In nonlinear control and stability theory, the Popov criterion is a stability criterion discovered 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.
The sub-class of Lur'e systems studied by Popov is described by:
where x ∈ Rn, ξ,u,y are scalars, and A,b,c and d have commensurate dimensions.
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 Consider the system described above and suppose then the system is globally asymptotically stable if there exists a number r > 0 such that