STEFANI
SHAH IAN
SAVANT f
1 I
,
\ \
\
I \
HOSTETTER
r
'.
~~..
-
Design .of Feedback Control Systems
Raymond T. Stefani California State University, Long Beach
Bahram Shahian California State University, Long Beach
Clement J. Savant, Jr. Gene H. Hostetter --
-
,\
-----NUST-EME COLLEGE LIBRARY
L11~!"2g*IU- - - - New York Oxford OXFORD UNIVERSITY PRESS
2002
Oxford University Press Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan
Copyright © 2002 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York, 10016 http://www.oup-usa.org Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.
Library of Congress Cataloging-in-Publication Data Design of feedback control systems I Raymond T. Stefani ... let al.l.-- 4th ed. p. em. -- (Oxford series in electrical and computer engineering) Includes bibliographical references and index. ISBN 0-19-514249-7 I. Feedback control systems. I. Stefani, Raymond T, II. Series. TJ 216 .0417 2001 629.8'3--dc21
,.,
..
,
Printing number: 9 8 7 6 5 4 3 Printed in the United States of America on acid-free paper
00-058913
TO Ted, Rick, and my Inspiration Saleh and Mahin; Farahnaz, Bita and Nima Barbara and the Savant family in memory of Clement Donna and the Hostetter family in memory of Gene
=ontents
Preface
xv
HAPTER 1 Continuous-Time System Description 1.1 1.2
Preview Basic Concepts 1.2.1 Control System Terminology The Feedback Concept 1.2.2
1.3 1.4
Modeling System Dynamics
1.5
Electrical Components 1.5.1 Mesh Analysis 1.5.2 State Variables 1.5.3 Node Analysis 1.5.4 Analyzing Operational Amplifier Circuits 1.5.5 Operational Amplifier Applications Translational Mechanical Components 1.6.1 Free-Body Diagrams 1.6.2 State Variables Rotational Mechanical Components 1.7.1 Free-Body Diagrams 1.7.2 Analogies 1.7.3 Gear Trains and Transformers
1.6
1.7
2 2 4 7 9 10
11 13
15 18 21
25 25 29 32 32 35 37
CONTENTS
1.8 1.9
Electromechanical Components Aerodynamics 1.9.1 Nomenclature 1.9.2 Dynamics 1.9.3 Lateral and Longitudinal Motion 1.1 0 Thennal Systems
vii
40 45 46 46 50 52
1.11 Hydraulics 1.12 Transfer Function and Stability 1.12.1 Transfer Functions 1.12.2 Response Terms 1.12.3 Multiple Inputs and Outputs 1.12.4 Stability
54 55
1.13 Block Diagrams 1.13.1 Block Diagram Elements 1.13.2 Block Diagram Reduction 1.13.3 Multiple Inputs and Outputs
73 73 75 78
1.14 Signal Flow Graphs 1.14.1 Comparison with Block Diagrams 1.14.2 Mason's Rule
79 79 83
1.15 A Positioning Servo 1.16 Controller Model of the Thyroid Gland 1.17 Stick-Slip Response of an Oil Well Drill
94
55 57 67
69
91
96
1.18 Summary References
101
Problems
105
CHAPTER 2 Continuous-lime System Response 2.1 2.2 2.3
Preview Response of FIrst-Order Systems Response of Second-Order Systems 2.3.1 Time Response 2.3.2 Overdamped Response 2.3.3 Critically Damped Response 2.3.4 Underdamped Response 2.3.5 Undamped Natural Frequency and Damping Ratio 2.3.6 Rise Time, Overshoot, and Settling Time
2.4
Higher-Order System Response
2.5
Stability Testing 2.5.1 Coefficient Tests 2.5.2 Routh-Hurwitz Testing 2.5.3 Significance of the Array Coefficients
103
119 119
120 126 126 127 128 128 129 136 141 143 143
145 147
CONTENTS
iii
2.5.4 2.5.5 2.5.6 2.5.7
148 150 154 155 159 159 163 165 168 171 173 174
Performance Specifications
183
Left-Column Zeros Row of Zeros Eliminating a Possible Odd Divisor Multiple Roots 2.6 Parameter Shifting 2.6.1 Adjustable Systems 2.6.2 Kharitonov's Theorem 2.7 An Insulin Delivery System 2.8 Analysis of an Aircraft Wing 2.9 Summary References Problems ~HAPTER 3
3.1 3.2
