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

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 andwe 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 highdimensional 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. © 2012 Elsevier Ltd. All rights reserved.