.  doi: 10.1007/s11424-010-9273-4.  Google Scholar

[18]

Z. Y. Meng, W. Ren and Z. You, Distributed finite-time attitude containment control for multiple rigid bodies, Autonmatica, 46 (2010), 2092-2099.  doi: 10.1016/j.automatica.2010.09.005.  Google Scholar

[19]

S. Mondal and R. Su, Disturbance observer based consensus control for higher order multi-agent systems with mismatched uncertainties, 2016 American Control Conference, (2016), 2826–2831. doi: 10.1109/ACC.2016.7525347.  Google Scholar

[20]

C. Peng, J. Zhang and Q.-L. Han, Consensus of multiagent systems with nonlinear dynamics using an integrated sampled-data-based event-triggered communication scheme, IEEE Transactions on Systems, Man and Cybernetics: Systems, 49 (2019), 589-599.  doi: 10.1109/TSMC.2018.2814572.  Google Scholar

[21]

Z. R. Qiu, L. H. Xie and Y. G. Hong, Quantized leaderless and leader-following consensus of high-order multi-agent systems with limited data rate, IEEE Transactions on Automatic Control, 61 (2016), 2432-2447.  doi: 10.1109/TAC.2015.2495579.  Google Scholar

[22]

M. Rehan, A. Jameel and C. K. Ahn, Distributed consensus control of one-sided Lipschitz nonlinear multiagent systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1297-1308.  doi: 10.1109/TSMC.2017.2667701.  Google Scholar

[23]

W. Ren and E. Atkins, Distributed multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control, 50 (2005), 655-661.   Google Scholar

[24]

C. Ren, Z. Shi and T. Du, Distributed observer-based leader-following consensus control for second-order stochastic multi-agent systems, IEEE Access, 6 (2018), 20077-20084.  doi: 10.1109/ACCESS.2018.2820813.  Google Scholar

[25]

L. N. Rong, J. W. Lu, S. Y. Xu and Y. M. Chu, Reference model-based containment control of multi-agent systems with higher-order dynamics, IET Control Theory and Applications, 8 (2014), 796-802.  doi: 10.1049/iet-cta.2013.0148.  Google Scholar

[26]

P. Shi and Q. Shen, Cooperative control of multi-agent systems with unknown state-dependent controlling effects, IEEE Transactions on Automation Science and Engineering, 12 (2015), 827-834.   Google Scholar

[27]

S. Z. Su and Z. L. Lin, Distributed synchronization control of multi-agent systems with switching directed communication topologies and unknown nonlinearities, 2015 54th IEEE Conference on Decision and Control, (2015), 5444–5449. doi: 10.1109/CDC.2015.7403072.  Google Scholar

[28]

X. H. Wang, Y. G. Hong and H. B. Ji, Distributed optimization for a class of nonlinear multiagent systems with disturbance rejection, IEEE Transactions on Cybernetics, 46 (2016), 1655-1666.  doi: 10.1109/TCYB.2015.2453167.  Google Scholar

[29]

X. Y. Wang, S. H. Li, X. H. Yu and J. Yang, Distributed active anti-disturbance consensus for leader-follower higher-order multi-agent systems with mismatched disturbances, IEEE Transactions on Automatic Control, 62 (2017), 5795-5801.  doi: 10.1109/TAC.2016.2638966.  Google Scholar

[30]

J. H. Wang, Y. L. Xu, Y. Xu and D. D. Yang, Time-varying formation for high-order multi-agent systems with external disturbances by event-triggered integral sliding mode control, Appl. Math. Comput., 359 (2019), 333-343.  doi: 10.1016/j.amc.2019.04.066.  Google Scholar

[31]

Z. Wu, Y. Xu, Y. Pan, P. Shi and Q. Wang, Event-triggered pinning control for consensus of multiagent systems with quantized information, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1929-1938.   Google Scholar

[32]

D. Yang, X. D. Li and J. L. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar

[33]

Z. Y. Yu, H. J. Jiang and C. Hu, Leader-following consensus of fractional-order multi-agent systems under fixed topology, Neurocomputing, 149 (2015), 613-620.  doi: 10.1016/j.neucom.2014.08.013.  Google Scholar

[34]

C. Zhang, Distributed ESO based cooperative tracking control for high-order nonlinear multiagent systems with lumped disturbance and application in multi flight simulators systems, The International Society of Automation Transactions, 74 (2018), 217-228.   Google Scholar

[35]

F. Zhang and W. Wang, Decentralized optimal control for the mean field LQG problem of multi-agent systems, International Journal of Innovative Computing Information and Control, 13 (2017), 55-66.   Google Scholar

[36]

D. Zhang, Z. H. Xu, D. Srinivasan and L. Yu, Leader-follower consensus of multiagent systems with energy constraints: A markovian system approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1727-1736.  doi: 10.1109/TSMC.2017.2677471.  Google Scholar

[37]

