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

The following relationships can be easily obtained

\[X_{k+1}-\hat{X}_{k+1|k+1}^{i}= \left[I+P_{k+1|k}^{i}\left(\bar{C}_{k+1}^{i}\right)^{T}\left(R_{k +1}^{i}\right)^{-1}\bar{C}_{k+1}^{i}\right]^{-1} \tag{13}\] \[\times \left[\hat{A}_{k}^{i}\left(X_{k}-\hat{X}_{k|k}^{i}\right)+\hat{B }_{2,k}^{i}\,w_{k}\right]\] \[- \left[\left(P_{k+1|k}^{i}\right)^{-1}+\left(\bar{C}_{k+1}^{i} \right)^{T}\left(R_{k+1}^{i}\right)^{-1}\bar{C}_{k+1}^{i}\right]^{-1}\] \[\times \left(\bar{C}_{k+1}^{i}\right)^{T}\left(R_{k+1}^{i}\right)^{-1}v _{k+1}^{i},\] \[X_{k+1}-\hat{X}_{k+1|k}^{i}= \hat{A}_{k}^{i}\left(X_{k}-\hat{X}_{k|k}^{i}\right)+\hat{B}_{2,k} ^{i}\,w_{k}\,.\]

According to the above equation, the explicit expressions for the three pseudo mutual covariance matrices \(P_{k+1|k+1}^{i,j}\), \(\bar{P}_{k+1|k}^{i,j}\), and \(P_{k+1|k}^{i,j}\) in (11) can be derived as follows

\[P_{k+1|k+1}^{i,j}= \left[ \begin{array}{c}I-P_{k+1|k}^{i}\left(\bar{C}_{k+1}^{i} \right)^{T}\\ \times\left(\bar{C}_{k+1}^{i}P_{k+1|k}^{i}\left(\bar{C}_{k+1}^{i}\right)^{T}+ R_{k+1}^{i}\right)^{-1}\bar{C}_{k+1}^{i}\end{array} \right]\] \[\times \left[\hat{A}_{k}^{i}P_{k|k}^{i,j}\left(\hat{A}_{k}^{j}\right)^{ T}+\hat{B}_{2,k}^{i}Q_{k}(\hat{B}_{2,k}^{j})^{T}\right]\] \[\times \left[ \begin{array}{c}I-P_{k+1|k}^{j}\left(\bar{C}_{k+1}^{j} \right)^{T}\\ \times\left(\bar{C}_{k+1}^{i}P_{k+1|k}^{j}\left(\bar{C}_{k+1}^{j}\right)^{T}+ R_{k+1}^{j}\right)^{-1}\bar{C}_{k+1}^{j}\end{array} \right]^{T},(i\
eq j)\] \[P_{k+1|k}^{i,j}= \left(I+P_{k+1|k}^{i}\left(\bar{C}_{k+1}^{i}\right)^{T}\left(R_{ k+1}^{i}\right)^{-1}\bar{C}_{k+1}^{i}\right)^{-1}\] \[\times \left[\hat{A}_{k}^{i}P_{k|k}^{i,j}\left(\hat{A}_{k}^{j}\right)^{ T}+\hat{B}_{2,k}^{i}Q_{k}(\hat{B}_{2,k}^{j})^{T}\right],\]\[P_{k+1|k}^{i,j}=\hat{A}_{k}^{i}P_{k|k}^{i,j}(\hat{A}_{k}^{i})^{T}+\hat{B}_{2,k}^{i }Q_{k}(\hat{B}_{2,k}^{j})^{T},\]

\[P_{k+1|k+1}^{i,j}= \left[ \begin{array}{c}I-P_{k+1|k}^{i}\big{(}\hat{C}_{k+1}^{i} \big{)}^{T}\\ \times\Big{(}\hat{C}_{k+1}^{i}P_{k+1|k}^{i}\big{(}\hat{C}_{k+1}^{i}\big{)}^{T} +R_{k+1}^{i}\Big{)}^{-1}\bar{C}_{k+1}^{i}\end{array} \right]\] \[\times \left[\hat{A}_{k}^{i}P_{k|k}^{i,j}\left(\hat{A}_{k}^{i}\right)^{T }+\hat{B}_{2,k}^{i}Q_{k}\left(\hat{B}_{2,k}^{i}\right)^{T}\right]\] \[\times \left[ \begin{array}{c}I-P_{k+1|k}^{i}\big{(}\hat{C}_{k+1}^{i} \big{)}^{T}\\ \times\Big{(}\hat{C}_{k+1}^{i}P_{k+1|k}^{i}\big{(}\hat{C}_{k+1}^{i}\big{)}^{T} +R_{k+1}^{i}\Big{)}^{-1}\bar{C}_{k+1}^{i}\end{array} \right]^{T}\] \[+ \left( \begin{array}{c}\big{(}\bar{C}_{k+1}^{i}\big{)}^{T}\Big{(} R_{k+1}^{i}\big{)}^{-1}\bar{C}_{k+1}^{i}\\ +\big{(}P_{k+1|k}^{i}\big{)}^{-1}\end{array} \right)^{-1}\Big{(}\bar{C}_{k+1}^{i }\Big{)}^{T}\Big{(}R_{k+1}^{i}\Big{)}^{-1}\] \[\times \bar{C}_{k+1}^{i}\left(\left( \begin{array}{c}\big{(}\bar{C}_{k+ 1}^{i}\big{)}^{T}\Big{(}R_{k+1}^{i}\big{)}^{-1}\bar{C}_{k+1}^{i}\\ +\big{(}P_{k+1|k}^{i}\big{)}^{-1}\end{array} \right)^{-1}\right)^{T},\]

in which \(P_{k+1|k}^{i}=P_{k+1|k}^{i}\), \(i,j=1,\cdots,N\). \(P_{k+1|k+1}^{i}\) is a pseudo-covariance matrix in robust state estimation. Thus, there is \(P_{k+1|k+1}^{i,j}\
eq P_{k+1|k+1}^{i}\). ## 4 Some Properties of the Fusion Estimator

This section has the goal of investigating the steady-state properties of event-triggered robust fusion estimators for multi-sensor systems with deterministic inputs, random packet drops, and state delays. Assume that the modelling errors \(\varepsilon_{k,j}\) in this section are within the set \(\boldsymbol{\mathcal{E}}\), \(\boldsymbol{\mathcal{E}}=\left\{\varepsilon\big{|}\big{|}\varepsilon_{k,j} \big{|}\leq 1,j=1,2,\cdots,l\right\}\). The matrices \(\left[ \begin{array}{cc}\mathrm{A}_{1,k}(0)&0_{n\times n(d-1)}&\mathrm{A}_{2,k} (0)\\ I_{nd}&0_{nd\times n}\end{array} \right]\), \(\left[ \begin{array}{c}B_{2,k}(0)\\ 0_{n\times dn}\end{array} \right]\), and \(\left[C_{k}^{i}(0)&0_{m\times dn}\right]\) are denoted as \(M_{k}\), \(F_{k}\), and \(O_{k}^{i}\), respectively. In addition, the following assumptions need to be made.