Classification of linear dynamical systems in the plane in admitting a stabilizing hybrid feedback control
type: Item
instance of: scholarly article
title: Classification of linear dynamical systems in the plane in admitting a stabilizing hybrid feedback control
zbMATH Open document ID: 0967.93077
DOI: 10.1023/A:1009548410223
author: Elena Litsyn
author: Yurii V. Nepomnyashchikh
author: Arcady Ponosov
published in: Journal of Dynamical and Control Systems
publication date: 2001-08-20T00:00:00Z
review text: The authors consider a stabilization problem for the following controlled system \[ \dot x = A x + B u,\qquad y = c x \tag{1} \] where \( x \in \mathbb{R}^2\), \(u \in \mathbb{R}^1\), \(y \in \mathbb{R}^1 \). For stabilization a linear output feedback control \( u = G y \) is used. A simple example, considered by Z. Artstein, \[ \dot x_1 = x_2, \quad \dot x_2 = -x_1 + u, \quad y = x_1, \quad u = u (y) \tag{2} \] shows that although system (2) is both controllable and observable, it cannot be stabilized by such feedback (even nonlinear and discontinuous). It was, however, shown by Z. Artstein that there exists a hybrid feedback control, under which system (2) becomes asymptotically stable. A hybrid feedback control procedure attaches a discrete time controller (an automaton) to the underlying continuous dynamical system and prescribes when and how a control \(u\) should be switched on and off. As Artstein's example shows, such a hybrid feedback may help when the ordinary feedback fails to stabilize the system. In the paper, a classification of linear controlled planar systems is suggested and all cases where the system admits a linear hybrid stabilization are described. In connection with the results obtained, the authors point at a preprint devoted to the same problem: \textit{C. Benassi} and \textit{A. Gavioli}, Hybrid stabilization of planar linear systems with one-dimensional outputs. Preprint, Dipartamento di Matematica Pura e Applicata, Università di Modena, Italy, August, 1999.
reviewed by: A. M. Kovalev
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 1545849
zbMATH Keywords: linear controlled system
zbMATH Keywords: stabilization via hybrid controls
zbMATH Keywords: planar systems
MaRDI profile type: Publication
full work available at URL: a:1009548410223
OpenAlex ID: W1725782210
description: scientific article; zbMATH DE number 1545849