Control Systems Engineering PDF 6th Edition

Apago PDF Enhancer This page intentionally left blank Apago PDF Enhancer E1IFC 10/27/2010 17:49:19 Page 1 Antenna Azimuth Position Control System Layout Potentiometer...

Apago PDF Enhancer This page intentionally left blank Apago PDF Enhancer E1IFC 10/27/2010 17:49:19 Page 1 Antenna Azimuth Position Control System Layout Potentiometer Antenna θ i(t) θ o(t) Desired Azimuth azimuth angle angle input output Differential amplifier and power amplifier Motor Apago PDF Enhancer Potentiometer Schematic +V n-turn potentiometer Fixed Differential Power Motor field –V preamplifier amplifier Ra vi(t) + vp(t) ea(t) K1 vo(t) – K s+a Ja kg-m2 N1 Da N-m s/rad Gear Kb V-s/rad Armature Kt N-m/A N2 JL kg-m2 –V Gear DL N-m-s/rad N3 n-turn potentiometer Gear +V E1IBC 10/27/2010 18:4:22 Page 1 Unmanned Free-Swimming Submersible Vehicle Pitch Control System Commanded Pitch elevator Elevator Elevator Vehicle command Pitch gain deflection actuator deflection dynamics Pitch θc(s) + + δec(s) 2 δe(s) –0.125(s + 0.435) θ (s) –K1 s+2 (s + 1.23)(s2 + 0.226s + 0.0169) – – Pitch rate sensor –K2s Apago PDF Enhancer Heading Control System Commanded Heading Heading Heading rudder Rudder Rudder Vehicle (yaw) command gain deflection actuator deflection dynamics rate Heading ψc(s) + + δrc(s) 2 δr(s) –0.125(s + 0.437) ψ (s) ψ (s) –K1 1 s+2 (s + 1.29)(s + 0.193) s – – Yaw rate sensor –K2s E1FFIRS 11/04/2010 13:38:30 Page 1 Apago PDF Enhancer E1FFIRS 11/04/2010 13:38:31 Page 2 Apago PDF Enhancer E1FFIRS 11/04/2010 13:38:31 Page 3 CONTROL SYSTEMS ENGINEERING Sixth Edition Norman S. Nise California State Polytechnic University, Pomona Apago PDF Enhancer John Wiley & Sons, Inc. E1FFIRS 11/04/2010 13:38:32 Page 4 To my wife, Ellen; sons, Benjamin and Alan; and daughter, Sharon, and their families. Vice President & Publisher Don Fowley Publisher Daniel Sayre Senior Editorial Assistant Katie Singleton Associate Director of Marketing Amy Scholz Marketing Manager Christopher Ruel Production Manager Dorothy Sinclair Production Editor Sandra Dumas Creative Director Harry Nolan Cover Designer James O’Shea Cover Photo Jim Stroup, Virginia Tech Photo Department Manager Hilary Newman Photo Editor Sheena Goldstein Executive Media Editor Thomas Kulesa Associate Media Editor Jennifer Mullin Production Management Services Integra Software Services Inc. This book was typeset in 10/12 TimesRoman at Thomson and printed and bound by R. R. Donnelley (Jefferson City). The cover was printed by R. R. Donnelley (Jefferson City). The paper in this book was manufactured by a mill whose forest management programs include sustained yield-harvesting of its timberlands. Sustained yield harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. This book is printed on acid-free paper.
1 On the cover: CHARLI, a 5-foot tall autonomous humanoid robot built by Dr. Dennis Hong and his students at RoMeLa (Robotics and Mechanisms Laboratory) in the College of Engineering of Virginia Tech. Apago PDF Enhancer Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work, in 2008, we launched a Corporate Citizenship initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. The software programs available with this book have been included for their instructional value. They have been tested with care but are not guaranteed for any particular purpose. The publisher and author do not offer any warranties or restrictions, nor do they accept any liabilities with respect to the programs. Copyright # 2011, 2006, 2003, 1996 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646- 8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. ISBN 13 978-0470-54756-4 ISBN 13 978-0470-91769-5 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 E1FTOC 10/27/2010 16:47:4 Page 5 Contents PREFACE, ix Problems, 98 Cyber Exploration Laboratory, 112 1. INTRODUCTION, 1 Bibliography, 115 1.1 Introduction, 2 1.2 A History of Control Systems, 4 3. MODELING IN THE TIME 1.3 System Configurations, 7 DOMAIN, 117 1.4 Analysis and Design Objectives, 10 3.1 Introduction, 118 Case Study, 12 3.2 Some Observations, 119 1.5 The Design Process, 15 3.3 The General State-Space 1.6 Computer-Aided Design, 20 Representation, 123 1.7 The Control Systems Engineer, 21 3.4 Applying the State-Space Summary, 23 Representation, 124 Review Questions, 23 3.5 Converting a Transfer Function Problems, 24 to State Space, 132 Cyber Exploration Laboratory, 30 3.6 Converting from State Space to a Bibliography, 31 Apago PDF Enhancer Transfer Function, 139 2. MODELING IN THE FREQUENCY 3.7 Linearization, 141 DOMAIN, 33 Case Studies, 144 Summary, 148 2.1 Introduction, 34 Review Questions, 149 2.2 Laplace Transform Review, 35 Problems, 149 2.3 The Transfer Function, 44 Cyber Exploration Laboratory, 157 2.4 Electrical Network Transfer Functions, 47 Bibliography, 159 2.5 Translational Mechanical System Transfer Functions, 61 2.6 Rotational Mechanical System 4. TIME RESPONSE, 161 Transfer Functions, 69 4.1 Introduction, 162 2.7 Transfer Functions for Systems 4.2 Poles, Zeros, and System Response, 162 with Gears, 74 4.3 First-Order Systems, 166 2.8 Electromechanical System 4.4 Second-Order Systems: Introduction, 168 Transfer Functions, 79 4.5 The General Second-Order System, 173 2.9 Electric Circuit Analogs, 84 4.6 Underdamped Second-Order Systems, 177 2.10 Nonlinearities, 88 4.7 System Response with 2.11 Linearization, 89 Additional Poles, 186 Case Studies, 94 4.8 System Response With Zeros, 191 Summary, 97 4.9 Effects of Nonlinearities Upon Review Questions, 97 Time Response, 196 v E1FTOC 10/27/2010 16:47:4 Page 6 vi Contents 4.10 Laplace Transform Solution Cyber Exploration Laboratory, 335 of State Equations, 199 Bibliography, 336 4.11 Time Domain Solution of State Equations, 203 7. STEADY-STATE ERRORS, 339 Case Studies, 207 Summary, 213 7.1 Introduction, 340 Review Questions, 214 7.2 Steady-State Error for Unity Problems, 215 Feedback Systems, 343 Cyber Exploration Laboratory, 228 7.3 Static Error Constants Bibliography, 232 and System Type, 349 7.4 Steady-State Error Specifications, 353 5. REDUCTION OF 7.5 Steady-State Error for Disturbances, 356 MULTIPLE SUBSYSTEMS, 235 7.6 Steady-State Error for Nonunity Feedback Systems, 358 5.1 Introduction, 236 7.7 Sensitivity, 362 5.2 Block Diagrams, 236 7.8 Steady-State Error for Systems in 5.3 Analysis and Design of State Space, 364 Feedback Systems, 245 Case Studies, 368 5.4 Signal-Flow Graphs, 248 Summary, 371 5.5 Mason’s Rule, 251 Review Questions, 372 5.6 Signal-Flow Graphs of Problems, 373 State Equations, 254 Cyber Exploration Laboratory, 384 5.7 Alternative Representations in State Space, 256 Apago PDF Enhancer Bibliography, 386 5.8 Similarity Transformations, 266 Case Studies, 272 8. ROOT LOCUS TECHNIQUES, 387 Summary, 278 8.1 Introduction, 388 Review Questions, 279 8.2 Defining the Root Locus, 392 Problems, 280 8.3 Properties of the Root Locus, 394 Cyber Exploration Laboratory, 297 8.4 Sketching the Root Locus, 397 Bibliography, 299 8.5 Refining the Sketch, 402 8.6 An Example, 411 6. STABILITY, 301 8.7 Transient Response Design 