Lecture Notes in Control and Information Sciences

Domain of Attraction

Analysis and Control via SOS Programming

Authors: Chesi, Graziano

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  • Offers instructors a concise and simple description of the main features of SOS programming which can be used in the classroom
  • Gives readers a systematic strategy for the estimation and control of the domain of attraction in dynamical systems
  •  Allows the reader to tackle stability problems through a thorough knowledge of the domain of attraction of equilibrium points
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About this book

The stability of equilibrium points plays a fundamental role in dynamical systems. For nonlinear dynamical systems, which represent the majority of real plants, an investigation of stability requires the characterization of the domain of attraction (DA) of an equilibrium point, i.e., the set of initial conditions from which the trajectory of the system converges to such a point. It is well-known that estimating the DA, or even more attempting to control it, are very difficult problems because of the complex relationship of this set with the model of the system.

The book also offers a concise and simple description of the main features of SOS programming which can be used in research and teaching. In particular, it introduces various classes of SOS polynomials and their characterization via LMIs and addresses typical problems such as establishment of positivity or non-positivity of polynomials and matrix polynomials, determining the minimum of rational functions, and solving systems of polynomial equations, in cases of both unconstrained and constrained variables. The techniques presented in this book are available in the MATLAB® toolbox SMRSOFT, which can be downloaded from http://www.eee.hku.hk/~chesi.

Domain of Attraction addresses the estimation and control of the DA of equilibrium points using the novel SOS programming scheme, i.e., optimization techniques that have been recently developed based on polynomials that are sums of squares of polynomials (SOS polynomials) and that amount to solving convex optimization problems with linear matrix inequality (LMI) constraints, also known as semidefinite programs (SDPs). For the first time in the literature, a means of dealing with these issues is presented in a unified framework for various cases depending on the nature of the nonlinear systems considered, including the cases of polynomial systems, uncertain polynomial systems, and nonlinear (possibly uncertain) non-polynomial systems. The methods proposed in this book are illustrated in a variety of real systems and simulated systems with randomly chosen structures and/or coefficients which includes chemical reactors, electric circuits, mechanical devices, and social models.

The book also offers a concise and simple description of the main features of SOS programming which can be used in research and teaching. In particular, it introduces various classes of SOS polynomials and their characterization via LMIs and addresses typical problems such as establishment of positivity or non-positivity of polynomials and matrix polynomials, determining the minimum of rational functions, and solving systems of polynomial equations, in cases of both unconstrained and constrained variables. The techniques presented in this book are available in the MATLAB® toolbox SMRSOFT, which can be downloaded from http://www.eee.hku.hk/~chesi.

About the authors

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.

Reviews

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)


Table of contents (7 chapters)

Table of contents (7 chapters)

Buy this book

eBook $109.00
price for USA in USD (gross)
  • ISBN 978-0-85729-959-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $139.00
price for USA in USD
  • ISBN 978-0-85729-958-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Domain of Attraction
Book Subtitle
Analysis and Control via SOS Programming
Authors
Series Title
Lecture Notes in Control and Information Sciences
Series Volume
415
Copyright
2011
Publisher
Springer-Verlag London
Copyright Holder
Springer London
eBook ISBN
978-0-85729-959-8
DOI
10.1007/978-0-85729-959-8
Softcover ISBN
978-0-85729-958-1
Series ISSN
0170-8643
Edition Number
1
Number of Pages
XVIII, 286
Number of Illustrations
46 b/w illustrations
Topics