Modern Control Engineering Fifth Edition Katsuhiko Ogata Prentice Hall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VP/Editorial Director, Engineering/Computer Science: Marcia J. Horton Assistant/Supervisor: Dolores Mars Senior Editor: Andrew Gilfillan Associate Editor: Alice Dworkin Editorial Assistant: William Opaluch Director of Marketing: Margaret Waples Senior Marketing Manager: Tim Galligan Marketing Assistant: Mack Patterson Senior Managing Editor: Scott Disanno Art Editor: Greg Dulles Senior Operations Supervisor: Alan Fischer Operations Specialist: Lisa McDowell Art Director: Kenny Beck Cover Designer: Carole Anson Media Editor: Daniel Sandin 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. Many of the designations by manufacturers and seller to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data on File 10 9 8 7 6 5 4 3 2 1 ISBN 10: 0-13-615673-8 ISBN 13: 978-0-13-615673-4
C Contents Preface ix Chapter 1 Introduction to Control Systems 1 1–1 Introduction 1 1–2 Examples of Control Systems 4 1–3 Closed-Loop Control Versus Open-Loop Control 7 1–4 Design and Compensation of Control Systems 9 1–5 Outline of the Book 10 Chapter 2 Mathematical Modeling of Control Systems 13 2–1 Introduction 13 2–2 Transfer Function and Impulse-Response Function 15 2–3 Automatic Control Systems 17 2–4 Modeling in State Space 29 2–5 State-Space Representation of Scalar Differential Equation Systems 35 2–6 Transformation of Mathematical Models with MATLAB 39 iii
2–7 Linearization of Nonlinear Mathematical Models 43 Example Problems and Solutions 46 Problems 60 Chapter 3 Mathematical Modeling of Mechanical Systems and Electrical Systems 63 3–1 Introduction 63 3–2 Mathematical Modeling of Mechanical Systems 63 3–3 Mathematical Modeling of Electrical Systems 72 Example Problems and Solutions 86 Problems 97 Chapter 4 Mathematical Modeling of Fluid Systems and Thermal Systems 100 4–1 Introduction 100 4–2 Liquid-Level Systems 101 4–3 Pneumatic Systems 106 4–4 Hydraulic Systems 123 4–5 Thermal Systems 136 Example Problems and Solutions 140 Problems 152 Chapter 5 Transient and Steady-State Response Analyses 159 5–1 Introduction 159 5–2 First-Order Systems 161 5–3 Second-Order Systems 164 5–4 Higher-Order Systems 179 5–5 Transient-Response Analysis with MATLAB 183 5–6 Routh’s Stability Criterion 212 5–7 Effects of Integral and Derivative Control Actions on System Performance 218 5–8 Steady-State Errors in Unity-Feedback Control Systems 225 Example Problems and Solutions 231 Problems 263 iv Contents
Chapter 6 Control Systems Analysis and Design by the Root-Locus Method 269 6–1 Introduction 269 6–2 Root-Locus Plots 270 6–3 Plotting Root Loci with MATLAB 290 6–4 Root-Locus Plots of Positive Feedback Systems 303 6–5 Root-Locus Approach to Control-Systems Design 308 6–6 Lead Compensation 311 6–7 Lag Compensation 321 6–8 Lag–Lead Compensation 330 6–9 Parallel Compensation 342 Example Problems and Solutions 347 Problems 394 Chapter 7 Control Systems Analysis and Design by the Frequency-Response Method 398 7–1 Introduction 398 7–2 Bode Diagrams 403 7–3 Polar Plots 427 7–4 Log-Magnitude-versus-Phase Plots 443 7–5 Nyquist Stability Criterion 445 7–6 Stability Analysis 454 7–7 Relative Stability Analysis 462 7–8 Closed-Loop Frequency Response of Unity-Feedback Systems 477 7–9 Experimental Determination of Transfer Functions 486 7–10 Control Systems Design by Frequency-Response Approach 491 7–11 Lead Compensation 493 7–12 Lag Compensation 502 7–13 Lag–Lead Compensation 511 Example Problems and Solutions 521 Problems 561 Chapter 8 PID Controllers and Modified PID Controllers 567 8–1 Introduction 567 8–2 Ziegler–Nichols Rules for Tuning PID Controllers 568 Contents v
8–3 Design of PID Controllers with Frequency-Response Approach 577 8–4 Design of PID Controllers with Computational Optimization Approach 583 8–5 Modifications of PID Control Schemes 590 8–6 Two-Degrees-of-Freedom Control 592 8–7 Zero-Placement Approach to Improve Response Characteristics 595 Example Problems and Solutions 614 Problems 641 Chapter 9 Control Systems Analysis in State Space 648 9–1 Introduction 648 9–2 State-Space Representations of Transfer-Function Systems 649 9–3 Transformation of System Models with MATLAB 656 9–4 Solving the Time-Invariant State Equation 660 9–5 Some Useful Results in Vector-Matrix Analysis 668 9–6 Controllability 675 9–7 Observability 682 Example Problems and Solutions 688 Problems 720 Chapter 10 Control Systems Design in State Space 722 10–1 Introduction 722 10–2 Pole Placement 723 10–3 Solving Pole-Placement Problems with MATLAB 735 10–4 Design of Servo Systems 739 10–5 State Observers 751 10–6 Design of Regulator Systems with Observers 778 10–7 Design of Control Systems with Observers 786 10–8 Quadratic Optimal Regulator Systems 793 10–9 Robust Control Systems 806 Example Problems and Solutions 817 Problems 855 vi Contents
Appendix A Laplace Transform Tables 859 Appendix B Partial-Fraction Expansion 867 Appendix C Vector-Matrix Algebra 874 References 882 Index 886 Contents vii
P 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, vector- matrix 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 pre- sented 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 analy- sis and design of control systems. This edition of Modern Control Engineering is organized into ten chapters.The outline of this book is as follows: Chapter 1 presents an introduction to control systems. Chapter 2 ix
