T t w t H 2 iI S

Figure 4.9.1: A white random sequence.

r(ife+i,ife) is referred to as the process noise transition matrix, and Eq. (4.9.47) is an n x m quadrature since $(ik+1,r) and b(t) are known functions. Using the defnit ion of the estimation error covariance matrix

Note that E[(Xk — xk)uT] = 0; that is, uk cannot affect the estimation error at since a fhite time must evolve for uk to affect the state. Finally, carrying out the expectation operation in Eq. (4.9.49) yields

Pfe+1 = $(ife+i,ife)Pfe$T(ife+i,ife) + r(ife+i,ife)QferT(ifc+i,ifc). (4.9.50)

The estimation error covariance matrix Pk+1 can be obtained by integrating the differential equation for P given by Eq. (4.9.35), or Pk+1 may be obtained by using the state and process noise transition matrices as indicated in Eq. (4.9.50). A comparison of Eq. (4.9.35) and Eq. (4.9.50) indicates the following:

1. Since P(t) is symmetric, only n(n + 1)/2 of the n x n system of equations represented by Eq. (4.9.35) must be integrated. However, the n(n + 1)/2 equations are coupled and must be integrated as a single fist-order system of dimension n(n + 1)/2.

2. The n x n system represented by Eq. (4.9.50) can be separated into an n x n system of differential equations for $ and an n x m quadrature for

0 0

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