Z. Q. Zhang, L. Zhang, F. Hao and L. Wang, Leader-following consensus for linear and Lipschitz nonlinear multiagent systems with quantized communication, IEEE Transactions on Cybernetics, 47 (2017), 1970-1982.  doi: 10.1109/TCYB.2016.2580163.  Google Scholar

show all references

References:
[1]

H. B. Du, Y. G. He and Y. Y. Cheng, Finite-time synchronization of a class of second-order nonlinear multi-agent systems using output feedback control, IEEE Transactions on Circuits and Systems I: Regular Papers, 61 (2014), 1778-1788.  doi: 10.1109/TCSI.2013.2295012.  Google Scholar

[2]

H. B. Du, G. H. Wen, G. R. Chen, J. D. Cao and F. E. Alsaadi, A distributed finite-time consensus algorithm for higher-order leaderless and leader-following multiagent systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1625-1634.  doi: 10.1109/TSMC.2017.2651899.  Google Scholar

[3]

M.-M. Duan, C.-L. Liu and F. Liu, Event-triggered consensus seeking of heterogeneous first-order agents with input delay, IEEE Access, 5 (2017), 5215-5223.  doi: 10.1109/ACCESS.2017.2696026.  Google Scholar

[4]

K. Fathian, T. H. Summers and N. R. Gans, Robust distributed formation control of agents with higher-order dynamics, IEEE Control Systems Letters, 2 (2018), 495-500.  doi: 10.1109/LCSYS.2018.2841941.  Google Scholar

[5]

L. Gao, J. Li, X. Zhu and W. Chen, Leader-following consensus of linear multi-agent systems with state-observer under switching topologies, 2012 12th International Conference on Control Automation Robotics and Vision, (2012), 572–577. Google Scholar

[6]

G. X. Gu, L. Marinovici and F. L. Lewis, Consensusability of discrete-time dynamic multiagent systems, IEEE Transactions on Automatic Control, 57 (2012), 2085-2089.  doi: 10.1109/TAC.2011.2179431.  Google Scholar

[7]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, 1952.  Google Scholar

[8]

B. Hou, F. C. Sun, H. B. Li and G. B. Liu, Consensus of second-order multi-agent systems with time-varying delays and antagonistic interactions, Tsinghua Science and Technology, 20 (2015), 205-211.  doi: 10.1109/TST.2015.7085634.  Google Scholar

[9]

H. H. Ji, H. T. Zhang, Z. Y. Ye, H. Zhang, B. W. Xu and G. R. Chen, Stochastic consensus control of second-order nonlinear multiagent systems with external disturbances, IEEE Transactions on Control of Network Systems, 5 (2018), 1585-1596.  doi: 10.1109/TCNS.2017.2736959.  Google Scholar

[10]

S. H. Li, H. B. Du and X. Z. Lin, Finite-time consensus algorithm for multiagent systems with double-integrator dynamics, Automatica J. IFAC, 47 (2011), 1706-1712.  doi: 10.1016/j.automatica.2011.02.045.  Google Scholar

[11]

Z. K. Li, Z. S. Duan and F. L. Lewis, Distributed robust consensus control of multi-agent systems with heterogeneous matching uncertainties, Automatica J. IFAC, 50 (2014), 883-889.  doi: 10.1016/j.automatica.2013.12.008.  Google Scholar

[12]

H. Q. Li, X. F. Liao, T. W. Huang, W. Zhu and Y. B. Liu, Second-order global consensus in multiagent networks with random directional link failure, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 565-575.  doi: 10.1109/TNNLS.2014.2320274.  Google Scholar

[13]

G. P. Li, X. Y. Wang and S. H. Li, Finite-time output consensus of higher-order multiagent systems with mismatched disturbances and unknown state elements, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 2751-2581.  doi: 10.1109/TSMC.2017.2759095.  Google Scholar

[14]

G. P. Li, X. Y. Wang and S. H. Li, Distributed composite output consensus protocols of higher-order multi-agent systems subject to mismatched disturbances, IET Control Theory and Applications, 11 (2017), 1162-1172.  doi: 10.1049/iet-cta.2016.0814.  Google Scholar

[15]

X. D. Li, X. Y. Yang and T. W. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar

[16]

C.-L. Liu, L. Shan, Y.-Y. Chen and Y. Zhang, Average-consensus filter of first-order multi-agent systems with disturbances, IEEE Transactions on Circuits and Systems II: Express Briefs, 65 (2018), 1763-1767.  doi: 10.1109/TCSII.2017.2762723.  Google Scholar

[17]

C. Q. Ma, T. Li and J. F. Zhang, Consensus control for leader-following multi-agent systems with measurement noise, Journal of Systems Science and Complexity, 23 (2010), 35-49.  doi: 10.1007/s11424-010-9273-4.  Google Scholar

[18]

Z. Y. Meng, W. Ren and Z. You, Distributed finite-time attitude containment control for multiple rigid bodies, Autonmatica, 46 (2010), 2092-2099.  doi: 10.1016/j.automatica.2010.09.005.  Google Scholar

[19]

