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 (that is, time invariant) systems.

System description

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

$\begin{array}{cc}\dot{x}& =Ax+bu\\ \dot{\xi }& =u\\ y& =cx+d\xi \end{array}$

$\begin{array}{c}u=-\phi (y)\end{array}$

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

$H(s)=\frac{d}{s}+c(sI-A{)}^{-1}b$

Criterion

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 $\underset{\omega \in ℝ}{\inf }\mathrm{Re}[(1+j\omega r)H(j\omega )]>0.$

See also

• Circle criterion

References

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