# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems
## 3 Results
### Output-Feedback-Based Distributed FD
Furthermore, for \(k=2,\cdots,|\bar{\mathcal{N}_{i}}|\), columns of \(C_{i}\bar{b}_{f_{i_{k}}}\) are always linearly independent, where the first condition holds. Then, we need to verify that \(\mathcal{M}_{ik}\) is of full column rank for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\), which is shown as follows:  
\[ \begin{bmatrix}sI-\Omega_{11i}&-\Omega_{12i}&-\Omega_{13i}&\mathbf{e}_{k} \otimes B_{f_{\tilde{t}_{k}}}\\ 0&sI-\Omega_{22i}&-\Omega_{23i}&\mathbf{e}_{k}\otimes B_{f_{\tilde{t}_{k}}}\\ 0&0&sI-(I_{N}\otimes S-c\mathcal{H}\otimes I_{q})&0\\ 0&\tilde{C}_{i}&0&0\end{bmatrix}  \tag{31}\]  
Consider (31), let \(\
u_{1},\
u_{2}\in\mathbb{R}^{\sum_{k=1}^{|\bar{\mathcal{N}_{i}}|}n_{\tilde{t} _{k}}},\
u_{3}\in\mathbb{R}^{Nq}\) and \(\
u_{4}\in\mathbb{R}^{k}\) satisfy (32). \[ \begin{bmatrix}(sI-\Omega_{11i})\cdot\
u_{1}-\Omega_{12i}\cdot\
u_{2}-\Omega_{ 13i}\cdot\
u_{3}+(\mathbf{e}_{k}\otimes B_{f_{\tilde{t}_{k}}})\cdot\
u_{4}\\ (sI-\Omega_{22i})\cdot\
u_{2}-\Omega_{23i}\cdot\
u_{3}+(\mathbf{e}_{k}\otimes B _{f_{\tilde{t}_{k}}})\cdot\
u_{4}\\ [sI-(I_{N}\otimes S-c\mathcal{H}\otimes I_{q})]\cdot\
u_{3}\\ \bar{C}_{i}\cdot\
u_{2}\end{bmatrix} =0 \tag{32}\]It has been proved in Theorem 2 that columns of \(sI-(I_{N}\otimes S-c\mathcal{H}\otimes I_{q})\) are linearly independent for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\), we have \(v_{3}=0\). Therefore, \(v_{2}=v_{4}=0\) if (33) is of full column rank for \(\forall s\in\mathbb{C},\mathrm{Re}(s)\geq 0\),  
\[ \begin{bmatrix}sI-\Omega_{22i}&\mathbf{e}_{k}\otimes B_{f_{i_{k}}}\\ \bar{C}_{i}&0\end{bmatrix}  \tag{33}\]  
where (33) could be transformed to (34) through appropriate row and column transformation, the rank of which is equal to (33),  
\[ \begin{bmatrix}sI-\mathrm{diag}(A_{i_{1}},\cdots,A_{i_{|i_{|i_{|i}}|}})& \mathbf{e}_{k}\otimes B_{f_{i_{k}}}\\ \mathrm{diag}(C_{m_{i_{1}}^{2}},\cdots,C_{m_{|i_{|i_{|i}}|}^{2}})&0\end{bmatrix}  \tag{34}\]  
according to Assumption 4, (34) is of full column rank for \(\forall s\in\mathbb{C},\mathrm{Re}(s)\geq 0\), i.e., \(v_{2}=v_{4}=0\). As \(\Omega_{11i}\) is Hurwitz stable, \(v_{1}=0\). Hence, (31) is of full column rank for \(\forall s\in\mathbb{C},\mathrm{Re}(s)\geq 0\). Finally, existence of observers (29) for any node in \(\mathcal{N}_{i}\) is established. **Remark 11**. : _In comparison with the output-feedback-based COR problem in Reference [4], where the authors design two control laws, which correspond to two kinds of follower agents in the MAS, the first kind are called informed agents, i.e., measurement output \(y_{mi}\) contains exogenous signal \(v\), then \(v\) is detectable from \(y_{mi}\). The second kind are called uninformed agents, where measurement output \(y_{mi}\) does not contain exogenous signal \(v\). As Section 2 just considers that all the follower agents are uninformed, i.e., a special case of literature [2], a single control law (21) will realise COR if no fault occurs, where \(c>0\), and the designed \(K_{1i}\) and \(L_{i}\) make \(A_{i}+B_{i}K_{1i}\) and \(A_{i}+L_{i}C_{mi}\) Hurwitz stable, which is not contradicted with the existence conditions of observers (29), i.e., control law (21) and observers (29) could run simultaneously._  
**Remark 12**. : _The faulty agent location algorithm for the output-feedback-based distributed FD scheme is the same as Algorithm 1 and omitted here._