Optimization Viewpoint on Kalman Smoothing with Applications to Robust and Sparse Estimation
In this chapter, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem,
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Optimization Viewpoint on Kalman Smoothing with Applications to Robust and Sparse Estimation Aleksandr Y. Aravkin, James V. Burke and Gianluigi Pillonetto
Abstract In this chapter, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.
8.1 Introduction Kalman filtering and smoothing methods form a broad category of computational algorithms used for inference on noisy dynamical systems. Over the last 50 years, these algorithms have become a gold standard in a range of applications, including A. Y. Aravkin (B) Numerical Analysis and Optimization, IBM T.J. Watson Research Center, Vancouver, BC, Canada e-mail: [email protected] J. V. Burke Department of Mathematics, University of Washington, Seattle, WA, USA e-mail: [email protected] G. Pillonetto Control and Dynamic Systems Department of Information Engineering, University of Padova, Padova, Italy e-mail: [email protected] A. Y. Carmi et al. (eds.), Compressed Sensing & Sparse Filtering, Signals and Communication Technology, DOI: 10.1007/978-3-642-38398-4_8, © Springer-Verlag Berlin Heidelberg 2014
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238 Fig. 8.1 Dynamic systems amenable to Kalman smoothing methods
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space exploration, missile guidance systems, general tracking and navigation, and weather prediction. In 2009, Rudolf Kalman received the National Medal of Science from President Obama for the invention of the Kalman filter. Numerous books and papers have been written on these methods and their extensions, addressing modifications for use in nonlinear systems, smoothing data over time intervals, improving algorithm robustness to bad measurements, and many other topics. The classic Kalman filter [29] is almost always presented as a set of recursive equations, and the classic Rauch-Tung-Striebel (RTS) fixed-interval smoother [42] is typically formulated as two coupled Kalman filters. An elegant derivation based on projections onto spaces spanned by random variables can be found in [2]. In this chapter, we use the terms ‘Kalman filter’ and ‘Kalman smoother’ much more broadly, including any method of inference on any dynamical system fitting
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