Digital control systems theory hardware software constantine h houpis gary b lamont

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.
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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.
3

4 5 6 7 8 9 0 CMO FC 9 5 4 3 2

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
1 .2

1 .3
1 .4
1 .5
1 .6
1 .7
1.8
1 .9
2

2.1
2.2
2.3
2.4

xix

Introduction

Introduction
Digital Control-System Modeling
1 .2. 1 Sampling Process
1 .2.2 System Terminology
1 .2.3 General Sampled-Data System Variables
1 .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
2
4
5
7
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

3.1
3.2
3.3

3.4
3.5
3.6
3.7
3.8
3.9
3.10
4

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. 1 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. 1 Partial-Fraction Method
4.5.2 Power-Series Method (Direct Division Method)
Limitations
Z Transform of System Equations
4.7 . 1 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
1 00
111
1 12
1 14
1 14
1 15
1 18
123
1 24
128
1 30
131
131
1 36
137
140
145
149

CONTENTS

5

5.1
5.2

5.3

5.4

5.5
5.6
5.7
5.8
6

6.1
6.2
6.3
6.4

6.5
6.6
6.7
7

7.1

7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7. 1 1

xiii

Discrete Control Analysis

151

Introduction
System Stability
5.2. 1 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. 1 Steady-State Error-Coefficient Formulation
5.3.2 Evaluation of Steady-State Error Coefficients
5 .3.3 Use of Steady-State Error Coefficients
Root-Locus Analysis
5.4. 1 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
1 76
1 77
1 80
183
1 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

1 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. 1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8. 1 1
8.12
8.13
8.14
8.15
8.16
8.17
9

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)
7 . 1 3 . 1 I/O Hardware Structure
7.1 3 .2 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. 1 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
8. 1 3 . 1 Real-Time Operating Systems Requirements
8.1 3.2 Simple Real-Time Operating System
Watchdog Timers
Sampling-Time Selection
8 . 1 5 . 1 Single- Rate Sampling
8 . 1 5.2 Multirate Sampling
User Interfaces to Real-Time Operating Systems
Summary

267
268
27 1
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
9. 5 . 1 Convolution Model
9.5.2 Difference Equation Model
9.5.3 Spectral Density of Linear Discrete Systems
Vector-Matrix Representation of Random Processes

31 1
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 1
348
352
356
358
367
368
369

10.1
10.2
10.3

1 0.4
10.5
10.6
10.7
10.8
10.9
10.10
11

1 1.1
1 1 .2
1 1 .3
1 1 .4

1 1 .5
1 1 .6
1 1 .7

1 1 .8
12

12.1
12.2
1 2.3
1 2.4
1 2.5
1 2.6
1 2.7
1 2.8
1 2.9

12.10
12. 1 1

Cascade Compensation-Digitization
(DIG) Technique

370

Introduction
Digitization (DIG) Design Technique
Guillemin-Truxal (GT) Compensation Method
Lead Cascade Compensation
1 1 .4. 1 s-Plane DIG Design
1 1 .4.2 w-Plane DIG Design
1 1 .4.3 s-Plane to w-Plane Correlation
Lag Compensation
Lag-Lead Compensation
Extensive Cascade DIG Example
1 1 .7.1 Analysis of the Basic System
1 1 .7.2 Guillemin-Truxal Approach
1 1 .7.3 Pseudo-Continuous Time (PeT) Approach
1 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)
1 2.9. 1 Lead Compensation
1 2.9.2 Lag Compensation
1 2.9.3 Controller Implementation
Deadbeat Response
Summary

394
394
396
397
400
400
404
406
408
408
409
410
41 1
412


xvi

CONTENTS

13

13.1
1 3.2
1 3.3
1 3.4
1 3.5
1 3.6
1 3.7

1 3.8
14

14.1
14.2
14.3
14.4

14.5
14.6
14.7
14.8
15

15.1
1 5.2
1 5.3
15.4
15.5
1 5.6
1 5 .7
15.8
1 5.9

15.10

Feedback Compensation

413

Introduction
General Analysis
DIR Technique for Feedback Control
1 3. 3 . 1 Guillemin-Truxal Approach
1 3.3.2 Root-Locus Approach
DIG Technique for Feedback Control
Controlling Unwanted Disturbances
13.5. 1 DIG Technique
1 3.5.2 DIR Technique
Extensive Digital Feedback Compensator Example
1 3.6.1 DIG Example
1 3.6.2 DIR Example
Software for Digital Controllers
Summary

