Paper:

# Finite-Time Consensus of Double-Integrator Multi-Agent Systems with Time-Varying Directed Topologies

## Fang Wang^{*}, Xin Chen^{**, †}, and Yong He^{**}

^{*}School of Information Science & Engineering, Central South University

Changsha, Hunan 410083, China

^{**}School of Automation, China University of Geosciences

Wuhan, Hubei 430074, China

^{†}Corresponding author

The finite-time consensus problem for double-integrator multi-agent systems (MASs) is studied using time-varying directed topologies. In detail, a distributed finite-time control protocol is designed to achieve the weighted average consensus on the basis of both relative position and relative velocity measurements by utilizing a homogeneous control technique. Then, on the basis of graph theory, homogeneity with dilation and LaSalle’s invariance principle, the designed finite-time consensus protocol ensures finite-time convergence to a consensus in the time-varying directed topologies without a global leader. Finally, some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.20, No.2, pp. 254-261, 2016.

- [1] Y. Pei and J. Sun, “Consensus of discrete-time linear multi-agent systems with Markov switching topologies and time-delay,” Neurocomputing, Vol.151, pp. 776-781, 2015.
- [2] A. Garulli, A. Giannitrapani and M. Leomanni, “Minimum switching control for systems of coupled double integrators,” Automatica, Vol.60, pp. 115-121, 2015.
- [3] J. Ghommam and M. Saad, “Backstepping-based cooperative and adaptive tracking control design for a group of underactuated AUVs in horizontal plan,” Int. J. of Control, Vol.87, No.5, pp. 1076-1093, 2014.
- [4] J. Wang and M. Xin, “Integrated optimal formation control of multiple unmanned aerial vehicles,” IEEE Trans. on Control Systems Technology, Vol.21, No.5, pp. 1731-1744, 2013.
- [5] X. Dong, J. Xi, G. Lu, et al., “Formation control for high-order linear time-invariant multi-agent systems with time delays,” IEEE Trans. on Control of Network Systems, Vol.1, No.3, pp. 232-240, 2014.
- [6] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” IEEE Trans. on Automatic Control, Vol.51, No.3, pp. 401-420, 2006.
- [7] W. Dong, “Flocking of multiple mobile robots based on backstepping,” IEEE Trans. on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol.41, No.2, pp. 414-424, 2011.
- [8] W. Ren, R. W. Beard and E. M. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Systems Magazine, Vol.2, No.27, pp. 71-82, 2007.
- [9] S. Li, H. Du, and P. Shi, “Distributed attitude control for multiple spacecraft with communication delays,” IEEE Trans. on Aerospace and Electronic Systems, Vol.50, No.3, pp. 1765-1773, 2014.
- [10] A. M. Zou, K. D. Kumar, and Z. G. Hou, “Attitude coordination control for a group of spacecraft without velocity measurements,” IEEE Trans. on Control Systems Technology, Vol.20, No.5, pp. 1160-1174, 2012.
- [11] Y. Zhao, Z. Duan, and G. Wen, “Finite-time consensus for second-order multi-agent systems with saturated control protocols,” IET Control Theory & Applications, Vol.9, No.3, pp. 312-319, 2014.
- [12] J. M. Hendrickx, G. Shi and K. H. Johansson, “Finite-time consensus using stochastic matrices with positive diagonals,” IEEE Trans. on Automatic Control, Vol.60, No.4, pp. 1070-1073, 2015.
- [13] S. Yu and X.Long, “Finite-time consensus for second-order multi-agent systems with disturbances by integral sliding mode,” Automatica, Vol.54, pp. 158-165, 2015.
- [14] Y. Wu, X. Yu, Z. Man, “Terminal sliding mode control design for uncertain dynamic systems,” Systems & Control Letters, Vol.34, No.5, pp. 281-287, 1998.
- [15] M. Franceschelli, A. Pisano, A. Giua, et al., “Finite-time consensus with disturbance rejection by discontinuous local interactions in directed graphs,” IEEE Trans. on Automatic Control, Vol.60, No.4, pp. 1133-1138, 2015.
