Pfc1 [xfei xfeixfei xfei

and substituting Eqs. (4.9.46) and (4.9.19) into (4.9.48) leads to r. Furthermore, the n x n system of equations represented by the solution for $(tk+1,tk) can be integrated as a sequence of n x 1 column vectors.

The comparison between the two methods indicates that integration of fewer equations is required to obtain the solution for P(t) with Eq. (4.9.35). However, the integration of these equations may be more difficult than the integration associated with the larger system represented by Eq. (4.9.50) since they are coupled.

The equations for determining x using the sequential processing algorithm are unchanged whenever a zero-mean process noise is included. However, as has been shown, the equations that propagate the estimation error covariance matrix do change; that is, the first of Eq. (4.7.31) is replaced by Eq. (4.9.50). Generally, the batch processing algorithm is not used with process noise because mapping of the process noise effects from the observation times to the epoch time is cumbersome. It further results in a normal matrix with an observation weighting matrix that will be nondiagonal and whose dimension is equal to m x to, where m is the total number of observations. Computation of the inverse of the normal matrix will be so cumbersome that the normal equation solution involving process noise in the data accumulation interval is impractical. For example, a one-day tracking period for the TOPEX/Poseidon satellite by the French tracking system, DORIS, typically yields 5000-7000 doppler observations.

The advantage of using the process noise compensated sequential estimation algorithm lies in the fact that the asymptotic value of P(t) will approach a nonzero value determined by the magnitude of Q(t). That is, for certain values of Q(t), the increase in the state error covariance matrix P(t) during the interval between observations will balance the decrease in the covariance matrix that occurs at the observation point. In this situation, the estimation procedure will always be sensitive to new observations.

The question of how to choose the process noise covariance matrix, Q(t), is complex. In practice, it is often chosen as a simple diagonal matrix and its elements are determined by trial and error. Although this method can be effective for a particular estimation scenario, such a process noise matrix is not generally applicable to other scenarios. The dynamic evolution of the true states of parameters estimated in a filter typically is affected by stochastic processes that are not modeled in the fite r's deterministic dynamic model. The process noise model is a characterization of these stochastic processes, and the process noise covari-ance matrix should be determined by this process noise model. Development of the process noise model will not be presented in depth here; however, extensive discussions are given by Cruickshank (1998), Ingram (1970), and Lichten (1990).

The Gauss-Markov process is used as a process noise model and will be introduced here. It is computationally well suited for describing unmodeled forces since it obeys Gaussian probability laws and is exponentially correlated in time.

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