# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 3 The Robust Fusion Estimation Procedure
Under the inequality condition in Theorem 1, it can be obtained as

\[\left\|\hat{X}^{i}_{k|k}-\bar{X}^{i}_{k|k}\right\|^{2}_{\Omega^{i}_{k}}\geq \omega^{i}\left\|\hat{X}^{i}_{k|k}-\bar{X}^{i}_{k|k}\right\|^{2}. \tag{18}\]

When \(t^{i}_{k}=0\), it means that there is \(\left\|\hat{X}^{i}_{k|k}-\bar{X}^{i}_{k|k}\right\|^{2}_{\Omega^{i}_{k}}\leq \theta^{i}\). Furthermore, it can be obtained that \(\left\|\hat{X}^{i}_{k|k}-\bar{X}^{i}_{k|k}\right\|^{2}\leq\theta^{i}\) / \(\omega^{i}\), then \(\left\|Y_{k}-\bar{Y}_{k}\right\|^{2}\leq\Sigma^{l}_{i=1}\theta^{i}\) / \(\omega^{i}\). The proof is done. To minimize the volume of the non-transported region, \(\hat{X}^{i}_{k|k}\) and \(\Omega^{i}_{k}\) can be appropriately denoted as

\[ \begin{split}\hat{X}^{i}_{k|k}&=\hat{X}^{i}_{k|k-1} \hat{A}^{i}_{k-1|k-1},\\ \Omega^{i}_{k}&=\biggr{(}\frac{1}{\text{tr}\left(P^{i }_{k|k-1}\right)}\;\bar{P}^{i}_{k|k-1}\biggr{)}^{-1},\end{split}  \tag{19}\]in which \(\hat{P}^{i}_{k|k-1}=\bar{A}_{k-1}\left[ \begin{array}{c}\sigma^{i}_{1,k-1}P^{i,i}_ {k-1|k-1}\\ +\sigma^{i}_{2,k-1}P^{i}_{k-1|k-2}\\ +\left(1-t^{i}_{k-1}\right)\frac{\phi^{i}}{n+2}\left(\Omega^{i}_{k-1}\right)^ {-1}\end{array} \right]\left(\bar{A}_{k-1}\right)^{T}+\hat{B}^{i}_{2,k-1}\hat{Q}^ {i}_{k-1}\left(\hat{B}^{i}_{2,k-1}\right)^{T}\). Two methods exist for determining the local prediction of \(X_{k}\) as per (19). The first method utilized in this paper is a local estimation based on sensor nodes. This fusion estimation method does not necessitate broadcasting but requires each sensor node to retain past information. The second method is based on the \(k-1\) moment fusion estimation \(\hat{X}_{k-1|k-1}\). ## 5 Numerical Simulations

This section cites the tractor-car system detailed in [34], shown in Figure 2, and extends it to a multi-sensor system for sample simulations. The performance of the derived robust fusion estimator is demonstrated through comparison with the fusion estimator for the Kalman filter based on actual and nominal parameters using the same fusion method across two distinct sets of numerical simulations with modelling errors (fixed or not) and varying transmission rates and packet drop rates. This numerical simulation consists of two sensors. For each set, 500 time experiments were conducted, with 200 moments designated for each set, generating 200 input-output data pairs. In the simulations, the overall average estimated error variance \(E\left\|X_{k}-\hat{X}_{k|k}\right\|^{2}\approx\frac{1}{500}\sum_{f=1}^{500} \left\|X_{k}-\hat{X}^{(f)}_{k|k}\right\|^{2}\) is computed for each moment, and the implementation of event-triggered and occurrence of packet drops are displayed. Since the vehicle steering and directional angles in the tractor-car system are nonlinear, they can be linearized and expressed as

\[\left\{ \begin{aligned} x_{k+1}^{1}=&\bigg{(}1.0000-\frac{vk}{L}\bigg{)}x_{k}^{1}+\bigg{(}\frac{vk}{L}-0.2296\bigg{)}x_{k-d} ^{1}+\bigg{(}0.1764+\frac{vk}{L}\bigg{)}x_{k}^{2}\\ &+\bigg{(}0.1764+\frac{vk}{L}\bigg{)}x_{k-d}^{2}+\bigg{(}0.9804+ \frac{vk}{L}\bigg{)}w_{k}^{1}+\bigg{(}0.9804+\frac{vk}{L}\bigg{)}u_{k}^{1},\\ x_{k+1}^{2}=&\bigg{(}1.0000-\frac{vk}{L}\bigg{)}x_{k}^{2 }+\bigg{(}\frac{vk}{L}-0.2296\bigg{)}x_{k-d}^{2}+\bigg{(}0.9804+\frac{vk}{L} \bigg{)}w_{k}^{2}\\ &+\bigg{(}0.9804+\frac{vk}{L}\bigg{)}u_{k}^{2},\end{aligned} \right. \tag{20}\]

in which \(x_{k}^{1}\), \(x_{k}^{2}\), \(u_{k}\), \(w_{k}\), \(x_{k-d}^{1}\), and \(x_{k-d}^{2}\) are the direction angle of the tractor, the direction angle of the car, the tractor steering angle, the process noise, d-step time-delay for state 1, and d-step time-delay for state 2, respectively. \(x_{k}\) is the state vector, \(x_{k}=\left[ \begin{array}{cc}x_{k}^{1}&x_{k}^{2}\end{array} \right]^{T}\). \(L\), \(k\), and \(v\) denote the length of the tractor, the sampling period, and the constant speed,

Figure 2: The tractor–car system. respectively.