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Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise

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Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise Linköping studies in science and technology. Licentiate Thesis. No. 1613 Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise Michael Roth LERTEKN REG IK AU L TO RO MATIC CONT LINKÖPING Division of Automatic Control Department of Electrical Engineering Linköping University, SE-581 83 Linköping, Sweden http://www.control.isy.liu.se [email protected] Linköping 2013 This is a Swedish Licentiate’s Thesis. Swedish postgraduate education leads to a Doctor’s degree and/or a Licentiate’s degree. A Doctor’s Degree comprises 240 ECTS credits (4 years of full-time studies). A Licentiate’s degree comprises 120 ECTS credits, of which at least 60 ECTS credits constitute a Licentiate’s thesis. Linköping studies in science and technology. Licentiate Thesis. No. 1613 Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise Michael Roth [email protected] www.control.isy.liu.se Department of Electrical Engineering Linköping University SE-581 83 Linköping Sweden ISBN 978-91-7519-535-3 ISSN 0280-7971 LIU-TEK-LIC-2013:47 Copyright © 2013 Michael Roth Printed by LiU-Tryck, Linköping, Sweden 2013 Dedicated to all vegetarians. Abstract This thesis is on filtering in state space models. First, we examine approximate Kalman filters for nonlinear systems, where the optimal Bayesian filtering recur- sions cannot be solved exactly. These algorithms rely on the computation of cer- tain expected values. Second, the problem of filtering in linear systems that are subject to heavy-tailed process and measurement noise is addressed. Expected values of nonlinearly transformed random vectors are an essential in- gredient in any Kalman filter for nonlinear systems, because of the required joint mean vector and joint covariance of the predicted state and measurement. The problem of computing expected values, however, goes beyond the filtering con- text. Insights into the underlying integrals and useful simplification schemes are given for elliptically contoured distributions, which include the Gaussian and Student’s t distribution. Furthermore, a number of computation schemes are discussed. The focus is on methods that allow for simple implementation and that have an assessable computational cost. Covered are basic Monte Carlo in- tegration, deterministic integration rules and the unscented transformation, and schemes that rely on approximation of involved nonlinearities via Taylor polyno- mials or interpolation. All methods come with realistic accuracy statements, and are compared on two instructive examples. Heavy-tailed process and measurement noise in state space models can be ac- counted for by utilizing Student’s t distribution. Based on the expressions for conditioning and marginalization of t random variables, a compact filtering algo- rithm for linear systems is derived. The algorithm exhibits some similarities with the Kalman filter, but involves nonlinear processing of the measurements in form of a squared residual in one update equation. The derived filter is compared to state-of-the-art filtering algorithms on a challenging target tracking example, and outperforms all but one optimal filter that knows the exact instances at which outliers occur. The presented material is embedded into a coherent thesis, with a concise intro- duction to the Bayesian filtering and state estimation problems; an extensive sur- vey of available filtering algorithms that includes the Kalman filter, Kalman filters for nonlinear systems, and the particle filter; and an appendix that provides the required probability theory basis. v Populärvetenskaplig sammanfattning När Kalman år 1960 presenterade sin lösning för filtrering och prediktion i dy- namiska system, så startade han en helt ny era inom signalbehandling. Kalman- filtret kom helt rätt i tiden och fick en ovanligt snabb spridning. Dels gick det att implementera på den tidens datorer, dels så löste det akuta problem i Apollo- projektet, vilket blev dess första tillämpning. Även om kalmanfiltret är anpassat för linjära dynamiska system, så kan det även användas då systemet är olinjärt. Detta kräver dock beräkning av vissa statistis- ka storheter, såsom väntevärden och kovariansmatriser. Olika lösningar till dessa delproblem ger upphov till en rad olinjära varianter av kalmanfiltret. I denna avhandling ges en översikt av olika beräkningsmetoder för de nödvändiga sta- tistiska storheterna, samt en jämförelse av dess fördelar och nackdelar. Flera nya insikter, samband och resultat ges. Kalmanfiltret är det optimala linjära filtret, även om brussignalerna i ett linjärt sy- stem inte är normalfördelade. Däremot kan det finnas olinjära filter som är bättre. Om det förekommer stora avvikande värden på brussignalerna (s.k. outliers) så får kalmanfiltret stora avvikelser i sina skattningar som tar tid att bli av med. I denna avhandling undersöks därför students t-fördelningen, som kan modellera outliers. Ett filter som baseras på denna fördelning härleds. Den nya algoritmen är olinjär och övertygar