This article investigates the $\eta$-exponential stabilization and positivity of the controlled discrete-time polynomial fuzzy model (DPFM) system with time delay. $\eta$-exponential polynomial copositive Lyapunov candidate is proposed to detect the system stability under the convergence (decay) rate. Static output feedback fuzzy controller is then synthesized to assure the DPFM closed-loop system with time delay. We propose Chebyshev membership functions (CMFs) to approach the primary membership function and attenuate the conservativeness of $\eta$-exponential stability formulation and positivity analysis. Chebyshev norm approximation error is utilized to introduce the error between CMFs and primary membership functions using slack matrices. A numerical example is presented to validate the proposed methods.