The problem of designing Luenberger-like observer for nonlinear Lipschitz systems is addressed. Based on using the coordinate transformation, we classify many practical approaches of observer synthesis into direct and indirect methods. A generalized Lipschitz condition and the related stability condition are then employed to establish a general framework in designing the Luenberger-like observer dealing with both methods. Moreover, the proposed framework is interpreted in terms of some linear matrix inequalities which allow using numerical techniques to obtain a reduced order observer. The results are then extended to the adaptive state observer for nonlinear systems with some unknown parameters. The simulation examples finally demonstrate the effectiveness of the proposed synthesis methods. Key Words Nonlinear Lipschitz system, Luenberger-like observer, General framework, Linear Matrix Inequality (LMI), adaptive state observer
