Graph-theoretic analysis of network input-output processes: Zero structure and its implications on remote feedback control

The control of dynamical processes in networks is considered, in the case where measurement and actuation capabilities are sparse and possibly remote. Specifically, we study control of a canonical network dynamics, when only one network component's state can be measured and only one (in general different) component can be actuated. To do so, we characterize the finite- and infinite-zeros of the resulting SISO system in terms of the graph topology. Using these results, we establish graph-theoretic conditions under which there are zeros in the closed right-half plane. These conditions depend on the length, strength, and number of the paths from the component where the input is applied to the component where the measurements are made. Then, we present the implications of these conditions on the controller design task focusing in stabilizations/destabilization of network processes under static negative feedback.