A differential geometric approach to nonlinear filtering: the projection filter

Abstract-The paper focuses on nonlinear state estimation assuming non-Gaussian distributions of the states and the disturbances. The posterior distribution and the aposteriori distribution is described by a chosen family of paramtric distributions. The state transformation then results in a transformation of the paramters in the distribution. This transformation is approximated by a neural network using offline training, which is based on monte carlo sampling. In the paper, there will also be presented a method to construct a flexible distributions well suited for covering the effect of the non-linearities. The method can also be used to improve other parametric methods around regions with strong nonlinearities by including them inside the network.

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“…We used the cross-entrophy as the distance between the two distributions. The aim is to minimize the integral given in (5).…”

Section: Trainingmentioning

confidence: 99%

“…Daum filters [4] are exact solutions to a restricted class of non-linear systems, and they use a member of the exponential family to represent the state distribution. Projection filters [5] on the other hand provide an approximation to exact non-linear filters. They can use a variety of parametric distributions.…”

Section: Introductionmentioning

confidence: 99%

Pre-trained Neural Networks Used for Non-linear State Estimation

Bayramoglu

Andersen

Ravn

et al. 2011

2011 10th International Conference on Machine Learning and Applications and Workshops

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“…Hence, the true posterior density, i.e., the result of the processing step that might not be explicitly available, is approximated by a density tractable in subsequent processing steps. Many types of generic analytic density representations are available for that purpose, including Gaussian mixtures [1], Edgeworth series expansions [2], and exponential densities [3].…”

Section: A Motivationmentioning

confidence: 99%

Dirac mixture trees for fast suboptimal multi-dimensional density approximation

Klumpp

Hanebeck

2008

2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems

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Abstract-We consider the problem of approximating an arbitrary multi-dimensional probability density function by means of a Dirac mixture density. Instead of an optimal solution based on minimizing a global distance measure between the true density and its approximation, a fast suboptimal anytime procedure is proposed, which is based on sequentially partitioning the state space and component placement by local optimization. The proposed procedure adaptively covers the entire state space with a gradually increasing resolution. It can be efficiently implemented by means of a pre-allocated tree structure in a straightforward manner. The resulting computational complexity is linear in the number of components and linear in the number of dimensions. This allows a large number of components to be handled, which is especially useful in high-dimensional state spaces.

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“…Here, the aim is to obtain the conditional probability density function of the model parameters, given the observations up to the current instant. While the Kalman filter provides the optimal solution to a linear inverse problem, nonlinear problems may be tackled either via variants of the optimal particle filter (PF; Doucet et al 2000), geometric filter (Brigo et al 1998) or via several suboptimal strategies, including extended or ensemble Kalman filters (EnKFs; Evensen 1994). A remarkable advantage of such filters, especially the ones that use Monte Carlo simulations through an ensemble of trajectories, is that they do not require explicit regularization.…”

Section: Introductionmentioning

confidence: 99%

A pseudo-dynamical systems approach to a class of inverse problems in engineering

2009

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A pseudo-dynamical approach for a class of inverse problems involving static measurements is proposed and explored. Following linearization of the minimizing functional associated with the underlying optimization problem, the new strategy results in a system of linearized ordinary differential equations (ODEs) whose steady-state solutions yield the desired reconstruction. We consider some explicit and implicit schemes for integrating the ODEs and thus establish a deterministic reconstruction strategy without an explicit use of regularization. A stochastic reconstruction strategy is then developed making use of an ensemble Kalman filter wherein these ODEs serve as the measurement model. Finally, we assess the numerical efficacy of the developed tools against a few linear and nonlinear inverse problems of engineering interest.

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A differential geometric approach to nonlinear filtering: the projection filter

Cited by 77 publications

References 10 publications

Pre-trained Neural Networks Used for Non-linear State Estimation

Pre-trained Neural Networks Used for Non-linear State Estimation

Dirac mixture trees for fast suboptimal multi-dimensional density approximation

A pseudo-dynamical systems approach to a class of inverse problems in engineering

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