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Transfer Operator-Based Approach for Domain of Attraction Computation and Experimental Data Analysis Kai Wang Iowa State University

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Transfer Operator-Based Approach for Domain of Attraction Computation and Experimental Data Analysis Kai Wang Iowa State University Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2010 Transfer operator-based approach for domain of attraction computation and experimental data analysis Kai Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Electrical and Computer Engineering Commons Recommended Citation Wang, Kai, "Transfer operator-based approach for domain of attraction computation and experimental data analysis" (2010). Graduate Theses and Dissertations. 11762. https://lib.dr.iastate.edu/etd/11762 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Transfer operator-based approach for domain of attraction computation and experimental data analysis by Kai Wang A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Electrical Engineering Program of Study Committee: Umesh Vaidya, Major Professor Baskar Ganapathysubramanian Hui Hu Iowa State University Ames, Iowa 2010 Copyright c Kai Wang, 2010. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my parents without whose support I would not have been able to complete this work. iii TABLE OF CONTENTS LIST OF FIGURES . vi PART I TRANSFER OPERATER-BASED APPROACH FOR DOMAIN OF ATTRACTION COMPUTATION 1 OVERVIEW . 2 CHAPTER 1. INTRODUCTION . 4 1.1 Preliminaries . 4 1.2 Motivations . 4 1.3 Literature Review . 5 CHAPTER 2. TRANSFER OPERATOR APPROACH FOR COMPUT- ING DOMAIN OF ATTRACTION . 8 2.1 Lyapunov Measure for Stability . 8 2.2 Finite Dimension Approximation . 12 2.3 Domain of Attraction Computation . 16 2.3.1 Procedure for Constructing Markov Matrix from the Sub-Markov Matrix 19 2.3.2 Iterative Algorithm for Computation of DA . 20 2.4 Simulations for Domain of Attraction . 21 2.4.1 Example for a 2 Dimensional Polynomial System . 21 2.4.2 Example for a 2 Dimensional Non-polynomial System . 22 2.4.3 Example for a 3 Dimensional Polynomial System . 22 2.4.4 Example for Henon Map . 23 iv 2.4.5 Example for Chua's Equation . 25 CHAPTER 3. CONCLUSIONS AND FUTURE RESEARCH . 27 3.1 Conclusions . 27 3.2 Future Research . 27 PART II EXPERIMENTAL DATA ANALYSIS 28 OVERVIEW . 29 CHAPTER 4. INTRODUCTION . 31 4.1 Preliminaries . 31 4.1.1 Proper Orthogonal Decomposition Mode . 31 4.1.2 Koopman Operator and Koopman Mode . 32 4.2 Motivations for Experimental Data Analysis . 33 4.2.1 Proper Orthogonal Decomposition Mode Analysis . 33 4.2.2 Spectral Analysis . 33 4.3 Literature Review . 34 4.3.1 Proper Orthogonal Decomposition Mode . 34 4.3.2 Koopman Mode . 34 4.4 Experimental Setup of Flapping Wings . 34 CHAPTER 5. PROPER ORTHOGONAL DECOMPOSITION MODE . 36 5.1 POD Mode of Data for Different Flight Speeds . 37 5.1.1 POD Mode of Non-synchronous Data . 37 5.1.2 POD Mode of Synchronous Data . 38 5.2 POD Mode of Data for Different Flapping Amplitudes . 41 5.2.1 POD Mode of Non-synchronous Data . 41 5.2.2 POD Mode of Synchronous Data . 42 CHAPTER 6. PERRON-FROBENIUS MODE . 45 6.1 P-F Operator and P-F Mode . 45 v 6.2 Algorithm for Generation of P-F Mode from Snapshots . 47 6.3 P-F Mode and Dominant Frequency of Experimental Data . 49 6.3.1 P-F Mode and Dominant Frequency of Data for Different Flight Speeds 50 6.3.2 P-F Mode and Dominant Frequency of Data for Different Flapping Am- plitudes . 54 CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH . 59 7.1 Conclusions . 59 7.2 Future Research . 60 BIBLIOGRAPHY . 61 vi LIST OF FIGURES Figure 2.1 Eigenvalues and the invariant measure . 13 Figure 2.2 Scheme for computation of domain of attraction . 16 Figure 2.3 Domain of attraction and phase portrait for Example 1 . 22 Figure 2.4 Domain of attraction and phase portrait for Example 2 . 23 Figure 2.5 Domain of attraction for Example 3 . 24 Figure 2.6 Domain of attraction for Example 4 . 25 Figure 2.7 Attractor set for Example 5 . 26 Figure 2.8 Domain of attraction for Example 5 . 26 Figure 5.1 POD modes of non-synchronous data for different flight speeds . 37 Figure 5.2 Eigenvalues of POD modes of non-synchronous data for different flight speeds . 38 Figure 5.3 POD modes of synchronous data for different flight speeds . 39 Figure 5.4 Eigenvalues of POD modes of synchronous data for different flight speeds 40 Figure 5.5 POD modes of non-synchronous data for different flapping amplitudes 41 Figure 5.6 Eigenvalues of POD modes of non-synchronous data for different flap- ping amplitudes . 42 Figure 5.7 POD modes of synchronous data for different flapping amplitudes . 43 Figure 5.8 Eigenvalues of POD modes of synchronous data for different flapping amplitudes . 