When the dynamics of the system of interest evolve in continuous time, but analysis and implementation are more convenient in discrete-time, we will require a means for determining a discrete-time model in the form of eqn. (4.65) which is equivalent to eqn. (4.57) at the discrete-time instants 
Specification of the equivalent discrete-time model requires computation of the discrete-time state transition matrix
for eqn. (4.65) and the process noise covariance matrix Qd for eqn. (4.67). These computations are discussed in the following two subsections for time invariant systems. A method to compute
and Qd over longer periods of time for which F or Q may not be constant is discussed in Section 7.2.5.2.
Calculation of $fc from F(t)
For equivalence at the sampling instants when F is a constant matrix, $ can be determined as in eqn. (3.61):
where
is the sample period. Methods for computing the matrix exponential are discussed in Section B.12.
Example 4.20 Assume that for a system of interest,
and the submatrices denoted by
and
are constant over the
interval
Then,
Expanding the Taylor series of
is straightforward, but tedious. The result is
which is the closed form solution. When F33 can be approximated as zero, the following reduction results
Eqn. (4.104) corresponds to the F matrix for certain INS error models after simplification.
If the state transition matrix is required for a time interval
of duration long enough that the F matrix cannot be considered constant, then it may be possible to proceed by subdividing the interval. When the interval can be decomposed into subintervals
where
and the F matrix can be considered constant over intervals of duration less than t, then by the properties of state transition matrices,
where
is defined as in eqn. (4.103) with F considered as constant for
The transition matrix
) is defined from previous iterations of eqn. (4.107) where the iteration is initialized at t = 
with
The iteration continues for the interval of time propagation to yield
Calculation of Qdfc from Q(t)
For equivalence at the sampling instants, the matrix
must account for the integrated effect of w(t) by the system dynamics over each sampling period. Therefore, by integration of eqn. (4.57) and comparison with eqn. (4.65), wk must satisfy
Comparison with eqn. (4.65) leads to the definition:
Then, with the assumption that w(t) is a white noise process, we can compute
as follows:
Therefore, the solution is
If F and Q are both time invariant, then Qd is also time invariant. A common approximate solution to eqn. (4.110) is
which is accurate only when the eigenvalues of F are very small relative to the sampling period T (i.e., ||FT|| ^ 1). This approximation does not account for any of the correlations between the components of the driving noise wk that develop over the course of a sampling period due to the integration of the continuous-time driving noise through the state dynamics. The reader should compare the results from Examples 4.21, 4.22, and 4.23.
Example 4.21 For the double integrator system with
and Qd using eqns. (4.103) and (4.111) for T = 1.
Using the Matlab function ‘expm’ we obtain
Using eqs.(4.111) we obtain
Given the state of computing power available, there is no reason why more accurate approximations for Qd are not used. A few methods for calculating the solution to eqn. (4.110) are described in the following subsections. Each of the following methods assumes that F and Q are both time invariant. If F and Q are slowly time varying, then these methods could be used to determine approximate solutions by recalculating Qd over each sampling interval.
Solution by Matrix Exponentials
It is shown in [133] that the exponential of the 2n x 2n matrix
is
where D is a dummy variable representing a portion of the answer that will not be used. Based on the expressions in the second column of eqn. (4.114),
are calculated as
where
denotes the the sub-matrix of
composed of the i through j-th rows and k through Z-th columns of matrix
Example 4.22 For the simple system defined in Example 4-21, after constructing
and computing its matrix exponential, we have
Therefore, using the lower right 2 x 2 block, we have that
Using the upper right 2×2 block we have that
which is significantly different from the result in eqn. (4.112) that was obtained from the approximate solution in eqn. (4.111).
Solution by Taylor Series
If the Taylor series approximation for
is substituted into eqn. (4.110), using k = 0 to simplify the notation, the result accurate to third order in F and 4th order in T is
Although the Taylor series approach is approximate for some implementations, eqn. (4.119) is sometimes a convenient means to identify a closed form solution for either
or Qd. Even an approximate solution in closed form such as eqn. (4.119) is useful in situations where F is time-dependent and eqns. (4.114-4.116) cannot be solved on-line.
Example 4.23 For the simple system defined in Example 4-21,
for
therefore, using eqn. (4.118)
Simplifying eqn. (4.119) for this specific example gives
which provides the same result as in Example 4.22.
