# zbMATH — the first resource for mathematics

Containment control of continuous-time linear multi-agent systems with aperiodic sampling. (English) Zbl 1330.93113
Summary: In this paper, the containment control problem of continuous-time linear multi-agent systems is investigated. An aperiodic sampled-data based protocol is induced by using neighboring information with uncertainly time-varying sampling intervals. By utilizing the proposed protocol and properties of Laplacian matrix, the containment control problem of continuous-time linear multi-agent systems is equivalently transformed into a stability problem of discrete-time linear systems. The stability analysis is based on the robustness of related discrete-time systems against perturbation caused by the variation of sampling intervals. By using small-gain theorem, sufficient conditions are obtained to guarantee the stability of uncertain discrete-time systems. Furthermore, two special cases are given to illustrate the method proposed in this paper. The theoretical results are verified by some simulations.

##### MSC:
 93C05 Linear systems in control theory 68T42 Agent technology and artificial intelligence 93C57 Sampled-data control/observation systems 93B17 Transformations 93C55 Discrete-time control/observation systems 93D09 Robust stability
Full Text:
##### References:
 [1] Biggs, N., Algebraic graph theory, (1974), Cambridge Univ. Press Cambridge, U.K. · Zbl 0284.05101 [2] Cao, Y., Stuart, D., Ren, W., & Meng, Z. (2010). Distributed containment control for double-integrator dynamics: algorithms and experiments. In 2010 American control conference (pp. 3830-3835). [3] Cortes, J.; Martinez, S.; Bullo, F., Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions, IEEE Transactions on Automatic Control, 51, 8, 1289-1298, (2006) · Zbl 1366.93400 [4] Cortés, J.; Martínez, S.; Karatas, T.; Bullo, F., Coverage control for mobile sensing networks, IEEE Transactions on Robotics and Automation, 20, 2, 243-255, (2004) [5] Dimarogonas, D. V., Egerstedt, M., & Kyriakopoulos, K. J. (2006). A leader-based containment control strategy for multiple unicycles. In proceedings of the 45th IEEE conference on decision and control (pp. 5968-5973). [6] Ding, W.; Yan, G.; Lin, Z., Collective motions and formations under pursuit strategies on directed acyclic graphs, Automatica, 46, 1, 174-181, (2010) · Zbl 1213.93073 [7] Do, K. D., Formation control of multiple elliptical agents with limited sensing ranges, Automatica, 48, 7, 1330-1338, (2012) · Zbl 1246.93005 [8] Dong, Y.; Huang, J., A leader-following rendezvous problem of double integrator multi-agent systems, Automatica, 49, 5, 1386-1391, (2013) · Zbl 1319.93005 [9] Galbusera, L.; Ferrari-Trecate, G.; Scattolini, R., A hybrid model predictive control scheme for containment and distributed sensing in multi-agent systems, Systems & Control Letters, 62, 5, 413-419, (2013) · Zbl 1276.93003 [10] Gao, Y.; Ma, J.; Zuo, M.; Mo, L.; Yu, J., Consensusability of continuous-time multi-agent systems with general linear dynamics and intermittent measurements, IET Control Theory & Applications, 7, 6, 842-847, (2012) [11] Horn, R. A.; Johnson, C. R., Matrix analysis, (1987), Cambridge Univ. Press Cambridge, U.K. [12] Jadbabaie, A.; Lin, J.; Morse, A. S., Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48, 6, 988-1001, (2003) · Zbl 1364.93514 [13] Ji, M.; Ferrari-Trecate, G.; Egerstedt, M.; Buffa, A., Containment control in mobile networks, IEEE Transactions on Automatic Control, 53, 8, 1972-1975, (2008) · Zbl 1367.93398 [14] Khargonekar, P. P.; Petersen, I. R.