Design Examples and Design Problems (DP) CHAPTER I PAGE Example Hybrid Fuel Vehicles 21 22 Example Wind Power Example Embedded Computers 23 25 Example Rotating Disk Speed Control 27 Example Insulin Delivery Control System Example Disk Drive Read System 28 38 CDP1.1 Traction Drive Motor Control Automobile Noise Control 38 DP1.1 38 DP 1.2 Automobile Cruise Control Dairy Farm Automation 38 DP 1.3 Welder Control 38 DPI.4 38 Automobile Traction Control DP1.5 39 Hubble Telescope Vibration DP1.6 Nanorobotics in Medicine 39 DPI.7 CHAPTER Example Fluid Flow Modeling Example Electric Traction Motor Control Example Mechanical Accelerometer Example Laboratory Robot Example Low-Pass Filter Example Disk Drive Read System CDP2.1 Traction Drive Motor Control DP2.1 Selection of Transfer Functions DP2.2 Television Beam Circuit DP2.3 Transfer Function Determination DP2.4 Op Amp Differentiating Circuit CHAPTER Example Modeling the Orientation of a Space Station Example Printer Bell Drive Example Disk Drive Read System CDP3.1 Traction Drive Motor Control DP3.1 Shock Absorber for Motorcycle DP3.2 Diagonal Matrix Differential Equation DP3.3 Aircraft Arresting Gear DP3.4 Bungi Jumping System DP3.5 State Variable Feedback CHAPTER Example English Channel Boring Machines Example Mars Rover Vehicle Example Blood Pressure Control Example Disk Drive Read System CDP4.1 Traction Drive Motor Control 83 93 95 98 99 117 139 139 139 139 139 176 183 192 21)8 208 209 209 209 209 232 235 237 251 270 DP4.1 DP4.2 DP4.3 DP4.4 DP4.5 DP4.6 Speed Control System Airplane Roll Angle Control Velocity Control System Laser Eye Surgery Pulse Generating Op Amp Hvdrobot 270 271 271 271 272 272 CHAPTER Example Hubble Telescope Pointing Example Attitude Control of an Airplane Example Disk Drive Read System CDP5.1 Traction Drive Motor Control DP5.1 Jet Fighter Roll Angle Control DP5.2 Welding Arm Position Control DP5.3 Automobile Active Suspension DP5.4 Satellite Orientation Control DP5.5 De-burring Robot for Machined Parts DP5.6 DC Motor Position Control 350 351 CHAPTER Example Tracked Vehicle Turning Example Robot-Controlled Motorcycle Example Disk Drive Read System CDP6.1 Traction Drive Motor Control DP6.1 Automobile Ignition Control DP6.2 Mars Guided Vehicle Control DP6.3 Parameter Selection DP6.4 Space Shuttle Rocket DP6.5 Traffic Control System DP6.6 State Variable Feedback DP6/7 Inner and Outer Loop Control DP6.8 PD Controller Design 373 375 390 402 402 403 403 403 403 403 404 404 CHAPTER Example Laser Manipulator Control Example Robot Control System Example Automobile Velocity Control Example Disk Drive Read System CDP7.1 Traction Drive Motor Control DP7.1 Pitch Rate Aircraft Control DP7.2 Helicopter Velocity Control DP7.3 Mars Rover DP7.4 Remotely Controlled Welder DP7.5 ' High-Performancc Jet Aircraft DP7.6 Control of Walking Motion DP7.7 OP Amp Control System DP7.8 Robot Arm Elbow Joint Actuator DP7.9 Four-Wheel-Steered Automobile 316 319 333 349 349 349 349 350 447 448 452 463 485 485 485 486 486 486 486 487 487 487 DP7.10 DP7.11 DP7.12 DP7.13 Pilot Crane Control Planetary Rover Vehicle Roll Angle Aircraft Autopilot PD Control of a Marginally Stable Process CHAPTER Example Engraving Machine Control Example Control of a Six-Legged Robot Example Disk Drive Read System CDP8.1 Traction Drive Motor Control DP8.1 Automobile Steering System DP8.2 Autonomous Planetary Explorer-Ambler DP8.3 Vial Position Control Under a Dispenser DP8.4 Automatic Anesthesia Control DP8.5 Black Box Control DP8.6 State Variable System Design CHAPTER Example Remotely Controlled Reconnaissance Vehicle Example Hot Ingot Robot Control Example Disk Drive Read System CDP9.1 Traction Drive Motor Control DP9.1 Mobile Robot for Toxic Waste Cleanup DP9.2 Control of a Flexible Arm DP9.3 Blood Pressure Regulator DP9.4 Robot Tennis Player DP9.5 Electrohydraulic Actuator DP9.6 Steel Strip-Rolling Mill DP9.7 Lunar Vehicle Control DP9.8 High-Speed