Question 8.2: Sketch the root locus for the system shown in Figure 8.11.

Let us begin by calculating the asymptotes. Using Eq. (8.27), the real-axis intercept is evaluated as

σ_{a} = \frac{\sum  finite  poles  −  \sum  finite  zeros }{\#finite  poles  −  \#finite zeros}     (8.27)

σ_{a} = \frac{(−1  −  2  −  4)  −  (−3)}{4  −  1} = –  \frac{4}{3}                     (8.29)

The angles of the lines that intersect at −4/3, given by Eq. (8.28), are

θ_{a} = \frac{(2k + 1) π }{\#finite  poles  −  \#finite  zeros}   (8.30a)

= π/3            for k = 0              (8.30b)

= π                 for k = 1              (8.30c)

= 5π/3            for k = 2              (8.30d)

If the value for k continued to increase, the angles would begin to repeat. The number of lines obtained equals the difference between the number of finite poles and the number of finite zeros.

Rule 4 states that the locus begins at the open-loop poles and ends at the open-loop zeros. For the example there are more open-loop poles than open-loop zeros. Thus, there must be zeros at infinity. The asymptotes tell us how we get to these zeros at infinity.

Figure 8.12 shows the complete root locus as well as the asymptotes that were just calculated. Notice that we have made use of all the rules learned so far. The real-axis segments lie to the left of an odd number of poles and/or zeros. The locus starts at the open-loop poles and ends at the open-loop zeros. For the example there is only one open-loop finite zero and three infinite zeros. Rule 5, then, tells us that the three zeros at infinity are at the ends of the asymptotes.