Preview Analyzing Tracking Systems 3.2.1 Importance of Tracking Systems 3.2.2 Natural Response, Relative Stability, and Damping 3.3 Forced Response 3.3.1 Steady State Error 3.3.2 Initial and Final Values 3.3.3 Steady State Errors to Power-of-Time Inputs 3.4 Power-of-Time Error Performance 3.4.1 System Type Number 3.4.2 Achieving a Given Type Number 3.4.3 Unity Feedback Systems 3.4.4 Unity Feedback Error Coefficients 3.5 Performance Indices and Optimal Systems 3.6 System Sensitivity 3.6.1 Calculating the Effects of Changes in Parameters 3.6.2 Sensitivity Functions 3.6.3 Sensitivity to Disturbance Signals 3.7 Time Domain Design 3.7.1 Process Control 3.7.2 Ziegler-Nichols Compensation 3.7.3 Chien-Hrones-Reswick Compensation 3.8 An Electric Rail Transportation System 3.9 Phase-Locked Loop for a CB Receiver 3.10 Bionic Eye 3.11 Summary References Problems
183 184 184 187 189 189 190 192 198 198 200 201 204 208
215 215 216 220 223
224 224 225
231 234 237 240
242 244
CONTENTS
CHAPTER 4 Root Locus Analysis
4.1 4.2
Preview Pole-Zero Plots 4.2.1 Poles and Zeros 4.2.2 Graphical Evaluation 4.3 Root Locus for Feedback Systems 4.3.1 Angle Criterion 4.3.2 High and Low Gains 4.3.3 Root Locus Properties 4.4 Root Locus Construction 4.5 More About Root Locus 4.5.1 Root Locus Calibration 4.5.2 Computer-Aided Root Locus 4.6 Root Locus for Other Systems 4.6.1 Systems with Other FomlS 4.6.2 Negative Parameter Ranges 4.6.3 Delay Effects 4.7 Design concepts (Adding Poles and Zeros) 4.8 A Light-Source Tracking System 4.9 An Artificial Limb 4.10 Control of a Flexible Spacecraft 4. J 1 Bionic Eye 4.12 Summary References Problems
CHAPTER 5 Root Locus Design
5.1 5.2 5.3
5.4 5.5 5.6
5.7 5.8 5.9
Preview Shaping a Root Locus Adding and Canceling Poles and Zeros 5.3.1 Adding a Pole or Zero 5.3.2 Canceling a Pole or Zero Second-Order Plant Models An Uncompensated Example System Cascade Proportional Plus Integral (PI) 5.6.1 General Approach to Compensator Design 5.6.2 Cascade PI Compensation Cascade Lag Compensation Cascade Lead Compensation Cascade Lag-Lead Compensation
ix
254 254 255 255 256 260 260 261 262 263 272 272 284 286 286 288 293 295 300 302 308 310 313 314 314 327 327 328 329 329 330 334 338 341 341 343 347 351 355
CONTENTS
5.10 Rate Feedback Compensation (PD) 5.11 Proportional-Integral-Derivative Compensation 5.12 Pole Placement 5.12.1 Algebraic Compensation 5.12.2 Selecting the Transfer Function 5.12.3 Incorrect Plant Transmittance 5.12.4 Robust Algebraic Compensation 5.12.5 Fixed-Structure Compensation
357 361 365 366 367 370 373 378
5.13 An Unstable High-Performance Aircraft 5.14 Control of a Flexible Space Station 5.15 Control of a Solar Furnace 5.16 Summary References Problems
381 385 388 393 394 395
'HAPTER 6 Frequency Response Analysis
405
6.1 6.2
6.3
6.4
6.5
6.6 6.7
Preview Frequency Response 6.2.1 Forced Sinusoidal Response 6.2.2 Frequency Response Measurement 6.2.3 Response at Low and High Frequencies 6.2.4 Graphical Frequency Response Methods Bode Plots 6.3.1 Amplitude Plots in Decibels Real Axis Roots 6.3.2 Products of Transmittance Terms 6.3.3 6.3.4 Complex Roots Using Experimental Data 6.4.1 Finding Models 6.4.2 Irrational Transmittances Nyquist Methods 6.5,1 Generating the Nyquist (polar) Plot 6.5.2 Interpreting the Nyquist Plot Gain Margin Phase Margin
6.8 Relations Between Closed-Loop and Open-Loop Frequency Response 6.9 Frequency Response of a Flexible Spacecraft 6.10 Summary References Problems
405 406 406
407 410 412 420 420 424 428 433 446
446 447 449 450 456 464 469
475 480 485 488 488
CONTENTS
CHAPTER 7 Frequency Response Design 7.1 7.2 7.3 7.4 7.5
Preview Relation Between Root Locus, Time Domain, and Frequency Domain Compensation Using Bode Plots Uncompensated System Cascade Proportional Plus Integral (PI) and Cascade Lag Compensations
