Kalman filtering (KF) is a wide class of algorithms designed, in words selected from this outstanding book, “to obtain an optimal estimate” of the state of a system from information in the presence of noise.
Applications of KF were, by 1980, endemic to the aerospace industry. Today their industrial strength descendants are endemic, period, and require much more than the passing insight that is derived from hands-on use of KF software packages. I see this excellent and extremely well-crafted book as the (literally) optimal remediation.
The preface puts forward the intended readership:
This book is designed to serve three purposes. It is written not only for self-study but also for use in a[n] ... introductory course on Kalman filtering theory for upper-division undergraduate or first-year graduate applied mathematics or engineering students.
It is also written to serve as a reference for engineers, and it succeeds more than admirably in achieving both purposes.
The first chapter expounds upon mathematical prerequisites, algorithms, and an application to real-time tracking. There is a very clear derivation and enumeration of real matrix properties, and what amounts to an efficient mini-course in the matrix realization of linear algebra. The lively paced sections on probability, (co)variances, white noise, and least-squares fitting support the notion of the filter being an unbiased, minimum-variance estimate of the (evolving) state of a system. As is true of the remainder of the book, the exercises are well designed and integral to the book’s logical and pedagogical flow. For example, the reader is asked in this chapter to prove that normal distributed random vectors X,Y are (statistically) independent if and only if their covariance is zero.
The second chapter treats the elements of KF. It makes the clarifying distinction among digital filtering, optimal prediction, and digital smoothing, and expresses the book’s scope as digital filtering only. It elaborates this chapter’s KF as a linear, unbiased, minimum-variance, weighted least-squares estimate of a system’s state vector at (discrete) time k using data from previous times. The important role of recursion in real-time storage saving is deemed very helpful in achieving computational space efficiency.
Chapters 3 through 5 served to confirm my hunch that this is a book on the basis of which one could confidently design accurate, efficient, and robust KF software. The scope of these chapters includes relaxation of matrix nonsingularity in deriving the KF algorithm, and the continued prominence of recursion; situations where system and measurement noise are not independent, that is, are correlated as in an aircraft on-board radar guidance system; the autoregressive moving-average model with exogenous (here a rare typographical error) inputs (ARMAX); and nonwhite noise.
Among the most confidence-inspiring attributes of this work are that the notation of the various KF equations is very strictly consistent throughout the book and that the KF equations reduce themselves to the simpler ones of earlier chapters when conditions are relaxed. For example, “[The authors] remark that if the colored [that is, nonwhite] noise processes become white ... then this Kalman filtering algorithm reduces to the one derived in [chapters] 2 and 3.”
Chapters 6 through 12 comprise special cases, refinements, limiting cases, and extensions--all well founded by earlier chapters. The limiting KF is characterized by known matrices that are time independent. The algebraic manipulations are clever to the point of being inspiring, and are explicit and amenable to correctness verification. The computational time and accuracy advantage of the square root algorithm is explicated as requiring inversion of only triangular matrices and “working with the square-root of possibly very large or very small numbers.” I very much applaud the authors for addressing this oft-neglected numerical analysis issue. Ditto regarding Exercise 7.4’s “equality after rounding in the computer.” The sections on the extended KF and on system(-parameter) identification impart real insight along several additional dimensions: linear algebra, probability and statistics, numerical analysis, linearization, and algorithms for parallel computation. Regarding this last, the book very usefully states that “extended Kalman filtering algorithms are ad hoc,” and that there is no “rigorous theory to guarantee the [extended or modified KF’s] optimality.” Chapter 9 treats decoupling an n-dimensional KF algorithm into n independent one-dimensional formulas. Chapter 10 on interval analysis, a time-honored [1] and important subject, is my favorite because numerical analysis issues pervade all of (floating-point) computation. The arsenal of theorems here is small, but very powerful. Chapter 11 is a good introduction to wavelet essentials and can be construed as being advanced in, for example, referring to B-spline functions and compact support. Chapter 12, on sensor networks, takes on extra relevance in this age of the Internet of Things (IoT); Boolean decision variables (active/inactive) are introduced here.
The final chapter comprises notes for further study, including mention of the continuous-time Kalman–Bucy filter; forward and backward estimation (Kalman smoother); and random-disturbance, stochastic optimal control (among other topics). The active, diligent reader will find the introduction to these advanced topics eminently useful.
The book has my highest recommendation, and it will reward readers for careful and iterative study of its text and well-designed exercises.
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