413
414
417
417
420
421
424
425
428
428
429
430
43 1
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. 1 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
1 5 .9. 1 Scalar Case Relationship
1 5.9.2 Vector-Matrix Formulation
1 5 .9.3 Accuracy Considerations of the w Transformation
1 5.9.4 Model Relationship as T � Zero
1 5 .9.5 Norrnal Form
Summary

477
477
479
479
483
486

443
448
448
453
459
462
464
464
47 1
474

476

490
493
499
500
502
503
505
506
506


16

16. 1
16.2
16.3

16.4
16.5

Discrete Quantitative Feedback Technique

16.9
16. 1 0

17

17.1

17.2
17.3
1 7.4
1 7.5
17.6
1 7.7
17.8
18

18.1
1 8.2
1 8.3
1 8.4
1 8.5
18.6
1 8.7

xvii

508

Introduction
Continuous MISO and MIMO QFf Approach
Non-Minimum-Phase Analog Plant
16.3.1 Analog QFI' Design Procedure for an nmp Plant
Discrete MISO Model with Plant Uncertainty
QFf w-Domain DIG Design
16.5 . 1 Closed-Loop System Specifications
16.5.2 Plant Templates
16.5.3 Bounds B(jv) on Lo(jv)

16.5.4 Nonminimum-Phase Lo(w)
16.5.5 Synthesizing Lmo(w)
16.5.6 Ws 1 20 Is Too Small
16.5.7 Error in the Design
16.5.8 Design of the Prefilter F(w)
Simulation
Basic Design Procedure for a MISO Sampled-Data Control System
Applicability of Design Techniques to Other Plants
16.8. 1 MIMO Plants
16.8.2 Nonlinear Plants
QFf Technique Applied to the PCT System
Summary
16. 10. 1 Minimum-Phase, Nonminimum Phase and Unstable P(s)
16. 1 0.2 Disturbance Attenuation
16. 1 0.3 Conclusions

508
509
513
514
517
518
520
523
524
524
527
528
532
535

535
538
542
542
543
543
543
543
544
544

Introduction
Basic System Structure
Discrete Controllability
State-Space Pole Placement
Discrete Observability
State Observers
17.6. 1 Full-State Observers
17.6.2 Reduced-Order and Current Observers
State-Space Stability
Summary

546
547
548
552
553
557
557
561

563
563

=

16.6
16.7
16.8

CONTENTS

Modem Discrete Control Theory

Discrete Optimal Control

Introduction
Optimal Control Concepts
Maximum Principle
Discrete Linear Regulator
1 8.4. 1 General Second-Order System Optimal Controllers
1 8.4.2 Discrete Riccati Equation Solution
Optimal Control Variations
Sampling Time
Summary

546

565
565
566

567
568
574
575
580
581
582


xviii

CONTENTS

19

19.5
1 9.6
19.7

583
583
584
590
590
591
592
597
597
597
598

600
603
605
606

Problems

19.4

Discrete Estimation and Stochastic Control

Introduction
Parameter Identification
Discrete Optimal Estimation
19.3. 1 Additive Noise Model
19.3.2 Estimation Problem Formulation
1 9.3.3 Discrete Optimal Estimation
19.3.4 Solution to Optimal Filter Equations
Discrete Stochastic Control
19.4. 1 Combined Model
19.4.2 Optimal Performance
Roo Robust Optimal Control
Discrete Adaptive Control
Summary
Bibliography

