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

Alexander N Churilov

doi:10.35470/2226-4116-2019-8-2-58-68

Abstract

A nonlinear system with a sector bound nonlinearity is considered. The system is subject to a stabilizing sampled feedback with finite width impulses. An impulsive counterpart of the circle criterion for absolute stability is obtained with the help of the Gelig’s averaging method.

Highlights

In the recent decades a great popularity was gained by the study of hybrid systems that combine continuoustime and impulsive dynamics
While processes in continuous physical and biological systems are often rather slow, the interactions between these systems sometimes exhibit fast behaviors that may be interpreted as an impulsive signal. (Signals are understood as some portions of energy that are used for interaction and information exchange [Basiladze, 2009].) An example is the processes in physiology, where body organs are governed by neural impulses of the brain, while some hormones secretion in these organs modulates impulsive activity of the brain
The majority of works in this field employ a zero-order hold (ZOH), when a control value calculated at the beginning of a sampling interval is kept constant throughout all this interval

Summary

Introduction

In the recent decades a great popularity was gained by the study of hybrid systems that combine continuoustime and impulsive dynamics. While processes in continuous physical and biological systems are often rather slow, the interactions between these systems sometimes exhibit fast behaviors that may be interpreted as an impulsive signal. A natural way to handle these so different time scales is to average the fast processes in time and simplify the analysis. Stabilization of a Lur’e system by a sampled ZOH feedback was addressed in [Seifullaev and Fradkov, 2015a; Seifullaev and Fradkov, 2015b; Seifullaev and Fradkov, 2015c; Seifullaev and Fradkov, 2016; Seifullaev et al, 2017; Zhang et al, 2017; Bryntseva and Fradkov, 2018], Some other types of nonlinear systems under a ZOH event-based control were treated in [Wang et al, 2018; Proskurnikov and Mazo Jr., 2018]. This requirement is motivated by applications to networked control (see [Hespanha et al, 2007; Lu et al, 2012; Liu et al, 2017; Liu et al, 2019]) and allows to save capacity of a communication channel

Objectives

Discussion

Conclusion