Computer Controlled Systems: Theory and Applications

491 Pages·1991·18.492 MB·English

by  George A. Perdikaris (auth.)

Computer Controlled Systems
International Series on
MICROPROCESSOR-BASED SYSTEMS ENGINEERING
VOLUME8
Editor
ProfessorS. G. Tzafestas, National Technical University, Athens, Greece
Editorial Advisory Board
Professor C. S. Chen, University ofA kron, Ohio, U.S.A.
Professor F. Harashima, University ofTokyo, Tokyo, Japan
Professor G. Messina, University ofCatania, Catania, Italy
Professor N. K. Sinha, McMaster University, Hamilton, Ontario, Canada
Professor D. Tabak, Ben Gurion University ofthe Negev, Beer Sheva, Israel
The tit/es published in this series are listed at the end ofthis volume.
Computer
Controlled Systems
Theory and Applications
by
GEORGE A. PERDIKARIS
University ofWisconsin-Parkside, Kenosha,
Wisconsin, U.S.A.
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Perdikaris, George A.
Computer controlled systems : theory and applications I George A.
Perdikaris.
p. cm. -- <Microprocessor based systems engineering : v. 8l
Inc 1u des 1n dex.
1. Automatie control. 2. M1croprocessors. I. Title.
II. Series: International series on mlcroprocessor-based systems
eng1neer1ng ; v. 8.
TJ223.M53P43 1991
629.8'95--dc20 91-29108
ISBN 978-90-481-4100-5 ISBN 978-94-015-7929-2 (eBook)
DOI 10.1007/978-94-015-7929-2
Printed on acid-free paper
All Rights Reserved
© 1991 Springer Science+Business Media New York
Originally published by K.luwer Academic Publishers 1991
Softcover reprint of the bardeover 1st edition 1991
No part of the material protected by this Copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
PREFACE ..................................................................................... ix
Chapter 1: INTRODUCTION ............................................................ 1
1.1 OVERVIEW ................................................................................. 1
1.2 CONTINUOUS (ANALOG) CONTROL SYSTEMS .................................2
1.2-1 Open-Loop Control Systetns ..................................................2
1.2-3 Closed-Loop (Feedback) Control Systems ..................................3
1.3 DISCRETE (DIGITAL) CONTROL SYSTEMS .......................................5
1.3-1 Open-Loop Digital Control Systems ..........................................6
1.3-2 Closed-Loop Digital Control Systems ........................................?
1.4 INTERFACE DEVICES ...................................................................9
1.4-1 Analog-to-Digital Converter (ADC) ..........................................9
1.4-2 Digital-to-Analog Converter (DAC) .......................................... 12
1.4-3 Incremental and Absolute Encoders .........................•................ 14
1.5 IMPLEMENTATION CONSIDERATIONS ............................................ 16
1.5-1 Sampling Frequency Selection ................................................ 16
1. 5-2 Quantization Errors and Effects ............................................... 17
1.5-3 Computer Wordlength .......................................................... 18
1.5-4 Computational Delays .......................................................... 18
Chapter 2: CONTINUOUS-TIME SYSTEMS ...................................... 19
2.1 INTR.ODUCTION .......................................................................... 19
2.1-1 Continuous Singularity Functions ............................................2 1
2.1-2 Systems Analysis and Design .................................................2 4
2.2 DIFFERENTIALEQUAT IONS ..........................................................2 5
2.2-1 Electrical Networks .............................................................2 6
2.2-2 Mechanical Systems ............................................................2 9
2.2-3 Control Systems ................................................................ 30
2.3 TIME RESPONSE USING CLASSICAL METIIODS ................................ 32
2.3-1 Solution of Homogeneous Differential Equations .......................... 32
2.3-2 Solution of Nonhomogeneous Differential Equations ......................4 0
2.4 THE LAPLACE TR.ANSFORM .......................................................... 53
2.4-2 TABLES OF LAPLACE TRANSFORMS .................................. 58
vi Computer Controiiet! Systems: %eory antl.91pp{ications
2.4-3 Partial Fraction Expansion .....................................................6 4
2.5 TIME RESPONSE USING TRANSFORM METHODS ..............................7 0
2.6 TRANSFER FUNCTIONS AND BLOCK DIAGRAMS .............................7 7
2.6-1 Transfer Functions and Differential Equations ..............................7 8
2.6-2 Transfer Functions of Electrical Networks ..................................7 9
2.6-3 Transfer Functions of Simple Block Diagrams .............................. 81
