Question 8.7: Sketch the root locus for the system shown in Figure 8.19(a)...

The problem solution is shown, in part, in Figure 8.19(b). First sketch the root locus. Using Rule 3, the real-axis segment is found to be between −2 and −4. Rule 4 tells us that the root locus starts at the open-loop poles and ends at the open-loop zeros. These two rules alone give us the general shape of the root locus.

a. To find the exact point where the locus crosses the ζ = 0.45 line, we can use the root locus program discussed in Appendix H.2 at www.wiley.com/go/Nise/ControlSystemsEngineering8e to search along the line

θ = 180° − cos ^{-1} 0.45 = 116.7°                    (8.52)

for the point where the angles add up to an odd multiple of 180°. Searching in polar coordinates, we find that the root locus crosses the ζ = 0.45 line at 3.4∠116.7° with a gain, K, of 0.417.

b. To find the exact point where the locus crosses the jω-axis, use the root locus program to search along the line

θ = 90°                  (8.53)

for the point where the angles add up to an odd multiple of 180°. Searching in polar coordinates, we find that the root locus crosses the jω-axis at ±j3.9 with a gain of K = 1.5.

c. To find the breakaway point, use the root locus program to search the real axis between −2 and −4 for the point that yields maximum gain. Naturally, all points will have the sum of their angles equal to an odd multiple of 180°. A maximum gain of 0.0248 is found at the point −2.88. Therefore, the breakaway point is between the open-loop poles on the real axis at −2.88.

d. From the answer to b, the system is stable for K between 0 and 1.5.

Students who are using MATLAB should now run ch8apB1 in Appendix B. You will learn how to use MATLAB to plot and title a root locus, overlay constant ζ and ωn curves, zoom into and zoom out from a root locus, and interact with the root locus to find critical points as well as gains at those points. This exercise solves Example 8.7 using MATLAB.