SCIENCE CHINA Information Sciences, Volume 61, Issue 9: 092208(2018) https://doi.org/10.1007/s11432-017-9256-3

Distributed filtering for time-varying networked systems with sensor gain degradation and energy constraint: a centralized finite-time communication protocol scheme

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  • ReceivedJul 29, 2017
  • AcceptedSep 22, 2017
  • PublishedAug 10, 2018


This paper focuses on the distributed filtering problem for a class of time-varying networked systems with sensor gain degradation and energy constrained communication protocol. To satisfy the requirement of power consumption and reduce the schedule computing complexity, centralized cyclic finite-time communication strategy optimization is taken into account. The networked system considered in this paper consists of spatially distributed sensors linked to their neighbor sensors, where each sensor node suffers from different gain degradation, and the transmission decisions of all the communication channels obey the centralized transmission schedule strategy identically. First, we present scattered communication action based on single-sensor transmission modeling with an energy constraint. Subsequently, an optimal communication protocol considering expected average error covariance is derived between the target system and each sensor node over the distributed sensor systems, based on a centralized finite-time scheme. Finally, by transforming the overall estimation error covariance of the systems at each sampling time into quadratic form, a conditionally unbiased least-square recursive distributed filtering technique over the networked system is designed at each sensor node. The system stability condition under such an optimal schedule is also accomplished with bounded covariance. A numerical example is provided to demonstrate the utility and effectiveness of the distributed filtering technique using the proposed optimized energy constrained communication protocol.


This work was supported by National Natural Science Foundation of China (Grant Nos. 61490701, 61210012, 61290324, 61473164), and Research Fund for the Taishan Scholar Project of Shandong Province of China (Grant No. LZB2015-162).



Proof of Theorem 1

Owing to limited space, the proof line is given below. We consider the permutation and combination theory and use the interpolation method. From Lemmas 1 and 2 in the main paragraph, we need to separate every 0 between two 1s in any configuration available. Because it satisfies the completely scattered condition, we are able to separate every 0 between two 1s. Intuitively, there are several communication schedules as long as two or more 0s are not adjacent.

Proof of Theorem 2

Owing to limited space, the proof line is given below and the process is similar with the proof of Theorem 1. We consider the permutation and combination theory and apply the interpolation method. From Lemmas 1 and 2 in the main paragraph, we want to separate 0s between two 1s on average. Here because it does not satisfy the completely scattered condition, there must be at least two adjacent 0s. We can obtain $(n+1-{\rm~Rm})$ segments where there are ${\rm~Qt}$ 0s in each while we can obtain ${\rm~Rm}$ segments where there are $({\rm~Qt}+1)$ 0s in each. Thus there are $\binom{n+1}{{\rm~Rm}}$ combinations, which is the number of optimal communication schedules. And the corresponding cost function is available when substituted into the formula.

Proof of Theorem 3

We have the form

\begin{align}\tilde{x}(k+1)=& \left[\bar{A}(k)-\sum_{i=1}^nE_iH(k)T_i(k)\bar{M}(k)\bar{C}(k)\right]\tilde{x}(k) \\ &-\sum_{i=1}^nE_iH(k)T_i(k)\left[\bar{\Lambda}(k)-\bar{M}(k)\right]\bar{C}(k)\bar{x}(k)-\sum_{i=1}^nE_iH(k)T_i(k)\bar{v}(k) +\bar{w}(k). \tag{31} \end{align}

We also have the definition

\begin{equation}P(k+1)={\rm E}\left\{\tilde{x}(k+1)\tilde{x}^{\rm T}(k+1)\right\}. \tag{32}\end{equation}

Then it follows that

\begin{align}P(k+1) =&\left[\bar{A}(k)-\sum_{i+1}^nE_i H(k)T_i(k)\bar{M}(k)\bar{C}(k)\right]P(k)\left[\bar{A}(k)-\sum_{i+1}^nE_i H(k)T_i(k)\bar{M}(k)\bar{C}(k)\right]^{\rm T} \\ &+\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]V(k)\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]^{\rm T} \\ &+\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]U(k)\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]^{\rm T} +W(k). \tag{33} \end{align}

