Simulation of Dynamic Systems with MATLAB and Simulink, 2nd Edition

the need to solve four simultaneous equations for the unknown constants c

1

, c

2

, d

1

, and d

2

.

Substituting the values for c

1

, c

2

, d

1

, and d

2

into Equation 4.99 yields

Y(s) ¼

1=15

s

þ

1=12

s þ 3

þ

(3=20)s  1=20

s

2

þ 2s þ 5

(4:104)

¼

1

15

1

s



þ

1

12

1

s þ 3





1

20

3s þ 1

s

2

þ 2s þ 5



(4:105)

The last term is inverted using Equation 4.97 with d

1

¼3, d

2

¼1, a ¼1, and b ¼2.

Y(t) ¼

1

15

þ

1

12

e

3t



1

20

e

t

3 cos 2t þ

3(1) þ 1

2



sin 2t



(4:106)

¼

1

15

þ

1

12

e

3t



1

20

e

t

(3 cos 2t  sin 2t)(4:107)

The second method for inverse Laplace transforming terms like the one in Equation 4.97 is based on

decomposing it into two terms that can be readily inverted. Starting with an expression containing