Precise Clock Synchronization in Industrial Internet of Things: Networked Control Perspective
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摘要:
本文针对物联网中时变的时钟参数, 运用网络化控制理论观点, 通过对时钟状态建模的本质分析, 区别于"相对时钟建模", 提出了全分布规模化时钟状态追踪卡尔曼滤波(Kalman filtering). 考虑量测的丢失, 则扩展为追踪时钟参数的修正Kalman filtering算法. 我们提出了以BMU (Basic measurement unit)构建新的MMSE (Minimum mean square error)等价变换下的能观测性状态解耦量测模型, 新的量测模型能够实现MMSE量测规模化扩展, 且理论上分析了时钟同步的条件和计算了统计时钟同步误差的相应上界, 并且在时钟同步精度与潜在的通信网络质量间作出了量化均衡.
Abstract:In this paper, aiming at the time-varying clock parameters in the internet of things, the fully distributed scaled Kalman filtering for clock state tracking, which is different from the "relative clock modeling", has been proposed by using the point of view of the networked control theory through the essential analysis of the clock state modeling. Considering the loss of measurement, it can be extended to the modified Kalman filtering algorithm for tracking the clock parameters. We have proposed an observable state decoupling measurement model under a new MMSE (minimum mean square error) equivalent transformation based on BMU (basic measurement unit). The new measurement model is able to realize the scaled expansion of MMSE measurement, and theoretically analyze the conditions of clock synchronization and calculate the corresponding upper bound of statistical clock synchronization error, and quantify the balance between synchronization accuracy of clock and potential quality of the communication network.
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图 1 一组时钟信息交换过程($z_{i, k}^{\{i, j\}}$和$z_{j, k}^{\{j, i\}}$互为对称量测信息)
Fig. 1 A set of clock information exchange processes ($z_{i, k}^{\{i, j\}}$ and $z_{j, k}^{\{j, i\}}$ are symmetrical measurement information each other)
图 2 可观性对BMU RAMSE系统稳定性的影响
Fig. 2 Effect of observability for system stability of a BMU-RAMSE
图 3 可观性对BMU-RMSE系统稳定性的影响
Fig. 3 Effect of observability for system stability of a BMU-RMSE
表 1 不可靠量测丢包变量定义
Table 1 Definition of unreliable packet loss variables
名称 定义 $\pi_{i, k}^{\{i, j\}}$ 描述$S_i$和$S_j$间第$k$次信息交换. 交换成功, 则$\pi_{i, k}^{\{i, j\}}=1$; 否则$\pi_{i, k}^{\{i, j\}}=0$. 多链路下, $S_i$与任意邻居$S_{m_j}$间则为$\pi_{i, k}^{\{i, m_j\}}$ $\phi_{i, j}$ 描述$S_i$和$S_j$间的接收概率, $P(\pi_{i, k}^{\{i, j\}}=1)=\phi_{i, j}$, $P(\pi_{i, k}^{\{i, j\}}$ $=0)=1-\phi_{i, j}$. 多链路下, $S_i$与任意邻居$S_{m_j}$间则为$\phi_{i, m_j}$ $\boldsymbol{\chi}$ 表示一个同步周期, $S_i$与$|\mathcal{N}_i|$个邻居的$|\mathcal{N}_i|$次信息交换中, 丢包组合$\boldsymbol{\chi}=\{\chi_1, \chi_2, \cdots, \chi_g, \cdots, \chi_{2^{|\mathcal{N}_i|}}|g\in \{1$, $2$, $\cdots$, $2^{|\mathcal{N}_i|}\}\}$, $\chi_g$为第$g$种丢包情况, 共$2^{|\mathcal{N}_i|}$种 $\pi_{i, g, k}^{\{i, m_j\}}$ 描述第$k$个采样周期第$g$种丢包情况时($\chi_g$), $S_i$与$S_{m_j}$信息交换是否成功地取值. 若成功, $\pi_{i, g, k}^{\{i, m_j\}}=1$, 否则$\pi_{i, g, k}^{\{i, m_j\}}$ $=$ $0$ $\boldsymbol{L}_{i, g, k}$ 如$\pi_{i, g, k}^{\{i, m_j\}}$, 则$\boldsymbol{L}_{i, g, k}={\rm diag}\{\pi_{i, g, k}^{\{i, m_1\}}, \cdots, \pi_{i, g, k}^{\{i, m_j\}}$, $\cdots$, $\pi_{i, g, k}^{\{i, m_{|\mathcal{N}_i|}\}}\}$, 表示$S_i$与所有邻居在$\chi_g$情况时的丢包系数矩阵 $\eta_{i, g, k}$ 与所有邻居在$\chi_g$时的概率$\eta_{i, g, k}=\prod_{j=1}^{|\mathcal{N}_i|} \alpha_{i, m_j, k}$ $\alpha_{i, m_j, k}$ 表示$S_i$与$S_{m_j}$信息交换概率 $\alpha_{i, m_j, k}=\begin{cases} \phi_{i, m_j}, &\mbox{若}~ \pi_{i, k}^{\{i, m_j\}}=1\\ 1-\phi_{i, m_j}, &\mbox{若}~ \pi_{i, k}^{\{i, m_j\}}=0 \end{cases}$ $\boldsymbol{\gamma}_{i, g, k}$ 第$k$步绝对量测信息$\boldsymbol{\gamma}_{i, k}$在$\chi_g$情况时的具体表达式, $\boldsymbol{\gamma}_{i, g, k}$为构造的值, 实际量测中不可获得 $\boldsymbol{z}_{i, g}(k)$ 第$k$步相对量测信息$\boldsymbol{z}_{i}(k)$在$\chi_g$情况时的具体表达式, $\boldsymbol{z}_{i, g}(k)$为一个实际中能够量测的值 
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