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stability and $$H_\infty$$ control theory, IEEE Transactions on Automatic Control, 35, 3, 356-361, (1990) · Zbl 0707.93060 [15] Li, J.; Ren, W.; Xu, S., Distributed containment control with multiple dynamic leaders for double-integrator dynamics using only position measurements, IEEE Transactions on Automatic Control, 57, 6, 1553-1559, (2012) · Zbl 1369.93030 [16] Lin, Z.; Francis, B.; Maggiore, M., Necessary and sufficient graphical conditions for formation control of unicycles, IEEE Transactions on Automatic Control, 50, 1, 121-127, (2005) · Zbl 1365.93324 [17] Liu, H., Cheng, L., Tan, M., & Hou, Z.-G. (2014). Containment control of double-integrator multi-agent systems with aperiodic sampling: a small-gain theorem based method. In proceedings of the 33rd Chinese control conference (pp. 1407-1412). [18] Liu, H.; Xie, G.; Wang, L., Necessary and sufficient conditions for solving consensus problems of double-integrator dynamics via sampled control, International Journal of Robust and Nonlinear Control, 20, 15, 1706-1722, (2010) · Zbl 1204.93080 [19] Liu, H.; Xie, G.; Wang, L., Containment of linear multi-agent systems under general interaction topologies, Systems & Control Letters, 61, 4, 528-534, (2012) · Zbl 1250.93010 [20] Liu, H.; Xie, G.; Wang, L., Necessary and sufficient conditions for containment control of networked multi-agent systems, Automatica, 48, 7, 1415-1422, (2012) · Zbl 1246.93008 [21] Lou, Y.; Hong, Y., Target containment control of multi-agent systems with random switching interconnection topologies, Automatica, 48, 5, 879-885, (2012) · Zbl 1246.93104 [22] Mei, J.; Ren, W.; Ma, G., Distributed containment control for Lagrangian networks with parametric uncertainties under a directed graph, Automatica, 48, 4, 653-659, (2012) · Zbl 1238.93009 [23] Notarstefano, G.; Egerstedt, M.; Haque, M., Containment in leader-follower networks with switching communication topologies, Automatica, 47, 5, 1035-1040, (2011) · Zbl 1233.93009 [24] Olfati-Saber, R., Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Transactions on Automatic Control, 51, 3, 401-420, (2006) · Zbl 1366.93391 [25] Olfati-Saber, R.; Murray, R. M., Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301 [26] Ren, W.; Beard, R. W., Consensus seeking in multiagent systems under dynamically changing interation topologies, IEEE Transactions on Automatic Control, 50, 5, 655-661, (2005) · Zbl 1365.93302 [27] Ren, W.; Beard, R. W.; Atkins, E. M., Information consensus in multivehicle cooperative control: collective group behavior through local interaction, IEEE Control Systems Magazine, 27, 2, 71-82, (2007) [28] Rockafellar, R. T., Convex analysis, (1970), Princeton University Press New Jersey · Zbl 0202.14303 [29] Ryan, A., Zennaro, M., Howell, A., Sengupta, R., & Hendrick, K.J. (2004). An overview of emerging results in cooperative uav control. In 43rd IEEE conference on decision and control (pp. 602-607). [30] Tomlin, C.; Pappas, G. J.; Sastry, S., Conflict resolution for air traffic management: A study in multiagent hybrid systems, IEEE Transactions on Automatic Control, 43, 4, 509-521, (1998) · Zbl 0904.90113 [31] Van Loan, C., The sensitivity of the matrix exponential, SIAM Journal of Numerical Analysis, 14, 6, 971-981, (1977) · Zbl 0368.65006 [32] Yoo, S., Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form, Automatica, 49, 7, 2145-2153, (2013) · Zbl 1364.93413 [33] Zhai, C.; Hong, Y., Decentralized sweep coverage algorithm for multi-agent systems with workload uncertainties, Automatica, 49, 7, 2154-2159, (2013) · Zbl 1364.93041 [34] Zhang, H.; Zhai, C.; Chen, Z., A general alignment repulsion algorithm for flocking of multi-agent systems, IEEE Transactions on Automatic Control, 56, 2, 430-435, (2011) · Zbl 1368.93233
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.