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This study focuses on the observer-based robust stabilisation problem of a class of non-linear fractional-order uncertain systems with admissible time-variant uncertainty and bounded perturbation on the observer gain. A sufficient condition for the robust asymptotic stability of the observer-based non-linear fractional-order uncertain systems is presented based on direct Lyapunov approach. Utilising the matrix's singular value decomposition method (SVD), the systematic robust stabilisation design algorithm is then proposed. All the results are obtained in terms of linear matrix inequalities. A numerical example is provided to illustrate the efficiency and advantage of the proposed algorithm.