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**Kalman filtering. Theory and practice with MATLAB.
4th ed.**
*(English)*
Zbl 1322.93001

Hoboken, NJ: John Wiley & Sons (ISBN 978-1-118-85121-0/hbk; 978-1-118-98498-7/ebook). xvii, 617 p. (2015).

The book “Kalman filtering. Theory and practice with MATLAB” is a well-written text with modern ideas which are expressed in a rigorous and clear manner. It is also a professional reference on Kalman filtering: fully updated, revised, and expanded.

The book consists of nine chapters and appendices and also contains a description of software. The first four chapters (introduction, linear dynamic systems, probability and expectancy, random processes) cover an informal introduction, the essential background material on the dynamic models used in Kalman filtering, especially those represented by systems of linear differential equations, mathematical foundations of probability theory, probability density functions, moments, optimal estimates, and random processes. The chapters 5 and 6 discuss the linear least-mean-squared estimation problems, filtering, prediction and smoothing performance, properties of optimal estimators. The chapter 7, implementation methods, concerns the following problems: how computer roundoff can degrade Kalman filter performance, alternative implementation methods that are more robust against roundoff errors, and the relative computational costs of these alternative implementations. Chapter 8, nonlinear approximations, is about some of the more successful nonlinear, including “extended” Kalman fiters for “quasilinear” problems and tests for assessing whether extended Kalman filtering is adequate for the proposed applications. Chapter 9, practical considerations, contains methods that have been found useful in understanding the observed behavior of Kalman filters and for detecting and correcting anomalous behavior. The material of the previous two chapters (square-root and nonlinear filtering) is evolved in this way and is a part of general subject. The discussion includes more matters of practice than nonlinearities. Chapter 10 presents applications of Kalman filtering for navigation problems. It is hard to imagine an application for Kalman filtering more fruitful than navigation. The book contains many illustrative examples including adaptations for nonlinear filtering, global navigation satellite systems, the error modeling of gyros and accelerometers, inertial navigation systems, and freeway traffic control. A lot of exercises at the end of chapters are a good argument to consider this book as a perfect supplement to any traditional course in systems theory. The book is very useful for people who use Kalman filters, as well as for beginners to study and practice in Kalman filtering.

In this fourth edition, the authors added a new chapter on the probability distributions, added two sections with easier derivations of the Kalman gain, added a section on a new Sigma Rho filter implementation, updated the treatment of nonlinear approximations to Kalman filtering, expanded coverage of applications in navigation, added many more derivations and implementations for satellite and inertial navigation error models, and included many new examples of sensor integration. The problem sets are also updated.

For more practical results and references see the book by B. P. Gibbs [Advanced Kalman filtering, least-squares and modeling: a practical handbook. Hoboken, New Jersey: John Wiley & Sons, Inc. (2011)]

P.S. A historical remark of the authors: “When the Kalman filter was published in 1960, it would find immediate applications in many military systems, including inertial and satellite navigation.” To shed light on the history of the development of the theory and methods of system analysis it is necessary to mention that in 1957 the famous mathematician Yurij Makarovich Berezanskyj wrote a 40 pages report (secret, declassified in 1998) “Investigation of the stability of movement of the missile R-12 in the active phase of flight” in which he described a criterion of stability (controllability) of dynamical systems which is similar to Kalman criterion for controllability.

The book consists of nine chapters and appendices and also contains a description of software. The first four chapters (introduction, linear dynamic systems, probability and expectancy, random processes) cover an informal introduction, the essential background material on the dynamic models used in Kalman filtering, especially those represented by systems of linear differential equations, mathematical foundations of probability theory, probability density functions, moments, optimal estimates, and random processes. The chapters 5 and 6 discuss the linear least-mean-squared estimation problems, filtering, prediction and smoothing performance, properties of optimal estimators. The chapter 7, implementation methods, concerns the following problems: how computer roundoff can degrade Kalman filter performance, alternative implementation methods that are more robust against roundoff errors, and the relative computational costs of these alternative implementations. Chapter 8, nonlinear approximations, is about some of the more successful nonlinear, including “extended” Kalman fiters for “quasilinear” problems and tests for assessing whether extended Kalman filtering is adequate for the proposed applications. Chapter 9, practical considerations, contains methods that have been found useful in understanding the observed behavior of Kalman filters and for detecting and correcting anomalous behavior. The material of the previous two chapters (square-root and nonlinear filtering) is evolved in this way and is a part of general subject. The discussion includes more matters of practice than nonlinearities. Chapter 10 presents applications of Kalman filtering for navigation problems. It is hard to imagine an application for Kalman filtering more fruitful than navigation. The book contains many illustrative examples including adaptations for nonlinear filtering, global navigation satellite systems, the error modeling of gyros and accelerometers, inertial navigation systems, and freeway traffic control. A lot of exercises at the end of chapters are a good argument to consider this book as a perfect supplement to any traditional course in systems theory. The book is very useful for people who use Kalman filters, as well as for beginners to study and practice in Kalman filtering.

In this fourth edition, the authors added a new chapter on the probability distributions, added two sections with easier derivations of the Kalman gain, added a section on a new Sigma Rho filter implementation, updated the treatment of nonlinear approximations to Kalman filtering, expanded coverage of applications in navigation, added many more derivations and implementations for satellite and inertial navigation error models, and included many new examples of sensor integration. The problem sets are also updated.

For more practical results and references see the book by B. P. Gibbs [Advanced Kalman filtering, least-squares and modeling: a practical handbook. Hoboken, New Jersey: John Wiley & Sons, Inc. (2011)]

P.S. A historical remark of the authors: “When the Kalman filter was published in 1960, it would find immediate applications in many military systems, including inertial and satellite navigation.” To shed light on the history of the development of the theory and methods of system analysis it is necessary to mention that in 1957 the famous mathematician Yurij Makarovich Berezanskyj wrote a 40 pages report (secret, declassified in 1998) “Investigation of the stability of movement of the missile R-12 in the active phase of flight” in which he described a criterion of stability (controllability) of dynamical systems which is similar to Kalman criterion for controllability.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93E03 | Stochastic systems in control theory (general) |

93E10 | Estimation and detection in stochastic control theory |

93E11 | Filtering in stochastic control theory |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

62M20 | Inference from stochastic processes and prediction |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |