Domain of Attraction - Analysis and Control via SOS Programming

For nonlinear dynamical systems, which represent the majority of real devices, any study of stability requires the investigation of the domain of attraction of an equilibrium point, i.e. the set of initial conditions from which the trajectory of the system converges to equilibrium. Unfortunately, both estimating and attempting to control the domain of attraction are very difficult problems, because of the complex relationship of this set with the model of the system. Domain of Attraction addresses the estimation and control of the domain of attraction of equilibrium points via SOS programming, i.e. optimization techniques based on the sum of squares of polynomials (SOS) that have been recently developed and that amount to solving convex problems with linear matrix inequality constraints. A unified framework for addressing these issues is presented for in various cases depending on the nature of the nonlinear systems considered, including the cases of polynomial, non-polynomial, certain and uncertain systems. The methods proposed are illustrated various example systems such as electric circuits, mechanical devices, and nuclear plants. Domain of Attraction also deals with related problems that can be considered within the proposed framework, such as characterizing the equilibrium points and bounding the trajectories of nonlinear systems, and offers a concise and simple description of the main features of SOS programming, which can be used for general purpose in research and teaching.

List of contents

SOS Polynomials.- Optimization with SOS Polynomials.- Preliminaries of Nonlinear Systems.- Domain of Attraction in Polynomial Systems.- Domain of Attraction in Uncertain Polynomial Systems.- Domain of Attraction in Non-polynomial Systems.- Domain of Attraction via Multiple Lyapunov Function.- Miscellaneous.

About the author

Graziano Chesi received the Laurea degree in Information Engineering from the University of Firenze (Italy) in 1997, and the Ph.D. in Systems Engineering from the University of Bologna (Italy) in 2001. He was a visiting scientist at the Department of Engineering of the University of Cambridge (UK) in the period 1999-2000. He held fellowships from the Japanese Society for the Promotion of Science and the European Union at the Department of Information Physics and Computing of the University of Tokyo (Japan) in the period 2001-2004. He was a Research Associate and an Assistant Professor at the Department of Information Engineering of the University of Siena (Italy) in the period 2000-2006. In 2006 he joined the Department of Electrical and Electronic Engineering of the University of Hong Kong (Hong Kong). Dr. Chesi was the recipient of the Best Student Award of the Faculty of Engineering of the University of Firenze in 1997. He is an Associate Editor of Automatica, an Associate Editor of the IEEE Transactions on Automatic Control, and a Guest Editor of the Special Issue of the IEEE Transactions on Automatic Control on “Positive Polynomials in Control”. His research interests include computer vision, nonlinear systems, optimization, robotics, robust control, and systems biology.

Summary

For nonlinear dynamical systems, which represent the majority of real devices, any study of stability requires the investigation of the domain of attraction of an equilibrium point, i.e. the set of initial conditions from which the trajectory of the system converges to equilibrium. Unfortunately, both estimating and attempting to control the domain of attraction are very difficult problems, because of the complex relationship of this set with the model of the system. Domain of Attraction addresses the estimation and control of the domain of attraction of equilibrium points via SOS programming, i.e. optimization techniques based on the sum of squares of polynomials (SOS) that have been recently developed and that amount to solving convex problems with linear matrix inequality constraints. A unified framework for addressing these issues is presented for in various cases depending on the nature of the nonlinear systems considered, including the cases of polynomial, non-polynomial, certain and uncertain systems. The methods proposed are illustrated various example systems such as electric circuits, mechanical devices, and nuclear plants. Domain of Attraction also deals with related problems that can be considered within the proposed framework, such as characterizing the equilibrium points and bounding the trajectories of nonlinear systems, and offers a concise and simple description of the main features of SOS programming, which can be used for general purpose in research and teaching.

Additional text

From the reviews:
“The manuscript deals with the important problem of estimating the domain of attraction of an equilibrium point (say, the origin) for solutions of smooth Ordinary Differential Equations (ODEs) … . The book is a timely and useful contribution to the systems control literature. It contains many explicitly worked examples, with nice and informative visual illustrations, and also pointers to public-domain Matlab codes written by the author.” (Didier Henrion, Zentralblatt MATH, Vol. 1242, 2012)

Report

From the reviews:
"The manuscript deals with the important problem of estimating the domain of attraction of an equilibrium point (say, the origin) for solutions of smooth Ordinary Differential Equations (ODEs) ... . The book is a timely and useful contribution to the systems control literature. It contains many explicitly worked examples, with nice and informative visual illustrations, and also pointers to public-domain Matlab codes written by the author." (Didier Henrion, Zentralblatt MATH, Vol. 1242, 2012)

Product details

• Authors: Graziano Chesi
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.10.2011
• EAN: 9780857299581
• ISBN: 978-0-85729-958-1
• No. of pages: 286
• Weight: 442 g
• Illustrations: XVIII, 286 p. 46 illus.
• Series: Lecture Notes in Control and Information Sciences; Lecture Notes in Control and Information Sciences; Lecture Notes in Control and I
• Subject: Natural sciences, medicine, IT, technology > Technology > Electronics, electrical engineering, communications engineering