DIGITAL CONTROL SYSTEMS THEORY, HARDWARE, SOFTWARE SECOND EDITION Constantine H Houpis, Ph.D Gary B Lamont, Ph.D Pro fessors ofElectrical Engineering School ofEngineering Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio McGraw-Hil� Inc New York Bogota Milan SI Louis Caracas Montreal Singapore San Francisco Lisbon Sydney London Auckland Madrid New Delhi Paris Tokyo Toronto Mexico San Juan DIGITAL CONTROL SYSTEMS INTERNATIONAL EDITION 1992 Theory, Hardware, Software Exclusive rights by McGraw-Hill Book Co - Singapore for manufacture and export This book cannot be re-exported from the country to which it is consigned by McGraw-Hill CMO FC Copyright © 1992, 1985 by McGraw-Hill, Inc All rights reserved Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher This bookwasset in Times Roman by Electronic Technical Publishing Services The editors were Roger L Howell, Anne T Brown, and Eleanor Castellano; the production supervisor was Louise Karam The cover was designed by Keithley and Assoc for Joseph A Piliero New drawings were done by J&R Services, Inc Library of Congress Cataloging-in-Publication Data Houpis, Constantine H Digital control systems: theory, hardware, software H Houpis, Gary B Lamont.-2nd ed p / Constantine cm.-(McGraw-Hill series in electrical engineering Control theory) Includes bibliographical references and index l Digital control systems ISBN 0-07-030500-5 I Lamont, Gary B III Series TJ223.M53H68 1992 629.8'95dc20 When ordering this title use 91-22736 ISBN 0-07-112637-8 II Title ABOUT THE AUTHORS Constantine H Houpis is a professor of electrical engineering at the Air Force Institute of Technology, where he has taught since 1952 He also supervises the doctoral program in electrical engineering at the Institute and is a consultant to the Air Force Flight Control Directorate at Wright-Patterson Air Force Base, Ohio Previously, he had taught at Wayne State University Professor Houpis received his Ph.D in electrical engineering at the University of Wyoming Dr Houpis is the author of numerous control theory technical articles and textbooks and is a member of Tau Beta Pi, Eta Kappa Nu, Sigma Xi, and ASEE, and is a Fellow of IEEE Gary B Lamont is a professor of electrical and computer engineering at the Air Force Institute of Technology In addition to teaching graduate courses in digital control theory, computer engineering, and computer science, he does consulting in these areas Prior to joining the faculty of the Institute in 1970, he was a systems analyst and engineer at the Aerospace Division of Honeywell Professor Lamont recieved his Bachelor of Physics, MSEE, and Ph.D degrees from the University of Minnesota Dr Lamont has authored numerous papers on automatic control (conventional, modem, and digital), expert systems, parallel processing algorithms, computer-aided design, and educational techniques He is a member of Eta Kappa Nu, Tau Beta Pi, ASEE, ACM, and a member of IEEE CONTENTS Preface 1.1 1.8 2.1 2.2 2.3 2.4 xix Introduction Introduction Digital Control-System Modeling Sampling Process 2.2 System Terminology 2.3 General Sampled-Data System Variables 2.4 Systems Modeling Why Use Digital Control? Control-System Analysis and Synthesis The Interdisciplinary Field of Digital Control Digital Control Development Nature of the Engineering Control Problem Text Outline Summary Continuous-Time Control-System Response Characteristics Introduction Background Simple Second-Order System Tracking-Response Characteristics Higher-Order System Tracking-Response Characteristics 2.4.1 Time-Response Characteristics of a Third-Order All-Pole Plant 2.4.2 Time-Response Characteristics of a Third-Order, One-Zero Plant 2.4.3 Time-Response Characteristics of a Sixth-Order