Probabilistic Performance Bounds for Randomized Sensor Selection in Kalman Filtering
We consider the problem of randomly choosing the sensors of a linear time-invariant dynamical system subject to process and measurement noise. We sample the sensors independently and from the same distribution. We measure the performance of a Kalman filter by its estimation error covariance. Using tools from random matrix theory, we derive probabilistic bounds on the estimation error covariance in the semi-definite sense. We indirectly improve the performance of our Kalman filter for the maximum eigenvalue metric and show that under certain conditions the optimal sampling distribution that minimizes the maximum eigenvalue of the upper bound is the solution to an appropriately defined convex optimization problem. Our numerical results show the efficacy of the optimal sampling scheme in improving Kalman filter performance relative to the trivial uniform sampling distribution and a greedy sampling $\textit{with replacement}$ algorithm.