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# An Introduction to Nonlinearity in Control Systems

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Language:  English
The book is concerned with the effects of nonlinearity in feedback control systems and techniques which can be used to design feedback loops containing nonlinear elements.
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The book is concerned with the effects of nonlinearity in feedback control systems and techniques which can be used to design feedback loops containing nonlinear elements. After a short introductory chapter on nonlinearity and its possible effects the use of phase plane methods for nonlinear second order systems is discussed. The next three chapters cover the describing function method and its applications in the determination of limit cycles, stability analysis and the prediction of unique nonlinear behaviour such as jump resonance and subharmonic generation. The following chapter shows using the Tsypkin approach how in feedback loops with a relay nonlinearity any limit cycles and their stability can be found exactly. Examples are given which cover some unique forms of limit cycles such as limit cycles with a sliding mode and multipulse limit cycles, which can be caused by resonant plant transfer functions. In recent years with the advent of microprocessor controllers there has been considerable interest in techniques for controller tuning based on limit cycle data and this is the topic covered in the next chapter. The analyses used make use of both the describing function method and the Tsypkin method presented in the previous chapters. The next topic covered is absolute stability methods for a nonlinear feedback loop with a single nonlinearity with emphasis placed on frequency domain methods to allow comparison with describing function results. The final chapter covers some methods for the design of nonlinear systems. The material is of an introductory nature but hopefully gives an overview and provides a basic understanding of various topics, many developed in recent years, and several of which have entire textbooks devoted to them.

1. Introduction
1. What is nonlinearity?
2. Forms of nonlinearity
3. Structure and Behaviour
4. Overview of contents
5. References
2. The Phase Plane Method
1. Introduction
2. Basic Principles
3. The Phase Plane for Systems with Linear Segmented Nonlinearities
4. Conclusions
5. References
6. Bibliography
3. The Describing Function
1. Introduction
2. The Sinusoidal Describing Function
3. Some Properties of the DF
4. The Evaluation of some DFs
5. Nonlinear Models and DFs
6. Harmonic Outputs
7. Sine plus Bias DF and the IDF
8. Conclusions
9. References
10. Bibliography
11. Appendix -Tables of Describing Functions
4. Stability and Limit Cycles using th e DF
1. Introduction
2. Limit Cycle Evaluation
3. Stability of a Predicted Limit Cycle
4. DF Accuracy
5. Some Examples of Limit Cycle Evaluation
6. More than one Nonlinear Element
7. Applications of SBOF to find Limit Cycles
8. Conclusions
9. Bibliography
5. The SSDF and Harmonically Forced Systems
1. Introduction
2. The Cubic Nonlinearity with Two Sinusoidal Inputs
3. Modified Nonllnearltles and the SSDF
4. The IDF for Related Signals
5. More Accurate Determination of Limit Cycles
6. Closed Loop Frequency Response
7. Jump Resonance
8. Conclusions
9. References
6. Limit cycles in relay systems
1. Introduction
2. The Frequency Domain Approach
3. Properties and Evaluation of A loci
4. Solving for Limit Cycles
5. Limit Cycle Stability
6. Some Interesting Limit Cycle Problems
7. Forced oscillations
8. Conclusions
9. References
10. Appendix
7. Controlle r Tuning from Relay Produ ced Limit Cycles
1. Introduction
2. Knowledge from the Limit Cycle
3. Tuning the Controller
4. Autotuning using the Relay in Parallel with the Controller
5. Conclusions
6. References
8. Absolute Stability Results
1. Introduction
2. Lyapunov's Method
3. Application of Lyapunov's Method
4. Definitions and Loop Transformations
5. Frequency Domain Criteria
6. Examples
7. Conclusions
8. References
9. Bibliography
9. Design of Nonlinear Control Systems
1. Introduction
2. Linearization
3. Frequency Response Shaping
4. Nonlinear Compensation
5. Compensation using DF Models
6. Time Optimal Control
7. Sliding Mode Control
8. Fuzzy Logic
9. Neural Networks
10. Exact Linearization
12. References

Derek Atherton

Professor Derek. P. Atherton
BEng, PhD, DSc, CEng, FIET, 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 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 four books, is a co-author of two others [1-6] 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.
6. Atherton, D.P. An Introduction to Nonlinearity in Control Systems. 2011Bookboon publications at www.bookboon.com

Derek P. Atherton.br />February 2012