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:

: $\begin \dot & = Ax+bu \\ \dot & = u \\ y & = cx+d\xi \end$

$\begin u = -\varphi (y) \end$

where ''x'' ∈ R^''n'', ''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 ''y''Φ(''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

:$H(s) = \frac + c(sI-A)^b$

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 \inf_ \operatorname \left (1+j\omega r) H(j\omega)\right> 0.

See also

* Circle criterion

References

* {{cite book , last1=Haddad , first1=Wassim M. , last2=Chellaboina , first2=VijaySekhar , title=Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. , date=2011 , publisher=Princeton University Press , isbn=9781400841042

Nonlinear control
Stability theory