Transformada de Laplace e Saída do Sistema

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Problem 3.12PP
Consider the standard second-order system
G(s) = -
^ + 2^Q)nS +
(a) Write the Laplace transform of the signal in Fig.
(b) What is the transform of the output if this signal is appiied to Gfs^?
(c) Find the output of the system for the input shown in Fig. 
Figure Plot of input signai
t “ ( ') I I I
Figure Plot of input signai
«(0
1 2 3
Time (sec)
Step-by-step solution
Step 1 of 8
(a)
The input signai is.
Define the signal, .
u{l) = t ; 0 £ / £ l 
=1 ; l £ ( £ 2 
= - / ; 2 S / S 3
Step 2 of 8
Write the formula for the Laplace transform. F W of a function, / ( ' )
Define the signal, . 
! / ( / ) » / ; 0 £ / £ l
= - / ; 2 S r S 3
Step 2 of 8
Write the formula for the Laplace transform. F W of a function, / ( ' )
Ri ih^titiitp thp fi inrtinn f / t \
The transform of the output Y ( j ) is.
Y{s) = G{s)U(s)
3— p ------ ^ f o r ')+ i( -3 e -* ' + 3e-*') f
S
U (s)
a i ^ [ l - e ' * - ( l + 3 5 )e ^ + ( l + 3 j)g '* * ]
5* + 2^01^ + )
j ' ( i " + 2 ,------ ---------- tt .
' j " ( i ’ +2fflV> + “ 2)
Apply inverse Laplace transform. 
ait O A
+ - T + -
s + C
s^[s^+2Ca>^ + a i) s + +
Step 5 of 8
Equate the coefficients on both sides. 
B + D ^ O 
A + C +2^o)^D — 0 
2 C (o ,A + o iD ^0 
A o i = a i 
A = l
Determine D.
2 (a ,( - l ) + a iD = 0
b = - K
Determine C.
l + C + 2