## Tfctfc J

along with the nonlinear equations, Xo = F(Xo,t), which defile the nominal trajectory, Xo(t). The matrix A(t) is evaluated on the reference trajectory,

where F(X*,t) is the time derivative of the state vector in the differential equations governing the time evolution of the system. The observation-state mapping matrix is given by

where G(X*,tj) are the observation-state relationships evaluated on the nominal or reference trajectory.

Notice that the solution for X0 involved inversion of the information matrix, Ao, where

Generally the normal equation would not be solved by a direct inversion of A0 but rather would be solved by an indirect but more accurate technique, such as the Cholesky decomposition described in Chapter 5, Section 5.2. The sequence of operations required to implement the batch estimation process is outlined in Fig. 4.6.1. Note that the algorithm in Fig. 4.6.1 assumes that there are no observations at t0. If observations exist at t0, set A = P0 + Hf R-1H0 and N = Hf R-1y0 in the initialization box, A. As previously stated, the entire sequence of computations are repeated until the estimation process has converged. If there are observations at t0, the state transition matrix for processing these observations is the identity matrix.

This procedure yields a minimum value of the performance index (see Eq. (4.3.24))

and ej is the best estimate of the observation error.

In practice, P0 is generally not a realistic representation of the accuracy of X0 and it is used only to better condition the estimation error covariance matrix, P. In this case, X0 usually is set to zero for each iteration and P0 is chosen to be a diagonal matrix with large diagonal values. Hence, the frs t term in Eq. (4.6.8) will be very small and the tracking data residuals will determine the value of J(x). The RMS of the observation residuals generally is computed and may be used as a measure of convergence; when the RMS no longer changes the solution is assumed to be converged. The RMS is computed from

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