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On computational methods for nonlinear estimation
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
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 [en]
Nonlinear estimation, Particle filter, Kalman filter, System identification, Convex optimization, Differential-algebraic equation
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-24325Local ID: 3951ISBN: 91-7373-759-3 (print)OAI: oai:DiVA.org:liu-24325DiVA: diva2:244643
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-27
List of papers
1. A Modeling and Filtering Framework for Linear Differential-Algebraic Equations
Open this publication in new window or tab >>A Modeling and Filtering Framework for Linear Differential-Algebraic Equations
2003 (English)In: Proceedings of the 42th IEEE Conference on Decision and Control, 2003, 892-897 vol.1 p.Conference paper, Published paper (Refereed)
Abstract [en]

General approaches to modeling, for instance using object-oriented software, lead to differential-algebraic equations (DAE). As the name reveals, it is a combination of differential and algebraic equations. For state estimation using observed system inputs and outputs in a stochastic framework similar to Kalman filtering, we need to augment the DAE with stochastic disturbances ("process noise"), whose covariance matrix becomes the tuning parameter. We will determine the subspace of possible causal disturbances based on the linear DAE model. This subspace determines all degrees of freedom in the filter design, and a Kalman filter algorithm is given. We illustrate the design on a system with two interconnected rotating masses.

Keyword
Implicit systems, Descriptor systems, Singular systems, White noise, Noise, Discretization, Kalman filters
National Category
Engineering and Technology Control Engineering
Identifiers
urn:nbn:se:liu:diva-13917 (URN)10.1109/CDC.2003.1272679 (DOI)000189434100154 ()0-7803-7924-1 (ISBN)
Conference
42nd IEEE Conference on Decision and Control, Maui, HI, USA, December, 2003
Available from: 2006-09-04 Created: 2006-09-04 Last updated: 2013-11-27
2. A Note on State Estimation as a Convex Optimization Problem
Open this publication in new window or tab >>A Note on State Estimation as a Convex Optimization Problem
2003 (English)In: Proceedings of the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003, Vol. 6, no 6-10, 61-64 vol.6 p.Conference paper, Published paper (Refereed)
Abstract [en]

The Kalman filter computes the maximum a posteriori (MAP) estimate of the states for linear state space models with Gaussian noise. We interpret the Kalman filter as the solution to a convex optimization problem, and show that we can generalize the MAP state estimator to any noise with a log-concave density function and any combination of linear equality and convex inequality constraints on the states. We illustrate the principle on a hidden Markov model, where the state vector contains probabilities that are positive and sum to one.

Keyword
State estimation, Kalman filter, Convex optimization, Hidden Markov Models
National Category
Engineering and Technology Control Engineering
Identifiers
urn:nbn:se:liu:diva-13918 (URN)10.1109/ICASSP.2003.1201618 (DOI)0-7803-7663-3 (ISBN)
Conference
2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, Hong Kong, China, April, 2003
Available from: 2006-09-04 Created: 2006-09-04 Last updated: 2013-11-27
3. Marginalized Particle Filters for Mixed Linear/Nonlinear State-Space Models
Open this publication in new window or tab >>Marginalized Particle Filters for Mixed Linear/Nonlinear State-Space Models
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
Keyword
Kalman filter, Marginalization, Navigation systems, Nonlinear systems, Particle filter, State estimation
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-11749 (URN)10.1109/TSP.2005.849151 (DOI)
Available from: 2008-05-07 Created: 2008-05-07 Last updated: 2017-12-13Bibliographically approved
4. Particle Filters for System Identification of State-Space Models Linear in Either Parameters or States
Open this publication in new window or tab >>Particle Filters for System Identification of State-Space Models Linear in Either Parameters or States
2003 (English)In: Proceedings of the 13th IFAC Symposium on System Identification, 2003, 1251-1256 vol.1 p.Conference paper, Published paper (Refereed)
Abstract [en]

The potential use of the marginalized particle filter for nonlinear system identification is investigated. The particle filter itself offers a general tool for estimating unknown parameters in non-linear models of moderate complexity, and the basic trick is to model the parameters as a random walk (so called roughening noise) with decaying variance. We derive algorithms for systems which are non-linear in either the parameters or the states, but not both generally. In these cases, marginalization applies to the linear part, which firstly significantly widens the scope of the particle filter to more complex systems, and secondly decreases the variance in the linear parameters/states for fixed filter complexity. This second property is illustrated on an example of chaotic model. The particular case of freely parametrized linear state space models, common in subspace identification approaches, is bi-linear in states and parameters, and thus both cases above are satisfied. One can then choose which one to marginalize.

Keyword
System identification, Nonlinear estimation, Recursive estimation, Particle filters, Kalman filters, Bayesian estimation
National Category
Engineering and Technology Control Engineering
Identifiers
urn:nbn:se:liu:diva-13919 (URN)978-0080437095 (ISBN)
Conference
13th IFAC Symposium on System Identification, Rotterdam, The Netherlands, August, 2003
Available from: 2006-09-04 Created: 2006-09-04 Last updated: 2013-11-27

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