IEEE - Aerospace and Electronic Systems - May 2020 - 50

The homogeneous state response [5] to this system is given
by
2
3 2
3
À2
À11et þ e2t þ 8eÀt
~
xðtÞ ¼ etA 4 1 5 ¼ 4 11et À 2e2t À 8eÀt 5:
4
À11et À e2t þ 16eÀt
A truncated Taylor series was used to compute the matrix
exponential and solve this initial value problem numerically for t ¼ f1; 2; 3; 4; 5g. Using a two-norm of A and
d ¼ 0:01; the number of terms needed in the summation
was found using (1) to be N ¼ 49: The percent error for
the resultant two-norm of the error vector for each time is
shown in Table 1 for 20 and 49 terms as well as Matlab's
default matrix exponential function, expm.m. While the
summation with 20 terms gives good precision, 49 terms
matches the precision of Matlab.
For further study, the forward and reverse summations
of the Taylor series were implemented in Matlab so as to
impose single-precision arithmetic for all operations.
Using that code, the matrix exponential was computed for

24 72
A¼
:
65 15
In the forward summation, the error accumulated, and calculation of the matrix exponential failed due to numerical
precision. On the other hand, the reverse summation of
Taylor terms mitigates the floating point error and the
matrix exponential is computed accurately. The errors in

Table 1.

Percent Differences Between the Numerical and Analytic
Solutions
t

N = 20

N = 49

expm

0.638%

0.104%

0.0205%

0.0106%

0.164%

0.0012%

Figure 3.
Norm of matrix differences between the forward and backward
Taylor series approximations minus the Matlab expm function.
= jjTaylor expm À Matlab expmjj. The reverse Taylor series
approximation converges to the same value as Matlab, whereas
the forward approximation eventually fails.

the approximations are shown in Figure 3 versus the number of terms in the truncated series. Figure 3 shows that
the forward summation fails at just under 100 terms.
When computing the exponentials of random 6 Â 6
matrices with Taylor series, a favorable trend in computational time was found for jjAjj < 40, as illustrated in
Figure 4. Using the forward summation for convenience of
automatic truncation and increasing the norm shows that
using the forward summation of the Taylor series reduces the
computation time to less than 50% of the Matlab function,
expm.m, for jjAjj < 40: For jjAjj ) 40, the Taylor series
was found to fail to converge to the correct answer. This is a
well-known difficulty of exponential matrix calculations for
nonnormal matrices. Normal matrices are most generally
defined as AÃ A ¼ AAÃ ; where Ã denotes the matrix adjoint.
Examples of normal matrices include orthogonal, symmetric,
skew-symmetric, unitary, Hermitian, and skew-Hermitian
matrices. Normal matrices are discussed further in [6].
While normal matrices may be ideal, it is not a necessary condition. For further illustration, Taylor series were
applied to compute the matrix exponential under drastically different conditions. Applying the algorithm to a set

IEEE A&E SYSTEMS MAGAZINE

MAY 2020

IEEE - Aerospace and Electronic Systems - May 2020

Table of Contents for the Digital Edition of IEEE - Aerospace and Electronic Systems - May 2020