Table Of ContentRolf Isermann
Digital
Control Systems
Volume 1:
Fundamentals, Deterministic Control
Second, Revised Edition
With 88 Illustrations
Springer-V erlag Berlin Heidelberg GmbH
Professor Dr.-Ing. Rolf Isermann
Institut fur Regelungstechnik
Technische Hochschule Darmstadt
SchloBgraben 1
D-6100 Darmstadt, West Germany
ISBN 978-3-642-86419-3
Library of Congress Cataloging-in-Publication Data
Isermann, Rolf. [Digitale Regelsysteme. English] Digital control systems/Rolf Isermann.
Rev. and enl. translation of: Digitale Regelsysteme. Includes bibliographical
references and index. Contents: v. 1. Fundamentals, deterministic control.
ISBN 978-3-642-86419-3 ISBN 978-3-642-86417-9 (eBook)
DOI 10.1007/978-3-642-86417-9
1. Digital control systems. 1. Title.
TJ213.I647\3 1989
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© Springer-Verlag Berlin Heidelberg 1989
Originally published by Springer-Verlag Berlin Heidelberg New York in 1989
Softcover reprint of the hardcover 2nd edition 1989
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Preface
The great advances made in large-scale integration of semiconductors and the
resulting cost-effective digital processors and data storage devices determine the
present development of automation.
The application of digital techniques to process automation started in about
1960. when the first process computer was installed. From about 1970 process
computers with cathodic ray tube display have become standard equipment for
larger automation systems. Until about 1980 the annual increase of process
computers was about 20 to 30%. The cost of hardware has already then shown a
tendency to decrease. whereas the relative cost of user software has tended to
increase. Because of the high total cost the first phase of digital process automation
is characterized by the centralization of many functions in a single (though
sometimes in several) process computer. Application was mainly restricted to
medium and large processes. Because of the far-reaching consequences of a
breakdown in the central computer parallel standby computers or parallel back-up
systems had to be provided. This meant a substantial increase in cost. The tendency
to overload the capacity and software problems caused further difficulties.
In 1971 the first microprocessors were marketed which, together with large-scale
integrated semiconductor memory units and input output modules, can be assem
bled into cost-effective microcomputers. These microcomputers differ from process
computers in fewer but higher integrated modules and in the adaptability of their
hardware and software to specialized, less comprehensive tasks. Originally, micro
processors had a shorter word length, slower operating speed and smaller operat
ional software systems with fewer instructions. From the beginning, however,
they could be used in a manifold way resulting in larger piecenumbers and lead
ing to lower hardware costs, thus permitting the operation with small-scale
processes.
By means of these process-microcomputers which exceed the efficiency of former
process computers decentralized automatic systems can be applied. To do so, the
tasks up to now been centrally processed in a process computer are delegated to
various process microcomputers. Together with digital buses and possibly placed
over computers many different hierarchically organized automatization structures
can be build up. They can be adapted to the corresponding process. Doing so the
high computer load of a central computer is avoided, as well as a comprehensive
and complex user-software and a lower reliability. In addition decentralized
systems can be easier commissioned, can be provided with mutual redundancy
vi Preface
(lower susceptibility to malfunctions) and can lead to savings in wiring. The second
phase of process automation is thus characterized by decentralization.
Besides their use as substations in decentralized automation systems process
computers have found increasing application in individual elements of automation
systems. Digital controllers and user-programmable sequence control systems,
based on microprocessors, have been on the market since 1975.
Digital controllers can replace several analog controllers. They usually require an
analog-digital converter at the input because of the wide use of analog sensors,
transducers and signal transmission, and a digital-analog converter at the output to
drive actuators designed for analog techniques. It is to be expected that, in the long
run, digitalization will extend to sensors and actuators. This would not only save
a-d and d-a converters, but would also circumvent certain noise problems, permit
the use of sensors with digital output and the reprocession of signals in digital
measuring transducers (for example choice of measurement range, correction of
nonlinear characteristics, computation of characteristics not measurable in a direct
way, automatic failure detection, etc.). Actuators with digital control will be
developed as well. Digital controllers not only are able to replace one or several
analog controllers they also succeed in performing additional functions, previously
exercised by other devices or new functions. These additional functions are such as
programmed sequence control of setpoints, automatic switching to various con
trolled and manipulated variables, feedforward adjusted controller parameters as
functions of the operating point, additional monitoring of limit values, etc.
Examples of new functions are: communication with other digital controllers,
mutual redundancy, automatic failure detection and failure diagnosis, various
additional functions, the possibility of choice between different control algorithms
and, in particular, selftuning or adaptive control algorithms. Entire control systems
such as cascade-control systems, multivariable control systems with coupling
controllers, control systems with feedforward control which can be easily changed
by configuration of the software at commissioning time or later, can be realized by
use of one digital controller. Finally, very large ranges ofthe controller parameters
and the sample time can be realized. It is because of these many advantages that,
presently various digital devices of process automation are being developed, either
completing or replacing the process analog control technique.
