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Unformatted text preview: Feedback Control of
Dynamic Systems Fourth Edition Gene F. Franklin
Stanford University J. David Powell
Stanford University Abbas Emami—Naeini
SC Solutions, Inc. Prentice Hall A Prentice Hall
Upper Saddle River, New Jersey 07458 Library of Congress CataloginginPublication Data Franklin, Gene F. Feedback control of dynamic systems / Gene F. Franklin. J. David Powell, Abbas EmamiNaeini.—4th ed. p. cm. Includes index. ISBN (P1370323934
1. Feedback control systems. 1. Powell, J. David. 11. EmamiNaeini, Abbas. 111. Title.
CIP Data available. Vice President and Editorial Director, ECS: Marcia J. Horton
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earth courtesy of NASA. Photograph of Boeing 747 jet aircraft courtesy of Tony Stone Images. To Gertrude, David, Carole
Valerie, Daisy, Annika, Davenport
Malahat, Sheila, Nima Prentice © 2002, 1994, 1991, 1986 by Prentice Hall
Hall Prentice—Hall, Inc.
' Upper Saddle River, New Jersey 07458 All rights reserved. No part of this book may be reproduced, in any format or by any means, without permission in writing
from the publisher. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the de—
velopment. research, and testing of the theories and programs to determine their effectiveness. The author and publisher
make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in
this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection
with, or arising out of, the furnishing, performance, or use of these programs. ‘ MATLAB and Simulink are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive. Natick. MA, 017602098.
Printed in the United States of America 109876543 ISBN III13433333344 Reprinted with corrections December, 2002. Pearson Education Ltd., London Pearson Education Australia Pty. Limited, Sydney
Pearson Education Singapore. Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong
Pearson Education Canada Inc., Toronto Pearson Education de Mexico, SA. de CV. Pearson Education—Japan, Tokyo Pearson Education Malaysia. Pte. Ltd. Pearson Education, Upper Saddle River, New Jersey Contents Preface xi An Overview and Brief History of Feedback Control 1 1.1
1.2
1.3
1.4 Chapter Overview 1 A Perspective on Feedback Control 1
A Simple Feedback System 2 A First Analysis of Feedback 6 A Brief History 9 An Overview of the Book 15
Summary 17 Review Questions 18 Problems 18 Dynamic Models 22 2.1
2.2
2.3
2.4
2.5
2.6 Chapter Overview 22 A Perspective on Dynamic Models 23
Dynamics of Mechanical Systems 24
Differential Equations in StateVariable Form 41
Models of Electric Circuits 46 Models of Electromechanical Systems 51
Heat and FluidFlow Models 57
Linearization and Scaling 68 Summary 77 Review Questions 78 Problems 78 Contents P} F} Dynamic Response 94 3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8 Chapter Overview 94 A Perspective on System Response 95
Review of Laplace Transforms 96 System Modeling Diagrams 123 Effect of Pole Locations 135
TimeDomain Speciﬁcations 144 Effects of Zeros and Additional Poles 150
Stability 157 Numerical Simulation 167 Obtaining Models from Experimental Data 173
Summary 179 Review Questions 181 Problems 181 Basic Properties of Feedback 200 4.1
4.2
4.3
4.4 The 5.1
5.2
5.3
5.4
5.5
5.6
5.7 Chapter Overview 200 A Perspective on Properties of Feedback 201
A Case Study of Speed Control 202 The Classical Three—Term Controller 215
SteadyState Tracking and System Type 230
Digital Implementation of Controllers 245
Summary 251 Review Questions 252 Problems 253 RootLocus Design Method 270 Chapter Overview 270 A Perspective on the Rootlocus Design Method 271
Root Locus of a Basic Feedback System 272
Guidelines for Sketching a Root Locus 277
Selected Illustrative Root Loci 293 Selecting the Parameter Value 307 Dynamic Compensation 310 A Design Example Using the Root Locus 322
Extensions of the RootLocus Method 329
Summary 341 Review Questions 343 Problems 344 >>>> >>> Contents The FrequencyResponse Design Method 364 6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11 Chapter Overview 364 A Perspective on the Frequency—response Design Method 365
