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The output-feedback stabilisation problem is solved for discrete-time systems with time-varying delay in the state. A stability condition is first proposed, which is dependent on the minimum and maximum delay bounds. Based on this easily verifiable stability condition, the problems of stabilisation by static and dynamic output-feedback controllers are solved within the linear matrix inequality (LMI) framework. Since the obtained conditions for the existence of admissible controllers are not expressed as strict LMI conditions, the cone complementary linearisation procedure is exploited to solve the nonconvex feasibility problem. In addition, the obtained results, including stability analysis, static output-feedback stabilisation and dynamic output-feedback stabilisation are further extended to discrete time-delay systems with norm-bounded uncertain parameters. Numerical examples are also presented to illustrate the applicability of the developed results.