On node controllability and observability in complex dynamical networks

We analyze in detail the subtle yet critical differences between the structural controllability and observability of the triplet $(A,B,C)$ in the two cases that this is viewed as a linear dynamical network of interconnected nodes or as a a single complex system. Investigating the controllability and observability properties of each single node when the network is not completely controllable and/or observable, we show that the first point of view requires the development of novel tools leading, ultimately, to a state space decomposition that is different from the one proposed in 1963 by R.E. Kalman for linear systems.