||It is conjectured that under weak asymmetry, one-dimensional exclusion processes universally converge to the Kardar-Parisi-Zhang (KPZ) equation. Based on Gaertner's discrete Cole-Hopf transformation, Bertini and Giacomin (1997) prove this convergence for the special case of simple (nearest-neighbor) exclusion. In this work we extend the discrete Cole-Hopf transformation, and eliminate the main nonlinearity using a gradient-type condition. We thus obtain the universal convergence to the KPZ equation of a class of non-simple exclusion processes with a hopping range of at most 3. This is joint work with Amir Dembo.