Nonlinear Algorithms for Controlling a Group of Mobile Agents on a Segment

Abstract

A group of mobile agents on a straight line is considered. Agents are understood as numbered points that can change their position. It is assumed that the dynamics of agents is modeled by second-order integrators, with each agent receiving information from one of its left and one of its right neighbors (not necessarily nearest neighbors). It is required to provide a given nonlinear-uniform (uniform with respect to a prescribed nonlinear function) deployment of agents on a straight line segment. It is worth mentioning that, in numerous applications such as optic, acoustics, physiology, information theory, thermodynamics, etc., instead of linear scale, nonlinear ones (for instance, logarithmic) are used. In addition, it should be noted that an important class of formation control problems is synchronization of processes with respect to certain functions of phase coordinates. To solve the stated problem, nonlinear decentralized protocols are proposed. The conditions on the control parameters are determined, under which the agents converge to the required positions. The robustness of the constructed control protocols with respect to communication delay and network topology switching (replacing chosen neighbors by the other ones) is investigated. In this case, it is assumed that information about the magnitude of the delay and about the switching law may be absent. It is shown that for any constant non-negative delay and any admissible law for switching connections, a given deployment of agents is guaranteed. The proofs of the stated statements are based on the application of the Lyapunov direct method and a special form of the decomposition method. Original constructions of Lyapunov functions and Lyapunov—Krasovskii functionals are used. The results of numerical simulation are presented, confirming the obtained theoretical conclusions.