Structural controllability and observability of complex network with output feedback

Control theory provides useful tools for steering engineered and natural systems to desired states. Despite recent progress, a framework to control and observe the network with output feedback is still missing. Here we propose some new concepts, such as feedback-stem, feedback-bud and feedback-cactus. From the perspective of feedback-cactus, we propose a graphic necessary and sufficient condition for the structural controllability and observability of the network with output feedback. This condition combined with maximum matching allows us to solve the minimum input and output problem. Applying our framework to real and model networks, we find that some nodes play a dual role, that is, they are both driver nodes and sensor nodes. The proportion of such dual-role nodes is higher in sparse and homogeneous networks and is encoded by the degree distribution. Statistics find that the power-law distribution of the length of the feedback-stem is ubiquitous in real and model networks. Further, we use the proportion of edges in the feedback-cactus to evaluate the edge participation, i.e., the extent to which edges participate in the control and observation of the network with output feedback. Numerical simulations and theoretical analysis show that edge participation is higher in sparse and homogeneous networks.