The aim of this book is to present graduate students with a thorough survey of reference probability models and their applications to optimal estimation and control. These new and powerful methods are particularly useful in signal processing applications where signal models are only partially known and are in noisy environments. Well-known results, including Kalman filters and the Wonheim filter emerge as special cases. The authors begin with discrete time and discrete state spaces. From there, they proceed to cover continuous time, and progress from linear models to non-linear models, and from completely known models to only partially known models. Readers are assumed to have basic grounding in probability and systems theory as might be gained from the first year of graduate study, but otherwise this account is self-contained. Throughout, the authors have taken care to demonstrate engineering applications which show the usefulness of these methods.

The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achiev able now than ever before, and the need has never been greater. Of course, when we use the term high peiformance control we are thinking of achieving this in the real world with all its complexity, uncertainty and variability. Since we do not expect to always achieve our desires, a more complete title for this book could be "Towards High Performance Control." To illustrate our task, consider as an example a disk drive tracking system for a portable computer. The better the controller performance in the presence of eccen tricity uncertainties and external disturbances, such as vibrations when operated in a moving vehicle, the more tracks can be used on the disk and the more memory it has. Many systems today are control system limited and the quest is for high performance in the real world."

The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achiev able now than ever before, and the need has never been greater. Of course, when we use the term high peiformance control we are thinking of achieving this in the real world with all its complexity, uncertainty and variability. Since we do not expect to always achieve our desires, a more complete title for this book could be "Towards High Performance Control." To illustrate our task, consider as an example a disk drive tracking system for a portable computer. The better the controller performance in the presence of eccen tricity uncertainties and external disturbances, such as vibrations when operated in a moving vehicle, the more tracks can be used on the disk and the more memory it has. Many systems today are control system limited and the quest is for high performance in the real world."

As more applications are found, interest in Hidden Markov Models continues to grow. Following comments and feedback from colleagues, students and other working with Hidden Markov Models the corrected 3rd printing of this volume contains clarifications, improvements and some new material, including results on smoothing for linear Gaussian dynamics.

In Chapter 2 the derivation of the basic filters related to the Markov chain are each presented explicitly, rather than as special cases of one general filter. Furthermore, equations for smoothed estimates are given. The dynamics for the Kalman filter are derived as special cases of the authors general results and new expressions for a Kalman smoother are given. The Chapters on the control of Hidden Markov Chains are expanded and clarified. The revised Chapter 4 includes state estimation for discrete time Markov processes and Chapter 12 has a new section on robust control.

This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys tems implementation, either in continuous time or discrete time, which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods."