DYNAMIC MODELING AND CONTROL OF ENGINEERING SYSTEMS
THIRD EDITION
This textbook is ideal for a course in Engineering System Dynamics and Controls. The work is a comprehensive treatment of the analysis of lumped-parameter physical systems. Starting with a discussion of mathematical models in general, and ordinary differential equations, the book covers input–output and state-space models, computer simulation, and modeling methods and techniques in mechanical, electrical, thermal, and fluid domains. Frequency-domain methods, transfer functions, and frequency response are covered in detail. The book concludes with a treatment of stability, feedback control (PID, lag–lead, root locus), and an introduction to discrete-time systems. This new edition features many new and expanded sections on such topics as Solving Stiff Systems, Operational Amplifiers, Electrohydraulic Servovalves, Using MATLAB^{®} with Transfer Functions, Using MATLAB with Frequency Response, MATLAB Tutorial, and an expanded Simulink^{®} Tutorial. The work has 40 percent more end-of-chapter exercises and 30 percent more examples.
Bohdan T. Kulakowski, Ph.D. (1942–2006) was Professor of Mechanical Engineering at Pennsylvania State University. He was an internationally recognized expert in automatic control systems, computer simulations and control of industrial processes, systems dynamics, vehicle–road dynamic interaction, and transportation systems. His fuzzy-logic algorithm for avoiding skidding accidents was recognized in 2000 by Discover magazine as one of its top 10 technological innovations of the year.
John F. Gardner is Chair of the Mechanical and Biomedical Engineering Department at Boise State University, where he has been a faculty member since 2000. Before his appointment at Boise State, he was on the faculty of Pennsylvania State University in University Park, where his research in dynamic systems and controls led to publications in diverse fields from railroad freight car dynamics to adaptive control of artificial hearts. He pursues research in modeling and control of engineering and biological systems.
J. Lowen Shearer (1921–1992) received his Sc.D. from the Massachusetts Institute of Technology. At MIT, between 1950 and 1963, he served as the group leader in the Dynamic Analysis & Control Laboratory, and as a member of the mechanical engineering faculty. From 1963 until his retirement in 1985, he was on the faculty of Mechanical Engineering at Pennsylvania State University. Professor Shearer was a member of ASME’s Dynamic Systems and Control Division and received that group’s Rufus Oldenberger Award in 1983. In addition, he received the Donald P. Eckman Award (ISA, 1965), and the Richards Memorial Award (ASME, 1966).
DYNAMIC MODELING AND CONTROL OF ENGINEERING SYSTEMS
THIRD EDITION
Bohdan T. Kulakowski
Deceased, formerly Pennsylvania State University
John F. Gardner
Boise State University
J. Lowen Shearer
Deceased, formerly Pennsylvania State University
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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www.cambridge.org
Information on this title: www.cambridge.org/9780521864350
© John F. Gardner 2007
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2007
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Kulakowski, Bohdan T.
Dynamic modeling and control of engineering systems / Bohdan T. Kulakowski, John F.
Gardner, J. Lowen Shearer. – 3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-521-86435-0 (hardback)
ISBN-10: 0-521-86435-6 (hardback)
1. Engineering – Mathematical models. 2. System engineering – Mathematical models.
I. Gardner, John F. (John Francis), 1958– II. Shearer, J. Lowen. III. Title.
TA342.S54 2007
620.001′1 – dc22 2006031544
Cambridge University Press has no responsibility for
the persistence or accuracy of URLs for external or
third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such
Web sites is, or will remain, accurate or appropriate.
MATLAB^{®} and Simulink^{®} are trademarks of The MathWorks, Inc. and are used with
permission. The MathWorks does not warrant the accuracy of the text or exercises in this
book. This book’s use or discussion of MATLAB^{®} and Simulink^{®} software or related
products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB^{®} and Simulink^{®} software.
