Question 10.14: Analysis Using Root Locus Refer to the root locus obtained i...

When K=0, which corresponds to having no control, the root locus starts from the poles of the loop gain L(s). As shown in Figure 10.51, all four open-loop poles are located in the left s-plane. This implies that the open-loop system is stable. As K increases, the four closed-loop poles move along four different branches of the root locus. For 0 \leq K<60.91, all of the poles are in the left-half s-plane, and thus the closed-loop system is stable. When K=60.91, two complex poles, \pm 3.44 \mathrm{j}, appear on the imaginary axis, and thus the closed-loop system becomes marginally stable. For K> 60.91, the two complex branches of the root locus cross the imaginary axis and enter the right-half s-plane and the closed-loop system becomes unstable.

When K varies between 0 and 60.91 , the pair of complex poles is always closer to the origin than the other poles. Consequently, they dominate the effects of all other poles and thus the closed-loop system exhibits underdamped oscillations. For example, the closed-loop poles are -3.29,-1.66, and -1.03 \pm 1.15 \mathrm{j} when K=1. Figure 10.52 is the corresponding closed-loop unit-step response, which has an overshoot of 21.1 \%. As K increases, the closed-loop system will exhibit severe oscillations because the dominant complex poles move toward the imaginary axis and the damping decreases. For example, the closed-loop poles are -5.16,-1.06, and -0.39 \pm 2.74 j when K=30. Figure 10.53 is the corresponding closed-loop unit-step response, in which the overshoot is as high as 110 \%.