Biomedical Engineering Reference
In-Depth Information
FIGURE 4-18
S-plane plot
showing the poles
and zeros.
The stability can be determined by considering how the output changes with time after an
impulse. For example, consider a first-order system with a pole at
−
3
1
s
+
3
G
(
s
)
=
The output and input are related by
Y
(
s
)
=
X
(
s
)
G
(
s
)
For a unit impulse,
X
(
s
)
=
1; therefore,
1
s
+
Y
(
s
)
=
G
(
s
)
=
3
Taking the inverse Laplace transform gives
e
−
3
t
y
(
t
)
=
which decreases to zero as time increases.
Now consider a similar system with the pole at
+
3
1
s
−
3
Repeating the previous analysis results in the following response:
G
(
s
)
=
e
+
3
t
y
(
t
)
=
This increases with time; therefore, the system is unstable.
In general terms, the output of a system excited by an impulse will be the sum of a
number of exponential terms. If any of those exponentials increases with time, the system
is unstable.
In terms of the poles, this translates into the following, as illustrated in Figure 4-19:
If all of the poles are in the left-hand half of the plane (shown shaded), then the system is
stable. If any lie on the y-axis, the system is said to be critically stable, and if any are in
the right-hand half of the plane the system will be unstable.
