Extension of Popov absolute stability criterion to non-autonomous systems with delays

This paper extends in a simple way the classical absolute stability Popov criterion to multivariable systems with delays and with time-varying memoryless non-linearities subject to sector conditions. The proposed sufficient conditions are expressed in the frequency domain, a form well-suited for robustness issues, and lead to simple graphical interpretations for scalar systems. Apart from the usual conditions, the results assume basically a generalized sector condition on the derivative of the non-linearities with respect to time. Results for local and global stability are given, the latter concerning in particular the linear time-varying ones. For rational transfers, the frequency conditions are equivalent to some easy-to-check linear matrix inequalities: this leads to a tractable method of numerical resolution by rational approximation of the transfer. As an illustration, a numerical example is provided.