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Marginalized Particle Filters for Mixed Linear/Nonlinear State-Space Models
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
2005 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 53, no 7, 2279-2289 p.Article in journal (Refereed) Published
Abstract [en]

The particle filter offers a general numerical tool to approximate the posterior density function for the state in nonlinear and non-Gaussian filtering problems. While the particle filter is fairly easy to implement and tune, its main drawback is that it is quite computer intensive, with the computational complexity increasing quickly with the state dimension. One remedy to this problem is to marginalize out the states appearing linearly in the dynamics. The result is that one Kalman filter is associated with each particle. The main contribution in this paper is the derivation of the details for the marginalized particle filter for a general nonlinear state-space model. Several important special cases occurring in typical signal processing applications will also be discussed. The marginalized particle filter is applied to an integrated navigation system for aircraft. It is demonstrated that the complete high-dimensional system can be based on a particle filter using marginalization for all but three states. Excellent performance on real flight data is reported.

Place, publisher, year, edition, pages
IEEE Signal Processing Society, 2005. Vol. 53, no 7, 2279-2289 p.
Keyword [en]
Kalman filter, Marginalization, Navigation systems, Nonlinear systems, Particle filter, State estimation
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:liu:diva-11749DOI: 10.1109/TSP.2005.849151OAI: oai:DiVA.org:liu-11749DiVA: diva2:18177
Available from: 2008-05-07 Created: 2008-05-07 Last updated: 2013-11-27Bibliographically approved
In thesis
1. Efficient Estimation and Detection Methods for Airborne Applications
Open this publication in new window or tab >>Efficient Estimation and Detection Methods for Airborne Applications
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The overall purpose with this thesis is to investigate and provide computationally efficient methods for estimation and detection. The focus is on airborne applications, and we seek estimation and detection methods which are accurate and reliable yet effective with respect to computational load. In particular, the methods shall be optimized for terrain-aided navigation andcollision avoidance respectively. The estimation part focuses on particle filtering and the in general much more efficient marginalized particle filter. The detection part focuses on finding efficient methods for evaluating the probability of extreme values. This is achieved by considering the, in general, much easier task to compute the probability of level-crossings.

The concept of aircraft navigation using terrain height information is attractive because of the independence of external information sources. Typicallyterrain-aided navigation consists of an inertial navigation unit supported by position estimates from a terrain-aided positioning (TAP) system. TAP integrated with an inertial navigation system is challenging due to its highly nonlinear nature. Today, the particle filter is an accepted method for estimation of more or less nonlinear systems. At least when the requirements on computational load are not rigorous. In many on-line processing applications the requirements are such that they prevent the use of theparticle filter. We need more efficient estimation methods to overcome this issue, and the marginalized particle filter constitutes a possible solution. The basic principle for the marginalized particle filter is to utilize linear and discrete substructures within the overall nonlinear system. These substructures are used for efficient estimation by applying optimal filters such as the Kalman filter. The computationally demanding particle filter can then be concentrated on a smaller part of the estimation problem.

The concept of an aircraft collision avoidance system is to assist or ultimately replace the pilot in order to to minimize the resulting collision risk. Detection is needed in aircraft collision avoidance because of the stochastic nature of thesensor readings, here we use information from video cameras. Conflict is declared if the minimum distance between two aircraft is less than a level. The level is given by the radius of a safety sphere surrounding the aircraft.We use the fact that the probability of conflict, for the process studied here, is identical to the probability for a down-crossing of the surface of the sphere. In general, it is easier to compute the probability of down-crossings compared to extremes. The Monte Carlo method provides a way forward to compute the probability of conflict. However, to provide a computationally tractable solution we approximate the crossing of the safety sphere with the crossing of a circular disc. The approximate method yields a result which is as accurate as the Monte Carlo method but the computational load is decreased significantly.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2009. 66 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1231
National Category
Signal Processing
Identifiers
urn:nbn:se:liu:diva-15826 (URN)978-91-7393-720-7 (ISBN)
Public defence
2009-01-30, Visionen, Hus B, Campus Valla, Linköping universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2009-01-13 Created: 2008-12-08 Last updated: 2009-03-02Bibliographically approved
2. Estimation of Nonlinear Dynamic Systems: Theory and Applications
Open this publication in new window or tab >>Estimation of Nonlinear Dynamic Systems: Theory and Applications
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with estimation of states and parameters in nonlinear and non-Gaussian dynamic systems. Sequential Monte Carlo methods are mainly used to this end. These methods rely on models of the underlying system, motivating some developments of the model concept. One of the main reasons for the interest in nonlinear estimation is that problems of this kind arise naturally in many important applications. Several applications of nonlinear estimation are studied.

