Abstarct: In this thesis, an optimal solution H_-⁄H_∞ to the problem of fault detection in non-linear time invariant systems is presented. First, the basic concepts of fault and fault types are defined. Then a formulated performance index baxsed on H_-⁄H_∞ is presented to minimize the effects of disturbance, and maximize the fault impact on the residual fault signal. Then an optimal filter to solve the fault problem by H_-⁄H_∞ method, via optimized LMI problem solving method is designed for tank reactor systems CSTR. Afterwards the simulation results are compared with other methods baxsed on the observer such as Lunberger observers, H_2 ,H_∞ and H_2⁄H_∞ .