Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks.

*(English)*Zbl 1268.93006Summary: In this paper, we study the problem of distributed containment control of a group of mobile autonomous agents with multiple stationary or dynamic leaders under both fixed and switching directed network topologies. First, when the leaders are stationary and all followers share an inertial coordinate frame, we present necessary and sufficient conditions on the fixed or switching directed network topology such that all followers will ultimately converge to the stationary convex hull formed by the stationary leaders for arbitrary initial states in a space of any finite dimension. When the directed network topology is fixed, we partition the (nonsymmetric) Laplacian matrix and explore its properties to derive the convergence results. When the directed network topology is switching, the commonly adopted decoupling technique based on the Kronecker product in a high-dimensional space can no longer be applied and we hence present an important coordinate transformation technique to derive the convergence results. The proposed coordinate transformation technique also has potential applications in other high-dimensional distributed control scenarios and might be used to simplify the analysis of a high-dimensional system to that of a one-dimensional system when the decoupling technique based on the Kronecker product cannot be applied. Second, when the leaders are dynamic and all followers share an inertial coordinate frame, we propose a distributed tracking control algorithm without velocity measurements. When the directed network topology is fixed, we derive conditions on the network topology and the control gain to guarantee that all followers will ultimately converge to the dynamic convex hull formed by the dynamic leaders for arbitrary initial states in a space of any finite dimension. When the directed network topology is switching, we derive conditions on the network topology and the control gain such that all followers will ultimately converge to the minimal hyperrectangle that contains the dynamic leaders and each of its hyperplanes is normal to one axis of the inertial coordinate frame in any high-dimensional space. We also show via some counterexamples that it is, in general, impossible to find distribute containment control algorithms without velocity measurements to guarantee that all followers will ultimately converge to the convex hull formed by the dynamic leaders under a switching network topology in a high-dimensional space. Simulation results are presented as a proof of concept.

##### Keywords:

containment control; multi-agent systems; cooperative control; consensus; network topology; Kronecker product; velocity measurements; decoupling technique; convex hull; Laplacian matrix; minimal hyperrectangle**OpenURL**

##### References:

[1] | Agaev, R.; Chebotarev, P., The matrix of maximum out forests of a digraph and its applications, Automation and remote control, 61, 9, 1424-1450, (2000) · Zbl 1057.05038 |

[2] | Alefeld, G.; Schneider, N., On square roots of \(M\)-matrices, Linear algebra and its applications, 42, 119-132, (1982) · Zbl 0483.15009 |

[3] | Cao, Y., & Ren, W. (2009). Containment control with multiple stationary or dynamic leaders under a directed interaction graph. In Proceedings of the IEEE conference on decision and control (pp. 3014-3019). |

[4] | Cao, Y.; Ren, W.; Li, Y., Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication, Automatica, 45, 5, 1299-1305, (2009) · Zbl 1162.93004 |

[5] | Cao, Y.; Ren, W.; Meng, Z., Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking, Systems and control letters, 59, 9, 522-529, (2010) · Zbl 1207.93103 |

[6] | Clarke, F.H., Optimization and nonsmooth analysis, (1990), SIAM · Zbl 0727.90045 |

[7] | Cortes, J., Discontinuous dynamical systems: a tutorial on solutions, nonsmooth analysis, and stability, IEEE control systems magazine, 28, 3, 36-73, (2008) · Zbl 1395.34023 |

[8] | DeGroot, M.H., Reaching a consensus, Journal of American statistical association, 69, 345, 118-121, (1974) · Zbl 0282.92011 |

[9] | Ferrara, G.F.-T.A., & Vecchio, C. (2007). Sliding mode control for coordination in multi-agent systems with directed communication graphs. In Proceedings of the European control conference (pp. 1477-1484). |

[10] | Filippov, A.F., Differential equations with discontinuous righthand sides, (1988), Kluwer Academic Publishers · Zbl 0664.34001 |

[11] | Gazi, V., Swarm aggregations using artificial potentials and sliding mode control, IEEE transactions on robotics, 21, 6, 1208-1214, (2005) |

[12] | Hong, Y.; Hu, J.; Gao, L., Tracking control for multi-agent consensus with an active leader and variable topology, Automatica, 42, 7, 1177-1182, (2006) · Zbl 1117.93300 |

[13] | 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 |

[14] | 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 |

[15] | Jin, Z., & Murray, R.M. (2006). Consensus controllability for coordinated multiple vehicle control. In The 6th international conference on cooperative control and optimization. |

[16] | Khalil, H.K., Nonlinear systems, (2002), Prentice Hall Upper Saddle River, NJ · Zbl 0626.34052 |

[17] | Lin, Z.; Broucke, M.; Francis, B., Local control strategies for groups of mobile autonomous agents, IEEE transactions on automatic control, 49, 4, 622-629, (2004) · Zbl 1365.93208 |

[18] | Moore, K.; Lucarelli, D., Decentralized adaptive scheduling using consensus variables, International journal of robust and nonlinear control, 17, 10-11, 921-940, (2007) · Zbl 1266.90110 |

[19] | Moreau, L. (2004a). Stability of continuous-time distributed consensus algorithms. In Proceedings of the IEEE conference on decision and control (pp. 3998-4003). |

[20] | Moreau, L. (2004b). Stability of continuous-time distributed consensus algorithms. URL http://www.citebase.org/abstract?id=oai:arXiv.org:math/0409010. |

[21] | Murray, R.M., Recent research in cooperative control of multivehicle systems, ASME journal of dynamic systems, measurement, and control, 129, 5, 571-583, (2007) |

[22] | 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 |

[23] | Paden, B.; Sastry, S., A calculus for computing filippovâ€™s differential inclusion with application to the variable structure control of robot manipulators, IEEE transactions on circuits and systems, CAS-34, 1, 73-82, (1987) · Zbl 0632.34005 |

[24] | Ren, W., Multi-vehicle consensus with a time-varying reference state, Systems & control letters, 56, 7-8, 474-483, (2007) · Zbl 1157.90459 |

[25] | Ren, W.; Beard, R.W., Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE transactions on automatic control, 50, 5, 655-661, (2005) · Zbl 1365.93302 |

[26] | Ren, W.; Beard, R.W.; Atkins, E.M., Information consensus in multivehicle cooperative control, IEEE control systems magazine, 27, 2, 71-82, (2007) |

[27] | Winkler, R.L., The consensus of subjective probability distributions, Manage science, 15, 2, B61-B75, (1968) |

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