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Soft wearable robots present a promising approach for elbow assistance and rehabilitation. However, most existing devices rely on open-loop control strategies, which renders individual customization cumbersome and incapable of adapting to dynamic interactions. The challenges in implementing closed-loop control arise from the complex nonlinearities of soft robots and the unpredictable external disturbances encountered during human-robot interactions. To achieve closed-loop control of elbow soft wearable robots, this paper proposes a disturbance observer (DO)-based nonsmooth feedback (NSF) method. Specifically, a tailored DO is designed to enhance feedforward compensation by analyzing the unique characteristics of the disturbances encountered in practical systems. A nonrecursive NSF is employed to suppress residual disturbances and nonlinearities, with the finite-time stability of the closed-loop system rigorously guaranteed. The proposed method balances efficacy and simplicity by leveraging the concise models of both the system and disturbances to enhance performance while avoiding intricate modeling. Moreover, its nonrecursive design results in a straightforward control law and facilitates implementation. Extensive comparison and ablation experiments validate the superiority of the proposed method over existing approaches. Human trials involving 8 healthy subjects and 7 stroke patients demonstrate that our method enhances task performance, reduces muscle strain during elbow assistance scenarios, and significantly improves elbow motor function in rehabilitation training.