18
Rolf 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 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
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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 assembled 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, microprocessors had a shorter word length, slower operating speed and smaller operational software systems with fewer instructions. From the beginning, however, they could be used in a manifold way resulting in larger piecenumbers and leading 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 controlled 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 behaviour 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 processes, 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 multivariable 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 self tuning 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 Forschungsgemeinschaft 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 simulations 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). . . . . . . . . . . . . . . . . . . . . ....
3 Fundamentals of Linear Sampled-data Systems (Discrete-time Systems) ............................ .
3.1 Discrete-time Signals . . . . . . . . . . . . . . . . . . 3.1.1 Discrete Functions, Difference Equations. . . 3.1.2 Impulse Trains . . . . . . . . . . . . . . . . . . 3.1.3 Fourier-Transform of the Impulse Train ....
3.2 Laplace-transformation of Discrete-time Functions and Shannon's Sampling Theorem. . . . 3.2.1 Laplace-transformation . . . . . 3.2.2 Shannon's Sampling Theorem . 3.2.3 Holding Element . . . . . . . .. 3.2.4 Frequency Response of Sampled Systems ....
3.3 z-Transform . . . . . . . . . . . . . 3.3.1 Introduction of z-Transform 3.3.2 z-Transform Theorems .... 3.3.3 Inverse z-Transform . . . . .
3.4 Convolution Sum and z-Transfer Function. . 3.4.1 Convolution Sum. . . . . . . . . . . . . . 3.4.2 Pulse Transfer Function and z-Transfer-Function. 3.4.3 Properties of the z-Transfer Functions and Difference
Equations ................ .
3.5 Poles and Zeros, Stability . . . . . . . . . . . . . 3.5.1 Location of Poles in the z-Plane. . . . . . 3.5.2 Stability Condition. . . . . . . . . . . . . . 3.5.3 Stability Analysis through Bilinear Transformation. . . .
13
17
17 17 22 23
27 27 28 30 32
33 33 34 37
40 40 41
45
51 52 56 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 15 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
Contents xi
5.5.1 Pole-assignment Design. . . . . . . . . . . . . . . . . . 138 5.5.2 Design as a Cancellation Controller. . . . . . . . . . . 140 5.5.3 Design of PID-Controllers Through Approximation of
other Controllers. . . . . . . . . . . . . . . . . . . . . . . 143
5.6 Tuning Rules for Parameter-optimized Control Algorithms. . 143 5.6.1 Tuning Rules for Modified PID-controllers . . . . . . . . 144 5.6.2 Tuning Rules Based on Measured Step Functions . . 144 5.6.3 Tuning Rules with Oscillation Tests. . . . . . . . . . 150
5.7 Choice of Sample Time for Parameter-optimized Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.8 Supplementary Functions of Digital PID-Controllers. . . . . . 154
6 General Linear Controllers and Cancellation Controllers. . . . . . . . 157
6.1 General Linear Controllers . . . . . . . . . . . . . . . . . . . . . 157 6.1.1 General Linear Controller Design for Specified Poles . 158 6.1.2 General Linear Controller Design Through Parameter
Optimization . . . . . 160
6.2 Cancellation Controllers . . . . . . . . . . . . . . . . . . . . . . 160
7 Controllers for Finite Settling Time. . . . . . . . . . . . . . . . . . . . 166
7.1 Deadbeat Controller Without Prescribed Manipulated Variable 166 7.2 Deadbeat Controller with Prescribed Manipulated Variable. 171 7.3 Choice of the Sample Time for Deadbeat Controllers . . 176 7.4 Approximation Through PID-Controllers . . . . . . . . . . . 177
8 State Controller and State Observers . . . . . . 180
8.1 Optimal State Controllers for Initial Values . . . . . . . . . 181 8.2 Optimal State Controllers for External Disturbances . . . . 189 8.3 State Controllers with a Given Characteristic Equation. . 195 8.4 Modal State Control. . . . . . . . . . . . . . . . . . . . . . 196 8.5 State Controllers for Finite Settling Time (Deadbeat). . . 201 8.6 State Observers . . . . . . . . . . . . . . 202 8.7 State Controllers with Observers. . . . . . . . . 206
8.7.1 An Observer for Initial Values. . . . . . . 206 8.7.2 Observer for External Disturbances. . . . 208 8.7.3 Introduction of Integral Action Elements into the
State Controller. . . . . . . . . . . . . . . . . . . . . . . 215 8.7.4, Measures to Minimize Observer Delays 219
8.8 State Observer of Reduced-Order. . . . . . . . . 219 8.9 State Variable Reconstruction. . . . . . . . . . . 223 8.10 Choice of Weighting Matrices and Sample Time. . . . . 225
8.10.1 Weighting Matrices for State Controllers and Observers 226 8.10.2 Choice of the Sample Time. . . . . . . . . . . . . . . . . 227
xii Con ten ts
9 Controllers for Processes with Large Deadtime . 228
9.1 Models for Processes with Deadtime. . . . 228 9.2 Deterministic Controllers for Deadtime Processes. 230
9.2.1 Processes with Large Deadtime and Additional Dynamics 230 9.2.2 Pure Deadtime Processes. . . . . . . . . . . . . . . . . . . 232
9.3 Comparison of the Control Performance and the Sensitivity of Different Controllers for Deadtime Processes. . . . . . . . . . . 235
10 Sensitivity and Robustness with Constant Controllers. 242
10.1 On the Sensitivity of Closed-loop Systems . . . 243 10.2 Insensitive Control Systems. . . . . . . . . . . . 246
10.2.1 Insensitivity through Additional Dynamic Feedback 247 10.2.2 Insensitivity through Variation of the Design of General
Controllers. . . . . . . . . . . . . 249
10.3 On the Robustness of Control Systems 253 10.4 Robust Control Systems . . . . . . . . . 258
11 Comparison of Different Controllers for Deterministic Disturbances. 263
11.1 Comparison of Controller Structures, Poles and Zeros . 11.1.1 General Linear Controller for Specified Poles 11.1.2 Low Order Parameter-optimized Controllers. 11.1.3 General Cancellation Controller 11.1.4 Deadbeat Controller 11.1.5 Predictor Controller . . . . . . .
