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:

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

${\begin{matrix}u=-\varphi (y)\end{matrix}}$

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

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

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 ${\textstyle \inf _{\omega \,\in \,\mathbb {R} }\operatorname {Re} \left[(1+j\omega r)H(j\omega )\right]>0.}$

See also

• Circle criterion

References

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