# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
The state of the multi-sensor system is shown in Table 1. For simplicity, the event-triggered is abbreviated as ET, and the success of the transmission is simplified as PD. The \(\hat{X}_{k|k-1}^{i}\) in (6) is the predicted values of the \(i\)-th sensor node for moment \(k\). \(\eta_{k}^{i}\) is the virtual measurement noise of the \(i\)-th virtual channel for moment \(k\), which can be derived by

\[ \begin{split}\hat{X}_{k|k-1}^{i}=\hat{A}_{k-1}^{i}\hat{X}_{k-1|k -1}^{i}+\hat{B}_{1k-1}^{i}u_{k-1},\\ \eta_{k}^{i}=\begin{cases}\hat{X}_{k|k}^{i}-X_{k},\dot{r}_{k}^{i} =1,r_{k}^{i}=1,\\ \hat{X}_{k|k-1}^{i}-X_{k},\dot{r}_{k}^{i}=1,r_{k}^{i}=0,\\ \hat{X}_{k|k}^{i}-X_{k}-g_{k}^{i},\dot{r}_{k}^{i}=0.\end{cases}\end{split}  \tag{7}\]

The fusion estimation with both random packet drops and event-triggered transmission strategies is investigated, and the following matrices are defined as

\[ \begin{split} Y_{k}&=\text{col}\bigg{\{}t_{k}^{i} \Big{(}r_{k}^{i}\hat{X}_{k|k}^{i}+\left(1-r_{k}^{i}\right)\hat{X}_{k|k-1}^{i} \Big{)}+\left(1-t_{k}^{i}\right)\hat{X}_{k|k}^{i}\Big{|}_{i=1}^{i}\bigg{\}},\\ \eta_{k}&=\text{col}\bigg{\{}\left(\dot{t}_{k}^{i}r_{k} ^{i}+\left(1-t_{k}^{i}\right)\right)\big{(}\hat{X}_{k|k}^{i}-X_{k}\big{)} \Big{|}_{i=1}^{i}\bigg{\}}\\ +\text{col}\bigg{\{}\dot{t}_{k}^{i}\Big{(}1-r_{k}^{i}\Big{)} \big{(}\hat{X}_{k|k-1}^{i}-X_{k}\big{)}\Big{|}_{i=1}^{i}\bigg{\}}+\text{col} \left\{\left(1-t_{k}^{i}\right)g_{k}^{i}\Big{|}_{i=1}^{i}\right\},\\ H&=\text{col}\bigg{(}\left.\dot{t}\right|_{i=1}^{i}\bigg{)}. \end{split}  \tag{8}\]

The information in the fusion center is obtained from the virtual measurement channel

\[Y_{k}=HX_{k}+\eta_{k}.\]

In accordance with the best linear unbiased criterion (BLUE) in [32], we can obtain the fusion estimate and its error covariance matrix. \[\begin{split}\hat{X}_{k|k}&=\left(H^{T}\bar{P}_{k} ^{-1}H\right)^{-1}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\



in which \(\Gamma_{k}=\Gamma_{k,1}+\Gamma_{k,2}+\Gamma_{k,2}^{T}+\Gamma_{k,3}\). The matrices \(\Gamma_{k,1}\), \(\Gamma_{k,2}\), and \(\Gamma_{k,3}\) in the formula are equal to

\[\Gamma_{k,1}=\left[ \begin{array}{cccc}\left(\sigma_{1,k}^{1}\right)^{2}P_{k|k }^{1,1}&\cdots&\sigma_{1,k}^{1}\sigma_{1,k}^{l}P_{k|k}^{1,l}\\ \vdots&\ddots&\vdots\\ \sigma_{1,k}^{l}\sigma_{1,k}^{1}P_{k|k}^{l,1}&\cdots&\left(\sigma_{1,k}^{l} \right)^{2}P_{k|k}^{l,l}\end{array} \right],\]

\[\Gamma_{k,2}=\left[ \begin{array}{cccc}0&\cdots&\sigma_{1,k}^{1}\sigma_{2,k}^ {l}P_{k|k-1}^{1,l}\\ \vdots&\ddots&\vdots\\ \sigma_{1,k}^{l}\sigma_{2,k}^{1}P_{k|k-1}^{1,1}&\cdots&0\end{array} \right], \tag{11}\]

\[\Gamma_{k,3}=\left[ \begin{array}{cccc}\left(\sigma_{2,k}^{1}\right)^{2}P_{k| k-1}^{1,1}&\cdots&\sigma_{2,k}^{1}\sigma_{2,k}^{l}P_{k|k-1}^{1,l}\\ \vdots&\ddots&\vdots\\ \sigma_{2,k}^{l}\sigma_{2,k}^{1}P_{k|k-1}^{l,1}&\cdots&\left(\sigma_{2,k}^{l} \right)^{2}P_{k|k-1}^{l,l}\end{array} \right],\]

\[\sigma_{1,k}^{i}=\left(i_{k}^{i}r_{k}^{i}+\left(1-i_{k}^{i}\right)\right), \sigma_{2,k}^{i}=i_{k}^{i}\left(1-r_{k}^{i}\right).\]

Then, we consider the state estimation errors of the following dynamic system. \[\left\{ \begin{array}{c}X_{k+1}=\hat{A}_{k}^{i}X_{k}+\hat{B}_{k,1}^{i}u_{k}+ \hat{B}_{2,k}^{i}w_{k},\\ y_{k}^{i}=\hat{C}_{k}^{i}X_{k}+g_{k}^{i},1\leq i\leq l.\end{array} \right.