Generalized Concentration-Based Performance Guarantees on Sensor Selection for State Estimation

Abstract

In this work, we apply concentration-based results to the problem of sensor selection for state estimation to provide us with meaningful guarantees on the properties of our selection. We consider a selection of sensors that is randomly chosen with replacement for a stochastic linear dynamical system, and we utilize the Kalman filter to perform state estimation. Our main contributions are four-fold. First, we derive novel matrix concentration inequalities (CIs) for a sum of positive semi-definite random matrices. Second, we provide two algorithms for specifying the parameters required to apply our matrix CIs, a novel statistical tool. Third, we propose two avenues for improving the sample complexity of this statistical tool. Fourth, we provide a procedure for optimizing the semi-definite bounds of our matrix CIs. When our matrix CIs are applied to the problem of sensor selection for state estimation, our final contribution is a procedure for optimizing the filtered state estimation error covariance matrix of the Kalman filter. Finally, we show through simulations that our bounds significantly outperform those of an existing matrix CI and are applicable for a larger parameter regime. Also, we demonstrate the applicability of our matrix CIs for the state estimation of nonlinear dynamical systems.