CONTROL SYSTEM FUNDAMENTALS edited by William S Levine Library of Congress Cataloging-in-Publication Data Levine, W S Control system fundamentals / William S Levine p em Includes bibliographical references ISBN 0-8493-0053-3 (alk paper) I Automatic control Control theory I Title TJ213.L419 1999 629.8-{\c21 99-048036 CIP This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe The material in this book was taken from, The Control Handbook Edited by William S Levine © 2000 by CRC Press LLC No claim to original U.S Government works International Standard Book Number 0-8493-0053-3 Library of Congress Card Number 99-048036 890 Printed in the United States of America Printed on acid-free paper Preface Control technology is remarkably varied; control system implementations range from float valves to microprocessors Control system fundamentals include regulating the amount of water in a toilet tank, controlling the flow and generation of electrical power over huge geographic regions, regulating the behavior of gasoline engines, controlling the thickness of rolled products as varied as paper and sheet steel, and hundreds of controllers hidden in consumer products of all kinds The different applications often require unique sensors and actuators, all of which call for an understanding of the fundamentals of control systems The fundamentals are just what the name implies: the basics of control engineering Note that this book includes major sections on digital control and modeling of dynamical systems There are also chapters on specification of control system nonlinearities, and digital implementation of control systems Contributors Anders Ahlen Richard C Dorf Christopher Department of Technology Uppsala University Uppsala, Sweden University of California Davis, California Department of Electrical and Electronic Engineering University of Wales Swansea, Singleton Park, Wales Hilding Elmqvist Karl J Astrom Department of Automatic Control Lund Institute of Technology Lund, Sweden Dynasim AB Research Park Ideon Lund, Sweden A Feuer Derek P Atherton School of Engineering University of Sussex Brighton, England Electrical Engineering Department Technion-Israel Institute ofTechnoloc1 Haifa, Israel Dean K Frederick P Jobling Jason C Jones Mrcbanical Engineering Department UniYersity of California Bertdey, California Edward W Kamen Sd100I of Electrical and Computer £illineering Georgia Institute of Technology Adanta, Georgia Mechanical Engineering Department University of California Berkeley,California Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute Troy,NY Robert H Bishop James T Gillis California State University Saaamento, California University of Texas Austin, Texas The Aerospace Corporation Los Angeles, California A David M Auslander Richard D Braatz G.C Goodwin University of Illinois Department of Chemical Engineering Urbana, Illinois Department of Electrical and Computer Engineering University of Newcastle Newcastle, NSW, Australia Fran~ois E Cellier Stefan F Graebe Department of Electrical and Computer Engineering University of Arizona Thcson, Arizona PROFACTOR GmbH Steyr, Austria Charles M Close The Aerospace Corporation El Segundo, California J Laub Department of Electrical and Computer Engineering University of California Santa Barbara, California William S Levine Department of Electrical Engineering University of Maryland College Park, Maryland Mohamed C.W Gray Electrical, Computer and Systems Engineering Department RensseIaer polytechnic Institute Troy, New York B P Lathi Tore Hagglund Department of Automatic Control Lund Institute of Technology Lund, Sweden Mansour Swiss Federal Institute of Technology (ETH) Zurich, Switzerland R H Middleton Gene H Hostetter Department of Electrical and Computer Engineering University of Newcastle Newcastle, NSW, Australia Bradley W Dickinson Constantine Norman Princeton U~ Princeton, New Jersey Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio John J D' Azzo Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio H Houpis S Nise California State Polytechnic Uoiftnily Pomona, California Katsuhiko Ogata john R Ridgely Raymond T Stefani University of Minnesota Minneapolis, Minnesota Mechanical Engineering Department University of California Berkeley,California Electrical Engineering Department California State University Long Beach, California C • Magnus Rimv all All en R St u bb eru d Gustaf Olsson Department of Industrial Electrical ' Engmeenng an dA utomation Lund Institute of Technology Lund, Sweden ABB Corporate Research and Development Heidelberg, Germany Martin Otter P M Van Dooren Institute for Robotics and System Dynamics German Aerospace Research Establishment Oberpfaffenhoren (DLR) Wesseling, Germany Z.j Palmor Umverslty of Callforma Irvine, California Faculty of Mechanical Engmeenng Technion-Israel Institute of Technology Haifa, Israel R V Patel Department of Electrical and Computer Engineering Concordia University Montreal, Quebec, Canada Charles E Rohrs Tellabs Mishawaka, Indiana Department of Mathematical Engineering Universite Catholique Louvain, Belgium Mohammed S Santina William A Wolovich The Aerospace Corporation Los Angeles California , Brown University Providence Rhode Island , Jeff S Shamma jiann-Shiou Yang Center for Control and Systems Research Department of Aerospace Engineering and Engineering Mechanics University of Texas Austin, Texas Department of Electrical and Computer Engineering University of Minnesota Duluth, Minnesota Contents SECTION I Mathematical Foundations Mathematical Foundations 1.1 1.2 Edward vv Kamen Introduction Fundamentals of the Fourier, Laplace, and z-Transforms Applications and Examples Matrices and Linear Algebra Bradley Complex Variables 4.1 4.2 4.3 4.4 SECTION II 17 - 33 33 33 37 42 43 45 46 47 -.• - • 51 65 • .• • 85 51 53 57 60 Input-Output Models William S Levine State Space James T Gillis Block Diagrams Dean K Frederick and Charles M Close Signal-Flow Graphs Norman S Nise • .• • • • • Modeling from Physical Principles Fram;ois E Cellier, Hilding Elmqvist tmtl MmMo.r _ System Identification When Noise Is Negligible William S Levine 65 72 85 93 Determining Models 7.1 7.2 17 17 25 Models for Dynamical Systems Graphical Models 6.1 6.2 vv Gray Standard Mathematical Models 5.1 5.2 C Complex Numbers Complex Functions Complex Integrals Conformal Mappings .• .• • • • .• vv Dickinson 3.1 Introduction 3.2 Matrices 3.3 Vector Spaces 3.4 Linear Equations 3.5 Eigenvalues and Eigenvectors 3.6 The Jordan Form and Similarity of Matrices 3.7 Singular Value Decomposition 3.8 Matrices and Multivariable Functions 3 The Fourier, Laplace, and z-Transforms 2.1 2.2 2.3 B.P Lathi Differential Equations Difference Equations 99 99 108 SECTION III Analysis and Design Methods for Continous- Time Systems Analysis Methods 8.1 8.2 Time Response of Linear Time-Invariant Systems Raymond T Suifani Controllability and Observability William A W%vich - - - 115 115 121 Stability Tests 9.1 9.2 9.3 9.4 131 The Routh-Hurwitz Stability Criterion Robert H Bishop and Richard C Dorf The Nyquist Stability Test Charles E Rohrs Discrete-Time and Sampled-Data Stability Tests Mohamed Mansour Gain Margin and Phase Margin Raymond T Stefani 10 Design Methods _ 157 10.1 Specification of Control Systems Jiann-Shiou Yang and WiI1imn S Levine 10.2 Design Using Performance Indices Richard C Dorf and Robert H Bishop 10.3 Nyquist, Bode, and Nichols Plots John] D'Azzo and Constmrtine H Houpis 10.4 The Root Locus Plot William S Levine 10.5 PID Control Karl] Astrom and Tore Hiigglund 10.6 State Space - Pole Placement Katsuhiko Ogata 10.7 Internal Model Control Richard D Braatz 10.8 Time-Delay Compensation - Smith Predictor and its Modifications z.] Palmor 10.9 Modified Predictors Richard D Braatz 10.10Time- Delay Compensation for Unstable Plants Z.] Palmor SECTION IV 158 169 173 192 198 209 215 224 231 235 Digital Control 11 Discrete-Time Systems Mohammed S Santina, Allen R StubbmuI and Gene H Hostetter 11.1 Discrete-Time Systems 12 Sampled Data Systems 12.1 12.2 12.3 12.4 12.5 131 135 146 152 A Feuer and G C Goodwin Introduction and Mathematical Preliminaries 'Sensitivity Functions' in Sampled Data Control Systems Sensitivity Consideration Examples Linear Quadratic Design of Sampled- Data Controllers 13 Discrete-Time Equivalents to Continuous-Time Systems Gene H Hostetter 241 241 255 255 256 258 259 262 MoIummred S Santina, Allen R Stubberud and 267 13.1 Introduction 13.2 Digitizing Analog Controllers 13.3 Discretization of Continuous- Time State Variable Models 14 Design Methods for Discrete-Time, Linear Time- Invariant Systems 267 267 275 Mohammed S Santina, Allen R Stub- berud and Gene H Hostetter 283 14.1 14.2 14.3 14.4 14.5 14.6 283 284 289 292 293 296 An Overview Classical Control System Design Methods Eigenvalue Placement with State Feedback Step-Invariant Discrete-Time Observer Design Tracking System Design Designing Between-Sample Response 15 Quantization Effects 15.1 15.2 15.3 15.4 15.5 15.6 15.7 Mohammed S Santina, Allen R Stubberud and Gene H Hostetter Overview Fixed-Point and Floating-Point Number Representations Truncation and Roundoff Effects of Coefficient Quantization Quantization Effects in AID Conversion Stochastic Analysis of Quantization Errors in Digital Processing Limit Cycle and Deadband Effects 303 303 303 305 306 308 309 312 16 Sample-Rate Selection 16.1 16.2 16.3 16.4 16.5 16.6 17 Mohammed S Santina, Allen R Stubberud and Gene H Hostetter 315 315 317 319 321 321 Introduction The Sampling Theorem Control System Response and the Sampling Period Control System Response to External Disturbances Measurement Noise and Prefiltering Effect of Sampling Rate on Quantization Error Real-Time Software for Implementation of Feedback Control David M Auslander, 325 John R Ridgely, and Jason C Jones 325 325 326 326 327 327 328 330 332 333 334 336 344 17.1 An Application Context 17.2 The Software Hierarchy 17.3 The Ground Rules 17.4 Portability 17.5 Software Structure: Scan Mode 17.6 Control Software Design 17.7 Design Formalism 17.8 Scheduling 17.9 Task Type Preferences 17.10 Intertask Communication 17.11Prototype Platform 17.12 Program Structure: The Anatomy of a Control Program 17.13Sample Problems and Utilities 18 Programmable Controllers 18.1 18.2 18.3 18.4 18.5 18.6 18.7 19 20 347 Gustaf Olsson 347 348 349 353 355 357 361 The Development of Programmable Controllers Binary Sensors and Actuators Elementary Switching Theory Ladder Diagrams Programmable Controllers PLC Programming PLCs as Part of Automation Systems SECTION V Analysis and Design Methods for Nonlinear Systems Analysis Methods 367 Derek P Atherton 19.1 The Describing Function Method 19.2 The Phase Plane Method 367 376 Design Methods 381 381 385 392 R H Middkton 20.1 Dealing with Actuator Saturation Stefan F Graebe and Anders Ahlin 20.2 Bumpless Transfer Jeff S Shamnul 20.3 Linearization and Gain-Scheduling SECTION VI 21 315 Software for Control System Analysis and Design Numerical and Computational Issues in Linear Control and System Theory R V Patel, and P.M Van Dooren 21.1 21.2 21.3 21.4 21.5 Introduction Numerical Background Fundamental Problems in Numerical Linear Algebra Applications to Systems and Control Mathematical Software A.] Laub, 403 403 405 407 410 416 21.6 Concluding Remarks 22 Software for Modeling and Simulating Control Systems 22.1 22.2 22.3 22.4 22.5 Martin Otter and Franfois E Cellier Introduction Special Demands of Control Engineers for a Simulation Tool Overview