Biomedical Engineering Reference
In-Depth Information
FIGURE 4-17
System block
diagram for the
disturbance input.
4.7
SYSTEM STABILITY
A system is considered to be stable if every bounded input produces a bounded output. An
alternative definition is that a system is stable if the impulse response dies away to zero.
A transfer function,
G
(
s
)
, can be written as the ratio of two polynomials that can be
factorized into their respective roots, where the roots can be real or can occur in complex
pairs,
K
)
(
s
+
p
1
)(
s
+
p
2
)
···
(
s
+
p
n
)
(
s
+
z
1
)(
s
+
z
2
)
···
(
s
+
z
m
G
(
s
)
=
(4.75)
The roots of the numerator
z
1
,
z
2
,...
z
m
, are called zeros, and the roots of the denominator
p
1
,
p
2
,...
p
n
, are called poles.
The zeros are the values of
s
for which the transfer function becomes zero, while the
poles are the values of
s
for which the transfer function becomes infinite.
Consider a simple transfer function
s
−
1
G
(
s
)
=
(4.76)
s
2
+
4
s
+
5
The denominator can be factorized to produce a complex pole pair and plotted on the
s
-plane in Figure 4-18.
s
−
1
G
(
s
)
=
(4.77)
(
s
−
[1
+
j
2]
)(
s
−
[1
−
j
2]
)
