## 2 Control Scheme Design
 \tag{9}\]  
where \(h_{i}=0,1,\cdots,\tau^{\rm cs}_{t-1,i}-\tau^{\rm cs}_{t,i}\), \(\hat{x}_{t,i}(t-\tau^{\rm cs}_{t-1,i}+h_{i})=\hat{x}_{t,i}(t-\tau^{\rm cs}_{t-1,i}+h_{i}|t-\tau^{\rm cs}_{t-1,i}+h_{i}-1)\) is the estimate of the state and actuator faults, and \(K_{i}=\lim\limits_{t\to\infty}K_{i}(t)\) is the gain of the Kalman filter with  
\[ \begin{cases}K_{i}(t)=P_{i}(t|t-1)C^{\rm T}_{t,i}(C_{t,i}P_{i}(t|t-1)C^{\rm T }_{t,i}+R_{i})^{-1}\\ P_{i}(t|t-1)=A_{t,i}P_{i}(t-1|t-1)A^{\rm T}_{t,i}+\overline{Q_{i}}\\ P_{i}(t|t)=(I_{n_{1}}-K_{i}(t)C_{t,i})P_{i}(t|t-1),\end{cases}  \tag{10}\]  
where \(n_{1}=n+p\), \(\overline{Q_{i}}=\text{diag}\{Q_{i},Q_{Ri}\}\) and \(Q_{Ri}\in\mathfrak{R}^{p\times p}>0\). When \(t-\tau^{\rm cs}_{t,i}\leq t-\tau^{\rm cs}_{t-1,i}-1\), the controller does not receive updated output data of the agent, and the estimate state and the actuator fault of the agent remain the same as the previous moment  
Figure 1: ENPFTC scheme. \[\hat{x}_{t,i}(t-\tau_{t,i}^{\text{es}}|t-\tau_{t,i}^{\text{es}}-1)=\hat{x}_{t,i}(t- \tau_{t-1,i}^{\text{es}}-1|t-\tau_{t-1,i}^{\text{es}}-2). \tag{11}\]  
Combining the state estimate of agent \(i\) and (4), the error and state prediction and output incremental prediction for agent \(i\) can be obtained as follows  
\[ \begin{split}&\hat{x}_{\text{e},1}(t-\tau_{t,1}^{\text{es}}+h_{1}|t -\tau_{t,1}^{\text{es}}-1)\\ &=A_{\text{e},1}\hat{x}_{\text{e},1}(t-\tau_{t,1}^{\text{es}}+h_{ 1}-1|t-\tau_{t,1}^{\text{es}}-1)\\ &+B_{\text{e},1}\Delta u_{1}(t-\tau_{t,1}^{\text{es}}+h_{1}-1)\\ &+F_{\text{e},1}\Delta\hat{f}_{1}(t-\tau_{t,1}^{\text{es}}|t- \tau_{t,1}^{\text{es}}-1)\\ &+E_{\text{e},i}\Delta r(t-\tau_{t,1}^{\text{es}}+h_{1}),\end{split}  \tag{12}\]  
\[ \begin{split}&\hat{x}_{\text{e},i}(t-\tau_{t,i}^{\text{es}}+h_{i}|t -\tau_{t,i}^{\text{es}}-1)\\ =& A_{\text{e},i}\hat{x}_{\text{e},i}(t-\tau_{t,i}^{ \text{es}}+h_{i}-1|t-\tau_{t,i}^{\text{es}}-1)\\ &+B_{\text{e},i}\Delta u_{i}(t-\tau_{t,i}^{\text{es}}+h_{i}-1)\\ &+F_{\text{e},i}\Delta\hat{f}_{1}(t-\tau_{t,i}^{\text{es}}|t- \tau_{t,i}^{\text{es}}-1)\\ &+E_{\text{e},i}\Delta\hat{y}_{1}(t-\tau_{t,i}^{\text{es}}+h_{i} |t-\tau_{t,i}^{\text{es}}),i\
eq 1,\end{split}  \tag{13}\]  
\[\Delta\hat{y}_{\text{e},i}(t-\tau_{t,i}^{\text{es}}+h_{i}|t-\tau_{t,i}^{\text {es}})=C_{\text{e},i}\hat{x}_{\text{e},i}(t-\tau_{t,i}^{\text{es}}+h_{i}|t- \tau_{t,i}^{\text{es}}-1), \tag{14}\]  
where \(h_{i}=1,2,\cdots,\tau_{t,i}^{\text{es}}+\overline{\tau}^{\text{ca}}\), \(\hat{x}_{\text{e},i}(t-\tau_{t,i}^{\text{es}}+h_{i}|t-\tau_{t,i}^{\text{es}} -1)=[\Delta\hat{x}_{i}^{\text{T}}(t-\tau_{t,i}^{\text{es}}+h_{i}|t-\tau_{t,i} ^{\text{es}}-1),\hat{x}_{i}^{\text{T}}(t-\tau_{t,i}^{\text{es}}+h_{i}|t-\tau_ {t,i}^{\text{es}})]^{\text{T}}\), \(\Delta\hat{f}_{i}(t-\tau_{t,i}^{\text{es}}|t-\tau_{t,i}^{\text{es}}-1)=I_{\text {x}}\Delta\hat{x}_{t,i}(t-\tau_{t,i}^{\text{es}}|t-\tau_{t,i}^{\text{es}}-1)\), \(I_{\text{x}}=[ \begin{array}{c}0\quad I\end{array} ]\). In order to realize the output tracking control of the system, the incremental control law of agent \(i\) is designed by combining (12)-(14) as follows  
\[ \begin{split}&\Delta u_{i}(t+\overline{\tau}^{\text{ca}})\\ =&-L_{i}^{\text{a}}\hat{x}_{\text{e},i}(t+\overline{\tau}^{\text{ca}}|t- \tau_{t,i}^{\text{es}}-1)-B_{\text{e},i}^{+}F_{\text{e},i}\Delta\hat{f}_{i}(t -\tau_{t,i}^{\text{es}}|t-\tau_{t,i}^{\text{es}}-1)\\ &+L_{i}^{\text{b}}\sum_{\begin{subarray}{c}j=1\\ j\
eq i\end{subarray}}^{N}a_{ij}(\hat{y}_{j}(t+\overline{\tau}^{\text{ca}}|t- \tau_{t,j}^{\text{es}})-\Delta\hat{y}_{i}(t+\overline{\tau}^{\text{ca}}|t- \tau_{t,j}^{\text{sc}})),\end{split}  \tag{15}\]  
\[u_{i}(t+\overline{\tau}^{\text{ca}})=u_{i}(t+\overline{\tau}^{\text{ca}}-1)+ \Delta u_{i}(t+\overline{\tau}^{\text{ca}}), \tag{16}\]  
where \(L_{i}^{\text{a}}\) and \(L_{i}^{\text{b}}\) are the controller to be determined, \(B_{\text{e},i}^{+}\) is the generalized inverse of \(B_{\text{e},i}\).