# Control Engineering

## An introduction with the use of Matlab

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## About the book

### Description

The book covers the basic aspects of linear single loop feedback control theory. Explanations of the mathematical concepts used in classical control such as root loci, frequency response and stability methods are explained by making use of MATLAB plots but omitting the detailed mathematics found in many textbooks. There is a chapter on PID control and two chapters provide brief coverage of state variable methods. The approach adopted allows more time to be devoted to controller design by different methods, to compare the results and also to examine the effects of plant parameter variations.

This free eBook can be read in combination with and in some cases instead of the following textbooks:

- Modern Control Engineering, by Katsuhiko Ogata
- Control Systems Engineering 6th edition, by Norman S Nise
- Modern Control Engineering 4th edition, by Katsuhiko Ogata
- Basic Control Systems Engineering 1st edition, by Paul H Lewis & Chang Yang
- Principles of Control Engineering 1st edition, by Fred White
- Modern Control Engineering 1st edition, by P N Paraskevopoulos

### Preface

Preface to the second edition

It is almost four years since the first edition of this book so it seemed appropriate to reread it carefully again and make any suitable changes. Also during the intervening period I have added two further bookboon books one on ‘An Introduction to Nonlinearity in Control Systems’ and another very recently on ‘Control Engineering Problems with Solutions’. This later book contains worked examples and some problems with answers only, which cover the material in this book and ‘An Introduction to Nonlinearity in Control Systems’. It is hoped that the relevant chapters of ‘Control Engineering Problems with Solutions’ will help the reader gain a better understanding and deeper knowledge of the topics covered in this textbook.

Minor changes have been made to this second edition mainly with respect to a few changes in wording, but sadly despite repeated reading a few minor technical errors were found and corrected, for which I apologise. These were Figure 3.6 which had some incorrect markings and was not very clear due to the numbers chosen giving lines almost on top of each other. This has been corrected by choosing a different frequency for illustrating the frequency response calculation procedure. Further, some negative signs were omitted from equation (2.14), the units of H on page 50 were given incorrectly as were the subscripts on the a’s and a matrix in the material in section 10.5.1, page 131, on transforming to the controllable canonical form. Finally the cover page has been changed to contain a picture which is more relevant to the book.

Derek P Atherton

Brighton , June 2013.

Preface to the first edition

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.

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

### Content

**1. Introduction**

1.1 What is Control Engineering?

1.2 Contents of the Book

1.3 References

**2. Mathematical Model Representations of Linear Dynamical Systems**

2.1 Introduction

2.2 The Laplace Transform and Transfer Functions

2.3 State space representations

2.4 Mathematical Models in MATLAB

2.5 Interconnecting Models in MATLAB

2.6 Reference

**3. Transfer Functions and Their Responses**

3.1 Introduction

3.2 Step Responses of Some Specific Transfer Functions

3.3 Response to a Sinusoid

**4. Frequency Responses and Their Plotting**

4.1 Introduction

4.2 Bode Diagram

4.3 Nyquist Plot

4.4 Nichols Plot

**5. The Basic Feedback Loop**

5.1 Introduction

5.2 The Closed Loop

5.3 System Specifications

5.4 Stability

**6. More on Analysis of the Closed Loop System**

6.1 Introduction

6.2 Time Delay

6.3 The Root Locus

6.4 Relative Stability

6.5 M and N Circles

**7. Classical Controller Design**

7.1 Introduction

7.2 Phase Lead Design

7.3 Phase Lag Design

7.4 PID Control

7.5 References

**8. Parameter Optimisation for Fixed Controllers**

8.1 Introduction

8.2 Some Simple Examples

8.3 Standard Forms

8.4 Control of an Unstable Plant

8.5 Further Comments

8.6 References

**9. Further Controller Design Considerations**

9.1 Introduction

9.2 Lag-Lead Compensation

9.3 Speed Control

9.4 Position Control

9.5 A Transfer Function with Complex Poles

9.6 The Effect of Parameter Variations

9.7 References

**10. State Space Methods**

10.1 Introduction

10.2 Solution of the State Equation

10.3 A State Transformation

10.4 State Representations of Transfer Functions

10.5 State Transformations between Different Forms

10.6 Evaluation of the State Transition Matrix

10.7 Controllability and Observability

10.8 Cascade Connection

**11. Some State Space Design Methods**

11.1 Introduction

11.2 State Variable Feedback

11.3 Linear Quadratic Regulator Problem

11.4 State Variable Feedback for Standard Forms

11.5 Transfer Function with Complex Poles

### About the Author

Professor Derek. P. Atherton

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

Derek Atherton studied at the universities of Sheffield ( BEng 1956) and Manchester, obtaining a PhD in 1962 and DSc in 1975 from the latter. He spent the period from 1962 to 1980 teaching and doing research in Canada, first at McMaster University until 1964 and then at the University of New Brunswick. Whilst in Canada 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, and as a member of the IFAC Council from 1990-96. He was an Editor of the IEE Proceedings on Control Theory and Applications (CTA) for several years until 2007 and was also 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 three books, is a co-author of two others [1-5] 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.

1. Atherton D P, *Nonlinear Control Engineering: Describing Function Analysis and Design*. London, Van Nostrand Reinhold, September 1975, 627 pages. (also abridged version) Atherton D P, *Nonlinear Control Engineering*. Van Nostrand Reinhold, 1982, student edition, 470 pages

2. Atherton D P, *Stability of Nonlinear Systems*. Research Studies Press, John Wiley,1981, 231 pages

3. Atherton, D.P. Control Engineering 2009 Bookboon publications at www.bookboon.com

4. Furuta K, Sano A and Atherton D P *State Variable Methods in Automatic Control*. John Wiley, 1988, 212 pages.

5. Xue, D , Chen,Y and Atherton,D.P Linear Feedback Control; *Analysis and Design with MATLAB* SIAM books, Philadelphia, USA, 2007, pp354.

Derek P. Atherton.

August 2010.