In this paper, a distributed Model Predictive Control (DMPC) strategy is developed for a multi-zone building plant with disturbances. The control objective is to maintain each zone's temperature at a specified level with the minimum cost of the underlying HVAC system. The distributed predictive framework is introduced with stability proofs and disturbances prediction, which have not been considered in previous related works. The proposed distributed MPC performed with 48% less computation time, 25.42% less energy consumption, and less tracking error compared with the centralized MPC. The controlled system is implemented in a smart building test bed.