19. 1
19.2
19.3

613

Appendixes

A
B
C
D
E
F
G
H
I
J
K

Fourier Transform
Convolution
Pade Approximation
Power Series
Computer-Aided-Design (CAD) Programs for Control
Matrix Manipulations
Signal Flow Graphs
w-Transformation Characteristics
Number Representations
Theory of Probability
Discrete QFT Design Process

66 1
667

669
672
674
678
689
695
696
701
709

Answers to Selected Problems

714

Index

739


PRE FACE ,

There is a need for a fundamental textbook on sampled-data control theory and
applications that emphasizes the use of the small digital computer as a controller. The
implementation of a digital controller allows design flexibility and system extendability
in an efficient and effective manner as compared to analog components. The real­
time adaptability of mode changes due to plant parameter variations, environmental
changes, and requirements modifications usually require a digital computer.
This text accomplishes the objective of providing sampled-data control system
topics mainly for upper-level undergraduate and first-year graduate students. It effec­
tively merges and interrelates the two general areas which are vital to a practicing

digital control engineer: discrete and sampled-data control theory and computer en­
gineering. This textbook provides a clear, understandable, logical development and
motivated account that spans sampled-data control theory with computer engineering
as an integrated entity.
The minimum background required for this book is a fundamental course in
continuous-time control systems. Some higher-order language programming expe­
rience would be useful but not necessary in appreciating the software engineering
sections. The textbook has been developed to achieve a minimal satisfactory under­
standing of discrete-data and sampled-data systems. This development is based on the
reader having a mathematical background in differential equations, integral calculus,
and Laplace transforms.
The authors have tried to exert meticuluous care with explanations, diagrams,
calculations, tables , and symbols. The text provides a strong, comprehensive, and
illuminating account of those elements of conventional control theory which have
relevance in the design and analysis of sampled-data control systems. The variety of
different design techniques presented contributes to the development of the student's
working understanding of what A. T. Fuller has called "the enigmatic control system."
To provide a coherent development of the subject, formal proofs and lemmas have


"X

PREFACE

been eschewed with an organization that draws the perceptive student steadily and
surely into the demanding theory of multirate, multivariable control systems.
The text, which is summarized in the following paragraphs, introduces the con­
cepts of sampled-data theory, relates it to continuous analysis methods as appropriate,
integrates computer engineering concepts with digital control implementation, and
presents both scalar and modem control design and synthesis techniques.

Chapter 1 introduces the sampling process model while the development of ideal
impulse sampling is detailed in Chapter 3. The use of linear difference equations to
model the performance of sampled-data control systems is also presented in Chapter 3.
Many of the analysis and design techniques of sampled-data control systems
emphasized are extensions of continuous control-theory methods that are reviewed in
Chapter 2, including the modeling of a desired system control ratio for tracking and
disturbance rejection.
Chapter 4 introduces the Z-transform (zee transform) as a method for the
analysis and design of sampled-data control systems in the z-plane. The correlation
between the pole-zero pattern in the s- and z-planes is presented with respect to time­
response characteristics. The properties and mathematical representations of open-loop
and closed-loop sampled-data control systems are developed, and their corresponding
block diagram and signal-flow representations are given.
Stability analysis as presented in Chapter 5 is accomplished by applying Jury' s
stability test o r Routh's stability criterion i n the z- and w-domains respectively.
The w' -transformation is developed as an approximation to the Laplace transform.
Chapter 5 binds together the s-, z-, and w'-domain analyses with respect to time­
response characteristics by means of the root-locus and frequency-response methods.
In Chapter 6 a general sampled-data control-system design technique is devel­
oped. The Pade approximation, the Tustin transformation, and the pseudo-continuous­
time (PCT) control system model are involved in this technique.
The basic organization of analog-to-digital (AID) and digital-to-analog (D/A)
converters and I/O programming is presented in Chapter 7. The objective is to develop
a detailed understanding of conversion processes in a digital control system. General
control transducers are also presented, providing insight into their construction, accu­
racy, and utilization in control systems.
A concise but integrated presentation of the fundamentals of computer engineer­
ing as related to digital control system is set forth in Chapter 8. This chapter provides
an introduction to logical operations, hardware architecture, software engineering, and
real-time operating systems as used in the detailed design of digital controllers and

