# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
\right)^{-1},\hat{X}_{0|0}^{ i}=P_{0|0}^{i}\!\left(\mathcal{C}_{0}^{i}(0)\right)^{T}\!\left(R_{0}^{i}\right)^{-1}\!y_{ 0}^{i}\), in which \(\hat{\Pi}_{0}^{i}=\left(\Pi_{0}^{-1}+\mu_{0}^{i}\sum_{j=1}^{i}\left(\frac{ \partial\mathcal{(}\mathcal{C}_{0}^{i}(\epsilon_{0}))^{T}}{\partial\epsilon_{ ij}}\right)\!\left(\frac{\partial\mathcal{C}_{0}^{i}(\epsilon_{0})}{ \partial\epsilon_{ij}}\right)\right|_{\epsilon_{0}=0}\right)^{-1}\);
2 Set design parameters \(\gamma_{k}^{i}\); for\(k=1\to n\)do
3\((a)\)Replace \(\mathcal{T}_{1,k}^{i},\mathcal{T}_{2,k}^{i},\mathcal{\tilde{A}}_{k}^{i}(0), \mathcal{\tilde{B}}_{1,k}^{i}(0),\mathcal{\tilde{B}}_{2,k}^{i}(0),P_{k|k^{ \prime}}^{i},Q_{k}\) by: \[\left(\mathcal{\tilde{B}}_{k|k}^{i}\right)^{-1}=\left(P_{k|k}^{i} \right)^{-1}+\mu_{k}^{i}\left(S_{k}^{i}\right)^{T}S_{k}^{i},\] \[\mathcal{\tilde{B}}_{2,k}^{i}=\mathcal{T}_{2,k}^{i}-\mu_{k}^{i} \mathcal{\tilde{S}}_{k}^{i}\mathcal{\tilde{B}}_{k|k}^{i}\left(S_{k}^{i} \right)^{T}\mathcal{T}_{2,k}^{i},\] \[\mathcal{\tilde{B}}_{2,k}^{i}(0)=\mathcal{\tilde{B}}_{2,k}(0)-\mu _{k}^{i}\mathcal{\tilde{A}}_{k}(0)\mathcal{\tilde{B}}_{k|k}^{i}\left(S_{k}^{i }\right)^{T}\mathcal{T}_{2,k}^{i},\] \[\mathcal{\tilde{A}}_{k}^{i}(0)=\left(\mathcal{\tilde{A}}_{k}(0)- \mu_{k}^{i}\mathcal{\tilde{B}}_{2,k}^{i}(0)\mathcal{\tilde{Q}}_{k}^{i}\left( T_{2,k}^{i}\right)^{T}S_{k}^{i}\right)\left(I-\mu_{k}^{i}\mathcal{\tilde{B}}_{ 4|k}^{i}\left(S_{k}^{i}\right)^{T}S_{k}^{i}\right),\] \[\mathcal{\tilde{B}}_{1,k}^{i}(0)=\mathcal{\tilde{B}}_{1,k}(0)-\mu _{k}^{i}\left(\mathcal{\tilde{A}}_{k}(0)\mathcal{\tilde{B}}_{k|k}^{i}\left(S_ {k}^{i}\right)^{T}+\mathcal{\tilde{B}}_{2,k}^{i}(0)\mathcal{\tilde{Q}}_{k}^{i }\left(T_{2,k}^{i}\right)^{T}\right)^{T}\mathcal{T}_{1,k}^{i};\]
4\((b)\) Update the priori pseudo-covariance and pseudo-covariance matrix: \[P_{k+1|k}^{i}= \mathcal{\tilde{A}}_{k}(0)\mathcal{\tilde{B}}_{k|k}^{i}\mathcal{ \tilde{A}}_{k}^{T}(0)+\mathcal{\tilde{B}}_{2,k}^{i}(0)\mathcal{\tilde{Q}}_{k} ^{i}\left(\mathcal{\tilde{B}}_{2,k}^{i}(0)\right)^{T}\,,\] \[P_{k+1|k+1}^{i}= P_{k+1|k}^{i}-P_{k+1|k}^{i}\Big{(}\mathcal{\tilde{C}}_{k+1}^{i} (0)\Big{)}^{T}\left(R_{k+1}^{i}+\mathcal{\tilde{C}}_{k+1}^{i}(0)P_{k+1|k}^{i} \big{(}\mathcal{\tilde{C}}_{k+1}^{i}(0)\big{)}^{T}\right)^{-1}\] \[\times\mathcal{\tilde{C}}_{k+1}^{i}(0)P_{k+1|k^{\prime}}^{i}\]
5\((c)\) Update the state of the local estimation: \[\hat{X}_{k+1|k+1}^{i}= \mathcal{\tilde{A}}_{k}^{i}(0)\mathcal{\tilde{A}}_{k|k}^{i}+ \mathcal{\hat{B}}_{1,k}^{i}(0)u_{k}+P_{k+1|k+1}^{i}\big{(}\mathcal{\tilde{C}}_ {k+1}^{i}(0)\big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}\] \[\times \Big{[}y_{k+1}^{i}-\mathcal{C}_{k+1}^{i}(0)\big{(}A_{k}^{i}(0) \mathcal{\tilde{X}}_{k|k}^{i}+\mathcal{\tilde{B}}_{1,k}^{i}(0)u_{k}\big{)}\Big{]}.\]

**Algorithm 1**The local robust state estimation based on sensitivity penalty (Appendix A)

Now, considering only the event-triggered transmission strategy, (4) corresponds to the measurements received by the fusion center from sensor node i, that is, \(t_{k}^{i}=1\). When sensor data are not transmitted, (4) will be replaced by

\[\mathcal{\tilde{B}}_{2,k}^{i}=\mathcal{T}_{2,k}^{i}-\mu_{k}^{i} \mathcal{\tilde{S}}_{k}^{i}\mathcal{\tilde{B}}_{k|k}^{i}\left(S_{k}^{i} \right)^{T}\mathcal{T}_{2,k}^{i},\]0

Here, \(v_{k}^{i}\) is uniformly distributed within the ellipsoid mentioned in (3) and is not correlated with the estimation error \(\mathcal{\hat{X}}_{k|k}^{i}-X_{k}\). According to the event-triggered transmission strategy, when there are packet drops in the communication channel from the estimator to the fusion center, the virtual measurement channel can be replaced with

\[\gamma_{k}^{i}=\left\{ \begin{array}{l}\mathcal{\hat{X}}_{k|k}^{i}t_{k}^{i}=1,r_ {k}^{i}=1\\ \mathcal{\hat{X}}_{k|k-1}^{i}t_{k}^{i}=1,r_{k}^{i}=0\\ \mathcal{\hat{X}}_{k|k}^{i},t_{k}^{i}=0\end{array} \right. \tag{6}\]where \(r_{k}^{i}\) is explicitly utilized in (6) to indicate whether packet drop occurs in sensor transmission to the fusion center and \(r_{k}^{i}=\{0,1\}\).