As compared to analog control systems, here are some of the characteristics of
digital control systems using process computers or process microcomputers:
- Feedforward and feedback control are realized in the form of software.
- Discrete-time signals are generated.
- The signals are quantized in amplitUde through the finite word length in a-d
converters, the central processor unit, and d-a converters.
- The computer can automatically perform the analysis of the process and the
synthesis of the control.
Because of the great flexibility of control algorithms stored in software, one is not
limited, as with analog control systems, to standardized modules with P-, 1- and
D-behaviour, but one can further use more sophisticated algorithms based on
mathematical process models. Many further functions can be added. It is especially
Preface vii
significant that on-line digital process computers permit the use of process
identification-, controller design-, and simulation methods, thus providing the
engineer with new tools.
Since 1958 several books have been published dealing with the theoretical
treatment and synthesis of linear sampled-data control, based on difference
equations, vector difference equations and the z-transform. Up to 1977, when the
first German edition of this book appeared, books were not available in which the
various methods of designing sampled-data control have been surveyed, compared
and presented so that they can be used immediately to design control algorithms for
various types of processes. Among other things one must consider the form and
accuracy of mathematical process models obtainable in practice, the computational
effort in the design and the properties ofthe resulting control algorithms, such as the
relationship between control performance and the manipulation effort, the behav
iour for various processes and various disturbance signals, and the sensitivity to
changes in process behaviour. Finally, the changes being effected in control
behaviour through sampling and amplitude quantization as compared with analog
control had also be studied.
Apart from deterministic control systems the first edition of this book dealt also
with stochastic control, multi variable control and the first results of digital adaptive
control. In 1983 this book was translated into Chinese. In 1981 the enlarged English
version entitled "Digital Control Systems" was edited, followed by the Russian
translation in 1984, and, again a Chinese translation in 1986. In 1987 the 2nd
edition appeared in German, now existing in two volumes. This book is now the 2nd
English edition.
As expected, the field of digital control has been further developed. While new
results have been worked out in research projects, the increased application
provided a richer experience, thus allowing a more profound evaluation of the
various possibilities. Further stimulation of how to didactically treat the material
has been provided by several years of teaching experience and delivering courses in
industry. This makes the second edition a complete revision of the first book,
containing many supplements, especially in chapters 1,3,5,6, 10,20,21,23,26,30,
31. Since, compared with the first edition, the size of the material has been
significantly increased, it was necessary to divide the book in two volumes.
Both volumes are directed to students and. engineers in industry desiring to be
introduced to theory and application of digital control systems. Required is only a
basic familiarity of continuous-time (analog) control theory and control technique
characterized by keywords such as: differential equation, Laplace-Transform,
transfer function, frequency response, poles, zeroes, stability criterions, and basic
matrix calculations. The first volume deals with the theoretical basics of linear
sampled-data control and with deterministic control. Compared with the first
edition the introduction to the basics of sampled-data control (part A) has been
considerably extended. Offering various examples and exercises the introduction
concentrates on the basic relationships required by the up-coming chapters and
necessary for the engineer. This is realized by using the input/output-, as well as the
state-design. Part B considers control algorithms designed for deterministic noise
signals. Parameter-optimized algorithms, especially with PID-behaviour are
viii Preface
investigated in detail being still the ones most frequently used in industry. The
sequel presents general linear controllers (higher order), cancellation controllers,
and deadbeat controllers characteristic for sampled-data control. Also state
controllers including observers due to different design principles and the required
supplements are considered. Finally, various control methods for deadbeat pro
cesses, insensitive and robust controllers are described and different control
algorithms are compared by simulation methods. Part C of the second volume is
dedicated to the control design for stochastic noise signals such as minimum
variance controllers. The design ofinterconnected control systems (cascade control,
feedforward control) are described in Part D while part E treats different multi
variable control systems including multivariable state estimation. Digital adaptive
control systems which have made remarkable progress during the last ten years are
thoroughly investigated in Part F. Following a general survey, on-line identification
methods, including closed loop and various parameter-adaptive control methods
are presented. Part G considers more practical aspects, such as the influence of
amplitude quantization, analog and digital noise filtering and actuator control.
Finally the computer-aided design of control with special program systems is
described, including various applications and examples of adaptive and selft uning
control methods for different technical processes. The last chapters show, that the
control systems and corresponding design methods, in combination with process
modeling methods described in the two volumes were compiled in program systems.
Most of them were tried out on our own pilot processes and in industry. Further
specification of the contents is given in chapter 1.
A course "Digital Control Systems" treats the following chapters: 1, 2, 3.1-3.5,
3.7,4,5,6,7,3.6,8,9,11. The weekly three hours lecture and one hour exercises is
given at the Technische Hochschule Darmstadt for students starting the sixth
semester. For a more rapid apprehension of the essentials for applications the
following succession is recommended: 2, 3.1 to 3.5 (perhaps excluding 3.2.4, 3.5.3,
3.5.4) 4, 5.1, 5.2.1, 5.6, 5.7, 6.2, 7.1, 11.2, 11.3 with the corresponding exercises.
