Popov criterion

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

: $$

\begin{align}

\dot{x} & = Ax+bu \\

\dot{\xi} & = u \\

y & = cx+d\xi

\end{align}

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

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

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

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

• Circle criterion

• {{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}}

Category:Nonlinear control

Category:Stability theory