# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
 \tag{12}\]

The initial state \(X_{0}\) is estimated such that \(\dot{e}_{0}^{i}(\varepsilon_{0})=y_{0}^{i}-\tilde{C}_{0}^{i}(\varepsilon_{0}) X_{0}\) and the cost function is \(J\big{(}\alpha_{0}^{i}\big{)}=\gamma_{0}^{i}\bigg{[}\|X_{0}\|_{\Pi_{0}^{-1}}^{2}+ \big{\|}y_{0}^{i}-\tilde{C}_{0}^{i}(\varepsilon_{0})X_{0}\|_{\big{(}R_{0}^{i} \big{)}^{-1}}^{2}\bigg{]}+\big{(}1-\gamma_{0}^{i}\big{)}\sum_{j=1}^{l}\Bigg{(} \Bigg{\|}\frac{\partial\dot{e}_{0}^{i}(\varepsilon_{0})}{\partial\alpha_{j}} \Bigg{\|}^{2}\bigg{)}_{\varepsilon_{0}=0}\). The following initial state estimate and initial estimation error covariance matrix can be obtained

\[ \begin{split} P_{0|0}^{i}=\Bigg{(}\begin{array}{c}\big{(}\tilde{C }_{0}^{i}(0)\big{)}^{T}\big{(}R_{0}^{i}\big{)}^{-1}\tilde{C}_{0}^{i}(0)\\ +\big{(}\tilde{T}_{0}^{i}\big{)}^{-1}\end{array}\\ \hat{X}_{0|0}^{i}=P_{0|0}^{i}\big{(}\tilde{C}_{0}^{i}\big{)}\big{)}^{T}\big{(}R_{0 }^{i}\big{)}^{-1}y_{0}^{i},\end{split} \]

in which \(\hat{\Pi}_{0}^{i}=\Bigg{(}\Pi_{0}^{-1}+\mu_{0}^{i}\sum_{j=1}^{l}\Big{(}\frac{ \partial\big{(}\tilde{C}_{0}^{i}(0)\big{)}^{T}}{\partial\varepsilon_{0j}} \Big{)}\Bigg{)}\Big{|}_{\varepsilon_{0}=0}\Bigg{)}^{-1}\). Define \(\hat{M}_{k}^{i}\), \(\hat{T}_{2,k}^{i}\), \(\hat{X}_{k|k+1}^{i}\), \(\alpha_{k,opt}^{i}\), \(\Big{(}\hat{P}_{k|k}^{i}\Big{)}^{-1}\), and \(\big{(}\hat{Q}_{k}^{i}\big{)}^{-1}\) as \(\hat{C}_{k+1}^{i}(0)\Big{[}\tilde{A}_{k}(0)\hat{B}_{2,k}^{i}(0)\Big{]}\), \(\hat{T}_{2,k}^{i}-\mu_{k}^{i}S_{k}^{i}\hat{P}_{k|k}^{i}(\xi_{k}^{i})^{T}T_{2,k}^ {i}\), \(\hat{X}_{k|k+1}^{i}+\mu_{k}^{i}\hat{P}_{k|k}^{i}(\xi_{k}^{i})^{T}T_{2,k}^{i} \hat{w}_{k|k+1}\), \(col\left\{\hat{X}_{k|k+1}^{i}-\hat{X}_{k|k}^{i}w_{k|k+1}\right\}\), and \(\Big{(}P_{k|k}^{i}\Big{)}^{-1}+\mu_{k}^{i}\big{(}\hat{S}_{k}^{i}\big{)}^{T}S_{k}^{i} \big{(}Q_{k}\big{)}^{-1}+\mu_{k}^{i}\big{(}\hat{T}_{2,k}^{i}\big{)}^{T}\Big{(}T +\mu_{k}^{i}S_{k}^{i}P_{k|k}^{i}(\xi_{k}^{i})^{T}\Big{)}^{-1}T_{2,k}^{i}\). It is known by the following algebraic relation

\[\left[ \begin{array}{cc}\left(P_{k|k}^{i}\right)^{-1}&0\\ 0&Q_{i}^{-1}\end{array} \right]+\mu_{k}^{i}\Big{[} \begin{array}{cc}S_{k}^{i}&T_ {2,k}^{i}\end{array} \Big{]}^{T}\Big{[} \begin{array}{cc}S_{k}^{i}&T_{2,k}^{i} \end{array} \Big{]}=\] \[\left[ \begin{array}{cc}I&0\\ \mu_{k}^{i}\Big{(}T_{2,k}^{i}\Big{)}^{T}S_{k}^{i}\dot{P}_{k|k}^{i}&I\end{array}  \right]\left[\begin{array}{cc}\left(\dot{P}_{k|k}^{i}\right)^{-1}&0\\ 0&\left(\dot{Q}_{k}^{i}\right)^{-1}\end{array



From (A3), we have

\[ \begin{cases}\hat{X}_{k,k+1}^{i}=&\hat{X}_{k|k}^{i}+\hat{P}_{k|k}^{i}\hat{A}_{k}^ {T}(0)\Big{(}C_{k+1}^{i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}\\ &\times\Big{[}y_{k+1}^{i}-C_{k+1}^{i}(0)\Big{(}\bar{B}_{1,k}(0)u_{k}+\bar{A}_{k }(0)\hat{X}_{k|k+1}^{i}+\hat{B}_{2,k}^{i}(0)\hat{v}_{k|k+1}\Big{)}\Big{]}\\ &-\mu_{k}^{i}\hat{P}_{k|k}^{i}\Big{(}\hat{S}_{k}^{i}\Big{)}^{T}\Big{(}S_{k}^{i} \hat{X}_{k|k}^{i}+T_{1,k}^{i}u_{k}\Big{)}\\ \hat{v}_{k|k+1}=&\hat{Q}_{k}^{i}\Big{(}\hat{B}_{2,k}^{i}(0)\Big{)}^{T}\Big{(} C_{k+1}^{i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}\\ &\times\Big{[}y_{k+1}^{i}-C_{k+1}^{i}(0)\Big{(}\bar{B}_{1,k}(0)u_{k}+\bar{A}_ {k}(0)\hat{X}_{k|k+1}^{i}+\hat{B}_{2,k}^{i}(0)\hat{v}_{k|k+1}\Big{)}\Big{]}\\ &-\mu_{k}^{i}\hat{Q}_{k}^{i}\Big{(}\hat{T}_{2,k}^{i}\Big{)}^{T}\Big{(}S_{k}^{i }\hat{X}_{k|k}^{i}+T_{1,k}^{i}u_{k}\Big{)}\end{cases} .\] (A4)

Define the variable \(\hat{X}_{k+1|k+1}^{i}=\bar{A}_{k}(0)\hat{X}_{k|k+1}^{i}+\hat{B}_{2,k}^{i}(0) \hat{w}_{k|k+1}+\bar{B}_{1,k}(0)u_{k}\). Bringing (A4) into \(\hat{X}_{k+1|k+1}^{i}\), we obtain

\[