NEW METHODS FOR ESTIMATION, MODELING AND VALIDATION OF DYNAMICAL SYSTEMS USING AUTOMATIC DIFFERENTIATION A Dissertation by DANIEL TODD GRIFFITH Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY December 2004 Major Subject: Aerospace Engineering NEW METHODS FOR ESTIMATION, MODELING AND VALIDATION OF DYNAMICAL SYSTEMS USING AUTOMATIC DIFFERENTIATION A Dissertation by DANIEL TODD GRIFFITH Submitted to Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved as to style and content by: ______________________________ John L. Junkins (Chair of Committee) ______________________________ ______________________________ Srinivas R. Vadali John E. Hurtado (Member) (Member) ______________________________ ______________________________ Alan B. Palazzolo Walter E. Haisler (Member) (Head of Department) December 2004 Major Subject: Aerospace Engineering iii ABSTRACT New Methods for Estimation, Modeling and Validation of Dynamical Systems Using Automatic Differentiation. (December 2004) Daniel Todd Griffith, B.S., Morehead State University; B.S., University of Kentucky; M.S., University of Kentucky Chair of Advisory Committee: Dr. John L. Junkins The main objective of this work is to demonstrate some new computational methods for estimation, optimization and modeling of dynamical systems that use automatic differentiation. Particular focus will be upon dynamical systems arising in Aerospace Engineering. Automatic differentiation is a recursive computational algorithm, which enables computation of analytically rigorous partial derivatives of any user-specified function. All associated computations occur, in the background without user intervention, as the name implies. The computational methods of this dissertation are enabled by a new automatic differentiation tool, OCEA (Object oriented Coordinate Embedding Method). OCEA has been recently developed and makes possible efficient computation and evaluation of partial derivatives with minimal user coding. The key results in this dissertation details the use of OCEA through a number of computational studies in estimation and dynamical modeling. Several prototype problems are studied in order to evaluate judicious ways to use OCEA. Additionally, new solution methods are introduced in order to ascertain the iv extended capability of this new computational tool. Computational tradeoffs are studied in detail by looking at a number of different applications in the areas of estimation, dynamical system modeling, and validation of solution accuracy for complex dynamical systems. The results of these computational studies provide new insights and indicate the future potential of OCEA in its further development. v To my parents Dan and Barbara and my brothers and sisters Christie, Ryan and Andy vi ACKNOWLEDGMENTS Setting upon any of life’s journeys requires the support and guidance of many people. Many people have shared equally the successes and disappointments which have arisen while on the path to completing this dissertation. Of course, there are many who have made technical contributions that aided me in meeting my educational objectives and, more specifically to completing this work, many have made meaningful contributions. I have benefited greatly, as well, from the support of many people in my personal life who have done so much to shape me into who I am today. I am greatly indebted to my graduate advisor, Dr. John Junkins. I thank him for welcoming me to study with him at Texas A&M in August 2000. I, of course, acknowledge the technical contributions he has made toward completing this dissertation. He has been a source of knowledge on many subjects, and has given me a glimpse into the fundamental principles which unify many technical areas. He has been a source of inspiration to me whether it be in the classroom, in a research meeting or in passing in the hallway, and gave his time freely to me whenever it was needed. I especially appreciate the guidance and words of encouragement he provided during my toughest times and most misguided moments. I appreciate the financial support he provided over my first year of study and during the latter months while I completed this dissertation. My earnest hope is that I take a great deal from my experiences with Dr. Junkins to my professional life since I feel the impact he has had on me is significant in many ways, not only for the technical knowledge and the large-scale perspective I have vii gained from him, but also for the passion he has for his work and students. I could not be more pleased with the time I spent with him at Texas A&M. My parents, Dan and Barbara, have always been my biggest asset in their constant support, by the example they set, and for the drive and confidence they have instilled in me my entire life. They have, most importantly, tempered my ambition by demonstrating the balance in life that I strive for in order to be