6.1 Introduction, 302 via Gain Adjustment, 415 6.2 Routh-Hurwitz Criterion, 305 8.8 Generalized Root Locus, 419 6.3 Routh-Hurwitz Criterion: 8.9 Root Locus for Positive-Feedback Special Cases, 308 Systems, 421 6.4 Routh-Hurwitz Criterion: 8.10 Pole Sensitivity, 424 Additional Examples, 314 Case Studies, 426 6.5 Stability in State Space, 320 Summary, 431 Case Studies, 323 Review Questions, 432 Summary, 325 Problems, 432 Review Questions, 325 Cyber Exploration Laboratory, 450 Problems, 326 Bibliography, 452 E1FTOC 10/27/2010 16:47:4 Page 7 Contents vii 9. DESIGN VIA ROOT LOCUS, 455 Review Questions, 609 Problems, 610 9.1 Introduction, 456 Cyber Exploration Laboratory, 621 9.2 Improving Steady-State Error via Bibliography, 623 Cascade Compensation, 459 9.3 Improving Transient Response via Cascade Compensation, 469 11. DESIGN VIA FREQUENCY 9.4 Improving Steady-State Error and RESPONSE, 625 Transient Response, 482 11.1 Introduction, 626 9.5 Feedback Compensation, 495 11.2 Transient Response via 9.6 Physical Realization of Compensation, 503 Gain Adjustment, 627 Case Studies, 508 11.3 Lag Compensation, 630 Summary, 513 11.4 Lead Compensation, 635 Review Questions, 514 11.5 Lag-Lead Compensation, 641 Problems, 515 Case Studies, 650 Cyber Exploration Laboratory, 530 Summary, 652 Bibliography, 531 Review Questions, 653 Problems, 653 10. FREQUENCY RESPONSE Cyber Exploration Laboratory, 660 TECHNIQUES, 533 Bibliography, 661 10.1 Introduction, 534 10.2 Asymptotic Approximations: 12. DESIGN VIA STATE SPACE, 663 Bode Plots, 540 Apago PDF Enhancer 10.3 Introduction to the Nyquist Criterion, 559 12.1 Introduction, 664 10.4 Sketching the Nyquist Diagram, 564 12.2 Controller Design, 665 10.5 Stability via the Nyquist Diagram, 569 12.3 Controllability, 672 10.6 Gain Margin and Phase Margin 12.4 Alternative Approaches to via the Nyquist Diagram, 574 Controller Design, 676 10.7 Stability, Gain Margin, and Phase Margin 12.5 Observer Design, 682 via Bode Plots, 576 12.6 Observability, 689 10.8 Relation Between Closed-Loop Transient 12.7 Alternative Approaches to and Closed-Loop Frequency Observer Design, 693 Responses, 580 12.8 Steady-State Error Design Via 10.9 Relation Between Closed- and Open-Loop Integral Control, 700 Frequency Responses, 583 Case Study, 704 10.10 Relation Between Closed-Loop Transient Summary, 709 and Open-Loop Frequency Responses, 589 Review Questions, 710 10.11 Steady-State Error Characteristics Problems, 711 from Frequency Response, 593 Cyber Exploration Laboratory, 719 10.12 Systems with Time Delay, 597 Bibliography, 721 10.13 Obtaining Transfer Functions 13. DIGITAL CONTROL SYSTEMS, 723 Experimentally, 602 Case Study, 606 13.1 Introduction , 724 Summary, 607 13.2 Modeling the Digital Computer, 727 E1FTOC 10/27/2010 16:47:5 Page 8 viii Contents 13.3 The z-Transform, 730 Summary, 885 13.4 Transfer Functions, 735 Bibliography, 886 13.5 Block Diagram Reduction, 739 13.6 Stability, 742 Glossary, 887 13.7 Steady-State Errors, 749 Answers to Selected Problems, 897 13.8 Transient Response on the z-Plane, 753 13.9 Gain Design on the z-Plane, 755 Credits, 903 13.10 Cascade Compensation via Index, 907 the s-Plane, 758 13.11 Implementing the Digital Appendix E MATLAB’s GUI Tools Tutorial Compensator, 762 (Online) Case Studies, 765 Summary, 769 Appendix F MATLAB’s Symbolic Review Questions, 770 Math Toolbox Tutorial (Online) Problems, 771 Appendix G Matrices, Determinants, Cyber Exploration Laboratory, 778 and Systems of Equations Bibliography, 780 (Online) Appendix A List of Symbols, 783 Appendix H Control System Computational Aids (Online) Appendix B MATLAB Tutorial, 787 Appendix I Derivation of a Schematic for a B.1 Introduction, 787 DC Motor (Online) B.2 MATLAB Examples, 788 Apago PDF Enhancer Appendix J Derivation of the Time Domain B.3 Command Summary, 833 Solution of State Equations Bibliography, 835 (Online) Appendix C MATLAB’s Simulink Appendix K Solution of State Equations for Tutorial, 836 t0 6¼ 0 (Online) C.1 Introduction, 836 Appendix L Derivation of Similarity C.2 Using Simulink, 836 Transformations (Online) C.3 Examples, 841 Appendix M Root Locus Rules: Derivations Summary, 855 (Online) Bibliography, 856 Control Systems Engineering Toolbox Appendix D LabVIEW Tutorial, 857 (Online) D.1 Introduction, 857 Cyber Exploration Laboratory Experiments D.2 Control Systems Analysis, Design, Covers Sheets (Online) and Simulation, 858 D.3 Using LabVIEW, 859 Lecture Graphics (Online) D.4 Analysis and Design Examples, 862 Solutions to Skill-Assessment Exercises D.5 Simulation Examples, 876 (Online) Online location is www.wiley.com/college/nise E1FPREF 10/27/2010 21:39:14 Page 9 Preface This book introduces students to the theory and practice of control systems engineer- ing. The text emphasizes the practical application of the subject to the analysis and design of feedback systems. The study of control systems engineering is essential for students pursuing degrees in electrical, mechanical, aerospace, biomedical, or chemical engineering. Control systems are found in a broad range of applications within these disciplines, from aircraft and spacecraft to robots and process control systems. Control Systems Engineering is suitable for upper-division college and univer- sity engineering students and for those who wish to master the subject matter through self-study. The student using this text should have completed typical lower- division courses in physics and mathematics through differential equations. Other required background material, including Laplace transforms and linear algebra, is Apago PDF Enhancer incorporated in the text, either within chapter discussions or separately in the appendixes or on the book’s Companion Web site. This review material can be omitted without loss of continuity if the student does not require it. Key Features The key features of this sixth edition are:  Standardized chapter organization  Qualitative and quantitative explanations  Examples, Skill-Assessment Exercises, and Case Studies throughout the text  WileyPLUS content management system for students and professors  Cyber Exploration Laboratory and Virtual Experiments  Abundant illustrations  Numerous end-of-chapter problems  Emphasis on design  Flexible coverage  Emphasis on computer-aided analysis and design including MATLAB11 and LabVIEW12 1 MATLAB is a registered trademark of The MathWorks, Inc. 2 LabVIEW is a registered trademark of National Instruments Corporation. ix E1FPREF 10/27/2010 21:39:14 Page 10 x Preface  Icons identifying major topics Let us look at each feature in more detail. Standardized Chapter Organization Each chapter begins with a list of chapter learning outcomes, followed by a list of case study learning outcomes that relate to specific student performance in solving a practical case study problem, such as an antenna azimuth position control system. Topics are then divided into clearly numbered and labeled sections containing explanations, examples, and, where appropriate, skill-assessment exercises with answers. These numbered sections are followed by one or more case studies, as