Target Controllability and Target Observability of Structured Network Systems

The duality between controllability and observability enables methods developed for full-state control to be applied for full-state estimation, and vice-versa. For applications in which control or estimation of all state variables is unfeasible, the generalized notions of output controllability and functional observability establish the minimal conditions for the control and estimation of a target subset of state variables, respectively. Given the seemly unrelated nature of these properties, thus far methods for target control and target estimation have been developed independently in the literature. Here, we characterize the graph-theoretic conditions for target controllability and target observability (special cases of output controllability and functional observability for structured systems, respectively). This allow us to rigorously establish a weak and strong duality between these generalized properties. When both properties are equivalent (strongly dual), we show that efficient algorithms developed for target controllability can be used for target observability, and vice-versa, for the optimal placement of sensors and drivers. These results are applicable to large-scale networks, in which control and monitoring are often sought for small subsets of nodes.