Additionally, $\Omega(k)$ can be calculated as follows:

\begin{align}\Omega(k+1)&={\rm E}\{x(k+1)x^{\rm T}(k+1)\} \\ &={\rm E}\left\{\left\{A(k)x(k)+w(k)\right\}\left\{A(k)x(k)+w(k)\right\}^{\rm T}\right\} \\ &=A(k)\Omega(k)A^{\rm T}(k)+S(k). \tag{34} \end{align}

According to the notation, it follows that

\begin{align}P(k+1)=&\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]Y(k)\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]^{\rm T} \\ &-Z^{\rm T}(k)\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]^{\rm T} \\ &-\left[\sum_{i=1}^nE_i H(k)T_i(k)\right]Z(k)+\bar{A}(k)P(k)\bar{A}^{\rm T}(k)+W(k). \tag{35} \end{align}

Because $Y(k)=Y^{\rm~T}(k)>0$, the previous equation can be written as

\begin{align}P(k+1) =&\left[\sum_{i=1}^nE_i H(k)T_i(k)-Z^{\rm T}(k)Y^{-1}(k)\right]Y(k)\left[\sum_{i=1}^nE_i H(k)T_i(k)-Z^{\rm T}(k)Y^{-1}(k)\right]^{\rm T} \\ &-Z^{\rm T}(k)Y^{-1}(k)Z(k)+\bar{A}(k)P(k)\bar{A}^{\rm T}(k)+W(k). \tag{36} \end{align}

In the case when the networked sensor systems can not communicate completely, we determine that $P(k+1)$ is minimized when

\begin{equation}\sum_{i=1}^nE_iH(k)T_i(k)=\mathcal{H}(k). \tag{37}\end{equation}

It is known that $T_i(k)$ is not invertible for $i\in\mathcal{V}$ for the reason of sparse topology and matrix irreversibility. Under this circumstance, a means to determine the filter gains is to calculate $H(k)$ as above. We denote $T_i^\dag(k)$ as the Moore-Penrose pseudo inverse of $T_i(k)$. Thus, it can be assured that

\begin{equation}[H_{i1}(k),\ldots,H_{in}(k)]=[\mathcal{H}_{i1}(k),\ldots,\mathcal{H}_{in}(k)]T_i^\dag(k). \tag{38}\end{equation}

Moreover, it can be easily proven that $P(k)$ can be calculated recursively. The proof is thus complete.


[1] Liu Z, Yu Q Y. Experiments on 3D indoor localization of smart phones. J Shandong Univ Sci Tech (Nat Sci Ed), 2015, 34: 90--95. Google Scholar

[2] Shi B, Lu X S, Wang D, et al. High-speed acquisition of data with sensors in vehicle-borne close-range 3D survey system. J Shandong Univ Sci Tech (Nat Sci Ed), 2007, 26: 24--26. Google Scholar

[3] Kang Y, Ding Y, Li Z. A networked remote sensing system for on-road vehicle emission monitoring. Sci China Inf Sci, 2017, 60: 043201 CrossRef Google Scholar

[4] Luo Z Q. Universal Decentralized Estimation in a Bandwidth Constrained Sensor Network. IEEE Trans Inform Theor, 2005, 51: 2210-2219 CrossRef Google Scholar

[5] Wei G, Wang Z, Shu H. Robust filtering with stochastic nonlinearities and multiple missing measurements. Automatica, 2009, 45: 836-841 CrossRef Google Scholar

[6] Moayedi M, Foo Y K, Soh Y C. Adaptive Kalman Filtering in Networked Systems With Random Sensor Delays, Multiple Packet Dropouts and Missing Measurements. IEEE Trans Signal Process, 2010, 58: 1577-1588 CrossRef ADS Google Scholar

[7] Xie L, Lu L, Zhang D. Improved robust H2 and H filtering for uncertain discrete-time systems. Automatica, 2004, 40: 873-880 CrossRef Google Scholar

[8] Gupta V, Chung T H, Hassibi B. On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage. Automatica, 2006, 42: 251-260 CrossRef Google Scholar

