# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
Considering the system errors at linearization in the form of modelling errors \(\varepsilon_{k}\) substituted into the system model, the matrix parameters are obtained as

\[ \begin{split} A_{1,k}(\varepsilon_{k})&=\left[\begin{array} []{cc}1.0000-\frac{\upsilon k}{L}&0.1764+\frac{\upsilon k}{L}+\varepsilon_{k} \\ 0.0000&1.0000-\frac{\upsilon k}{L}\end{array}\right],\\ A_{2,k}(\varepsilon_{k})&=\left[\begin{array}{cc}\frac{\upsilon k}{L}-0.2296 &0.1764+\frac{\upsilon k}{L}+\varepsilon_{k}\\ 0.0000&\frac{\upsilon k}{L}-0.2296\end{array}\right],\\ B_{1,k}(\varepsilon_{k})&=\left[\begin{array}{cc}0.9804+\frac{\upsilon k}{L} &0.0000\\ 0.0000&0.9804+\frac{\upsilon k}{L}\end{array}\right],\\ B_{2,k}(\varepsilon_{k})&=\left[\begin{array}{cc}0.9804+\frac{\upsilon k}{L} &0.0000\\ 0.0000&0.9804+\frac{\upsilon k}{L}\end{array}\right].\end{split}  \tag{21}\]

In the numerical simulation, each parameter is taken as \(L=500\) cm, \(k=0.1\) s, and \(v=98\) cm/s, and a two-step state delay system was used. The matrix parameters are as follows

\[ \begin{split} A_{1,k}(\varepsilon_{k})&=\left[\begin{array}{ cc}0.9804&0.196+1.99\varepsilon_{k}\\ 0.0000&0.9804\end{array}\right],A_{2,k}(\varepsilon_{k})&=\left[\begin{array} []{cc}-0.2100&0.196+1.99\varepsilon_{k}\\ 0.0000&-0.2100\end{array}\right],\\ B_{1,k}(\varepsilon_{k})&=\left[\begin{array}{cc}1.0000&0.0000\\ 0.0000&1.0000\end{array}\right],B_{2,k}(\varepsilon_{k})&=\left[\begin{array} []{cc}1.0000&0.0000\\ 0.0000&1.0000\end{array}\right],\\ C_{k}^{1}(\varepsilon_{k})&=\left[\begin{array}{cc}1.0000&-1.0000\end{array} \right],&C_{k}^{2}(\varepsilon_{k})&=\left[\begin{array}{cc}0.4000&-0.5000 \end{array}\right],\\ R_{k}^{1}&=1.0000,R_{k}^{2}&=1.0000,\\ Q_{k}&=\left[\begin{array}{cc}1.9608&0.0195\\ 0.0195&1.9605\end{array}\right],\Pi_{0}&=\left[\begin{array}{cc}1.0000&0.00 00\\ 0.0000&1.0000\end{array}\right],u_{k}&=\left[\begin{array}{cc}1.0000\\ 0.1000\end{array}\right].\end{split} \]

The packet drop process \(r_{k}^{i}\) is assumed to be a stationary Bernoulli process. A constant value of 0.7300 is assigned to the filter design parameter \(\gamma_{k}^{i}\). In Case 1, the modeling errors \(\varepsilon_{k}\) are assumed to be a fixed value of \(-0.8508\). The transmission and packet drop rates for both sensors are set to 0.8 and 0.2, respectively. Figure 3a illustrates the fusion estimation error over time, demonstrating that the robust fusion estimator proposed in this study outperforms the fusion estimator for the Kalman filter based on nominal parameters by approximately 7.800 dB. Figure 3b,c depict the transmission of the two sensors and the packet drops of the communication channel, respectively. To clearly reflect the execution of the event-triggered, \(t_{k}^{i}\) is inverted, and \(r_{k}^{i}\) is treated similarly. Note that the plots of event-triggered realizations and packet drops here are from one of the 500 experiments used. The modelling errors \(\varepsilon_{k}\) are generated randomly and independently, conforming to a normal distribution with a truncation. The mean, standard variance, and truncation values of the normal distribution are set to 0.0000, 1.0000, and 1.0000, respectively. Figure 4a illustrates that the derived estimator surpasses the performance of the Kalman filter based on nominal parameters, and it can be seen from the 200 th moment that the estimator derived in this paper is 5.8600 (dB) lower than the nominal parameter-based Kalman filter. Figure 4b,c show the realization of the sensor transmission and the channel packet drop over 200 moments, respectively. In Case 2, the derived robust fusion estimator is tested using different transmission and packet drop rates. The modelling errors are the same as in Case 1 with a truncated normal distribution. Based on the analysis of Figure 5, it is evident that the derived estimator exhibits effective and reliable operation even under diverse transmission rates generated by the employed event-triggered transmission strategy.