Optimal Sensor Placement for Kalman Filtering in Stochastically Forced Consensus Networks

Given a linear dynamical system affected by noise, we consider the problem of optimally placing sensors (at design-time) subject to certain budget constraints to minimize the trace of the steady-state error covariance of the Kalman filter. Previous work has shown that this problem is NP-hard in general. In this paper, we impose additional structure by considering systems whose dynamics are given by a stochastic matrix corresponding to an underlying consensus network. In the case when there is a single input at one of the nodes in a tree network, we provide an optimal strategy (computed in polynomial-time) to place the sensors over the network. However, we show that when the network has multiple inputs, the optimal sensor placement problem becomes NP-hard.