deals with mathematical modeling of control systems. A linearization technique for non- linear 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 MAT- LAB is discussed in detail. Design of lead, lag, and lag-lead compensators with the root- locus 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 di- agram 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, par- ticularly with respect to satisfying requirements for step-response characteristics. Chapter 9 treats basic analyses of control systems in state space. Concepts of con- trollability 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 dis- cussion 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 presen- tation 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 self- study 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 Camp- bell, 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 in- volved in this publishing project, for the speedy yet superb production of this book. Katsuhiko Ogata x Preface
1 Introduction to Control Systems 1–1 INTRODUCTION Control theories commonly used today are classical control theory (also called con- ventional 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, mod- ern manufacturing systems, and any industrial operations involving control of temper- ature, 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 col- lege or university. All necessary background materials are included in the book. Math- ematical 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 gover- nor for the speed control of a steam engine in the eighteenth century. Other significant works in the early stages of development of control theory were due to 1
Minorsky, Hazen, and Nyquist, among many others. In 1922, Minorsky worked on automatic controllers for steering ships and showed how stability could be deter- mined 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 servomechanisms for position control systems, discussed the design of relay servomechanisms capable of closely following a chang- ing 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 closed- loop 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 con- trol theory, lead to systems that are stable and satisfy a set of more or less arbitrary per- formance 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. Clas- sical 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 sys- tems, 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 in- dustrial applications. During the years from 1960 to 1980, optimal control of both deterministic and sto- chastic systems, as well as adaptive and learning control of complex systems, were fully investigated. From 1980s to 1990s, developments in modern control theory were cen- tered 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 means that 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 con- troller in such a way that, if the error of the system stays within the assumed range, the designed control system will stay stable. The design method based on this principle is called robust control theory. This theory incorporates both the frequency- response approach and the time-domain approach. The theory is mathematically very complex. 2 Chapter 1 / Introduction to Control Systems
Because this theory requires mathematical background at the graduate level, inclu- sion 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. The control signal or manipulated variable 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. Control means 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 neces- sary to describe control systems. Plants. A plant may be a piece of equipment, perhaps just a set of machine parts functioning together, the purpose of which is to perform a particular operation. In this book, we shall call any physical object to be controlled (such as a mechanical device, a heating furnace, a chemical reactor, or a spacecraft) a plant. Processes. The Merriam–Webster Dictionary defines a process to be a natural, pro- gressively continuing operation or development marked by a series of gradual changes that succeed one another in a relatively fixed way and lead toward a particular result or end; or an artificial or voluntary, progressively continuing operation that consists of a se- ries of controlled actions or movements systematically directed toward a particular re- sult or end. In this book we shall call any operation to be controlled a process. Examples are chemical, economic, and biological processes. Systems. A system is a combination of components that act together and perform a certain objective. A system need not be physical. The concept of the system can be applied to abstract, dynamic phenomena such as those encountered in economics. The word system should, therefore, be interpreted to imply physical, biological, economic, and the like, systems. Disturbances. A disturbance is a signal that tends to adversely affect the value of the output of a system. If a disturbance is generated within the system, it is called internal, while an external disturbance is generated outside the system and is an input. Feedback Control. Feedback control refers to an operation that, in the presence of disturbances, tends to reduce the difference between the output of a system and some reference input and does so on the basis of this difference. Here only unpredictable dis- turbances are so specified, since predictable or known disturbances can always be com- pensated for within the system. Section 1–1 / Introduction 3
