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magnitudes and phase angles of the image points A'-F' are as follows: I V;-I -. 00,
zvk -. o; lvhl = oo, /Vh = -90 deg; iv/i - o, /V/ = -180 deg; ] v/i = o,
zv/ = o; iv~i = o, /vL = 180 deg; and iv/i = oo, / V~ = 90 deg.
The Nyquist plot is shown in Fig. 5.19d, Observe that the small circle of "zero"
radius encircles the originin the F-planein the counterclockwise direction because
the phase angle changes from -180 deg at B' to +180 deg at D'.
Example 5.8
Determine the stability of a unity feedback system given by
k(s + 2)
G(S)=(S_2) .s 3~
Solution. Here, H (s) -. 1. Furthermore, assume that the gain k is a variable.
We use MATLAB4 and the root-locus is drawn as shown in Fig. 5.20a. We observe
that the root locus starts in the right half of the s-plane, implying that the closed-
loop system is unstable for small values of the gzun k and, for k > 4.9420, the
480 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
a) Root-Iocus
b) Nyquist plot for k = 1 '
Fiig. 5.20 Root-Iocus and Nyquist plots for Example 5.8.
root-locus crosses over to the left halfofthe s-plane, indicating that the closed-loop
system becomes stable for k > 4.9420.
Now using MATLAB,4 let us draw the Nyquist plot as shown in Fig. 5.20b
assuming k = 1. We observe that the Nyquist plot in the F -plane does not encircle
the point -1, which means that N -. 0. Instead, it intersects the negative real
axis at s -. -0.2. We have P -. 2 because the poles s = 2 and s - 3 of G(s)
are located in the right half of the s-plane. According to Nyquist criterion, we
get Z = P - N -.2. In other words, the Nyquist criterion predicts that there are
two poles of the closed-loop transfer function T(s) located in the right half of the
s-plane and, therefore, thc~ystem is unstable. From the root-locus of Fig. 5.20a,
we find this to be true.
Suppose we increase the gain k beyond unity. Then the Nyquist plot will expand
and eventually touch the critical point -1. When this happens, the value of k is
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 481
c) Nyquist plot for k - 6
Fig. 5.20 Root~locus and Nyquist plots for Example 5.8, continued.
equal to 1l0.2 - 5, which is quite close to that predjcted by the root-locus method.
For higher values of gain k, the Nyquist plot will expand further and will encircle
the critical point -1 twice in a counterclockwise drreection as shown in Fig. 5.20c
for k -. 6. We then have N - 2 and Z - P - N - 2 - 2 - O, which indicates that
the closed-loop system has become stable.
This example has illustrated an important concept that the stability of closed-
loop systems depends on the value of the gain. Feedback systems that are unstable
for low values of gain can become stable for higher values of gain, and those that
are stable for low values of gain can become unstable for higher values of gain.
The Nyquist criterion can be used to determine the gain at the crossover point.
This kind of dependence of the system stability on thEeavalue of the gain leads to
the concepts of gain and phase margins as discussed in the next section.
The Nyquist stability criterion enables us to define two quantities that are mea-
sures of the level of stability of a given closed-loop system. These quantities are
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