New controllability conditions for multi-layer networked systems

The controllability of networked system with high-dimensional nodes is far lack of understanding. From the perspective of the eigenvectors, based on the decoupled idea and dimensionality reduction technique, this paper investigates the controllability of multi-layer networks with high-dimensional nodes where the network topology is described by multiple connection structures with different coupling modes, and proposes some new necessary and sufficient conditions of the controllability for two intra-layer topology cases of simultaneously block diagonalization (SBD) and simultaneously diagonalization (SD), which are represented by the small-size units of intra-layer topologies and node dynamics. Furthermore, an easy-to-use algorithm is developed to check the controllability of multi-layer networked system, and two examples are used for illustration and verification, and then these conditions are specified for four cases of special intra-layer topologies. These findings provide a new and insight understandings for the controllability of multi-layer networks, and reveal the affect of multi- layer connections (outer topologies and inner couplings) on network controllability, particularly the impressive result that either one of two-layer networks being uncontrollable implies the two-layer networked system being uncontrollable when the two-layer network structures have the common left eigenvectors.