Abstract: The sufficient conditions for the existence of the full-order and the reduced-order observers about Lipschitz nonlinear systems are given and proved.At the same time,a new form of reduced-order state observer is designed.Based on the solving of linear matrix inequality(LMI),the gain matrix of observer can be obtained completely and be easily designed by LMI tool box.The approach avoids the blindness in selecting the gain matrix.The simulation result of a practical model shows that the two kinds of state estimation errors asymptotically converge to zero.It indicates that the approach is efficient.