7.6 Cascade Lead Compensation 7.7 Cascade Lag-Lead Compensation 7.8 Rate Feedback Compensation 7.9 Proportional-Integral-Derivative Compensation 7.10 An Automobile Driver as a Compensator 7.11 Summary References Problems
CHAPTER 8 State Space Analysis 8.1
8.2
8.3
8.4
8.5
8.6
Preview State Space Representation 8.2.1 Phase-Variable Form 8.2.2 Dual Phase- Variable Form 8.2.3 Multiple Inputs and Outputs 8.2.4 Physical State Variables 8.2.5 Transfer Functions State Transformations and Diagonalization 8.3.1 Diagonal Forms 8.3.2 Diagonalization Using Partial Fraction Expansion 8.3.3 Complex Conjugate Characteristic Roots 8.3.4 Repeated Characteristic Roots Time Response from State Equations 8.4.1 Laplace Transform Solution Time Domain Response of First-Order Systems 8.4.2 8.4.3 Time Domain Response of Higher-Order Systems 8.4.4 System Response Computation Stability 8.5.1 Asymptotic Stability BIBO Stability 8.5.2 Internal Stability 8.5.3 Controllability and Observability 8.6.1 The Controllability Matrix The Observability Matrix 8.6.2 8.6.3 Controllability, Observability, and Pole-Zero Cancellation
xi
501 501 501 505 507 509 514 517 520 523 525
529 530 530 535
535 536 537 540 542 547
551 554 558 562
564 567
575 575 576 577
579 584 584
585 587
589 592 594 595
CONTENTS
8.7 8.8
8.6.4 Causes of Uncontrollability Inverted Pendulum Problems Summary
References
612 614
Problems HAPTER 9
9.1 9.2
9.3 9.4
9.5
9.6 9.7
596 603 610
State Space Design
Preview State Feedback and Pole Placement 9.2.1 Stabilizability 9.2.2 Choosing Pole Locations 9.2.3 Limitations of State Feedback Tracking Problems 9.3.1 Integral Control Observer Design 9.4.1 Control Using Observers 9.4.2 Separation Property 9.4.3 Observer Transfer Function Reduced-Order Observer Design 9.5.1 Separation Property 9.5.2 Reduced-Order Observer Transfer Function A Magnetic Levitation System Summary
626 626 626 630 632 635 637 638 640
644 646 647 650 653
654 657 667
References
668
Problems
669
HAPTER 10 Advanced State Space Methods
10.1 Preview 10.2 The Linear Quadratic Regulator Problem 10.2.1 Properties of the LQR Design 10.2.2 Return Difference Inequality 10.2.3 Optimal Root Locus 10.3 Optimal Observers-the Kalman Filter 10.4 The Linear Quadratic Gaussian (LQG) Problem 10.4.1 Critique of LQG 10.5 Robustness 10.5.1 Feedback Properties 10.5.2 Uncertainty Modeling 10.5.3 Robust Stability 10.6 Loop Transfer Recovery (LTR)
675 675 676 680 680 682
685 687 690 692 693
695 698
705
CONTENTS
10.7 Roo Control 10.7.1 10.7.2 10.7.3 10.7.4 10.8 Summary References Problems
A Brief History Some Preliminaries Hoo Control: Solution Weights in Hoo Control Problems
CHAPTER 11 Digital Control 11.1 Preview 11.2 Computer Processing 11.2.1 Computer History and Trends 11.3 AID and DJA Conversion 11.3.1 Analog-to-Digital Conversion 11.3.2 Sample and Hold 11.3.3 Digital-to-Analog Conversion 11.4 Discrete-Time Signals 11.4.1 Representing Sequences 11.4.2 z-Transformation and Properties 11.4.3 Inverse z Transform 1l.5 Sampling 11.6 Reconstruction of Signals from Samples 11.6.1 Representing Sampled Signals with Impulses 11.6.2 Relation Between the z Transform and the Laplace Transform 11.6.3 The Sampling Theorem 11.7 Discrete-Time Systems 11.7.1 Difference Equations and Response 11.7.2 z-Transfer Functions 11.7.3 Block Diagrams and Signal Flow Graphs 11.7.4 Stability and the Bilinear Transformation 11.7.5 Computer Software 11.8 State-Variable Descriptions of Discrete-Time Systems 11.8.1 Simulation Diagrams and Equations 11.8.2 Response and Stability 11.8.3 Controllability and Observability 11.9 Digitizing Control Systems 11.9.1 Step-Invariant Approximation 11.9.2 z-Transfer Functions of Systems with Analog Measurements 11.9.3 A Design Example 11.10 Direct Digital Design 11.10.1 Steady State Response
