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 H∞ 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.

Yuriy S. Shmaliy, IEEE Life Fellow, AAIA Fellow, AIIA Fellow (UFIR founder, Shmaliy’s Discrete Orthogonal Polynomials originator), earned BS, MS, and PhD degrees in Electrical Engineering from Kharkiv Aviation Institute, Kharkiv, Ukraine, in 1974, 1976, and 1982, respectively, and the Dr.Sc. degree in Electrical Engineering from USSR Government, in 1992. Since 1986, he has been a Full Professor. From 1985 to 1999, he was with Kharkiv Military University, Kharkiv, Ukraine. In 1992, he founded the Scientific Center Sichron and was the Director by 2002. Since 1999, he has been with the Universidad de Guanajuato, Guanajuato, Mexico, and from 2012 to 2015, he headed the Department of Electronics Engineering in this University.

He has 564 journal and conference papers and holds 81 patents. He has authored the books Continuous-Time Signals (Springer, 2006), Continuous-Time Systems (Springer, 2007), GPS-Based Optimal FIR Filtering of Clock Models (Nova Science Publ., 2009), and Optimal and Robust State Estimation: Finite Impulse response (FIR) and Kalman Approaches (Wiley & Sons, 2022)--One of the best estimation theory books of all time by BookAuthority. He also edited the book Probability: Interpretation, Theory and Applications (Nova Science Publ., 2012).

Prof. Shmaliy has pioneered the theory of optimal and robust Finite Impulse Response (FIR) state estimation and coined the Unbiased FIR (UFIR) State Estimator, which is now widely used by the filtering research community to solve diverse estimation problems as a robust alternative to Kalman filter. He discovered a new class of discrete orthogonal polynomials (DOP). To recognize his pioneering contributions, the DOP named after him are called "Discrete Shmaliy Moments" or “Shmaliy DOP,” and developed to “Discrete Shmaliy Transform.” He was rewarded a title, Honorary Radio Engineer of the USSR, in 1991, was with the Ukrainian State Award Committee on Science and Technology, in 1998-1999, and has been IEEE Fellow Committee Member, in 2023-2026. He was the recipient of the Royal Academy of Engineering Newton Research Collaboration Program Award, in 2015, IEEE Latin America Eminent Engineer Award, in 2021, and several best conference paper awards. He was invited many times to give tutorial, seminar, and plenary lectures.