Plant 2.4.4 Correlation between Frequency and Time Domains 2 11 12 13 14 16 17 18 19 19 20 23 27 28 28 31 32 xii CONTENTS 2.4.5 2.5 2.6 2.7 2.8 2.9 2.10 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Correlation of the Control Ratio with Frequency and Time Responses Nichol's Chart (NC) Analysis Cascade-Compensator Design Procedures Synthesizing a Desired Tracking Control Ratio with a Unit-Step Input Feedback Compensation Disturbance Rejection 2.9 Second-order Disturbance-Rejection Model 2.9.2 Single-Input Single-Output (SISO) Design Principles for Disturbance Rejection 2.9.3 Examples Summary 34 37 39 43 49 51 51 55 56 62 Linear Systems and the Sampling Process 63 Introduction Linear Time-Invariant (LTI) System Solution of Linear Difference Equations Sampling Process (Frequency Domain Analysis) Ideal Sampler Shannon ' s Sampling Theorem Sampling-Time Selection 3.7.1 Single-Rate Sampling 3.7.2 Multirate Sampling Weighting Sequence Data Conversion Introduction 3.9.1 Zero-Order Hold (ZOH) Summary 63 63 68 73 77 83 84 85 85 86 94 94 98 Discrete Systems Modeling 99 Introduction Definition and Determination of the z-Plane Transform Z Mapping between s and z Domains 4.3.1 Mapping of the Primary Strip 4.3.2 Mapping of the Constant Frequency Loci 4.3.3 Mapping of the Constant Damping-Coefficient Loci 4.3.4 Mapping of the Constant Damping-Ratio Loci Z-Transform Theorems The Inverse Z-Transform, Z - I 4.5 Partial-Fraction Method 4.5.2 Power-Series Method (Direct Division Method) Limitations Z Transform of System Equations 4.7 Open-Loop Hybrid Sampled-Data Control System 4.7.2 Open-Loop Discrete-Input-Data Control System 4.7.3 Closed-Loop Sampled-Data Control System 4.7.4 Signal Flow Graphs for Hybrid Systems (HSFG) Digital-Computer Transfer Function Summary 99 00 111 12 14 14 15 18 123 24 128 30 131 131 36 137 140 145 149 CONTENTS 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 1 xiii Discrete Control Analysis 151 Introduction System Stability 5.2 z-Plane Stability 5.2.2 z-Domain Nyquist Stability 5.2.3 Polar Plot Analysis 5.2.4 Extended z-Domain Stability Analysis: Jury 's Stability Test Steady-State Error Analysis for Stable Systems 5.3 Steady-State Error-Coefficient Formulation 5.3.2 Evaluation of Steady-State Error Coefficients 3.3 Use of Steady-State Error Coefficients Root-Locus Analysis 5.4 Procedure Outline 5.4.2 Root-Locus Construction Rules for Negative Feedback 5.4.3 Examples Bilinear Transformations 5.5.1 s-Plane and w-Plane Relationship 5.5.2 Routh Stability Criterion in w-Plane Time-Response Correlation between Planes (s, z, and w) Frequency Response Summary 151 152 153 155 156 157 160 162 163 164 166 167 168 170 76 77 80 183 88 194 Discrete Transfonn Analysis (Approximations) 198 Introduction Folding or Aliasing Transformation Methods between Planes (s, z, and w) Mapping Approximations of Z-Transform (Or Numerical Solution of Differential Equations) 6.4.1 First-Backward Difference 6.4.2 Tustin Transformation Pseudo-Continuous-Time (PCT) Control System Analysis of a Basic (Uncompensated) System 6.6.1 PCT Control System Model 6.6.2 Sampled-Data Control System Summary 98 200 201 Principles of Signal Conversion and Measurement 220 Introduction Timing Considerations B inary Coding of Information Conversion Systems Digital-to-Analog (D/A) Conversion Structures General Analog-to-Digital (AID) Conversion Structures Specific Analog to Digital Conversion Systems Measures of" Converter Performance Sample-and-Hold Operation Multiplexing Integrated AID and D/A Interfaces 220 220 222 224 225 229 234 243 245 247 248 206 206 209 213 215 215 216 219 xiv CONTENTS 7.12 7.13 7.14 8 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 1 8.12 8.13 8.14 8.15 8.16 8.17 9.1 