their implementations.
Chapter 9 develops the foundation for the statistical analysis of finite word length
discussed in Chapter 1 0 and for optimal estimation considered in Chapter 19, and
presents the fundamentals of continuous and discrete random processes. Using this sta­
tistical background, Chapter 1 0 focuses on modeling the effects of finite word length.
Chapters 1 1 and 12 discuss two approaches of analyzing and designing a sampled­
data, cascade-compensated control system: in the direct (DIR) techniques analysis and
synthesis are done in the z-plane and for the digitization (DIG) technique all work is


PREFACE

xxi

accomplished in either the s- or Wi -plane. The interrelations and comparison of the
results obtained by using these design methods and the effect of T on these results are
thoroughly discussed with the PID (proportional integral derivative) controller used •
as an example.
Feedback-compensated digital systems are discussed in Chapter 13. The pre­
sentation deals with the tracking problem requiring that the system output follow the
system input and the disturbance rejection model.
Chapter 14 introduces the state-variable methods of representing a sampled­
data control system and includes the analysis of system performance by state-space
representation. Chapter 1 5 develops a design method for minimizing the effect of plant
parameter variations on the system output.
Chapter 1 6 presents for the first time in any textbook the quantitative feedback
theory (QFf) design technique for sampled-data control systems. This is a power­
ful technique in designing a robust control system for plants with plant parameter
uncertainty and disturbances.
Chapter 1 7 develops the concepts of controllability and observability with re­

duced state controllers and observers. Chapter 1 8 presents the development of discrete
optimal controllers along with their digital-computer implementation. Chapter 1 9
discusses the design of estimators for multi-inputs, multi-output systems with emphasis
on identification, and also addresses the concepts of digital adaptive control and
stochastic control systems. Various appendices support the understanding of various
models and design techniques.
The two important and major features of Digital Control Systems : Theory, Hard­
ware, Software are uniqueness and flexibility of use. The unique features are integration
of discrete and sampled-data control theory with computer engineering; degree of
accuracy needed as T becomes smaller because of its impact on calculations and
digital implementation, with emphasis on computer-aided design (CAD); discussion
of text examples by integrating control theory and digital computer implementation;
extensive development of system analysis and design by root-locus techniques in
various domains; development of the pseudo-continuous-time (PCT) control system
design and analysis; development of discrete optimal control and discrete optimal
estimation methods; and emphasis of control law (algorithm) implementation in a
digital computer. Various design approaches are presented and compared in terms of
advantages and disadvantages in order to provide the digital control engineer with
appropriate criteria for selecting an applicable method.
The text is designed so that, depending on the course and the instructor, chapters
and/or sections can be presented in a variety of sequences. It may be used in ad­
vanced undergraduate and first-year graduate courses (two quarters or two semesters
in length); for a short course (40 lecture hours); as a self-study text; and for a single
course restricted to only scalar sampled-data control theory, Chapter 1 to 6 and 1 1
to 1 3 . Of course, the appropriate elements of computer engineering and transducer
modeling as developed in Chapters 7 through 10 are critical for detailed design and
implementation of digital controllers. Chapters 14 to 1 9 are included for modem digital
control and the estimation theory.

xxii

PREFACE

Use of a CACSD (computer-aided control system design) package is advan­
tageous in providing insight to design and analysis of digital control systems by
employing the various examples and exercises in the text. Also, a physical laboratory
for student implementation of various digital controller designs is of considerable
educational benefit (see the Instructor' s Manual for a series of suggested experiments
and facilities and for case studies).
We feel that with the mastery of the text material the student should be able to
analyze and design sampled-data control systems and implement digital controllers as
well as to provide the background for more extensive studies in the area.
The goals of achieving an integrated sampled-data theory and computer en­
gineering text have been extended to this second edition. Additional clarification of
design processes is included and applied to real-world design problems. Modem digital
control techniques have been added that employ quantitative feedback, optimal control,
stochastic control, and adaptive control theory methods.
COMPUTER-AIDED DIGITAL CONTROL
SYSTEM DESIGN