263 264 264 264 265 266
11.1.6 State Controller. . . . . . . . . . 267
11.2 Characteristic Values for Performance Comparison. 270 11.3 Comparison of the Performance of the Control Algorithms 272 11.4 Comparison of the Dynamic Control Factor. . . . . . . 283 11.5 Conclusions for the Application of Control Algorithms. 288
Appendix A
Al Tables of z-Transforms and Laplace-Transforms. . . . . . . . . . . 291 A2 Table of Some Transfer Elements with Continuous and Sampled
Systems . . . . . . . . . . . . . . . . . . . . . . . . 292 A3 Test Processes for Simulation. . . . . . . . . . . 292 A4 On the Differentiation of Vectors and Matrices 301
Appendix B. 301 Problems. . . 301 Appendix C. . 310 Results of the Problems. . 310 References.. . . . 321 Subject Index. . . . . . . . 329
Graphic Outline of Contents (Volume I)
Design of Control Systems Structure
Design of Control Algorithms
Informotion on Process and Signals
12 Control with Digital Computers
3 Fundamentals of Linear Sampled-Data Systems
4 Deterministic Control Systems (Survey)
5 -11 Single -input! Single-output Control Systems
5 Parameteroptimized Controllers (PlO)
6 General Linear and Cancellation Controllers
7 Deadbeat Controllers
1 8 State Controllers and Observers
9 Controllers for Processes with Large Deadtime
10 Robust Controllers
11 Comparison of Control Algorithms
Realization with Digital Computers
Summary of Contents Volume 11
C Control Systems for Stochastic Control Systems
12 Stochastic Control Systems (Introduction) 13 Parameter-optimized Controllers for Stochastic Disturbances 14 Minimum Variance Controllers for Stochastic Disturbances 15 State Controllers for Stochastic Disturbances
D Interconnected Control Systems
16 Cascade Control Systems 17 Feedforward Control
E Multivariable Control Systems
18 Structures of Multivariable Processes 19 Parameter-optimized M ultivariable Control Systems 20 Multivariable Matrix Polynomial Systems 21 Multivariable State Control Systems 22 State Variable Estimation
F Adaptive Control Systems
23 Adaptive Control Systems.-A Short Review 24 On-line Identification of Dynamic Processes and Stochastic Signals 25 Closed-loop On-line Identification 26 Parameter-adaptive Controllers
G Digital Control with Process Computers and Microcomputers
27 The Influence of Amplitude Quantization on Digital Control 28 The Filtering of Disturbances 29 Adaptation of Control Algorithms to Various Actuators 30 Computer Aided Design of Control Algorithms Using Process Identification
Methods 31 Adaptive and Self-tuning Control Using Microcomputers and Process
Computers
Graphic Outline of Contents (Volume 11)
Design of Control Systems Structure
Design of Control Algorithms
Information on Process and Signals
12 Stochastic Control Systems (Survey)
13-15 Single-input! Single -output Control Systems
15-17 Interconnected Control Systems
18-21 Multivariable Control Systems
13 Parameter -optimized opte. Controllers
14 Minimum Variance Controllers
115 State Controllers
15 Cascode Control Systems
17 Feedforward Control
19 Parameter -optimized Multivariable Control Systems
20 Matrix Polynomial ContrOllers
I 21 State Controllers
18 Structures of Multivoriable Processes
122 State Estima~ion
I 23 Adaptive Control Systems (Survey)
24125 Process Identification
25 Parameter -adaptive Control Systems
30 Computer Aided DeSign of Control Algorithms with Process Identification
31 Adaptive Control with Microcomputers
Realization with Digital Computers
27 Amplitude Quantization
I 28 Singnal Filtering
I 29 Actuator Control
List of Abbreviations and Symbols
This list defines commonly occurring abbreviations and symbols:
ab} parameters of the difference equations of the process
cd} parameters of the difference equations of stochastic signals
d deadtimed=Tt/To=I,2, ... e control deviation e = w - y (also ew = w - y); or equation error for par-
ameter estimation; or the number e = 2.71828 ... f frequency,f= I/Tp (Tp period), or parameter g impulse response (weighting function) h parameter
integer; or index; or i2 = - 1 k discrete time unit k=t/To=O, 1,2, ... I integer; or parameter m order of the polynomials A ( ), B( ), C( ), D( ) n disturbance signal (noise) p parameters of the difference equation of the controller, or integer p( ) probability density q parameters of the difference equation of the controller r weighting factor of the manipulated variable; or integer s variable of the Laplace transform s = b + iw; or signal
continuous time u input signal of the process, manipulated variable u(k) = U(k) - U 00