of Modeling and Simulation Software Shortcomings of Current Simulation Software Conclusions • • • • • • • • • • 23 Computer-Aided Control Systems Design 23.1 23.2 23.3 23.4 23.5 417 Introduction A Brief History of CACSD The State of the Art in CACSD CACSD Block Diagram Tools Conclusions Index C Magnus Rimvall and Christopher P.fabling 419 419 420 422 431 431 433 433 433 435 440 445 447 452 Cauchy-Riemann differential, 53 dllIerence, 9,29, 243 dllIerential, 28, 96, 267 discrete-time state, 249 instantaneous, 432 Laplace's, 53,61 linear, 42 linear algebraic, 409 Lyapunov,415 matrix, 49 normal, 42 Poisson's, 61 power continuity, 102 Riccati, 415 state, 96, 251 Sylvester, 415 unforced rotational link, 394 equilibrium, 394 equilibrium point, 81 equipotentialline, 62 equivalence relationship, 127 equivalent state-space representation, 125, 127 error backward, 408 input, 408 integral absolute (iae), 218 integral square (ise), 218 latency, 330 quantization, 304, 321 worst case (wce), 218 error analysis backward, 408 forward, 408 error matrix, 408 error variance (ev), 218 errors quantization, 309 essential singularity, 55, 60 estimate Cauchy's, 59 Euclidean norm, 40 Euler's backward method, 268 Euler's forward method, 268 event task, 337 exclusive or (xor) circuit, 351 execution asynchronous, 354 execution shell, 326 exit functions, 330 expansion Laurent, 59 expansIOns Laplace, 35 exponent, 304 exponential input, 6, 12 expressIOn INDEX differential, extern, 336 external disturbances, 319 externs, 334 f -invariant, 39 factor area dilation, 61 factorization Cholesky, 44 plu, 35, 36 qr,41 factors + jwt , 176 invariant, 46 jw, 176 quadratic, 177 feedback, 297, 299 continuous-time, 297 nondifferentiable, 397 observer, 293 output, 290 state, 210, 289 feedback compensation, 291 feedback control system type 0, 180 type 1, 180 type 2, 180 feedback forms, 93 feedback linearization, 429, 430 feedback loop, 373, 374 feedback system unity, 189 feedback systems, 88 feedback transfer function, 88 feedforward, 208 feedforward control, 208 feedforward sp, 230 fets (field-effect transistors), 353 fictitious binary point, 303 field-programmable logic arrays (fpla), 353 fieldbus, 361 fields, 33 filter all pass, 80 digital, 267 inverse, 294 notch, 197 reference input, 294 filter tuning, 386 filtering analog plant input, 297 higher rate digital, 299 finite-state machine, 328 first order subsystems, 311 fixed-point arithmetic, 303, 309 453 INDEX fixed-point binary arithmetic, 303 fixed-point equation, 49 fixed-point number, 303 flip-flop, 352 jk,352 toggle (t), 352 flip-flops, 352, 354 floating-point arithmetic, 304, 312 floating-point binary arithmetic, 303 floating-point number, 303 flop set-reset (sr) flip, 352 flow graphs signal, 93 flowcharts, 443 folding frequency, 272 forced components, 244 forced response, 4, 5, 115 form advance operator, 10 block jordan, 252 controllable, 252 controllable canonical, 76, 128 delay operator, 10 diagonal, 252 differential operator, 125 dynamical, 235 imc filter, 219 integral, 235 isa (instrument society of america), 200 Jordan, 45 linear, 393 modal canonical, 121 nondynamical, 236 observable, 252 observable canonical, 76, 126 parallel, 200, 307 rational canonical, 46 reduced, 55 series, 200, 203 Smith canonical, 36 standard, 200 form input, 441 formality, 328 forms cascaded,93 feedback, 93 parallel, 93 quadratic, 47 formula Ackermann's, 289 Cauchy integral, 58 de Moivre's, 52 Hadamard's, 56 heaviside's expansion, 55 inversion, 59 forward error analysis, 408 forward transfer function, 88 forward-path gains, 95 Fourier transform, 17 numerical computation, 27 fourier transform (ft), 255 fpla, 353 fractional transformations bilinear, 61 linear, 61 frequencies natural, frequency folding, 272 gain crossover, 155,288 gain-margin, 184 phase crossover, 152, 288 phase-margin, 184 resonant, 186 frequency and time responses, 190 frequency domain, 188,424 frequency domain criterion, 148 frequency domain methods, 288 frequency domain performance, 161 frequency prewarping, 270 frequency response, 186, 414 frequency transferfunction, 175 frequency-domain method, frequency-response curves, 173 Frobenius norm, 40 ftime function, 335 full-order observers, 292 function closed-loop transfer, 88 complementary sensitivity, 216 constructor, 340 describing, 368 feedback transfer, 88 forward transfer, 88 ftime, 335 irrational, 21 linear, 38 main, 336 open-loop transfer, 88, 181 orthogonal, 41 proper rational, 20 rational, 20 simulation, 343 transfer, 86, 93,96, 147, 179,244 transfer feedback, 88 virtual,336 z-transfer,244 function charts, 360 function exchange method, 334 functionals linear, 39 454 functions complex, 53 exit, 330 harmonic, 53 holomorphic analytic, 53 homogeneous scalar quadratic, 47 linear, 38, 40 logarithm, 57 matrix-valued, 48 meromorphic, 56 multiple-valued complex, 53 multivariable, 47 nonlinear, 47 orthogonal, 41 polynomial, 47 public, 337 rational, 55 regular analytic, 53 sensitivity, 256 single-valued complex, 53 transfer, 55,70,75,413 fundamental theorem, 51 fundamental theorem of algebra, 54 fundamental transformations, 61 g.a & t.m Korn, 427 gain high-frequency, 390 gain and phase margins, 162 gain block, 86 gain crossover frequency, 155, 288 gain margin, 152, 184,288 gain scheduling, 397 gain-margin frequency, 184 gain-scheduling, 392, 398 control design, 398 gains forward-path, 95 loop, 95 non touching-loop, 95 gate nand,351 nor, 350 universal, 350 Gauss' mean value theorem, 59 general linear group, 34 generalized eigenvalue problems, 410 generalized Nyquist's stability criterion, 182 generalized predictive control, 224 geometric series, 56 global scope, 426 grafcet,358 computer implementation of, 359 diagram, 358 grafcet (graphe de commande etape-transition), 358 gram matrix, 42 INDEX Gram-Schmidt orthogonalization, 41 Gram-Schmidt procedure, 41 grammian controllability, 122 observability, 123 graph bond, 102 graphical displays, 436 graphical input, 442 graphs bond, 102 signal-flow, 93 signal-flow, 96 group general linear, 34 guideposts, 188 gyrator (gy), 103 Hadamard's formula, 56 hardware computing, 436 harmonic functions, 53 header file, 336 heat flow, 101 heaviside's expansion formula, 55 heavyweight threads, 328 hermitian symmetric, 36 hermitian transpose, 36 hierarchical modeling, 105, 422 high-frequency gain, 390 high-level languages, 436 higher index models, 429 higher order holds, 299 higher rate digital filtering, 299 holomorphic analytic functions, 53 homogeneous difference equation, 244 homogeneous scalar quadratic functions, 47 hypermedia, 437 hysteresis, 369 ics, 353 idcom, 224 ideal tracking, 294 identity matrix, 34 IEEE, 428 image, 39 imaginary axis, 52 imaginary number, 51 imc design, 218 imc filter forms, 219 imc structure, 217 impulse response, 68 impulse-invariant approximation, 274 include, 336 independence linear, 38 455 INDEX index controllability, 253 ise, 170 itae, 170 performance, 169 step, 242 induced euclidean norm, 40 industrial control, 360 inequality cauchy's, 52 Schwarz, 41 triangle, 39, 52 inheritance class, 428 initial conditions, 4, 10 inner product spaces, 41 inner products, 39,40 input constant, 7, 13 exponential, 6, 12 exponential complex, form, 441 graphical, 442 menu driven, 441 sinusoidal, 7, 13 input error, 408 input-output linearization, 430 input-output models, 65 input-output stable, 244 inputs reference, 283 sinusoidal, 109 instability practical, 227 instantaneous equations, 432 instantiating tasks, 336 instructions meta, 445 target-code, 445 integral convolution, integral absolute error (iae), 218 integral control, 199 integral form, 235 integral square error (ise), 218 integral theorems, 57 integrals complex, 57 integrated systems, 432 integrated systems inc., 425 integration conditional, 202 integrator, 86 integrator block, 86 interchangeability, 362 interface algorithmic, 441 operator, 336 interfaces user, 437 interior point, 55 internal cancellation, 231 internal model control, 215, 217 internal model control (imc), 215 internal model controllers (IMC), 386 internal model principle, 235 internal stability, 216 interoperability, 362 interrupt, 328 interrupt scheduling, 331 interrupt signals, 356 interrupts, 331 intertask communication, 333 invariant factors, 46 invariant polynomials, 36 inverse matrix, 34 inverse discrete time fourier transform (idtft), 255 inverse filter, 294 inverse fourier transform (ift), 255 inverse Laplace transform, 55, 117 inverse models, 430 inverse polar plot, 179 inverses matrix, 35 inversion, 61 inversion formula, 59 invertibility, 35 irrational function, 21 irrational number, 51 irrational transforms, 21 isa (instrument society of america) form, 200 ise index, 170 isolated singularity, 55 itae index, 170 iterative solution, 10 jacobian matrix, 393 jk flip-flop, 352 jw factors, 176 Jordan blocks, 45 Jordan form, 45 jordan representation, 78 junction, 102 summing, 91 kernel, 39 Kronecker products, 47 ladder diagrams, 353 ladder logic, 443 lag compensation, 196 456 lead, 197 lag-lead compensator, 180 lan, 362 Ian (local area network), 362 Ian (local area networks), 362 language, 327 languages high-level, 436 modeling, 427 simulation, 426, 437 Laplace expansions, 35 Laplace transform inverse, 117 Laplace transforms, 17, 18 irrational, 21 rational, 20, 25 Laplace's equation, 53, 61 latch, 352 latched relay, 354 latency minimum, 331 latency error, 330 latent controller, 389 Laurent expansion, 59 Laurent series, 59 law, control, 198 lead compensation, 196 lead lag compensation, 197 least significant bit (Isb), 303 left coprime, 130 left eigenvector, 44 left half-plane (lhp), 152 level quantization, 308 Lie algebra, 45 Lie bracket, 45 Lie brackets, 45 Lie product, 45 life cycle, 326 lightweight threads, 328 limit, 53 limit cycle, 312 limit cycles, 312, 370 limit switches, 348 line equipotential, 62 linear algebraic equations, 409 linear combination, 96 linear equations, 42 numerical conditioning, 42 solution, 42 linear form, 393 linear fractional transformation, 61 linear function, 38 linear functionals, 39 linear functions, 38, 40 INDEX linear independence, 38 linear least squares, 41 linear least squares problems, 409 linear matrix equations, 49 linear multivariable control, 414 linear ordinary differential equations, 415 linear parameter varying systems, 399 linear quadratic design, 262 linear step-invariant discrete-time system, 244 linear step-invariant discrete-time systems, 249 linear system, 241 linear systems, 93, 147 linear time-invariant (LTI), 3, linear time-invariant (lti) systems, linear time-invariant systems, 115 linear time-varying dynamics, 397 linearization, 81, 82, 392, 394, 423 of functions, 393 about a trajectory, 397 about an equilibrium, 394 feedback, 430 input-output, 430 limitations, 396 linearizations, 400 linearizing, 422 linearly dependent, 38 linearly independent, 38 UNPACK,419 Liouville's theorem, 59 1m and phase diagram, 184 1m angle diagram, 185 1m curves, 178 1m diagram, 175 1m-angle diagram, 190 load disturbances, 219 loading effects, 86 local area networks (lans), 362 log magnitude, 174 log magnitude-angle diagram, 175 logarithm function, 57 logarithmic arithmetic, 303 logarithmic plots, 174 logic ladder, 443 positive, 349 state-transition, 329 transistor-transistor (ttl), 349 ttl,353 loop primary, 207 secondary, 207 loop gains, 95 low order polynom:als, 150 low-power schottky ttl (Is-ttI) elements, 353 lse optimal performance, 218 LTI, linear time-invariant, 3, 457 INDEX Lund Institute of Technology, 427 Lyapunov equations, 415 Lyapunov theory, 150 Lyapunov's linearization method, 82 m circles, 188 M curves, 190 m vectors, 33 machine epsilon, 407 magnitude spectrum, 18 main function, 336 make-before-break (mbb), 348 mantissa, 304 manual control, 386 manufacturing automation protocol (map), 362 manufacturing message specification (mms), 362 map, 362 conformal, 139 mappings conformal, 60, 61 margin gain, 152, 184,288 gain and phase, 162 phase, 152, 155, 184,288 Markov stability criterion, 150 masking, 350 mason's rule, 93 matching pole-zero, 274 matching step, 273 mathematical software, 418, 419 matrices, 33, 45, 47 block, 37 commuting, 45 coprime polynomial, 123 polynomial, 36 similar, 39 square, 33 symmetric, 44 unimodular, 36 matrix, 33 addition, 34 algebra, 34 block, 37 characteristic equation, 43 companion, 46, 98 controllability, 122, 124, 253 elements, 33 environments, 439, 442 error, 408 exponentials, 415 gram, 42 identity, 34 index, 45 inverse, 34 inverses, 35 inversion, 35 invertible, 34 jacobian, 393 linear equations, 49 multiplication, 34 non invertible, 34 nonnegative definite, 44 nonsingular, 34 normal, 43 observability, 123, 124 orthogonal, 37, 42 perturbed, 408 polynomials, 37 positive definite, 44 positive definite symmetric, 149 powers, 37 regular, 34 representations, 39 resolvant, 116, 121 resolvent, 48 singular, 34 skew-symmetric, 36 state transition, 116, 117, 122 symmetric, 36 transposition, 36 unimodular, 61, 130 unitary, 37 zero, 34 matrix equations, 49 matrix polynomial, 37 matrix transposition, 36 matrix-valued functions, 48 maximal eigenspace, 44 maximum modulus theorem, 59 maximum selector, 208 measurement noise, 111, 321 mechanical system, 102 menu driven input, 441 meromorphic function, 56 meta instructions, 445 method classical, 4, 14 describing function, 367 Euler's backward, 268 Euler's forward, 268 frequency-domain, function exchange, 334 phase plane, 376 trapezoidal, 269 tustin's, 269 method of convolution, 8, 13 methods approximation, 278 control system design, 284 frequency domain, 288 rectangular, 269 458 root locus design, 285 time-domain, z-transfer function, 244 MIMO, 75, 128 MIMO case, 124 mimo lti systems, 166, 168 MIMO state-space system, 122 mimo transfer functions, 78 minimal polynomial, 37, 46 minimum latency, 331 minimum modulus theorem, 59 minimum selector, 208 minimum-phase system, 179 Mitchell & Gauthier assoc., 426 modal canonical form, 121 mode scan, 327 mode switches, 203 model current-flow-causer, 101 quantization noise, 311, 312 scan, 327 voltage-drop-causer, 101 model inverse based, 220 model poles, 231 model predictive control, 224 model predictive heuristic control, 224 modeling, 99 energy, 102 hierarchical, 105, 422 object-oriented, 104,427,432 plant, 422 modeling languages, 427 modeling software, 424 models continuous-time state variable, 275 discrete-time, 275 higher index, 429 input-output, 65 inverse, 430 second-source, 432 state space, 75 state variable, 249 state-space, 99 modes characteristic,S, 11 natural,S, 11 oscillatory, 207 system, 125 uncontrollable, 122, 129 modulus, 52 monic polynomial, 36, 54 monomials, 47 monostable actuator, 349 most significant bit (msb), 303 multi-input-multi-output (MIMO), 75 INDEX multi-input/multi-output (MIMO) cases, 121 multiple-valued complex function, 53 multiplication boolean, 349 matrix, 34 scalar, 34 multivariable functions, 47 mutual exclusion, 334 nand, 351,357 nand gate, 351 natural components, 244 natural frequencies,S natural modes,S, 11 natural response, 4, 11, 115, 120 negative encirclement, 57 networking, 105 networks combinatorial, 352 local area, 362 plant wide, 362 Neumann's problem, 61 new operator, 336, 337 Nichols chart, 190 Nichols plot, 185 Nichols plots, 173 nodes, 93, 105 noise, 108 measurement, Ill, 321 white input, III noise suppression, 164 nonblocking code, 327 nondifferentiable feedback, 397 nondynamical form, 236 nonlinear functions, 47 nonlinear systems, 398 nonlinearity cubic, 368 saturation, 369 nonminimum-phase system, 179 nonnegative definite matrix, 44 nonrecursive difference equation, 10 nontouching-Ioop gains, 95 nor, 350, 351, 353, 357 nor gate, 350 norm, 39 Euclidean, 40 Frobenius, 40 induced euclidean, 40 uniform, 40 normal equations, 42 normal matrix, 43 normalized time, 170 norms, 39, 40 vector, 39 not, 349, 350,357 459 INDEX notation operational, 10 notch filter, 197 nullity, 42 nullspace, 39, 42 number complex, 51 condition, 43 fixed-point, 303 floating-point, 303 imaginary, 51 irrational, 51 transcendental, 51 winding, 58 numerical conditioning, 42 numerical differentiation, 273 numerical linear algebra, 409 numerical stability, 407 Nyquist, 173 Nyquist contour, 136 Nyquist criterion, 135 Nyquist diagram, 137 Nyquist diagrams, 140 Nyquist plot, 145, 152, 155 Nyquist plots, 173 Nyquist stability criterion, 55 Nyquist stability test, 135 Nyquist theorem, 135, 138 Nyquist's stability criterion, 181, 183 generalized, 182 object diagrams, 427, 428 object instantiation, 105 object-oriented modeling, 104,427,432 object-oriented modeling languages, 427 object-oriented modeling paradigm, 427 object-oriented modeling systems, 432 objects, 105 observability, 121, 123,252,297 grammian, 123 matrix, 123, 124 rank, 123 observable, 123 completely, 253 observable canonical form, 76, 126 observable form, 252 observer, 289 deadbeat, 292 reduced-order state, 293 observer design, 417 observer feedback, 293 observer-based controllers, 386 observer-predictor, 234 observers full-order, 292 reduced-order state, 293 off point pick, 90 omola, 427 one-degree-of-freedom tracking controller, 389 one-sided Laplace transform, 18 one-sided z transform, 23 open left half-plane (olhp), 21 open region, 