Based on a new special co-inner-outer factorization, a factorization approach for design fault detection observer for LSFDJ was proposed. It is a simple state-space method and can deal with time-varying LSFDJ with sensor noise and sensor faults. The performance of the fault detection observer is optimized in an H∞ setting,where the ratio between the gains from disturbance and fault to residual respectively is minimized. The design parameters of the detection observer were given in terms of the solution to the Riccati differential equation with jumps.