# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
\begin{split}\hat{X}_{k+1|k+1}^{i}&=\bar{B}_{1,k}(0)u _{k}+\bar{A}_{k}(0)\hat{X}_{k|k}^{i}+\bigg{(}\bar{A}_{k}(0)\hat{P}_{k|k}^{i} \bar{A}_{k}^{T}(0)+\hat{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(}\hat{B}_{2,k}^{i }(0)\Big{)}^{T}\bigg{)}\\ &\times\Big{(}C_{k+1}^{i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^ {-1}\Big{[}y_{k+1}^{i}-C_{k+1}^{i}(0)\hat{X}_{k+1|k+1}^{i}\Big{]}\\ &-\mu_{k}^{i}\Big{[}\bar{A}_{k}(0)\hat{P}_{k|k}^{i}\Big{(}S_{k}^{i }\Big{)}^{T}+\bar{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(}T_{2,k}^{i}\Big{)}^{T} \Big{]}\Big{(}S_{k}^{i}\hat{X}_{k|k}^{i}+T_{1,k}^{i}u_{k}\Big{)}.\end{split} \] (A5)

Letting \(P_{k+1|k}^{i}=\bar{A}_{k}(0)\hat{P}_{k|k}^{i}\bar{A}_{k}^{T}(0)+\bar{B}_{2,k} ^{i}(0)\hat{Q}_{k}^{i}\big{(}\hat{B}_{2,k}^{i}(0)\big{)}^{T}\), (A5) can be transformed into the form of (A6)

\[ \begin{split}&\bigg{(}I+P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0)\Big{)}^ {T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}C_{k+1}^{i}(0)\Big{)}\hat{X}_{k+1|k+1}^{i}= \bar{B}_{1,k}(0)u_{k}+\bar{A}_{k}(0)\hat{X}_{k|k}^{i}\\ &+P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i} \Big{)}^{-1}y_{k+1}^{i}-\mu_{k}^{i}\bigg{[}\bar{A}_{k}(0)\hat{P}_{k|k}^{i} \Big{(}S_{k}^{i}\Big{)}^{T}+\hat{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(}T_{2,k}^{ i}\Big{)}^{T}\bigg{]}\Big{(}S_{k}^{i}\hat{X}_{k|k}^{i}+T_{1,k}^{i}u_{k}\Big{)}. \end{split} \] (A6)

According to the matrix inverse lemma \((A+BCD)^{-1}=A^{-1}-A^{-1}B\big{(}DA^{-1}B+C^{-1}\big{)}^{-1}\)\(DA^{-1}\), the following procedure can be obtained

\[P_{k+1|k+1}^{i}=P_{k+1|k}^{i}-P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0)\Big{)}^{T} \bigg{(}R_{k+1}^{i}+\bar{C}_{k+1}^{i}(0)P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0) \Big{)}^{T}\bigg{)}^{-1}\bar{C}_{k+1}^{i}(0)P_{k+1|k}^{i}.\] (A6)

(A6) can be changed to

\[ \begin{split}&\bigg{(}I+P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0)\Big{)}^{T} \Big{(}R_{k+1}^{i}\Big{)}^{-1}C_{k+1}^{i}(0)\bigg{)}\hat{X}_{k+1|k+1}^{i}\\ &=\bigg{[}\bar{B}_{1,k}(0)-\mu_{k}^{i}\bigg{[}\bar{A}_{k}(0)\hat{P} _{k|k}^{i}\big{(}S_{k}^{i}\big{)}^{T}+\bar{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(} T_{2,k}^{i}\Big{)}^{T}\bigg{]}T_{1,k}^{i}\bigg{]}u_{k}\\ &+\bigg{[}\bar{A}_{k}(0)-\mu_{k}^{i}\bigg{[}\bar{A}_{k}(0)\hat{P} _{k|k}^{i}\big{(}S_{k}^{i}\big{)}^{T}+\bar{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(} T_{2,k}^{i}\Big{)}^{T}\bigg{]}S_{k}^{i}\bigg{]}\hat{X}_{k|k}^{i}\\ &+P_{k+1|k}^{i}\Big{(}C_{k+1}^{i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i} \Big{)}^{-1}y_{k+1}^{i}.\end{split} \]

Thus, the matrices \(\hat{A}_{k}^{i}(0)\) and \(\hat{B}_{1,k}^{i}(0)\) can be defined as

\[ \begin{split}&\hat{A}_{k}^{i}(0)=\bigg{(}\bar{A}_{k}(0)-\mu_{k}^{i} \bar{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i}\Big{(}T_{2,k}^{i}\Big{)}^{T}S_{k}^{i} \bigg{)}\Big{(}I-\mu_{k}^{i}\hat{P}_{k|k}^{i}\big{(}S_{k}^{i}\big{)}^{T}S_{k}^{i} \Big{)},\\ &\hat{B}_{1,k}^{i}(0)=\bar{B}_{1,k}(0)-\mu_{k}^{i}\bigg{(}\bar{A}_{k} (0)\hat{P}_{k|k}^{i}\big{(}S_{k}^{i}\big{)}^{T}+\bar{B}_{2,k}^{i}(0)\hat{Q}_{k}^{i} \Big{(}T_{2,k}^{i}\Big{)}^{T}\bigg{)}T_{1,k}^{i}.\end{split} \]Thus (A5) can be simplified as

\[ \begin{split}\hat{X}_{k+1|k+1}^{i}&=\hat{A}_{k}^{i}(0) \hat{X}_{k|k}^{i}+\hat{B}_{1,k}^{i}(0)u_{k}+P_{k+1|k+1}^{i}\Big{(}\hat{C}_{k+1}^ {i}(0)\Big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}\\ &\times\Big{[}y_{k+1}^{i}-\hat{C}_{k+1}^{i}(0)\Big{(}\hat{A}_{k}^ {i}(0)\hat{X}_{k|k}^{i}+\hat{B}_{1,k}^{i}(0)u_{k}\Big{)}\Big{]}.\end{split} \] (A7)

(A7) is similar to the form described in [29], so that \(\hat{X}_{k+1|k+1}^{i}\) can be specified as \(\hat{X}_{k+1|k+1}^{i}\). The derivation is complete. ## References

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