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# Modern Control Engineering OGATA 5th Ed

(Parte **1** de 6)

Modern Control Engineering

Fifth Edition

Katsuhiko Ogata

Prentice Hall

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Credits and acknowledgments borrowed from other sources and reproduced,with permission,in this textbook appear on appropriate page within text.

MATLAB is a registered trademark of The Mathworks,Inc.,3 Apple Hill Drive,Natick MA 01760-2098.

Copyright © 2010,2002,1997,1990,1970 Pearson Education,Inc.,publishing as Prentice Hall,One Lake Street,Upper Saddle River,New Jersey 07458.All rights reserved.Manufactured in the United States of America.This publication is protected by Copyright,and permission should be obtained from the publisher prior to any prohibited reproduction,storage in a retrieval system,or transmission in any form or by any means,electronic,mechanical,photocopying,recording,or likewise.To obtain permission(s) to use material from this work,please submit a written request to Pearson Education,Inc.,Permissions Department,One Lake Street,Upper Saddle River,New Jersey 07458.

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Library of Congress Cataloging-in-Publication Data on File

ISBN 10: 0-13-615673-8 ISBN 13: 978-0-13-615673-4

Contents

Preface ix

Chapter 1Introduction to Control Systems1

Chapter 2Mathematical Modeling of Control Systems 13

2–5State-Space Representation of Scalar Differential Equation Systems35

Example Problems and Solutions46 Problems 60

Chapter 3Mathematical Modeling of Mechanical Systems and Electrical Systems63

Example Problems and Solutions86 Problems 97

Chapter 4Mathematical Modeling of Fluid Systems and Thermal Systems100

Example Problems and Solutions140 Problems 152

Chapter 5Transient and Steady-State Response Analyses159

5–7Effects of Integral and Derivative Control Actions on System Performance218

5–8Steady-State Errors in Unity-Feedback Control Systems225

Example Problems and Solutions231 Problems 263 iv Contents

Chapter 6Control Systems Analysis and Design by the Root-Locus Method269

Example Problems and Solutions347 Problems 394

Chapter 7Control Systems Analysis and Design by the Frequency-Response Method 398

7–8Closed-Loop Frequency Response of Unity-Feedback Systems 477

Example Problems and Solutions521 Problems 561

Chapter 8PID Controllers and Modified PID Controllers567

Contents v

8–3Design of PID Controllers with Frequency-Response Approach 577

8–4Design of PID Controllers with Computational Optimization Approach 583

8–7Zero-Placement Approach to Improve Response Characteristics 595

Example Problems and Solutions614 Problems 641

Chapter 9Control Systems Analysis in State Space648

9–2State-Space Representations of Transfer-Function Systems 649

Example Problems and Solutions688 Problems 720

Chapter 10Control Systems Design in State Space722

Example Problems and Solutions817 Problems 855 vi Contents

Appendix ALaplace Transform Tables859 Appendix BPartial-Fraction Expansion867 Appendix CVector-Matrix Algebra874 References 882 Index 886

Contents vii

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Preface

This book introduces important concepts in the analysis and design of control systems. Readers will find it to be a clear and understandable textbook for control system courses at colleges and universities.It is written for senior electrical,mechanical,aerospace,or chemical engineering students.The reader is expected to have fulfilled the following prerequisites:introductory courses on differential equations,Laplace transforms,vectormatrix analysis,circuit analysis,mechanics,and introductory thermodynamics. The main revisions made in this edition are as follows:

•The use of MATLAB for obtaining responses of control systems to various inputs has been increased.

•The usefulness of the computational optimization approach with MATLAB has been demonstrated. •New example problems have been added throughout the book.

•Materials in the previous edition that are of secondary importance have been deleted in order to provide space for more important subjects.Signal flow graphs were dropped from the book.A chapter on Laplace transform was deleted.Instead, Laplace transform tables,and partial-fraction expansion with MATLAB are presented in Appendix A and Appendix B,respectively.

•A short summary of vector-matrix analysis is presented in Appendix C;this will help the reader to find the inverses of n x n matrices that may be involved in the analysis and design of control systems.

This edition of Modern Control Engineeringis organized into ten chapters.The outline of this book is as follows:Chapter 1 presents an introduction to control systems.Chapter 2 deals with mathematical modeling of control systems.A linearization technique for nonlinear mathematical models is presented in this chapter.Chapter 3 derives mathematical models of mechanical systems and electrical systems.Chapter 4 discusses mathematical modeling of fluid systems (such as liquid-level systems,pneumatic systems,and hydraulic systems) and thermal systems.

Chapter 5 treats transient response and steady-state analyses of control systems.

MATLAB is used extensively for obtaining transient response curves.Routh’s stability criterion is presented for stability analysis of control systems.Hurwitz stability criterion is also presented.

Chapter 6 discusses the root-locus analysis and design of control systems,including positive feedback systems and conditionally stable systems Plotting root loci with MATLAB is discussed in detail.Design of lead,lag,and lag-lead compensators with the rootlocus method is included.

Chapter 7 treats the frequency-response analysis and design of control systems.The

Nyquist stability criterion is presented in an easily understandable manner.The Bode diagram approach to the design of lead,lag,and lag-lead compensators is discussed.

Chapter 8 deals with basic and modified PID controllers.Computational approaches for obtaining optimal parameter values for PID controllers are discussed in detail,particularly with respect to satisfying requirements for step-response characteristics.

Chapter 9 treats basic analyses of control systems in state space.Concepts of controllability and observability are discussed in detail.

Chapter 10 deals with control systems design in state space.The discussions include pole placement,state observers,and quadratic optimal control.An introductory discussion of robust control systems is presented at the end of Chapter 10.