Steel-Rolling Mill DP9.9 Two-Tank Temperature Control DP9.10 State Variable Feedback Control CHAPTER 10 Example Rotor Winder Control System Example The X-Y Plotter Example Milling Machine Control System Example Disk Drive Read System CDP10.1 Traction Drive Motor Control DP10.1 Two Cooperating Robots DPI 0.2 Heading Control of a Bi-Wing Aircraft DP10.3 Mast Flight System DP10.4 Robot Control Using Vision DP10.5 High-Speed Train Tilt Control DP10.6 Large Antenna Control DPI 0.7 Tape Transport Speed Control DP10,8 Automobile Engine Control DP10.9 Aircraft Roll Angle Control 488 488 489 DP10.10 Windmill Radiometer DP10.11 Control with Time Delay DP10.12 Loop Shaping 751 752 752 489 CHAPTER 11 Example Automatic Test System Example Diesel Electric Locomotive Example Disk Drive Read System CDP11.1 Traction Drive Motor Control DPI LI Levitation of a Steel Ball DPI 1.2 Automobile Carburetor DPI 1.3 Sta te Variable Compensation DP11.4 Helicopter Control DP1L5 Manufacturing of Paper DPI 1.6 Coupled-Drive Control DPI 1.7 Tracking a Reference Input 795 798 810 821 821 821 821 822 822 823 823 523 526 540 561 561 561 561 561 563 563 607 610 629 659 659 659 659 659 659 659 662 662 663 707 711 714 726 747 747 747 747 749 749 749 750 750 751 CHAPTER 12 Example Aircraft Autopilot Example Space Telescope Control Example Robust Bobbin Drive Example Ultra-Precision Diamond Turning Machine Example Digital Audio Tape Controller Example Disk Drive Read System CDP12.1 Traction Drive Motor Control DP12.1 Turntable Position Control DP12.2 Robust Parameter Design DP12.3 Dexterous Hand Master DP12.4 Microscope Control DP12.5 Microscope Control DP12.6 Artificial Control of Leg Articulation DP 12.7 Elevator Position Control DP12.8 Electric Ventricular Assist Device DP12.9 Space Robot Control DP12.10 Solar Panel Pointing Control DP12.11 Magnetically Levitated Train DP12,12 Mars Guided Vehicle Control DP12.13 Benchmark Mass-Spring CHAPTER 13 Example Worktable Motion Control Example Fly-by-wire Aircraft Control Example Disk Drive Read System CDP13.1 Traction Drive Motor Control DP13.1 Temperature Control System DP13.2 Disk Drive Read-Write HeadPositioning System DP13.3 Vehicle Traction Control DP13.4 Machine-Tool System DP13.5 Polymer Extruder Control DP13.6 Sampled-Data System 853 853 856 858 861 876 891 891 891 891 892 893 893 894 894 895 896 896 896 896 926 928 940 947 947 947 947 948 948 948 Modern Control Systems ELEVENTH EDITION Richard C Dorf University of California, Davis Robert H Bishop The University of Texas at Austin Pearson Education International If you purchased this book within the United States or Canada you should be aware that it has been wrongfully imported without the approval of the Publisher or the Author Vice President and Editorial Director, ECS: Marcia L Horton Acquistions Editor: Michael McDonald Senior Managing Editor: Scott Disanno Senior Production Editor: Irwin Zucker Art Editor: Greg Dulles Manufacturing Manager: Alexis Heydt-Long Manufacturing Buyer: Lisa McDowell Senior Marketing Manager: Tim Galligan © 2008 Pearson Education, Inc Pearson Prentice Hall Pearson Education, Inc Upper Saddle River, NJ 07458 All rights reserved No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Pearson Prentice Hall® is a trademark of Pearson Education, Ina MATLAB is a registered trademark of The Math Works, Inc., 24 Prime Park Way, Natick, MA 01760-1520 The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs Printed in Singapore 10 ISBN 0-13-20L710-2 ^-0-13-201,710-2 Pearson Education Ltd., London Pearson Education Australia Pty Ltd., Sydney Pearson Education Singapore, Pte Ltd Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educacion de Mexico, S.A de C.V Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte Ltd Pearson Education, Inc., Upper Saddle River, New Jersey Of the greater teachers— when they are gone, their students will say: we did it ourselves Dedicated to Lynda