Kalman filtering is an optimal state estimation process applied to a dynamic system that involves random perturbations. This book studies Kalman filtering for discrete-time models. One of its main objectives is to present a thorough discussion of the mathematical theory and its applications to various elementary real-time problems.

An elementary derivation of the filtering equations is first presented. By assuming that certain matrices are non-singular, this approach enables the reader to easily understand the optimality of the Kalman filter. In the next chapter, these assumptions are dropped by using the well-known method of orthogonal projection. The authors extend this approach first to take care of correlated system and measurement noises, and then to handle colored noise processes. They also discuss Kalman filtering for nonlinear systems, with an application to adaptive system identification. In addition, the book studies steady-state Kalman filtering  theory  (including decoupling of filter equations) and efficient computational schemes for real-time application purposes.

The writing is informal. The mathematical argument is elementary and rigorous throughout, and will be easily readable by anyone with a minimal knowledge of linear algebra and system theory. A preliminary chapter on matrix theory, probability, and least squares is incorporated in an attempt to ensure that this text is self-contained. Each chapter contains a variety of exercises to improve the reader’s understanding of the material. Answers and hints are given at the end of the text. A collection of notes (concerning the Kalman smoother, adaptive Kalman filtering, systolic array implementation, and wavelet analysis) and about 100 references are included for further study. The book is written not only for self-study but also for use in a one-semester introductory course on Kalman filtering theory for first-year graduate students in applied mathematics or engineering.

This edition makes a number of minor corrections to the first edition and updates the references. The section on real-time system identification has been expanded, and a brief introduction to wavelet analysis has been added. This monograph entirely fulfills its objective of giving a rigorous but elementary treatment of Kalman filtering theory and briefly introducing some of its real-time applications.