will be outlined in a few paragraphs. Each chapter ends with a brief summary, several review questions requiring short answers, a set of homework problems, and experiments. Qualitative and Quantitative Explanations Explanations are clear and complete and, where appropriate, include a brief review of required background material. Topics build upon and support one another in a logical fashion. Groundwork for new concepts and terminology is carefully laid to avoid overwhelming the student and to facilitate self-study. Although quantitative solutions are obviously important, a qualitative or Apago PDF Enhancer intuitive understanding of problems and methods of solution is vital to producing the insight required to develop sound designs. Therefore, whenever possible, new concepts are discussed from a qualitative perspective before quantitative analysis and design are addressed. For example, in Chapter 8 the student can simply look at the root locus and describe qualitatively the changes in transient response that will occur as a system parameter, such as gain, is varied. This ability is developed with the help of a few simple equations from Chapter 4. Examples, Skill-Assessment Exercises, and Case Studies Explanations are clearly illustrated by means of numerous numbered and labeled Examples throughout the text. Where appropriate, sections conclude with Skill- Assessment Exercises. These are computation drills, most with answers that test comprehension and provide immediate feedback. Complete solutions can be found at www.wiley.com/college/nise. Broader examples in the form of Case Studies can be found after the last numbered section of every chapter, with the exception of Chapter 1. These case studies arc practical application problems that demonstrate the concepts introduced in the chapter. Each case study concludes with a ‘‘Challenge’’ problem that students may work in order to test their understanding of the material. One of the case studies, an antenna azimuth position control system, is carried throughout the book. The purpose is to illustrate the application of new material in each chapter to the same physical system, thus highlighting the continuity of the design process. Another, more challenging case study, involving E1FPREF 10/27/2010 21:39:14 Page 11 Preface xi an Unmannered Free-Swimming Submersible Vehicle, is developed over the course of five chapters. WileyPLUS Content Management System for Students and Professors WileyPLUS is an online suite of resources, including the full text, for students and instructors. For the sixth edition of Control Systems Engineering, this suite offers professors who adopt the book with WileyPLUS the ability to create homework assignments based on algorithmic problems or multi-part questions, which guide the student through a problem. Instructors also have the capability to integrate assets, such as the simulations, into their lecture presentations. Students will find a Read, Study, and Practice zone to help them work through problems based on the ones offered in the text. Control Solutions (prepared by JustAsk) are included in the WileyPLUS platform. The student will find simulations and Control Solutions in the Read, Study, and Practice zone. The Control Solutions are highlighted in the text with a WileyPLUS icon. A new addition to the WileyPLUS platform for this edition are National Instruments and Quanser Virtual Laboratories. You will find references to them in sidebar entries throughout the textbook. Visit www.wiley.com or contact your local Wiley representative for information. Apago PDF Enhancer Cyber Exploration Laboratory and Virtual Experiments Computer experiments using MATLAB, Simulink13 and the Control System Toolbox are found at the end of the Problems sections under the sub-heading Cyber Exploration Laboratory. New to this edition is LabVIEW, which is also used for experiments within the Cyber Exploration Laboratory section of the chapters. The experiments allow the reader to verify the concepts covered in the chapter via simulation. The reader also can change parameters and perform ‘‘what if’’ explora- tion to gain insight into the effect of parameter and configuration changes. The experiments are written with stated Objectives, Minimum Required Software Pack- ages, as well as Prelab, Lab, and Postlab tasks and questions. Thus, the experiments may be used for a laboratory course that accompanies the class. Cover sheets for these experiments are available at www.wiley.com.college/nise. In addition, and new to this sixth edition, are Virtual Experiments. These experiments are more tightly focused than the Cyber Exploration Laboratory experiments and use LabVIEW and Quanser virtual hardware to illustrate immediate discussion and examples. The experiments are referenced in sidebars throughout some chapters. 3 Simulink is a registered trademark of The MathWorks, Inc. E1FPREF 10/27/2010 21:39:14 Page 12 xii Preface Abundant Illustrations The ability to visualize concepts and processes is critical to the student’s under- standing. For this reason, approximately 800 photos, diagrams, graphs, and tables appear throughout the book to illustrate the topics under discussion. Numerous End-of-Chapter Problems Each chapter ends with a variety of homework problems that allow students to test their understanding of the material presented in the chapter. Problems vary in degree of difficulty and complexity, and most chapters include several practical, real- life problems to help maintain students’ motivation. Also, the homework problems contain progressive analysis and design problems that use the same practical systems to demonstrate the concepts of each chapter. Emphasis on Design This textbook places a heavy emphasis on design. Chapters 8, 9, 11, 12 and 13 focus primarily on design. But. even in chapters that emphasize analysis, simple design examples are included wherever possible. Throughout the book, design examples involving physical systems are identi- fied by the icon shown in the margin. End-of-chapter problems that involve the Apago PDF Enhancer design of physical systems are included under the separate heading Design Problems, and also in chapters covering design, under the heading Progressive Analysis and Design Problems. In these examples and problems, a desired response is specified, and the student must evaluate certain system parameters, such as gain, or specify a system configuration along with parameter values. In addition, the text includes numerous design examples and problems (not identified by an icon) that involve purely mathematical systems. Because visualization is so vital to understanding design, this text carefully relates indirect design specifications to more familiar ones. For example, the less familiar and indirect phase margin is carefully related to the more direct and familiar percent overshoot before being used as a design specification. For each general type of design problem introduced in the text, a methodology for solving the problem is presented—in many cases in the form of a step-by-step