Ferrera Bishop and Joy MacDonald Dorf In grateful appreciation Contents Preface xiii About the Authors CHAPTER Introduction to Control Systems 1.1 1.2 1.3 1.4 1.5 1.6 1-7 1.8 1.9 1.10 CHAPTER xxv Introduction Brief History of Automatic Control Examples of Control Systems Engineering Design 16 Control System Design 17 Mechatronic Systems 20 The Future Evolution of Control Systems 24 Design Examples 25 Sequential Design Example: Disk Drive Read System 28 Summary 30 Exercises 30 Problems 31 Advanced Problems 36 Design Problems 38 Terms and Concepts 39 Mathematical Models of Systems 41 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Introduction 42 Differential Equations of Physical Systems 42 Linear Approximations of Physical Systems 47 The Laplace Transform 50 The Transfer Function of Linear Systems 57 Block Diagram Models 71 Signal-Flow Graph Models 76 Design Examples 82 The Simulation of Systems Using Control Design Software 102 Sequential Design Example: Disk Drive Read System 117 Summary 119 Exercises 120 Problems 126 Advanced Problems 137 Design Problems 139 Computer Problems 140 Terms and Concepts 142 v vi CHAPTER Contents State Variable Models 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 CHAPTER Introduction 145 The State Variables of a Dynamic System 145 The State Differential Equation 149 Signal-Flow Graph and Block Diagram Models 154 Alternative Signal-Flow Graph and Block Diagram Models 165 The Transfer Function from the State Equation 170 The Time Response and the State Transition Matrix 172 Design Examples 176 Analysis of State Variable Models Using Control Design Software Sequential Design Example: Disk Drive Read System 192 Summary 196 Exercises 197 Problems 199 Advanced Problems 207 Design Problems 208 Computer Problems 210 Terms and Concepts 211 189 Feedback Control System Characteristics 212 41 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 CHAPTER 144 Introduction 213 Error Signal Analysis 215 Sensitivity of Control Systems to Parameter Variations 217 Disturbance Signals in a Feedback Control System 220 Control of the Transient Response 225 Steady-State Error 228 The Cost of Feedback 231 Design Examples 232 Control System Characteristics Using Control Design Software 246 Sequential Design Example: Disk Drive Read System 251 Summary 255 Exercises 257 Problems 261 Advanced Problems 267 Design Problems 270 Computer Problems 273 Terms and Concepts 276 The Performance of Feedback Control Systems 5.1 5.2 5.3 Introduction 278 Test Input Signals 278 Performance of Second-Order Systems 281 277 Contents 5.4 5.5 5.6 5*7 5.8 5.9 5.10 5.11 5.12 CHAPTER Effects of a Third Pole and a Zero on the Second-Order System Response 287 The s-Plane Root Location and the Transient Response 293 The Steady-State Error of Feedback Control Systems 295 Performance Indices 303 The Simplification of Linear Systems 312 Design Examples 315 System Performance Using Control Design Software 329 Sequential Design Example: Disk Drive Read System 333 Summary 337 Exercises 337 Problems 341 Advanced Problems 346 Design Problems 348 Computer Problems 350 Terms and Concepts 353 The Stability of Linear Feedback Systems 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 CHAPTER VII The Concept of Stability 356 The Routh-Hurwitz Stability Criterion 360 The Relative Stability of Feedback Control Systems 368 The Stability of State Variable Systems 370 Design Examples 373 System Stability Using Control Design Software 382 Sequential Design Example: Disk Drive Read System 390 Summary 393 Exercises 394 Problems 396 Advanced Problems 400 Design Problems 402 Computer Problems 404 Terms and Concepts 406 The Root Locus Method 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 355 407 Introduction 408 The Root Locus Concept 408 The Root Locus Procedure 413 Parameter Design by the Root Locus Method 431 Sensitivity and the Root Locus 437 Three-Term (PID) Controllers 444 Design