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of the root-locus is equal to the number of closed-loop poles.
   2) Symmetry. For all physical systems, the coefficients of the characteristic
equation are real. As a result, if any of its roots are complex, then they occur in
pairs as complex conjugates. All the real roots lie on either the positive or negative
real axes. Hence, the root-locus of a physical. system is always symmefflic with
respect to the real axis.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REViEW       469
     3) Realaxrs segments. Whether a given segment of the real axis forms a part
of the root-locus depends on the angle condition shown in Eq. (5.124), i.e., the
algebraic sum of the angles subtended at that point because all the poles and
zeros must be equal to an odd multiple of 180 deg. The net angle contribution of
 the complex poles or zeros is zero because they always occur as complex conjugate
 pairs. Furthermore, the angle contribution of a real axis pole or zero located to the
 right ofa point on the real axis is zero. rfhe angle contribution to a point on the root-
locus comes only from those real axis poles and zeros that are located on the left
side and is equal to -180 deg for poles and 180 deg for zeros. Because the sum
of all such c7ntributions has to be an odd multiple of 180 deg, it is clear that only
that part of the real axis segment forms a branch of the root-locus that lies to the
left of odd number of poles andtor zeros.
       4) Startin,g and ending pomis of root-locus.  To understand where the root~locus
begins and where it ends as the parameter k is varied from zero to infinit)r, let      '
G(s) = ~V [ ;                             (5.128)
H(s) = g;{:;                             (5.129)
Note that Ng = 0 and Dg = 0 give us, respectively, the zeros and poles ofthe open-
loop transfer function  G(s). Similarly,  Nh (s) = O and Dh (s) = O give, respectively,
the zeros and poles of H (s).
Then,
                                       k Ng(s)Dh (s)
T(s) = D8~s~D (s) + k N~s~N (s~                    (5.130)
When the parameter k ~ O, the closed-loop transfer function T(s) can be approx-
imated as
                                  T(s) = kNg(s)Dh(s)
             Dg(s)Dh(s)       (5.131)
i.e., when k -* 0, the poles of T(s) coincide with the combined open-loop poles
of G(s) and H(s). Therefore, the root-locus starts at the open-loop poles of the
system.
     When k ~ oo, we have
                                    T(s) = N8(s)Dh(s)
             Ng(s)Nti(s)       (5.132)
That is, when k -*  oo, the poles of T(s) approach the combined zeros of G(s)
and  H (s).ln other words, the root-locus ends at the open-loop zeros of the system.
Summarizing, the root-locus starts at the open-loop poles and ends at the open-loop
zeros. This statement implies that the system should have equal number of poles
and zeros, which is true if we assume that the missing zeros and poles are located
at infinity. To understand this point, consider
                                   k
                              G(s) = s~s+3~s+5~                     (5.133)
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470             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
　
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