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 systems.

System description

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. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to open sector, that is, Φ = 0 and yΦ > 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
  1. A is Hurwitz
  2. is controllable
  3. is observable
  4. d > 0 and
  5. Φ ∈
then the system is globally asymptotically stable if there exists a number r > 0 such that