Stability Analysis of Multi-Sensor Kalman Filtering over Lossy Networks
"> Figure 1
<p>Distributed systems.</p> "> Figure 2
<p>Diagram of a networked filtering system under distributed sensing.</p> "> Figure 3
<p>Stable and unstable regions.</p> "> Figure 4
<p>The error covariance matrix <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>k</mi> </msub> </semantics> </math> and channel state <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>k</mi> </msub> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>.</mo> <mn>8</mn> <mo>,</mo> <mspace width="4pt"/> <mn>0</mn> <mo>.</mo> <mn>9</mn> </mfenced> </mrow> </semantics> </math>: (<b>a</b>) The error covariance; (<b>b</b>) the associated channel state.</p> "> Figure 5
<p>The error covariance matrix <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>k</mi> </msub> </semantics> </math> and channel state <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>k</mi> </msub> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>.</mo> <mn>2</mn> <mo>,</mo> <mspace width="4pt"/> <mn>0</mn> <mo>.</mo> <mn>2</mn> </mfenced> </mrow> </semantics> </math>: (<b>a</b>) The error covariance; (<b>b</b>) the associated channel state.</p> "> Figure 6
<p>The error covariance matrix <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>k</mi> </msub> </semantics> </math> and channel state <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>k</mi> </msub> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>.</mo> <mn>6</mn> <mo>,</mo> <mspace width="4pt"/> <mn>0</mn> <mo>.</mo> <mn>8</mn> </mfenced> </mrow> </semantics> </math>: (<b>a</b>) The error covariance; (<b>b</b>) the associated channel state.</p> "> Figure 6 Cont.
<p>The error covariance matrix <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>k</mi> </msub> </semantics> </math> and channel state <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>k</mi> </msub> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>q</mi> <mn>2</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>.</mo> <mn>6</mn> <mo>,</mo> <mspace width="4pt"/> <mn>0</mn> <mo>.</mo> <mn>8</mn> </mfenced> </mrow> </semantics> </math>: (<b>a</b>) The error covariance; (<b>b</b>) the associated channel state.</p> ">
Abstract
:1. Introduction
2. Problem Formulation
3. Stability Conditions for Error Covariance
3.1. Necessary Condition for Covariance Stability
3.2. Stability for Non-Degenerate Systems
4. Discussion on Different Stability Criteria
4.1. Comparison with Stability Results under i.i.d. Packet Losses
4.2. Comparison with Existing Stability Results under Markovian Packet Losses
5. Numerical Examples
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Aubert, B.; Regnier, J.; Caux, S.; Alejo, D. Kalman-filter-based indicator for online interturn short circuits detection in permanent-magnet synchronous generators. IEEE Trans. Ind. Electron. 2015, 62, 1921–1930. [Google Scholar] [CrossRef]
- Cseko, L.H.; Kvasnica, M.; Lantos, B. Explicit MPC-based RBF neural network controller design with discrete-time actual Kalman filter for semiactive suspension. IEEE Trans. Control Syst. Technol. 2015, 23, 1736–1753. [Google Scholar] [CrossRef]
- Chen, P.P.; Ma, H.L.; Gao, S.W.; Huang, Y. Modified extended Kalman filtering for tracking with insufficient and intermittent observations. Math. Probl. Eng. 2015, 2015. [Google Scholar] [CrossRef]
- Nahi, N.E. Optimal recursive estimation with uncertain observation. IEEE Trans. Inf. Theory 1969, 15, 457–462. [Google Scholar] [CrossRef]
- Hadidi, M.T.; Schwartz, S.C. Linear recursive state estimators under uncertain observations. IEEE Trans. Autom. Control 1979, 24, 944–948. [Google Scholar] [CrossRef]
- Tugnait, J.K. Asymptotic stability of the MMSE linear filter for systems with uncertain observations. IEEE Trans. Autom. Control 1981, 27, 247–250. [Google Scholar] [CrossRef]
- Rezaei, H.; Esfanjani, R.M.; Sedaaghi, M.H. Improved robust finite-horizon Kalman filtering for uncertain networked time-varying systems. Inf. Sci. 2015, 293, 263–274. [Google Scholar] [CrossRef]
- Rezaei, H.; Esfanjani, R.M.; Farsi, M. Robust filtering for uncertain networked systems with randomly delayed and lost measurements. IET Signal Process. 2015, 9, 320–327. [Google Scholar] [CrossRef]
- Mo, Y.; Sinopoli, B. Towards Finding the Critical Value for Kalman Filtering with Intermittent Observations. Available online: http://arxiv.org/abs/1005.2442 (accessed on 19 February 2016).
