Controllability of consensus heterogeneous multi-agent networks over continuous time scale

The research, presented in this paper, concernes the controllability of a multi-agent network with a directed, unweighted, cooperative, and time-invariant communication topology. The network’s agents follow linear and heterogeneous dynamics, encompassing first-order, second-order, and third-order differential equations over continuous time. Two classes of neighbour-based linear distributed control protocols are considered: the first one utilises average feedback from relative velocities/relative accelerations, and the second one utilises feedback from absolute velocities/absolute accelerations. Under both protocols, the network’s agents achieve consensus in their states asymptotically. We observe that both of the considered dynamical rules exploit the random-walk normalised Laplacian matrix of the network’s graph. By categorising the agents of the network into leaders and followers, with leaders serving as exogenous control inputs, we analyse the controllability of followers within their state space through the influence of leaders. Specifically, matrix-rank conditions are established to evaluate the leader–follower controllability of the network under both control protocols. These matrix-rank conditions are further refined in terms of the system matrices’ eigenvalues and eigenvectors. The inference diagrams presented in this work provide deeper insights into how leader–follower interactions impact the network controllability. The efficacy of the theoretical findings is validated through numerical examples.