CYBERNETICS AND PHYSICS, VOL. 9, NO. 3, 2020, 129–135 GELIG’S AVERAGING METHOD FOR LOCAL STABILIZATION OF A CLASS OF NONLINEAR SYSTEMS BY A PULSE-WIDTH MODULATED CONTROL Alexander N. Churilov Laboratory of Control of Multi-agent, Distributed and Networked Systems, ITMO University. St. Petersburg, Russia a churilov@mail.ru Article history: Received 26.10.2020, Accepted 20.11.2020 Abstract A stabilization problem for a nonlinear system with a sector bound nonlinearity and a pulse-width modulated (PWM) feedback is considered. The linear matrix in- equalities (LMI) technique is used to estimate the do- main of attraction for the zero equilibrium of the closed system. Key words Nonlinear systems, hybrid systems, networked sys- tems. 1 Introduction The subject of this study is a controlled nonlinear sys- tem with a pulse-width modulator in the feedback. The system comprises a nonlinear continuous-time subsys- tem with a sector bound nonlinearity and a modulator, which transforms a continuous signal into a train of rect- angular pulses that are emitted at a given constant fre- quency, but their widths vary (are modulated). The pulse signal can take only three discrete values −1, 0 and 1. The idea of a PWM control goes back to the end of the XIXth century (see [Gouy, 1897]). Because of its sim- ple engineering implementation, PWM gained a great popularity and has been a subject of many mathemati- cal studies (see, among others, [Skoog and Blankenship, 1970; Kuntsevich and Chekhovoi, 1970; Kuntsevich and Chekhovoi, 1971; Tsypkin and Popkov, 1973; Kipnis, 1992; Gelig and Churilov, 1998; Yuan et al., 1998; Zhusubaliyev and Mosekilde, 2003; Massioni et al., 2019]). The two main mathematical approaches can be mentioned. In the case when the continuous part of a system is linear, the reduction to discrete-time equations is frequently applied (see, e. g., [Kadota and Bourne, 1961; Hou and Michel, 2001; De Koning, 2003; Asai, 2006; Tomita and Asai, 2006; Alm´ er et al., 2007; S´ ıra- Ramirez et al., 2015]). Another approach relies on av- eraging of the impulsive signal in sampling periods (see [Andeen, 1960a; Andeen, 1960b; S´ ıra-Ramirez, 1989; Taylor, 1992; Sakamoto and Hori, 2002; Sakamoto et al., 2002]). The Gelig’s method of averaging based on the absolute stability theory was proposed and devel- oped 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 in- tegral quadratic constraints (IQC) theory, see [Chauden- son et al., 2013; Chaudenson, 2013; Fetzer and Scherer, 2016; Fetzer, 2017]. However, the PWM systems con- sidered in the latter works had linear continuous parts. Since a PWM control signal is bounded, we can rarely achieve stabilization for all the initial data. However, the stabilization problem can be solved locally, in a vicin- ity of the zero point (cf. [Asai, 2006; Tomita and Asai, 2006]). A domain of attraction of the zero equilibrium is estimated using the computational technique of linear matrix inequalities (LMI. see [Boyd et al., 1994]). In this paper we will explore stability of a PWM sys- tem with the help of the Gelig’s version of the averaging method. We develop further the technique proposed pre- viously in [Churilov, 2019c] to extend it to the case of unsaturated behavior. 2 Problem Setting Consider a controlled nonlinear system ˙ x = Ax(t)+ B 0 f (t)+ Bu(t), (1) η(t)= C 0 x(t), σ(t)= Cx(t). (2)