Laura Menini - Algebraic Geometry for Robotics and Control Theory

Laura Menini

University of Rome Tor Vergata, Italy

Corrado Possieri

IASI-CNR, Italy

Antonio Tornamb

University of Rome Tor Vergata, Italy

Published by

World Scientific Publishing Europe Ltd.

57 Shelton Street, Covent Garden, London WC2H 9HE

Head office: 5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

Library of Congress Cataloging-in-Publication Data

Names: Menini, Laura, author. | Possieri, Corrado, author. | Tornamb, Antonio, author.

Title: Algebraic geometry for robotics and control theory / Laura Menini, University of Rome Tor Vergata, Italy, Corrado Possieri, IASI-CNR, Italy, Antonio Tornamb, University of Rome Tor Vergata, Italy.

Description: New Jersey : World Scientific, [2022] | Includes bibliographical references and index.

Identifiers: LCCN 2021021130 | ISBN 9781800610453 (hardcover) | ISBN 9781800610460 (ebook for institutions) | ISBN 9781800610477 (ebook for individuals)

Subjects: LCSH: Geometry, Algebraic. | Robotics--Mathematical models. | Control theory--Mathematical models.

Classification: LCC QA564 .M44 2022 | DDC 516.3/520285--dc23

LC record available at https://lccn.loc.gov/2021021130

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Copyright 2022 by World Scientific Publishing Europe Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/Q0308#t=suppl

Desk Editors: Balamurugan Rajendran/Michael Beale/Shi Ying Koe

Typeset by Stallion Press
Email:

Printed in Singapore

The purpose of this book is to show how algebraic geometry methods can be used to solve different problems in the fields of robotics and control theory, including in the last topic both analysis of dynamic systems and control design.

It is perhaps reductive to see the link between such disciplines based on the fact that algebraic geometry provides algorithms to solve, mostly in an exact manner and often parameterizing sets of solutions, systems of polynomial equalities and inequalities, which are very useful for the design of control laws and observers; as it will be seen throughout the book, algebraic geometry also provides the notions and concepts, such as ideals and varieties, to properly formalize some of the problems under study.

The various applications presented in this book are naturally originated from our previous research experience, and certainly do not exhaust the richness of results that can be obtained in robotics and control theory by using the techniques typical of algebraic geometry; with this in mind, just to help the reader willing to use the same methodologies, we have tried to detail as much as possible our derivations, illustrating with small examples also some of the relatively easy steps, with the purpose of being as much introductory as possible. In some cases, the examples are very simple: this is done on purpose to better emphasize the meaning of the concepts, together with the algorithmic aspects.

It is not assumed that the reader already knows algebraic geometry, only some basic knowledge of calculus and linear algebra is required; therefore, Chapter 1 is devoted to a concise but rigorous exposition of all the basic notions needed in this book; an effort has been made to render the description self contained and to illustrate every concept by examples, most of which simple enough so that computations can be performed by hand. However, such a first chapter is not aimed at a thorough presentation of the topic, so that proofs and also many results that do not find application here are not reported; for a deeper understanding of algebraic geometry the reader is referred to the cited texts, especially Cox et al. (2015).

To further guide the reader to effectively use the proposed methodologies and algorithms, in Chapter it is illustrated how the computations referred to in the examples of Chapter 1 can be performed on Macaulay2, a free CAS (computer algebra system). It is worth stressing that most of the computations described in the book can be performed by other CAS, Macaulay2has been preferred because it is free and complete as for the functionalities needed here.

After the first two chapters, the order of exposition is mostly based on the rationale of postponing further developments of the theory until the chapter or section in which they are used, this means that specifications of the general concepts exposed in Chapter 1 can be found in subsequent sections, just at the point in which they are needed.

As for robotics and control theory, it is assumed that the reader knows the fundamentals of dynamic systems theory and some standard tools in control design, such as Lyapunov stability basic theorems and, as for linear systems, structural properties and observer design principles; however, the problems have been mostly formulated in a self-contained manner, and, only in some cases where to recall more background would have been distracting, references are given to books or papers where the required concepts are introduced. Apart from such references, which are needed to complete or deepen what reported in the book, and others that are at some point necessary to better define our approach, we did not systematically include survey sections in the different chapters; this is due to the number and heterogeneity of the applications considered, it would have been very difficult to obtain a reasonably complete overview on each of the topics touched. We apologize for such an omission and we hope that the references included sufficiently complete this aspect.

Laura Menini received the Laurea degree in 1993 and the Ph.D. in 1997, from the University of Rome Tor Vergata, Italy, where she is currently Full Professor. She co-authored the books Symmetries and Semi-invariants in the Analysis of Nonlinear Systems (Springer, 2011) and Mathematical Methods for System Theory (World Scientific, 1998). She is Chair of the IFAC Coordinating Committee 2 Design Method and Senior Editor of the IEEE Control Systems Letters.

Corrado Possieri