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Bayesian Estimation and Tracking: A Practical Guide by Anton J. Haug

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3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance

Introducing a notation that will be used throughout the remainder of this text, let an estimate of xn conditioned on all observations up to time tp be written as img with

(3.25) equation

Thus, an estimate of xn that uses all observations, including the current one at time tn, is written as

(3.26) equation

while one that predicts xn based on all but the current observation is given by

(3.27) equation

The same can be done for the covariance matrix, writing

(3.28) equation

with prediction form of the covariance given by

(3.29) equation

3.4.1 State Vector Prediction

Using the Chapman–Kolmogorov equation (3.24) in (3.27) and (3.29) results in

(3.30) equation

and

(3.31)

From the system dynamic equation (3.17), (3.30) can be rewritten as

(3.32)

(3.33)

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