In the understanding of digital-control system-design methods the use of computer­
aided design (CAD) packages can provide a rich educational environment. The student
can validate textbook examples as well as various textbook exercises. There are various
commercial CAD packages (CNTLC, MATLAB, MATRIXx, SIMON ) for control that
could be used. ICECAP-PC (Interactive Control Engineering Computer Aided Package
for the Personal Computer) has been developed at the Air Force Institute of Tech­
nology and in conjunction with a number of universities to provide a public domain
CACSD package. The purpose of ICECAP-PC is to provide an educational CAD
program for control engineering students to analyze, design, synthesize and simulate

control systems (continuous and discrete). The public domain MS-DOS ICECAP-PC
executable program is available via the Instructor's Manual. Also included is a series of
macro files for various examples and problems in this text and a users manual (ASCII
file). If you are interested, an ICECAP source code can be provided for enhancements
as part of a university/industry consortium.
We express our thanks to the students who have used this book and to the
faculty who have reviewed it for their helpful comments and corrections. Particular
appreciation is expressed to Dr. J. J. D'Azzo, Head of the Electrical Engineering De­
partment; Professor Emeritus R. B . Fontana, and the digital control students of the Air
Force Institute of Technology for their inspiration. The continual encouragement and
review of the text by Dr. T. 1. Higgins, Professor Emeritus, University of Wisconsin,
has been a very important catalyst in the completion of this second edition. Special
thanks to Dr. Donald McLean, Senior Lecturer, University of Southhampton, England,
formerly a visiting Professor at the Air Force Institute of Technology, who provided a
detailed review of the first edition. The following reviewers also gave many valuable
comments: Richard K. Blandford, Evansville University; Christos G. Cassandras,
University of Massachusetts-Amherst; John F. Dorsey, Georgia Institute of Technol­
ogy; Warren J. Guy, Jr., Lafayette College; Nairn A. Kheir, Oakland University; and


PREFACE

xxiii

Renjeng Su, University of Colorado-Boulder. The perception and insight of all these
individuals has contributed extensively to the clarity and rigor of the presentation. Our
association with them has been a enlightening and refreshing experience. Finally, we
thank McGraw-Hill, especially Eleanor Castellano for her help and encouragement.
Constantine H. Houpis
Gary B . Lamont

CHAPTER .

1

INTRODUCTION

1.1

INTRODUCTION

The past five decades have witnessed the finn establishment of conventional and
modem control theory for continuous-time control systems. ! ,99 Not only has this theory
revolutionized industrial and medical processes, but it has also enabled humanity to
initiate the exploration of the universe, Sampled-data practical systems were initially
slow due to the inherent theoretical and physical implementation problems. During
this past decade many of these problems have been overcome and new sampled­
data control-system design techniques have emerged. A sampled-data control system,
utilizing a digital computer as a control element, can play a vital role in control-system
applications such as robotics, aircraft control, automobile control, medical operations,
process control, biomedical systems, and satellite operation.
Engineers and scientists attempt to design control systems to perfection so that
ideal system perfonnance is achieved. For a practical control system, physical realiz­
ability of components limits the extent to which the ideal system perfonnance can be
achieved. The advent of the digital computer as a computational device penn its more
accurate control in general but also constrains the speed of operation. However, this
accuracy has proven to be a critical element in the success of modem space explo­
ration and intricate process control. The advent of the microprocessor and its use as a
control element have provided the impetus, not only to enhance the theoretical anal­

ysis and synthesis techniques for many systems, but also to motivate control-system
designers to progress closer to their goal of "ideal system perfonnance. "
The purpose o f this text i s to present an extensive discussion of digital-control­
system tenninology, sampled-data control-system analysis and synthesis, and practical


