Popov criterion
In nonlinear control and stability theory, the Popov criterion is a stability criterion by Vasile M. Popov for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an open-sector condition. While the circle criterion can be applied to nonlinear time-varying systems, the Popov criterion is applicable only to autonomous (that is, time invariant) systems. System description. The sub-class of Lur'e systems studied by Popov is described by: formula_2 where "x" ∈ R, "ξ","u","y" are scalars, and "A","b","c" and "d" have commensurate dimensions. The nonlinear element Φ: R → R is a  time-invariant nonlinearity belonging to "open sector" (0, ∞), that is, Φ(0) = 0 and "y"Φ("y") > 0 for all "y" not equal to 0. Note that the system studied by Popov has a pole at the origin and there is no direct pass-through from input to output, and the transfer function from "u" to "y" is given by Criterion. Consider the system described above and suppose then the system is globally asymptotically stable if there exists a number "r" > 0 such that formula_4