xiii
709 709 710 713 715 722 723 724 733 733 734 734 737 737 739 741 741 741 744 749 751 753 753 756 757 760 760 762 763 764
768
771 771 774 777 779 779 782 785 788 788
CONTENTS
References
789 790 798 800
Problems
802
ii.1O.2 Deadbeat Systems 11.10.3 A Design Example 11.11 Summary
JENDIX A Matrix Algebra
812
A.l
Preview
812
A.2
Nomenclature
812
A.3
Addition and Subtraction
812
AA
Transposition
A.5
Multiplication
813 813
A.6
Determinants and Cofactors
814
A.7
Inverse
816
A.8
Simultaneous Equations
817
A.9
Eigenvalues and Eigenvectors
819
A.lO Derivative of a Scalar with Respect to a Vector
821
A.ll Quadratic Forms and Symmetry
823
A.12 Definiteness
824
A.13 Rank
826
A.14 Partitioned Matrices
827
Problems
830
")ENDIX B Laplace Transform
834
B.l
Preview
834
B.2
Definition and Properties
834
B.3
Solving Differential Equations
835
BA
Partial Fraction Expansion
837
B.5
Additional Properties of the Laplace Transform
841 842. 842 843 844
B.5.1 B.5.2 B.5.3 B.5.4 lex
Real Translation Second independent Variable Final- Value and initial-Value Theorems Convolution integral
845
Preface
As the new millennium begins, we look back in gratitude to the many faculty and students who have used the three earlier editions of this textbook and made many helpful suggestions to the authors. In those earlier editions we introduced comprehensive design examples, drill problems, and wide margins with notes. Other texts followed our lead and emulated those items. What other texts cannot emulate, we believe, is the clear and understandable exposition we bring to the field of control system science. Throughout this book we try to make complicated methodology accessible to a spectrum of students with widely varying backgrounds. Detail is there for those who want to know "why." Summaries and marginal comments are there for those who simply want to know "how." Revisions The most obvious change in this edition is the comprehensive keying of this text to MATLAB. We created sections of "Computer-Aided Learnirig" by which each student can learn how the MATLAB platfonn can be used to verify all figures and tables included in the text. We selected a small group of MATLAB commands to efficiently focus the use of that computational package. In a basic course such as this, it is essential that every student use the computer as an aid to learning and not as the primary source of information. The student should learn all basics and should be able to sketch (albeit roughly)· time response plots, root locus plots, and BodeJNyquist plots manually. MATLAB (or any other computer tool) may then be used to fine-tune understanding and to obtain results of high accuracy. But, those results must be critically reviewed by a knowledgeable user; otherwise the computer becomes the master and the user becomes the slave. Chapter I has been substantially revised. Linearization is introduced by which models may be generated. Operational amplifier applications are included for the various types of compensator designed later in the text. Substantive coverage is made of aerodynamics, thermal systems, and hydraulic systems. Drill problems cover those topics. Stability is covered in more detail. Signal flow graphs are better compared to block diagrams. Design examples are added for the human thyroid gland as a controller and for oil well drill dynamics. For Chapter 2, we include the significance of Routh array coefficients and the stability implication of multiple roots occurring as even divisors. An example of Kharitonov's theorem is added.