Examples 447 The Root Locus Using Control Design Software 458 Sequential Design Example: Disk Drive Read System 463 viii Contents 7*10 CHAPTER Frequency Response Methods 8.1 8.2 83 8.4 8.5 8.6 8.7 8.8 8.9 CHAPTER Summary 465 Exercises 469 Problems 472 Advanced Problems 482 Design Problems 485 Computer Problems 490 Terms and Concepts 492 493 Introduction 494 Frequency Response Plots 496 Frequency Response Measurements 517 Performance Specifications in the Frequency Domain 519 Log Magnitude and Phase Diagrams 522 Design Examples 523 Frequency Response Methods Using Control Design Software Sequential Design Example: Disk Drive Read System 540 Summary 541 Exercises 546 Problems 549 Advanced Problems 558 Design Problems 560 Computer Problems 564 Terms and Concepts 566 Stability in the Frequency Domain 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 534 567 Introduction 568 Mapping Contours in the s-Plane 569 The Nyquist Criterion 575 Relative Stability and the Nyquist Criterion 586 Time-Domain Performance Criteria in the Frequency Domain 594 System Bandwidth 601 The Stability of Control Systems with Time Delays 601 Design Examples 606 PID Controllers in the Frequency Domain 620 Stability in the Frequency Domain Using Control Design Software 621 Sequential Design Example: Disk Drive Read System 629 Summary 632 Exercises 640 Problems 646 Advanced Problems 656 Design Problems 659 Computer Problems 664 Terms and Concepts 665 Index Absolute stability, A system description that reveals whether a system is stable or not stable without consideration of other system attributes such as degree of stability, 356,406 Acceleration error constant, Ka The constant evaluated as lim[j ,2 G(5)l The sreadys-*0 state error for a parabolic input, r(t) = At1/!, is equal toA/Ka, 298 Acceleration input, steady-state error, 297-298 Accelerometer, 71,83 Ackermann's formula, 756, 767-768,772,777-778, 809-810,816 Across-variable, 43,45 Actuator, The device that causes the process to provide the output; the device that provides the motive power to the process, 62,142 Additive perturbation, A system perturbation model expressed in the additive form Ga{s) = G(s) + A(s) where G(s) is the nominal plant, A(s) is the perturbation that is bounded in magnitude, and Ca(s) is the family of perturbed plants, 834,899 Agricultural systems, 13 Aircraft, and computer-aided design, 19 unmanned, 15 Aircraft autopilot, 853 Aircraft attitude control, 319 Airplane control, 266,474-475, 482,747-748 All-pass network, A nonminimum phase system that passes all frequencies with equal gain, 513-514,566 Alternative signal flow graph, and block diagram models, 165-170 Amplidyne, 127 Amplifier, feedback, 219 Amplitude quantization error, 906-907, 950 Analogous variables, 47 Analog-to-digital converter, 902,906 Analysis of robustness, 834-836 Angle of departure, The angle at which a locus leaves a complex pole in the s-plane, 422-423,426,441^143,491 Angle of the asymptotes, The angle that the asymptote makes with respect to the real axis,4>A, 415,418,491 Armature-controlled motor, 64-65,69,81,94,117,127,137, 139 Array operations in MathScript, 979 Array operations in MATLAB, 959-960 Artificial hand, 11,14,36 Assumptions, Statements that reflect situations and conditions that are taken for granted and without proof In control systems, assumptions are often employed to simplify the physical dynamical models of systems under consideration to make the control design problem more tractable, 42,83-84,142 Asymptote, The path the root locus follows as the parameter becomes very large and approaches infinity, 415 of root locus, 415 Asymptote centroid, The center of the linear asymptotes, 416 aA, Asymptotic approximation for a Bode diagram, 502 Automatic control, history of, 4-8 Automatic fluid dispenser, 200,202 Automatic test system, 795-797 Automation, The control of an industrial process by automatic means, 6,39 Automobile steering