Biomedical Engineering Reference
In-Depth Information
FIGURE 4-23
Proportional controller for a second-order system with unity feedback.
In this case the system transfer function is
K
/
s
(
s
+
3
)
T
(
s
)
=
1
+
K
/
s
(
s
+
3
)
K
=
+
3
s
+
K
The characteristic equation is now a quadratic,
s
2
s
2
+
3
s
+
K
=
0, and the roots are
±
√
b
2
−
b
−
4
ac
s
=
2
a
2
√
9
−
4
K
3
2
±
1
=−
When
K
=
0, the roots are at
3
2
±
3
2
s
=−
=
0or3
This is as expected.
As
K
increases from 0 to 9
4, the root at 0 becomes more negative, moving toward
−
3
/
2, whereas the root at
−
3 becomes more positive, also moving toward
−
3
/
2.
As
K
continues to increase from 9
/
/
4, the roots become complex and split. By the
time
K
=
3, the roots are
2
−
j
3
3
s
=−
4
and
j
3
4
The root locus plot can easily be plotted, as shown in Figure 4-24.
In this case, the response moves from being overdamped to critical damping to being
underdamped for a step input, as shown in Figure 4-25.
As systems become more complicated, it becomes less convenient to plot root loci
manually. Fortunately, MATLAB includes scripts that automate the process, and the pre-
vious example is shown in Figure 4-26.
Note that the open-loop transfer function is represented by the coefficients of the
polynomials describing the numerator and the denominator. The root locus command
assumes that the system includes unity gain feedback with a proportional controller gain
3
2
+
s
=−