9.2 9.3 9.4 9.5 9.6 Measurement in Digital Control Systems 7.12.1 Temperature Measurements 7.1 2.2 Pressure and Related Measurements 7.12.3 Motion Measurement 7.1 2.4 Position Measurement 7.12.5 Transducer Signal Conditioning 7.12.6 Saturation Analysis Programming Input and Output (I/O) I/O Hardware Structure 7.1 Programmed I/O Mode 7.1 3.3 Interrupt I/O Devices Summary 250 250 253 256 258 260 260 260 261 261 263 265 Digital-Control-S ystem Implementation 267 Introduction Control Logic Computer Architecture for Control Software for Control 8.4 Algorithms 8.4.2 Language Hierarchy Software Engineering in Digital Control Systems Requirements-Data Flow Diagrams (DFDs) Real-Time Design for Digital Control Software Design in Control Systems Direct Design Method Structured Programming and Implementation Software Testing Real-Time Scheduling Real-Time Operating Systems for Digital Control Real-Time Operating Systems Requirements 8.1 3.2 Simple Real-Time Operating System Watchdog Timers Sampling-Time Selection Single- Rate Sampling 5.2 Multirate Sampling User Interfaces to Real-Time Operating Systems Summary 267 268 27 274 274 277 278 281 285 289 292 293 293 295 299 301 302 304 305 306 307 308 309 Random Processes in Digital Control Systems 311 Introduction Digital Control of Random Inputs Random Processes (Stochastic Processes) Random Process Time Averages Linear System Response to Random Signals Convolution Model 9.5.2 Difference Equation Model 9.5.3 Spectral Density of Linear Discrete Systems Vector-Matrix Representation of Random Processes 31 312 313 317 319 319 322 324 326 CONTENTS XV 9.7 Summary 329 10 Finite Word Length and Compensator Structure 330 Introduction Quantization Errors Compensator Structure and Arithmetic Errors Compensator Coefficient Representation Sensitivity of Coefficients Scaling Limit-Cycle Phenomenon Due to Quantization Simulation and Tuning Detailed Design/lmplementation Process Summary 330 331 34 348 352 356 358 367 368 369 10.1 10.2 10.3 0.4 10.5 10.6 10.7 10.8 10.9 10.10 11 1.1 1 1 1 1 1 1 1 12 12.1 12.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 12.10 12 1 Cascade Compensation-Digitization (DIG) Technique 370 Introduction Digitization (DIG) Design Technique Guillemin-Truxal (GT) Compensation Method Lead Cascade Compensation 1 s-Plane DIG Design 1 4.2 w-Plane DIG Design 1 4.3 s-Plane to w-Plane Correlation Lag Compensation Lag-Lead Compensation Extensive Cascade DIG Example 1 7.1 Analysis of the Basic System 1 7.2 Guillemin-Truxal Approach 1 7.3 Pseudo-Continuous Time (PeT) Approach 1 7.4 w-Domain DIG Design Summary 370 372 374 379 379 381 384 385 385 387 387 388 389 391 392 Cascade Compensation-Direct (DIR) Technique 394 Introduction Direct (DIR) Design Technique Lead Compensation (DIR) Lag Compensation (DIR) Lag-Lead Compensation-DIR Frequency-Response Characteristics Proportional Integral Derivative (PID) Controller Set-Point PID Controllers Extensive Cascade Example (DIR) 2.9 Lead Compensation 2.9.2 Lag Compensation 2.9.3 Controller Implementation Deadbeat Response Summary 394 394 396 397 400 400 404 406 408 408 409 410 41 412 xvi CONTENTS 13 13.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 15 15.1 5.2 5.3 15.4 15.5 5.6 15.8 5.9 15.10 Feedback Compensation 413 Introduction General Analysis DIR Technique for Feedback Control 3 Guillemin-Truxal Approach 3.3.2 Root-Locus Approach DIG Technique for Feedback Control Controlling Unwanted Disturbances 13.5 DIG Technique 3.5.2 DIR Technique Extensive Digital Feedback Compensator Example 3.6.1 DIG Example 3.6.2 DIR Example Software for Digital Controllers Summary 413 414 417 417 420 421 424 425 428 428 429 430 43 435 Discrete State-Variable Model 436 Introduction State-Variable