v nonmeasurable, virtual disturbance signal w reference value, command variable, setpoint w(k)= W(k)- Woo x state variable y output signal of the process, controlled variable y(k) = Y(k) - Yoo z variable of the z-transformation z = eTos
ab} parameters of the differential equations of the process
A(s) denominator polynomial of G(s) B(s) numerator polynomial of G(s)
List of Abbreviations and Symbols xvii
A (z) denominator polynomial of the z-transfer function of the process model B(z) numerator polynomial of the z-transfer function of the process model C(z) denominator polynomial of the z-transfer function of the noise model D(z) numerator polynomial of the z-transfer function of the noise model G(s) transfer function for continuous-time signals G(z) z-transfer function H( ) transfer function of a holding element I control performance criterion K gain L word length M integer N integer or discrete measuring time P(z) denominator polynomial of the z-transfer function of the controller Q(z) numerator polynomial of the z-transfer function of the controller R( ) dynamical control factor S power density or sum criterion T time constant T95 settling time of a step response until 95% of final value To sample time Tt dead time U process input (absolute value) V loss function W reference variable (absolute value) Y process output variable (absolute value) b control vector c output vector k parameter vector of the state controller n noise vector (r xl) u input vector (p xl) v noise vector (p x 1) w reference variable vector (r xl) x state variable vector (m x 1) y output vector (r xl) A system matrix (m x m) B input matrix (m x p) C output matrix, observation matrix (r x m) D input-output matrix (r x p), or diagonal matrix F noise matrix or F = A - BK G matrix of transfer functions I unity matrix K parameter matrix of the state controller Q weighting matrix of the state variables (m x m) R weighting matrix of the inputs (p x p); or controller matrix .91 (z) denominator polynomial of the z-transfer function, closed loop ~(z) numerator polynomial of the z-transfer function, closed loop fj Fourier-transform
xviii List of Abbreviations and Symbols
:3 £( ) 3( ) ~( ) IX
f3 y b /:: , '1 K
A J1 v n (J
r w A o n L Q
x Xo
X x,Ax X Xoo
information Laplace-transform z-transform correspondence G(s)-+G(z) coefficient coefficient coefficient; or state variable of the reference variable model deviation, or error coefficient state variable of the noise model state variable of the noise model; or noise/signal ratio coupling factor; or stochastic control factor standard deviation of the noise v(k) order of P(z) order of Q(z); or state variable of the reference variable model 3.14159 ... standard deviation, (J2 variance, or related Laplace variable time shift angular frequency w = 2n/ Tp (Tp period) deviation; or change; or quantization unit parameter product sum related angular frequency =dx/dt exact quantity estimated or observed variable = x - Xo estimation error average value in steady state
Mathematical abbreviations
exp(x) E { } var[ ] cov [ ] dim tr adj det
Indices P Pu Pv
=eX
expectation of a stochastic variable variance covariance dimension, number of elements trace of a matrix: slim of diagonal elements adjoint determinant
process process with input u process with input v
R or C S or C o 00
List of Abbreviations and Symbols XIX
feedback controller, feedback control algorithm, regulator feedforward controller, feedforward control algorithm exact value steady state, d.c.-value
Abbreviations for controllers or control algorithms (C)
i-PC-j
DB LC-PA PREC MV SC
parameter optimized controller with i parameters and j parameters to be optimized Deadbeat-controller linear controller with pole assignment predictor controller minimum variance controller state controller (usually with an observer)
Abbreviations for parameter estimation methods
COR-LS IV LS ML STA
correlation analysis with LS parameter estimation instrumental variables least squares maximum-likelihood stochastic approximation
The letter R means recursive algorithm, i.e. RIV, RLS, RML.
Other abbreviations
ADC CPU DAC PRBS
Remarks
analog-digital converter central processing unit digital-analog converter pseudo-random binary signal
The vectors and matrices in the Figures are roman types and underlined. Hence it corresponds e.g. x--+",; K--+K. The symbol for the dimension of time t in seconds is usually s and sometimes sec in order to avoid misinterpretation as the Laplace variable s.
