As mentioned at the end of Section 11.5, the phase and gain margins may provide sufficie...

As mentioned at the end of Section 11.5, the phase and gain margins may provide sufficient conditions to ensure that a stable feedback system remains stable. For example, we showed that a stable feedback system will remain stable as the gain is increased, until we reach a limit specified by the gain margin. This does not imply (a) that the feedback system cannot be made unstable by decreasing the gain or (b) that the system will be unstable for all values of gain greater than the pin margin limit. In this problem, we illustrate these two points.

(a) Consider a continuous-time feedback system with

Sketch the root locus for this system for K > 0. Use the properties of the root locus described in the text and in Problem 11.34 to help you draw the locus accurately. Once you do so, you should see that for small values of the gain K the system is unstable, for larger values of K the system is stable, while for still larger values of K the system again becomes unstable. Find the range of values of K for which the system is stable. Hint: Use the same method as is employed in Example 11.2 and Problem 11.35 to determine the values of K at which branches of the root locus pass through the origin and cross the jω-axis

If we set our gain somewhere within the stable range that you have just found, we can increase the gain somewhat and maintain stability, but a large enough increase in gain causes the system to become unstable. This maximum amount of increase in gain at which the closed-loop system just becomes un-stable is the gain margin. Note that if we decrease the gain too much, we can also cause instability.

b) Consider the feedback system of part (a) with the gain K set at a value of 7. Show that the closed-loop system is stable. Sketch the log magnitude-phase plot of this system, and show that there are two nonnegative values of w for which ?G (jω)H(jω) = —π. Further, show that, for one of these values 7| G(jω)H(jω) |< 1, and for the other 7|G(jω)H(jω)| > 1. The first value provides us with the usual gain margin—that is, the factor 1/|7G(jω)H(jω | by which we can increase the gain and cause instability. The second provides us with the factor 1/|7G(jω)H(jω)| by which we can decrease the gain and just cause instability.

(c) Consider a feedback system with

Sketch the root locus for K > 0. Show that two branches of the root locus begin in the left-half plane and, as K is increased, move into the right-half plane and then back into the left-half plane. Do this by examining the equation

Specifically, by equating the real and imaginary parts of this equation, show that there are two values of K ≥ 0 for which the closed-loop poles lie on the jω-axis.

Thus, if we set the gain at a small enough value so that the system is stable, then we can increase the gain up until-the point at which the two branches of the root locus intersect the jω-axis. For a range of values of gain beyond this point, the closed-loop system is unstable. However, if we continue to increase the gain, the system will again become stable for K large enough.

(d) Sketch the Nyquist plot for the system of part (c), and confirm the conclusions reached in part (c) by applying the Nyquist criterion. (Make sure to count the net number of encirclements of —1/K.)

Systems such as that considered in parts (c) and (d) of this problem are often referred to as being conditionally stable systems, because their stability properties may change several times as the gain is varied.