Frequency Response 366 Neutral Stability 387 The Nyquist Stability Criterion 390 Stability Margins 403 Bode’s GainPhase Relationship 411 ClosedLoop Frequency Response 415 Compensation 417 Alternate Presentations of Data 443 Speciﬁcations in Terms of the Sensitivity Function 448 Time Delay 457 Obtaining a PoleZero Model from FrequencyResponse Data 459
Summary 462 Review Questions 464 Problems 465 State—Space Design 492 7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12 Chapter Overview 492 A Perspective on StateSpace Design 493 Advantages of State Space 494 Analysis of the State Equations 495 Control—Law Design for FullState Feedback 515 Selection of Pole Locations for Good Design 529 Estimator Design 541 Compensator Design: Combined Control Law and Estimator 555
Loop Transfer Recovery (LTR) 569 Introduction of the Reference Input with the Estimator 574
Integral Control and Robust Tracking 586 Direct Design with Rational Transfer Functions 604 Design for Systems with Pure Time Delay 608 Lyapunov Stability 614 Summary 618 Review Questions 622 Problems 622 Digital Control 646 8.1
8.2 Chapter Overview 646 A Perspective on Digital Control 647
Digitization 648 Dynamic Analysis of Discrete Systems 650 vii Contents ix
V111 Contents C .2 Elementary Operations on Matrices 838
C .3 Trace 839 C .4 Transpose 839 (1.5 Determinant and Matrix Inverse 839 C .6 Properties of the Determinant 840 C7 Inverse of Block Triangular Matrices 841
C8 Special Matrices 841 C .9 Rank 842 C .10 Characteristic Polynomial 842 CH Cayley—Hamilton Theorem 843 C. 12 Eigenvalues and Eigenvectors 843 C13 Similarity Transformations 844 C .14 Matrix Exponential 845 C. 1 5 Fundamental Subspaces 845 C .16 SingularValue Decomposition 846
C17 Positive Deﬁnite Matrices 847 C18 Matrix Identity 847 8.3 Design by Emulation 658
8.4 Discrete Design 668
8.5 StateSpace Design Methods 676
8.6 Hardware Characteristics 683
8.7 WordSize Effects 686
8.8 SampleRate Selection 689
 Summary 692
Review Questions 694
Problems 694 9 ControlSystem Design: Principles and Case Studies 706 Chapter Overview 706
A Perspective on Design Principles 707
9.1 An Outline of Control Systems Design 708
9.2 Design of a Satellite’s Attitude Control 714
9.3 Lateral and Longitudinal Control of a Boeing 747 733
9.4 Control of the Fuel—Air Ratio in an Automotive Engine 752
9.5 Control of a Digital Tape Transport 759
9.6 Control of the Read/Write Head Assembly of a Hard Disk 770
9.7 Control of Rapid Thermal Processing (RTP) Systems in Semiconductor
Wafer Manufacturing 781
Summary 795
Review Questions 797
Problems 797 Appendix D Controllability and Observabiiity 849 D.1 Controllability 850
D2 Observability 855 Appendix E Ackermann’s Formula for Pole Placement 857 AppendixF MATLAB Commands 861
AppendixA Laplace Transforms 811 Al The DC Laplace Transform 811 Appendix G Solutions to Review Questions 863 Appendix B A Review of Complex Variables 825 8.1 Deﬁnition of a Complex Number 825 8.2 Algebraic Manipulations 827 B3 Graphical Evaluation of Magnitude and Phase 829
B4 Differentiation and Integration 830 B5 Euler’s Relations 831 B6 Analytic Functions 832 B.7 Cauchy’s Theorem 832 B8 Singularities and Residues 832 B9 Residue Theorem 833 B. 10 The Argument Principle 834 References 879 index 887 Appendix C Summary of Matrix Theory 837
C1 Matrix Deﬁnitions 837 Preface In this fourth edition we again had the objectives of retaining the best of the
previous editions, to rewrite key sections where we felt it was possible to im
prove the presentations and enhance the book’s pedagogical effectiveness, and
to take better advantage of the wide use of computers in control design, espe
cially the toolboxes of MATLAB and Simulink, from The Mathworks, Inc. The basic structure of the book is unchanged and we continue to combine
analysis with design using the three approaches of the root locus, frequency
response, and state variable equations. The text continues to include carefully
worked out examples, many of them new to this edition, to illustrate the mate
rial. As a new feature, to assist the students in verifying that they have learned
the material, we provide a set of review questions at the end of each chap
ter with answers in the back of the book. While modest changes were made
throughout the entire book, special attention was given to the introduction of
