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In nonlinear control and stability theory, the Popov criterion is a
stability criterion In control theory, and especially stability theory, a stability criterion establishes when a system is stable. A number of stability criteria are in common use: *Circle criterion *Jury stability criterion *Liénard–Chipart criterion *Nyquist st ...
discovered by
Vasile M. Popov Vasile Mihai Popov (born 1928) is a leading systems theorist and control engineering specialist. He is well known for having developed a method to analyze stability of nonlinear dynamical systems, now known as Popov criterion. Biography He was b ...
for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an open-sector condition. While the
circle criterion In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for linear time-invariant (LTI) systems. Overvi ...
can be applied to nonlinear time-varying systems, the Popov criterion is applicable only to autonomous (that is,
time invariant In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
) systems.


System description

The sub-class of Lur'e systems studied by
Popov Popov (; masculine), or Popova (; feminine), is a common Russian, Bulgarian, Macedonian and Serbian surname. Derived from a Slavonic word ''pop'' (, " priest"). The fourth most common Russian surname, it may refer to: *Alek Popov (born 1966), ...
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 Hurwitz is one of the variants of a surname of Ashkenazi Jewish origin (for historical background see the Horowitz page). Notable people with the surname include: *Adolf Hurwitz (1859–1919), German mathematician **Hurwitz polynomial **Hurwitz ma ...
#(''A'',''b'') is controllable #(''A'',''c'') is observable #''d'' > 0 and #Φ ∈ (0,∞) then the system is
globally asymptotically stable In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s s ...
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 In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for linear time-invariant (LTI) systems. Overvi ...


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