Estimation, control, and the discrete Kalman filter.

*(English)*Zbl 0685.93001
Applied Mathematical Sciences, 71. New York etc.: Springer-Verlag. xiii, 274 p. DM 98.00 (1989).

The author has written a very readable book, providing a good introduction to linear estimation and Kalman filtering and smoothing. Mathematical details are not always taken care of, but the book is certainly readable. For example, in the first chapter, where basic probability is reviewed, there is a mixture of elementary undergraduate probability together with a section on the Radon Nikodym theorem, which is used to establish conditional expectations. A brief outline of measure theory is sketched in Appendices A to D, but probability theory is not really developed in a measure theoretic way. The second short chapter provides a motivation and description of minimum variance estimation while the third chapter gives a nice elementary treatment of entropy. Topics from linear algebra such as adjoints, projections and pseudo- inverses of matrices are treated in chapter four, and these are then combined with earlier ideas to discuss linear minimum variance estimation. Bayesian ideas of recursive linear estimation are introduced before the Kalman filter, in discrete time, is treated in chapter seven. Applications and extensions, such as the linear tracking problem and fixed interval smoothing, are covered in the final two chapters. Remaining appendices discuss Hilbert space and the diagonalization of symmetric matrices. By restricting the treatment of the Kalman filter to discrete time all problems of stochastic calculus are avoided and the material is accessible to a wider audience. However, the notation in discrete time is a little messy.

In conclusion the book is, as the author intended, certainly suitable for graduate engineers who have a background which includes some measure theory. Students in applied probability who have studied a measure theoretic based treatment of their subject should be exposed to stochastic differential calculus and a more mathematical approach to filtering.

In conclusion the book is, as the author intended, certainly suitable for graduate engineers who have a background which includes some measure theory. Students in applied probability who have studied a measure theoretic based treatment of their subject should be exposed to stochastic differential calculus and a more mathematical approach to filtering.

Reviewer: R.Elliott

##### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93E10 | Estimation and detection in stochastic control theory |

93E11 | Filtering in stochastic control theory |

93E14 | Data smoothing in stochastic control theory |

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