1–2 EXAMPLES OF CONTROL SYSTEMS In this section we shall present a few examples of control systems. Speed Control System. The basic principle of a Watt’s speed governor for an en- gine is illustrated in the schematic diagram of Figure 1–1. The amount of fuel admitted to the engine is adjusted according to the difference between the desired and the actual engine speeds. The sequence of actions may be stated as follows: The speed governor is ad- justed such that, at the desired speed, no pressured oil will flow into either side of the power cylinder. If the actual speed drops below the desired value due to disturbance, then the decrease in the centrifugal force of the speed governor causes the control valve to move downward, supplying more fuel, and the speed of the engine increases until the desired value is reached. On the other hand, if the speed of the engine increases above the desired value, then the increase in the centrifu- gal force of the governor causes the control valve to move upward. This decreases the supply of fuel, and the speed of the engine decreases until the desired value is reached. In this speed control system, the plant (controlled system) is the engine and the controlled variable is the speed of the engine. The difference between the desired speed and the actual speed is the error signal. The control signal (the amount of fuel) to be applied to the plant (engine) is the actuating signal. The external input to dis- turb the controlled variable is the disturbance. An unexpected change in the load is a disturbance. Temperature Control System. Figure 1–2 shows a schematic diagram of tem- perature control of an electric furnace. The temperature in the electric furnace is meas- ured by a thermometer, which is an analog device. The analog temperature is converted Power cylinder Oil under pressure Pilot valve Close Engine Load Open Figure 1–1 Fuel Speed control Control system. valve 4 Chapter 1 / Introduction to Control Systems
Thermometer A/D Interface converter Controller Electric furnace Programmed input Figure 1–2 Temperature control Relay Amplifier Interface system. Heater to a digital temperature by an A/D converter. The digital temperature is fed to a con- troller through an interface.This digital temperature is compared with the programmed input temperature, and if there is any discrepancy (error), the controller sends out a sig- nal to the heater, through an interface, amplifier, and relay, to bring the furnace tem- perature to a desired value. Business Systems. A business system may consist of many groups. Each task assigned to a group will represent a dynamic element of the system. Feedback methods of reporting the accomplishments of each group must be established in such a system for proper operation. The cross-coupling between functional groups must be made a mini- mum in order to reduce undesirable delay times in the system. The smaller this cross- coupling, the smoother the flow of work signals and materials will be. A business system is a closed-loop system. A good design will reduce the manageri- al control required. Note that disturbances in this system are the lack of personnel or ma- terials, interruption of communication, human errors, and the like. The establishment of a well-founded estimating system based on statistics is manda- tory to proper management. It is a well-known fact that the performance of such a system can be improved by the use of lead time, or anticipation. To apply control theory to improve the performance of such a system, we must rep- resent the dynamic characteristic of the component groups of the system by a relative- ly simple set of equations. Although it is certainly a difficult problem to derive mathematical representations of the component groups, the application of optimization techniques to business sys- tems significantly improves the performance of the business system. Consider, as an example, an engineering organizational system that is composed of major groups such as management, research and development, preliminary design, ex- periments, product design and drafting, fabrication and assembling, and tesing. These groups are interconnected to make up the whole operation. Such a system may be analyzed by reducing it to the most elementary set of com- ponents necessary that can provide the analytical detail required and by representing the dynamic characteristics of each component by a set of simple equations. (The dynamic performance of such a system may be determined from the relation between progres- sive accomplishment and time.) Section 1–2 / Examples of Control Systems 5
