Geometric Control Theory Over Networks: Minimal Node Cardinality Disturbance Decoupling Problems

In this paper we show how to formulate and solve disturbance decoupling problems over networks while choosing a minimal number of input and output nodes. Feedback laws that isolate and eliminate the impact of disturbance nodes on specific target nodes to be protected are provided using state, output, and dynamical feedback. For that, we leverage the fact that when reformulated in terms of sets of nodes rather than subspaces, the controlled and conditioned invariance properties admit a simple graphical interpretation. For state and dynamical feedback, the minimal input and output cardinality solutions can be computed exactly in polynomial time, via min-cut/max-flow algorithms.