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This paper addresses the problems of discrete-time state and unknown input/fault estimation for continuous-time nonlinear systems with multiple unknown inputs. Taylor series expansion and a nonlinear transformation are used to convert the nonlinear continuous-time system into a discrete-time model. The conditions for the observability of unknown inputs w.r.t. outputs are discussed. The novelty lies in the formulation of multiple sliding-mode estimator for the states that are directly influenced by unknown inputs, which cannot be decoupled by nonlinear transformation. This framework allows for the estimation of unknown inputs from the multiple sliding modes. The existence of discrete-time sliding mode is guaranteed, and the relation between the boundary layer thickness and the sliding-mode gain design that will eliminate chattering and the boundedness conditions is obtained. The proposed technique can be applied for fault detection and isolation. Simulation results with application to three-phase motor are given to demonstrate the effectiveness of the proposed method.