Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 163

This series is a convergent series for all values of t and can be computed using
different techniques such as Cayley Hamilton and Laplace transform. The derivative

-

of e At with respect to t is

d e At
d t ¼ A þ

2A 2 t 2
1

A 3 t 2
2

A n t n 1
( n

þ

! þþ )! þ

! 1

A 2 t 2
2

A 3 t 3
3

¼ AI þ At þ

! þ ! þ

¼ Ae At ¼ e At A

( 3 : 193 )

The integral of the exponential function e At is

ð

t

e A t dt ¼ A 1 ( e At I ) ( 3 : 194 )

0

The above integral is valid if and only if matrix A is nonsingular. If A is singular,
there is no simple closed-form solution.

Example 3.48
Find Ð 0 e At d t if

01

(a) A ¼

2 3

2 2

(b) A ¼

1 1

S OLUTION
(a) First we nd e At using Cayley Hamilton technique:

-

¼

jlI Aj¼ l

1

( l þ 1)( l þ 2) ¼ 0 ! l 1 ¼ 1, l 2 ¼ 2

2 l þ 3

The corresponding eigenvectors are

T

Ax 1 ¼ l 1 x 1 , therefore x 1 ¼ ½ 1 1

T

Ax 2 ¼ l 2 x 2 , therefore x 2 ¼ ½ 1 2