procedure, beginning with a statement of design objectives. Example problems serve to demonstrate the methodology by following the procedure, making simplifying assumptions, and presenting the results of the design in tables or plots that compare the performance of the original system to that of the improved system. This comparison also serves as a check on the simplifying assumptions. Transient response design topics are covered comprehensively in the text. They include:  Design via gain adjustment using the root locus  Design of compensation and controllers via the root locus  Design via gain adjustment using sinusoidal frequency response methods  Design of compensation via sinusoidal frequency response methods E1FPREF 10/27/2010 21:39:14 Page 13 Preface xiii  Design of controllers in state space using pole-placement techniques  Design of observers in state-space using pole-placement techniques  Design of digital control systems via gain adjustment on the root locus  Design of digital control system compensation via s-plane design and the Tustin transformation Steady-state error design is covered comprehensively in this textbook and includes:  Gain adjustment  Design of compensation via the root locus  Design of compensation via sinusoidal frequency response methods  Design of integral control in state space Finally, the design of gain to yield stability is covered from the following perspectives:  Routh-Hurwitz criterion  Root locus  Nyquist criterion  Bode plots Flexible Coverage Apago The material in this book can be adapted PDF orEnhancer for a one-quarter a one-semester course. The organization is flexible, allowing the instructor to select the material that best suits the requirements and time constraints of the class. Throughout the book, state-space methods are presented along with the classical approach. Chapters and sections (as well as examples, exercises, review questions, and problems) that cover state space are marked by the icon shown in the margin and can be omitted without any loss of continuity. Those wishing to add a basic introduction to state-space modeling can include Chapter 3 in the syllabus. In a one-semester course, the discussions of slate-space analysis in Chapters 4, 5, 6 and 7, as well as state-space design in Chapter 12, can be covered along with the classical approach. Another option is to teach state space separately by gathering the appropriate chapters and sections marked with the State Space icon into a single unit that follows the classical approach. In a one-quarter course, Chapter 13, ‘‘Digital Control Systems,’’ could be eliminated. Emphasis on Computer-Aided Analysis and Design Control systems problems, particularly analysis and design problems using the root locus, can be tedious, since their solution involves trial and error. To solve these problems, students should be given access to computers or programmable calcula- tors configured with appropriate software. In this sixth edition, MATLAB continues to be integrated into the text as an optional feature. In addition, and new to this E1FPREF 10/27/2010 21:39:14 Page 14 xiv Preface edition, we have included LabVIEW as an option to computer-aided analysis and design. Many problems in this text can be solved with either a computer or a hand-held programmable calculator. For example, students can use the programmable calcu- lator to (1) determine whether a point on the s-plane is also on the root locus, (2) find magnitude and phase frequency response data for Nyquist and Bode diagrams, and (3) convert between the following representations of a second-order system:  Pole location in polar coordinates  Pole location in Cartesian coordinates  Characteristic polynomial  Natural frequency and damping ratio  Settling time and percent overshoot  Peak time and percent overshoot  Settling time and peak time Handheld calculators have the advantage of easy accessibility for homework and exams. Please consult Appendix H, located at www.wiley.com/college/nise, for a discussion of computational aids that can be adapted to handheld calculators. Personal computers are better suited for more computation-intensive appli- cations, such as plotting time responses, root loci, and frequency response curves, as well as finding state-transition matrices. These computers also give the student a real-world environment in which to analyze and design control systems. Those not using MATLAB or LabVIEW can write their own programs or use other programs, such as Program CC. Please consult Appendix H at www.wiley.com/college/nise for a Apago PDF Enhancer discussion of computational aids that can be adapted for use on computers that do not have MATLAB or LabVIEW installed. Without access to computers or programmable calculators, students cannot obtain meaningful analysis and design results and the learning experience will be limited. Icons Identifying Major Topics Several icons identify coverage and optional material. The icons are summarized as follows: Control Solutions for the student are identified with a WileyPLUS icon. These problems, developed by JustAsk, are worked in detail and offer explanations of every facet of the solution. The MATLAB icon identifies MATLAB discussions, examples, exercises, and problems. MATLAB coverage is provided as an enhancement and is not required to use the text. The Simulink icon identifies Simulink discussions, examples, exercises, and problems. Simulink coverage is provided as an enhancement and is not required to use the text. The GUI Tool icon identifies MATLAB GUI Tools discussions, examples, exercises, and problems. The discussion of the tools, which includes the LTI Viewer, the Simulink LTIViewer, and the SISO Design Tool, is provided as an enhancement and is not required to use the text. E1FPREF 10/27/2010 21:39:14 Page 15 Preface xv The Symbolic Math icon identifies Symbolic Math Toolbox discussions, exam- ples, exercises, and problems. Symbolic Math Toolbox coverage is provided as an enhancement and is not required to use the text. The LabVIEW icon identifies LabVIEW discussions, examples, exercises, and problems. LabVIEW is provided as an enhancement and is not required to use the text. The State Space icon highlights state-space discussions, examples, exercises, and problems. State-space material is optional and can be omitted without loss of continuity. The Design icon clearly identifies design problems involving physical systems. New to This Edition The following list describes the key changes in this sixth edition End-of-chapter problems More than 20% of the end-of-chapter problems are either new or revised. Also, an additional Progressive Analysis and Design Problem has been added at the end of the chapter problems. The new progressive problem analyzes and designs a hybrid electric vehicle. MATLAB The use of MATLAB for computer-aided analysis and design con- tinues to be integrated into discussions and problems as an optional feature in the sixth edition. The MATLAB tutorial has been updated to MATLAB Version 7.9 (R 2009b), the Control System Toolbox Version 8.4, and