# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems
## 2 Preliminaries and Problem Formulation
### Problem Formulation  
The considered heterogeneous MAS consists of the following \(N\) agents:  
\[ \begin{split}\dot{x}_{i}&=A_{i}x_{i}+B_{i}u_{i}+E_{i}v+B _{f_{i}}f_{i}\\ e_{i}&=C_{i}x_{i}+D_{i}u_{i}+F_{i}v\\ y_{mi}&=C_{mi}x_{i},i=1,2,\cdots,N\end{split}  \tag{5}\]  
where \(x_{i}\in\mathbb{R}^{n_{i}}\), \(e_{i}\in\mathbb{R}^{p_{i}}\), \(y_{mi}\in\mathbb{R}^{p_{m}}\), \(u_{i}\in\mathbb{R}^{m_{i}}\) and \(f_{i}\in\mathbb{R}^{l_{i}}\) are the state, tracking error, measurement output, control input and fault signal of the \(i\)-th agent, respectively. Columns of \(B_{f_{i}}\) are linearly independent, and \((A_{i},B_{i},C_{mi})\) are stabilisable and detectable. \(v\in\mathbb{R}^{q}\) is the reference input to be tracked or the disturbance to be rejected, which is assumed to be generated by the following exosystem:  
\[\hat{v}=Sv\]  
In this paper, the leader labelled as \(0\) could represent the exosystem. System (5) could be seen as follower agents, where only some follower agents of the MAS could utilise \(v\) directly, i.e., the leader is their neighbour in the graph \(\mathcal{G}\), and the remaining follower agents just have paths to the leader. The above two facts mean that the leader has paths to all follower agents in the graph \(\mathcal{G}\). System (5) also needs to satisfy the following assumptions. **Assumption 1**. : \(Re[\lambda_{i}(S)]=0\)_, \(i=1,\cdots,q\)._  
**Assumption 2**. : _The following equations_  
\[ \begin{split}& X_{i}S=A_{i}X_{i}+B_{i}U_{i}+E_{i}\\ & 0=C_{i}X_{i}+D_{i}U_{i}+F_{i},i=1,2,\cdots,N\end{split}  \tag{6}\]  
_have solution pairs \((X_{i},U_{i})\), respectively._  
**Remark 2**. : _According to Theorem 1.9 in [27], solution pairs \((X_{i},U_{i})\) are determined to exist if \(rank \begin{bmatrix}\lambda_{i}(S)I-A_{i}&B_{i}\\ C_{i}&D_{i}\end{bmatrix} =n_{i}+p_{i}\) for any \(\lambda_{i}(S)\). In fact, \( \begin{bmatrix}sI-A_{i}&B_{i}\\ C_{i}&D_{i}\end{bmatrix} \) is of full row rank for \(\forall s\in\mathbb{C},Re(s)\geq 0\) if System (5) are minimum phase systems, combined with Assumption 1, matrix Equation (6) will always have solutions if System (5) are minimum phase systems. Furthermore, the solution pair \((X_{i},U_{i})\) will be unique if \(p_{i}=m_{i}\)._  
**Assumption 3**. : _There exists only one faulty agent in each \(\bar{\mathcal{N}}_{i}\), \(i=1,\cdots,N\)._  
For System (5), a state-feedback-based distributed control law is proposed in [1]. \[ \begin{split} u_{i}&=K_{1i}x_{i}+K_{2i}\eta_{i}\\ \dot{\eta}_{i}&=S\eta_{i}-c\sum_{j=1}^{N}[a_{ij}(\eta_{i}- \eta_{j})+a_{i0}(\eta_{i}-v)]\end{split}  \tag{7}\]  
where \(\eta_{i}\) is a dynamic compensator, \(c\) is a positive scalar to design, \(K_{1i}\) and \(K_{2i}\) are parameter matrices to design. Let \(K_{2i}\) be as follows:  
\[K_{2i}=U_{i}-K_{1i}X_{i}\]  
\(X_{i}\) and \(U_{i}\) are determined by matrix equations in Assumption 2. If \(A_{i}+B_{i}K_{1i}\) are Hurwitz stable, \(c>0\), and the leader has paths to all follower agents, control law (7) will realise COR when there exists no faulty agent [3], i.e., tracking errors \(e_{i}\) converge to zero asymptotically. However, the given exosystem under Assumption 1 just has an unforced purely oscillatory solution, which cannot include some kinds of solutions such as those of a damped differential system or the solution of a forced differential system or dynamic system, i.e., some kinds of practical signals cannot be generated by the given exosystem, where the application prospect is limited. This is due to the fact that the given exosystem needs to ensure that matrix Equation (6) has solutions, and the following Theorems 2 and 3 hold. Furthermore, COR will not realise if some agent suffers from fault signals. Hence, there exists a need to detect the faulty agent. This paper aims at adopting a distributed control law for the MAS, and designing distributed observers in some agents to detect the possibly faulty agent.