transforms in Chapter 3, to the introduction to feedback in Chapter 4, and to
the organization and statements of the problems appearing at the end of each
chapter. In the three central chapters on the design methods, we continue to ex
pect the students to learn how to perform the basic calculations by hand in
order to be able to guide a design by understanding (and frequently by a quick
sketch) rather than by computer rote. However, more than in previous edi
tions, we deemphasize the manual work and introduce computer tools early
on in recognition of the universal use of these tools in control analysis and de
sign. For example, we no longer mark certain problems as requiring a computer
but, rather, expect that the student has access to a computer in every case, as
needed. Furthermore, in recognition of the fact that, increasingly, controllers are
implemented in embedded computers, we introduce digital control in Chapter
4 and in a number of cases compare the responses of feedback systems using
analog controllers with those having a digital “equivalent” controller. As be
fore, we have prepared a collection of all the MATLAB “.m” ﬁles used to produce
the ﬁgures in the book and these are available at the companion web site for
this title: http://www.prenhall.com/franklin xii Preface or at the homepage for SC Solutions, Inc.:‘
http://scsolutlons.com/scsolutions.control.htm As representative applications of control, we again present extensive case stud
ies in Chapter 9. In this edition we have added new studies of the control of
the read~write head assembly of a computer hard disk and the temperature
control of a silicon wafer in a Rapid Thermal Processor used in the fabrication
of integrated circuits. We feel that this fourth edition presents the material with good pedagogical
support, provides strong motivation for the study of control, and represents a solid foundation for meeting the educational challenges of a study of feedback
control. Addressing the Educational Challenges Some of the educational challenges facing students of feedback control are
longstanding; others have emerged in recent years. Some of the challenges re
main for students across their entire engineering education; others are unique
to this relatively sophisticated course. Whether they are old or new, general or
particular, the educational challenges we perceived were critical to the evolu
tion of this text. Here we will state several educational challenges and describe
our approaches to each of them. 0 CHALLENGE: Students must master design as well as analysis techniques. Design is central to all of engineering and especially to control systems. Stu
dents ﬁnd that design issues, with their corresponding opportunities to tackle
practical applications, particularly motivating. But students also ﬁnd design
problems difﬁcult because design problem statements are usuallypoorly posed
and lack unique solutions. Because of both its inherent importance for and its
motivational effect on students, design is emphasized throughout this text so
that conﬁdence in solving design problems is developed from the start. The emphasis on design begins in Chapter 4, following the development
of modeling and dynamic response. The basic idea of feedback is introduced
ﬁrst, showing its inﬂuence on disturbance rejection, tracking accuracy, and ro—
bustness to parameter changes. The design orientation continues with uniform
treatments of the root locus, frequency response, and state variable feedback
techniques. All of the treatments are aimed at providing the knowledge neces
sary to ﬁnd a good feedback control design with no more complex mathematical
development than is essential to clear understanding. Throughout the text, examples are used to compare and contrast the design
techniques afforded by the different design methods and, in the capstone case
studies of Chapter 9, complex realworld design problems are tackled using all
of the methods in a uniﬁed way. 0 CHALLENGE: New ideas continue to be introduced into control. Preface Xiii Control is an active ﬁeld of research and hence there is a steady inﬂux of new
concepts, ideas, and techniques. In time, some of these elements develop to
the point where they join the list of things every control engineer must know.
