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Control Engineering - An introduction with the use of Matlab

155 Pages · 2010 · 3.69 MB · English

  • Control Engineering - An introduction with the use of Matlab

    DEREK ATHERTON


    CONTROL ENGINEERING


    AN INTRODUCTION WITH THE USE OF MATLAB


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    BOOKBOON.COM Derek Atherton


    Control Engineering


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    2 Control Engineering


    © 2009 Derek Atherton & Ventus Publishing ApS


    ISBN 978-87-7681-466-3


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    3 Control Engineering Contents


    Contents


    Preface 8


    About the author 10


    1. Introduction 11


    1.1 What is Control Engineering? 11


    1.2 Contents of the Book 13


    1.3 References 15


    2. Mathematical Model Representations of Linear Dynamical Systems 16


    2.1 Introduction 16


    2.2 The Laplace Transform and Transfer Functions 17


    2.3 State space representations 20


    2.4 Mathematical Models in MATLAB 23


    2.5 Interconnecting Models in MATLAB 26


    2.6 Reference 28


    3. Transfer Functions and Their Responses 29


    3.1 Introduction 29


    3.2 Step Responses of Some Specifi c Transfer Functions 30


    3.3 Response to a Sinusoid 36


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    Control Engineering Contents


    40


    4. Frequency Responses and Their Plotting 40


    4.1 Introduction 40


    4.2 Bode Diagram 46


    4.3 Nyquist Plot 48


    4.4 Nichols Plot


    50


    5. The Basic Feedback Loop 50


    5.1 Introduction 51


    5.2 The Closed Loop 52


    5.3 System Specifi cations 55


    5.4 Stability


    60


    6. More on Analysis of the Closed Loop System 60


    6.1 Introduction 60


    6.2 Time Delay 61


    6.3 The Root Locus 64


    6.4 Relative Stability 68


    6.5 M and N Circles


    70


    7. Classical Controller Design 70


    7.1 Introduction 70


    7.2 Phase Lead Design 77


    7.3 Phase Lag Design 80


    7.4 PID Control 90


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    Control Engineering Contents


    7.5 References 92


    8. Parameter Optimisation for Fixed Controllers 92


    8.1 Introduction 93


    8.2 Some Simple Examples 96


    8.3. Standard Forms 101


    8.4 Control of an Unstable Plant 103


    8.5 Further Comments 104


    8.6 References


    105


    9. Further Controller Design Considerations 105


    9.1 Introduction 105


    9.2 Lag-Lead Compensation 107


    9.3 Speed Control 109


    9.4 Position Control 110


    9.5 A Transfer Function with Complex Poles 114


    9.6 The Effect of Parameter Variations 120


    9.7 References


    121


    10. State Space Methods 121


    10.1 Introduction 121


    10.2 Solution of the State Equation 124


    10.3 A State Transformation 124


    10.4 State Representations of Transfer Functions 130


    10.5. State Transformations between Different Forms 131


    10.6 Evaluation of the State Transition Matrix


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    Control Engineering Contents


    133


    10.7 Controllability and Observability 134


    10.8 Cascade Connection


    135


    11. Some State Space Design Methods 135


    11.1 Introduction 136


    11.2 State Variable Feedback 138


    11.3 Linear Quadratic Regulator Problem 139


    11.4 State Variable Feedback for Standard Forms 144


    11.5 Transfer Function with Complex Poles


    147


    Appendix A


    152


    Appendix B


    154


    Appendix C


    155


    Appendix D


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    7


    Control Engineering Preface


    Preface



    Control engineering courses have been given in universities for over fifty years. In fact it is just


    fifty years since I gave my first lectures on the subject. The basic theoretical topics taught in what


    is now often referred to as classical control have changed little over these years, but the tools


    which can be used to support theoretical analysis and the technologies used in control systems


    implementation have changed beyond recognition. I was lucky enough in the early days to have


    access to one of the first digital computers in a UK university, but programming was elementary,


    input was paper tape and output results, obtained often after a considerable delay, were just


    numbers on paper, which had to be laboriously plotted if one needed a graph. Simulations were


    done on analogue computers, which although having some nice features, had many deficiences.


    Today there are powerful digital simulation languages and specialised numerical software


    programs, which can be used on a desk top or lap top computer with excellent interaction and


    good graphical output. Although this book is not concerned with the technological


    implementation of control systems the technology has changed from components such as the


    vacuum tube, individual resistors and capacitors, and d.c commutator motors to integrated


    circuits, microprocessors, solid state power electronics and brushless machines. All of these are


    orders of magnitude cheaper, more robust, reliable and efficient.



