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:
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
Criterion
Consider the system described above and suppose
- A is Hurwitz
- (A,b) is controllable
- (A,c) is observable
- d > 0 and
- Φ ∈ (0,∞)
then the system is globally asymptotically stable if there exists a number r > 0 such that
See also
References
- Haddad, Wassim M.; Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press. ISBN 9781400841042.