Disturbances　are　inevitable　in　control　practice.　Many　methods　have　been　reported　for　disturbance　rejection.　Among　them,　equivalent-input-disturbance　(EID)　approach　is　effective　for　both　matched　and　mismatched　disturbances.　However,　since　the　disturbances　are　usually　unknown,　conventional　methods　cannot　be　used　to　analyze　the　control　performance　of　disturbance　rejection,　especially　the　dynamic　performance.　In　this　paper,　by　treating　the　Luenberger　observer　as　the　role　of　an　ideal　system,　an　auxiliary　variable,　i.e.,　the　output　error　betwween　the　disturbed　system　and　the　ideal　system,　is　introduced　to　aid　the　disturbance-rejection　performance　analysis　of　EID　approaches.　The　relation　between　the　control　performance　and　the　output　error　are　revealed.　Then,　a　nonlinear　EID　estimator　is　constructed　to　speed　up　the　disturbance-rejection　control.　The　finite-time　dynamic　performance　and　uniformly　ultimately　bounded　steady-state　performance　are　guaranteed.　Further,　a　design　algorithm　is　developed　for　the　NEID-based　closed-loop　control　system.　Finally,　by　comparing　with　a　conventional　linear　EID-based　method,　simulation　results　illustrate　the　validity　and　superiority　of　the　developed　method.　©　2022　IEEE.