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.

System description

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The sub-class of Lur'e systems studied by Popov is described by:

 

 

where xRn, ξ,u,y are scalars, and A,b,c and d have commensurate dimensions. The nonlinear element Φ: RR 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

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Consider the system described above and suppose

  1. A is Hurwitz
  2. (A,b) is controllable
  3. (A,c) is observable
  4. d > 0 and
  5. Φ ∈ (0,∞)

then the system is globally asymptotically stable if there exists a number r > 0 such that  

See also

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References

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  • Haddad, Wassim M.; Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press. ISBN 9781400841042.