    The majority of students graduating from engineering courses in universities will go on to work


    in industry where employers, if the company is to survive, will provide their employees doing


    analytical control system design with computers with appropriate computational software. The


    role of the university lecturer should therefore be to teach courses in such a way that the student


    knows enough detail about the concepts used that he can see whether results obtained are


    plausible, whilst leaving the computer to do the detailed analytical calculations. This has the


    advantage that more realistic problems can be studied, comparisons can easily be made between


    the results produced by alternative design approaches and hopefully the student can learn more


    about control engineering than worrying about doing mathematics. Many students, without doubt,


    are ‘turned off’ control engineering because of the perceived mathematical content and whilst


    further study on the theoretical aspects is required for prospective research students, they will be


    a small proportion of the class in a first course on control engineering. There are difficulties in


    this approach, as I am strongly of the opinion that student’s weaknesses in algebra have been


    caused by them not having carried out traditional procedures in arithmetic due to the adoption of


    calculators. However, I’m also sure there is a ‘happy medium’ somewhere. The use of modern


    software with simulation facilities allows the student to practice the interesting philosophy about


    doing engineering put forward in the book ‘Think, Play, Do’ by Dodgson et al OUP,2005.



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    8


    Control Engineering Preface


    The material presented in this book has been set out with this philosophy in mind and it is hoped


    that it will enable the reader to obtain a sound knowledge of classical control system analytical


    design methods. Several software packages could have been used to support this approach but


    here MATLAB, which is the most widely used, has been employed. Sadly, however, if


    universities continue to use outdated examining methods where students are required to plot root


    locus, Nyquist diagrams etc. the reader may have to spend some additional time doing


    computations best done by a computer! Because I want to ‘get over’ ideas, understanding and


    concepts without detailed mathematics I have used words such as ‘it can be shown that’ to


    shorten some of the mathematical detail. This provides the reader interested in theory with the


    opportunity to do additional calculations.



    The first chapter provides a brief introduction to feedback control and then has a section


    reviewing the contents of the book, which will therefore not be repeated here. I am indebted to


    my recent former students Ali Boz and Nusret Tan for providing me with some diagrams,


    assistance with computations, reading the text and doing some of the research which has


    provided information and results on some of the topics covered. For over forty years I have


    benefitted greatly from discussions with and input from many research students, who are too


    numerous to name here but have all helped to enrich the learning experience. Finally, I would


    like to acknowledge the efforts of my friend Dr Karl Jones in reading through the manuscript and


    providing me with constructive feedback. I trust that few errors remain in the text and I’d


    appreciate feedback from any reader who finds any or has any questions on the contents.



    Derek P. Atherton


    Brighton,


    February 2009



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    9


    Control Engineering About the author


    About the author



    Professor Derek. P. Atherton


    BEng, PhD, DSc, CEng, FIEE, FIEEE, HonFInstMC, FRSA



    Derek Atherton studied at the universities of Sheffield and Manchester, obtaining a PhD in 1962


    and DSc in 1975 from the latter. He spent the period from 1962 to 1980 teaching in Canada


    where he served on several National Research Council committees including the Electrical


    Engineering Grants Committee.



    He took up the post of Professor of Control Engineering at the University of Sussex in 1980 and


    is currently retired but has an office at the university, gives some lectures, and has the title of


    Emeritus Professor and Associate Tutor. He has been active with many professional engineering


    bodies, serving as President of the Institute of Measurement and Control in 1990, President of the


    IEEE Control Systems Society in 1995, being the only non North American to have held the


    position, and as a member of the IFAC Council from 1990-96. He served as an Editor of the IEE


    Proceedings on Control Theory and Applications (CTA) for several years until 2007 and was


    formerly an editor for the IEE Control Engineering Book Series. He has served EPSRC on


    research panels and as an assessor for research grants for many years and also served as a


    member of the Electrical Engineering Panel for the Research Assessment Exercise in 1992.



    His major research interests are in non-linear control theory, computer aided control system


    design, simulation and target tracking. He has written two books, is a co-author of two others and


    has published more than 350 papers in Journals and Conference Proceedings. Professor Atherton


    has given invited lectures in many countries and supervised over 30 Doctoral students.



    Derek P. Atherton.


    February 2009.



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    10



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