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Generalization of Polynomial Chaos for Estimation of Angular Random Variables Scholars' Mine Doctoral Dissertations Student Theses and Dissertations Summer 2020 Generalization of polynomial chaos for estimation of angular random variables Christine Louise Schmid Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Aerospace Engineering Commons Department: Mechanical and Aerospace Engineering Recommended Citation Schmid, Christine Louise, "Generalization of polynomial chaos for estimation of angular random variables" (2020). Doctoral Dissertations. 2919. https://scholarsmine.mst.edu/doctoral_dissertations/2919 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. GENERALIZATION OF POLYNOMIAL CHAOS FOR ESTIMATION OF ANGULAR RANDOM VARIABLES by CHRISTINE LOUISE SCHMID A DISSERTATION Presented to the Graduate Faculty of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY in AEROSPACE ENGINEERING 2020 Approved by: Kyle J. DeMars, Advisor Jacob E. Darling Serhat Hosder Robert Paige Henry Pernicka Copyright 2020 CHRISTINE LOUISE SCHMID All Rights Reserved iii ABSTRACT The state of a dynamical system will rarely be known perfectly, requiring the vari­ able elements in the state to become random variables. More accurate estimation of the uncertainty in the random variable results in a better understanding of how the random variable will behave at future points in time. Many methods exist for representing a random variable within a system including a polynomial chaos expansion (PCE), which expresses a random variable as a linear combination of basis polynomials. Polynomial chaos expansions have been studied at length for the joint estimation of states that are purely translational (i.e. described in Cartesian space); however, many dynamical systems also include non-translational states, such as angles. Many methods of quantifying the uncertainty in a random variable are not capable of representing angular random variables on the unit circle and instead rely on projections onto a tangent line. Any element of any space V can be quantified with a PCE if V is spanned by the expansion’s basis polynomials. This implies that, as long as basis polynomials span the unit circle, an angular random variable (either real or complex) can be quantified using a PCE. A generalization of the PCE is developed allowing for the representation of complex valued random variables, which includes complex representations of angles. Additionally, it is proposed that real valued polynomials that are orthogonal with respect to measures on the real valued unit circle can be used as basis polynomials in a chaos expansion, which reduces the additional numerical burden imposed by complex valued polynomials. Both complex and real unit circle PCEs are shown to accurately estimate angular random variables in independent and correlated multivariate dynamical systems. iv ACKNOWLEDGMENTS First, I would like to thank my advisor Dr. Kyle J. DeMars for taking me in as an undergraduate and introduce me to the complex* world of state estimation. Without his guidance*, I would not be the researcher, engineer, or person that I am today. I must also thank him for his endless supply of revisions: I think, I can finally say, I’ve got the hang of, where to correctly place all of my commas. I would also like to thank my committee members, Drs. Jacob Darling, Serhat Hosder, Robert Paige, and Henry Pernicka. Whether through traditional teaching, or less formal methods of instruction, each of you has been instrumental in my education and in the preparation of this dissertation. In addition to my committee, I would like to thank all of the other students throughout my graduate career that have kept me sane during this entire process. You helped me work through all of my roadblocks, even when my main contributions to the lab were groan inducing puns. A specific thank you goes to Matt and Kari; it hasn’t been the easiest of semesters, some might even say it’s been the most challenging of semesters, but we made it through, and everything is fine, because life, uh, finds a way. Finally, I’d like to thank all my friends and family for supporting me through this entire process. Rob, you and Kodiak have been my rocks; I don’t know where I’d be without you two. To my parents, Amy and Charlie Schmid, I have never felt like there was something I couldn’t do, even when I decided to challenge the cliche “the sky’s the limit”. pun intended v TABLE OF CONTENTS Page ABSTRACT............................................................................................................................ iii ACKNOWLEDGMENTS..................................................................................................... iv LIST OF ILLUSTRATIONS....................................................................................................viii LIST OF TABLES................................................................................................................... x SECTION 1. INTRODUCTION................................................................................................................ 1 1.1. MOTIVATION AND LITERATURE SURVEY............................................... 1 1.2. CONTRIBUTIONS................................................................................................ 5 1.3. ORGANIZATION................................................................................................. 7 2. MATHEMATICAL PRELIMINARIES............................................................................ 9 2.1. MATHEMATICAL BASES.................................................................................. 9 2.1.1. Inner Product ............................................................................................. 11 2.1.2. Orthogonal B ases..................................................................................... 12 2.2. POLYNOMIAL BASES........................................................................................ 14 2.2.1. Askey Polynomials................................................................................... 15 2.2.2. Szego Polynomials................................................................................... 16 2.2.3. Arbitrary Polynomials ............................................................................. 21 2.2.3.1. General polynomial formulation.......................................... 21 2.2.3.2. Gram-Schmidt........................................................................ 23 2.3. PROBABILITY DENSITY FUNCTIONS........................................................ 25 vi 2.3.1. Densities in R ........................................................................................... 26 2.3.2. Densities on S ........................................................................................... 26 2.3.2.1. Wrapped normal density ....................................................... 27 2.3.2.2. von Mises density................................................................... 29 2.3.2.3. Bingham density..................................................................... 29 2.3.3. Statistics.................................................................................................... 30 3. UNCERTAINTY QUANTIFICATION......................................................................... 33 3.1. UNSCENTED TRANSFORM............................................................................. 34 3.2. POLYNOMIAL CHAOS EXPANSION............................................................. 36 3.2.1. Statistics.................................................................................................... 38 3.2.2. Coefficient Approximation - Least Squares.......................................... 44 3.2.2.1. Sampling m ethods.................................................................. 45 3.2.2.2. Implementation procedure .................................................... 48 3.2.3. Coefficient Solution - Inner Product ..................................................... 49 3.2.4. Expansion Truncation.............................................................................. 53 3.2.5. Coefficient Analysis.................................................................................. 57 4. CHAOS ON THE UNIT CIRCLE.................................................................................. 59 4.1. COMPLEX EXPANSION..................................................................................... 59 4.1.1. Statistics.................................................................................................... 60 4.1.2. Rogers-Szego-Chaos................................................................................ 63 4.1.3. Function Evaluations and Numerical Complexity............................... 65 4.2. POLYNOMIAL CHAOS ON THE REAL UNIT CIRCLE.............................. 67 4.3. POLYNOMIAL DENSITY .................................................................................. 70 5. RESULTS AND DISCUSSION...................................................................................... 72 5.1. ANGLES ONLY UNCERTAINTY PROPAGATION 73 vii 5.1.1. Assumptions.............................................................................................
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