53 open systems, 441 open systems interconnection (osi) scheme, 361 open unit disk, 24 open-loop transfer function, 88, 181 operating system, 334 operating systems, 437 operational notation, 10 operations complete, 351 operator differential, 125 operator interface, 336 optimum control system, 170 o~350,351,353,354,356,357 or block (orb), 356 ordinary differential equation, 3, 66, 74 orthogonal basis, 41 orthogonal complement, 41 orthogonal function, 41 orthogonal functions, 41 orthogonal matrix, 37, 42 orthogonal projection, 41 orthogonal projections, 41 orthogonal vectors, 41 orthogonality, 40 orthonormal basis, 41 oscillatory modes, 207 osi, 361, 362 output feedback, 290 outputs, 358 overflow, 304 Pade approximations, 81 Pade approximation, 80 pal, 353 paradigm object-oriented modeling, 427 parallel combination, 87 parallel form, 200, 307 parallel forms, 93 parallel systems, 87 parallelism, 327 parameter changes, 203 parametric resonance, 82 part analytic, 60 principal, 60 partial differential equation, 460 INDEX partial state, 125 particular solution, 5, 12 partitions conformable, 37 Paynter, Henry, 102 percent overshoot, 119 performance index, 169 period sampling, 242, 317 perturbed matrix, 408 petri-nets, 443 phase crossover frequency, 152,288 phase diagram, 175 phase margin, 152, 155, 184,288 phase plane method, 376 phase spectrum, 18 phase-margin frequency, 184 physical principles, 99 pick-off point, 90 pid,361 pid control, 198, 206 pid controllers, 347 plane complex, 52, 188 plant, 297 composite continuous-time model, 297 multiple-inputs, 290 plant input signals, 297 plant modeling, 422 plant wide networks, 362 plants unstable, 235 pic, 347, 348, 355-357,359-362 programming, 357 pIc system, 356 pld, 355 plot Bode, 152, 155, 173-175 direct, 188 direct polar, 179, 186, 189 inverse polar, 179 logarithmic, 174 Nichols, 173, 185 Nyquist, 145, 152, 173 nyquist, 155 polar direct, 179, 181 root locus, 192, 197 plots Bode, 146 pIs (pulse instruction), 357 plu factorization, 35, 36 point binary, 303 equilibrium, 81 fictitious binary, 303 interior, 55 pick-off, 90 radix, 303 singular, 55 point sensors, 348 Poisson's equation, 61 pole, 248 pole assignment, 417 pole placement, 209, 212, 291 arbitrary, 210 pole-zero matching, 274 poles, 20, 55, 118, 121, 138, 181,413 model, 231 polling, 356 polynomial, 129 annihilating, 37 characteristic, 5,11,37,115, 120, 125 matrix, 37 minimal, 37, 46 monic, 36 polynomial approximation, 268 polynomial functions, 47 polynomial matrices, 36, 123 polynomial nonlinearity, 374 polynomials, 54 coprime denominator, 291 coprime numerator, 291 delta-operator, 150 invariant, 36 low order, 150 portability, 326 positive definite matrix, 44 positive definite symmetric matrix, 149 positive encirclement, 57 positive logic, 349 potential theory, 61 power, 353 power continuity equations, 102 power series, 56 power series expansions, 56 practical instability, 227 precision relative machine, 407 preemptable scheduler, 332 preemptable tasks, 332 preemptive scheduling, 332 prefiltering, 321 prewarping,272 primary controller, 225 primary loop, 207 principal part, 60 principle internal model, 235 principle of the argument, the, 136 principles physical, 99 problem 461 INDEX boundary-value, 61 Dirichlet's, 61 Neumann's, 61 problems generalized eigenvalue, 410 linear least squares, 409 procedure Gram-Schmidt, 41 process control, 373 processing digital, 309 product Lie, 45 tensor, 47 product rule, 35 products inner, 40 program structure, 336 programmable array logic (pal), 353 programmable controllers, 347, 355 programmable logic devices (pld), 353 programmable logical controllers (pIc), 347 programming pIc, 357 projection complementary orthogonal, 41 orthogonal, 41 projections orthogonal,41 proper rational function, 20 proportional band, 203 proportional control, 198 proportional feedback gain, 195 proportional-integral-derivative (pid) controllers, 221 protected variables, 340 protocol manufacturing automation (map), 362 technical and office (top), 362 pseudo-inverse, 43 public functions, 337 public section, 337 purely imaginary complex numbers, 51 qr factorization, 41 quadratic factors, 177 quadratic forms, 47 quantization effects, 308 quantization error, 304, 321 quantization errors, 309 quantization level, 308 quantization noise model, 311, 312 quantized amplitudes, 241 'iJ -equivalence, 36 radius spectral, 44 radius of convergence, 56 radix point, 303 ramp-invariant approximation, 274 range, 39, 42 rank, 42 raster graphics displays, 437 rate sampling, 297 ratio signal-to-noise, 309 rational canonical form, 46 rational function, 20 rational functions, 55 rational Laplace transforms, 20, 25 rational z transforms, 23 real axis, 52 receptivity, 358 rectangular methods, 269 recursive difference equation, 10 reduced form, 55 reduced-order state observer, 293 reduced-order state observers, 293 reference conditioner, 383 reference governor, 383 reference input filter, 294 reference input model system, 295 reference inputs, 283 reference model tracking system design, 295 region open, 53 region of analyticity, 53 region of convergence, 19 regular analytic functions, 53 regulator, 283 relationship complete equivalence, 126 equivalence, 127 relative machine precision, 407 relay, 369, 373 electromechanical, 349 latched,354 self-holding, 354 removable singularity, 55 repeated roots, 5, 11 representation equivalent state-space, 127 signed-magnitude, 304 two's complement, 304 reset state, 354 residue, 60 residue arithmetic, 303 residue theorem, 59, 60 residues, 20 resistive source (rs) elements, 104 resistors, 429 resolvant matrix, 116, 121 INDEX 462 resolvent matrix, 48 resonance parametric, 82 resonant frequency, 186 response between-sample, 296 