A randomized approach to sensor placement with observability assurance

Given a linear dynamical system, we provide a probabilistic treatment to the classic problem of placing sensors in a set of candidate locations such that the observability Gramian of the resulting placement is sufficiently non-singular. Our contributions are as follows: First, we present a randomized algorithm that samples the sensor locations with replacement as per a specified distribution. Second, we derive a high probability bound on two measures of non-singularity, viz. the minimum eigenvalue and the trace of the inverse of the observability Gramian of the resulting placement, relative to that of placing one sensor at every location. Our analysis yields upper and lower bounds on any eigenvalue-based metric used in sensor placement and characterizes the tradeoff between the number of samples required by the algorithm and the two measures of the observability Gramian. We supplement the claims with insightful numerical studies and comparisons with multiple competing approaches.