Necessary and sufficient characterisation for output synchronisation of linear heterogeneous agents using reflexive generalised inverses

Abstract

This work studies the conditions on which the outputs of linear heterogeneous (i.e. non-identical) agents are asymptotically synchronised, where the dynamics of the agents and the dimensions of the state variables might widely differ among themselves. With the constraint that the relative outputs of the agents are the only measurements, it is shown that the reflexive generalised inverses play a key role in characterising the conditions, and give rise to the synchronisation stability of the agents. Together with the internal model principle for synchronisation, this results in a necessary and sufficient condition for reaching the asymptotic output synchronisation. This necessary and sufficient characterisation has its own geometric interpretation as the reflexive generalised inverses are related to the projections onto subspaces. Finally, it is shown that the condition is in fact a natural generalisation of the state synchronisability of homogeneous (i.e. identical) linear agents. A computer simulation is also provided to verify the result.