control system, Automobiles, hybrid fuel vehicles, 21, 40 Auxiliary polynomial, The equation that immediately precedes the zero entry in the Routh array, 365,496 Avemar ferry hydrofoil, 736 Axis shift, 369 Backward difference rule, A computational method of approximating the time derivative of a function given by x(kT) x(kT) - x((k l)r) T where t = kT,T is the sample time, and k = 1,2, , 925,950 Bandwidth, The frequency at which the frequency response has declined dB from its low-frequency value, 520, 566,596,665 Bellman, R., Biological control system, 14 Black, H.S., 5-6,8,130,830 Block diagram, Unidirectional, operational block that represents the transfer 1007 1008 Index functions of the elements of the system, 71,72 Block diagram models, 71-76, 107-116 alternative signal-flow graphs, 165-170 signal-flow graphs, 154-165 Block diagram transformations, 73-74 Bobbin drive, 356 Bode,H.W., 500,830 Bode plot, The logarithm of magnitude of the transfer function is plotted versus the logarithm of o>, the frequency The phase, cj>, of the transfer function is separately plotted versus the logarithm of the frequency, 500-501, 541,567 asymptotic approximation, 502 Boring machine system, 232 Bounded response, 356 Branch on signal-flow graph, 76 Break frequency, The frequency at which the asymptotic approximation of the frequency response for a pole (or zero) changes slope, 502, 505,566 Breakaway point The point on the real axis where the locus departs from the real axis of the 5-plane, 418-420,491 Bridge,Tacoma Narrows, 357-359 Camera control, 308-312,341 Canonical form, A fundamental or basic form of the state variable model representation, including phase variable canonical form, input feedforward canonical form, diagonal canonical form, and Jordan canonical form, 211 Capek, Karel, 10 Cascade compensation network, A compensator network placed in cascade or series with the system process, 671-675,755 Cauchy's theorem If a contour encircles Z zeros and P poles of F( s), the corresponding contour encircles the origin of the F(s)-plane N = Z - P times clockwise, 568, 571-575,665 Characteristic equation, The relation formed by equating to zero the denominator of a transfer function, 52,142,387 Circles, constant, 596 Closed-loop feedback control system, A system that uses a measurement of the output and compares it with the desired output, 3,39,214 Closed-loop sampled-data system, 912 Closed-loop transfer function, A ratio of the output signal to the input signal for an interconnection of systems when all the feedback or feedfoward loops have been closed or otherwise accounted for Generally obtained by block diagram or signal flow graph reduction, 74,142, 387 388 Command following A n important aspect of control system design wherein a nonzero reference input is tracked, 779, 826 Compensation, The alteration or adjustment of a control system to provide a suitable performance, 668 using a phase-lag network on the Bode diagram, 691 using a phase-lag network on the s-plane, 692 using a phase-lead network on the Bode diagram, 675 using a phase-lead network on the s-plane, 681 using analytical methods, 700 using integration networks, 688 using state-variable feedback, 757 Compensator, An additional component or circuit that is inserted into the system to equalize or compensate for the performance deficiency, 477,668,755.757 Compensator design, full-state feedback and observer, 773 Complementary sensitivity function, The function C(s) = w „ , , ^ , that + Gc(s)G(s) satisfies the relationship S(s) + C(s) = 1, where 5(5) is the sensitivity function The function C(s) = T(s) is the closed-loop transfer function, 216,834,899 Complexity, A measure of the structure, intricateness, or behavior of a system that characterizes the relationships and interactions between various components, 16,39,276 in cost of feedback, 231 Complexity of design The intricate pattern of interwoven parts and