
(c) The unit step response is (see Equation 2.23)
x
1
(t) ¼ K 1 e
zv
n
t
cos v
d
t þ
zv
n
v
d
sin v
d
t
, t 0(3:172)
The damped natural frequency v
d
is computed from its definition
v
d
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 z
2
q
v
n
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 0:5
2
p
0:4 ¼
ffiffiffi
3
p
5
rad=s
Substituting the system parameter values into Equation 2.23 and simplifying lead to
x
1
(t) ¼ 21 e
t=5
cos
ffiffiffi
3
p
5
t þ
ffiffiffi
3
p
3
sin
ffiffiffi
3
p
5
t
, t 0(3:173)
(d) The continuous-time state vector at steady state
x(1) is obtained from
_
x(1) ¼ A
x(1) þ Bu(1) ¼ 0(3:174)
x(1) ¼A
1
Bu(1)(3:175)
where
A ¼
01
v
2
n
2zv
n
¼
01
(0:4)
2
2(0:5)(0:4)
¼
01
0:16 0:4
B ¼
0
Kv
2
n
¼
0
2(0:4)
2
¼
0
0:32
)
x(1) ¼
01
0:16 0:4
1
0
0:32
[1] ¼
2