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respect to the real axis. That segment of the real axis forms a part of the root-locus,
which lies to the left of the odd number of poles ancVor zeros. Thus, the real axis
segment between the poles at 0 and -1 and between the pole at -2 and zero at
-4 and all the real axis that is to the left of the pole at -5 forms the branches of
the root-locus.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       471
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
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