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Sensitivity and Stability Analysis of Nonlinear Kalman Filters with Application to Aircraft Attitude Estimation

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Sensitivity and Stability Analysis of Nonlinear Kalman Filters with Application to Aircraft Attitude Estimation Graduate Theses, Dissertations, and Problem Reports 2013 Sensitivity and stability analysis of nonlinear Kalman filters with application to aircraft attitude estimation Matthew Brandon Rhudy West Virginia University Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Rhudy, Matthew Brandon, "Sensitivity and stability analysis of nonlinear Kalman filters with application ot aircraft attitude estimation" (2013). Graduate Theses, Dissertations, and Problem Reports. 3659. https://researchrepository.wvu.edu/etd/3659 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. SENSITIVITY AND STABILITY ANALYSIS OF NONLINEAR KALMAN FILTERS WITH APPLICATION TO AIRCRAFT ATTITUDE ESTIMATION by Matthew Brandon Rhudy Dissertation submitted to the Benjamin M. Statler College of Engineering and Mineral Resources at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Approved by Dr. Yu Gu, Committee Chairperson Dr. John Christian Dr. Gary Morris Dr. Marcello Napolitano Dr. Powsiri Klinkhachorn Department of Mechanical and Aerospace Engineering Morgantown, West Virginia 2013 Keywords: Attitude Estimation, Extended Kalman Filter, GPS/INS Sensor Fusion, Stochastic Stability Copyright 2013, Matthew B. Rhudy Abstract SENSITIVITY AND STABILITY ANALYSIS OF NONLINEAR KALMAN FILTERS WITH APPLICATION TO AIRCRAFT ATTITUDE ESTIMATION by Matthew B. Rhudy State estimation techniques are important tools for analyzing systems that contain states that are not directly measureable. If the estimated states are used, for example, in place of the true states in a feedback controller, the accuracy and stability of the estimates becomes crucial for the safe and effective execution of the controller. This is especially important in aircraft control applications, where safety is an essential concern. Because of this, the stability characteristics of the state estimation are investigated. Additionally, two different nonlinear Kalman filters are considered and compared with respect to various design parameters. This work considers the sensitivity and stability characteristics of nonlinear state estimation through the aircraft attitude estimation problem. This problem is approached using sensor information from Global Positioning System (GPS) and Inertial Navigation System (INS) in order to obtain estimates of the aircraft attitude angles. This case study uses experimentally collected flight data from subscale aircraft to derive estimation results. The goal of this work is to obtain a better understanding of the properties of nonlinear Kalman filters in order to make more informed decisions regarding the selection and tuning of these filters for different real-world applications. ACKNOWLEDGMENTS This research was partially supported by NASA grant # NNX07AT53A, grant # NNX10AI14G, grant # NNX12AM56A, and NASA West Virginia Space Grant Consortium Graduate Fellowship. I would like to thank my research advisor, Dr. Yu Gu, for his guidance and support throughout this project. Dr. Gu, your insight and technical advice provided me with great opportunities to succeed and really learn to think about various research problems. I greatly appreciate your time and effort spent providing me with both technical and career advice. I would like to thank the other faculty members of the Flight Control Systems Laboratory, Dr. Marcello Napolitano, Dr. Srikanth Gururajan, Dr. Brad Seanor, and Dr. Haiyang Chao. Your assistance, particularly with the flight testing program, was invaluable to my research and professional growth. I would also like to thank my committee members Dr. John Christian, Dr. Powsiri Klinkhachorn, and Dr. Gary Morris. Your discussions and feedback regarding this work were very helpful in improving its quality. Current and former students of the research group, Frank Barchesky, Matteo Dariol, Giovanni DeNunzio, Jason Gross, Matteo Guerra, Tanmay Mandal, Kyle Lassak, Kerri Phillips, Daniele Tancredi, Amanda McGrail, your collaborative effort has helped me to become a better group member, leader, and mentor. Thank you. I would particularly like to acknowledge the contribution of Jason Gross, who offered a fantastic introduction into this research area, as well as many productive discussions allowing us to both further our understanding. Finally, I would like to thank my wife, Stephanie, for supporting me throughout this process. Thank you for your love and support. iii TABLE OF CONTENTS ACKNOWLEDGMENTS ............................................................................................... iii 1.0 INTRODUCTION.................................................................................................... 