2.6-4 Block Diagram Reduction ...................................................... 85
2.6-5 Steady-State Error Analysis ................................................... 92
2.7 TRANSFER FUNCTIONS AND SIGNAL FLOW GRAPHS ....................... 94
2.7-1 Signal Flow Graph Definitions ................................................9 5
2.7-2 Mason's Gain Formula .........................................................9 6
2.8 CONTINUOUS SYSTEMS POLES, ZEROS, AND STABll..ITY .................. 102
2.8-1 Routh's Stability Criterion ..................................................... 105
2.8-2 Root Locus Dia gram ............................................................ 110
2. 9 FREQUENCY RESPONSE CONCEPTS ............................................... 114
2.9-1 Frequency Response Plots. .................................................... 114
2.9-2 Frequency Plots of the Open-Loop Transfer Function ..................... 125
2.9-3 Correlation ofTransient Time and Frequency Responses ................. 128
2.10 DIGITAL SIMULATIONMETHODS ................................................... 131
2.10-1 System Simulation by Programming Block Diagrams ..................... 132
2.10-2 System Simulation Using ACSL. ............................................. 134
2.10-2 System Simulation Using SIMULAB ........................................ 135
Chaper 3: DISCRETE-TIME SYSTEMS ............................................ 139
3.1 INTRODUCTION .......................................................................... 139
3.1-1Discrete Singularity Functions .................................................... 141
3.2 DIFFERENCE EQUATIONS ............................................................. 144
3.3 TIME RESPONSE USING CLASSICAL METHODS ................................ 147
3.3-1 Solution of Homogeneous Difference Equations ........................... 148
3.3-2 Solution ofNonhomogeneous Difference Equations ....................... 153
3.4 SAMPLING THEORY ..................................................................... 157
3.4-1 The Zero-Order-Hold (ZOH) Element. ....................................... 164
3.5 THE Z TRANSFORM ..................................................................... 167
3.5-1 Z Transform Theorems ......................................................... 168
3.5-2 Tables of Z Transforms ........................................................ 173
3.5-3 Method of Long Division ...................................................... 176
3.5-4 Partial Fraction Expansion ..................................................... 179
3.6 TIME RESPONSE USING Z TRANSFORM METHODS ........................... 182
3. 7 TRANSFER FUNCTIONS AND BLOCK DIAGRAMS ............................. 186
3. 7-1 The Pulse Transfer Function .................................................. 187
3.8 DISCRETE SYSTEMS POLES, ZEROS, AND STABILITY ........................ 201
3.8-1 Bilinear Transformation ........................................................2 05
3.8-2 Routh's Stability Criterion .....................................................2 06
3.8-3 The Jury Stability Criterion ....................................................2 08
3.8-4 Root Locus Diagram for Discrete systems ...................................2 11
3.9 FREQUENCY RESPONSE OF DISCRETE SYSTEMS ..............................2 12
3.10 DIGITAL SIMULATION AND Z TRANSFORMS ....................................2 18
3.10-1 Numerical Differentiation and Integration ....................................2 18
3.10-2 Numerical Solution of Differential Equations ...............................2 27
3.10-3 Simulation ofDiscrete Systems Using SIMULAB .........................2 31
Chapter 4: ST ATE VARIABLE METHODS .........................................2 35
4.1 IN1RODUCTION ..........................................................................2 35
Co11f1?Uter Contro{fd S!Jstems: rtneoru ana JiW!Cications vii
4.2 STATE SPACE CONCEPTS CONTINUOUS SYSTEMS ...........................2 36
4.2-1 State Transition Equation ......................................................2 41
4.2-2 State Equations and Transfer Functions ......................................2 49
4.2-3 From Transfer Functions to State Equations ................................2 50
4.2-4 Phase-variable Canonical Form ...............................................2 55
4.3 STATE SPACE CONCEPTSDISCRETE SYSTEMS .................................2 58
4.3-1 Solution of Discrete State Equations ..........................................2 61
4.3-2 Discrete State Equations and Transfer Functions ...........................2 64
4.3-3 From Discrete Transfer Functions to State Equations ......................2 65
4.4 STAT EEQUAT IONS DISCRETE FROM CONTINUOUS ..........................2 71