closed-loop, 140 control system, 317, 319 deadbeat, 170, 171 discrete-frequency, 249 forced, 4, 5, 115 frequency, 414 impulse, 68 natural, 4, 11, 115, 120 sinusoidal and time, 187 system, 249 transient, 158 zero-input, 115 zero-state, 115, 250 response model design, 294 response model tracking, 294 response terms, 244 response, natural, responses frequency and time, 190 return statement, 332 Riccati equations, 415 right coprime, 130 ring, 33 rise time, 119 robust bidirectional transfer, 389 robust control, 417 root loci, 195 root locus design methods, 285 root locus plot, 192, 197 roots characteristic,S, 11 repeated,S, 11 rotation, 61 Rouche's theorem, 59 rounding, 304 roundoff, 305 Routh-Hurwitz array, 132 Routh-Hurwitz criterion, 132 Routh -Hurwitz stability criterion, 131 Routh- Hurwitz stability criterion, the, 131 row vectors, 33 rule Cramer, 35 Cramer's, 35 product, 35 sampled data sampled-data sampled-data sampled-data control systems, 256 controllers, 262 stability tests, 146 system, 147 samples, 242 sampling, 297 sampling period, 242, 317 sampling rate, 297, 321 sampling theorem, 315 saturation nonlinearity, 369 scalability, 441 scalar, 34 scalar differential equation, 115 scalar multiplication, 34 scan mode, 327 scan model, 327 scanf statement, 327, 336, 344 scanned code, 327 scheduled controllers, 388 scheduler, 360 scheduling, 330 cooperative, 330 gain, 397 interrupt, 331 preemptive, 332 time slice, 332 scheduling variable, 398 scheme open systems interconnection (osi), 361 schottky diodes, 353 Schur-Cohn criterion, 148 Schwarz inequality, 41 scope global, 426 second order subsystems, 312 second -order response, 119 second-source models, 432 secondary loop, 207 selector control, 208 self-holding relay, 354 sensitivity analysis, 307 sensitivity functions, 216, 256 sensors binary, 348 digital, 348 point, 348 sequence alternative parallel, 359 simple, 359 simultaneous parallel, 359 sequential circuits, 354 series geometric, 56 Laurent, 59 power, 56 Taylor's, 56 series combination, 86 series form, 203 series forms, 200 set 463 INDEX spanning, 38 set point weighting, 199 set-reset (sr) flip flop, 352 settling time, 119 signal discrete-time, 242 signal flow diagrams, 443 signal switching, 353 signal-flow graphs, 93, 96 signal-to-noise ratio, 309 signals, 100,353 discrete-time, 241, 242 interrupt, 356 plant input, 297 sustained input, 355 signed-magnitude representation, 304 similar matrices, 39 similarity transformation, 39, 76, 252 simnon, 426 simple closed curve, 57 simple pole, 55 simple second-order system, 186 simple sequence, 359 simstruct, 432 simulation digital, 437 simulation function, 343 simulation languages, 426, 437 simulation software, 424 simulators block diagram, 425 simulink,425 simultaneous parallel sequence, 359 single-input systems, 289 single-input-single-output (SISO), 75 single-input/single-output (SISO) linear, time-invariant system, 121 single-pole double-throw (spdt), 348 single-pole, single-throw (spst) mechanical switch, 348 single-valued complex function, 53 singular point, 55 singular value, 40 singular value decomposition, 46, 410 singular value decomposition (svd) , 46 singular values, 46 singularity, 53 essential, 55, 60 isolated, 55 removable, 55 sinusoidal and time response, 187 sinusoidal describing function, 367 sinusoidal input, 7, 13 sinusoidal inputs, 109 sinusoidal responses, 173 SISO,75 siso lti systems, 158 skew-symmetric matrix, 36 Smith canonical form, 36 Smith predictor, 224 smith predictor, 221 Smith predictor (dtc), 225 software mathematical,418 modeling, 424 simulation, 424 system, 436 software hierarchy, 325 solution classical, 4, 11 complementary, 4, 5, 11 iterative, 10 particular, 5, 12 space dual, 39 state, 72, 121,209 vector, 37 spaces inner product, 41 vector, 37 span, 38 spanning set, 38 spdt switch, 348 specification manufacturing message (mms), 362 spectral radius, 44 spectral representation, 44 spectrum magnitude, 18 phase, 18 square matrices, 33 sr,354 sspa systems, 426 sstosys, 439 stability, 131, 150, 184,244,394 asymptotic, 394 backward, 408 internal,216 numerical, 407 weak,416 stabilization, 395 stand alone feedback control, 326 standard basis vectors, 38 standard form, 200 state controllable, 122 partial, 125 reset, 354 uncontrollable, 122 waiting, 340 state equations, 251 signal-flow graphs, 96 state feedback, 210, 289 464 INDEX state space, 72, 73, 121,209 state space approach, 72 state space models, 75 state transition matrix, 116, 117, 122 state variable models, 249 state variables, 116 state-space models, 99 state-space system, 123, 127, 128 state-transition diagrams, 443 state-transition logic, 329 states, 328 static system, 352 statics, 334 steady-state accuracy, 160 steady-state components, 244 step index, 242 step invariant, 241, 250 step reconstruction, 242 step-invariant approximation, 274 step-invariant discrete-time observer design, 292 step-varying systems, 251 stochastic analysis, 309 streamline, 62 strictly proper, 20 strong data typing, 442 structure control, 216 imc,217 program, 336 submodel statement, 106 subspace, 38 subspaces, 38 subsystems first order, 311 second order, 312 sum convolution, 14 summer, 85 summing junction, 85, 91 superposition, 241 sustained input signals, 355 switch single-pole, single-throw (spst) mechanical, 348 spdt, 348 switch statement, 330 switches, 349 switches, mode, 203 switching, 353 Sylvester equations, 415 symmetric matrices, 44 symmetric matrix, 36 synchronous charts, 354 system complex, 207 control, 158 digital control, 241 discrete-time, 146, 147, 150 linear, 147,241 linear step-invariant discrete-time, 244 mechanical, 102 mimo hi, 166, 168 MIMO state-space, 122 minimum-phase, 179 nonminimum-phase, 179 optimum control, 170 pIc, 