[9] Han D, Wu J, Mo Y. On Stochastic Sensor Network Scheduling for Multiple Processes. IEEE Trans Automat Contr, 2017, 62: 6633-6640 CrossRef Google Scholar

[10] Wu J, Johansson K H, Shi L. A Stochastic Online Sensor Scheduler for Remote State Estimation With Time-Out Condition. IEEE Trans Automat Contr, 2014, 59: 3110-3116 CrossRef Google Scholar

[11] Yang F, Wang K C, Huang Y. Energy-Neutral Communication Protocol for Very Low Power Microbial Fuel Cell Based Wireless Sensor Network. IEEE Senss J, 2015, 15: 2306-2315 CrossRef Google Scholar

[12] Mo Y, Ambrosino R, Sinopoli B. Sensor selection strategies for state estimation in energy constrained wireless sensor networks. Automatica, 2011, 47: 1330-1338 CrossRef Google Scholar

[13] Yang W, Shi H. Power allocation scheme for distributed filtering over wireless sensor networks. CrossRef Google Scholar

[14] Han D, Wu J, Zhang H. Optimal sensor scheduling for multiple linear dynamical systems. Automatica, 2017, 75: 260-270 CrossRef Google Scholar

[15] Shi L, Cheng P, Chen J. Sensor data scheduling for optimal state estimation with communication energy constraint. Automatica, 2011, 47: 1693-1698 CrossRef Google Scholar

[16] Han D, Mo Y, Wu J. Stochastic Event-Triggered Sensor Schedule for Remote State Estimation. IEEE Trans Automat Contr, 2015, 60: 2661-2675 CrossRef Google Scholar

[17] Leong A S, Dey S, Quevedo D E. Sensor Scheduling in Variance Based Event Triggered Estimation With Packet Drops. IEEE Trans Automat Contr, 2017, 62: 1880-1895 CrossRef Google Scholar

[18] Shi D, Chen T, Shi L. An event-triggered approach to state estimation with multiple point- and set-valued measurements. Automatica, 2014, 50: 1641-1648 CrossRef Google Scholar

[19] Liu Q, Wang Z, He X. Event-Based Recursive Distributed Filtering Over Wireless Sensor Networks. IEEE Trans Automat Contr, 2015, 60: 2470-2475 CrossRef Google Scholar

[20] Dong H, Wang Z, Ding S X. Event-Based $H_{\infty}$ Filter Design for a Class of Nonlinear Time-Varying Systems With Fading Channels and Multiplicative Noises. IEEE Trans Signal Process, 2015, 63: 3387-3395 CrossRef ADS Google Scholar

[21] Salvo Rossi P, Ciuonzo D, Kansanen K. On Energy Detection for MIMO Decision Fusion in Wireless Sensor Networks Over NLOS Fading. IEEE Commun Lett, 2015, 19: 303-306 CrossRef Google Scholar

[22] Shirazinia A, Dey S, Ciuonzo D. Massive MIMO for Decentralized Estimation of a Correlated Source. IEEE Trans Signal Process, 2016, 64: 2499-2512 CrossRef ADS Google Scholar

[23] Ciuonzo D, Rossi P S, Willett P. Generalized Rao Test for Decentralized Detection of an Uncooperative Target. IEEE Signal Process Lett, 2017, 24: 678-682 CrossRef ADS arXiv Google Scholar

[24] Ciuonzo D, Rossi P S. Distributed detection of a non-cooperative target via generalized locally-optimum approaches. Inf Fusion, 2017, 36: 261-274 CrossRef Google Scholar

[25] Xiao He , Zidong Wang , Donghua Zhou . Robust $H_{\infty}$ Filtering for Time-Delay Systems With Probabilistic Sensor Faults. IEEE Signal Process Lett, 2009, 16: 442-445 CrossRef ADS Google Scholar

[26] Liu Y, He X, Wang Z. Optimal filtering for networked systems with stochastic sensor gain degradation. Automatica, 2014, 50: 1521-1525 CrossRef Google Scholar

[27] Yalcin H, Collins R, Hebert M. Background estimation under rapid gain change in thermal imagery. Comput Vision Image Understanding, 2007, 106: 148-161 CrossRef Google Scholar

