(a) Consider again the feedback system of Example 11.2: The root locus for K &l...

(a) Consider again the feedback system of Example 11.2:

The root locus for K < 0 is plotted in Figure 11.14(b). For some value of K, the closed-loop poles are on the jω -axis. Determine this value of K and the: corresponding locations of the closed-loop poles by examining the real and imaginary parts of the equation

which must be satisfied if the point s = jω is on the root locus for any given values of K. Use this result plus the analysis in Example 11.2 to find the full range of values of K (positive and negative) for which the closed-loop system is stable.

(b) Note that the feedback system is unstable for | K | sufficiently large. Explain why this is true in general for continuous-time feedback systems for which G(s)H(s) has a zero in the right-half plane and for discrete-time feedback systems for which G(z)H(z) has a zero outside the unit circle.