Required product Research Preliminary Product Fabrication Product Management and design Experiments design and and Testing development drafting assembling Figure 1–3 Block diagram of an engineering organizational system. A functional block diagram may be drawn by using blocks to represent the func- tional activities and interconnecting signal lines to represent the information or product output of the system operation. Figure 1–3 is a possible block diagram for this system. Robust Control System. The first step in the design of a control system is to obtain a mathematical model of the plant or control object. In reality, any model of a plant we want to control will include an error in the modeling process.That is, the actual plant differs from the model to be used in the design of the control system. To ensure the controller designed based on a model will work satisfactorily when this controller is used with the actual plant, one reasonable approach is to assume from the start that there is an uncertainty or error between the actual plant and its mathematical model and include such uncertainty or error in the design process of the control system. The control system designed based on this approach is called a robust control system. Suppose that the actual plant we want to control is G(s) and the mathematical model of the actual plant is G(s), that is, G(s)=actual plant model that has uncertainty ¢(s) G(s)=nominal plant model to be used for designing the control system G(s) and G(s) may be related by a multiplicative factor such as G(s) = G(s)[1 + ¢(s)] or an additive factor G(s) = G(s) + ¢(s) or in other forms. Since the exact description of the uncertainty or error ¢(s) is unknown, we use an estimate of ¢(s) and use this estimate, W(s), in the design of the controller. W(s) is a scalar transfer function such that ¢(s)q 6 W(s)q = max W(jv) 0vq where W(s)q is the maximum value of W(jv) for 0 v q and is called the H infinity norm of W(s). 6 Chapter 1 / Introduction to Control Systems
Using the small gain theorem, the design procedure here boils down to the deter- mination of the controller K(s) such that the inequality W(s) ß ß 6 1 1 + K(s)G(s) q is satisfied, where G(s) is the transfer function of the model used in the design process, K(s) is the transfer function of the controller, and W(s) is the chosen transfer function to approximate ¢(s). In most practical cases, we must satisfy more than one such inequality that involves G(s), K(s), and W(s)’s. For example, to guarantee robust sta- bility and robust performance we may require two inequalities, such as Wm(s)K(s)G(s) ß ß 6 1 for robust stability 1 + K(s)G(s) q Ws(s) ß ß 6 1 for robust performance 1 + K(s)G(s) q be satisfied. (These inequalities are derived in Section 10–9.) There are many different such inequalities that need to be satisfied in many different robust control systems. (Robust stability means that the controller K(s) guarantees internal stability of all systems that belong to a group of systems that include the system with the actual plant. Robust performance means the specified performance is satisfied in all systems that be- long to the group.) In this book all the plants of control systems we discuss are assumed to be known precisely, except the plants we discuss in Section 10–9 where an introduc- tory aspect of robust control theory is presented. 1–3 CLOSED-LOOP CONTROL VERSUS OPEN-LOOP CONTROL Feedback Control Systems. A system that maintains a prescribed relationship between the output and the reference input by comparing them and using the difference as a means of control is called a feedback control system.An example would be a room- temperature control system. By measuring the actual room temperature and comparing it with the reference temperature (desired temperature), the thermostat turns the heat- ing or cooling equipment on or off in such a way as to ensure that the room tempera- ture remains at a comfortable level regardless of outside conditions. Feedback control systems are not limited to engineering but can be found in various nonengineering fields as well. The human body, for instance, is a highly advanced feed- back control system. Both body temperature and blood pressure are kept constant by means of physiological feedback. In fact, feedback performs a vital function: It makes the human body relatively insensitive to external disturbances, thus enabling it to func- tion properly in a changing environment. Section 1–3 / Closed-Loop Control versus Open-Loop Control 7
Closed-Loop Control Systems. Feedback control systems are often referred to as closed-loop control systems. In practice, the terms feedback control and closed-loop control are used interchangeably. In a closed-loop control system the actuating error signal, which is the difference between the input signal and the feedback signal (which may be the output signal itself or a function of the output signal and its derivatives and/or integrals), is fed to the controller so as to reduce the error and bring the output of the system to a desired value. The term closed-loop control always implies the use of feedback control action in order to reduce system error. Open-Loop Control Systems. Those systems in which the output has no effect on the control action are called open-loop control systems. In other words, in an open- loop control system the output is neither measured nor fed back for comparison with the input. One practical example is a washing machine. Soaking, washing, and rinsing in the washer operate on a time basis. The machine does not measure the output signal, that is, the cleanliness of the clothes. In any open-loop control system the output is not compared with the reference input. Thus, to each reference input there corresponds a fixed operating condition; as a result, the accuracy of the system depends on calibration. In the presence of disturbances, an open-loop control system