vid ett mer utmanade scenario där vi jämför filtret med existerande algoritmer i litteraturen. vii Acknowledgments First, I wish to acknowledge my supervisor Fredrik Gustafsson. Fredrik keeps track of what his students are working on, encourages new ideas, and gives im- mediate feedback upon request — usually via e-mail around midnight. While being amazingly efficient, he still understands the challenges of being a parent. I would also like to thank Svante Gunnarsson as representative and head of the automatic control group for accepting me as PhD student. Svante does a very good job in leading the control group, yet maintains a down-to-earth and friendly attitude towards all group members. Furthermore, Mikael Norrlöf is acknowl- edged for replying to my e-mail inquiry in early 2010, which eventually led to my employment in Linköping. The group coordinators Åsa Karmelind (former) and Ninna Stensgård (current) have helped with all administrative issues. Furthermore, Åsa arranged a lovely apartment for my family and me. Thank you both! Being a PhD student in the automatic control group means being well advised. I would like to thank Emre Özkan for his positive attitude towards research and confidence in my problem solving capabilities. Gustaf Hendeby has always shown interest in what I have been working on, which is very much appreciated. Additionally, I would like to thank Umut Orguner and Saikat Saha for their ideas and input. Since I started in 2010, I have always had good company in my office. Therefore, I would like to acknowledge my office mates Henrik Ohlsson and Johan Dahlin. While Henrik had a tendency to drag large amounts of sand into the office, Johan keeps his half of the room rather tidy. Sometimes I wonder if he is not a bit more German than I am. The first winter in Sweden was immensely brightened up by welcoming new friends. Tohid Ardeshiri is a very considerate person, a generous host, and has an amazing family. Niklas Wahlström does not only impress with his optimization skills: he can whistle like no one else can, and does not mind switching between Swedish, English, and German. Both never cease to give a helping hand. I think that being a PhD student is sometimes challenging. After all, it is not easy to find a good trade-off between research, teaching, graduate courses, and life beyond university. All the more important is a good atmosphere among fellow PhD students. Expressed in the words of writer Mark Vonnegut, "We are here to help each other get through this thing, whatever it is." Therefore, I would like to acknowledge the PhD students in the group, some of which already graduated. In addition to the already mentioned, these are Christian Andersson Naesseth; Daniel Ankelhed; Patrik Axelsson; Jonas Callmer; André Carvalho Bittencourt, for the hint with the blue ink; Niclas Evestedt; Karl Granström; Ylva Jung; Sina Khoshfetrat Pakazad, for still asking me to join for a beer after my frequent declines; Manon Kok; Roger Larsson; Jonas Linder; ix x Acknowledgments Fredrik Lindsten; Christian Lundquist; Christian Lyzell, for quality TikZ figures that I will include in the next thesis; George Mathai; Isak Nielsen; Hanna Nyqvist; Daniel Petersson; Peter Rosander; Maryam Sadeghi; Daniel Simon; Zoran Sjanic; Per Skoglar; Marek Syldatk, for sharing the enjoyable experience, together with Ylva and Daniel P., of writing a thesis in summer; Martin Skoglund; Lubos Vaci; Klas Veibeck; Johanna Wallén. Gustaf, Fredrik, Emre, Manon, Niklas, Johan, and Tohid are acknowledged for their valuable comments that have significantly improved the manuscript. Niklas and Fredrik helped out with the Swedish summary. Furthermore, I would like to thank Gustaf and Henrik Tidefelt for developing and David Törnqvist for main- taining the thesis template. I am lucky enough to have a number of very good friends in Linköping, Berlin, Braunschweig, and selected other places. Even though we do not get to see each other that often nowadays, I always look forward to the next meeting. My parents Ingrid and Christian Roth, and my brothers Thomas and Andreas have always supported me in several ways. It would not have been easier with- out you, and I always enjoy being back at home. Also the Neupert family is gratefully acknowledged. Finally, I want to thank my amazing wife Kora Neupert. Without your love and constant support, I would not have been able to finish this thesis. We have the privilege of watching Clara grow up, and that is something that I would not want to share with anyone else. Clara is gratefully acknowledged for making me smile everyday, and for the deep conversations that we share over breakfast. Over the last few weeks I could only spend little time with both of you, and I hope I can make up for that soon. Financial support for this thesis has been provided by the MC Impulse project, a European commission FP7 Marie Curie initial training network under grant agreement number 238710. Linköping, Sweden Michael Roth Contents 1 Introduction 1 1.1 Considered Problems . 3 1.2 Contributions . 4 1.3 Publications . 5 2 State Estimation 7 2.1 Stochastic State Space Models .
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