# Event-Triggered Robust Fusion Estimation for Multi-Sensor Time-Delay Systems with Packet Drops
## 2 Problem Formulation and Some Preliminaries  
Consider the following discrete-time uncertain linear stochastic system with deterministic inputs and d-steps state delay  
\[ \begin{cases}x_{k+1}=A_{1,k}(\varepsilon_{k})x_{k}+A_{2,k}(\varepsilon_{k})x_{ k-d}+B_{1,k}(\varepsilon_{k})u_{k}+B_{2,k}(\varepsilon_{k})w_{k}\\ y^{i}_{k}=C^{i}_{k}(\varepsilon_{k})x_{k}+g^{i}_{k},1\leq i\leq L,k\geq 0 \end{cases}  \tag{1}\]  
where \(k\) represents the discrete-time and \(i\) represents the sensor label. Furthermore, \(x_{k}\) is the state, \(y^{i}_{k}\) is the measurement, \(w_{k}\) represents the process noise, \(u_{k}\) is the deterministic control input, and \(g^{i}_{k}\) is the compound effect of measurement and communication errors. The following assumptions need to be made to guarantee the fitness of the state estimation problem. (a) \(w_{k}\) and \(g_{k}^{i}\) are normally distributed with white noise, \(x_{0}\), \(w_{k}\), and \(g_{k}^{i}\) are mutually independent random variables. \[\mathbf{E}(w_{k})=0,\mathbf{E}\big{(}g_{k}^{i}\big{)}=0,\] \[\mathbf{E}\Bigg{(}\left[ \begin{array}{c}x_{0}-\mathbf{E}(x_{0} )\\ w_{k}\\ g_{k}^{i}\end{array} \right](*)^{\mathsf{T}}\Bigg{)}=\left[ \begin{array}{cc} \Pi_{0}\\ &Q_{k}\delta_{kj}\\ &R_{k}^{i}\delta_{kj}\end{array} \right],\]  
where \(\Pi_{0}\), \(Q_{k}\), and \(R_{k}^{i}\) are known positive definite matrices and \(\delta_{kj}\) denotes the Kronecker symbolic function. (b) The elements in the matrices \(A_{1,k}(\varepsilon_{k})\), \(A_{2,k}(\varepsilon_{k})\), \(B_{1,k}(\varepsilon_{k})\), \(B_{2,k}(\varepsilon_{k})\) and \(C_{k}^{i}(\varepsilon_{k})\) are known differentiable functions of the modelling errors, and the modelling errors \(\varepsilon_{k}\) consist of \(l\) mutually independent real-valued scalar bounded uncertainties \(\varepsilon_{k,j}\), \(j=1,...,l\). In the process of transmitting the measurement value \(Y\) from the sensor node to the fusion center, the channel may experience packet drops. A random variable \(r\) is defined to indicate the success or failure of the communication between the sensor node and the fusion center, taking the value of \(1\) for the successful transmission and \(0\) when the communication channel fails. The aim of this paper is to develop a fusion algorithm based on local estimates from each sensor node for multi-sensor systems with parameter uncertainty, state delay, random packet drops, and communication rate limitations. To balance communication cost and estimation performance, an event-triggered transmission strategy like in [15] is used in this paper. Consider the following measurement channel  
\[Y=H\phi+g\]  
where \(Y\in\mathbf{R}^{m}\) is the measurement output, \(h\in\mathbf{R}^{m\times n}\) is the measurement matrix of the system, \(\phi\in\mathbf{R}^{n}\) represents the state, and \(g\in\mathbf{R}^{m}\) represents the measurement noise. A binary variable is denoted by \(t\), and when \(t=1\) indicates that the sensor node sends a measurement \(Y\) and the other way around. The specific form of the event-triggered transmission strategy is as follows. \[t_{k}^{i}=\left\{ \begin{array}{c}0,Y-\tilde{Y}\in\Xi,\\ 1,\text{ others },\end{array} \right.\]  
in which \(\tilde{Y}\in\mathbf{R}^{m}\) and \(\Xi\in\mathbf{R}^{m}\) are measurable sets. Generally, the center of mass of \(\Xi\) is at the origin, that is, \(\int_{\Xi}\varphi d\varphi=0\). Note that the decision transmission in the event-triggered transmission strategy is actually when the difference between the measured value and the determined measured value is greater than a threshold value. The transmission rate, \(a^{i}\in(0,1)\) for each sensor node is derived by \(\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\sum_{k=1}^{T}\{t_{k}^{i}\}=a^{i}\). In addition, for any given desired transmission rate \(a^{i}\), a threshold \(\Xi\) can be easily determined. Based on Lemma 1 in [15], a virtual measure \(Y=\tilde{Y}=H\varepsilon+g-v\) is now defined, where it is uniformly distributed over, and is independent of, \(X\) and \(g\).