knowledge required, 16 Components, The parts, subsystems, or subassemblies that comprise a total system, 276 in cost of feedback, 231 Computer control systems, 901,902 for electric power plant, 13 Computer-aided design, 19 Computer-aided engineering (CAE), 21 Conditionally stable system, 475 Conformal mapping, A contour mapping that retains the angles on the s-plane on the F(s)-plane, 570,655 Congress, 14 Constant M circles, 597 Constant N circles, 597 1009 Index Continuous design problem, 38, 139,208,270,349,402,485, 561,659,747,821,891,947 Contour map, A contour or trajectory in one plane is mapped into another plane by a relation F(s), 569 Contours in the s-plane, 569 Control engineering, 2,8-9 Control system An interconnection of components forming a system configuration that will provide a desired response, 2,39 characteristics using m-files, 246 design, 17 modern examples, 8-16 Controllability, 757-763 Controllability matrix, A linear system is (completely) controllable if and only if the controllability matrix P c = [B AB A B A"B] has full rank, where A is an nxn matrix; for single-input, single-output linear systems, the system is controllable if and only if the determinant of the nxn controllability matrix P c is nonzero, 758,826 Controllable system, A system with unconstrained control input u that transfers any initial state x(0) to any other state x(r), 758,826 conv function, 105,968 Convolution integral, 280 Corner frequency See Break frequency Cost of feedback, 231-232 Coulomb damper, 45 Critical damping, The case where damping is on the boundary between underdamped and overdamped, 54,142 Critically damped system, 103 Damped oscillation An oscillation in which the amplitude decreases with time, 56,142 Dampers, 45 Damping ratio, A measure of damping; a dimensionless number for the second-order characteristic equation, 54, 142,292 estimation of, 292 DC motor, An electric actuator that uses an input voltage as a control variable, 142 armature controlled, 64,81 field controlled, 63 Deadbeat response, 755 Decade, A factor of ten in frequency (e.g., the range of frequencies from rad/sec to 10 rad/sec is one decade), 502,566 of frequencies, 502 Decibel (dB), The units of the logarithmic gain, 566 Decoupled state variable format, 166 Design, The process of conceiving or inventing the forms, parts, and details of a system to achieve a reasoned purpose, 16-17,39 Design of a control system, The arrangement or the plan of the system structure and the selection of suitable components and parameters, 755 robot control, 396 in the time domain, 757 using a phase-lag network on the Bode diagram, 696 using a phase-lag network on the s-plane, 691 using a phase-lead network on the Bode diagram, 675 using a phase-lead network on the s-plane, 681 using integration networks, 688 using state-feedback, 756 Design specifications, 278 Detectable, A system in which the states that are unobservable are naturally stable., 761,826 Diagonal canonical form, A decoupled canonical form displaying the n distinct system poles on the diagonal of the state variable representation A matrix, 166,211 Differential equations, An equation including differentials of a function, 42,143 Differential operator, 50 Differentiating circuit, 68 Digital computer compensator, A system that uses a digital computer as the compensator element, 918-921 Digital control system, A control system using digital signals and a digital computer to control a process, 901-950 Digital control systems using control design software, 935 Digital controllers, implementation of, 925 Digital to analog converter, 905 Direct system, 213 See also Open-loop control system Discrete-time approximation, An approximation used to obtain the time response of a system based on the division of the time into small increments A ; , 211 Disk drive read system See Sequential design example Disturbance rejection property, 221-224 Disturbance signal