Representation Time-Domain State and Output Equations for Sampled-Data Control Systems State-Variable Representation of a Discrete-Time SISO System 14.4 Phase-Variable Method 14.4.2 Canonical-Variable Method 14.4.3 Physical-Variable Method 14.4.4 State Transition Equation 14.4.5 State-Variable Representation Summary State-Variable Representation in the z Domain System Stability Time Response between Sampling Instants Summary 436 437 State-Space Design Methods 477 Introduction State-Feedback Pole Placement State-Variable Feedback: Parameter Insensitivity State Feedback Using Digitization State-Feedback H -Equivalent Digital Control System Design Procedure Frequency- Domain Compensation Design Using Mean-Square Error Minimization Digital Filters Direct s-Plane to w-Plane Transformation Scalar Case Relationship 5.9.2 Vector-Matrix Formulation 9.3 Accuracy Considerations of the w Transformation 5.9.4 Model Relationship as T � Zero 9.5 Norrnal Form Summary 477 477 479 479 483 486 443 448 448 453 459 462 464 464 47 474 476 490 493 499 500 502 503 505 506 506 738 ANSWERS TO SELEcrED PROBLEMS (a) for Q = 0, Pc = [ 0559 0.27 1 237 0.27 1 ] Kc = [0.5517 0.32 8] and the closed-loop eigenvalues are 0.4343 and -0.7676: (c) for Q = 51: P c Kc = = [ 65.455 682 6.1682 89 ] [0.4090 0.3404] with the eigenvalues of the closed-loop transfer function -0.0360 and 493 The value of J is �Pcxo As Q increases so does Pc and Kc becomes relatively smaller In this case the state moves toward zero with faster decay from the initial conditions as higher Q implicitly directs Chapter 19 19.3 (a) The steady-state covariance of the error is P = 0.5428 with the Kalman filter gain and the estimator eigenvalue 1577 (b) The steady-state covariance of the error is P = 0279 with the Kalman filter gain 0.2229 and the estimator eigenvalue 0.0542 19.4 Kc = 0.46, u(k) = - 0.46x(k), and Kf = 0.44 using the CAD package INDEX Abstract data types, 291 Analog-to-digital converter (AID) (Cant.) Acceleration error coefficient, 26 conversion time, 234 Accuracy, I , 1 data-transfer organization, 230 scaling, 356-358 Adaptive control, 603-604 errors, 243-245 flow structure, 230 Admissible controls, 566 integrated interface, 248 Advance operator, 103 methods for: Aerospace vehicle, 14 continuous, 237 Aircraft, 1 counter, 236 Aircraft digital controller, 14 feedback, 236 Algorithms, 274-278 logical process, 268 realization, 275 Aliasing, 200 01 Nyquist frequency, 201 All-pass filter (apf), 3-514, 525 lag characteristic, Alphanumeric encoding (information), 223 American Standard Code for Information Exchange (ASCII), 223 Amplitude modulation, Analog system, 4, 3-5 17 Analog-to-digital converter (AID), 2, 4, 7, 1 , simultaneous (flash), 235 successive approximation, 240 performance, 243 reference, 229 sampling time, 234 scaling, 232 time delay, 221 timing, 230-23 Analysis, stochastic, 13, 1-329, 593-605 Analytical model, 7-1 Architecture, computer system, 4, 27 -274 non-minimum-phase QFr design, 4-5 Arithmetic logic unit (manipulation unit), 272 234-243, 245-250 accuracy, 234 aperture time, 234 comparator operation, 23 Assembler, 278 Arithmetic process, 696-700 Assembly language, 278 Autocorrelation, 2, Autocovariance, 2, 16, 327 740 INDEX Autoregressive (AR), 585 Autoregressive moving-average with exogenous inputs (ARMAX), 585, 595 Backward difference, 68 (See also Difference) Bandwidth, 33 Base, 6, 696 (See also Number systems conversion; Radix) Basic system, 20, 5-2 18, 387-388 Bilinear transfonnation, 526 characteristic of, 526 confonnal mapping, 178 Tustin, 201-206 w, 176 w', 176-194, 377, 509, 537, 695 Binary-coded decimal (BCD), 223 Binary coding, 222 Binary number system, 696-700 (See also Number systems conversion) Bipolar