Question 8.5: For the system of Figure 8.11, find the frequency and gain, ...
For the system of Figure 8.11, find the frequency and gain, K, for which the root locus crosses the imaginary axis. For what range of K is the system stable?
The closed-loop transfer function for the system of Figure 8.11 is
T (s) = \frac{K (s + 3)}{s^{4} + 7s³ + 14s² + (8 + K) s + 3K} (8.40)
Using the denominator and simplifying some of the entries by multiplying any row by a constant, we obtain the Routh array shown in Table 8.3.
TABLE 8.3 Routh table for Eq (8.40)
| s^{4} | 1 | 14 | 3K |
| s³ | 7 | 8 + K | |
| s² | 90 – K | 21K | |
| s¹ | \frac{− K² − 65K + 720}{90 − K} | ||
| s^{0} | 21K |
A complete row of zeros yields the possibility for imaginary axis roots. For positive values of gain, those for which the root locus is plotted, only the s¹ row can yield a row of zeros. Thus,
−K² − 65K + 720 = 0 (8.41)
From this equation K is evaluated as
K = 9.65 (8.42)
Forming the even polynomial by using the s² row with K = 9.65, we obtain
(90 − K) s² + 21K = 80.35s² + 202.7 = 0 (8.43)
and s is found to be equal to ±j1.59. Thus the root locus crosses the jω-axis at ±j1.59 at a gain of 9.65. We conclude that the system is stable for 0 ≤ K < 9.65.