This text is devoted to supporting students equally in their need to grasp both
traditional and more modern topics. In each of our previous editions we have tried to give equal time to root
locus, frequency response, and state variable methods for design. In this edition
we have shifted the emphasis from manual design methods augmented with
computer tools to an emphasis on computeraided methods augmented with a
solid mastery of the underlying techniques. Included in this reemphasis is the
early introduction of sampling, which enables one to design digital controllers.
While this material can be skipped to save time without disruption of the ﬂow
of the text, we feel that it is very important for students to recognize that digital
control is being used increasingly and that the most basic techniques of digital
control are easily mastered. With regret we acknowledge that we are not able at this time to introduce
the important topics of hybrid control or designs based on various optimization
methods. 0 CHALLENGE: Students need to manage a great deal of information. The vast array of systems to which feedback control is applied and the grow—
ing variety of techniques available for the solution of control problems means
that today’s student of feedback control must learn many new ideas. How do
students keep their perspective as they plow through lengthy and complex tex
tual passages? How do they identify highlights and draw conclusions? How do
they review for exams? Helping students with these tasks was a criterion for the
fourth edition. We outline these features in the accompanying table on page xiv. 0 CHALLENGE: Students offeedback control come from a wide range of
disciplines. Feedback control is an interdisciplinary ﬁeld in that control is applied to sys—
tems in every conceivable area of engineering. Consequently, some schools
have separate introductory courses for control Within the standard disciplines
and some, such as Stanford University, have a single set of courses taken by
students from many disciplines. However, to restrict the examples to one ﬁeld
is to miss much of the range and power of feedback; but to cover the whole
range of applications is overwhelming. In this book we develop the interdisci
plinary nature of the ﬁeld and provide review material for several of the most
common technologies so that students from many disciplines will be comfort
able with the presentation. For electrical engineering students who typically
have a good background in transform analysis, we include an introduction to
writing equations of motion for mechanical mechanisms in Chapter 2. For me—
chanical engineers, we include in Chapter 3 a review of the Laplace Transform
and dynamic response as needed in control. In addition, we introduce other
technologies brieﬂy and, from time to time, we present the equations of motion
of a physical system Without derivation but with enough physical description to XiV Preface FEATURE REFERENCE
EXAMPLE
Chapter openers offer perspective and overview. They place the speciﬁc Chapter 3 opener,
chapter topic in the context of the discipline as a whole and they pp. 94—95
brieﬂy overview the chapter sections.
Margin. notes help students scan for chapter highlights. They point to pp. 49—50
important deﬁnitions, equations, and concepts.
Boxed highlights identify key concepts within the running text. They Advantage of feedback,
also function to summarize important design procedures. p. 206;
compensation design,
p. 440
Bulleted chapter summaries help with student review and prioritization. Chapter 2 summary,
These summaries brieﬂy reiterate the key concepts and conclusions pp. 77—78
of the chapter.
Synopsis of design aids. Relationships used in design and throughout Inside back cover
the book are collected in one place for easy reference.
The color blue is used (1) to highlight useful; pedagogical features; Fig. 5.43, p. 330
(2) to highlight components under particular scrutiny within block Fig. 2.9, p. 32
diagrams; (3) to distinguish curves on graphs; and (4) to lend a more
realistic look to ﬁgures of physical systems.
Review questions at the end of each chapter with solutions in the back Chapter 2, p. 78 guide the student in self—study. be understood from a response point of View. Examples of some of the physical
systems represented in the text include the readwrite head for a computer disk
drive, a satellite tracking system, the fuelair ratio in an automobile engine, and
an airplane autopilot system. Outline of the Book The contents of the book is organized into nine chapters and seven appendixes.
