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The one-sided Lipschitz non-linear system is a generalisation of its well-known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. In this study, the authors deal with the problem of observer design for one-sided Lipschitz non-linear systems by using the linear matrix inequality (LMI) approach. Sufficient conditions that ensure the existence of observers for one-sided Lipschitz non-linear systems are established and expressed in terms of linear matrix inequalities (LMIs), which are easily and numerically tractable via standard software algorithms. It is shown that the proposed conditions are less conservative and more simpler than some existing results in recent literature. Simulation results on two examples are given to illustrate the effectiveness and advantages of the proposed design.