Sensor Selection via Randomized Sampling

Given a linear dynamical system, we consider the problem of constructing an approximate system using only a subset of the sensors out of the total set such that the observability Gramian of the new system is approximately equal to that of the original system. Our contributions are as follows. First, we present a randomized algorithm that samples the sensors with replacement as per specified distributions. For specific metrics of the observability Gramian such as the trace or the maximum eigenvalue, we derive novel bounds on the number of samples required to yield a high probability lower bound on the metric evaluated on the approximate Gramian. Second, with a different distribution, we derive high probability bounds on other standard metrics used in sensor selection, including the minimum eigenvalue or the trace of the Gramian inverse. This distribution requires a number of samples which is larger than the one required for the trace and the maximum eigenvalue, but guarantees non-singularity of the approximate Gramian if the original system is observable with high probability. Third, we demonstrate how the randomized procedure can be used for recursive state estimation using fewer sensors than the original system and provide a high probability upper bound on the initial error covariance. We supplement the theoretical results with several insightful numerical studies and comparisons with competing greedy approaches.