The chapters include some sections of advanced or enrichment material marked
with a triangular blue icon that can be omitted without interfering with the ﬂow
of the material. Examples and problems based on this material are also marked
with these icons. The appendixes include background and reference material
such as Laplace transform tables, a review of complex variables, a review of
matrix theory, and answers to the end—ofchapter review questions. In Chapter 1, the essential ideas of feedback and some of the key design
issues are introduced. The chapter also contains a brief history of control, from Preface XV the ancient beginnings of process control to the contributions of ﬂight control
and electronic feedback ampliﬁers. It is hoped that this brief history will give a
context for the ﬁeld, introduce some of the key ﬁgures who contributed to its
development, and provide motivation to the student for the studies to come. Chapter 2 is a short presentation of dynamic modeling and includes me
chanical, electrical, electro—mechanical, ﬂuid, and thermodynamic devices. It
also discusses the state variable formulation of differential equations. This ma
terial can be omitted, used as the basis for review homework to smooth out the
usual nonuniform preparation of students, or covered in depth. Chapter 3 covers dynamic response as used in control. Again, much of this
material may have been covered previously, especially by electrical engineering
students. For many students, the correlation between pole locations and tran
sient response and the effects of extra zeros and poles on dynamic response is
new material, as is the notion of stability of a closedloop system. This material
needs to be covered carefully. Chapter 4 introduces feedback in the most elementary context, permitting
concentration on the essential effects of feedback on tracking accuracy, distur
bance rejection, and sensitivity to model errors. The basic equation and transfer
functions of feedback are introduced along with the deﬁnitions of the sensitivity
and complementary sensitivity functions. In the context of a ﬁrstorder model
for speed control, the concepts of proportional, integral, and derivative (PID)
control are introduced. In this way, the student gets the idea of what control
is all about before the tedious rules of root locus or the Nyquist Stability Cri
terion are developed. Finally, in this chapter the basic issues of digital control
are introduced, along with the idea of a digital equivalent controller. In this ap
proach, the central issues of control design are brought forward and can remain
in the foreground during the development of the necessary analysis that goes
with construction of sophisticated design tools. The concepts of steady—state
tracking error and system type are also treated here. Following the overview of feedback, the core of the book presents the
design methods based on root locus, frequency response, and state variable
feedback in Chapters 5, 6, and 7, respectively. Chapter 8 develops in more detail the tools needed to design feedback
control for implementation in a digital computer. However, for a complete
treatment of feedback control using digital computers, the reader is referred to
the companion text, Digital Control of Dynamic Systems, by Franklin, Powell,
and Workman (Prentice Hall, 1998). In Chapter 9, the three primary approaches are integrated in several case
studies and a framework for design is described that includes a touch of the
realworld context of practical control design. Course Configurations The material in this text can be covered ﬂexibly. Most ﬁrstcourse students
in controls will have some background in dynamics and Laplace transforms. xvi Preface Therefore, Chapter 2 and most of Chapter 3 would be a review for those stu
dents. In a 10week quarter, it is possible to review Chapter 3, and cover all of
Chapters 1, 4, 5, and 6. Most optional sections noted with a blue triangle should
be omitted. In the second quarter, Chapters 7 and 9 can be covered comfortably
including these optional sections. Alternatively, some optional sections could
be omitted and selected portions of Chapter 8 included. A semester course
should comfortably accommodate Chapters 1—7, including the review material
of Chapters 2 and 3, if needed. If time remains after this core coverage, se
lected case studies from Chapter 9 or some introduction of digital control from
Chapter 8 may be added. The entire book can also be used for a threequarter sequence of courses
consisting of modeling and dynamic response (Chapters 2 and 3), classical con
trol (Chapters 4—6), and modern control (Chapters 7—9). Two basic 10week courses are offered at Stanford and are taken by seniors
and ﬁrstyear graduate students who have not had a course in control, mostly
in the Departments of Aeronautics and Astronautics, Mechanical Engineering,
and Electrical Engineering. The ﬁrst course reviews Chapters 2 and 3 and cov
ers Chapters 4—6. The more advanced course is intended for graduate students
and reviews Chapters 4—6 and covers Chapters 7—9. This sequence comple—
ments a graduate course in linear systems and is the prerequisite to courses
in digital control, optimal control, ﬂight control, and smart product design.
Several of the subsequent courses include extensive laboratory experiments.
