## 2 Control Scheme Design
The considered actuator faults are slow time-varying and can therefore be approximated as the following equation  
\[f_{i}(t+1)=f_{i}(t). \tag{6}\]  
With (1) and (6), it can be obtained that  
\[ \begin{cases}x_{t,i}(t+1)=A_{t,i}x_{t,i}(t)+B_{t,i}u_{i}(t)+I_{\text{f}}\omega _{i}(t)\\ y_{i}(t)=C_{t,i}x_{t,i}(t)+v_{i}(t),\end{cases}  \tag{7}\]  
\[\text{where }x_{t,i}(t)=\left[ \begin{array}{c}x_{i}(t)\\ f_{i}(t)\end{array} \right]\in\mathfrak{R}^{q}\), \(A_{t,i}=\left[ \begin{array}{cc}A_{i}&F_{i}\\ 0&I\end{array} \right]\), \(B_{t,i}=\left[ \begin{array}{c}B_{i}\\ 0\end{array} \right]\), \(C_{t,i}=\left[ \begin{array}{cc}C_{i}&0\end{array} \right]\), \(I_{\text{f}}=\left[ \begin{array}{c}I\\ 0\end{array} \right]\). Furthermore, the two assumptions below are also necessary. **Assumption 1**. : _The sensor, controller, and actuator of each agent operate in the time-driven and synchronous manner._  
**Assumption 2**. : _Random network delays \(\tau^{\text{ca}}_{t,i}\) and \(\tau^{\text{sc}}_{t,i}\) in both the forward and feedback channels of each agent are bounded, i.e., \(0\leqslant\tau^{\text{ca}}_{t,i}\leqslant\overline{\tau}^{\text{ca}}\) and \(0\leqslant\tau^{\text{sc}}_{t,i}\leqslant\overline{\tau}^{\text{sc}}\), where \(\overline{\tau}^{\text{ca}}\) and \(\overline{\tau}^{\text{sc}}\) are non-negative integers._  
An event-triggered networked predictive fault-tolerant control (ENPFTC) scheme is designed to reduce unnecessary data transmission in the feedback channel, and compensate for actuator faults and the random communication constraints in the feedback and forward channels, as shown in Figure 1.  
Event triggers use a static triggering mechanism with the following event triggering conditions\[T_{\varphi+1}=\min\{t|(y_{i}(t)-y_{i}(T_{\varphi}))^{\rm T}N_{i}(y_{i}(t)-y_{i}(T _{\varphi})) \tag{8}\] \[>\eta_{i}y_{i}(t)^{\rm T}N_{i}y_{i}(t)\},\]  
where \(T_{\varphi}\) is the trigger moment, \(\eta_{i}\) is a scalar parameter, and \(N_{i}\) is a matrix to be designed. In addition, to avoid event triggers going un-triggered for long periods of time, it is necessary to make \(T_{\varphi+1}\leq T_{\varphi}+W_{i}\), which \(W_{i}\) is the maximum trigger interval. Upon satisfying condition (8), the information of the agent is transmitted to the Kalman filter through the network, and then generates a set of control sequences by the predictive fault-tolerant controller, otherwise not transmitted as a way to conserve network resources. Furthermore, event triggers will introduce an additional delay which is defined as \(\tau^{\rm c}_{t,i}\), where \(\tau^{\rm c}_{t,i}\leq W_{i}\). The feedback channel's total delay of the agent is then denoted as \(\tau^{\rm cs}_{t,i}\), where \(\tau^{\rm cs}_{t,i}=\tau^{\rm cs}_{t,i}+\tau^{\rm c}_{t,i}\), and where \(0\leq\tau^{\rm cs}_{t,i}\leq\tau^{\rm sc}+W_{i}\). **Remark 1**. : _If trigger conditions are too lenient, delayed responses may occur. If too strict, resource consumption rises. By tuning \(\eta_{i}\) and \(N_{i}\) while monitoring response latency and trigger frequency, an optimal balance point can be identified, determining the parameter values of \(\eta_{i}\) and \(N_{i}\)._  
Predictive fault-tolerant controllers are designed in the control loop of each agent, event triggers are designed after the sensors, and delay compensators are designed in the actuators. Using the Kalman filter can estimate both the state and the actuator faults of the agent simultaneously  
\[ \begin{split}&\hat{x}_{t,i}(t-\tau^{\rm cs}_{t-1,i}+h_{i}|t- \tau^{\rm cs}_{t-1,i}+h_{i}-1)\\ =& A_{t,i}\hat{x}_{t,i}(t-\tau^{\rm cs}_{t-1,i}+h_{i }-1|t-\tau^{\rm cs}_{t-1,i}+h_{i}-2)\\ &+B_{t,i}u_{i}(t-\tau^{\rm cs}_{t-1,i}+h_{i}-1)+K_{i}(y_{i}(t- \tau^{\rm cs}_{t-1,i}+h_{i})\\ &-C_{t,i}(A_{t,i}\hat{x}_{t,i}(t-\tau^{\rm cs}_{t-1,i}+h_{i}-1|t- \tau^{\rm cs}_{t-1,i}+h_{i}-2)\\ &+B_{t,i}u_{i}(t-\tau^{\rm cs}_{t-1,i}+h_{i}-1))),\end{split}