coding, 232 Block diagrams, 1-140 Bode plots, 54 Boolean logic (boolean algebra), 268-270 AND, 269 COINCIDENCE (COMPARISON), 27 connectives, 270 EXCLUSIVE OR, 27 expression, 270 logic circuit diagrams, 271 NAND (not-AND), 271 NEGATION, 269 NOR (not-OR), 27 OR, 269 Bounds, 524 disturbance BD , 14, 541 optimal, B�, 14, 541 stability, Bs , 528, 534, 541 tracking BR, 514, 541 lower BL , 14, 54 upper Bu, 514, 541 Breakpoint (comer frequency), 54 Butterworth filter, 493-496, 498-499 CAD (see Computer-aided design) Canonical state equations, 568 Canonical-variable method, 453-459 diagram, 457 Cascade compensation (see Compensation, cascade) Central processing unit (CPU), 272 Characteristic equation, 53 Characteristic equation (Cont.) closed-loop system, 153-155, 472 complex roots, standard fonn, 1 from control ratio, 153- 55 open-loop system, 153 roots fonn root locus, 154 Routh stability criterion, 80-1 83 stability, 54-160, 80-183 Chebyshev filter, 493, 496-498 Clock signal, 304 Closed-loop control system, 2, 137-145, 508-545 characteristic equation, 53-155, 472 environmental conditions, 413 transfer function, 139, 53, 468 Coding, 223 alphanumeric infonnation, 223 ASCII, 223 binary-coded decimal (BCD), 223 cyclic codes, 223 decimal numbers, 696-700 gray code, 223 Coefficients, 9, 14, 267 error (see Error coefficients) realization, 267, 348-349 sensitivity to finite word length, 352-356 Cohesion, module, 291 Companion fonn, 450 Companion matrix, 440 Compensation, 12, 19-23 cascade, 12, 370-412, 1-545 deadbeat response, 1-412 design procedure, 370-372 DIG, 372-394 DIR, 394-404 versus feedback compensation, 3-4 Guillemin-Truxal method, 23, 46-49 objectives of, 20 QFT, 508-545 (See also Quantitative feedback theory) root locus, 39-49, 378-385, 390, 396-400, 408-409 set-point PID, 404-406 feedback, 49-50, 370-37 , 413-431 versus cascade compensation, 3-4 14 disturbance rejection, 1-62, 424-428 frequency domain, 3-498 general analysis, 414-417 Guillemin-Truxal method, 417-421 objectives of, 20 root locus, 414-417, 420-424, 427, 429-4 , 482, 485 Compensator, 12, 1 , 17, 528 cascade, Guillemin-Truxal method, 23, 46-49 coefficient realization, 348-352 INDEX Compensator (Cant.) continuous-data, finite-word-length implementation, 330-362, 410 lag (see Lag compensator) lead (see Lead compensator) lead-lag (see Lead-lag compensator) passive, PID, 23, 404-408 realization, 341 -352, 375, 410 cascade, 343 direct, 343 error sources, 348-352 parallel, 343 scaling, 356-358 (See also Digital controller) Compiler 277 Complements, 696-697 number, 696-697 Complex translation theorem, 79, 19 Computational devices, I Computational time delay, 7, , 221 Computer(s), I I arithmetic operations, 341-348 numbers, 696-700 programming, 267-3 10 word length, 348-356, 696-700 Computer-aided control system design (CACSD), 2, 674 67 Computer-aided design (CAD), for control-system analysis and synthesis, 532, 674 67 QFT (w-domain and discrete), 709-7 Computer arithmetic, 696-700 Computer engineering, Computer logic, 268-270 (See also Boolean logic) Computer operating system, 277, 299-304 Computer program, 2, 12, 275 Computer system architecture, 272 Constant coefficients, 9, 14 Construction process, 96 continuous signal, 96 Continuous amplitude-continuous time control systems, 7-8 Continuous amplitude-discrete time control systems, 7-8 Continuous control theory, 2, 19 62, 508 Continuous-data control systems, 2, 12, 58 response characteristics, 19 62 state-variable representation of, 436 443, 477-483 Continuous signal, 96 construction process, 96 reconstruction process, 96 741 Continuous-time system, 1-73, 508 linear difference equation model of, 68-73 Control law, algorithm, 198, 274-277 difference