Space-Time Sampling for Network Observability

Abstract

Abstract We will investigate under what conditions taking coarse samples from a network will contain the same information as a finer set of samples. Our goal is to estimate the initial state of a linear network of subsystems, which are distributed in a spatial domain, from noisy measurements. We develop a framework to produce feasible sets of spatio-temporal samples for the estimation problem, which essentially have a non-uniform space-time sampling pattern. If the number of sampling locations is comparable to the size of the network, the sampling pattern will have a high degree of redundancy. For these cases, using an efficient algorithm, we present a method for finding a feasible subset of samples that have a sparser space-time sampling pattern. It is shown that spatial samples can be traded for time samples: choosing to sample from a smaller set of subsystems must be compensated by taking more frequent time samples from those subsystems. Furthermore, we apply the Kadison-Singer paving solution to sparsify a feasible redundant sampling strategy with guaranteed estimation bounds. We support our theoretical findings via several numerical examples.