For each H1(s) in Figure E.12 is there a finite positive value of K for which this system is unstable?

Question

Answer 1

In the root locus, the closed-loop pole starts on the loop transfer function pole at s = -10 for low K and as K increases the closed-loop pole moves toward the loop transfer function zero at s = 4 in the right half-plane. Therefore at some finite positive value of K the system is unstable.



In the root locus, the closed loop poles start at the loop transfer function double pole at s = - 1 in the open left-half plane. As K increases the poles migrate straight up and down and therefore never cross into the right half-plane. Therefore for all K the system is stable.



There are three real loop transfer function poles in the left half-plane. In the root locus, at some value of K the closed-loop poles will enter the right half-plane. Therefore at some finite positive value of K the system is unstable.