The Extended Sequential Estimation Algorithm

To minimize the effects of errors due to the neglect of higher order terms in the linearization procedure leading to Eq. (4.2.6), the extended form of the sequential estimation algorithm is sometimes used. This algorithm is often referred to as the Extended Kalman Filter (EKF). The primary difference between the sequential and the extended sequential algorithm is that the reference trajectory for the extended sequential algorithm is updated after each observation to refect the best estimate of the true trajectory. For example, after processing the kth observation, the best estimate of the state vector at is used to provide new initial conditions for the reference trajectory,

Using Xk for the reference trajectory leads to xk = 0, which will result in xk+1 = 0. The integration for the reference trajectory and the state transition matrix is reinitialized at each observation epoch, and the equations are integrated forward from to tfc+1. The estimate for Xk+1 is then computed from

where Kk+1 and yk+1 are computed based on the new reference orbit. Then, the reference orbit is updated at time ik+1 by incorporating Xk+1 and the process proceeds to tk+2. The process of incorporating the estimate at each observation point into the reference trajectory for propagating to the next observation epoch leads to the reference trajectory being the prediction of the estimate of the nonlinear state; for example, X*(i) = X(t).

In actual practice, it is not a good idea to update the reference trajectory using the fist observations. This is particularly true if the observations contain signifi cant noise. After a few observations have been processed, the estimates of X will stabilize, and the trajectory update can be initiated.

The advantage of the extended sequential algorithm is that convergence to the best estimate will be more rapid because errors introduced in the linearization process are reduced. In addition, because the state estimate deviation, X(t), need not be mapped between observations, it is not necessary to compute the state transition matrix. The estimation error covariance matrix, P(t), can be mapped by integrating the matrix differential equation (4.9.35) discussed in Section 4.9.

The major disadvantage of the extended sequential algorithm is that the differential equations for the reference trajectory must be reinitialized after each observation is processed.

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