- [16] J. Zhang, Z. Han and H. Wu, “Robust finite-time stability and stabilisation of switched positive systems,” IET Control Theory & Applications, Vol.8, No.1, pp. 67-75, 2014.
- [17] S. Kamal, A. Raman and B. Bandyopadhyay, “Finite-Time Stabilization of Fractional Order Uncertain Chain of Integrator: An Integral Sliding Mode Approach,” IEEE Trans. on Automatic Control, Vol.58, No.6, pp. 1597-1602, 2013.
- [18] Y. Wu, F. Gao and Z. Liu, “Finite-time state-feedback stabilisation of non-holonomic systems with low-order non-linearities,” IET Control Theory & Applications, Vol.9, No.10, pp. 1553-1560, 2015.
- [19] X. Liu, W. Yu, J. Cao, et al., “Finite-time synchronisation control of complex networks via non-smooth analysis,” IET Control Theory & Applications, Vol.9, No.8, pp. 1245-1253, 2015.
- [20] B. Zhang, Y. Jia, J. Du, et al., “Finite-time synchronous control for multiple manipulators with sensor saturations and a constant reference,” IEEE Trans. on Control Systems Technology, Vol.22, No.3, pp. 1159-1165, 2014.
- [21] B. Zhang, Y. Jia and F. Matsuno, “Finite-time observers for multi-agent systems without velocity measurements and with input saturations,” Systems & Control Letters, Vol.68, pp. 86-94, 2014.
- [22] Y. Zhao, Z. Duan, G. Wen, et al., “Distributed finite-time tracking for a multi-agent system under a leader with bounded unknown acceleration,” Systems & Control Letters, Vol.81, pp. 8-13, 2015.
- [23] Y. Zhang and Y. Yang, “Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements,” Physics Letters A, Vol.377, No.3, pp. 243-249, 2013.
- [24] X. Wang, S. Li and P. Shi, “Distributed finite-time containment control for double-integrator multiagent systems,” IEEE Trans. on Cybernetics, Vol.44, No.9, pp. 1518-1528, 2014.
- [25] Y. Zhao, Z. S. Duan, G. H. Wen and Y. J. Shi, “Distributed finite-time tracking control for multiagent systems: a observer-based approach,” Systems & Control Letters, Vol.62, pp. 22-28, 2013.
- [26] J. Wen, L. Peng and S. K. Nguang, “Finite-time analysis and design for discrete-time switching dynamics Markovian jump linear systems with time-varying delay,” IET Control Theory & Applications, Vol.8, No.17, pp. 1972-1985, 2014.
- [27] X. Lin, H. Du, S. Li, et al., “Finite-time stability and finite-time weighted
*L*_{2}-gain analysis for switched systems with time-varying delay,” IET Control Theory & Applications, Vol.7, No.7, pp. 1058-1069, 2013. - [28] F. Wang, X. Chen and Y. He, “Finite-time consensus of second-order multi-agent systems with jointly-connected topologies,” IEEE 2014 33
^{rd}Chinese Control Conf. (CCC), pp. 1662-1667, 2014. - [29] F. Sun, J. Chen, Z. H. Guan, et al., “Leader-following finite-time consensus for multi-agent systems with jointly-reachable leader,” Nonlinear Analysis: Real World Applications, Vol.13, No.5, pp. 2271-2284, 2012.
- [30] X. Lu, S. Chen and J. Lüu, “Finite-time tracking for double-integrator multi-agent systems with bounded control input,” IET Control Theory & Applications, Vol.7, No.11, pp. 1562-1573, 2013.
- [31] C. Li and Z. Qu, “Distributed finite-time consensus of nonlinear systems under switching topologies,” Automatica, Vol.50, No.6, pp. 1626-1631, 2014.
- [32] T. Chu, L. Wang, T. Chen, et al., “Complex emergent dynamics of anisotropic swarms: convergence vs oscillation,” Chaos, Solitons & Fractals, Vol.30, No.4, pp. 875-885, 2006.
- [33] N. Rouche, P. Habets and M. Laloy, “Stability Theory by Liapunov’s Direct Method,” Springer-Verlag, New York, 1977.
- [34] L. Rosier, “Homogeneous Lyapunov function for homogeneous continuous vector field,” Systems & Control Letters, Vol.9, pp. 467-473, 1992.
- [35] X. Wang and Y. Hong, “Distributed finite-time χ-consensus algorithms for multi-agent systems with variable coupling topology,” J. of Systems Science and Complexity, Vol.23, pp. 209-218, 2010.