1 1.1 OBJECTIVE ....................................................................................................... 6 1.2 ORGANIZATION .............................................................................................. 6 2.0 BACKGROUND INFORMATION ....................................................................... 8 2.1 STABILITY OF CONTINUOUS-TIME SYSTEMS ...................................... 8 2.1.1 Stability of Continuous-Time Linear Systems ..................................... 11 2.1.2 Lyapunov’s Linearization Method for Continuous-Time Systems .... 15 2.1.3 Lyapunov’s Direct Method for Continuous-Time Systems ................ 17 2.2 STABILITY OF DISCRETE-TIME SYSTEMS .......................................... 19 2.2.1 Stability of Discrete-Time Linear Systems ........................................... 20 2.2.2 Lyapunov’s Direct Method for Discrete-Time Systems ...................... 21 2.3 NONLINEAR STATE ESTIMATION .......................................................... 23 2.4 DISCRETE-TIME LINEAR KALMAN FILTER ........................................ 24 2.4.1 Discrete-Time Linear Kalman Filter Stability ..................................... 26 2.5 EXTENDED KALMAN FILTER .................................................................. 28 2.5.1 Extended Kalman Filter Stability ......................................................... 29 2.6 UNSCENTED KALMAN FILTER ................................................................ 35 2.6.1 Linearization of the UKF for Non-Additive Noise ............................... 37 2.6.2 Unscented Kalman Filter Stability ........................................................ 38 3.0 AIRCRAFT ATTITUDE ESTIMATION ........................................................... 40 3.1 GPS/INS SENSOR FUSION ........................................................................... 40 3.2 LOOSELY-COUPLED GPS/INS SENSOR FUSION FORMULATIONS 42 3.2.1 Inertial Navigation Equations ................................................................ 42 3.2.2 Acceleration Vector Attitude Estimation .............................................. 44 3.2.3 3-State GPS/INS Sensor Fusion Formulation ...................................... 46 3.2.4 6-State GPS/INS Sensor Fusion Formulation ...................................... 50 3.2.5 9-State GPS/INS Sensor Fusion Formulation ...................................... 51 3.2.6 12-State GPS/INS Sensor Fusion Formulation .................................... 51 3.2.7 15-State GPS/INS Sensor Fusion Formulation .................................... 52 3.3 ALTERNATIVE ATTITUDE ESTIMATION METHODS ........................ 53 4.0 EXPERIMENTAL PLATFORM ......................................................................... 57 4.1 FLIGHT DATA SELECTION ........................................................................ 60 5.0 SENSITIVITY ANALYSIS................................................................................... 63 5.1 PERFORMANCE EVALUATION METRICS ............................................ 65 5.2 EXPERIMENTAL SENSITIVITY ANALYSIS RESULTS ........................ 66 5.2.1 Comparison of Baseline Results for Individual Data Sets .................. 69 5.2.2 Sensitivity to Tuning of Assumed Noise Covariance Matrices ........... 71 5.2.3 Sensitivity to Sampling Rate .................................................................. 73 5.2.4 Sensitivity to Initialization Error .......................................................... 76 iv 5.2.5 Sensitivity to GPS Outages ..................................................................... 77 5.2.6 Robustness to Uncertainty in IMU Measurements .............................. 80 5.2.7 Comparison of Linearization Techniques ............................................ 82 5.2.8 Sensitivity to GPS Time Offset .............................................................. 84 5.2.9 Sensitivity to Acceleration due to Gravity ............................................ 86 5.2.10 Experimental Sensitivity Analysis Conclusions ................................... 87 5.3 MATRIX SQUARE ROOT OPERATIONS FOR UKF
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