xv
PREFACE
xvi
Hurwitz determinants are now presented in Chapter 3. It is now shown how coefficients of the transfer function may be selected to force a given type number to occur. An interesting biomedical design example is added, that of a bionic eye for the blind. Time response examples are added to illustrate time domain design. The main change to Chapter 4 is inclusion of computer-aided means for calculating breakaway points, entry points, departure angles, and approach angles. The MATLAB command rltool is introduced. Delay effects are evaluated as a function of 1/ T where T is the delay in seconds. The bionic eye example is again used, this time to illustrate use of the root locus. Chapter 5 is revised comprehensively. Root locus design methods are now more general and more flexible. The effect of adding or canceling poles or zeros is covered in detail. The MATLAB command rl tool is suggested as a primary computer aid in that the effect of each root locus design point may be evaluated in terms of step response and the Bode plot. A new design example is introduced for a solar furnace. Chapter 6 now begins with an introduction to all frequency response plots. It is argued that frequency response data are complex vectors, hence can be plotted in a variety of ways resulting in Bode, Nyquist, and Nichols plots. There is a new section that discusses the relation between open-loop and closed-loop frequency response plots. Closed-loop frequency response data such as bandwidth and peak resonance are introduced more formally. Nichols plots, Nichols charts, and constant loci M and N circles are also discussed. Chapter 7 on frequency domain design remains unchanged. Chapter 8 now includes a design example of the classic inverted pendulum problem and several variations. This famous problem has become a benchmark for testing novel control design techniques and provides an excellent tool for introducing the important concepts of controllability, observability, pole-zero cancellation, and practical issues such as sensor placement. Appropriate MATLAB commands for state space modeling, transforniation, analysis, and simulation are also discussed. Chapters 9-11 have minor corrections along with the introduction of MATLAB commands for digital control. Use of This Textbook The text can be divided into six areas: Classical analysis including modeling (Chapters 1-4,6) Classical design (Chapters 5 and 7) State-variable analysis (Chapter 8) State-variable design (Chapter 9) Advanced topics (Chapter 10) Digital control (Chapters 11) These six areas represent building blocks to construct a course. We have purposely included more material than a three-semester unit course or a four-quarter unit course would normally cover. The extra material is intended to give ~he instructor flexibility in structuring a course to meet the needs of the program, the university, and the community served. We suggest that it is better to cover a smaller number of units well than to cover a larger number poorly. For example, a two-course sequence could be created where the first course covers classical analysis (Chapters 1-4 and 6) followed by a second course including state variables, design, and advanced topics (Chapters 5 and 7-10). Chapter 11 is often used as reference material, introducing the student to digital control and providing a comparison with analog methods. The possibilities are endless.