[28] Solomon I S D, Knight A J. Spatial processing of signals received by platform mounted sonar. IEEE J Ocean Eng, 2002, 27: 57-65 CrossRef Google Scholar

[29] Zhou D H, He X, Wang Z. Leakage Fault Diagnosis for an Internet-Based Three-Tank System: An Experimental Study. IEEE Trans Contr Syst Technol, 2012, 20: 857-870 CrossRef Google Scholar

[30] Olfati-Saber R, Shamma J S. Consensus filters for sensor networks and distributed sensor fusion. In: Proceedings of the 44th IEEE Conference on Decision and Control, Seville, 2005. 6698--6703. Google Scholar

[31] Carli R, Chiuso A, Schenato L. Distributed Kalman filtering based on consensus strategies. IEEE J Sel Areas Commun, 2008, 26: 622-633 CrossRef Google Scholar

[32] Schizas I D, Mateos G, Giannakis G B. Distributed LMS for Consensus-Based In-Network Adaptive Processing. IEEE Trans Signal Process, 2009, 57: 2365-2382 CrossRef ADS Google Scholar

[33] Khan U A, Moura J M F. Distributing the Kalman Filter for Large-Scale Systems. IEEE Trans Signal Process, 2008, 56: 4919-4935 CrossRef ADS arXiv Google Scholar

[34] Schizas I D, Ribeiro A, Giannakis G B. Consensus in Ad Hoc WSNs With Noisy Links-Part I: Distributed Estimation of Deterministic Signals. IEEE Trans Signal Process, 2008, 56: 350-364 CrossRef ADS Google Scholar

[35] Cattivelli F S, Sayed A H. Diffusion Strategies for Distributed Kalman Filtering and Smoothing. IEEE Trans Automat Contr, 2010, 55: 2069-2084 CrossRef Google Scholar

[36] Calafiore G C, Abrate F. Distributed linear estimation over sensor networks. Int J Control, 2009, 82: 868-882 CrossRef Google Scholar

[37] Speranzon A, Fischione C, Johansson K. A distributed minimum variance estimator for sensor networks. IEEE J Sel Areas Commun, 2008, 26: 609-621 CrossRef Google Scholar

[38] Caballero-águila R, Hermoso-Carazo A, Linares-Pérez J. Distributed fusion filters from uncertain measured outputs in sensor networks with random packet losses. Inf Fusion, 2017, 34: 70-79 CrossRef Google Scholar

[39] Tian T, Sun S, Li N. Multi-sensor information fusion estimators for stochastic uncertain systems with correlated noises. Inf Fusion, 2016, 27: 126-137 CrossRef Google Scholar

[40] Liu Y, Wang Z, He X. Minimum-Variance Recursive Filtering Over Sensor Networks With Stochastic Sensor Gain Degradation: Algorithms and Performance Analysis. IEEE Trans Control Netw Syst, 2016, 3: 265-274 CrossRef Google Scholar

[41] Dong H, Wang Z, Gao H. Distributed Filtering for a Class of Time-Varying Systems Over Sensor Networks With Quantization Errors and Successive Packet Dropouts. IEEE Trans Signal Process, 2012, 60: 3164-3173 CrossRef ADS Google Scholar

[42] Zhang H, Cheng P, Shi L. Optimal DoS Attack Scheduling in Wireless Networked Control System. IEEE Trans Contr Syst Technol, 2016, 24: 843-852 CrossRef Google Scholar

[43] Zhang H, Cheng P, Shi L. Optimal Denial-of-Service Attack Scheduling With Energy Constraint. IEEE Trans Automat Contr, 2015, 60: 3023-3028 CrossRef Google Scholar

[44] Anderson B D O, Moore J B. Optimal Filtering. Englewood: Prentice-Hall, 1979. Google Scholar

[45] Wang L, Guo G, Zhuang Y. Stabilization of NCSs by random allocation of transmission power to sensors. Sci China Inf Sci, 2016, 59: 067201 CrossRef Google Scholar

[46] Gupta V, Hassibi B, Murray R M. Optimal LQG control across packet-dropping links. Syst Control Lett, 2007, 56: 439-446 CrossRef Google Scholar

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