4.6 CONTROLLABILITY AND OBSERVABILITY .......................................2 75
4. 7 CASE STUDY ANALYSIS OF A MOTOR DRIVE SYSTEM .......................2 80
Chapter 5: DIGIT AL SIGNAL PROCESSING ....................................2 91
5.1 INTRODUCTION ..........................................................................2 91
5.2 DIGITAL FILTER CONCEPTS ..........................................................2 92
5.3 ANALOG FILTER CONCEPTS .........................................................2 95
5.3-1 Butterworth Filter Forms ......................................................2 96
5.3-2 Binomial Filter Forms ..........................................................2 99
5.3-4 Other Types of Analog Filters .................................................3 00
5.4 DESIGN OF IIR DIGITAL FILTERS ................................................... 302
5.4-1 IMPULSEINVARIANCEMETIIOD ....................................... 302
5.4-2 Step Invariance or Zero-Order-Hold (ZOH) Method ....................... 303
5.4-3 Pole-Zero Matehing Method ...................................................3 04
5.4-4 Bilinear Transformation Method ..............................................3 05
5.5 DESIGN OF FINITE IMPULSE RESPONSE (FIR) FILTERS .....................3 11
5.5-1 Lagrangian Multipliers Method ................................................ 312
5.5-2 Least-Squares Method ..........................................................3 14
5.5-3 FIR Filters for Differentiation, Smoothing, and Prediction .................. 315
5.6 DIGITA L FILTER IMPLEMENTATION CONSIDERA TIONS .....................3 16
Chapter 6: COMPUTER CONTROLLED SYSTEMS ............................ 322
6.1 INTRODUCTION .......................................................................... 322
6.2 OPEN-LOOP CONTROL Deadbeat Algorithms ........................................ 323
6.3 THE LAPLACE TRANSFORM ME1HOD .............................................3 24
6.3-1 First-Order Plant ................................................................ 324
6.3-2 Second-Order PlantOverdamped Case ....................................... 326
6.3-3 Second-Order Plant Underdamped Case ..................................... 330
6.3-4 "Positive Cast" or "Posicast" Control ........................................3 34
6.3-5 Second-Order Plant Critically-Damped Case ................................3 35
6.4 THE Z TRANSFORM METHOD ........................................................ 337
6.5 THE STA TE VARIABLE METHOD ..................................................... 340
6.5-1 The nth-Order Plant Case ...................................................... 343
6.6 CLOSED-LOOP CONTROL Deatbeat Algorithms .....................................3 48
6.6-1 The Plain (Free) IntegratorPlant. ............................................. 349
6.6-2 First-Order Plant Case ..........................................................3 51
6.6-3 Second-Order Plant Case ...................................................... 353
6.6-4 The nth-Order Case ............................................................. 356
6. 7 DEADBEAT ALGORITHMS FOR RAMP INPUTS .................................. 358
6. 7-1 The Plain Integrator Plant ...................................................... 359
6.7-2 First-Order Plant Case .......................................................... 361
6.8 ALGORITHMS FOR CONTROLLING DISTURBANCES .......................... 365
6. 8-1 The Z Transform Method ...................................................... 366
6.8-2 The State Variable Method .................................................... 370
viii Computer Cantrotlei Systems: 'IIieory antf .9lpp!ic.ations
6.9 A CONTROL SCHEME FOR REFERENCE AND/OR DISTURBANCE. ......... 374
6.9-1 The Plain Integrator Plant Step Inputs. .......................................3 78
6.9-2 The First-Order Plant Step Inputs. ............................................ 379
6.9-3 The Plain Integrator PlantRamp Inputs ...................................... 381
6.10 SIMULATION OF COMPUTER CONI'ROL SYSTEMS ............................ 383
6.10-1 Simulation Languages .......................................................... 383
Chapter 7: COMPUTER CONTROLLED SYSTEMS ........................... 392
7.1 INIRODUCI10N .......................................................................... 392
7.2 TUE ANALOG AND DIGITA L PID ALGORITHMS ................................. 395
7.3 TUE ANALOG AND DIGITAL PDF AND PDFF ALGORITHMS ................. 397
7.3-1 PDFF Control ofthe Plain Integrator Plant ..................................4 01
7.3-2 PDFF Control of the Double Integrator Plant ...............................4 03
7.3-3 Tuning of Control Algorithms by Standard Filter Forms ..................4 03
7.4 CASE STUDY DIGITAL CONTROL OF A MOTOR DRIVE PLANT .............4 08