Raymond T. Stefani Bahram Shahian Clement J. Savant Jr. (late) Gene H. Hostetter (late)
Continuous-Time System Description
The first conscious use of feedback control of a physical system by mankind lives in prehistory. Possibly it was a spillway in an irrigation network, where excess water was automatically drained. Development of a mathematical framework for the description, analysis, and design of control systems dates from the introduction of James Watt's flyball governor (1760), which was used to regulate the speed of steam engines, and the subsequent work by James Clerk Maxwell (ca. 1868) and others to improve the design and extend its applicability. Since that era, the theory and practice of control system design advanced rapidly. Important new concepts and tools were developed in connection with telephone and radio communications in the 1920s and 1930s. Rather poorly performing electronic devices, including amplifiers and modulators, were dramatically improved by feedback. World War II further accelerated the development of classical control theory and practice. Heavy guns had to be rapidly and accurately positioned. Precise navigation and target tracking were increasingly important, and aircraft performance was improved greatly with the incorporation of complex control systems to aid the pilot. Latter, automation became a household word as industry began to depend more and more upon automatically controlled machinery. Today, feedback control systems are pervasive in industry and in our everyday lives. They range from governmental regulation (such as that governing monetary policy) to automated and highly flexible manufacturing plants to sophisticated automobiles, household appliances, and entertainment systems. It is our purpose to learn to design feedback control systems for a wide variety of applications. 1
2
CONTINUOUS-TIME SYSTEM DESCRIPTION
Control system designers find that block diagrams provide a particularly useful way to visualize the interconnections of system components, thus revealing the system structure. Successful design begins by creating a mathematical model of the system to be stabilized. Next, the contentS of the blocks within a diagram must be identified. Finally, values must be selected for those parameters that are adjustable, and sometimes additional components must be added to provide acceptable performance. This chapter begins by defining basic control system terminology. Since design requires a model of each system of interest, the behaviors of many typical electrical, mechanical, and electromechanical systems are described. The resulting differential equations must be rendered into a forin useful to the controls engineer. The goal can be accomplished by Laplace-transforming each differential equation and then generating a relationship, the transmittance, between the input and output of each block of the control system block diagram. In Appendix B, a summary of the Laplace transform method is presented. The block diagram can be reduced to just one input-output relationship, the system overall transfer function. By converting the block diagram into an equivalent fo~, the signal flow graph can be developed. Subsequent chapters will describe the design steps that follow once the block diagram has been defined and the transfer function has become available. All the chapters of this text conclude with examples that are intended to reinforce the key points of the chapter in an interesting and informative manner. Chapter 1 concludes with discussion of a positioning servo, analysis of the thyroid gland, and design of an oil well drilling system. While the material in the first chapter involves subjects already known to the reader from previous experience, the text provides a coherent review. The emphasis here is on using rather than proving results.
1.2.1 Control System Terminology
The plant (process), inputs, and outputs are defined.
Controller and open-loop control are defined.
Control systems influence each facet of modem life. Automatic washers and dryers, microwave ovens, chemical processing plants, navigation and guidance systems, space satellites, pollution control, mass transit, and economic regulation are a few examples. In the broadest sense, a control system is any interconnection of components to provide a desired function. The portion of a system that is to be controlled is called the plant or the process. It is affected by applied signals, called inputs, and produces signals of particular interest, called outputs, as indicated in Figure 1.l(a). The plant is fixed insofar as the control system designer is concerned. Whether the plant is an automobile engine, an electri!=al generator, or a nuclear reactor, it is the designer's job to ensure that the plant operates as required. Other components must be specially created and connected as a means to an end. A controller may be used to produce a desired behavior of the plant, as shown in Figure 1.1(b). The controller generates plant input signals designed to produce
BASIC CONCEPTS
Inputs
Outputs
3
Desired Disturbance plant inputs behavior , . - - - - - - ,
(a)
(b)
Pump load Shalt speed
30·
(e)
60· 90· 120· Throttle position (angle)
ISO·
(d)
Figure 1.1 (a) A plant or process to be controlled. (b) An open-loop control system. (c) Example of an open-loop control system. (d) Engine speed versus throttle angle curves.