2

INTRODUCTION

implementation techniques and considerations from a software and hardware point of
view including measurement technology. Various aspects of a general digital control
system are discussed in this text from various levels of observation (theory, hardware,
software).
The three simplest control-system configurations or architectures are shown in
the block diagram of Fig. 1 . 1 . Figure 1 . 1 (a) is an open-loop control system representing
many industrial process structures. The other three closed-loop configurations represent
the most commonly used control systems where the performance specifications are
more restrictive. The process block in Fig. 1 . 1 can represent a dc motor speed or
position system (translational or rotational), a thermal system, a hydraulic system, etc.
These block diagrams represent the fundamental control-system notation: The process
to be controlled is called the plant, and the controller or compensator. The process
may also contain sensors for measurement of plant dynamics (measured variables).
Figure 1 . 1 (d) represents a general sampled-data feedback control system with a digital
controller consisting of three elements. The analog-to-digital (AID) converter samples
the analog signal and converts the sampled signal to digits for numerical processing
by the digital computer. The digital-to-analog (D/A) converter transforms the resulting
discrete or digital signal values to an analog control signal.
Acceptable system performance requires the control system output c(t) (con­
trolled variables) to track an input ret) (manipulated variables) despite various system

disturbances. If the input is zero, then the system is defined as a regulator. Satisfac­
tory regulation or tracking with disturbances is associated with a disturbance rejection
system. If the plant parameters vary but the system retains acceptable regulation or
tracking characteristics with disturbances, then the system is defined as robust. Var­
ious design approaches presented in the following chapters depend upon the desired
performance using these general system definitions.
1.2 DIGITAL CONTROL·SYSTEM
MODELING

A digital-control-system model can be viewed from many different levels including
the control law (algorithm), the computer program, conversion between analog and
digital signal domains, and system performance. One of the most important aspects
leading to the understanding of digital control systems is at the sampling process
level, which is introduced in this section. The associated system terminology, which
is critical in understanding digital control concepts, is also presented.
1.2.1

Sampling Process 1 , 2

In continuous control systems, all system variables are continuous signals as repre­
sented by Fig. 1 .2(a). That is, whether the system is linear or nonlinear, all variables
are continuously present and are therefore known at all times. (This text deals only with
linear or linearized systems.) Another category of control systems is one in which one
signal e(t) is sampled at intervals of time T. This is depicted in Fig. 1 .2(c) so that the
sampled signal appears as a pulse train of varying amplitudes, as shown in Fig. 1 .2(d)


1.2 DIGITAL CONTROL-SYSTEM MODELING

Input

Controller
(digital computer
system)

3

Output

(0)

'"p"'

Y

c
L__

rO
__

H

p
L__

....J

;

S
_

..
----I� Out P ut

(b)

�--__-_ Output

Input --_�

+

(c)

Di sturbanc e

Digital controller

Distu r bance

e(t)

r(t)

Di stur banc e

FIGURE

1.1

(d)

General architecture of digital control systems: (a) Feedforward or open-loop system; (b) unity-feedback
system; (e) nonunity-feedback system; (d) general digital control system.

with the sampled output e;(t)==e(t)p(t). That is, the pulse train of Fig. 1 .2(b) is mod­
ulated with the continuous-time signal of Fig. 1 .2(a) to yield the sampled function
of Fig. 1 .2(d). Such sampling may be an inherent characteristic of the system. For
example, a radar tracking system supplies information on a vehicle 's position at dis­
crete periods of time. This information is therefore available at a succession of time
intervals as data levels.


4

INTRODUCfION

pIt )

e(t)

•

I

y

o

(a)
Pulse train
generator

e(r)
Continuous
amplitude (analog) signal

_

....__
--L-__
----,

Pulse modulator

e(t)

(e)

i

Pulse
(analog) signal

e;(r) '" e(t)p(r)
o
(d)

Continuous and discrete signals: (a) Continuous analog signal e(t); (b) sampling pulse train pet); (e)
sampling device; (d) sampled function.
FIGURE 1.2

