Nonlinear Kalman Filtering in the Absence of Direct Functional Relationships Between Measurement and State

This letter introduces a Kalman Filter framework for systems with process noise and measurements characterized by state-dependent, nonlinear conditional means and covariances. Estimating such general nonlinear models is challenging because traditional methods, such as the Extended Kalman Filter, linearize only functions – not noise – and require state-independent covariances. These limitations often necessitate Bayesian approaches that rely on specific distribution assumptions. To address these challenges, we propose a framework that employs a recursive least squares method that relies solely on conditional means and covariances, eliminating the need for explicit probability distributions. By applying first-order linearizations and incorporating targeted modifications to manage state dependence, the filter simplifies implementation, reduces computational demands, and provides a practical solution for systems that deviate from the assumptions underlying traditional Kalman filters. Simulation results on a compartmental model demonstrate performance comparable to sequential Monte Carlo methods while significantly lowering computational costs, effectively addressing real-world challenges of scalability and precision.