Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 165

and

F( s ) ¼ ( sI A ) 1

( 3 : 198 )

Therefore,

w( t ) ¼ L 1 [( sI A ) 1

] ( 3 : 199 )

Example 3.49

Find e At if

3 1

A ¼

0 4

S OLUTION
Using Laplace transform, we have

1

1

s þ 31 s þ 4 1

(sI A) 1 F (s) ¼ ¼ ¼

0 s þ 4

s 2

0 s þ 3

þ 7s þ 12

or

2
4

3
5

s þ 4 1

(s þ 3)(s þ 4) (s þ 3)(s þ 4)

F (s) ¼

s þ 3

0

(s þ 3)(s þ 4)

Partial fraction of the entries of matrix F (s) yields

2
4

3
5

1
s þ

1
s þ

1
s þ 3 þ 3 4

F (s) ¼

1
s þ

0

4

Hence,

e 3t e 3t þ e 4t

e At ¼ L 1 (sI A) 1 ¼

e 4t

0

3.11.7 M ATRIX E XPONENTIAL F UNCTION A k
Matrix exponential A k has applications in design of discrete time-control systems. It
is the state-transition matrix of LTI discrete systems in state-space form. It can be