the Symbolic Math Toolbox Version 5.3 In addition, MATLAB code continues to be incorporated in the chapters in the form of sidebar boxes entitled TryIt. Virtual Experiments Virtual experiments, developed by National Instru- ments and Quanser, are included via sidebar references to experiments on Wiley- PLUS. The experiments are performed with 3-D simulations of Quanser hardware Apago PDF Enhancer using developed LabVIEW VIs. Virtual Experiments are tightly focused and linked to a discussion or example. Cyber Exploration Laboratory Experiments using LabVIEW have been added. Cyber Exploration Laboratory experiments are general in focus and are envisioned to be used in an associated lab class. MATLAB’s Simulink The use of Simulink to show the effects of nonlinear- ities upon the time response of open-loop and closed-loop systems appears again in this sixth edition. We also continue to use Simulink to demonstrate how to simulate digital systems. Finally, the Simulink tutorial has been updated to Simulink 7.4 Chapter 11 Lag-lead compensator design using Nichols charts has been added to Section 11.5. LabVIEW New to this edition is LabVIEW. A tutorial for this tool is included in Appendix D. LabVIEW is used in Cyber Exploration Laboratory experiments and other problems throughout the textbook. Book Companion Site (BCS) at www.wiley.com/college/nise The BCS for the sixth edition includes various student and instructor resources. This free resource can be accessed by going to www.wiley.com/college/nise and clicking on Student Companion Site. Professors also access their password-protected re- sources on the Instructor Companion Site available through this url. Instructors should contact their Wiley sales representative for access. E1FPREF 10/27/2010 21:39:14 Page 16 xvi Preface For the Student:  All M-files used in the MATLAB, Simulink, GUI Tools, and Symbolic Math Toolbox tutorials, as well as the TryIt exercises  Copies of the Cyber Exploration Laboratory experiments for use as experi- ment cover sheets  Solutions to the Skill-Assessment Exercises in the text  LabVIEW Virtual Experiments and LabVIEW VIs used in Appendix D For the Instructor;  PowerPoint14 files containing the figures from the textbook  Solutions to end-of-chapter problem sets  Simulations, developed by JustAsk, for inclusion in lecture presentations Book Organization by Chapter Many times it is helpful lo understand an author’s reasoning behind the organization of the course material. The following paragraphs hopefully shed light on this topic. The primary goal of Chapter 1 is to motivate students. In this chapter, students learn about the many applications of control systems in everyday life and about the advantages of study and a career in this field. Control systems engineering design objectives, such as transient response, steady-state error, and stability, are intro- duced, as is the path to obtaining these objectives. New and unfamiliar terms also are included in the Glossary. Apago PDF Enhancer Many students have trouble with an early step in the analysis and design sequence: transforming a physical system into a schematic. This step requires many simplifying assumptions based on experience the typical college student does not yet possess. Identifying some of these assumptions in Chapter 1 helps to fill the experience gap. Chapters 2, 3, and 5 address the representation of physical systems. Chapters 2 and 3 cover modeling of open-loop systems, using frequency response techniques and state- space techniques, respectively. Chapter 5 discusses the representation and reduction of systems formed of interconnected open-loop subsystems. Only a representative sample of physical systems can be covered in a textbook of this length. Electrical, mechanical (both translational and rotational), and electromechanical systems are used as examples of physical systems that are modeled, analyzed, and designed. Linearization of a nonlinear system—one technique used by the engineer to simplify a system in order to represent it mathematically—is also introduced. Chapter 4 provides an introduction to system analysis, that is, finding and describing the output response of a system. It may seem more logical to reverse the order of Chapters 4 and 5, to present the material in Chapter 4 along with other chapters covering analysis. However, many years of teaching control systems have taught me that the sooner students see an application of the study of system representation, the higher their motivation levels remain. Chapters 6, 7, 8, and 9 return to control systems analysis and design with the study of stability (Chapter 6), steady-state errors (Chapter 7), and transient response of higher-order systems using root locus techniques (Chapter 8). Chapter 9 covers design of compensators and controllers using the root locus. 4 PowerPoint is a registered trademark of Microsoft Corporation. E1FPREF 10/27/2010 21:39:14 Page 17 Preface xvii Chapters 10 and 11 focus on sinusoidal frequency analysis and design. Chapter 10, like Chapter 8, covers basic concepts for stability, transient response, and steady- state-error analysis. However, Nyquist and Bode methods are used in place of root locus. Chapter 11, like Chapter 9, covers the design of compensators, but from the point of view of sinusoidal frequency techniques rather than root locus. An introduction to state-space design and digital control systems analysis and design completes the text in Chapters 12 and 13, respectively. Although these chapters can be used as an introduction for students who will be continuing their study of control systems engineering, they are useful by themselves and as a supplement to the discussion of analysis and design in the previous chapters. The subject matter cannot be given a comprehensive treatment in two chapters, but the emphasis is clearly outlined and logically linked to the rest of the book. Acknowledgments The author would like to acknowledge the contributions of faculty and students, both at California State Polytechnic University, Pomona, and across the country, whose suggestions through all editions have made a positive impact on the new edition. I am deeply indebted to my colleagues, Elhami T. Ibrahim, Salomon Oldak, and Norali Pernalete at California State Polytechnic University, Pomona for author- ing the creative new problems you will find at the end of every chapter. Dr. Pernalete created the LabVIEW experiments and problems you will find in this new edition. The new progressive problem, hybrid vehicle, that is at the end of every chapter is the creation of Dr Ibrahim. In addition to his busy schedule as Electrical and Computer Apago PDF Enhancer Engineering Department Chairman and author of many of the new problems, Professor Oldak also error checked new additions to the book and prevented glitches from ever reaching you, the reader. I would like to express my appreciation to contributors to this sixth edition who participated in reviews, accuracy checking, surveys, or focus groups. They are: Jorge Aravena, Louisiana State University; Kurt