Selected Problems of Fractional Systems Theory
This monograph covers some selected problems of positive fractional 1D and 2D linear systems. It is an extended and modified English version of its preceding Polish edition published by Technical University of Bialystok in 2009. This book is based on the
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Tadeusz Kaczorek
Selected Problems of Fractional Systems Theory
ABC
Series Advisory Board P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis
Author Prof. Tadeusz Kaczorek Bialystok University of Technology Faculty of Electrical Engineering ul. Wiejska 45A 15-351 Bialystok Poland E-mail: [email protected]
ISBN 978-3-642-20501-9
e-ISBN 978-3-642-20502-6
DOI 10.1007/978-3-642-20502-6 Lecture Notes in Control and Information Sciences
ISSN 0170-8643
Library of Congress Control Number: 2011928068 c 2011 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com
Preface
This monograph covers some selected problems of positive fractional 1D and 2D linear systems. It is an extended and modified English version of its preceding Polish edition published by Technical University of Bialystok in 2009. This book is based on the lectures delivered by the author to the Ph.D. students of the Faculty of Electrical Engineering of Bialystok University of Technology and of Warsaw University of Technology and on invited lectures in several foreign universities in the last three years. The monograph consists of 12 chapters. 4 appendices and a list of references. Chapter 1 is devoted to the fractional discrete-time systems. The solution to the state equations of fractional discrete-time linear systems is derived. Necessary and sufficient conditions for the positivity, the reachability and the controllability to zero are established. The minimum energy control problem for the positive fractional systems is formulated and solved. The solution to the fractional different orders discrete-time linear systems is also derived. The fractional continuous-time linear systems are considered in Chapter 2. The definitions of fractional derivatives are recalled and the solutions to the fractional state equations of continuous-time linear systems are derived. Necessary and sufficient conditions for the internal and external positivity and the reachability of the systems without and with delays are established. The fractional positive 2D linear systems are addressed in
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