### “Subspace System Identification of the Kalman Filter”

**Authors:**David Di Ruscio,

**Affiliation:**Telemark University College

**Reference:**2003, Vol 24, No 3, pp. 125-157.

**Keywords:**Identification methods, Subspace methods Stochastic systems, Sampled data systems, Linear systems

**Abstract:**Some proofs concerning a subspace identification algorithm are presented. It is proved that the Kalman filter gain and the noise innovations process can be identified directly from known input and output data without explicitly solving the Riccati equation. Furthermore, it is in general and for colored inputs, proved that the subspace identification of the states only is possible if the deterministic part of the system is known or identified beforehand. However, if the inputs are white, then, it is proved that the states can be identified directly. Some alternative projection matrices which can be used to compute the extended observability matrix directly from the data are presented. Furthermore, an efficient method for computing the deterministic part of the system is presented. The closed loop subspace identification problem is also addressed and it is shown that this problem is solved and unbiased estimates are obtained by simply including a filter in the feedback. Furthermore, an algorithm for consistent closed loop subspace estimation is presented. This algorithm is using the controller parameters in order to overcome the bias problem.

PDF (3644 Kb) DOI: 10.4173/mic.2003.3.1

**DOI forward links to this article:**

[1] David Di Ruscio (2009), doi:10.4173/mic.2009.2.3 |

[2] David Di Ruscio (2009), doi:10.4173/mic.2009.4.2 |

[3] Bernt Lie, David Di Ruscio, Rolf Ergon, Bjørn Glemmestad, Maths Halstensen, Finn Haugen, Saba Mylvaganam, Nils-Olav Skeie and Dietmar Winkler (2009), doi:10.4173/mic.2009.3.4 |

[4] A. Sæther, C. Arakaki, C. Ratnayake and D. Di Ruscio (2009), doi:10.1080/02726350902991007 |

[5] Geir W. Nilsen and David Di Ruscio (2005), doi:10.4173/mic.2005.3.3 |

[6] Jan-Willem van Wingerden, Marco Lovera, Marco Bergamasco, Michel Verhaegen and Gijs van der Veen (2013), doi:10.1049/iet-cta.2012.0653 |

[7] Martin Grossl (2013), doi:10.1109/SysTol.2013.6693941 |

[8] Morten Bakke, Tor A. Johansen and Sigurd Skogestad (2010), doi:10.3182/20100705-3-BE-2011.00099 |

[9] Geir Werner Nilsen and David Di Ruscio (2004), doi:10.1016/S1474-6670(17)31893-1 |

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**BibTeX:**

@article{MIC-2003-3-1,

title={{Subspace System Identification of the Kalman Filter}},

author={Di Ruscio, David},

journal={Modeling, Identification and Control},

volume={24},

number={3},

pages={125--157},

year={2003},

doi={10.4173/mic.2003.3.1},

publisher={Norwegian Society of Automatic Control}

};