Biomedical Engineering Reference
In-Depth Information
4.7.1 Root Locus
The stability of a system based on the positions of the poles as the controller gains are
changed is known as the root locus method. It is one way to design controllers and to
determine the limits in the gains that can be applied by a controller before instability
results.
The simplest controller that can be implemented is one that includes a gain,
K
, in the
forward path, as shown in Figure 4-21.
The system transfer function is
G
(
s
)
T
(
s
)
=
1
+
G
(
s
)
)
1
+
K
/(
s
+
3
)
K
/(
s
+
3
=
K
s
+
3
+
K
=
The characteristic equation is
s
0,
and this will be at
s
=−
3. However, when
K
>
0, then the pole will occur at
s
=−
(
3
+
K
)
,
so as
K
increases, the pole will become more negative, as shown in Figure 4-22.
The system response can be solved for an impulse or a step response, and it will be
found that the speed of response increases as
K
increases.
In a second-order system, the response is completely different. Consider the propor-
tional controller shown in Figure 4-23, also with gain
K
.
+
3
+
K
=
0, so the open-loop pole occurs when
K
=
FIGURE 4-21
Proportional
controller for a
first-order system
with unity feedback.
FIGURE 4-22
Root locus plot for a
simple controller.