7.4-1 Velocity and Position Plant Models ...........................................4 08
7.4-2 Digital Control of Velocity and Position Plants ............................ .410
7.4-3 Fittering the Derivative (Kd) Path ............................................ .417
7.4-4 A Second-Order Derivative (Kj) in the Feedforward Path .................4 19
7.4-5 Time and Frequency Analysis of Digital Control Systems ................. 421
7.4-6 Steady-State Error Analysis of Digital Control Systems ...................4 25
7.5 PHASE-LEAD CONTROL ALGORITHMS ............................................4 28
7.6 CONTROL ALGORITHMS FOR PLANTS WITH DEAD TIME ...................4 30
7.7 ZIEGLER-NICHOLS TUNING METHOD FOR PID ALGORITHMS ............. 433
7.8 DIGITAL CONTROLLER IMPLEMENTATION CONSIDERATIONS ............4 36
Chapter 8: SYSTEM IDENTIFICATION ............................................4 40
8.1 INI'RODUCI10N ..........................................................................4 40
8.2 System Identification A Time Domain Approach ........................................4 40
8.2-1 Identifying the Discrete Transfer Function G(z) ............................4 41
8.2-2 Determining G(s) from G(z) ...................................................4 43
8.2-3 Identifying the Pulse Transfer Function GhG(z). ...........................4 47
8.3 A MODEL REDUCTION SCHEME .....................................................4 48
8.3-1 Model Reduction from Transfer Function Models ..........................4 48
8.3-2 Model Reduction from a State Space Model.. .............................. .451
8.4 CASE STUDY REDUCED-ORDER MODELS FORA MOTOR DRIVE .......... .455
APPENDIX A:. REFERENCES ..............................................................4 61
APPENDIX B: REAL-TIME PROGRAMS ................................................4 64
INDEX ..............................................................................................4 74
Preface
The primary objective of the book is to provide advanced undergraduate or frrst-year
graduate engineering students with a self-contained presentation of the principles
fundamental to the analysis, design and implementation of computer controlled systems. The
material is also suitable for self-study by practicing engineers and is intended to follow a first
course in either linear systems analysis or control systerns.
A secondary objective of the book is to provide engineering and/or computer science
audiences with the material for a junior/senior-level course in modern systems analysis.
Chapters 2, 3, 4, and 5 have been designed with this purposein rnind. The emphasis in such
a course is to develop the rnathernatical tools and methods suitable for the analysis and design
of real-time systems such as digital filters. Thus, engineers and/or computer scientists who
know how to program computers can understand the mathematics relevant to the issue of
what it is they are programrning. This is especially important for those who may work in
engineering and scientific environments where, for instance, programrning difference
equations for real-time applications is becorning increasingly common. A background in
linear algebra should be an adequate prerequisite for the systems analysis course.
Chapter 1 of the book presents a brief introduction to computer controlled systems. It
describes the general issues and terminology relevant to the analysis, design, and
implementation of such systems.
Chapter 2 lays the foundation of systems analysis using the more farniliar continuous
concepts. It develops and applies the classical and Laplace transform methods to analyze,
model, and simulate systems, both in time and frequency domains. Many of the systems
analyzed in Chapter 2 are designed in later chapters as practical control systems.
Chapter 3 parallels the development of Chapter 2, but for discrete-time or discrete
systems. The important theories of sampling and Z transforms are introduced and applied to
model, simulate, and analyze discrete systems. Numerical integration and differentiation
methods are developed because of their important role in discretizing continuous filters and
controllers.
Chapter 4 is devoted to the topic of state-space techniques. The mathematics of
modern systems theory are presented and integrated for continuous and discrete systems.
The state equations are formulated and solved using time as well as transform techniques.
The problern of obtaining discrete state equations from continuous is also considered, with
emphasis on algorithms suitable for computerization.
The discrete and continuous mathematics of Chapters 2, 3, and 4 are used throughout
the book. Furthermore, these mathematics form the necessary background for advanced
studies in areas such as adaptive control, optimization theory , and for Iiterature
investigation.
