3. Since n -m = 2, there am two asymptotes, and the center of gravity is

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y = (-0.05 - 0.1 - 2) - (-0.5 - 1.0) =

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The angles that the asymptotes make with the real axis are *&I. These are shown in Fig. 15.7.

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At this stage, we can sketch part of the root-locus diagram. Since the locus is on the real axis between -0.1 and -0.5 and between -1 and -2, it should be evident that one branch moves from the pole at -2 to the zero at -1 and another branch moves from the pole at -0.1 to the zero at -0.5. The remaining two branches move from the poles at 0 and -0.05 toward each other along the teal axis until they meet, at which point they must break away from the real axis and move in some way toward the vertical asymptotes that intersect the real axis at -0.325.

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With the information now available, it is difficult to continue the sketch with confidence, for the breakaway point is so close to the origin that there is some likelihood that the loci will move into the right half plane before approaching the asymptote. If this should occur, each locus would have to cross the imaginary axis twice, in which case there would be an intermediate range of K over which the system is unstable. On either end of this range of K, the system is stable. This condition is called conditional stability. The possibility of the locus crossing the imaginary axis twice is suggested by the analog from potential theory that was mentioned earlier. This can be explained as follows: immediately after the locus leaves the real axis at the breakaway point, it has a tendency to move to the right half plane because the pole at -0.1 repels the locus. However, after the locus moves to a point sufficiently far from this repelling pole, it is attracted mote strongly by the two zetas at -0.5 and - 1 and has the tendency to return to the left half plane where we know it must eventually approach the vertical asymptote. Actually to determine whether or not the locus moves into the right half plane requires that the points at which the loci cross the imaginary axis be determined. This can be done by use of the Routh test as illustrated in Example 15.1. The details of the calculation will not be given here; however, the reader can show that there are two values of gain K which give a pair of roots of the characteristic equation that lie on the imaginary axis. These gains and corresponding roots are approximately

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K = 0.004 K = 2.4

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K, = 0.6 KC = 360

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s = rjO.1 s = kjl.1

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From these results, we conclude that the system will oscillate with constant amplitude with a frequency w = 0.1 radltime when K, = 0.6; it will also oscillate at constant amplitude with w = 1.1 when K, = 360. The system is unstable for 0.6 < K, < 360. The system is stable for K, < 0.6 and for K, > 360. The complete root-locus diagram is sketched in Fig. 15.7.

SUMMARY In this chapter, the rules for plotting root-locus diagrams have been presented and applied to several control systems. It should be emphasized that the basic advantage of this method is the speed and ease with which a rough sketch of the loci can be obtained. This sketch frequently gives much of the desired information on stability. A few further calculations of points on the locus are usually all that are necessary to obtain accurate, quantitative behavior of the roots. The root locus for variation of parameters other than K,, such as ~0, has not been discussed here. The method of constructing this type of diagram is similar to that presented here and is discussed in detail in other texts [see Wilts