Fundamentals of Kalman Filtering: A Practical Approach, Second Edition
Containing numerous real-life problems and examples presented in detail, this text demonstrates the many ways in which Kalman filters can be designed.
TABLE OF CONTENTS Fundamentals of Kalman Filtering: A Practical Approach, Second Edition
Appendix B: Key Formula and Concept Summary
The purpose of this Appendix is to summarize key concepts and formulas that have already been presented in the text in one convenient location.
Overview of Kalman-Filter Operation Principles
If there are no known disturbances, the Kalman filtering equation is simply given by
where, 
is the estimate of the vector of states at the current (kth) time, ? k propagates previous estimates 
to a current estimate based solely on the known system model, H transforms the propagated states into a prediction of what the measurement should be, z k is the actual current vector of measurements, and K k is the Kalman filter gain that converts the discrepancy between actual and predicted measurements into a correction to the states that were propagated using only the known system model.
Kalman-Filter Gains and the Riccati Equations
The Riccati equations are a set of discrete recursive matrix equations that are used to solve for the Kalman gains. The first matrix Riccati equation is given by
where M k is the current covariance matrix of the system predicted states propagated from the covariance of previous state estimates P k ? 1 based solely on the system model incorporated in ? k. Q k is the process noise matrix that represents our uncertainty in our model of the real world. The second matrix Riccati equation is
where K k is the Kalman gain matrix used in the Kalman filter and R k
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