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S4 :        1
S3 :       f
S2: -50,000
SI :       5
SO :       2
There are two sign changes. Hence, there will be two roots that are either positiye
or have positive real parts. Hence, the given system is unstable. Note that if e
appears in any expression, we have to evaluate the value of that expression by
taking the limit as e tends to zero.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW      467
Fig. 5.13    Feedback control system.
5-7.2  Root-Locus Method
  The root-locus is a powerful method of determining the nature of transient
response and stability ofcontrol systems.lt is a graphical method and is particularly
well suited for application to those problems where any parameter or the loop-gain
is a variable.
    Consider a closed-loop system as shown in Fig. 5.13. The closed-loop transfer
function is given by
rfhe equation
T(s)
kG(s)
1 + kG(s)H (s)
1+kG(s)H(s) : 0
(5.121)
(5.122)
is known as the characteristic equation of the given closed-loop system. The roots
of this equation are also called the eigenvalues o.f the closed-loop system. In
other words, the poles of T(s) are the eigenvalues of the closed-loop system. The
root-locus is a plot of the variation of roots of Eq. (5.122) as the parameter k is
varied from zero to infinity. Using Eq. (5.122), we can deduce the following two
conditions for a given point to lie on the root-locus.
     1) Magnitude condition. If a given point s js to lie on the root- locus, we must
have
k G (s)H (s)l  =
(5.123)
2) Phase condition. For a given point to lie on the root-locus, we must have
l.kG(s)H (s) -. (2n + 1)180
(5.124)
where n ~ O,:1:1,:-J:2,.... Note that the expression on the right-hand side of
Eq. (5.124) is an odd multiple of 180 deg with either positive or negative sign.
These two conditions form the basis of sketching the root-locus as the paramcter
k varies from zero to infinity.
     To understand the meaning ofEqs. (5.123) and (5.124), let us refer to Fig. 5.14.
Suppose P is to be a point on the root-locus; then according to the magniLude
121
340
~10
2! 0
 1
22
50
20
O
468           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
                                           
                                           
┏━┳━━━━━━━━━━━━━━━━━━┓
┃  ┃                      x    Poles of ┃
┃  ┃         o Zerosof                  ┃
┃  ┃                                    ┃
┃  ┃--                                  ┃
┣━╋━━━━━━━━━━━━━━━━━━┫
┃  ┃-24 Root-~ocus                      ┃
┗━┻━━━━━━━━━━━━━━━━━━┛
(s) H(s)
(s)H(s)
Fiig. 5.14    Magnitudes and angles ofvectors for a point on the root-locus.
condition
   1  nli,p      (5.125)
k = IG s~H~s l = nl,z
where I,,P  and li,z are the magnitudes of the vectors drawn from each of the poles
 and zeros to the given point on the root-locus. For the root-locus shown in Fig. 5.14,
the magnitude and phase conditions are
k = ; ;                               (5.126)
　
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