# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure  
Taking into account the impact of modelling errors on estimation performance, we adopt a robust state estimation algorithm based on sensitivity penalization [29] to obtain local estimates for multi-sensor systems. A design parameter \(\gamma_{k}^{i}\), \(0<\gamma_{k}^{i}<1\), is defined to compromise between nominal estimation performance and performance deterioration due to modelling errors. Derived from the foundation of the Kalman filter, this robust state estimation algorithm utilizes sensitivity penalization of model uncertainty estimation errors. It shares a similar form and comparable computational complexity with the standard Kalman filter. When \(\gamma_{k}^{i}=1\), this estimator degenerates to the standard Kalman filter. By introducing the augmentation matrix \(X\) and augmenting the original system (1) with states, the system becomes  
\[\left\{ \begin{aligned} & X_{k+1}=\bar{A}_{k}(\varepsilon_{k})X_{k}+ \bar{B}_{1,k}(\varepsilon_{k})u_{k}+\bar{B}_{2,k}(\varepsilon_{k})w_{k},\\ & y_{k}^{i}=\bar{C}_{k}^{i}(\varepsilon_{k})X_{k}+v_{k}^{i},1 \leq i\leq L,k\geq 0,\end{aligned} \right. \tag{2}\]  
in which,  
\[\bar{A}_{k}(\varepsilon_{k})=\left[ \begin{array}{cccc}A_{1,k}(\varepsilon_ {k})&0_{n\times m}&\cdots&0_{n\times n}&A_{2,k}(\varepsilon_{k})\\ &\text{I}_{n}&&0_{n\times n}\\ &&\ddots&&\vdots\\ &&&\text{I}_{n}&0_{n\times n}\end{array} \right],\]  
\[\bar{B}_{1,k}(\varepsilon_{k})=\left[ \begin{array}{cc}(B_{1,k}(\varepsilon _{k}))^{T}&0_{n\times dn}^{T}\end{array} \right]^{T},\]  
\[\bar{B}_{2,k}(\varepsilon_{k})=\left[ \begin{array}{cc}(B_{2,k}(\varepsilon _{k}))^{T}&0_{n\times dn}^{T}\end{array} \right]^{T},\]  
\[\bar{C}_{k}^{i}(\varepsilon_{k})=\left[ \begin{array}{cc}C_{k}^{i}(\varepsilon _{k})&0_{n\times dn}\end{array} \right].\]  
As can be seen from the above transformation, the re-modeled system is a discrete linear uncertain system without state delay. Following the transformation of the system model from (1) to (2), it is evident that the system matrix dimension changes from \(n\) to \(n(d+1)\). **Remark 2**: _In this paper, the system is considered only for constant state delays. Based on the state augmentation method, only the system matrix, input (control) matrix, and output matrix of the system need to be changed. The method transforms the original system into a non-time-delay system, but the system dimension will increase from the original \(n\) to \(n(n+d)\). The state augmentation method is simple and suitable when the delay step is low because the computational burden will  
Figure 1: Block diagram of the multi-sensor system with state delay. increase when it is significant. However, the time delay step is generally manageable in practical production so the problem could be more influential._  
To obtain the locally robust state estimate for the \(i-\)th sensor node, we first define several important matrices \(S_{k}^{i}\), \(T_{1,K}^{i}\), and \(T_{2,K}^{i}\), which play a key role in the parameter modification process, as follows:  
\[S_{k}^{i}=\left[\left(S_{k,1}^{i}(0,0)\right)^{T},\cdots,\left(S _{k,l}^{i}(0,0)\right)^{T}\right]^{T},\] \[T_{1,k}^{i}=\left[\left(T_{1,k,1}^{i}(0,0)\right)^{T},\cdots, \left(T_{1,k,l}^{i}(0,0)\right)^{T}\right]^{T},\] \[T_{2,k}^{i}=\left[\left(T_{2,k,1}^{i}(0,0)\right)^{T},\cdots, \left(T_{2,k,l}^{i}(0,0)\right)^{T}\right]^{T},\] \[S_{k,j}^{i}(\varepsilon_{k},\varepsilon_{k+1})=\left[ \begin{array} []{c}\frac{\partial C_{k+1}^{i}(\varepsilon_{k+1})}{\partial\varepsilon_{k+ 1}}\bar{A}_{k}(\varepsilon_{k})\\ \bar{C}_{k+1}^{i}(\varepsilon_{k+1})\frac{\partial\bar{A}_{k}(\varepsilon_{k })}{\partial\varepsilon_{k}j}\end{array} \right],\] \[T_{1,k,j}^{i}(\varepsilon_{k},\varepsilon_{k+1})=\left[ \begin{array} []{c}\frac{\partial\bar{C}_{k+1}^{i}(\varepsilon_{k+1})}{\partial\varepsilon _{k+1}}\bar{B}_{1,k}(\varepsilon_{k})\\ \bar{C}_{k+1}^{i}(\varepsilon_{k+1})\frac{\partial\bar{B}_{k}(\varepsilon_{k })}{\partial\varepsilon_{k}j}\end{array} \right],\] \[T_{2,k,j}^{i}(\varepsilon_{k},\varepsilon_{k+1})=\left[