Prerequisites for the course sequence include dynamics or circuit analysis and
Laplace transforms. Prerequisites to this Feedback Control Course This book is for a ﬁrst course at the senior level for all engineering majors.
For the core topics in Chapters 4—7, prerequisite understanding of modeling
and dynamic response is necessary. Many students will come into the course
with sufﬁcient background in those concepts from previous courses in physics,
circuits, and dynamic response. For those needing review, Chapters 2 and 3
should ﬁll in the gaps. An elementary understanding of matrix algebra is necessary to understand
the statespace material. While all students will have much of this in prerequi
site math courses, a review of the basic relations is given in Appendix C and a
brief treatment of particular material needed in control is given at the start of
Chapter 7. The emphasis is on the relations between linear dynamic systems
and linear algebra. Supplements An Instructor’s Manual with complete solutions to homework problems is avail
able to faculty who adopt the fourth edition. The web sites mentioned above
include the .m ﬁles used to generate all of the MATLAB ﬁgures in the book. Preface xvii Acknowledgments Finally, we wish to acknowledge our great debt to all those who have contributed
to the development of feedback control into the exciting ﬁeld it is today and
speciﬁcally to the considerable help and education we have received from our
students and our colleagues. In particular, we have beneﬁted in this effort by
many discussions with the following individuals, who have taught introductory
control at Stanford: A. E. Bryson, Jr., R. H. Cannon, J r., D. B. DeBra, S. Rock,
C. Tomlin, S. Boyd, and P. Enge. Special thanks go to Prof. Dan DeBra for
his careful reading of the manuscript and many useful comments. In addition,
our colleagues M. Spong, L. Pao, D. Meyer, K. Pasino, P. Dorato, and M. Saif
have provided valuable feedback. We also appreciate the help of D. de Roover,
G. van der Linden, J. Ebert, R. Kosut, M. Tao, and A. Rahimi. Furthermore,
L. Kobayashi, HT. Lee, E. Thuriyasena, and M. Matsuoka provided valuable
help in proofreading the chapters as well as in the preparation of the solutions
manual. We would also like to thank the following reviewers: Hy D. Tran at the
University of New Mexico; Paul 1. R0 at North Carolina State University, Lucy
Y. Pao at University of Colorado, Arnold Lee Swindlehurst at Brigham Young
University, and E. Harry Law at Clemson University. GENE F. FRANKLIN J. DAVID POWELL
ABBAS EMAMINAEINI
Stanford, California Table of
Laplace Transforms WQQMAWN—I \O 10 ll 12 13 14 15 16 17
18 19 20 21 1 Us
US2
2!/s3
3!/s4
mY/s
1/(s + a)
l/(s + a)2 m+l 1/(s + (1)3 1/(s + a)“ a
s(s+a)
a
m
ba
m
s (s + a)2 a2 s(s + a)2 (b — a)s
(s + a)(s + b)
a/(s2 + a2)
s/(s2 + (12) s + a b (s + a)2 + b2 a2 + b2 s[(s + (1)2 + b2] _1_,
(m — 1)! m—le—at 1— e‘“ l
—(at— 1+ e_‘")
a e—at bt  e
(1 — at)e_‘”
1— e‘a'a + at) be'bt — ae‘“t sin at
cos at e_‘”cos bt e‘msin bt l — (3.“ (cos bt + gsin bt) Chronological History
of Feedback Control Drebble, Incubator 'l 728
Watt, Flyball governor I 868
Maxwell, Flyball stability analysis 'I 8 77
Routh, Stability 189 Liapunov, Nonlinear stability ll Sperry, Gyroscope and autopilot
I 92 Black, Feedback electronic ampliﬁer: Bush, Differential analyzer ll Nyquist, Nyquist stability criterion Bode, Frequency response methods Wiener, Optimal filter design
ZieglerNichols PID tuning Hurewicz, Sampled data systems; Nichols, Nichols chart 1948 Evans, Root locus Kochenberger, Nonlinear analysis Pontryagin, Maximum principle Bellman, Dynamic programming Draper, Inertial navigation; Kalman, Optimal estimation ...
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