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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

Abstract

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.


Acknowledgment

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).


Supplement

Appendix

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.


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