Biomedical Engineering Reference
In-Depth Information
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
T
r
T
p
T
s
0
0.02
0.04
0.06
0.08
0.1
t (s)
FIGURE 13.67
Sample second-order response with graphical estimates of rise time, time to peak overshoot,
and settling time.
Rise time,
T
r
. The time for the response to rise from 10 to 90 percent of steady state.
Settling time,
T
s
. The time for the response to settle within
5 percent of the steady-state
value.
Peak overshoot time,
T
p
. The time for the response to reach the first peak overshoot.
Using values graphically determined from the data for the preceding quantities, as
shown in Figure 13.67, provides estimates for the parameters of the model. The time to peak
overshoot,
T
p
, is found by first calculating
e
zo
n
t
@
y
@
t
¼
g
p
1
½
zo
n
cos o
d
t
þ
ð
f
Þ
o
d
sin o
d
t
þ
ð
f
Þ
ð
13
:
74
Þ
z
2
K
from
@
@
t
using the chain rule on Eq. (13.74) and then determining
T
p
t
¼
T
p
¼
0
, yielding
p
T
p
¼
p
1
ð
13
:
75
Þ
z
2
o
n
The response at
T
p
is
!
p
1
p
z
z
2
yT
ðÞ¼
C
1
þ
e
ð
13
:
76
Þ
and is found as follows
2
3
0
1
p
1
o
n
p
zo
n
z
2
þ
e
1
p
4
5
@
A
y
ð
T
p
Þ¼
C
1
p
cos o
d
p
1
þ
f
z
2
z
2
o
n
