Digital control systems. Theoretical problems and simulation tools

450 Pages·2018·3.661 MB·English

by  Miridakis, Nikolaos I, Veloni, Anastasia

Digital Control Systems

Digital Control Systems
Theoretical Problems and
Simulation Tools
Anastasia Veloni
Nikolaos I. Miridakis
MATLAB® and Simulink® are a trademark of The MathWorks, Inc. and is used with permission.
The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s
use or discussion of MATLAB® and Simulink® software or related products does not constitute
endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB® and Simulink® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-138-03920-9 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher can-
not assume responsibility for the validity of all materials or the consequences of their use. The
authors and publishers have attempted to trace the copyright holders of all material reproduced in
this publication and apologize to copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write and let us know so we
may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information
storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC),
222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that
provides licenses and registration for a variety of users. For organizations that have been granted a
photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Professor Anastasia Veloni dedicates this book to the
honor of her parents, Nikolaos and Eystathia.
Dr. Nikolaos I. Miridakis dedicates this book to his daughter, Ioanna,
his wife, Melpomeni, and his parents, Ioannis and Panayota.

Contents
Preface ......................................................................................................................xi
Authors .................................................................................................................xiii
1. Introduction .....................................................................................................1
1.1 Introduction ...........................................................................................1
1.2 Description of Analog and Digital Control Systems .......................2
1.3 Advantages of Digital Control Systems and Applications ..............5
2. z-Transform ......................................................................................................7
2.1 Introduction ...........................................................................................7
2.2 From Laplace Transform to z-Transform ...........................................7
2.2.1 Comparison of s- and z-Planes in the Region
of Convergence .......................................................................10
2.3 z-Transform Properties .......................................................................11
2.3.1 Time Shift ................................................................................11
2.3.2 Linearity ..................................................................................12
2.3.3 Time Reverse ..........................................................................12
2.3.4 Convolution ............................................................................12
2.3.5 Differentiation in z-Domain .................................................12
2.3.6 Initial and Final Value Theorem ..........................................12
2.4 Inverse z-Transform ............................................................................13
2.4.1 Division Method ....................................................................13
2.4.2 Method of Partial Fraction Expansion ................................13
2.4.3 Method of Complex Integration ..........................................14
2.5 Formula Tables ....................................................................................15
2.6 Solved Exercises ..................................................................................15
3. Transfer Function .........................................................................................53
3.1 Introduction .........................................................................................53
3.2 Open-Loop Sampled-Data Control System .....................................53
3.3 Closed-Loop Sampled-Data Control System ...................................54
3.4 Signal Flow Graphs .............................................................................55
3.5 Mason’s Formula .................................................................................57
3.6 Difference Equations ..........................................................................58
3.7 Formula Tables ....................................................................................60
3.8 Solved Exercises ..................................................................................61
vii
viii Contents
4. Transfer Function Discretization ..............................................................89
4.1 Introduction .........................................................................................89
4.2 Discretization Methods ......................................................................90
4.2.1 Impulse-Invariance Method or z-Transform Method .......91
4.2.2 Step-Invariance Method or z-Transform Method
with Sample and Hold ..........................................................92
4.2.3 Backward Difference Method ..............................................94
4.2.4 Forward Difference Method ................................................96
4.2.5 Bilinear or Tustin Method ....................................................96
4.2.6 Frequency Prewarping Method ...........................................98
4.2.7 Matched Pole–Zero Method .................................................99
4.3 Comparison of Discretization Methods ........................................101
4.4 Formulas .............................................................................................102
4.5 Solved Exercises ................................................................................104
5. State-Space Representation ......................................................................159
5.1 Introduction .......................................................................................159
5.2 Discrete-Time State-Space Equations .............................................160
5.2.1 Eigenvalues and Eigenvectors ............................................161
5.3 Solution of State Equations ..............................................................161
5.4 State-Space Representation ..............................................................162
5.4.1 Direct Form ...........................................................................162
5.4.2 Canonical Form ....................................................................164
5.4.3 Controllable Canonical Form .............................................165
5.4.4 Observable Canonical Form ...............................................167
5.4.5 Jordan Canonical Form .......................................................168
5.5 Controllability and Observability ..................................................169
5.6 State-Space Discretization................................................................171
5.6.1 Discretization of Continuous-Time Systems in the
State Space with Numerical Integration Methods ..........171
5.6.2 Discretization of Continuous Time Systems in
State-Space with Numerical Differentiation Methods ....173
5.6.3 Discretization with the Zero-Order Hold Method .........174
5.6.4 Discretization with the First-Order Hold Method ..........174
5.7 Formula Tables ..................................................................................175
5.8 Solved Exercises ................................................................................175
6. Stability of Digital Control Systems ......................................................217
6.1 Stability ...............................................................................................217
6.2 Unit-Circle Criterion .........................................................................219
6.3 Routh Criterion Using the Bilinear Mobius Transformation ......219
6.4 Jury Criterion (Eliahu I. Jury 1955) .................................................221
6.5 Root Locus Method ...........................................................................221
6.5.1 Rules for Approximate Establishment of Root Locus ....222
6.6 Nyquist Stability Criterion ...............................................................224
Contents ix
6.7 Bode Stability Criterion (H.W. Bode—1930) ..................................226
6.8 Formula Table ....................................................................................227
6.9 Solved Exercises ................................................................................229
7. Time and Harmonic Response Analysis Steady-State Errors ...........275
7.1 Time Response ...................................................................................275
7.1.1 Impulse Time Response of First-Order Systems .............276
7.1.2 Impulse Time Response of Second-Order Systems ........277
7.1.3 Step Response of First-Order System ................................278
7.1.4 General Form for the Step Response of Discrete
Time System ..........................................................................278
7.1.5 Correlation between Analog and Discrete Time
Response ...............................................................................281
7.2 Steady-State Errors ............................................................................283
7.3 Harmonic Response of Discrete Systems ......................................287
7.4 Formula Tables ..................................................................................288
7.5 Solved Exercises ................................................................................290
8. Compensation of Digital Control Systems ...........................................325
8.1 Introduction .......................................................................................325
8.2 Indirect Design Methods .................................................................326
8.2.1 Selection of Sampling Rate .................................................328
8.3 Direct Design Methods ....................................................................330
8.3.1 Design via Analytical Methods .........................................330
8.3.2 Design Using the Root Locus Method
of the Characteristic Equation ............................................331
8.3.3 Design in the Frequency Domain .....................................332
8.4 PID Digital Controller ......................................................................333
8.4.1 Digital PID Controller Tuning ...........................................336
8.5 Deadbeat Digital Controller ............................................................338
8.6 Phase Lead/Lag Digital Compensators .........................................338
8.7 Formula Table ....................................................................................339
8.8 Solved Exercises ................................................................................340
9. Simulation Tools: MATLAB, Simulink, LabVIEW,
Comprehensive Control ............................................................................399
9.1 Introduction .......................................................................................399
9.2 Control Systems Simulation Using MATLAB ...............................399
9.2.1 Analysis and System Modeling .........................................400
9.2.2 Control Systems Design ......................................................400
9.2.3 Simulation of Digital Control Systems
Using MATLAB ....................................................................400
9.3 Simulink .............................................................................................406
9.3.1 Introduction ..........................................................................406
9.3.2 Model Creation .....................................................................407
9.4 LabVIEW.............................................................................................411