Dedicated to the memories of Professor Bohdan T. Kulakowski (1942–2006), the victims of the April 16, 2007 shootings at Virginia Tech, and all who are touched by senseless violence. May we never forget and always strive to learn form history.
Contents
Preface | page xi | ||
1 | INTRODUCTION | 1 | |
1.1 Systems and System Models | 1 | ||
1.2 System Elements, Their Characteristics, and the Role of Integration | 4 | ||
Problems | 9 | ||
2 | MECHANICAL SYSTEMS | 14 | |
2.1 Introduction | 14 | ||
2.2 Translational Mechanical Systems | 16 | ||
2.3 Rotational–Mechanical Systems | 30 | ||
2.4 Linearization | 34 | ||
2.5 Synopsis | 44 | ||
Problems | 45 | ||
3 | MATHEMATICAL MODELS | 54 | |
3.1 Introduction | 54 | ||
3.2 Input–Output Models | 55 | ||
3.3 State Models | 61 | ||
3.4 Transition Between Input–Output and State Models | 68 | ||
3.5 Nonlinearities in Input–Output and State Models | 71 | ||
3.6 Synopsis | 76 | ||
Problems | 76 | ||
4 | ANALYTICAL SOLUTIONS OF SYSTEM INPUT–OUTPUT EQUATIONS | 81 | |
4.1 Introduction | 81 | ||
4.2 Analytical Solutions of Linear Differential Equations | 82 | ||
4.3 First-Order Models | 84 | ||
4.4 Second-Order Models | 92 | ||
4.5 Third- and Higher-Order Models | 106 | ||
4.6 Synopsis | 109 | ||
Problems | 111 | ||
5 | NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS | 120 | |
5.1 Introduction | 120 | ||
5.2 Euler’s Method | 121 | ||
5.3 More Accurate Methods | 124 | ||
5.4 Integration Step Size | 129 | ||
5.5 Systems of Differential Equations | 133 | ||
5.6 Stiff Systems of Differential Equations | 133 | ||
5.7 Synopsis | 138 | ||
Problems | 139 | ||
6 | SIMULATION OF DYNAMIC SYSTEMS | 141 | |
6.1 Introduction | 141 | ||
6.2 Simulation Block Diagrams | 143 | ||
6.3 Building a Simulation | 147 | ||
6.4 Studying a System with a Simulation | 150 | ||
6.5 Simulation Case Study: Mechanical Snubber | 157 | ||
6.6 Synopsis | 164 | ||
Problems | 165 | ||
7 | ELECTRICAL SYSTEMS | 168 | |
7.1 Introduction | 168 | ||
7.2 Diagrams, Symbols, and Circuit Laws | 169 | ||
7.3 Elemental Diagrams, Equations, and Energy Storage | 170 | ||
7.4 Analysis of Systems of Interacting Electrical Elements | 175 | ||
7.5 Operational Amplifiers | 179 | ||
7.6 Linear Time-Varying Electrical Elements | 186 | ||
7.7 Synopsis | 188 | ||
Problems | 189 | ||
8 | THERMAL SYSTEMS | 198 | |
8.1 Introduction | 198 | ||
8.2 Basic Mechanisms of Heat Transfer | 199 | ||
8.3 Lumped Models of Thermal Systems | 202 | ||
8.4 Synopsis | 212 | ||
Problems | 213 | ||
9 | FLUID SYSTEMS | 219 | |
9.1 Introduction | 219 | ||
9.2 Fluid System Elements | 220 | ||
9.3 Analysis of Fluid Systems | 225 | ||
9.4 Electrohydraulic Servoactuator | 228 | ||
9.5 Pneumatic Systems | 235 | ||
9.6 Synopsis | 243 | ||
Problems | 244 | ||
10 | MIXED SYSTEMS | 249 | |
10.1 Introduction | 249 | ||
10.2 Energy-Converting Transducers and Devices | 249 | ||
10.3 Signal-Converting Transducers | 254 | ||
10.4 Application Examples | 255 | ||
10.5 Synopsis | 261 | ||
