Recursive State Estimation

Since 1960-1961, when Rudolf E. Kalman has published his seminal work on discrete-time recursive optimal filtering and, together with Richard S. Bucy, on continuous-time optimal filtering of linear nonstationary stochastic processes with white Gaussian noise, recursive filtering and smoothing algorithm have become and still remain a key tool for real-time state estimation. This is despite the fact that Bayesian and convolution-based approaches say that optimal recursions are only available for white Gaussian and colored Gauss-Markov noise. Otherwise, since non-Gaussian noise has high-order statistics, the hypothetical recursive forms seem to be so complex that it is hardly practical to use them instead of batch forms. Therefore, recursive forms are widely used, although this often calls into question their accuracy in harsh environments.

When solving state estimation problems for signal processing and control using recursive algorithms, researchers traditionally associate them with Kalman filtering, even when not using all its recursive forms. This is even though some solutions, such as the robust iterative UFIR filter, as well as the transfer function-based H8 filter, generalized H2 filter, L1 filter, etc., have nothing to do with Kalman filtering. Moreover, data-driven and AI-aided model-based filtering algorithms also lose connection to it. This leads to the idea that instead of thinking of recursive algorithms as Kalman-like, it is worth focusing on the general recursive form and cover all available recursive state estimators under one umbrella, treating Kalman filter is a special case.

This book attempts to do this by describing 53 pseudo codes and other forms of optimal, suboptimal, and robust recursive state estimation algorithms.