## 2 Control Scheme Design  
The topology between agents can be represented by a directed graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), where \(\mathcal{V}=\{v_{1},v_{2},\cdots,v_{N}\}\) and \(\mathcal{E}=\{(v_{i},v_{j}):v_{i},v_{j}\in\mathcal{V}\}\) denote the vertex set and the edge set, respectively. \(\mathcal{A}=[a_{ij}]\) represents a non-negative adjacency matrix with \(a_{ii}=0\). Consider a second-order discrete-time NMAS with a virtual leader and \(N\) followers  
\[ \begin{cases}x_{i}(t+1)=A_{i}x_{i}(t)+B_{i}u_{i}(t)+F_{i}f_{i}(t)+\omega_{i}(t) \\ y_{i}(t)=C_{i}x_{i}(t)+v_{i}(t),\end{cases}  \tag{1}\]  
where \(x_{i}\in\mathfrak{R}^{n},u_{i}\in\mathfrak{R}^{m},f_{i}\in\mathfrak{R}^{q},y_ {i}\in\mathfrak{R}^{p}\) are the state, the input, the actuator fault and the system output of agent \(i\), \(i=1,2,\cdots,N\), respectively. The system matrices are represented by \(A_{i},\ B_{i},\ C_{i}\) and \(F_{i}\) which have the corresponding dimension. Suppose that \((A_{i},B_{i})\) is controllable, and \((A_{i},C_{i})\) is observable. \(\omega_{i}(t)\sim N(0,Q_{i})\) and \(v_{i}(t)\sim N(0,R_{i})\) are process noise and measurement noise, respectively. The purpose of the designed control method is to allow the leader to track the reference signal \(r(t)\), which provides a common benchmark for all agents. Furthermore, to enable the follower to track the leader and maintain the desired output tracking error, the tracking error of agent \(i\) is defined as  
\[e_{i}(t)= \begin{cases}r(t)-y_{i}(t),&i=1\\ y_{i}(t)-y_{1}(t),&i\
eq 1,\end{cases}  \tag{2}\]  
where \(e_{1}(t)\) denotes the error of the leader in tracking \(r(t)\), and \(e_{i}(t)(i\
eq 1)\) denotes the tracking error of the follower in tracking the leader. From (1) and (2), it can be deduced that the output tracking error of agent \(i\) has the following relationship  
\[e_{i}(t+1)= \begin{cases}e_{i}(t)+\Delta r(t+1)-C_{i}A_{i}\Delta x_{i}(t)\\ -C_{i}B_{i}\Delta u_{i}(t)-C_{i}F_{i}\Delta f_{i}(t)\\ -C_{i}\Delta\omega_{i}(t)-\Delta v_{i}(t+1),&i=1\\ e_{i}(t)+\Delta y_{1}(t+1)-C_{i}A_{i}\Delta x_{i}(t)\\ -C_{i}B_{i}\Delta u_{i}(t)-C_{i}F_{i}\Delta f_{i}(t)\\ -C_{i}\Delta\omega_{i}(t)-\Delta v_{i}(t+1),&i\
eq 1,\end{cases}  \tag{3}\]where the symbol \(\Delta\) represents the forward difference. For example, \(\Delta r(t+1)=r(t+1)-r(t)\). Then, combining (1) and (3), the following tracking error augmentation system can be obtained  
\[x_{\text{e},i}(t+1)= \begin{cases}A_{\text{e},i}x_{\text{e},i}(t)+B_{\text{e},i }\Delta u_{i}(t)\\ +F_{\text{e},i}\Delta f_{i}(t)+E_{\text{e},i}\Delta r(t+1)\\ +I_{\text{e},i}\Delta\omega_{i}(t)-E_{\text{e},i}\Delta v_{i}(t+1),\quad i=1\\ A_{\text{e},i}x_{\text{e},i}(t)+B_{\text{e},i}\Delta u_{i}(t)\\ +F_{\text{e},i}\Delta f_{i}(t)+E_{\text{e},i}\Delta y_{1}(t+1)\\ +I_{\text{e},i}\Delta\omega_{i}(t)-E_{\text{e},i}\Delta v_{i}(t+1),\quad i\
eq 1 \end{cases}  \tag{4}\]  
\[\Delta y_{i}(t)=C_{\text{e},i}x_{\text{e},i}(t), \tag{5}\]  
where \(x_{\text{e},i}(t)=\left[ \begin{array}{c}\Delta x_{i}(t)\\ e_{i}(t)\end{array} \right]\in\mathfrak{R}^{n+p}\), \(A_{\text{e},i}=\left[ \begin{array}{cc}A_{i}&0\\ -C_{i}A_{i}&I\end{array} \right]\), \(B_{\text{e},i}=\left[ \begin{array}{cc}B_{i}\\ -C_{i}B_{i}\end{array} \right]\), \(C_{\text{e},i}=\left[ \begin{array}{cc}C_{i}&0\end{array} \right]\), \(E_{\text{e},i}=\left[ \begin{array}{c}0\\ I\end{array} \right]\), \(F_{\text{e},i}=\left[ \begin{array}{c}F_{i}\\ -C_{i}F_{i}\end{array} \right]\), \(I_{\text{e},i}=\left[ \begin{array}{c}I\\ -C_{i}\end{array} \right]\). \(I\) represent the identity matrix with suitable dimensions, and similarly, 0 is a zero matrix with appropriate dimensions. When the NMAS (1) is in a steady state, the tracking error broadening system exists with \(\lim_{t\rightarrow\infty}x_{\text{e},i}(t)=0\). Therefore, the tracking control issue can be reframed as a common asymptotic stabilization problem.