equation, 267 digital, 70, 198 Control problem, general nature of, 16-17 Control ratio, 32, 43, 55 acceptable closed-loop, 1-5 approximate, 55 desired model, 43-49, 374, 8-419, 48 frequency response, 32-39 pseudo-, 43, 531 pseudo-disturbance, 531, 537 pseudo-tracking, 165, 431 synthesizing disturbance, 508, 2, 540 synthesizing tracking, 43-49, 508, 2, 520, 538-545 Control system, 1-18 analysis and synthesis of, 12- approximation techniques, closed-loop (see Closed-loop control system) design aids, design approach, 12-13 digital (see Digital control systems) discrete-data, 1-145 (See also Sampled-data control systems) disturbance rejection, 62, 424-428, 508-545 MIMO, 5, 508-5 10, 13-517, 542 MISO, 19, 50, 508-545 open-loop (see Open-loop control system) performance specifications, 19-2 , 23-25, 32-34 SISO, 20 62 Control unit, 27 1-272 Controllability, 546, 548-552 complete, 549 complete output, 549 complete state, 549 matrix, 550 strong, 549 weak , 549 Controller, 2, 12, 410 (See also Compensator) Conventional control theory, 1-2, 19 62 Convergence, 101 Conversion of number systems, 696-700 Conversion time, 7, 1 , 221 Converters, 2, 224-250 AID (see Analog-to-digital converter) D/A (see Digital-to-analog converter) errors, 33 1-341 Convolution, 667-668 frequency-domain, 76 742 INDEX Convolution (Cont.) integral, 79, 87, 667 summation, 94 Correlation, 34, 1 , of sinusoidal and time response, 34-38 between time and z domains, 1 1- 17 Countable infinite sequence, 703 Coupling, module, 29 Covariance matrices, 595 Covariance tuning, 595 Cross-correlation sequence, Cross-covariance function, Cumulative distribution function (CDF), 316 Cyclic codes, 223 gray, 223 Dahlin algorithm, 376 Damping ratio, 25, 32 of dominant poles, 25, 27 effective, 32 relationship of, to resonant frequency, 32 Data conversion process, 94-98 Data dictionary (DD), 281 Data-flow diagram (DFD), 285 Data-hold device, 5, 94 Data structure, 277 Deadbeat response, 321 , 326, 41 -4 2, 653 Deadlock, 286 Delay operator, 102 Delay time, 221 Difference, 68 first-backward, 68, 206-209 first-forward, 70 rth-backward, 68 second-backward, 68 Difference equation, 90, 322 control law, 267 digital computer, 14, 90 Differential equation, Differentiation, 65 numerical, approximation for, 65, 70 Differentiation theorem, 20 Digital-coded signals, 221-223 Digital computer, I algorithm, 147-148, 274-277 transfer function, 145-149 Digital-computer solution, CAD, , 674-677 Digital control, \3-14 Digital control development, 1-18 aircraft, 14 model transfer function, 9-10 process control, 14 robotics, root locus, 2, 166 75 software, 43 1-434 time domain, 2, 32-34 Digital control law, 70 Digital control systems, 1-18, 274-277, 508-545 advantages of, I I architecture, 2, 271 disadvantages of, 1 quantization error, 1 , 331 state-feedback H -equivalent, 483-490 Digital controller (compensator), 12-13, 43 arithmetic operations, 341-348 realization, 375, 381, 489 Digital filter, 493-499 s-plane to w-plane transformation, 499-506 Digital flight control systems, 144 Digital signal processing, 13-14 Digital systems, Digital-to-analog converter (D/A), 4-7, I I , 225-228 binary weighted, 226 integrated interface, 248 ladder network, 225 R-2R ladder, 226 227 resolution, 228 time delay, 221 Digital transducers, 1 , 250 260 Digitization (DIG) technique, , 199, 3-21 , 370 392, 509-545 Direct digital control (DOC), 5, 14 Direct (DIR) technique, 151, 199, 4, 218, 370, 3-431 Discrete amplitude-continuous time, 7-8 Discrete amplitude-discrete time, 7-8 Discrete control analysis, 1-197 Discrete-data control system, 14-15 (See also Sampled-data control systems) Discrete-data systems, Discrete quantitative feedback technique (see Quantitative feedback