Problems | 261 | ||
11 | SYSTEM TRANSFER FUNCTIONS | 273 | |
11.1 Introduction | 273 | ||
11.2 Approach Based on System Response to Exponential Inputs | 274 | ||
11.3 Approach Based on Laplace Transformation | 276 | ||
11.4 Properties of System Transfer Functions | 277 | ||
11.5 Transfer Functions of Multi-Input, Multi-Output Systems | 283 | ||
11.6 Transfer Function Block-Diagram Algebra | 286 | ||
11.7 MATLAB Representation of Transfer Function | 293 | ||
11.8 Synposis | 298 | ||
Problems | 299 | ||
12 | FREQUENCY ANALYSIS | 302 | |
12.1 Introduction | 302 | ||
12.2 Frequency-Response Transfer Functions | 302 | ||
12.3 Bode Diagrams | 307 | ||
12.4 Relationship between Time Response and Frequency Response | 314 | ||
12.5 Polar Plot Diagrams | 317 | ||
12.6 Frequency-Domain Analysis with MATLAB | 319 | ||
12.7 Synopsis | 323 | ||
Problems | 323 | ||
13 | CLOSED-LOOP SYSTEMS AND SYSTEM STABILITY | 329 | |
13.1 Introduction | 329 | ||
13.2 Basic Definitions and Terminology | 332 | ||
13.3 Algebraic Stability Criteria | 333 | ||
13.4 Nyquist Stability Criterion | 338 | ||
13.5 Quantitative Measures of Stability | 341 | ||
13.6 Root-Locus Method | 344 | ||
13.7 MATLAB Tools for System Stability Analysis | 349 | ||
13.8 Synopsis | 351 | ||
Problems | 352 | ||
14 | CONTROL SYSTEMS | 356 | |
14.1 Introduction | 356 | ||
14.2 Steady-State Control Error | 357 | ||
14.3 Steady-State Disturbance Sensitivity | 361 | ||
14.4 Interrelation of Steady-State and Transient Considerations | 364 | ||
14.5 Industrial Controllers | 365 | ||
14.6 System Compensation | 378 | ||
14.7 Synopsis | 383 | ||
Problems | 383 | ||
15 | ANALYSIS OF DISCRETE-TIME SYSTEMS | 389 | |
15.1 Introduction | 389 | ||
15.2 Mathematical Modeling | 390 | ||
15.3 Sampling and Holding Devices | 396 | ||
15.4 The z Transform | 400 | ||
15.5 Pulse Transfer Function | 405 | ||
15.6 Synopsis | 407 | ||
Problems | 408 | ||
16 | DIGITAL CONTROL SYSTEMS | 410 | |
16.1 Introduction | 410 | ||
16.2 Single-Loop Control Systems | 410 | ||
16.3 Transient Performance | 412 | ||
16.4 Steady-State Performance | 418 | ||
16.5 Digital Controllers | 421 | ||
16.6 Synopsis | 423 | ||
Problems | 424 | ||
APPENDIX 1. Fourier Series and the Fourier Transform | 427 | ||
APPENDIX 2. Laplace Transforms | 432 | ||
APPENDIX 3. MATLAB Tutorial | 438 | ||
APPENDIX 4. Simulink Tutorial | 463 | ||
Index | 481 |
Preface
From its beginnings in the middle of the 20th century, the field of systems dynamics and feedback control has rapidly become both a core science for mathematicians and engineers and a remarkably mature field of study. As early as 20 years ago, textbooks (and professors) could be found that purported astoundingly different and widely varying approaches and tools for this field. From block diagrams to signal flow graphs and bond graphs, the diversity of approaches, and the passion with which they were defended (or attacked), made any meeting of systems and control professionals a lively event.
