Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances

Wenjuan Qi; Zunbing Sheng

doi:doi:10.11648/j.mcs.20180306.11

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Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances

Wenjuan Qi, Zunbing Sheng

Published in Mathematics and Computer Science (Volume 3, Issue 6)

Received: 7 September 2018 Accepted: 19 September 2018 Published: 4 January 2019

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Abstract

This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.

Published in	Mathematics and Computer Science (Volume 3, Issue 6)
DOI	10.11648/j.mcs.20180306.11
Page(s)	113-128
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Uncertain System, Uncertain Noise Variance, Robust Kalman Filtering, Minimax Estimator, Robust Accuracy, Lyapunov Equation Approach, Convergence

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APA Style

Wenjuan Qi, Zunbing Sheng. (2019). Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Mathematics and Computer Science, 3(6), 113-128. https://doi.org/10.11648/j.mcs.20180306.11

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ACS Style

Wenjuan Qi; Zunbing Sheng. Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Math. Comput. Sci. 2019, 3(6), 113-128. doi: 10.11648/j.mcs.20180306.11

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AMA Style

Wenjuan Qi, Zunbing Sheng. Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Math Comput Sci. 2019;3(6):113-128. doi: 10.11648/j.mcs.20180306.11

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@article{10.11648/j.mcs.20180306.11,
  author = {Wenjuan Qi and Zunbing Sheng},
  title = {Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances},
  journal = {Mathematics and Computer Science},
  volume = {3},
  number = {6},
  pages = {113-128},
  doi = {10.11648/j.mcs.20180306.11},
  url = {https://doi.org/10.11648/j.mcs.20180306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20180306.11},
  abstract = {This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.},
 year = {2019}
}

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TY  - JOUR
T1  - Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances
AU  - Wenjuan Qi
AU  - Zunbing Sheng
Y1  - 2019/01/04
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20180306.11
DO  - 10.11648/j.mcs.20180306.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 113
EP  - 128
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20180306.11
AB  - This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.
VL  - 3
IS  - 6
ER  -

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Author Information

Wenjuan Qi

School of Mechnical and Electronical Engineering, Heilongjiang University, Harbin, China
Zunbing Sheng

School of Mechnical and Electronical Engineering, Heilongjiang University, Harbin, China

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APA Style

Wenjuan Qi, Zunbing Sheng. (2019). Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Mathematics and Computer Science, 3(6), 113-128. https://doi.org/10.11648/j.mcs.20180306.11

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ACS Style

Wenjuan Qi; Zunbing Sheng. Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Math. Comput. Sci. 2019, 3(6), 113-128. doi: 10.11648/j.mcs.20180306.11

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AMA Style

Wenjuan Qi, Zunbing Sheng. Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances. Math Comput Sci. 2019;3(6):113-128. doi: 10.11648/j.mcs.20180306.11

Copy | Download

@article{10.11648/j.mcs.20180306.11,
  author = {Wenjuan Qi and Zunbing Sheng},
  title = {Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances},
  journal = {Mathematics and Computer Science},
  volume = {3},
  number = {6},
  pages = {113-128},
  doi = {10.11648/j.mcs.20180306.11},
  url = {https://doi.org/10.11648/j.mcs.20180306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20180306.11},
  abstract = {This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances
AU  - Wenjuan Qi
AU  - Zunbing Sheng
Y1  - 2019/01/04
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20180306.11
DO  - 10.11648/j.mcs.20180306.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 113
EP  - 128
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20180306.11
AB  - This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.
VL  - 3
IS  - 6
ER  -

Copy | Download