An unwanted input signal that affects the system's output signal, 220-225,276 Dominant roots, The roots of the characteristic equation that represent or dominate the closed-loop transient response, 288,353, 427,491,521,566 Dynamics of physical systems, 41 Electric power industry, Electric traction motor, 13 93-95 • 1010 Index Electrohydraulic actuator, 66, 69,129 Engineering design, The process of designing a technical system, 16-17,40 English channel tunnel boring system, 232 Engraving machine 523-526, 537-538 Epidemic disease, model of, 167-168,372 Equilibrium state, 167 Error, steady-state, 228 Error constants, acceleration, 296 ramp, 297 step, 295 Error signal, The difference between the desired output, R{s), and the actual output Y(s); therefore E(s) = R(s) - Y(s), 110,143,276 Estimation error The difference between the actual state and the estimated state e(r) = x(t) - x(t), 769,826 Euler's method, A first-order explicit integration method utilized to obtain numerical solutions of differential equations, 211 Evans, W.R., 408 Examples of control systems, Exponential matrix function, 150 Extender, 135,206,742 Federal Reserve Board, 14 Feedback, amplifier, 219 control system, 3,9-11, 720-726 cost of, 231-232 full-state control design, 763 negative, 3,6 positive, 32 of state variables, 782,784 Feedback control system, and disturbance signals, 220-225 feedback function, 111-113,968 Feedback signal, A measure of the output of the system used as feedback to control the system, 3,40,110 Feedback systems, history of, Final value The value that the output achieves after all the transient constituents of the response have faded Also referred to as the steady-state value, 54 of response of y(t), 54 Final value theorem, The theorem that states that lim y(t) t—*co = lim 5Y(5), where Y(s) is the Laplace transform of y ( , 54 Flow graph See Signal-flow graph Flyball governor, A mechanical device for controlling the speed of a steam engine, 4-5,40 Forward rectangular integration, A computational method of approximating the integration of a function given by x(kT)**x({k-l)T) + Tx((k-1)T), where t = kT, T is the sample time, and k = 1,2, , 925,950 Fourier transform The transformation of a function of time, / ( f ) into the frequency domain, 496 Fourier transform pair, A pair of functions, one in the time domain, denoted by f(t), and the other in the frequency domain, denoted by F{j Yw) s(s + 2^,,) — CLOSED-LOOP MAGNITUDE PLOT UNIT STEP RESPONSE Overshoot -• Z)\ogMpM ^ '/; Rise time Pea t me ^ * Settling time (to within % of the final value) J Percent overshoot J + e -ivP/l-f J and = P J Maximum magnitude (f s 0.7) pto = io&r^' Vw2 Time-to-peak r C0n Settling time J M„ 0), Time J Resonant frequency {£ :£ 0.7) &), = w„Vl - 2£2 *„Vi - f2 Rise time (time to rise from 10% to 90% of final value) J Bandwidth (0.3 < f < 0.8) a>fi = (-1.19« + 1.85K ^ = 2,16f + 0.60 (0.3 < £ < 0.8) P1D Controller: < « * ) = = K r + KDs *i + s U + Zi)(S + 22) s TABLE PAGE 5.5 Summary of Steady-State Errors 298 5.6 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Step Input 308 5.7 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Ramp Input 312 10.2 Coefficients and Response Measures of a Deadbeat System 706 10.7 A Summary of the Characterislics of Phase-Lead and Phase-Lag Compensation Networks 729 ... understood systems such as chemical process systems The present challenge to control engineers is the modeling and control of modern, complex, interrelated systems such as traffic control systems, ... navigation, and control of aerospace vehicles XXV Introduction to Control Systems 1.1 Introduction 1.2 Brief History of Automatic Control 1.3 Examples of Control Systems 1.4 Engineering Design 1.5 Control. .. control Most physiological control systems are closed-loop systems However, we find not one controller but rather control loop within control loop, forming a hierarchy of systems The modeling of