The parabola test for absolute stability

In applying the Popov stability test, a certain straight line is drawn; the Popov locus must lie on one side of this line. Thus geometric considerations alone indicate that the interesting sectors of absolute stability for conditionally stable systems cannot be found directly. However, the straight line of the Popov test may be replaced by a certain parabola and the conditionally stable sectors of absolute stability may then be discovered. The test has its best application for conditionally stable systems but can be used whenever the Popov test can be used. In fact, the Popov straight line may be obtained as a limiting form of the parabola. The test is ordinarily weaker than the Popov test, but nonlinear sectors having a nonzero lower bound may be found more directly.