The models most commonly used for estimation are based on stochastic difference equations, referred to as state-space models. This thesis is mainly concerned with models of this kind. However, there will be a brief digression from this, in the treatment of the mathematically more intricate differential-algebraic equations. Here, the purpose is to write these equations in a form suitable for statistical signal processing.

The nonlinear state estimation problem is addressed using sequential Monte Carlo methods, commonly referred to as particle methods. When there is a linear sub-structure inherent in the underlying model, this can be exploited by the powerful combination of the particle filter and the Kalman filter, presented by the marginalized particle filter. This algorithm is also known as the Rao-Blackwellized particle filter and it is thoroughly derived and explained in conjunction with a rather general class of mixed linear/nonlinear state-space models. Models of this type are often used in studying positioning and target tracking applications. This is illustrated using several examples from the automotive and the aircraft industry. Furthermore, the computational complexity of the marginalized particle filter is analyzed.

The parameter estimation problem is addressed for a relatively general class of mixed linear/nonlinear state-space models. The expectation maximization algorithm is used to calculate parameter estimates from batch data. In devising this algorithm, the need to solve a nonlinear smoothing problem arises, which is handled using a particle smoother. The use of the marginalized particle filter for recursive parameterestimation is also investigated.

The applications considered are the camera positioning problem arising from augmented reality and sensor fusion problems originating from automotive active safety systems. The use of vision measurements in the estimation problem is central to both applications. In augmented reality, the estimates of the camera’s position and orientation are imperative in the process of overlaying computer generated objects onto the live video stream. The objective in the sensor fusion problems arising in automotive safety systems is to provide information about the host vehicle and its surroundings, such as the position of other vehicles and the road geometry. Information of this kind is crucial for many systems, such as adaptive cruise control, collision avoidance and lane guidance.

Place, publisher, year, edition, pages
Institutionen för systemteknik, 2006
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 998
Series
Keyword
Nonlinear estimation, system identification, Kalman filter, particle filter, marginalized particle filter, expectation maximization, automotive applications
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-7124 (URN)91-85497-03-7 (ISBN)
Public defence
2006-02-02, Visionen, Hus B, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2006-09-04 Created: 2006-09-04 Last updated: 2009-06-04
3. On computational methods for nonlinear estimation
Open this publication in new window or tab >>On computational methods for nonlinear estimation
2003 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The Bayesian approach provides a rather powerful framework for handling nonlinear, as well as linear, estimation problems. We can in fact pose a general solution to the nonlinear estimation problem. However, in the general case there does not exist any closed-form solution and we are forced to use approximate techniques. In this thesis we will study one such technique, the sequential Monte Carlo method, commonly referred to as the particle filter. Some work on linear stochastic differential-algebraic equations and constrained estimation using convex optimization will also be presented.

The sequential Monte Carlo method offers a systematic framework for handling estimation of nonlinear systems subject to non-Gaussian noise. Its main drawback is that it requires a lot of computational power. We will use the particle filter both for the nonlinear state estimation problem and the nonlinear system identification problem. The details for the marginalized (Rao-Blackwellized) particle filter applied to a general nonlinear state-space model will also be given.

General approaches to modeling, for instance using object-oriented software, lead to differential-algebraic equations. One of the topics in this thesis is to extend the standard Kalman filtering theory to the class of linear differential-algebraic equations, by showing how to incorporate white noise in this type of equations.

There will also be a discussion on how to use convex optimization for solving the estimation problem. For linear state-space models with Gaussian noise the Kalman filter computes the maximum a posteriori estimate. We interpret the Kalman filter as the solution to a convex optimization problem, and show that we can generalize the maximum a posteriori state estimator to any noise with log-concave probability density function and any combination of linear equality and convex inequality constraints.

Place, publisher, year, edition, pages
Linköping, Sweden: Linköping University, 2003. 62 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1047
Keyword
Nonlinear estimation, Particle filter, Kalman filter, System identification, Convex optimization, Differential-algebraic equation
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-24325 (URN)3951 (Local ID)91-7373-759-3 (ISBN)3951 (Archive number)3951 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-27

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Schön, ThomasGustafsson, FredrikNordlund, Per-Johan

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