# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems
## 3 Results
### Output-Feedback-Based Distributed FD
\begin{split}&\dot{\psi}_{i}=A_{i}\psi_{i}+\bar{B}_{f_{i}}\tilde{ f}_{i}\\ & A_{i}=\begin{bmatrix}\Omega_{11i}&\Omega_{12i}&\Omega_{13i}\\ 0&\Omega_{22i}&\Omega_{23i}\\ 0&0&I_{N}\otimes S-cH\otimes I_{q}\end{bmatrix},\bar{B}_{f_{i}}=\begin{bmatrix} \Omega_{3i}\\ \Omega_{3i}\\ 0\end{bmatrix}\\ &\Omega_{11i}=\text{diag}(A_{\bar{i}_{1}}+B_{\bar{i}_{1}}K_{1\bar{i}_{1}}, \cdots,A_{\bar{i}_{|\mathcal{K}_{i}|}}+B_{\bar{i}_{|\mathcal{K}_{i}|}}K_{1\bar{ i}_{|\mathcal{K}_{i}|}})\\ &\Omega_{12i}=\text{diag}(-B_{\bar{i}_{1}}K_{1\bar{i}_{1}},\cdots,-B_{\bar{i}_{| \mathcal{K}_{i}|}}K_{1\bar{i}_{|\mathcal{K}_{i}|}})\\ &\Omega_{13i}=\left[\mathbf{e}_{\bar{i}_{1}}\otimes(B_{\bar{i}_{1}}K_{2\bar{ i}_{1}})^{T},\cdots,\mathbf{e}_{\bar{i}_{|\mathcal{K}_{i}|}}\otimes(B_{\bar{i}_{| \mathcal{K}_{i}|}}K_{2\bar{i}_{|\mathcal{K}_{i}|}})^{T}\right]^{T}\\ &\Omega_{22i}=\text{diag}(A_{\bar{i}_{1}}+L_{\bar{i}_{1}}C_{m\bar{i}_{1}}, \cdots,A_{\bar{i}_{|\mathcal{K}_{i}|}}+L_{\bar{i}_{|\mathcal{K}_{i}|}}C_{m\bar{ i}_{|\mathcal{K}_{i}|}})\\ &\Omega_{23i}=\left[-\mathbf{e}_{\bar{i}_{1}}\otimes E_{\bar{i}_{1}}^{T}, \cdots,-\mathbf{e}_{\bar{i}_{|\mathcal{K}_{i}|}}\otimes E_{\bar{i}_{|\mathcal{K }_{i}|}}^{T}\right]^{T}\\ &\Omega_{3i}=\text{diag}(B_{\bar{f}_{1}},\cdots,B_{\bar{f}_{|\mathcal{K}_{i}| }})\end{split}  \tag{26}\]  
and (24) could be changed to the following form:  
\[ \begin{split}& y_{i}=C_{i}\dot{\psi}_{i}\\ & C_{i}=\begin{bmatrix}0&\bar{C}_{i}&0\end{bmatrix},\bar{C}_{i}= \text{diag}(C_{m\bar{i}_{1}},\cdots,C_{m\bar{i}_{|\mathcal{K}_{i}|}})\end{split}  \tag{27}\]Combine (26) and (27), we obtain \(|\bar{\mathcal{N}_{i}}|\) systems as follows:  
\[\left\{ \begin{array}{l}\dot{\psi}_{i}=A_{i}\psi_{i}+\bar{b}_{f_{i_{k}}}f_{i_{k} }+\bar{B}_{f_{-i_{k}}}\bar{f}_{-i_{k}}\\ y_{i}=C_{i}\psi_{i},k=1,\cdots,|\bar{\mathcal{N}_{i}}|\end{array} \right. \tag{28}\]  
where \(\bar{b}_{f_{i_{k}}}= \begin{bmatrix}\mathbf{e}_{k}^{T}\otimes B_{f_{i_{k}}}^{T }&\mathbf{e}_{k}^{T}\otimes B_{f_{i_{k}}}^{T}&0\end{bmatrix} ^{T}\), \(\bar{B}_{f_{-i_{k}}}\) is the component of \(\bar{B}_{f_{i}}\) after deleting \(\bar{b}_{f_{i_{k}}}\). Follow the same steps in Section 2.1, \(|\bar{\mathcal{N}_{i}}|\) observers are designed in agent \(i\). \[ \begin{array}{l}\dot{z}_{i_{k}}=G_{i_{k}}z_{i_{k}}+R_{i_{k}}y_{i}\\ \dot{\bar{w}}_{i_{k}}=z_{i_{k}}+Q_{i_{k}}y_{i},k=1,\cdots,|\bar{\mathcal{N}_{ i}}|\end{array}  \tag{29}\]  
where the design of parameter matrices is the same as (16). Next we just need to prove existence of observers for FD. **Remark 10**. : _In comparison with \(y_{i}\) in (8), (22) contains measurement output and state estimate of one agent's and its neighbours'. In the following, Theorem 3 will prove that existence of the above observers is still ensured, where \(y_{i}\) does not contain state information. The objective proposed in Section 1 will realise under the condition that state information is unmeasurable._  
**Theorem 3**. : _If Assumptions 1 and 4 are satisfied, \(c>0\), \(A_{i_{k}}+B_{i_{k}}K_{i_{k}}\) are Hurwitz stable, \(k=1,\cdots,|\bar{\mathcal{N}_{i}}|\), and each follower agent has paths to the leader in the graph \(\mathcal{G}\), there exists an observer (29) for any agent \(\tilde{t}_{k}\in\bar{\mathcal{N}_{i}}\)._  
Proof of Theorem 3.: For \(\tilde{t}_{k}\in\bar{\mathcal{N}_{i}}\), Lemma 2 indicates that there exists an observer (29) if \(rank(C_{i}\bar{b}_{f_{i_{k}}})=rank(\bar{b}_{f_{i_{k}}})\), and \(\mathcal{M}_{ik}= \begin{bmatrix}sI-A_{i}&\bar{b}_{f_{i_{k}}}\\ C_{i}&0\end{bmatrix} \) is of full column rank for \(\forall s\in\mathbb{C}\), \(\mathrm{Re}(s)\geq 0\). Firstly, we verify the first condition, for \(k=1\), we have  
\[C_{i}\bar{b}_{f_{i_{1}}}= \begin{bmatrix}0&\mathrm{diag}(C_{m\tilde{t}_{1}}, \cdots,C_{m\tilde{t}_{|\bar{\mathcal{N}_{i}}|}})&0\end{bmatrix} \cdot \begin{bmatrix} \mathbf{e}_{1}\otimes B_{f_{\tilde{t}_{1}}}\\ \mathbf{e}_{1}\otimes B_{f_{\tilde{t}_{1}}}\\ 0\end{bmatrix} =C_{m\tilde{t}_{1}}B_{f_{\tilde{t}_{1}}} \tag{30}\]  
where \(C_{m\tilde{t}_{1}}B_{f_{\tilde{t}_{1}}}\) is of full column rank according to Assumption 4.