Mathematics of Kalman-Bucy Filtering

The second edition has not deviated significantly from the first. The printing of this edition, however, has allowed us to make a number of corrections which escaped our scrutiny at the time of the first printing, and to generally improve and tighten our presentation of the material. Many of these changes were suggested to us by colleagues and readers and their kindness in doing so is greatly appreciated. Delft, The Netherlands and P. A. Ruymgaart Buffalo, New York, December, 1987 T. T. Soong Preface to the First Edition 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 as all also in the social sciences, biological sciences, medical sciences, as well 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 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.

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 Hilbert 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.- 2.6.1 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.