Control of Continuous Linear Systems Control of Continuous Linear Systems Kaddour Najim First published in Great Britain and the United States in 2006 by ISTE Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd ISTE USA 6 Fitzroy Square 4308 Patrice Road London W1T 5DX Newport Beach, CA 92663 UK USA www.iste.co.uk © ISTE Ltd, 2006 The rights of Kaddour Najim to be identified as the author of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Najim, K. Control of continuous linear systems / Kaddour Najim. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-905209-12-5 ISBN-10: 1-905209-12-6 1. Linear systems--Automatic control--Mathematics. 2. Linear control systems. 3. Calculus, Operational. I. Title. TJ220.N35 2006 629.8'32--dc22 2006009140 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 10: 1-905209-12-6 ISBN 13: 978-1-905209-12-5 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 1. On Process Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2. Model classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1. Heat and mass balances . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.2. Mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.2.3. Electrical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.3. Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 2. Laplace Transforms and Block Diagrams . . . . . . . . . . . . . 67 2.1. The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2. Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3. Laplace transform calculations . . . . . . . . . . . . . . . . . . . . . . . 72 2.4. Differential and integral equations . . . . . . . . . . . . . . . . . . . . . 87 2.5. Block diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.6. Feedback systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 3. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.2. Step responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.3. System identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.4. Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Chapter 4. Stability and the Root Locus . . . . . . . . . . . . . . . . . . . . 205 4.1. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.1.1. The Routh–Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . 207 4.1.2. Revers’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5 6 Control of Continuous Linear Systems 4.2. The root locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Chapter 5. Regulation and PID Regulators . . . . . . . . . . . . . . . . . . . 255 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.2. Direct design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 5.3. PID tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 A. On Theoretical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.1. The Dirac impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.1.1. Residence time . . . . . . . . . . . . . . . . . . . . . . . . . . 310 A.2. The unit step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 A.3. The Routh–Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . 313 A.4. The Nyquist criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 320 A.5. The root locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 A.6. Computation of integrals of the form J 2 . . . . . . . . . . . . . . . 335 A.7. On non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 336 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Control of Continuous Linear Systems This page intentionally left blank Control of Continuous Linear Systems Kaddour Najim First published in Great Britain and the United States in 2006 by ISTE Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd ISTE USA 6 Fitzroy Square 4308 Patrice Road London W1T 5DX Newport Beach, CA 92663 UK USA www.iste.co.uk © ISTE Ltd, 2006 The rights of Kaddour Najim to be identified as the author of this work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Najim, K. Control of continuous linear systems / Kaddour Najim. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-905209-12-5 ISBN-10: 1-905209-12-6 1. Linear systems--Automatic control--Mathematics. 2. Linear control systems. 3. Calculus, Operational. I. Title. TJ220.N35 2006 629.8'32--dc22 2006009140 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 10: 1-905209-12-6 ISBN 13: 978-1-905209-12-5 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire. . references and index. ISBN-13: 97 8-1 -9 0520 9-1 2-5 ISBN-10: 1-9 0520 9-1 2-6 1. Linear systems- -Automatic control- -Mathematics. 2. Linear control systems. 3. Calculus,. references and index. ISBN-13: 97 8-1 -9 0520 9-1 2-5 ISBN-10: 1-9 0520 9-1 2-6 1. Linear systems- -Automatic control- -Mathematics. 2. Linear control systems. 3. Calculus,