Although the various tools of the field still exist, there appears to be a consensus forming that the tools are secondary to the insight they provide. The field of system dynamics is nothing short of a unique, useful, and utterly different way of looking at natural and manmade systems. With this in mind, this text takes a rather neutral approach to the tools of the field, instead emphasizing insight into the underlying physics and the similarity of those physical effects across the various domains.
This book has its roots as lecture notes from Lowen Shearer’s senior-level mechanical engineering course at Penn State in the 1970s with additions from Bohdan Kulakowski’s and John Gardner’s experiences since the 1980s. As such, it reveals those roots by beginning with lumped-parameter mechanical systems, engaging the student on familiar ground. The following chapters, dealing with types of models (Chapter 3) and analytical solutions (Chapter 4), have seen only minimal revisions from the original version of this text, with the exception of modest changes in order of presentation and clarification of notation. Chapters 5 and 6, dealing with numerical solutions (simulations), were extensively rewritten for the second edition and further updated for this edition. Although we made a decision to feature the industry-standard software package (MATLAB^{®}) in this book (Appendices 3 and 4 are tutorials on MATLAB and Simulink^{®}), the presentation was specifically designed to allow other software tools to be used.
Chapters 7, 8, and 9 are domain-specific presentations of electric, thermal, and fluid systems, respectively. For the third edition, these chapters have been extensively expanded, including operational amplifiers in Chapter 7, an example of lumped approximation of a cooling fin in Chapter 8, and an electrohydraulic servovalve in Chapter 9. Those using this text in a multidisciplinary setting, or for nonmechanical engineering students, may wish to delay the use of Chapter 2 (mechanical systems) to this point, thus presenting the four physical domains sequentially. Chapter 10 presents some important issues in dealing with multidomain systems and how they interact.
Chapters 11 and 12 introduce the important concept of a transfer function and frequency-domain analysis. These two chapters are the most revised and (hopefully) improved parts of the text. In previous editions of this text, we derived the complex transfer function by using complex exponentials as input. For the third edition, we retain this approach, but have added a section showing how to achieve the same ends using the Laplace transform. It is hoped that this dual approach will enrich student understanding of this material. In approaching these, and other, revisions, we listened carefully to our colleagues throughout the world who helped us see where the presentation could be improved. We are particularly grateful to Sean Brennan (of Penn State) and Giorgio Rizzoni (of Ohio State) for their insightful comments.
This text, and the course that gave rise to it, is intended to be a prerequisite to a semester-long course in control systems. However, Chapters 13 and 14 present a very brief discussion of the fundamental concepts in feedback control, stability (and algebraic and numerical stability techniques), closed-loop performance, and PID and simple cascade controllers. Similarly, the preponderance of digitally implemented control schemes necessitates a discussion of discrete-time control and the dynamic effects inherent in sampling in the final chapters ( 15 and 16). It is hoped that these four chapters will be useful both for students who are continuing their studies in electives or graduate school and for those for which this is a terminal course of study.
Supplementary materials, including MATLAB and Simulink files for examples throughout the text, are available through the Cambridge University Press web site (http://www.cambridge.org/us/engineering) and readers are encouraged to check back often as updates and additional case studies are made available.
Outcomes assessment, at the program and course level, has now become a fixture of engineering programs. Although necessitated by accreditation criteria, many have discovered that an educational approach based on clearly stated learning objectives and well-designed assessment methods can lead to a better educational experience for both the student and the instructor. In the third edition, we open each chapter with the learning objectives that underlie each chapter. Also in this edition, the examples and end-of-chapter problems, many of which are based on real-world systems encountered by the authors, were expanded.
This preface closes on a sad note. In March of 2006, just as the final touches were being put on this edition, Bohdan Kulakowski was suddenly and tragically taken from us while riding his bicycle home from the Penn State campus, as was his daily habit. His family, friends, and the entire engineering community suffered a great loss, but Bohdan’s legacy lives on in these pages, as does Lowen’s. As the steward of this legacy, I find myself “standing on the shoulders of giants” and can take credit only for its shortcomings.
JFG
Boise, ID
May, 2007