Chapter 5
Root Locus Synthesis
Xmath Interactive Control Design Module
5-8
ni.com
Interpreting the Nonstandard Contour Plots
The Root Locus
window can display phase contours other than the standard
180
°
as well as various magnitude contour plots. The meaning of these
curves is simple: if
L
(
s
) =
a
, then
s
would be a closed-loop pole if the loop
transfer function were multiplied by –1/
a
at the frequency
s
. For example,
a point
s
labeled
⏐
L
(
s
)
⏐
= –3 dB on one of the 170
°
curves would be a
closed-loop pole if the loop transfer function at the frequency were to
increase in magnitude by 3 dB and increase in phase by 10
°
.
This simple observation works two ways. Continuing the previous
example, to have a pole at
s
, try to change the current controller to achieve
the required phase shift +10
°
and gain increase (+3 dB) at (for example,
by adding an appropriate pole-zero pair).
On the other hand, if the complex number is a poor place for a closed-loop
pole (for example, very lightly damped or unstable), then the current
compensator is not robust, since only a 10
°
phase shift along with 3 dB
of gain change in loop gain (most likely, the plant) would result in a
closed-loop pole at
s
. In this case, you turn to the problem of synthesizing
new compensation which decreases the phase and magnitude of the loop
transfer function at the frequency
s
. This has the effect of making the
closed-loop system less likely to have a pole at
s
when the plant transfer
function is changed; that is, it results in a more robust design.
Figure 5-3 shows the Root Locus
window with the phase contours turned
off and the 0 dB magnitude contour turned on. The locus shows the set
of all possible closed-loop poles for the modified loop transfer function
L
(
s
) =
e
j
θ
L
(
s
) as
θ
varies from zero to 2
π
. By data-viewing the contour,
you can find the phase shift (value of
θ
) that corresponds to any point on
the locus.
