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Asymptotc
.b/_*. _
-5
a)
Fig. 5-15 Root-Iocus for Example 5.5.
472 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We should have three asymptotes corresponding to three branches of the root-
locus, which seek zeros at infinitjt. We have
<To - Epoles - E zeros
tlp - Flz
(0 - 1 - 2 - 5) - (-4)
~~
4-1
4
= -3
M - tan(2n +l)7r
llp - FlZ
- tan (2n + l)Jr
3
=tan; n-0
- tan7r n -. 1
57r
= tan 3 n:2
With this information, the root-locus can be sketched as shown in Fig. 5.15a
The root-locus crosses the imaginary axis from the left half to the right half of the
s-plane, i.e., the closed-loop system becomes unstable as the value of the gain k
is increased beyond this point.
The value of the gain k and frequency co where the root-locus crosses the imag-
mary axes can be obtained as follows.
The characteristic equation is
s4+8s3+l7s2+s(10+k)+4k - 0
Substituting s - jco, we obtain
co4 _ 17c.o2 + 4k + j(-8co3 + co[l0 +k]) = O
Equating real and imaginary parts to zero, we get k - 8,4856 and co : +1.5201.
MATLAB4 is a convenient tool for plotting the root-locus. The MATLAB com-
mand RLOCUS sketches the root-locus, and the command RLOCFIND enables us
to find the value ofthe gain k and the location of the closed-loop poles correspond-
ing to any point on the root-locus. Using RLOCFIND, we find that k - 8.8 and
co : 1.55 when the root-locus crosses theimaginary axis. These values are in good
agreement with the analytical values. Furthermore, the corresponding locations of
the closed-loop poles are -5.15, -2.88, and 0 + 1.55.
The root-locus obtained using MATLAB4 is shown in Fig. 5.15b.
We can also find other information using MATLAB.4 For example, we can find
the value of the gain k so that the closed-loop system is staole and operates with
a damping ratio < of 0.4. Using RLOCFIND, we obtaLn k - 2.0 and closed-loop
poles p = -5.04, -2.4, and :s0:3 + j0.8.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW 473
5.7.3 Nyquist Stab17ity Criterion
Concept of mapping. Before we discuss the Nyquist stability criterion, let
us briefly review the concept of mapping. Suppose W(S) :giv2en a'ontour-A in the
the s-plane as shown in Fig. 5.16a and a function F(s s2 + 2s + 1. Consider
a point P on the contour A in the s-plane, and let the coordinates of point P be
4 + j3. If we substitute this complex number into the given function F(s), we get
another complex number
F(s) = (4 + j3)2 + 2(4 + j3) +1 : 16 + j30 (5.137)
Z
a) s-plane
z
ContourA
b) F-plane
Fig.5.16 Conceptofmapping.
L:
474 PERFORMANCE, STABILI-fY, DYNAMICS, AND CONTROL
@
┏━┓
┃ ┃
┗━┛
Z
a) F(s) -s - zi
┏━━┳━┓
┃jco ┃ ┃
┃-/ ┃ ┃
┗━━┻━┛
0
Z
o
b) F(s) = 1/(s - pi)
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