Frequency criterion for nonlinear pulse systems stability

Gelig’s Averaging Method For Local Stabilization Of A Class Of Nonlinear Systems By A Pulse-Width Modulated Control

“…For brevity we will use notation t n = nT . Our analysis will be based on the Gelig's version of the averaging method [Gelig, 1982;Gelig and Churilov, 1998] and some additional mathematical technique from [Churilov, 2018;Churilov, 2019a;Churilov, 2019c]. The square of the nth pulse (taking the sign into account) can be calculated by the formula…”

“…Nonlinearity (8) presents a saturation function (see [Tarbouriech et al, 2011]). From (6) the following statement is valid (see [Gelig, 1982]):…”

“…Another approach relies on averaging of the impulsive signal in sampling periods (see [Andeen, 1960a;Andeen, 1960b;Síra-Ramirez, 1989;Taylor, 1992;). The Gelig's method of averaging based on the absolute stability theory was proposed and developed in [Gelig, 1982;Gelig and Churilov, 1993b;Gelig and Churilov, 1996;Gelig and Churilov, 1998;Gelig, 2009]). Further it was refined with the help of the integral quadratic constraints (IQC) theory, see [Chaudenson et al, 2013;Chaudenson, 2013;Fetzer and Scherer, 2016;Fetzer, 2017].…”

Stability analysis of Lur’e systems with a pulse-modulated feedback

“…The ideas of averaging were often used for digital redesign when a continuous-time impulsive system is converted to a discrete-time system that captures the main properties of the original model (see e. g. [Friedland, 1976;Ieko et al, 2001]). Further we will employ the technique of pulse averaging introduced in [Gelig, 1982] and refined in [Gelig and Churilov, 1993;Gelig and Churilov, 1998].…”

“…Stability considerations are based on the Gelig's averaging method introduced by [Gelig, 1982] (see also [Gelig and Churilov, 1998]) that will be discussed in detail in Section 4. The main idea of this method is a substitution of the initial train of pulses for a sequence of the average values of these pulses, with a supposition that these averages satisfy sectoral constraints not everywhere, but at some discrete time instants.…”