In　this　article,　based　on　the　concept　of　an　equivalent-input-disturbance　(EID),　a　tunable　nonlinear-function-based　estimator　is　developed　for　fast　and　efficient　control　of　unknown　and　mismatched　disturbances　of　a　tracking　control　system,　and　the　control　performances　are　analyzed.　A　great　superiority　over　the　conventional　EID　estimators　is　that　the　gain　is　varying　and　tunable　with　respect　to　an　intermediate　disturbance-rejection　performance　indicator.　First,　the　control　task　is　divided　into　the　control　of　reference-tracking　and　disturbance-rejection,　and　the　corresponding　control　mechanisms　are　clarified.　For　the　disturbance-rejection　control,　explanation　of　the　intermediate　performance　indicator　is　provided　and　a　tunable　nonlinear-function-based　EID　estimator　is　constructed.　Then,　the　tunability　condition　for　the　varying　gain　of　the　developed　EID　estimator　is　given.　A　relation　between　control　performances　and　control　parameters　is　clarified　a　theoretical　analysis.　An　input-to-state　stability　condition　is　derived　for　the　closed-loop　control　system.　Finally,　comparisons　with　other　methods　through　simulations　and　experiments　demonstrate　fast　disturbance-rejection　response,　weak　noise　impact,　and　strong　robustness　of　the　developed　method.