Comments on 'Controllability analysis of multi-agent systems using relaxed equitable partitions'

We want to comment on a recent paper by Martini et al. [see Int. J. Systems, Control and Communication, Vol. 2, Nos. 1-3, pp.100-121, 2010]. The above paper analyses controllability of a class of multi-agent systems using the leader-invariant relaxed equitable partition (LEP) of their underlying graphs. Two of its main results rely on the complete characterisation of the controllable subspace of the system in terms of the characteristic vectors obtained from the LEP. By providing counterexamples for these results, this note shows that LEP cannot fully characterise the controllable subspace. Therefore, finding graphical necessary and sufficient conditions for controllability of this class of systems is still an open problem.