Biomedical Engineering Reference
In-Depth Information
From the graph, t
¼
0.17 s. Thus, we have
u
ð
t
Þ:
t
y
ð
t
Þ¼
2
:
88 1
e
0
:
17
13.10.3 Identification of a Linear Second-Order System
Consider estimating the parameters of a second-order system, such as
M
‥
þ
B y
þ
Ky
¼
f
ð
t
Þ
.
It is often convenient to rewrite the original differential equation in the standard form for ease
in analysis as
‥
o
2
þ
2zo
n
y
þ
n
y
¼
f
ð
t
Þ
ð
13
:
71
Þ
where o
n
is the undamped natural frequency and z is the damping ratio. The roots of the
characteristic equation are
q
z
2
q
1
z
2
s
1
,
2
¼
zo
n
o
n
1
¼
zo
n
j
o
n
¼
zo
n
j
o
d
ð
13
:
72
Þ
A system with 0
<
z
<
1 is called underdamped, with z
¼
1 critically damped and with
z
>
1 overdamped.
Step Response
The complete response of the system in Eq. (13.72) to a step input with arbitrary initial
conditions is given in Table 13.5, where
is the steady-state value of
(
), and
A
1
,
A
2
,
B
1
,
y
ss
y
t
f
C
2
are the constants that describe the system evaluated from the initial condi-
tions. For the underdamped system, one can estimate z and o
n
,C
1
,and
to a step input with magni-
tude g from data as follows. Consider the step response in Figure 13.67. A suitable model
for the system is a second-order underdamped model (i.e., 0
<
z
<
1), with solution
"
#
e
zo
n
t
1
y
ð
t
Þ¼
C
1
þ
p
cos o
d
t
þ
ð
f
Þ
ð
13
:
73
Þ
z
2
p
1
z
1
z
2
tan
1
where
C
is the steady-state response,
y
ss
, and o
d
¼
o
n
and f
¼
p
þ
p
.
z
2
The following terms illustrated in Figure 13.67 are typically used to describe, quantita-
tively, the response to a step input:
TABLE 13.5
Step Response for a Second-Order System
Damping
Natural Response Equation
e
s
t
þ
A
e
s
t
Overdamped
y
ð
t
Þ¼
y
ss
þ
A
1
2
1
2
e
zo
n
t
cos o
d
t
þ
Underdamped
y
ðÞ¼
y
ss
þ
B
ð
f
Þ
1
Þ
e
zo
n
t
Critically damped
y
ð
t
Þ¼
y
ss
þ
C
1
þ
C
2
t
ð
