# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
(A) \(A_{1,k}(0)\), \(A_{2,k}(0)\), \(B_{1,k}(0)\), \(B_{2,k}(0)\), \(C_{k}^{i}(0)\), \(R_{k}^{i}\), \(Q_{k}\), \(S_{k}^{i}\), \(T_{1,k}^{i}\), \(T_{2,k}^{i}\), and \(\gamma_{k}^{i}\) are time-invariant. (B) The uncertain linear system of (1) is exponentially stable in the sense of Lyapunov and the matrices \(A_{1,k}(\varepsilon_{k})\), \(A_{2,k}(\varepsilon_{k})\), \(B_{1,k}(\varepsilon_{k})\), \(B_{2,k}(\varepsilon_{k})\), \(C_{k}^{i}(\varepsilon_{k})\), \(\Pi_{k}\), \(R_{k}^{i}\), \(Q_{k}\) are bounded whenever \(k>0\) and \(\varepsilon_{k}\in\mathcal{E}\). (C) For every sensor node, \(\big{(}M_{k},N_{k}^{i}\big{)}\) is detectable and the following matrix pair is detectable

\[\left( \begin{array}{c}M_{k}^{T}-\lambda_{k}^{i}\big{(}S_{k}^{i}\big{)}^{T} \bigg{(}I_{n(d+1)}+\lambda_{k}^{i}T_{2,k}^{i}Q_{k}\Big{(}T_{2,k}^{i}\Big{)}^{T }\bigg{)}^{-1}T_{2,k}^{i}Q_{k}(F_{k})^{T}\\ \qquad\qquad\bigg{(}I_{n(d+1)}+\lambda_{k}^{i}Q_{k}^{\frac{1}{2}}\Big{(}T_{2,k }^{i}\Big{)}^{T}T_{2,k}^{i}Q_{k}^{\frac{1}{2}}\Big{)}^{-\frac{1}{2}}Q_{k}^{ \frac{1}{2}}(F_{k})^{T}\end{array} \right)^{T},\]

where \(N_{k}^{i}=\left[ \begin{array}{c}\big{(}R_{k}^{i}\big{)}^{-\frac{1}{2}}O_{k}^ {i}\\ \sqrt{\lambda_{k}^{i}}S_{k}^{i}\end{array} \right]\). **Theorem 1** ([15]). : _Suppose that Assumptions (A), (B), and (C) hold and that each sensor transmits local estimate \(\hat{A}_{k|k}^{i}\) according to the event-triggered transmission strategy. If the weight matrix \(\Omega_{k}^{i}\) of the sensor node satisfies the condition_

\[\Omega_{k}^{i}\geq\omega^{i}\,I \tag{14}\]_for some positive real number_ \(\omega^{i}\)_, the estimation error_ \(X_{k}-\hat{X}_{k|k}\) _is consistently bounded for any possible choice of_ \(\left\{\hat{X}^{i}_{k|k^{\prime}}k\in Z_{+}\right\}\)_, which means_

\[\lim_{k\rightarrow\infty}\;\sup E\left\{\left\|X_{k}-\hat{X}_{k|k}\right\|^{2} \right\}<+\infty.\]

Proof of Theorem 1.: Let \(\hat{X}_{k|k}\) be the estimate obtained at time \(k\) through \(\tilde{Y}_{k}\) instead of \(\gamma_{k}\), \(\tilde{Y}_{k}=\mathrm{col}\bigg{\{}\sigma^{i}_{1,k}\hat{X}^{i}_{k|k}+\left. \sigma^{i}_{2,k}\hat{X}^{i}_{k|k-1}\right|^{l}_{i=1}\bigg{\}}\), which gives

\[\hat{X}_{k|k}=\hat{X}_{k|k}+\left(H^{T}\;\bar{P}^{-1}_{k}\;H\right)^{-1}H^{T} \;\bar{P}^{-1}_{k}\;\big{(}\gamma_{k}-\tilde{Y}_{k}\big{)},\]

so we have

\[E\left\{\left\|X_{k}-\hat{X}_{k|k}\right\|^{2}\right\}\leq 2E\Big{\{} \left\|X_{k}-\bar{X}_{k|k}\right\|^{2}\Big{\}}\] \[+2\Big{\|}\big{(}H^{T}\bar{P}^{-1}_{k}H\big{)}^{-1}\;H^{T}\;\bar{ P}^{-1}_{k}\Big{\|}^{2} \tag{15}\] \[\times E\Big{\{}\left\|Y_{k}-\tilde{Y}_{k}\right\|^{2}\Big{\}}.\]

Taking into account the first term on the right-hand side in (15), since \(\hat{X}_{k|k}\) is based on the vector \(\tilde{y}_{k}\), the following inequality can be obtained

\[E\left\{\left\|X_{k}-\bar{X}_{k|k}\right\|^{2}\right\}\leq tr\;\big{(}H^{T}\; \bar{P}^{-1}_{k}\;H\big{)}^{-1}. \tag{16}\]

According to Assumptions (A), (B), and (C), then \(P^{i,i}_{k|k}\) is convergent, and \(\bar{P}^{i,j}_{k|k-1}(i\
eq j)\) and \(P^{i,i}_{k|k-1}\) are also convergent [33]. The estimation error has a bounded covariance matrix at each \(k\). This indicates that \(\Gamma_{k}\) is converged, and the estimation error covariance matrix is bounded. From the inequality condition in Theorem 1 and the remainder of \(\bar{P}_{k}\), we can obtain

\[ \begin{split} tr\bigg{(}\big{(}1-t^{i}_{k}\big{)}\frac{\theta^{ i}}{n+2}\;\big{(}\Omega^{i}_{k}\big{)}^{-1}\bigg{)}&=\big{(}1-t^{i}_{k} \big{)}\frac{\theta^{i}}{n+2}tr\bigg{(}\Big{(}\Omega^{i}_{k}\Big{)}^{-1}\bigg{)} \\ &\leq\big{(}1-t^{i}_{k}\big{)}\frac{\theta^{i}}{\left(n+2\right) \omega^{i}}.\end{split}  \tag{17}\]

Hence, the uniform boundedness of \(E\bigg{\{}\left\|X_{k}-\bar{X}_{k|k}\right\|^{2}\bigg{\}}\) can be obtained by (16). Now it is only necessary to prove that the second part of the right-hand side of inequality (15) is uniform boundedness.