Cooperative control of double-integrator agents with heterogeneous control gains: Exponential consensus conditions and the heterogeneity-metric

Abstract

This paper considers cooperative control of multiple inertial (i.e., double-integrator) agents, with the couplings of the agents represented as a digraph. Particularly, the control inputs can be a general setting of heterogeneous control gains for the agents’ couplings. The main contributions in this paper are as follows: First, for the digraph being any unidirectional digraph (i.e., the corresponding Laplacian of the digraph can be expressed as an upper or lower triangular matrix if with proper numbering sequence of the agents) that contains a directed spanning-tree, we show that the necessary and sufficient condition for exponential convergence to consensus is that these gains can be merely arbitrarily positive. Second, for the digraph being generally weighted and strongly-connected (this type of the digraph includes all undirected and connected graphs), we provide the analytical lower-bounds of the heterogeneous gains for exponential convergence to consensus, whose bounds are less conservative than existing analytical results. Further, we propose the notion of the heterogeneity-metric to characterize the heterogeneity-degree of the heterogeneous gains and the notion of the maximum-heterogeneity-degree (MHD), and for the heterogeneous gains with the constraint of a designated MHD, we provide the conditions for exponential convergence to consensus, whose results are novel. Finally, we show the relation between the MHD constraint and the lower-bounds of the gains for exponential consensus.