On root locus traps

We consider a linear time invariant plant under constant gain feedback control. We derive necessary and sufficient conditions under which a closed loop characteristic root remains trapped on the real axis of the right half plane for positive, negative or all values of feedback gains. These conditions are stated in terms of the real axis right half plane poles and zeros of the plant transfer function G(s) and those of G'(s). The nonoccurrence of these conditions is a new necessary condition for constant gain feedback stabilizability. They provide a new lower bound on the dynamic order of stabilizing controllers. The condition which is similar to but stronger than the parity interlacing condition for strong stabilizability is illustrated with examples.