S. Mondal and R. Su, Disturbance observer based consensus control for higher order multi-agent systems with mismatched uncertainties, 2016 American Control Conference, (2016), 2826–2831. doi: 10.1109/ACC.2016.7525347.  Google Scholar

[20]

C. Peng, J. Zhang and Q.-L. Han, Consensus of multiagent systems with nonlinear dynamics using an integrated sampled-data-based event-triggered communication scheme, IEEE Transactions on Systems, Man and Cybernetics: Systems, 49 (2019), 589-599.  doi: 10.1109/TSMC.2018.2814572.  Google Scholar

[21]

Z. R. Qiu, L. H. Xie and Y. G. Hong, Quantized leaderless and leader-following consensus of high-order multi-agent systems with limited data rate, IEEE Transactions on Automatic Control, 61 (2016), 2432-2447.  doi: 10.1109/TAC.2015.2495579.  Google Scholar

[22]

M. Rehan, A. Jameel and C. K. Ahn, Distributed consensus control of one-sided Lipschitz nonlinear multiagent systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1297-1308.  doi: 10.1109/TSMC.2017.2667701.  Google Scholar

[23]

W. Ren and E. Atkins, Distributed multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control, 50 (2005), 655-661.   Google Scholar

[24]

C. Ren, Z. Shi and T. Du, Distributed observer-based leader-following consensus control for second-order stochastic multi-agent systems, IEEE Access, 6 (2018), 20077-20084.  doi: 10.1109/ACCESS.2018.2820813.  Google Scholar

[25]

L. N. Rong, J. W. Lu, S. Y. Xu and Y. M. Chu, Reference model-based containment control of multi-agent systems with higher-order dynamics, IET Control Theory and Applications, 8 (2014), 796-802.  doi: 10.1049/iet-cta.2013.0148.  Google Scholar

[26]

P. Shi and Q. Shen, Cooperative control of multi-agent systems with unknown state-dependent controlling effects, IEEE Transactions on Automation Science and Engineering, 12 (2015), 827-834.   Google Scholar

[27]

S. Z. Su and Z. L. Lin, Distributed synchronization control of multi-agent systems with switching directed communication topologies and unknown nonlinearities, 2015 54th IEEE Conference on Decision and Control, (2015), 5444–5449. doi: 10.1109/CDC.2015.7403072.  Google Scholar

[28]

X. H. Wang, Y. G. Hong and H. B. Ji, Distributed optimization for a class of nonlinear multiagent systems with disturbance rejection, IEEE Transactions on Cybernetics, 46 (2016), 1655-1666.  doi: 10.1109/TCYB.2015.2453167.  Google Scholar

[29]

X. Y. Wang, S. H. Li, X. H. Yu and J. Yang, Distributed active anti-disturbance consensus for leader-follower higher-order multi-agent systems with mismatched disturbances, IEEE Transactions on Automatic Control, 62 (2017), 5795-5801.  doi: 10.1109/TAC.2016.2638966.  Google Scholar

[30]

J. H. Wang, Y. L. Xu, Y. Xu and D. D. Yang, Time-varying formation for high-order multi-agent systems with external disturbances by event-triggered integral sliding mode control, Appl. Math. Comput., 359 (2019), 333-343.  doi: 10.1016/j.amc.2019.04.066.  Google Scholar

[31]

Z. Wu, Y. Xu, Y. Pan, P. Shi and Q. Wang, Event-triggered pinning control for consensus of multiagent systems with quantized information, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1929-1938.   Google Scholar

[32]

D. Yang, X. D. Li and J. L. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar

[33]

Z. Y. Yu, H. J. Jiang and C. Hu, Leader-following consensus of fractional-order multi-agent systems under fixed topology, Neurocomputing, 149 (2015), 613-620.  doi: 10.1016/j.neucom.2014.08.013.  Google Scholar

[34]

C. Zhang, Distributed ESO based cooperative tracking control for high-order nonlinear multiagent systems with lumped disturbance and application in multi flight simulators systems, The International Society of Automation Transactions, 74 (2018), 217-228.   Google Scholar

[35]

F. Zhang and W. Wang, Decentralized optimal control for the mean field LQG problem of multi-agent systems, International Journal of Innovative Computing Information and Control, 13 (2017), 55-66.   Google Scholar

[36]

D. Zhang, Z. H. Xu, D. Srinivasan and L. Yu, Leader-follower consensus of multiagent systems with energy constraints: A markovian system approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1727-1736.  doi: 10.1109/TSMC.2017.2677471.  Google Scholar

[37]

Z. Q. Zhang, L. Zhang, F. Hao and L. Wang, Leader-following consensus for linear and Lipschitz nonlinear multiagent systems with quantized communication, IEEE Transactions on Cybernetics, 47 (2017), 1970-1982.  doi: 10.1109/TCYB.2016.2580163.  Google Scholar

Figure 1.  The topology of the multi-agent system
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Figure 2.  Position states of agents
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Figure 3.  Velocity states of agents
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Figure 4.  Acceleration states of agents
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