# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
; formal analysis, X.D. ; investigation, X.D. ; resources, H.Y. and H.L. ; data curation, H.Y. ; writing--original draft preparation, X.D. ; writing--review and editing, X.D. ; visualization, H.Y. and H.L. ; supervision, H.Y. and H.L. ; project administration, H.Y. and H.L. ; funding acquisition, H.Y. and H.L. All authors have read and agreed to the published version of the manuscript. This work is supported by the National Natural Science Foundation of China under grant number 61873138; by the Shandong Provincial Natural Science Foundation under grant number ZR2019 MF063 and ZR2020 MF064. Not applicable. The authors declare no conflict of interest. Figure 6: \(\longrightarrow\): packet drop rate 0.8; \(\longrightarrow\): packet drop rate 0.4. Transmission rate: 0.2. The modelling errors \(\varepsilon_{k}\) are taken to a normal distribution with truncations. ## Abbreviations

The following abbreviations are used in this manuscript:

\[ \begin{split} col[*]&\text{The stacking vector or matrix}\\ E[*]&\text{The mathematical expectation}\\ f[*]&\text{The probability density function}\\ M^{\top}&\text{The stacking vector or matrix}\\ N(:,\sigma,\Omega)&\text{The notation for the Gaussian probability density function with mean $\sigma$ and covariance $\Omega$}\\ tr[*]&\text{The trace of the matrix}\end{split} \]

## Appendix A Derivation of Robust State Estimation

In order to reduce the sensitivity of the estimation performance to the modelling error, the following cost function can therefore be minimized

\[ \begin{split} J\Big{(}\alpha_{k}^{i}\Big{)}=&\gamma_{ k}^{i}\bigg{[}\Big{\|}\alpha_{k}^{i}\Big{\|}_{\Phi_{k}^{i}}^{2}+\Big{\|}H_{k}^{i}(0,0) \alpha_{k}^{i}-\delta_{k}^{i}(0,0)\Big{\|}_{\Psi_{k}^{i}}^{2}\bigg{]}\\ &+\Big{(}1-\gamma_{k}^{i}\Big{)}\sum_{j=1}^{l}\Bigg{(}\Bigg{\|} \frac{\partial e_{k}^{i}(\varepsilon_{k},\varepsilon_{k+1})}{\partial \varepsilon_{k,j}}\Bigg{\|}^{2}+\Bigg{\|}\frac{\partial e_{k}^{i}(\varepsilon _{k},\varepsilon_{k+1})}{\partial\varepsilon_{k+1,j}}\Bigg{\|}^{2}\Bigg{)}_{ \begin{subarray}{c}\varepsilon_{k}=0\\ \varepsilon_{k+1}=0\end{subarray}},\end{split} \]

in which \(\Psi_{k}^{i}=\Big{(}R_{k+1}^{i}\Big{)}^{-1}\), \(H_{k}^{i}(\varepsilon_{k},\varepsilon_{k+1})=\tilde{C}_{k+1}^{i}(\varepsilon_ {k+1})[\tilde{A}_{k}(\varepsilon_{k})\quad\tilde{B}_{2,k}(\varepsilon_{k})]\), \(\theta_{k}^{i}(\varepsilon_{k},\varepsilon_{k+1})=y_{k+1}^{i}-\tilde{C}_{k+1}^ {i}(\varepsilon_{k+1})\Big{(}\tilde{A}_{k}(\varepsilon_{k})\hat{X}_{k|k}^{i}+ \tilde{B}_{1,k}(\varepsilon_{k})u_{k}\Big{)}\), \(\Phi_{k}^{i}=\text{diag}\bigg{\{}\Big{(}P_{k|k}^{i}\Big{)}^{-1},Q_{i}^{-1} \bigg{\}}\), \(\alpha_{k}^{i}=col\{X_{k}-\tilde{X}_{k|k}^{i}w_{k}\}\), \(e_{k}^{i}(\varepsilon_{k},\varepsilon_{k+1})=y_{k+1}^{i}-\tilde{C}_{k+1}^{i}( \varepsilon_{k+1})\Big{(}\tilde{A}_{k}(\varepsilon_{k})\hat{X}_{k|k}^{i}+ \tilde{B}_{1,k}(\varepsilon_{k})u_{k}\Big{)}-\tilde{C}_{k+1}^{i}(\varepsilon _{k+1})[\tilde{A}_{k}(\varepsilon_{k})\] \(\tilde{B}_{2,k}(\varepsilon_{k})|\alpha_{k}^{i}\). From \(\Phi_{k}^{i}\) and \(\Psi_{k}^{i}\), \(J\big{(}\alpha_{k}^{i}\big{)}\) is a strictly convex function when \(0<\gamma_{k}^{i}\leq 1\). Letting \(\frac{\delta J(\alpha_{k}^{i})}{\delta\alpha_{k}^{i}}=0\), the global unique minimum is obtained

\[ \begin{split}&\Bigg{(}\Phi_{k}^{i}+\Big{(}H_{k}^{i}(0,0)\Big{)}^{T} \Psi_{k}^{i}H_{k}^{i}(0,0)+\frac{1-\gamma_{k}^{i}}{\gamma_{k}^{i}}\Big{[}\ \ S_{k}^{i}\quad T_{2,k}^{i}\ \Big{]}^{T}\Big{[}\ \ S_{k}^{i}\quad T_{2,k}^{i}\ \Big{]} \Big{)}\alpha_{k,opt}^{i}=\\ &\Big{(}H_{k}^{i}(0,0)\Big{)}^{T}\Psi_{k}^{i}\beta_{k}^{i}(0,0)- \frac{1-\gamma_{k}^{i}}{\gamma_{k}^{i}}\Big{[}\ \ S_{k}^{i}\quad T_{2,k}^{i}\ \Big{]}^{T}\Big{(}S_{k}^{i}\hat{X}_{k|k}^{i}+T_{1,k}^{i}u_{k}\Big{)}.\end{split}