desired outputs. Some of the plant inputs are accessible to the designer and some are generally not available. The inaccessible input signals are often disturbances to the plant. The double lines in the figure indicate that several signals of each type may be involved. Arrows indicate direction of flow. This system is termed open-loop because the control inputs are not influenced by the plant outputs: that is, there is no feedback around the plant. Such an open-loop control system has the advantage of simplicity, but its performance is highly dependent upon the properties of the plant, which may vary with time. The disturbances to the plant may also create an unwanted response, which it would be desirable to reduce. As an example, suppose that a gasoline engine is used to drive a large pump, as depicted in Figure l.l(c). The carburetor and the engine comprise a common type of control system wherein a large-power output is controlled with a small-power input. The carburetor is the controller in this case, and the engine is the plant. The desired plant output, a certain engine shaft speed, may be obtained by adjusting the throttle angle. Two plots of engine speed versus throttle angle are shown in Figure 1.1 (d). If the nominal curve is used, a throttle angle of 80° produces an engine speed of 2300 rpm. Suppose that a disturbances occurs, consisting of a change in engine load. For the new curve, a throttle angle of 80° produces an engine speed of only 1000 rpm. In some cases open-loop control may be acceptable. In other cases, it may not be acceptable to have system output change when other values change. In these more critical cases, the closed-loop procedure of the next section may be needed. Table 1.1 shows five examples. The first two examples are for open-loop systems in that no measurements are taken to adjust controller influence on the plant. Each of the two controllers is specified when a manual setting is made of temperature and speed respectively. Hair dampness and the type of material being drilled are
Open-loop examples 'are presented.
4
CONTINUOUS-TIME SYSTEM DESCRIPTION
disturbances affecting desired performance. In these two cases, the user simply alters the total time until the job is done. In the case of the hair dryer, output air temperature remains constant while drying time for hair will vary according to wetness. In the case of the drill, output speed may vary while the drilling requirement remains constant. Figure l.l(b) describes these systems.
1.2.2 The Feedback Concept Closed-loop control is distinguished from open-loop control.
If the requirements of the system cannot be satisfied with an open-loop control system, a closed-loop or feedback system is desirable. A path (or loop) is provided from the output back to the controller. Some or all of the system outputs are measured and used by the controller, as indicated in Figure 1.2(a). The controller may then compare a desired plant output with the actual output and act to reduce the difference between the two. Let us return to Table 1.1 and consider the third and fourth examples. Temperatures and speed are the system outputs, as was the situation for the first two examples, but now measurements are used to keep the outputs constant in the presence of dis- , turbances. If outside temperatures drops, a thermostat determines that the room is becoming too cold. The thermostat causes furnace heat to increase which, in tum, causes the room temperature to increase to the predetermined value. Changes in driving conditions represent disturbances affecting an automobile's speed. One possible feedback control configuration is shown in Figure 1.2(b). A tachometer produces a voltage proportional to the engine shaft speed. The input voltage, which is proportional to the desired speed, is set with a potentiometer. The tachometer voltage is subtracted from the input voltage, giving an error voltage that is proportional to the difference between the actual speed and the desired speed. The error voltage is then amplified and used to position the throttle. The throttle actuator could be a reversible electric motor, geared to the throttle arm. When the engine shaft speed is equal to the desired speed (when the difference or error is zero), the throttle remains fixed. If a change in load or a change in the engine components
Table 1.1 Examples of Open-Loop and Closed-Loop Systems ~"l~y"q\'~~i-
"Q~'!tppt
Input
' Controller
Heat setting
Dial
Hair dryer
Hair dampness
Speed setting
Dial
Drill
Type of material
Desired temperature Desired speed
Thermostat
Furnace
Cruise control
Auto engine
Desired performance
Electorate
President
Outside temperature Driving conditions Economy
Hot air temperature Rotating drill bit speed Hot air temperature Car speed
None
Decisions
Evaluation
None Room temperature Enginerprn