truly happy and completely successful. My most joyous moments have come since meeting the most special person in my life, my fiancé Loraine. Her support has inspired me to work diligently and focus on the completion of this dissertation. She has sacrificed a great deal by understanding the importance of this dissertation to me and has supported me in more ways than I could ever expect or deserve. I could not have completed this without her love and support. I acknowledge the tremendous impact that Dr. James Turner has had on technical merit of this dissertation. I thank him for the countless instances he has given his time to share his technical expertise. I appreciate his sense of humor and the passion he has for his work. It was entirely my pleasure to have the opportunity to work with him. I am very thankful for the insights and valuable time of my committee members Dr S. Rao Vadali, Dr. John E. Hurtado, and Dr. Alan Palazzolo. Dr. Vadali has provided me with a fine example in the thoughtfulness of his classroom teaching. I am thankful for the time he devoted to me as I learned much about optimization. I am indebted to Dr. Hurtado for many inspiring lectures and conversations regarding advanced topics in dynamics. I am thankful for his practical, informal style of teaching and mentoring. viii I am thankful for the support of the American Society for Engineering Education in awarding three years of financial assistance to support my doctoral studies through the National Defense Science and Engineering Fellowship Program from 2001-2004. The Aggie campus is unlike any other in this nation. I appreciate the caring nature, togetherness and values of the people of Texas A&M and College Station, Texas. In my time in College Station, I am quite thankful for the community of St. Mary’s Catholic Church, especially Father Mike Sis, for providing me with great words of wisdom and perspectives on living. I shared many good times with many new friends and colleagues at Texas A&M. I am thankful to have had two wonderful friends as office mates. Chad Searcy has been and is a friend who has truly enhanced my experience at Texas A&M by showing me the ropes when I arrived in College Station. I appreciate his kindness and generosity, and our many discussions on politics. Rajnish Sharma has been an exceptional person to share a workspace with in my last days at Texas A&M. I appreciate his warm personality and have taken a great deal from his relaxed attitude regarding life and work. I would like to acknowledge two friends who have made the times away from my desk very enjoyable. Lisa Biggs has been a great friend, and someone who has many times given me a good perspective on what is truly important. I thank her for her friendship. I am also very thankful for the times I have spent with Andy Sinclair. I had the pleasure of enjoying many discussions with him on Southeastern Conference (SEC) sports. Although he leans toward the University of Florida Gators (and I the University of Kentucky Wildcats), I am very happy for the sense of home I received from our ix discussions. I am also very happy for the significant technical discussions we had, many of which led us to hours of research on the dynamics encountered in the game of golf. The match play feature of the Tiger Woods golf game executed on the PlayStation II game system proved to be an ideal testbed for this study. I would also like to thank Art and Rocio Fano for welcoming me into their family. Art and Rocio have made the last year of my studies very enjoyable through their warmth and thoughtful attention. There are many others who have I have not mentioned here by name who I would like to thank. I sincerely thank them as well. x TABLE OF CONTENTS Page ABSTRACT…………………………………………………………………………. iii DEDICATION………………………………………………………………………. v ACKNOWLEDGMENTS…………………………………………………………… vi TABLE OF CONTENTS……………………………………………………………. x LIST OF TABLES………………………………….……………………………….. xii LIST OF FIGURES………………………………………………………………….. xiv CHAPTER I INTRODUCTION…………………………………………………… 1 II OVERVIEW OF COMPUTERIZED DIFFERENTIATION……….. 5 2.1 Symbolic Differentiation………………………………… 5 2.2 Automatic Differentiation……………………………….. 6 2.3 Overview of OCEA……………………………………… 7 III ESTIMATION AND CONTROL OF DYNAMICAL SYSTEMS…. 14 3.1 Review of First-order Algorithms……………………….. 15 3.2 Higher-order Generalized Sensitivity Calculations……… 22 3.3 Higher-order Algorithms………………………………… 38 3.4 Numerical Examples…………………………………….. 50 3.5 Summary…………………………………………………. 74 IV MODELING OF DYNAMICAL SYSTEMS……………………….. 75 4.1 Overview………………………………………………… 76 4.2 Equations of Motion Formulation……………………….. 77 4.3 Formulation via Lagrange’s Equation…………………… 81 4.4 Numerical Integration of Equations of Motion………….. 94 4.5 Numerical Examples…………………………………….. 96 4.6 Comparison with Hard-coding Equations of Motion……. 114 4.7 Summary…………………………………………………. 118