Behpour, Cal Poly San Luis Obispo; Bill Diong, Texas Christian University; Sam Guccione, Eastern Illinois University; Pushkin Kachroo, Virginia Tech; Dmitriy Kalantarov, Cal State San Diego; Kamran Iqbal, University of Arkansas, Little Rock; Pushkin Kachroo, Virginia Tech; Kevin Lynch, Northwestern University; Tesfay Meressi, University of Massachusetts, Dartmouth; Luai Najim, University of Alabama at Birmingham; Dalton Nelson, University of Alabama at Birmingham; Marcio S. de Queiroz, Louisiana State University; John Ridgely, Cal Poly San Luis Obispo; John Schmitt, Oregon State University; Lili Tabrizi, California State University, Los Angeles; Raman Unnik- rishnan, Cal State Fullerton; Stephen Williams, Milwaukee School of Engineering; Jiann-Shiou Yang, University of Minnesota, Duluth; and Ryan Zurakowski, Uni- versity of Delaware. The author would like to thank John Wiley & Sons, Inc. and its staff for once again providing professional support for this project through all phases of its development. Specifically, the following are due recognition for their contributions: Don Fowler, Vice President and Publisher, who gave full corporate support to the project; Daniel Sayre, Publisher, with whom I worked closely and who provided guidance and leadership throughout the development of the sixth edition; and Katie Singleton, Senior Editorial Assistant, who was always there to answer my questions and respond to my concerns in a professional manner. There are many others who E1FPREF 10/27/2010 21:39:14 Page 18 xviii Preface worked behind the scenes, but who should be thanked never the less. Rather than repeating their names and titles here, I refer the reader to the copyright page of this book where they are listed and credited. I am very thankful for their contributions. Next, I want to acknowledge Integra Software Services, Inc. and its staff for turning the sixth edition manuscript into the finished product you are holding in your hands. Specifically, kudos go out to Heather Johnson, Managing Editor, who, once again, was always there to address my concerns in a timely and professional manner. My sincere appreciation is extended to Erik Luther of National Instruments Corporation and Paul Gilbert and Michel Levis of Quanser for conceiving, coor- dinating, and developing the Virtual Experiments that I am sure will enhance your understanding of control systems. Finally, last but certainly not least, I want to express my appreciation to my wife, Ellen, for her support in ways too numerous to mention during the writing of the past six editions. Specifically though, thanks to her proofing final pages for this sixth edition, you the reader hopefully will find comprehension rather than apprehension in the pages that follow. Norman S. Nise Apago PDF Enhancer E1C01 10/21/2010 10:32:23 Page 1 1 Introduction Apago PDF Enhancer Chapter Learning Outcomes After completing this chapter, the student will be able to:
Define a control system and describe some applications (Section 1.1)
Describe historical developments leading to modern day control theory (Section 1.2)
Describe the basic features and configurations of control systems (Section 1.3)
Describe control systems analysis and design objectives (Section 1.4)
Describe a control system’s design process (Sections 1.5–1.6)
Describe the benefit from studying control systems (Section 1.7) Case Study Learning Outcomes
You will be introduced to a running case study—an antenna azimuth position control system—that will serve to illustrate the principles in each subsequent chapter. In this chapter, the system is used to demonstrate qualitatively how a control system works as well as to define performance criteria that are the basis for control systems analysis and design. 1 E1C01 10/21/2010 10:32:23 Page 2 2 Chapter 1 Introduction 1.1 Introduction Control systems are an integral part of modern society. Numerous applications are all around us: The rockets fire, and the space shuttle lifts off to earth orbit; in splashing cooling water, a metallic part is automatically machined; a self-guided vehicle delivering material to workstations in an aerospace assembly plant glides along the floor seeking its destination. These are just a few examples of the automatically controlled systems that we can create. We are not the only creators of automatically controlled systems; these systems also exist in nature. Within our own bodies are numerous control systems, such as the pancreas, which regulates our blood sugar. In time of ‘‘fight or flight,’’ our adrenaline increases along with our heart rate, causing more oxygen to be delivered to our cells. Our eyes follow a moving object to keep it in view; our hands grasp the object and place it precisely at a predetermined location. Even the nonphysical world appears to be automatically regulated. Models have been suggested showing automatic control of student performance. The input to the model is the student’s available study time, and the output is the grade. The model can be used to predict the time required for the grade to rise if a sudden increase in study time is available. Using this model, you can determine whether increased study is worth the effort during the last week of the term. Control System Definition A control system consists of subsystems and processes (or plants) assembled for the purpose of obtaining a desired output with desired performance, given a specified input. Figure 1.1 shows a control system in its simplest form, where the Input; stimulus Output; response Apago PDF Enhancer input represents a desired output. Control Desired response system Actual response For example, consider an elevator. When the fourth-floor button is pressed on the first floor, the elevator rises to the fourth floor with a FIGURE 1.1 Simplified description of a speed and floor-leveling accuracy designed for passenger comfort. The control system push of the fourth-floor button is an input that represents our desired output, shown as a step function in Figure 1.2. The performance of the elevator can be seen from the elevator response curve in the figure. Two major measures of performance are apparent: (1) the transient response and (2) the steady-state error. In our example, passenger comfort and passenger patience are dependent upon the transient response. If this response is too fast, passenger comfort is sacrificed; if too slow, passenger patience is sacrificed. The steady-state error is another important performance specification since passenger safety and convenience would be sacrificed if the elevator did not properly level. Input command 4 Elevator location (floor) Transient response Steady-state Steady-state response error Elevator response 1 FIGURE 1.2 Elevator response Time E1C01 10/21/2010 10:32:23 Page 3 1.1 Introduction 3 FIGURE 1.3 a. Early elevators were controlled by hand ropes or an elevator operator. Here a rope is cut to demonstrate the safety brake, an innovation in early elevators (# Bettman/ Corbis); b. One of two modern Duo-lift elevators makes its way up the Grande Arche in Paris. Two elevators are