The linear matrix inequality (LMI) based residual generator design approach is presented in this paper. Design conditions are expressed in the terms of LMIs with the matrix rank constraints, implying from Lyapunov equation, which correspond to a feasible solution. Obtained formulation is the convex LMI problem for the full order residual generator design. This handles the optimized structure of the residual generator for the discrete-time linear systems. Given method is demonstrated using a structural system model example, which includes the actuator and sensor fault vectors, respectively.