FAULT DETECTION IN DYNAMIC SYSTEMS USING THE LARGEST LYAPUNOV EXPONENT A Thesis by YIFU SUN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2011 Major Subject: Mechanical Engineering FAULT DETECTION IN DYNAMIC SYSTEMS USING THE LARGEST LYAPUNOV EXPONENT A Thesis by YIFU SUN Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Approved by: Chair of Committee, Alexander G. Parlos Committee Members, Alan B. Palazzolo Jose Silva-Martinez Head of Department, Dennis O’Neal May 2011 Major Subject: Mechanical Engineering iii ABSTRACT Fault Detection in Dynamic Systems Using the Largest Lyapunov Exponent. (May 2011) Yifu Sun, B.S, Beijing Institute of Technology Chair of Advisory Committee: Dr. Alexander G. Parlos A complete method for calculating the largest Lyapunov exponent is developed in this thesis. For phase space reconstruction, a time delay estimator based on the average mutual information is discussed first. Then, embedding dimension is evaluated according to the False Nearest Neighbors algorithm. To obtain the parameters of all of the sub-functions and their derivatives, a multilayer feedforward neural network is applied to the time series data, after the time delay and embedding dimension are fixed. The Lyapunov exponents can be estimated using the Jacobian matrix and the QR decomposition. The possible applications of this method are then explored for various chaotic systems. Finally, the method is applied to some real world data to demonstrate the general relationship between the onset and progression of faults and changes in the largest Lyapunov exponent of a nonlinear system. iv To my beloved parents and wife v ACKNOWLEDGEMENTS Foremost, I would like to express my gratitude to my advisor Dr. Alexander G. Parlos. Without his patience and guidance, I could not have finished this thesis work. Besides my advisor, I would like to thank the rest of my thesis committee: Dr. Alan B. Palazzolo and Dr. Jose Silva-Martinez for their encouragement, insightful comments, and hard questions. Last but not the least, I would like to thank my family: my parents Dahong Sun and Hong Ren, my wife Chunliu Mao. They always support me through my life. vi TABLE OF CONTENTS Page ABSTRACT .............................................................................................................. iii DEDICATION .......................................................................................................... iv ACKNOWLEDGEMENTS ...................................................................................... v TABLE OF CONTENTS .......................................................................................... vi LIST OF FIGURES ................................................................................................... ix LIST OF TABLES .................................................................................................... xii CHAPTER I INTRODUCTION ................................................................................ 1 1.1 Research Motivation ...................................................................... 1 1.2 Literature Review ........................................................................... 2 1.3 Problem Statement ......................................................................... 8 1.4 Research Contribution .................................................................... 9 1.5 Organization ................................................................................... 9 II INTRODUCTION OF CHAOS AND LYAPUNOV EXPONENTS .. 11 2.1 Introduction .................................................................................... 11 2.2 Description of Chaos ...................................................................... 12 2.3 Categories of Dynamical Systems .................................................. 14 2.4 Sensitive Dependence upon Initial Conditions .............................. 17 2.5 Indication of Chaos ........................................................................ 18 2.6 Chapter Summary ........................................................................... 21 III COMPUTING THE LYAPUNOV EXPONENTS FROM MEASUREMENTS ............................................................................. 22 3.1 Introduction .................................................................................... 22 3.2 Problems with Computing the Lyapunov Exponents ..................... 23 3.3 Phase Space Reconstruction ........................................................... 26 vii CHAPTER Page 3.3.1 Average Mutual Information ................................................. 27 3.3.2 False Nearest Neighbors Method .......................................... 29 3.4 Jacobian Matrix and the QR Decomposition ................................. 30 3.5 Artificial Neural Networks ............................................................. 35 3.5.1 Introduction of Artificial Neural Networks ........................... 35 3.5.2 Neural Network Learning ...................................................... 37 3.5.3 Multilayer Feedforward Networks ........................................ 40 3.6 Chapter Summary ........................................................................... 44 IV SIMULATION RESULTS ................................................................... 45 4.1 Introduction .................................................................................... 45 4.2 The Lorenz Attractor ...................................................................... 45 4.2.1 Introduction of the Lorenz Attractor and Its Phase Space Reconstruction ....................................................................... 45 4.2.2 The Largest Lyapunov Exponent Calculation for the Lorenz System ................................................................................... 49 4.2.3 Chaotic Dynamics of the Lorenz System .............................. 64 4.3 The Hénon Map .............................................................................. 69 4.4 The Rössler Attractor ..................................................................... 71 4.5 Chapter Summary ........................................................................... 75 V APPLICATION OF LARGEST LYAPUNOV EXPONENT TO FAULT DETECTION .......................................................................... 76 5.1 Introduction .................................................................................... 76 5.2 Proposed Method for Incipient Fault Detection ............................. 76 5.3 A Real World Example .................................................................. 77 5.4 Experimental Results ...................................................................... 79 5.5 Chapter Summary ........................................................................... 85 VI SUMMARY AND CONCLUSION ..................................................... 86 6.1 Summary ........................................................................................ 86 6.2 Conclusion ...................................................................................... 87 6.3 Limitation and Future work ............................................................ 87 REFERENCES .......................................................................................................... 89 APPENDIX A ........................................................................................................... 94 viii APPENDIX B ........................................................................................................... 98 VITA ......................................................................................................................... 99 ix LIST OF FIGURES FIGURE Page 2.1 The oscillation of the water at the tip before and after the water drop....... 13 2.2 Sketch of magnet, superconductor, and elastic beam support and excitation apparatus .................................................................................... 14 2.3 The real space and state space of a pendulum with and without friction ........................................................................................................ 15 2.4 The appearance of a torus attractor ............................................................ 16 2.5 The phase space of a Van der Pol attractor ................................................ 16 3.1 Flow chart for calculating Lyapunov exponents ........................................ 25 3.2 Flow chart for distinguishing false neighbors and real neighbors.............. 30 3.3 The structure of an artificial neuron ........................................................... 36 3.4 The structure of a multilayer neural network ............................................. 36 3.5 A single neuron structure ........................................................................... 37 3.6 Neuron activation functions ....................................................................... 42 4.1 Time series plots for the three states of the Lorenz system ....................... 46 4.2 Three-dimensional plot of the Lorenz system ............................................ 47 4.3 Average mutual information for the Lorenz system .................................. 48 4.4 The result of applying the False Nearest Neighbors method to the Lorenz system ........................................................................................................