# Difference between revisions of "NCS: Kalman Filtering"

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<!-- Enter a 1 paragraph description of the contents of the lecture. Make sure to include any key concepts, so that the wiki search feature will pick them up --> | <!-- Enter a 1 paragraph description of the contents of the lecture. Make sure to include any key concepts, so that the wiki search feature will pick them up --> | ||

In this lecture, we study the Kalman filter for discrete-time linear systems. In particular, we see under what assumptions and in what senses the Kalman filter is an optimal estimator. To prove the results we use some results about conditional expectations and Gaussian probabiliy distributions. We show that the filter contains one prediction step and one correcter step that takes the most recent measurement into account. | In this lecture, we study the Kalman filter for discrete-time linear systems. In particular, we see under what assumptions and in what senses the Kalman filter is an optimal estimator. To prove the results we use some results about conditional expectations and Gaussian probabiliy distributions. We show that the filter contains one prediction step and one correcter step that takes the most recent measurement into account. How the filter deals with sensor fusion is discussed and an example is used to illustrate the results. | ||

== Lecture Materials == | == Lecture Materials == | ||

<!-- Include links to materials that you used in your lecture. At a minimum, this should include a link to your lecture presentation. You might also include links to MATLAB scripts or other source code that students would find useful --> | <!-- Include links to materials that you used in your lecture. At a minimum, this should include a link to your lecture presentation. You might also include links to MATLAB scripts or other source code that students would find useful --> | ||

<!-- Sample lecture link: * [[Media:L1-1_Intro.pdf|Lecture: Networked Control Systems: Course Overview]] --> | <!-- Sample lecture link: * [[Media:L1-1_Intro.pdf|Lecture: Networked Control Systems: Course Overview]] --> | ||

* [[Media:L4-1_Kalman.pdf|Lecture: Kalman Filtering]] | |||

* [[Media:Stateestim.pdf|Lecture notes: State estimation]] | |||

== Reading == | == Reading == | ||

* <p>[http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf An Introduction to the Kalman Filter], G. Welch and G. Bishop</p> | * <p>[http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf An Introduction to the Kalman Filter], G. Welch and G. Bishop. A brief introduction to the Kalman filter in discrete time. No proofs are given, but it is a good first read.</p> | ||

* <p>[http://en.wikipedia.org/wiki/Kalman_filter Wikipedia: Kalman Filter]</p> | * <p>[http://en.wikipedia.org/wiki/Kalman_filter Wikipedia: Kalman Filter] A webpage that gives a proof and some applications.</p> | ||

* <p>[http://www.cs.unc.edu/~welch/kalman/kalmanPaper.html A New Approach to Linear Filtering and Prediction Problem], R.E. Kalman. ''Transactions of the ASME'', Series D, 1960. </p> | * <p>[http://www.cs.unc.edu/~welch/kalman/kalmanPaper.html A New Approach to Linear Filtering and Prediction Problem], R.E. Kalman. ''Transactions of the ASME'', Series D, 1960. A classical paper. Still very readable. It uses different notation than the lecture, and present a different and more general proof. </p> | ||

<!-- A reading list for the lecture. This will typically be 3-5 articles or book chapters that are particularly relevant to the material being presented. The reading list should be annotated to explain how the articles fit into the topic for the lecture. --> | <!-- A reading list for the lecture. This will typically be 3-5 articles or book chapters that are particularly relevant to the material being presented. The reading list should be annotated to explain how the articles fit into the topic for the lecture. --> | ||

== Additional Resources == | == Additional Resources == | ||

* <p>[http://www.cs.unc.edu/~welch/kalman/ The Kalman Filter], G. Welch and G. Bishop. A webpage with many links on Kalman filter.</p> | |||

* <p>[http://www.amazon.com/gp/product/0486439380/102-3301256-1504117?v=glance&n=283155 Optimal Filtering], B.D.O Anderson and J.B. Moore. Dover Books on Engineering, 2005. A reissue of a book from 1979. It contains a detailed mathematical presentation of filtering problems and the Kalman filter. A very good book.</p> | |||

<!-- Links to additional information. If there are good sources of additional information for students interested in exploring this topic further, these should go at the bottom of the page. --> | <!-- Links to additional information. If there are good sources of additional information for students interested in exploring this topic further, these should go at the bottom of the page. --> |

## Latest revision as of 04:51, 1 May 2006

Prev: Alice Planner | Course Home | Next: MHE |

In this lecture, we study the Kalman filter for discrete-time linear systems. In particular, we see under what assumptions and in what senses the Kalman filter is an optimal estimator. To prove the results we use some results about conditional expectations and Gaussian probabiliy distributions. We show that the filter contains one prediction step and one correcter step that takes the most recent measurement into account. How the filter deals with sensor fusion is discussed and an example is used to illustrate the results.

## Lecture Materials

## Reading

An Introduction to the Kalman Filter, G. Welch and G. Bishop. A brief introduction to the Kalman filter in discrete time. No proofs are given, but it is a good first read.

Wikipedia: Kalman Filter A webpage that gives a proof and some applications.

A New Approach to Linear Filtering and Prediction Problem, R.E. Kalman.

*Transactions of the ASME*, Series D, 1960. A classical paper. Still very readable. It uses different notation than the lecture, and present a different and more general proof.

## Additional Resources

The Kalman Filter, G. Welch and G. Bishop. A webpage with many links on Kalman filter.

Optimal Filtering, B.D.O Anderson and J.B. Moore. Dover Books on Engineering, 2005. A reissue of a book from 1979. It contains a detailed mathematical presentation of filtering problems and the Kalman filter. A very good book.