- Zhang, H.S.; Song, X.M.; Shi, L. Convergence and mean square stability of suboptimal estimator for systems with measurement packet dropping. IEEE Trans. Autom. Control 2012, 57, 1248–1253. [Google Scholar] [CrossRef]
- Sui, T.; You, K.; Fu, M.; Marelli, D. Stability of MMSE state estimators over lossy networks using linear coding. Automatica 2015, 51, 167–174. [Google Scholar] [CrossRef]
- You, K.; Fu, M.; Xie, L. Mean square stability for Kalman filtering with Markovian packet losses. Automatica 2011, 47, 2647–2657. [Google Scholar] [CrossRef]
- You, K.; Xie, L. Minimum data rate for mean square stabilizability of linear systems with Markovian packet losses. IEEE Trans. Autom. Control 2011, 56, 772–785. [Google Scholar] [CrossRef]
- Mo, Y.; Sinopoli, B. Kalman filtering with intermittent observations: Tail distribution and critical value. IEEE Trans. Autom. Control 2012, 57, 677–689. [Google Scholar]
- Rohr, E.R.; Marelli, D.; Fu, M. Kalman filtering with intermittent observations: On the boundedness of the expected error covariance. IEEE Trans. Autom. Control 2014, 59, 2724–2738. [Google Scholar] [CrossRef]
- Wu, J.; Shi, L.; Xie, L.; Johansson, K.H. An improved stability condition for Kalman filtering with bounded Markovian packet losses. Automatica 2015, 62, 32–38. [Google Scholar] [CrossRef]
- Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.I.; Sastry, S.S. Kalman filtering with intermittent observations. IEEE Trans. Autom. Control 2004, 49, 1453–1464. [Google Scholar] [CrossRef]
- Huang, M.; Dey, S. Stability of Kalman filtering with Markovian packet losses. Automatica 2007, 43, 598–607. [Google Scholar] [CrossRef]
- Liu, X.; Goldsmith, A. Kalman filtering with partial observation losses. In Proceedings of the IEEE Conference on Decision and Control, Nassau, Bahamas, 14–17 December 2004; pp. 4180–4186.
- Wang, B.F.; Guo, G. Kalman filtering with partial Markovian packet losses. Int. J. Autom. Comput. 2009, 6, 395–400. [Google Scholar] [CrossRef]
- Sui, T.; You, K.; Fu, M. Stability conditions for multi-sensor state estimation over a lossy network. Automatica 2015, 53, 1–9. [Google Scholar] [CrossRef]
- Plarre, K.; Bullo, F. On Kalman filtering for detectable systems with intermittent observations. IEEE Trans. Autom. Control 2009, 54, 1–9. [Google Scholar] [CrossRef]
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Gao, S.; Chen, P.; Huang, D.; Niu, Q. Stability Analysis of Multi-Sensor Kalman Filtering over Lossy Networks. Sensors 2016, 16, 566. https://doi.org/10.3390/s16040566
Gao S, Chen P, Huang D, Niu Q. Stability Analysis of Multi-Sensor Kalman Filtering over Lossy Networks. Sensors. 2016; 16(4):566. https://doi.org/10.3390/s16040566
Chicago/Turabian StyleGao, Shouwan, Pengpeng Chen, Dan Huang, and Qiang Niu. 2016. "Stability Analysis of Multi-Sensor Kalman Filtering over Lossy Networks" Sensors 16, no. 4: 566. https://doi.org/10.3390/s16040566