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

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

by Harold Klee, Randal Allen
April 2016
Intermediate to advanced content levelIntermediate to advanced
815 pages
31h 29m
English
CRC Press
Content preview from Simulation of Dynamic Systems with MATLAB and Simulink, 2nd Edition
(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 denition
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) ¼A
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
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Publisher Resources

ISBN: 9781439836736