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:
| \begin{align} x |
&=Ax+bu\\
| \xi |
&=u\\ y&=cx+d\xi\end{align}
\begin{matrix}u=-\varphi(y)\end{matrix}
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 (0, ∞), that is, Φ(0) = 0 and yΦ(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
H(s)=
| d | |
| s |
+c(sI-A)-1b
Consider the system described above and suppose
then the system is globally asymptotically stable if there exists a number r > 0 such that