Dichotomic differential inequalities and multi-agent coordination

Distributed algorithms of multi-agent coordination have attracted substantial attention from the research communities. The most investigated are Laplacian-type dynamics over time-varying weighted graphs, whose applications include, but are not limited to, the problems of consensus, opinion dynamics, aggregation and containment control, target surrounding and distributed optimization. While the algorithms solving these problems are similar, for their analysis different mathematical techniques have been used. In this paper, we propose a novel approach, allowing to prove the stability of many Laplacian-type algorithms, arising in multi-agent coordination problems, in a unified elegant way. The key idea of this approach is to consider an associated linear differential inequality with the Laplacian matrix, satisfied by some bounded outputs of the agents (e.g. the distances to the desired set in aggregation and containment control problems). Although such inequalities have many unbounded solutions, under natural connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the original protocol. The differential inequality thus admits only convergent but not “oscillatory” bounded solutions. This property, referred to as the dichotomy, has been long studied in the theory of differential equations. We show that a number of recent results from multi-agent control can be derived from the dichotomy criteria for Laplacian differential inequalities, developed in this paper, discarding also some technical restrictions

Intrinsic Reduced Attitude Formation with Ring Inter-Agent Graph

This paper investigates the reduced attitude formation control problem for a group of rigid-body agents using feedback based on relative attitude information. Under both undirected and directed cycle graph topologies, it is shown that reversing the sign of a classic consensus protocol yields asymptotical convergence to formations whose shape depends on the parity of the group size. Specifically, in the case of even parity the reduced attitudes converge asymptotically to a pair of antipodal points and distribute equidistantly on a great circle in the case of odd parity. Moreover, when the inter-agent graph is an undirected ring, the desired formation is shown to be achieved from almost all initial states

Necessary and Sufficient Conditions for Circle Formations of Mobile Agents with Coupling Delay via Sampled-Data Control

A circle forming problem for a group of mobile agents governed by first-order system is investigated, where each agent can only sense the relative angular positions of its neighboring two agents with time delay and move on the one-dimensional space of a given circle. To solve this problem, a novel decentralized sampled-data control law is proposed. By combining algebraic graph theory with control theory, some necessary and sufficient conditions are established to guarantee that all the mobile agents form a pregiven circle formation asymptotically. Moreover, the ranges of the sampling period and the coupling delay are determined, respectively. Finally, the theoretical results are demonstrated by numerical simulations