# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems
## 1 Introduction
[PERSON] and [PERSON] proposed a distributed anti-disturbance FD scheme for disturbed second-order MAS, where the influence of exogenous disturbances for FD results was actively rejected [21]. Besides MAS with double-integrator dynamics, [PERSON] et al. considered linear MAS with Lipschitz nonlinearity [22], and adopted a reduced-order observer design to lower the computational burden of each agent [23]. To the best of the authors' knowledge, there has been little published research on distributed FD schemes for heterogeneous MAS. [PERSON] et al. designed a distributed FD scheme for heterogeneous MAS [24], where the observer in one agent judged whether a fault occurred in itself and its neighbours or not. However, the above observer was not decoupled from control inputs, which meant that the FD results would be influenced by control inputs. In [25], the authors considered a heterogeneous MAS with sensor faults, where fault estimators were designed in each agent to detect the fault. However, the considered agent dynamics did not contain exogenous disturbances, which were assumed to affect each agent and unknown for most agents in the MAS [3; 4; 5; 6], and it seemed costly that all agents were equipped with fault estimators. Motivated by existing works mentioned above, we design a distributed FD scheme for disturbed MAS with heterogeneous dynamics based on UIO. As the exogenous disturbance is unknown for most agents of the MAS [3; 4; 5; 6], above agents need to estimate the exogenous disturbance through cooperating with its neighbours, therefore, a bank of UIOs are designed according to the formed closed-loop system with the information of communication topology, which utilise exogenous disturbance estimate as feedback information, i.e., avoids utilising unknown exogenous disturbance. The contributions of this paper are listed as follows: First, a state-feedback-based distributed FD scheme for heterogeneous MAS is proposed in this paper, as well as an output-feedback-based distributed FD scheme. Second, unlike [24], the FD results will not be influenced by agents' control inputs, since FD observers are designed for closed-loop systems. Third, in comparison with [25], the considered agent dynamics contain exogenous disturbances in this paper. FD observers could still be designed for those agents, which cannot obtain exogenous signal directly, and only partial agents need to be equipped with observers. The rest of this paper is organised as follows: Section 2 gives the preliminaries and problem formulation, where Sections 2.1 and 2.2 introduce relevant knowledge of graph theory and UIO, and Section 2.3 gives the problem formulation. Section 3 gives results of this paper, state-feedback-based and output-feedback-based distributed FD schemes are proposed in Sections 3.1 and 3.2. Section 4 gives two simulation examples to verify that the proposed FD schemes are effective. Finally, Section 5 gives conclusions and future directions. _Notation:_ Some standard notation will be adopted in this paper. \(\mathbb{C}\) and \(\mathbb{R}^{n}\) denote the set of complex numbers and \(n\)-dimensional Euclidean space, respectively. \(I_{N}\) and \(I\) denote an identity matrix with dimension \(N\) and appropriate dimension, respectively. \(\otimes\) denotes the Kronecker product. \(\mathbf{e}_{i}\) represents a column with only one nonzero entry '1', which locates in the \(i\)-th row. \(\mathrm{Re}(\zeta)\) represents the real part of \(\zeta\), where \(\zeta\in\mathbb{C}\). \(\mathrm{diag}(A_{1},A_{2},\cdots,A_{N})\) represents a block-diagonal matrix with matrices \(A_{i}\), \(i=1,2,\cdots,N\). \(\lambda_{i}(A)\) denotes the \(i\)-th eigenvalue of \(A\). \(A>0\) means that \(A\) is positive definite. '\(\dagger\)' denotes the factorial of a non-negative integer, and \(C_{a}^{b}=\frac{a!}{(a-b)!b! }\), where \(a,b\) are non-negative integers, and \(b\leq a\). The superscript '\(T\)' represents the transpose of a matrix.