Chapter 5 presents methods for transforrning analog prototype filters into digital
filters by applying mathematical tools developed in Chapters 2, 3, and 4. The design and
implementation of digital filters is viewed as an intermediate step between systems in general
and computer controlled systems in particular. The types of filters studied are used in later
chapters in the design of digital control systems.
Chapters 6 and 7 deal with the digital control problem. These chapters deal with the
design and implementation of algorithms used in various control schemes or strategies.
Transform as well as state variable methods are employed so that the reader can associate the
new material with what he/she already knows. This approach adds to the reader's
circumspection of the entire computer control problem.
The coverage of Chapter 6 emphasizes the computer control of plants for which a
mathematical model is known or can be readily obtained. The concepts of open-loop and
closed-loop control using finite-settling-time algorithms, also called deadbeat algorithms, are
X Computer Contro«ea Systems: 7'fteory antf .91pp{ications
presented. A general scheme for controlling plants when reference and/or external
disturbance inputs enter the controlloop is developed. Applications are illustrated by digital
simulation as wen as by laboratory measurement.
Chapter 7 deals with the computer control of plants for which a mathematical model
may or may not be readily available. It presents and applies the proportional-integral
derivative (PID), and the integral with proportional-derivative feedback (PDF) control
schemes. The PDF control scheme with added feedforward control action (PDFF) is also
considered. The control of more complex processes as wen as applications emphasizing the
computer control of motor drive systems are included.
Chapter 8 presents a practical method that can be used to identify a process by a
suitable mathematical model from its transient response. The problern of mathematical model
reduction by a systematic approach is also included.
The Appendices include sample programs for real-time signal processing and control
applications. Information about author's programs used in this book is also included.
The book was written by the author on an Apple Macintosh computer using Microsoft's
Word. The block diagrams were drawn using Deneba System's Canvas, and most of the
plots were generated using Synergy Software's Kaleidagraph.
The experimental charts of Chapter 6 were recorded in the laboratory. These charts
were subsequently scanned and included in the manuscript. The plant models were
prograrnmed on an analog computer which was controned by a digital microcomputer.
The experimental results of Chapter 7 were plotted from data "captured in real-time" as
the industrial drives were controned by a VME-based Motorola 68020 microcomputer. The
real-time programs are written in the C language using cross-compiler software by
lntermetrics.
Some computer simulations were performed using the Advanced Continuous
Simulation Language (ACSL) by Mitehen & Gauthier Associates, and some using MATLAB
and SIMULAB by The MathWorks. Most of the system analysis and design results were
obtained using the author's own set of programs. Data from these programs can be plotted
using a spreadsheet or a plotting program such as Kaleidagraph. For information and
availability of either an IBM PC or an Apple Macintosh version of the author's programs,
please contact the author.
This book project has been influenced by many technical works and people. Of those
works listed in the references, special thanks to the "pioneer" textbook writers C. L. Smith,
and particularly Cadzow & Martens on whose deadbeat control ideas much of Chapter 6 is
based. My special thanks to my esteemed coneague Dr. A.S. Rao, whose model reduction
scheme is highlighted in Chapter 8. Many thanks to Dr. Ron Brown and Tim Ruchti of
Marquette University for their valuable suggestions and contributions, and to Dr. Tom
Higgins, University of Wisconsin-Madison, and Dr. Spyros Tzafestas, National Technical
University of Athens, for prodding me to complete this book.
I am indebted to Tom Beck, Bill Hammel, Mike Lucas, Keith VanPatten, Mark Sevey
and many others at UNICO, Inc., for their continued support of UW-Parkside's Real Time
Systems (RTS) laboratory and related research projects.
I would also like to thank the fonowing people for their helpful comments and
encouragement over the years: Chuck Rohrer, TechniSoft; Ken Swanson and Gerald
O'Callaghan, Eaton-Dynamatic; Rabah Rennarre and Tom Rehm, Allen-Bradley; Dr. Steven
Bomba, Johnson Controls; and Bill Erickson and David Gerlach, Rexroth-Indramat. Thanks
to all of my colleagues at UW-Parkside and especially to Dr. Walter Feldt, Paul Sorensen,
Tom Labus, Padmini Varanasi, and to James Raymaker who literally "manned" the controls
in the RTS laboratory during the data collection phase of the book.
'Deaicatetf to 'Iia, Cfiristina, antf 'Tommy {eavaa1Js}
V.niversity of Wisconsin-Park§iae (jeorge PeraiKfl.ris