# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems
## 3 Results
### State-Feedback-Based Distributed FD
For \(h=|\mathcal{N}_{i}|-1\), if all the residual generators will not converge to zero, there exists no healthy agent in \(\mathcal{N}_{i}\)._  
**Theorem 2**. : _If Assumption 1 is satisfied, \(c>0\), and each follower agent has paths to the leader in the graph \(\mathcal{G}\), there exists an observer (15) for any \(\tilde{i}_{k}\in\mathcal{N}_{i}\)._  
Proof of Theorem 2.: For \(\tilde{i}_{k}\in\mathcal{N}_{i}\), Lemma 2 indicates that there exists an observer (15) if \(rank(C_{i}\tilde{b}_{f_{k}})=rank(\tilde{b}_{f_{k}})\), and \(\mathcal{M}_{i_{\overline{i_{k}}}}= \begin{bmatrix}sI-A_{i}&\tilde{b}_{f_{i_{k}}} \\ C_{i}&0\end{bmatrix} \) is of full column rank for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\). Firstly, we verify the first condition, where  
\[C_{i}\bar{b}_{f_{k}}= \begin{bmatrix}I&-C_{i1}\end{bmatrix} \cdot \begin{bmatrix} \mathbf{e}_{k}\otimes B_{f_{k}}\\ 0\end{bmatrix} =\mathbf{e}_{k}\otimes B_{f_{k}} \tag{18}\]  
It is easy to verify that coulms of \(C_{i}\bar{b}_{f_{k}}\) are linearly independent, where the first condition holds. Then, we need to verify that \(\mathcal{M}_{i\bar{i}_{k}}\) is of full column rank for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\), which is shown as follows:  
\[ \begin{bmatrix}sI-\Omega_{1}&-\Omega_{2}&\mathbf{e}_{k}\otimes B_{f_{k}}\\ 0&sI-\left(I_{N}\otimes S-c\mathcal{H}\otimes I_{q}\right)&0\\ I&-C_{i1}&0\end{bmatrix}  \tag{19}\]  
Let \(\
u_{1}\in\mathbb{R}^{\sum_{k=1}^{|V_{i}|}n_{\bar{i}_{k}}},\
u_{2}\in\mathbb{ R}^{Nq}\) and \(\
u_{3}\in\mathbb{R}^{l_{k}}\) satisfy (20). \[ \begin{bmatrix}(sI-\Omega_{1})\cdot\
u_{1}-\Omega_{2}\cdot\
u_{2}+(\mathbf{e} _{k}\otimes B_{f_{k}})\cdot\
u_{3}\\ \left[sI-\left(I_{N}\otimes S-c\mathcal{H}\otimes I_{q}\right)\right]\cdot \
u_{2}\\ \
u_{1}-C_{i1}\cdot\
u_{2}\end{bmatrix} =0 \tag{20}\]  
\(\mathcal{H}\) is positive definite according to Lemma 1, then, there exists a \(Y_{1}\) to make \(Y_{1}\mathcal{H}Y_{1}^{-1}=\Lambda_{1}=\mathrm{diag}[\lambda_{1}(\mathcal{H}), \cdots,\lambda_{N}(\mathcal{H})]\), where \(\lambda_{1}(\mathcal{H}),\cdots,\lambda_{N}(\mathcal{H})>0\). For the exosystem matrix \(S\), a unitary matrix \(Y_{2}\) could make \(Y_{2}SY_{2}^{H}=\Lambda_{2}\), where \(\Lambda_{2}\) is an upper triangular matrix with diagonal elements \(\{\lambda_{1}(S),\cdots,\lambda_{q}(S)\}\)[29]. Then, we obtain eigenvelus of \(I_{N}\otimes S-c\mathcal{H}\otimes I_{q}\) via pre-multiplying by \((I_{N}\otimes Y_{2})\cdot(Y_{1}\otimes I_{q})\) and post-multiplying by \((Y_{1}^{-1}\otimes I_{q})\cdot(I_{N}\otimes Y_{2}^{H})\) for \(I_{N}\otimes S-c\mathcal{H}\otimes I_{q}\), which are shown as follows:  
\[\{\lambda_{i}(S)-c\lambda_{j}(\mathcal{H}):i=1,\cdots,q,j=1,\cdots,N\}\]  
According to Assumption 1, we have \(\mathrm{Re}[\lambda_{i}(S)]=0\), i.e., \(\mathrm{Re}[\lambda_{i}(I_{N}\otimes S-c\mathcal{H}\otimes I_{q})]<0\). Hence, columns of \(sI-(I_{N}\otimes S-c\mathcal{H}\otimes I_{q})\) are linearly independent for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\), i.e., \(\
u_{2}=0\), as well as \(\
u_{1}\). As columns of \(B_{f_{k}}\) are linearly independent, we have \(\
u_{3}=0\), therefore, (19) is of full column rank for \(\forall s\in\mathbb{C},\mathrm{Re}(s)\geq 0\). Above conditions guarantee existence of observers (15) for any node in \(\bar{\mathcal{N}}_{i}\). Remark 5: _If compensator state \(\eta_{i_{1}},\eta_{i_{2}},\cdots,\eta_{i_{|V_{i}|}}\) in \(y_{i}\) are replaced by exosystem state \(v\), existence of the observers for FD is still ensured. In practical operation, only partial follower agents could obtain exosystem state \(v\), but each follower agent has access to itself and its neighbours' compensator state, i.e., the estimate of \(v\), therefore, feedback information \(y_{i}\) is designed as (8)._  
Remark 6: _Establishment of Theorem 1 depends on Assumption 3 and existence of observers for FD, where existence of the observers for FD is proved in Theorem 2.