In this paper, a generalized non-quadratic Lyapunov function and non-parallel distributed compensation (non-PDC) law are proposed for robust H∞ fuzzy control of discrete-time uncertain nonlinear systems. The focus is on designing a robust output feedback observer such that the observer error system is robustly asymptotically stable and has a guaranteed H∞ performance. A new basisdependent idea is introduced to solve the fuzzy observer-based control problem, which is different from the quadratic framework that entails fixed matrices for the entire membership function, or the linearly basis-dependent framework that uses linear convex combinations of s matrices. This idea is realized by carefully selecting the structure of the matrices involved in the products with system matrices. Linear matrix inequality (LMI) conditions are obtained for the existence of observer and based on these the results are casted into a convex optimization problem, which can be readily solved via standard numerical software. The effectiveness and the superiority of the proposed approach are demonstrated by two examples borrowed from the literature.