The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations

This paper considers the problems of the posterior Cramér–Rao bound and sensor selection for multi-sensor nonlinear systems with uncertain observations. In order to effectively overcome the difficulties caused by uncertainty, we investigate two methods to derive the posterior Cramér–Rao bound. The first method is based on the recursive formula of the Cramér–Rao bound and the Gaussian mixture model. Nevertheless, it needs to compute a complex integral based on the joint probability density function of the sensor measurements and the target state. The computation burden of this method is relatively high, especially in large sensor networks. Inspired by the idea of the expectation maximization algorithm, the second method is to introduce some 0–1 latent variables to deal with the Gaussian mixture model. Since the regular condition of the posterior Cramér–Rao bound is unsatisfied for the discrete uncertain system, we use some continuous variables to approximate the discrete latent variables. Then, a new Cramér–Rao bound can be achieved by a limiting process of the Cramér–Rao bound of the continuous system. It avoids the complex integral, which can reduce the computation burden. Based on the new posterior Cramér–Rao bound, the optimal solution of the sensor selection problem can be derived analytically. Thus, it can be used to deal with the sensor selection of a large-scale sensor networks. Two typical numerical examples verify the effectiveness of the proposed methods.