356 reference input model, 295 sampled-data, 147 simple second-order, 186 single-input/single-output (5150) linear, time-invariant, 121 siso hi, 158 state-space, 123, 127, 128 static, 352 tracking, 283 type 0, 178 type 1, 178 type 2, 179 uncontrollable, 122 system dynamics, 394 system identification, 108 system modes, 125 system response, 249 system software, 436 systembuild,425 systems automation, 361 autonomous, 82 continuous, 55 continuous-time, 275 control, 383 discrete, 55 discrete-time, 241 feedback, 88 linear, 93 linear parameter varying, 399 linear step-invariant discrete-time, 249 linear time-invariant, 115 linear time-invariant (hi), nonlinear, 398 object-oriented modeling, 432 open, 441 operating, 437 parallel, 87 sampled data control, 256 single-input, 289 step-varying, 251 unity-feedback, 88 systoss, 439 tables truth, 349 465 INDEX target-code instructions, 445 task,332 assignment, 333 class definition, 336 continuous, 333 event, 337 functions, 337 instantiating, 336 intermittent, 332 intertask communication, 333 tasks, 328 preemptable, 332 synchronization of, 360 unitary, 332 Taylor's series, 56 technical and office protocol (top), 362 temperature, 101 tensor product, 47 the mathworks, 432 The Mathworks Inc., 425 theorem Cauchy's, 57 Cayley-Hamilton, 37 contraction mapping, 49 discrete Hermite-Bieler, 148 fundamental, 51 Gauss' mean value, 59 integral, 57 Liouville's, 59 maximum modulus, 59 minimum modulus, 59 Nyquist, 135, 138 residue, 59, 60 Rouche's, 59 sampling, 315 theorems de morgan, 350 theory Lyapunov, 150 potential, 61 threads, 328 heavyweight, 328 lightweight, 328 time delays, 225 time domain, 110 time response, 115 time responses, 173 time slice scheduling, 332 time slicing scheduler, 332 time-delay compensation, 224, 235 time-domain methods, time-invariant, linear (LTI), 3, timekeeping, 335 timers, 358 toggle (t) flip-flop, 352 topological interconnection capability, 105 tracking, 202 closed-loop, 186 ideal, 294 response model, 294 zero-state, 294 tracking system, 283 tracking system design, 293, 294 transcendental number, 51 transfer bump less, 390 transfer contact, 348 transfer function, 86, 93, 147,244 closed-loop, 88 open-loop, 88 signal-flow graphs, 96 transfer functions, 55, 70, 75, 179,413 mimo, 78 transform discrete-time fourier (dtft), 255 fourier (ft), 255 inverse discrete time fourier (idtft), 255 inverse fourier Oft), 255 inverse Laplace, 55, 117 w,288 z,250 transformation bilinear, 61, 269, 270 bilinear fractional, 61 congruence, 48 fractional linear, 61 fundamental, 61 linear fractional, 61 similarity, 39, 76, 252 Tustin, 61 z,242 transformer (tf), 103 transforms bilateral Laplace, 18 bilateral z, 23 discrete time Fourier, 22 Fourier, 17 irrational Laplace, 21 Laplace, 17, 18 one-sided Laplace, 18 one-sided z, 23 rational Laplace, 20, 25 rational z, 23 two-sided Laplace, 18 two-sided z, 23 unilateral Laplace, 18 unilateral z, 23 z, 17,23 transient components, 244 transient response, 158 transistor- transistor logic (ttl), 349 transition logic, 329 466 transition matrix state, 117 transitions, 358 translation, 61 transparency, 441 transpose hermitian, 36 transposition matrix, 36 trapezoidal method, 269 triangle inequality, 39, 52 trigger, 352 truncation, 304, 305 truth tables, 349 ttl circuits, 353 ttllogic, 353 tuning controller, 386 filter, 386 Tustin transformation, 61 tustin's method, 269 two's complement representation, 304 two-degree-of-freedom tracking loop, 389 two-sided Laplace transform, 18 two-sided z transform, 23 type feedback control system, 180 type system, 178 type feedback control system, 180 type system, 178 type feedback control system, 180 type system, 179 uncontrollable, 122 mode, 122, 129 state, 122 system, 122 unforced rotational link equations, 394 uniform norm, 40 unilateral Laplace transform, 18 unilateral z transform, 23 unimodular matrix, 61, 130 unit vector, 38 unitary matrix, 37 unitary scheduler, 332 unitary tasks, 332 unity feedback system, 189 unity-feedback system, 88 universal gate, 350 unobservable, 123 unstable plants, 235 user code block (DCB), 432 user interfaces, 437 value absolute, 52 singular, 40 values characteristic,S, 11 singular, 46 variable scheduling, 398 variables, 252 protected, 340 state, 116 variance, 308 vector coordinate, 38 cyclic, 46 unit, 38 vector norms, 39 vector refresh displays, 437 vector space, 37 vector spaces, 37 vector storage displays, 437 vectors column m, 33 orthogonal,41 row, 33 standard basis, 38 velocity algorithms, 201, 203 velocity error constant, 134 VHDL-a, 428 virtual function, 336 virtual functions, 340, 343 voltage, 101 voltage-drop-causer model, 101 w transform, 288 waiting state, 340 weak stability, 416 while loop, 327 white input noise, III winding number, 58 windup, 390 word length, 303 worst case error (wce), 218 xor, 351, 353, 357 z transform, 250 z transformation, 242 z transforms, 17,23 z-transfer function, 244 z-transfer function methods, 244 zero matrix, 34 zero of order, 54 zero-input response, 115 zero-input response components, 250 zero-input stable, 252 zero-state response, 115,250 zero-state stable, 252 zero-state tracking, 294 zeros, 20,54, 118, 125, 129,413 .. .CONTROL SYSTEM FUNDAMENTALS edited by William S Levine Library of Congress Cataloging-in-Publication Data Levine, W S Control system fundamentals / William S Levine... Printed on acid-free paper Preface Control technology is remarkably varied; control system implementations range from float valves to microprocessors Control system fundamentals include regulating... control systems The fundamentals are just what the name implies: the basics of control engineering Note that this book includes major sections on digital control and modeling of dynamical systems