The book has been arranged toward facilitating the student’s gradual understanding of control theory.Highly mathematical arguments are carefully avoided in the presentation of the materials.Statement proofs are provided whenever they contribute to the understanding of the subject matter presented.

Special effort has been made to provide example problems at strategic points so that the reader will have a clear understanding of the subject matter discussed.In addition, a number of solved problems (A-problems) are provided at the end of each chapter, except Chapter 1.The reader is encouraged to study all such solved problems carefully; this will allow the reader to obtain a deeper understanding of the topics discussed.In addition,many problems (without solutions) are provided at the end of each chapter, except Chapter 1.The unsolved problems (B-problems) may be used as homework or quiz problems.

If this book is used as a text for a semester course (with 56 or so lecture hours),a good portion of the material may be covered by skipping certain subjects.Because of the abundance of example problems and solved problems (A-problems) that might answer many possible questions that the reader might have,this book can also serve as a selfstudy book for practicing engineers who wish to study basic control theories.

I would like to thank the following reviewers for this edition of the book:Mark Campbell,Cornell University;Henry Sodano,Arizona State University;and Atul G.Kelkar, Iowa State University.Finally,I wish to offer my deep appreciation to Ms.Alice Dworkin, Associate Editor,Mr.Scott Disanno,Senior Managing Editor,and all the people involved in this publishing project,for the speedy yet superb production of this book.

Katsuhiko Ogata x Preface

Introduction to Control Systems

1–1 INTRODUCTION

Control theories commonly used today are classical control theory (also called conventional control theory),modern control theory,and robust control theory.This book presents comprehensive treatments of the analysis and design of control systems based on the classical control theory and modern control theory.A brief introduction of robust control theory is included in Chapter 10.

Automatic control is essential in any field of engineering and science.Automatic control is an important and integral part of space-vehicle systems,robotic systems,modern manufacturing systems,and any industrial operations involving control of temperature,pressure,humidity,flow,etc.It is desirable that most engineers and scientists are familiar with theory and practice of automatic control.

This book is intended to be a text book on control systems at the senior level at a college or university.All necessary background materials are included in the book.Mathematical background materials related to Laplace transforms and vector-matrix analysis are presented separately in appendixes.

Brief Review of Historical Developments of Control Theories and Practices.

The first significant work in automatic control was James Watt’s centrifugal governorfor the speed control of a steam engine in the eighteenth century.Other significantworks in the early stages of development of control theory were due to

2Chapter 1/Introduction to Control Systems

Minorsky,Hazen,and Nyquist,among many others.In 1922,Minorsky worked on automatic controllers forsteering ships and showed how stability could be determined from the differential equations describing the system.In 1932,Nyquist developed a relatively simple procedure for determining the stability of closed-loop systems on the basis of open-loop response to steady-state sinusoidal inputs.In 1934, Hazen,who introduced the term servomechanismsfor position control systems, discussed the design of relay servomechanisms capable of closely following a changing input.

During the decade of the 1940s,frequency-response methods (especially the Bode diagram methods due to Bode) made it possible for engineers to design linear closedloop control systems that satisfied performance requirements.Many industrial control systems in 1940s and 1950s used PID controllers to control pressure,temperature,etc. In the early 1940s Ziegler and Nichols suggested rules for tuning PID controllers,called Ziegler–Nichols tuning rules.From the end of the 1940s to the 1950s,the root-locus method due to Evans was fully developed.

The frequency-response and root-locus methods,which are the core of classical control theory,lead to systems that are stable and satisfy a set of more or less arbitrary performance requirements.Such systems are,in general,acceptable but not optimal in any meaningful sense.Since the late 1950s,the emphasis in control design problems has been shifted from the design of one of many systems that work to the design of one optimal system in some meaningful sense.

As modern plants with many inputs and outputs become more and more complex, the description of a modern control system requires a large number of equations.Classical control theory,which deals only with single-input,single-output systems,becomes powerless for multiple-input,multiple-output systems.Since about 1960,because the availability of digital computers made possible time-domain analysis of complex systems,modern control theory,based on time-domain analysis and synthesis using state variables,has been developed to cope with the increased complexity of modern plants and the stringent requirements on accuracy,weight,and cost in military,space,and industrial applications.

During the years from 1960 to 1980,optimal control of both deterministic and stochastic systems,as well as adaptive and learning control of complex systems,were fully investigated.From 1980s to 1990s,developments in modern control theory were centered around robust control and associated topics.

Modern control theory is based on time-domain analysis of differential equation systems.Modern control theory made the design of control systems simpler because the theory is based on a model of an actual control system.However,the system’s stability is sensitive to the error between the actual system and its model.This meansthat when the designed controller based on a model is applied to the actual system,the system may not be stable.To avoid this situation,we design the control system by first setting up the range of possible errors and then designing the controller in such a way that,if the error of the system stays within the assumed range,thedesignedcontrol system will stay stable.The design method based on this principle is called robust control theory.This theory incorporates both the frequencyresponse approach and the time-domain approach.The theory is mathematically very complex.

Section 1–1 / Introduction 3

Because this theory requires mathematical background at the graduate level,inclusion of robust control theory in this book is limited to introductory aspects only.The reader interested in details of robust control theory should take a graduate-level control course at an established college or university.

Definitions.Before we can discuss control systems,some basic terminologies must be defined.

Controlled Variable and Control Signal or Manipulated Variable.The controlled variable is the quantity or condition that is measured and controlled.Thecontrol signal ormanipulatedvariable is the quantity or condition that is varied by the controller so as to affect the value of the controlled variable.Normally,the controlled variable is the output of the system.Controlmeans measuring the value of the controlled variable of the system and applying the control signal to the system to correct or limit deviation of the measured value from a desired value.

In studying control engineering,we need to define additional terms that are necessary to describe control systems.

(Parte **1** de 6)