theory, discrete MISO) Discrete signal flow graph, 142-145 Discrete state transition equation, 445, 462-464 Discrete state-variable model, 436-490 algorithms, 2, 274-277 Discrete time control theory, 4, 63-21 interdisciplinary field of, 13-14 real-time scheduling, 295 Discrete-time systems, , 443-490 SISO state-variable representation, 448-464 software for, 274 timing, 221 canonical variables, 453-459 phase variables, 448-453 Discrete-time systems (Cont.) physical variables, 459-464 state transition equation, 462-464 Discrete transfonn analysis, 19S-219, 370-43 Disnubance, 33, 424-428 uncertainty of, 1 Disnubance rejection, 2, 1-62, 424-428, 508-545 control system, 615-616, 629 design procedure, 55-56, 49�99 frequency domain, 54-58, 60-6 , 49�94, 508-545 model transfer function, , 508 root locus, 425-427 time domain, 52�, 424-428 Domain, 12 continuous, 12 discrete, 12 frequency, 12 time, 12 Dominant poles, 23-24, 27, 72 Dominant roots, 23-24, 28 Wn, INDEX 743 Figures of merit, 19-20 conventional control system, 24-25 Filtering problem, 591 Filters, 14, 49�98 all-pass, 3-5 14, 525 Butterworth, 493-499 Chebyshev, 493, 496-498 design of prefilter-QFr F(w), , 535, 542 digital, 14 forward-frequency transfer function, 83 holding device, 94-98 ideal low-pass, 82 input, 1 recursive, 331 smoothing, 94 squared-magnitude minimization, 493-496 transfer function, 1 Final-value theorem, 18, 1 9-122 Finite word length, 6, 14, 33(}"362, 420 First-backward difference, 68, 206-209 First-difference transfonnation, 206-209 First-order hold, 98, 245-247 Fixed-point representation, 699 Elastance, 127 Fly-by-wire (FBW) control system, 14 Folding, 200-201 Forward transfer function, 83, , 135 Ensemble, 314 Fourier analysis, 74-8 Effective 32 Effective (, 32 Eigenvalues, 471-472 Environmental conditions in closed-loop systems, 413 Ergodic random process, Error coefficients, static, 25-26 conventional systems, 25-27, 39, 42, 163-165 parabolic (acceleration), 26, 163-165 ramp velocity, 26, 162-166 step (position), 26, 62-165 use of, 160-166 w domain, 186 z domain, 162-163, 219 Errors, 25-27 steady-state, 25-27, 160-166 system, 25-27 (See also Observer) Estimation, 583-605 Examples, extensive, 387-392, 408-412, 428-431 Experiment, probability, 701-703 Extended Kalman filter, 595 Feedback compensation (see Compensation, feedback) Feedback control system, frequency response, 32-39, 508-545 Feedback gain matrix, optimal, 570 Flowchart, 276 Fourier coefficients, 75 Fourier series, 75, 664 Fourier transfonn, 75, 661-666 discrete (OFf), 665 fast (FFr), 665 Frequency, 23 damped natural, 23-24, 32 undamped natural, 32 Frequency domain, 12 compensation, 493-498, 508-545 convolution, 75 sampling process, 73-84 time-dornain correlation, 32-34, 88-194 Frequency response, 32 disturbance rejection, 54-58, 50S-545 effect of additional poles and zeros on, 34-37 method, 12, 50S-545 from pole-zero diagram, 32-37 resonant frequency, 32-33 sarnpled-data systems, 400-404 second-order systems, 32-34 system performance, 32-39 Gaussian density (normal), 705 Guidance landing system 744 INDEX Guidance systems, Guillemin-Truxal method, 23, 42, 47-49, 372-379, 388-389, 7-420 Heq(s), 479-493 H00 , 6QO- {)()2 Half-power point, 33 Hamiltonian function, 568 Hexidecimal number system, 697 Higher-order languages (HOL), 277 Higher-order system response, 24 Hold device, 5, 7, 3, 63, 98 and sampling, 5, , 3, 63, 98, 245-247 zero-order hold (see Zero-order hold) Homogeneous equation, 153, 445 Hybrid signal flow graph, 140 148 basic structures, procedure for, 14 1-142 Hybrid systems, sampled-data control systems, 5, 1-140 ICECAP (computer-aided design), 674, 676 Ideal sampler, 5, 77-83, 30 complementary (folded) spectra, frequency