driven by one motor, with each car acting as a counterbalance to the other. Today, elevators are fully auto- matic, using control systems to regulate position and velocity. Advantages of Control Systems With control systems we can move large equipment with precision that would otherwise be impossible. We can point huge antennas toward the farthest reaches of the universe to pick up faint radio signals; Apago PDF Enhancer controlling these antennas by hand would be impossible. Because of control systems, elevators carry us quickly to our destination, auto- matically stopping at the right floor (Figure 1.3). We alone could not provide the power required for the load and the speed; motors provide the power, and control systems regulate the position and speed. We build control systems for four primary reasons: 1. Power amplification 2. Remote control 3. Convenience of input form 4. Compensation for disturbances For example, a radar antenna, positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. A control system can produce the needed power amplifica- tion, or power gain. Robots designed by control system principles can compensate for human disabilities. Control systems are also useful in remote or dangerous locations. For example, a remote-controlled robot arm can be used to pick up material in a radioactive environment. Figure 1.4 shows a robot arm designed to work in contaminated environments. Control systems can also be used to provide convenience by FIGURE 1.4 Rover was built to work in changing the form of the input. For example, in a temperature control contaminated areas at Three Mile Island in system, the input is a position on a thermostat. The output is heat. Middleton, Pennsylvania, where a nuclear Thus, a convenient position input yields a desired thermal output. accident occurred in 1979. The remote-controlled Another advantage of a control system is the ability to compensate robot’s long arm can be seen at the front of the for disturbances. Typically, we control such variables as temperature in vehicle. E1C01 10/21/2010 10:32:29 Page 4 4 Chapter 1 Introduction thermal systems, position and velocity in mechanical systems, and voltage, current, or frequency in electrical systems. The system must be able to yield the correct output even with a disturbance. For example, consider an antenna system that points in a commanded direction. If wind forces the antenna from its commanded position, or if noise enters internally, the system must be able to detect the disturbance and correct the antenna’s position. Obviously, the system’s input will not change to make the correction. Conse- quently, the system itself must measure the amount that the disturbance has repositioned the antenna and then return the antenna to the position commanded by the input. 1.2 A History of Control Systems Feedback control systems are older than humanity. Numerous biological control systems were built into the earliest inhabitants of our planet. Let us now look at a brief history of human-designed control systems.1 Liquid-Level Control The Greeks began engineering feedback systems around 300 B.C. A water clock invented by Ktesibios operated by having water trickle into a measuring container at a constant rate. The level of water in the measuring container could be used to tell time. For water to trickle at a constant rate, the supply tank had to be kept at a constant level. This was accomplished using a float valve similar to the water-level control in today’s flush toilets. Soon after Ktesibios, the idea of liquid-level control was applied to an oil lamp by Philon of Byzantium. The lamp consisted of two oil containers configured vertically. The lower pan was open at the top and was the fuel supply for the flame. The closed upper bowl was the fuel reservoir for the pan below. The containers were Apago PDF Enhancer interconnected by two capillary tubes and another tube, called a vertical riser, which was inserted into the oil in the lower pan just below the surface. As the oil burned, the base of the vertical riser was exposed to air, which forced oil in the reservoir above to flow through the capillary tubes and into the pan. The transfer of fuel from the upper reservoir to the pan stopped when the previous oil level in the pan was reestablished, thus blocking the air from entering the vertical riser. Hence, the system kept the liquid level in the lower container constant. Steam Pressure and Temperature Controls Regulation of steam pressure began around 1681 with Denis Papin’s invention of the safety valve. The concept was further elaborated on by weighting the valve top. If the upward pressure from the boiler exceeded the weight, steam was released, and the pressure decreased. Ifitdid not exceed the weight, the valve didnotopen,and thepressureinsidethe boiler increased. Thus, the weight on the valve top set the internal pressure of the boiler. Also in the seventeenth century, Cornelis Drebbel in Holland invented a purely mechanical temperature control system for hatching eggs. The device used a vial of alcohol and mercury with a floater inserted in it. The floater was connected to a damper that controlled a flame. A portion of the vial was inserted into the incubator to sense the heat generated by the fire. As the heat increased, the alcohol and mercury expanded, raising the floater, closing the damper, and reducing the flame. Lower temperature caused the float to descend, opening the damper and increasing the flame. Speed Control In 1745, speed control was applied to a windmill by Edmund Lee. Increasing winds pitched the blades farther back, so that less area was available. As the wind 1 See Bennett (1979) and Mayr (1970) for definitive works on the history of control systems. E1C01 10/21/2010 10:32:29 Page 5 1.2 A History of Control Systems 5 decreased, more blade area was available. William Cubitt improved on the idea in 1809 by dividing the windmill sail into movable louvers. Also in the eighteenth century, James Watt invented the flyball speed governor to control the speed of steam engines. In this device, two spinning flyballs rise as rotational speed increases. A steam valve connected to the flyball mechanism closes with the ascending flyballs and opens with the descending flyballs, thus regulating the speed. Stability, Stabilization, and Steering Control systems theory as we know it today began to crystallize in the latter half of the nineteenth century. In 1868, James Clerk Maxwell published the stability criterion for a third-order system based on the coefficients of the differential equation. In 1874, Edward John Routh, using a suggestion from William Kingdon Clifford that was ignored earlier by Maxwell, was able to extend the stability criterion to fifth-order systems. In 1877, the topic for the Adams Prize was ‘‘The Criterion of Dynamical Stability.’’ In response, Routh submitted a paper entitled A Treatise on the Stability of a Given State of Motion and won the prize. This paper contains what is now known as the Routh-Hurwitz criterion for stability, which we will study in Chapter 6. Alexandr Michailovich Lyapunov also contributed to the development and formulation of today’s theories and practice of control system stability. A student of P. L. Chebyshev at the University of St. Petersburg in Russia, Lyapunov extended the work of Routh to nonlinear systems in his 1892 doctoral thesis, entitled The General Problem of Stability of Motion. During the second half of the 1800s, the development of control systems focused on the steering and stabilizing of ships. In 1874, Henry Bessemer, using a gyro to sense a ship’s motion and applying power generated by the ship’s hydraulic system, moved the ship’s saloon to keep it stable (whether this made a difference to Apago PDF Enhancer the patrons is doubtful). Other efforts were made to stabilize platforms for guns as well as to stabilize entire ships, using pendulums to sense the motion. Twentieth-Century Developments It was not until the early 1900s that automatic steering of ships was achieved. In 1922, the Sperry Gyroscope Company installed an automatic steering system that used the elements of compensation and adaptive control to improve performance. However, much of the general theory used today to improve the performance of automatic control systems is attributed to Nicholas Minorsky, a Russian born in 1885. It was his theoretical development applied to the automatic steering of ships that led to what we call today proportional-plus-integral-plus-derivative (PID), or three-mode, con- trollers, which we will study in Chapters 9 and 11. In the late 1920s and early 1930s, H. W. Bode and H. Nyquist at Bell Telephone Laboratories developed the analysis of feedback amplifiers. These contributions evolved into sinusoidal frequency analysis and design techniques currently used for feedback control system, and are presented in Chapters 10 and 11. In 1948, Walter R. Evans, working in the aircraft industry, developed a graphical technique to plot the roots of a characteristic equation of a feedback system whose parameters changed over a particular range of values. This technique, now known as the root locus, takes its place with the work of Bode and Nyquist in forming the foundation of linear control systems analysis and design theory. We will study root locus in Chapters 8, 9, and 13. Contemporary Applications Today, control systems find widespread application in the guidance, navigation, and control of missiles and spacecraft, as well as planes and ships at sea. For example, E1C01 10/21/2010 10:32:29 Page 6 6 Chapter 1 Introduction modern ships use a combination of electrical, mechanical, and hydraulic components to develop rudder commands in response to desired heading commands. The rudder commands, in turn, result in a rudder angle that steers the ship. We find control systems throughout the process control industry, regulating liquid levels in tanks, chemical concentrations in vats, as well as the thickness of fabricated material. For example, consider a thickness control system for a steel plate finishing mill. Steel enters the finishing mill and passes through rollers. In the finishing mill, X-rays measure the actual thickness and compare it to the desired thickness. Any difference is adjusted by a screw-down position control that changes the roll gap at the rollers through which the steel passes. This change in roll gap regulates the thickness. Modern developments have seen widespread use of the digital computer as part of control systems. For example, computers in control systems are for industrial robots, spacecraft, and the process control industry. It is hard to visualize a modern control system that does not use a digital computer. The space shuttle contains numerous control systems operated by an onboard computer on a time-shared basis. Without control systems, it would be impossible to guide the shuttle to and from earth’s orbit or to adjust the orbit itself and support life on board. Navigation functions programmed into the shuttle’s computers use data from the shuttle’s hardware to estimate vehicle position and velocity. This informa- tion is fed to the guidance equations that calculate commands for the shuttle’s flight control systems, which steer the spacecraft. In space, the flight control system gimbals (rotates) the orbital maneuvering system (OMS) engines into a position that provides thrust in the commanded direction to steer the spacecraft. Within the earth’s atmosphere, the shuttle is steered by commands sent from the flight control system to the aerosurfaces, such as the elevons. Apago PDF Enhancer Within this large control system represented by navigation, guidance, and control are numerous subsystems to control the vehicle’s functions. For example, the elevons require a control system to ensure that their position is indeed that which was commanded, since disturbances such as wind could rotate the elevons away from the commanded position. Similarly, in space, the gimbaling of the orbital maneu- vering engines requires a similar control system to ensure that the rotating engine can accomplish its function with speed and accuracy. Control systems are also used to control and stabilize the vehicle during its descent from orbit. Numerous small jets that compose the reaction control system (RCS) are used initially in the exoatmo- sphere, where the aerosurfaces are ineffective. Control is passed to the aerosurfaces as the orbiter descends into the atmosphere. Inside the shuttle, numerous control systems are required for power and life support. For example, the orbiter has three fuel-cell power plants that convert hydrogen and oxygen (reactants) into electricity and water for use by the crew. The fuel cells involve the use of control systems to regulate temperature and pressure. The reactant tanks are kept at constant pressure as the quantity of reactant diminishes. Sensors in the tanks send signals to the control systems to turn heaters on or off to keep the tank pressure constant (Rockwell Interna- tional, 1984). Control systems are not limited to science and industry. For example, a home heating system is a simple control system consisting of a thermostat containing a bimetallic material that expands or contracts with changing temperature. This expansion or contraction moves a vial of mercury that acts as a switch, turning the heater on or off. The amount of expansion or contraction required to move the mercury switch is determined by the temperature setting. E1C01 10/21/2010 10:32:29 Page 7 1.3 System Configurations 7 Reflective Protective layer layer (aluminum) Transparent plastic substrate Objective lens (acrylic resin) (a) Disc Photodiode Fixed mirror Objective lens Toric lens Tangential mirror FIGURE 1.5 Optical playback system: a. objective lens read- ing pits on an optical disc; b. optical path for playback, Coupling lens Prism 1/4-wavelength plate showing tracking mirror Cylindrical Grating rotated by a control system to lens Tracking mirror keep the laser beam positioned Laser diode on the pits (Pioneer Electronics Apago (b)PDF Enhancer (USA), Inc.) Home entertainment systems also have