Popov criterion
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 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- A is Hurwitz
- is controllable
- is observable
- d > 0 and
- Φ ∈