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

Area of Science:

• Signal Processing
• Estimation Theory

Background:

• Multi-sensor nonlinear systems with uncertain observations present significant challenges in state estimation and sensor management.
• The posterior Cramér-Rao bound (PCRB) is a crucial metric for evaluating the performance of state estimators.
• Existing methods for PCRB calculation in uncertain systems can be computationally intensive, especially for large-scale networks.

Purpose of the Study:

• To develop novel methods for deriving the posterior Cramér-Rao bound in multi-sensor nonlinear systems with uncertain observations.
• To address the computational complexity associated with traditional PCRB calculation methods.
• To enable analytical solutions for the sensor selection problem in large-scale sensor networks.

Main Methods:

• A recursive formula-based method utilizing Gaussian mixture models (GMMs) was investigated, though it involves complex integral computations.
• A second method, inspired by the expectation maximization algorithm, introduces latent variables to approximate the GMM, reducing computational burden.
• Continuous variable approximation and a limiting process were employed to derive a new PCRB for discrete uncertain systems, avoiding complex integrals.

Main Results:

• The second proposed method significantly reduces the computational complexity compared to the GMM-based approach.
• The derived PCRB facilitates an analytical solution to the sensor selection problem.
• Numerical examples demonstrate the effectiveness of both proposed methods, particularly for large sensor networks.

Conclusions:

• The developed methods provide efficient and accurate ways to compute the posterior Cramér-Rao bound for uncertain nonlinear systems.
• The analytical sensor selection approach based on the new PCRB is suitable for large-scale applications.
• This work contributes to improved performance and resource management in multi-sensor estimation systems.