Many of the described methods, development and results have been worked out
in a research project funded by the Bundesminister fUr Forschung und Technologie
(DV 5.505) within the project "ProzeBlenkung mit DV-Anlagen (PDV)" from
1973-1981 and in research projects funded by the Deutsche Forschungsgemein
schaft in the Federal Republic of Germany. The author is very grateful for this
support.
His thanks also go to his coworkers,-who had a significant share in the
generation of the results through several years of joint effort-for developing
methods, calculating examples, assembling program packages, performing simu
lations on digital and on-line computers, doing practical trials with various
processes and, finally, for proofreading.
The book was translated by my wife, Helge Isermann. We thank Dr. Ron Patton,
University of York, U.K., for screening the translation.
Darmstadt, June 1989 Rolf Isermann
Contents
1 Introduction..................................
A Fundamentals
2 Control with Digital Computers (Process Computers,
Microcomputers). . . . . . . . . . . . . . . . . . . . . .... 13
3 Fundamentals of Linear Sampled-data Systems (Discrete-time
Systems) ............................ . 17
3.1 Discrete-time Signals . . . . . . . . . . . . . . . . . . 17
3.1.1 Discrete Functions, Difference Equations. . . 17
3.1.2 Impulse Trains . . . . . . . . . . . . . . . . . . 22
3.1.3 Fourier-Transform of the Impulse Train .... 23
3.2 Laplace-transformation of Discrete-time Functions and
Shannon's Sampling Theorem. . . . 27
3.2.1 Laplace-transformation . . . . . 27
3.2.2 Shannon's Sampling Theorem . 28
3.2.3 Holding Element . . . . . . . .. 30
3.2.4 Frequency Response of Sampled Systems .... 32
3.3 z-Transform . . . . . . . . . . . . . 33
3.3.1 Introduction of z-Transform 33
3.3.2 z-Transform Theorems .... 34
3.3.3 Inverse z-Transform . . . . . 37
3.4 Convolution Sum and z-Transfer Function. . 40
3.4.1 Convolution Sum. . . . . . . . . . . . . . 40
3.4.2 Pulse Transfer Function and z-Transfer-Function. 41
3.4.3 Properties of the z-Transfer Functions and Difference
Equations ................ . 45
3.5 Poles and Zeros, Stability . . . . . . . . . . . . . 51
3.5.1 Location of Poles in the z-Plane. . . . . . 52
3.5.2 Stability Condition. . . . . . . . . . . . . . 56
3.5.3 Stability Analysis through Bilinear Transformation. . . . 56
x Contents
3.5.4 Schur-Cohn-Jury Criterion . . . . . . . . . . . . . . . . . . 59
3.5.5 Location of Zeros in the z-Plane. . . . . . . . . . . . . . . 62
3.6 State Variable Representation. . . . . . . . . . . . . . . . . . . . 64
3.6.1 The Vector Difference Equation Based on Vector
Differential Equation. . . . . . . . . . . . . . . . . . . . . . 64
3.6.2 The Vector Difference Equation Based on Difference
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6.3 Canonical Forms. . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.4 Processes with Deadtime . . . . . . . . . . . . . . . . . . . 74
3.6.5 Solution of Vector Difference Equation. . . . . . . . 76
3.6.6 Determination of the z-Transfer Function . . . . . . 77
3.6.7 Determination of the Impulse Response . . 78
3.6.8 Controllability and Observability . . . . . . . . . . . 78
3.7 Mathematical Models of Processes. . . . . . . . . . . . . . . . 82
3.7.1 Basic Types of Technical Processes . . . . . . . . . . . . 82
3.7.2 Determination of the Process Model-Modelling and
Identification . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.7.3 Calculation of z-Transfer Functions from s-Transfer
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7.4 Simplification of Process Models for Discrete-time Signals 90
B Control-Systems for Deterministic Disturbances
4 Deterministic Control Systems. . 97
5 Parameter-optimized Controllers. t 03
5.1 Discretizing the Differential Equations of Continuous
PID-Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Parameter-optimized Discrete Control Algorithms of Low-Order 106
5.2.1 Control Algorithms of First and Second Order . . . . . . 108
5.2.2 Control Algorithms with Prescribed Initial Manipulated
Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5
5.2.3 PID-Control Algorithm through z-Transformation. . . . 116
5.3 Modifications to Discrete PID-Control Algorithms. . . . . . . 118
5.3.1 Different Evaluation of Control Variable and Reference
Variable . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3.2 Different Discretizations of the Derivative Term. . . . . 120
5.3.3 Delayed Differential Term. . . . . . . . . . . . . . . . 120
5.4 Design Through Numerical Parameter Optimization. . 122
5.4.1 Numerical Parameter Optimization. . . . . . . . . 122
5.4.2 Simulation Results for PID-Control Algorithms. 124
5.5 PID-Controller Design Through Pole-Assignment,
Compensation and Approximation. . . . . . . . . . . . . . . . . 138