43 Figure 6.1 P-F modes of non-synchronous data for different flight speeds . 51 vii Figure 6.2 Frequencies of P-F modes of non-synchronous data for different flight speeds . 51 Figure 6.3 Frequencies of non-synchronous data for different flight speed . 52 Figure 6.4 P-F modes of synchronous data for different flight speeds . 53 Figure 6.5 Frequencies of P-F modes of synchronous data for different flight speeds 53 Figure 6.6 Frequencies of synchronous data for different flight speeds . 54 Figure 6.7 P-F modes of non-synchronous data for different flapping amplitudes . 55 Figure 6.8 Frequencies of P-F modes of non-synchronous data for different flapping amplitudes . 56 Figure 6.9 Frequencies of non-synchronous data for different flapping amplitudes . 57 Figure 6.10 P-F modes of synchronous data for different flapping amplitudes . 57 Figure 6.11 Frequencies of P-F modes of synchronous data for different flapping amplitudes . 58 Figure 6.12 Frequencies of synchronous data for different flapping amplitudes . 58 1 PART I TRANSFER OPERATER-BASED APPROACH FOR DOMAIN OF ATTRACTION COMPUTATION 2 OVERVIEW The problem of computation of domain of attraction (DA) is of great practical interest given the importance of the problem in various engineering and physical systems. Examples include DA computation in power systems, chemical process, aircraft control, biological systems, and ecology. Most of the existing methods for the computation of DA are Lyapunov functions based and are restricted to DA computation of an equilibrium point or at best periodic solution. There exists no method for DA computation of complex steady state such as chaotic attractor or sets consisting of quasi-periodic motion. Such steady states are often exhibited by complex dynamical systems. Examples include biological systems and synchronized state of coupled oscillators. Existing methods for DA computation are either developed for polynomial vector fields or relies on the point-wise description of the Lyapunov functions that describes DA. However complex dynamics can often be described most efficiently on sets as opposed to on points. In this part, we propose a transfer operator-based method for the computation of DA. This proposed approach is used for almost everywhere global stability verification of an attractor set in nonlinear dynamical system. The basic idea behind the transfer operator approach is to replace the finite dimensional nonlinear evolution of dynamical system with a linear, albeit infinite dimensional, evolution described by linear transfer operator. This linear transfer operator is called as Perron-Frobenius (P-F) and is used to propagate sets or measure supported on sets in the phase space. The finite dimensional approximation of the P-F operator can be obtained using set-oriented numerical techniques developed by Dellnitz and group. We extend the application of P-F operator along with set-oriented numerical methods for 3 the computation of DA. We show that the iterative algorithm based on P-F operator is efficient to compute the DA for any nonlinear system since it can not only characterize the DA for the nonlinear system with equilibrium dynamics but also system with complex non-equilibrium dynamics. 4 CHAPTER 1. INTRODUCTION This introduction is divided into three sections. Section 1.1 briefly introduces P-F operator. Section 1.2 describes the motivations for computation of DA. And section 1.3 offers a review of existing papers. 1.1 Preliminaries In this section, we present an introduction of P-F operator. This helps the reader to get an intuitive understanding of P-F operator before we define it strictly. For a dynamical system, one can associate two different linear operators, called Koopman operator and P-F operator. While the dynamical system describes the evolution of an initial condition, the P-F operator describes the evolution of uncertainty in initial conditions. Set oriented numerical methods have recently been used to approximate the infinite dimensional P- F operator using their finite dimension counterpart [1]. In addition, the spectral analysis of the P-F operator gives a description of the asymptotic dynamics of nonlinear dynamical systems. In particular, the eigenfunction with eigenvalue 1 characterizes the invariant set capturing the long term asymptotic behavior of the system. Spectrum of the P-F operator on the unit circle has the information about the cyclic behavior of the system. 1.2 Motivations In this section, we present the motivations for computation of DA of a nonlinear system. When the origin x = 0 is asymptotically stable, we are often interested in determining how far from the origin the trajectory can be and still converge to the origin as t approaches 1.
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