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For now let it suffice to point out that the Kalman filter maintains the first two moments of the state distribution, The a posteriori state estimate (1.7) reflects the mean (the first moment) of the state distribution¯ it is normally distributed if the conditions of (1.3) and (1.4) are met.The a posteriori estimate error covariance (1.6) reflects the variance of the state distribution (the second non-central moment).As such, the equations for the Kalman filter fall into two groups: time update equations and measurement update equations.The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step.The Kalman filter instead recursively conditions the current estimate on all of the past measurements.Figure 1-2 below offers a complete picture of the operation of the filter, combining the high-level diagram of Figure 1-1 with the equations from Table 1-1 and Table 1-2.Again notice how the time update equations in Table 1-1 project the state and covariance estimates from time step k to step k 1. Initial conditions for the filter are discussed in the earlier references.The first task during the measurement update is to compute the Kalman gain, .

The next step is to actually measure the process to obtain , and then to generate an a posteriori state estimate by incorporating the measurement as in (1.12).

We define (note the "super minus") to be our a priori state estimate at step k given knowledge of the process prior to step k, and to be our a posteriori state estimate at step k given measurement .

We can then define a priori and a posteriori estimate errors as The a priori estimate error covariance is then and the a posteriori estimate error covariance is In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that computes an a posteriori state estimate as a linear combination of an a priori estimate and a weighted difference between an actual measurement and a measurement prediction as shown below in (1.7).

On the other hand, as the a priori estimate error covariance approaches zero the actual measurement is trusted less and less, while the predicted measurement is trusted more and more.

The justification for (1.7) is rooted in the probability of the a priori estimate conditioned on all prior measurements (Baye's rule).

The Kalman filter addresses the general problem of trying to estimate the state of a first-order, discrete-time controlled process that is governed by the linear difference equation with a measurement that is The random variables and represent the process and measurement noise (respectively).

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