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Necessary and sufficient conditions for the controllability of chain-shape networks

This paper focuses on the controllability of chain-shape networks, where the nodes are multi-input/multi-output (MIMO) linear time-invariant (LTI) systems with higher-dimensional states. The necessary and sufficient conditions for the controllability of undirected and directed chain networks are established, respectively. The conditions are easier to verify, which explicitly show the effects of the node-system dynamics, external control inputs and inner interactions on the controllability of the whole networked system. Moreover, some necessary conditions are further derived, which are more straightforward and intuitive. The effectiveness of the conditions is demonstrated through several examples.