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. Each sensor is sampled independently and from the same distribution for the purpose of state estimation by Kalman filtering. Due to our randomized sampling procedure, the estimation error covariance cannot be bounded in a deterministic sense. Using tools from random matrix theory, we derive probabilistic bounds on the steady-state estimation error covariance in the semi-definite sense for an arbitrary sampling distribution. Our bounds are functions of several tunable parameters of interest, such as the number of sampled sensors and the likelihood that our bounds hold. We indirectly improve the performance of our Kalman filter for the maximum eigenvalue metric by finding the optimal sampling distribution. By minimizing the maximum eigenvalue of the upper bound, we are able to minimize the maximum eigenvalue of the steady-state estimation error covariance, the actual metric of interest. We identify the subset of sensors to sample with high frequency through the optimal sampling distribution. We illustrate our findings through several insightful simulations and comparisons with multiple sampling policies.