spectrum, 80 fundamental (primary) frequency, 80 Laplace transform, 79 Identification, 584 nonparametric, 589 Impulse, relationship to pulse, 80 response method, of unit strength, , Impulse sampler, 5, 77 Fourier transform, 78 Impulse train, 5, 73, 78 ideal, 78 Impulse transfer function, 91 Impulse weighting sequence, , Independent random variables, 706 Inertia, 27 Information hiding, 291 Information processing device, 7, 27 Inherent sampling, Initial condition (values), 437, 444 Initial value theorem, 19, 22 Input-output, 271 algorithm, 261 direct memory access (DMA), 260 management of operating system, 301 programming, 261 task driven scheduling, 295-302 Input-output devices, 271 interrupt, 263 Input-output hardware structure, 261 Input-output interfaces, 248-250 Input signals, 27 deterministic, 1 disturbance, 1 parabolic, 27 ramp, 27 random, standardized, 25 tracking, 1 unit step function, 19-20, 23-37 Integral: convolution, 667 superposition, 667 Integration, 68 numerical, 68-70 rectangular, 68, 69 trapezoidal, 90 Interdisciplinary field of digital control, 3-14 Interrupts, 263-265 event driven, 263 input-output, 299-304 (See also Operating systems) Inverse z transform (see z transform, inverse) Jacobi elliptic function, 498 Joint probability, 306 distribution function, 6, 704 cumulative, 306 mass function, 705 Jordan canonical form, 455 Jury-Blanchard array, 157 Jury's stability test, 157-160 Kalman estimator gain, 592 Kalman filter, 592-597 Kalman gain, 570 Kronecker delta function, Lag compensator, 21 application rules, 39, 395 cascade, , 39, 385, 397-400, 409-41 frequency-response characteristics, 403-404 network, 22 passive, 22, 396 root locus, 39 Lagrange multiplier, 567 Language, 271 machine, 277 INDEX Laplace transform, Mapping between s and z domains 745 (Cont.) constant frequency loci, 14 application of, 29 Lead compensator, primary strip, 1 1-1 14 application rules, 40-4 , 395 Marginal probability distributions, 706 cascade, , 379-384, 389-392, 408-409 Markov process, 324 frequency-response characteristics, 400-403 Mason's gain formula, 140, 693 tJ94 general design, 40-42 Matrix, 678-688 network, 22 canonical form, 453-459 passive, 22, 396 characteristic values, 680 root locus, 41-42 companion, 440 complex form, 683 tJ88 Lead-lag compensator, 22 cascade, 22, 385-387, 400 control, 438 network, 22 eigenValues and eigenvectors, 680 passive, 22, 396 feedforward, 438 inverse, 438, 681 Least significant bit (LSB), 225 Least significant digit (LSD), 697 linear algebra, 678 tJ79 Least-squares error estimate, nonsingular, 438 Least-squares identification, 586 output, 438 recursive, 588 plant, 438 Limit cycles, 358-366 quadratic form, 683 tJ88 Linear difference equations, resolvent, 438 continuous-time system model, 68-73 similarity transforms, 683 generation and solution, 68-73 state transition equation, 439 sampled-data systems, 68 transfer function: Linear optimal regulator problem (LQR), 566, closed-loop, 468 open-loop, 468 568-574 Linear quadratic (LQR) control problem, 566, 575 Linear quadratic gaussian (LQG) control problem, transpose of, 440 Matrix norms, 688 Matrix operations, 680 tJ83 584 Maximum frequency response Linear regression, 585 Mm, 24, 32 determination of, 32 Linear system, 7-8, 508 random input signals, 2, 9-322 relationship of: sampling process, 63-88 to damping ratio, 32 spectral density, 325 to peak overshoot, 32 Linear time-invariant (LTI) system, 63 tJ7, 508, Logic, boolean Maximum likelihood, 587 Maximum principle, 567-568 19 (see Boolean logic) Mean square value, 707 Mean-squared error minimization, 493-496, 586, Logical processes, 268 59 Logical reasoning, 268 Measurements, 250 260 Logical statements, 268 Loop transmission frequency (w