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Computer Science - Theoretical Computer Science | Mathematics of Kalman-Bucy Filtering

Mathematics of Kalman-Bucy Filtering

Ruymgaart, P.A., Soong, Tsu T.



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ISBN 978-3-642-96842-6

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  • About this textbook

Since their introduction in the mid 1950s, the filtering techniques developed by Kalman, and by Kalman and Bucy have been widely known and widely used in all areas of applied sciences. Starting with applications in aerospace engineering, their impact has been felt not only in all areas of engineering but also in the social sciences, biological sciences, medical sciences, as well as all other physical sciences. Despite all the good that has come out of this devel­ opment, however, there have been misuses because the theory has been used mainly as a tool or a procedure by many applied workers without them fully understanding its underlying mathematical workings. This book addresses a mathematical approach to Kalman-Bucy filtering and is an outgrowth of lectures given at our institutions since 1971 in a sequence of courses devoted to Kalman-Bucy filters. The material is meant to be a theoretical complement to courses dealing with applications and is designed for students who are well versed in the techniques of Kalman-Bucy filtering but who are also interested in the mathematics on which these may be based. The main topic addressed in this book is continuous-time Kalman-Bucy filtering. Although the discrete-time Kalman filter results were obtained first, the continuous-time results are important when dealing with systems developing in time continuously, which are hence more appropriately mod­ eled by differential equations than by difference equations. On the other hand, observations from the former can be obtained in a discrete fashion.

Content Level » Research

Keywords » Filtering - Mathematics

Related subjects » Probability Theory and Stochastic Processes - Statistics - Theoretical Computer Science

Table of contents 

1. Elements of Probability Theory.- 1.1 Probability and Probability Spaces.- 1.1.1 Measurable Spaces, Measurable Mappings and Measure Spaces.- 1.1.2 Probability Spaces.- 1.2 Random Variables and “Almost Sure” Properties.- 1.2.1 Mathematical Expectations.- 1.2.2 Probability Distribution and Density Functions.- 1.2.3 Characteristic Function.- 1.2.4 Examples.- 1.3 Random Vectors.- 1.3.1 Stochastic Independence.- 1.3.2 The Gaussian N Vector and Gaussian Manifolds.- 1.4 Stochastic Processes.- 1.4.1 The Hlibert Space L2(?).- 1.4.2 Second-Order Processes.- 1.4.3 The Gaussian Process.- 1.4.4 Brownian Motion, the Wiener-Lévy Process and White Noise.- 2. Calculus in Mean Square.- 2.1 Convergence in Mean Square.- 2.2 Continuity in Mean Square.- 2.3 Differentiability in Mean Square.- 2.3.1 Supplementary Exercises.- 2.4 Integration in Mean Square.- 2.4.1 Some Elementary Properties.- 2.4.2 A Condition for Existence.- 2.4.3 A Strong Condition for Existence.- 2.4.4 A Weak Condition for Existence.- 2.4.5 Supplementary Exercises.- 2.5 Mean-Square Calculus of Random N Vectors.- 2.5.1 Conditions for Existence.- 2.6 The Wiener-Lévy Process.- 2.61 The General Wiener-Lévy N Vector.- 2.6.2 Supplementary Exercises.- 2.7 Mean-Square Calculus and Gaussian Distributions.- 2.8 Mean-Square Calculus and Sample Calculus.- 2.8.1 Supplementary Exercise.- 3. The Stochastic Dynamic System.- 3.1 System Description.- 3.2 Uniqueness and Existence of m.s. Solution to (3.3).- 3.2.1 The Banach Space L2N(?).- 3.2.2 Uniqueness.- 3.2.3 The Homogeneous System.- 3.2.4 The Inhomogeneous System.- 3.2.5 Supplementary Exercises.- 3.3 A Discussion of System Representation.- 4. The Kalman-Bucy Filter.- 4.1 Some Preliminaries.- 4.1.1 Supplementary Exercise.- 4.2 Some Aspects of L2 ([a, b]).- 4.2.1 Supplementary Exercise.- 4.3 Mean-Square Integrals Continued.- 4.4 Least-Squares Approximation in Euclidean Space.- 4.4.1 Supplementary Exercises.- 4.5 A Representation of Elements of H (Z, t).- 4.5.1 Supplementary Exercises.- 4.6 The Wiener-Hopf Equation.- 4.6.1 The Integral Equation (4.106).- 4.7 Kalman-Bucy Filter and the Riccati Equation.- 4.7.1 Recursion Formula and the Riccati Equation.- 4.7.2 Supplementary Exercise.- 5. A Theorem by Liptser and Shiryayev.- 5.1 Discussion on Observation Noise.- 5.2 A Theorem of Liptser and Shiryayev.- Appendix: Solutions to Selected Exercises.- References.

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