Consider a continuous-time feedback system with (a) Sketch the root locus for K...

Consider a continuous-time feedback system with

(a) Sketch the root locus for K > 0 and for K < 0. (Him: The results of Problem 11.34 are useful here.)

(b) If you have sketched the locus correctly, you will see that for K > 0, two branches of the root locus cross the jω-axis, passing from the left-half plane into the right-half plane. Consequently, we can conclude that the closed-loop system is stable for , where is the value of the gain for which the two branches of the root locus intersect the jai-axis. Note that the sketch of the root locus does not by itself tell us what the value of is or the exact point on the jω-axis where the branches cross. As in Problem 11.35, determine by solving the pair of equations obtained as the real and imaginary. parts of

Determine the corresponding two values of to (which are the negatives of each other, since poles occur in complex-conjugate pairs).

From your root-locus sketches in part (a), note that there is a segment of the real axis between two poles which is on the root locus for K > 0, and a different segment is on the locus for K < 0. In both cases, the root locus breaks off from the real axis at some point. In the next part of this problem, we illustrate how one can calculate these breakaway points.

(c) Consider the equation denoting the closed-loop poles:

"break-in" points, where two branches of the root locus merge onto the real These methods plus the one illustrated are described in advanced texts Such as those listed in the bibliography at the end of the book.