Spectral analysis of extended consensus algorithms for multiagent systems

We analyze an extension of the well-known linear consensus protocol for agents moving in two dimensions, where the standard consensus feedback is multiplied with a rotation matrix. This leads to a richer family of trajectories, and if only the new feedback term is applied, periodic solutions emerge. For special configurations of the controller gains, the form of the system trajectories is given in terms of the eigenvalues and eigenvectors of the closed-loop system matrix. We characterize the resulting closed-loop trajectories for specific choices of the controller gains and of the communication graph topology. Furthermore, the control strategy is extended to agents with double integrator dynamics. It is shown that stability is achieved with sufficiently large velocity feedback. The effect of this feedback on the overall system performance is further investigated. We finally provide simulations to illustrate the theoretical results.