Optimal Control and Estimation for Partially Nested Interconnected Systems

In this paper, we study distributed estimation and control problems over graphs under partially nested information patterns. We show a duality result that is very similar to the classical duality result between state estimation and state feedback control with a classical information pattern, under the condition that the disturbances entering different systems on the graph are uncorrelated. The distributed estimation problem decomposes into $N$ separate estimation problems, where $N$ is the number of interconnected subsystems over the graph, and the solution to each subproblem is simply the optimal Kalman filter. This also gives the solution to the distributed control problem due to the duality of distributed estimation and control under partially nested information pattern. We then consider a weighted distributed estimation problem, where we get coupling between the estimators, and separation between the estimators is not possible. We propose a solution based on linear quadratic team decision theory, which provides a generalized Riccati equation for teams. We show that the weighted estimation problem is the dual to a distributed state feedback problem, where the disturbances entering the interconnected systems are correlated.