Sensor Selection in Presence of Random Failures

We analyze the observability of a dynamical system when each sensor is subject to random failure. In particular, we model the fact that each sensor may fail independently of the other and this failure is assumed to be a Bernoulli random variable with known parameter. We leverage results from random matrix theory to obtain probabilistic bounds on three metrics of observability, viz. the (negative of) maximum eigenvalue, the minimum eigenvalue and the (negative of) trace of the inverse of the observability Gramian. The goal is to perform sensor selection to maximize the expected value of the metric, which we show becomes equivalent to optimizing the metric evaluated over the expected value of the Gramian, with a known probability. A greedy algorithm is then used to perform the selection for which we characterize the factor of sub-optimality relative to the optimal corresponding to each metric. For a specific class of systems, the analytic bounds are reasonable for the extreme eigenvalues, but can be very conservative for the trace of the inverse Gramian.