will not perform the desired task. Open-loop control can be used, in practice, only if the relationship between the input and output is known and if there are neither internal nor external disturbances. Clearly, such systems are not feed- back control systems. Note that any control system that operates on a time basis is open loop. For instance, traffic control by means of signals operated on a time basis is another example of open-loop control. Closed-Loop versus Open-Loop Control Systems. An advantage of the closed- loop control system is the fact that the use of feedback makes the system response rela- tively insensitive to external disturbances and internal variations in system parameters. It is thus possible to use relatively inaccurate and inexpensive components to obtain the accurate control of a given plant, whereas doing so is impossible in the open-loop case. From the point of view of stability, the open-loop control system is easier to build be- cause system stability is not a major problem. On the other hand, stability is a major problem in the closed-loop control system, which may tend to overcorrect errors and thereby can cause oscillations of constant or changing amplitude. It should be emphasized that for systems in which the inputs are known ahead of time and in which there are no disturbances it is advisable to use open-loop control. Closed-loop control systems have advantages only when unpredictable disturbances and/or unpredictable variations in system components are present. Note that the output power rating partially determines the cost, weight, and size of a control system. The number of components used in a closed-loop control system is more than that for a corresponding open-loop control system. Thus, the closed-loop control system is generally higher in cost and power. To decrease the required power of a system, open- loop control may be used where applicable. A proper combination of open-loop and closed-loop controls is usually less expensive and will give satisfactory overall system performance. Most analyses and designs of control systems presented in this book are concerned with closed-loop control systems. Under certain circumstances (such as where no disturbances exist or the output is hard to measure) open-loop control systems may be 8 Chapter 1 / Introduction to Control Systems
desired. Therefore, it is worthwhile to summarize the advantages and disadvantages of using open-loop control systems. The major advantages of open-loop control systems are as follows: 1. Simple construction and ease of maintenance. 2. Less expensive than a corresponding closed-loop system. 3. There is no stability problem. 4. Convenient when output is hard to measure or measuring the output precisely is economically not feasible. (For example, in the washer system, it would be quite ex- pensive to provide a device to measure the quality of the washer’s output, clean- liness of the clothes.) The major disadvantages of open-loop control systems are as follows: 1. Disturbances and changes in calibration cause errors, and the output may be different from what is desired. 2. To maintain the required quality in the output, recalibration is necessary from time to time. 1–4 DESIGN AND COMPENSATION OF CONTROL SYSTEMS This book discusses basic aspects of the design and compensation of control systems. Compensation is the modification of the system dynamics to satisfy the given specifi- cations. The approaches to control system design and compensation used in this book are the root-locus approach, frequency-response approach, and the state-space ap- proach. Such control systems design and compensation will be presented in Chapters 6, 7, 9 and 10. The PID-based compensational approach to control systems design is given in Chapter 8. In the actual design of a control system, whether to use an electronic, pneumatic, or hydraulic compensator is a matter that must be decided partially based on the nature of the controlled plant. For example, if the controlled plant involves flammable fluid, then we have to choose pneumatic components (both a compensator and an actuator) to avoid the possibility of sparks. If, however, no fire hazard exists, then electronic com- pensators are most commonly used. (In fact, we often transform nonelectrical signals into electrical signals because of the simplicity of transmission, increased accuracy, increased reliability, ease of compensation, and the like.) Performance Specifications. Control systems are designed to perform specific tasks. The requirements imposed on the control system are usually spelled out as per- formance specifications. The specifications may be given in terms of transient response requirements (such as the maximum overshoot and settling time in step response) and of steady-state requirements (such as steady-state error in following ramp input) or may be given in frequency-response terms. The specifications of a control system must be given before the design process begins. For routine design problems, the performance specifications (which relate to accura- cy, relative stability, and speed of response) may be given in terms of precise numerical values. In other cases they may be given partially in terms of precise numerical values and Section 1–4 / Design and Compensation of Control Systems 9
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