1.2.2

System Terminology

It should be noted that in the control literature the terms digital systems, discrete-data
systems, digital control systems, sampled-data systems, and discrete-time systems have
been and are being used interchangeably. In the early development of this technology
an analog system not containing a digital device in which some of the signals were
sampled (pulse or amplitude modulated) was referred to as a sampled-data system.
In this text the term sampled-data control system is used to describe a system that
contains at least one sampled signal. With the advent of the digital computer, the term
discrete-time system denoted a system in which all its signals were in a digital coded
form (digitized). Most practical systems today are of a hybrid nature, i.e., contain both
analog and digital components.
Digital computers are available for performing the computations necessary in
a complex control system. The AID conversion device samples the device's input
signal at some sampling frequency Ws or equivalently at sampling instants t = kT
where T is defined as the sampling time. The process is completed with the sam­
pled analog signal value being converted to a discrete digital value (digital encoding).
The D/A converter performs the reverse conversion process generating an analog
control signal. The details of this conversion process are discussed in later chap­
ters. A system in which the digital computer is utilized as a control device is re­
ferred to as a digital control system. If the digital computer interfaces directly with


I.2 DIGITAL CONTROL-SYSTEM MODELING

5

the plant (or process), the system is referred to as a direct digital control (DDC)
system .
In practice, the output of the pulse modulator of Fig. 1 .2(c) is generally fed into
a data-hold device (a device that converts a discrete signal into a continuous signal), as
shown in Fig. 1 .3(a). The simplified representation of the sampling and hold devices
is shown in Fig. 1 .3(b), where the "ideal sampler" represents the unit impulse train
of Fig. 1 .3(c) and the output of the sampler is the amplitude-varying impulse train
e* (t) of Fig. 1 .3(d). The * indicates a sampled continuous signal such as e* (t) with b
defined as the kronecker delta function:

bet

-

kT) ==

{�

t = kT
t =I kT

( 1 . 1)

A train of these delta functions, bT==L::O bet kT), multiplied times the continuous
signal e(t), is a syntactical method of representing a mathematical model of the sam­
pled continuous signal, e* (t) as shown in Fig. 1.3(d). The utilization of a hold device,
which is shown in Fig. 1 .3(e), simplifies the mathematical analysis of the sampled

signal. The hold process in this case is defined as a "zero-order hold" since the output
at time kT is held constant over the next sampling interval. Thus, the structures of
Fig. 1 . 1 , where the controller contains the sampler and the hold device of Fig. 1 .3(a),
are sampled-data control systems. An example of a sampled-data control system is that
of a human, which can be illustrated in a setting of observing the environment while
performing a function such as driving a vehicle or dialing a phone. The human eye
samples the appropriate environmental signals; processes the information in the brain;
and controls the hands, arms, and legs accordingly. Another example is that of an air­
port automated aircraft landing system that uses radar information, an inherent sampled
process, defining aircraft lateral and vertical positions. These data are used as input to
the controller (an autopilot) for aircraft pitch, yaw, and roll control. These examples
represent multiple-input-multiple-output (MIMO) control systems. Such systems can
be negatively influenced by unmodeled disturbances such as noise, wind, and light.
1.2.3

-

General Sampled-Data System Variables

The variables of a sampled-data system can be described in terms of their time and am­
plitude characteristics grouped in four general categories: discrete amplitude-discrete
time (D-D), discrete amplitude-continuous time (D-C), continuous amplitude-discrete
time (C-D), and continuous amplitude-continuous time (C-C). The first letter refers
to the amplitude characteristic and the second letter refers to the time characteristic.
Table 1 . 1 summarizes these categories.
A general example of the classification of sampled-data control systems is shown
in Fig. 1 .4. Here the continuous (amplitude and time) input signal e(t) is impulse­
sampled, generating a continuous-amplitude-discrete-time signal by definition. The
hold circuit of Fig. 1 .4 generates a piecewise-continuous step function ("staircase").
The sampled data e*(t) can be any value (an infinite number of values) within some

predefined range of amplitude values.

