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A novel distributed fault detection and isolation (FDI) methodology for a network of nonhomogeneous multi-agent systems is proposed in this paper. An FDI filter is designed such that the effects of the disturbances and control inputs on the residual signals are minimized (for the purpose of fault detection) subject to the constraint that the transfer matrix function from the faults to the residuals is equal to a pre-assigned diagonal transfer matrix (for the purpose of fault isolation). It is shown that by using this approach each agent not only can detect and isolate its own faults but also is able to detect and isolate faults of its neighboring agents. Moreover, the method covers isolation of possibly simultaneous occurring faults in the system. Sufficient conditions for solvability of the problem are obtained in terms of linear matrix inequality (LMI) feasibility conditions. Extended LMIs characterization is used to reduce the conservativeness of the solution by introducing additional variables, to eliminate the coupling of the Lyapunov matrices with the system matrices. Simulation results illustrate the effectiveness of the proposed design methodology.