Duality Between Controllability and Observability for Target Control and Estimation in Networks

Output controllability and functional observability are properties that enable, respectively, the control and estimation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">part</i> of the state vector. These notions are of utmost importance in applications to high-dimensional systems, such as large-scale networks, in which only a target subset of variables (nodes) is sought to be controlled or estimated. Although the duality between full-state controllability and observability is well established, the characterization of the duality between their generalized counterparts remains an outstanding problem. Here, we establish both the weak and the strong duality between output controllability and functional observability. Specifically, we show that functional observability of a system implies output controllability of a dual system (weak duality), and that under a certain geometric condition the converse holds (strong duality). As an application of the strong duality, we derive a necessary and sufficient condition for target control via static feedback. This allow us to establish a separation principle between the design of target controllers and the design of functional observers in closed-loop systems. These results generalize the classical duality and separation principles in modern control theory.