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This paper presents a design method of a discrete-time reduced-order observer using the duality to discrete-time sliding-surface design. First, the duality between the coefficients of the discrete-time reduced-order observer and the sliding-surface design is established, and then, the design method for the observer using the Riccati equation is explained. A discrete-time sliding-mode controller based on the proposed observer is designed and tested on a laboratory-type experimental servo system. The results show the efficacy of the reduced-order observer designed by the duality concept.