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We construct systematically a class of uncontrollable diffusively coupled multiagent systems with a single leader and multichain topologies. For studying the controllability of diffusively coupled multiagent systems, such identified uncontrollable systems serve as counterexamples that prove the need to modify the existing sufficient condition using graph partitioning characterization. The uncontrollability of the constructed multichain structures can be preserved when the structures are further augmented to get better connected. The paper also provides an algorithm to obtain the minimal leader-invariant relaxed equitable partition for the graph associated with any diffusively coupled multiagent system guided by a single leader.
Cao et al. (Fri,) studied this question.