•


6

INTRODUcnON

_I'(I_ ) _ _">..·....
.... �

1';(1)

T

•

m(_ I)
L_"_O _\ _...JI- _ -device
.
m(l)

o T 2T

4T

6T

8T

(a)

•

•

(b)

(c)

(k+2)T
kT
(k+ \)T

('*(1) = 1'(1) 6�1) = l: l'(kT) 6(1- kT)
k=O

(d)

m(t)

FIGUR.E 1.3

L-o

'--

'--

'"'--6T

-'----f/f-1'kT

- - - -

-

-••

- -

t

(e)

Sampling process: (a) Pulse­
sampling and data-hold devices;
(b) simplified representation
of (a); (c) ideal sampler
representation; (d) output of
ideal sampler, (e) output of hold
device.

Figure 1 .5 presents the various signal classifications in a digital-computer control
system. Here again the impulse sampler generates a continuous-amplitude-discrete­
time C-D signal. The output eD(kT) of the AID quantizes e*(t) such that it can
be only one of a finite number of values within a specific value range. This phe­
nomenon is due to the finite word length of the computer. The digital computer itself
manipulates the quantized value (a base 2 number) into another discrete-amplitude

discrete-time D-D signal f(kT). Finally. the D/A transforms f(kT) into a discrete­
amplitude-continuous-time D-C signal m(t). Observe that the m(t) in this case as


1.2 DIGITAL CONTROL·SYSTEM MODELING

7

TABLE 1.1
The nature of sampled-data-system
variables
Time (temporal)
Amplitude
(spatial)

Discrete, D

Continuous, C

Discrete , D

D-D

D-C

Continuous , C

CoD

C-C

compared to the output of the hold device in Fig. 1 .4 has a discrete-amplitude char­
acteristic because of the quantization of the AID which is not present in Fig. 1 .4. The
D/A, however, does include a hold device and is modeled in Fig. 1 .6. The "fictitious"
sampler in this D/A model is needed to represent the fact that the input value f*(t) is
immediately converted (AID) and manipulated by the digital computer, resulting in a
value at the hold device input during the same sampling instant. That is, there is no
computational time delay TD in the computer. This instantaneous computation is, of
course, impossible. If the computational time delay is much less than the sampling
time T, then the assumption that TD � 0 is appropriate, resulting in a simplified
analytical model. This model is used in the theoretical development of later chapters.
1.2.4

Systems Modeling

The generation of a plant model for which a controller is realized as a digital in­
formation processing device is of primary importance. Generally, in an academic
environment the plant model is given in the form of a linear system with constant
coefficients. This type of model is usually easy to analyze with proper techniques. In

>;

_�e�(t�_
)

c-c

e(t)

o

c-c

T

2T

T

m(tJ

e*(t)

o

T

2T

o

Example of analog signal classifications in a sampled-data system (see Table 1 .1).
FIGURE 1.4

c-c

COD

T

2T


8

e(t)

INTRODUCTION

e*(t)

c-c

0

T

D-D

C-D

(digital signal)

(analog signal)

(analog signal)

0

2T

!c,

T

0

f'2
o

2T
m(t)

l( k T)

D-D

Cj

I
I

T

If1 ('3
I
I

• t

2T

T

2T

I
I

I

I

• t

I

'j i s a quantized

value

D-C

(analog signal)

(digital signal)

2
T('

)

T'
I
I

is a quantized value

o

m is quantized

T

2T

3T

FIGURE 1.5

Example of signal classifications in a digital control system.

the real world most models are nonlinear in nature and are time-varying due to the
physics of the system and its environment. With various assumptions (which may not
be appropriate) the nonlinear system is linearized and analyzed, and a control law is
generated to meet given specifications. Testing is then initiated to validate the system
model within a constrained domain.
Many control problems tum out to be directly related to the use of an incorrect
plant model. The development of the original plant model is probably the most im­
portant and probably the most difficult aspect of control engineering. The application

of the various analysis and design techniques usually proceeds in a straightforward
manner after the model is obtained.

i-----------l

\ >r
D-D

I( k T l

I

I

T

0-0

[*(t )

_I

Hold

I

l m(t�
ID-C
I

L- ___________ -.J

oI

A converter

[*(t)

.

D-D

(analog signal)

o T2T

D/A converter with a hold device.

FIGURE 1.6