This paper proposes a new framework to distributed model predictive control (MPC) design for linear time- and space-invariant (LTSI) distributed systems subject to constraints. Given a two-dimensional, input-output model that describes the distributed dynamics among the subsystems, it is shown that a non-minimal state space realization leads to numerically tractable linear matrix inequality (LMI) based terminal state feedback controller design. The local online optimization problem is defined at the subsystem level with subsystems exchanging predictions through coupled states and can be solved in parallel at all subsystems non-iteratively. Stability and recursive feasibility are guaranteed in the presence of one-step delayed exchanging information among subsystems by imposing consistency constraints and terminal constraints. Attributed to the non-minimal state space realization, input-output properties are preserved in the MPC formulation, and hence no state estimator is needed for the online implementation. Simulation results using a heat equation demonstrate a satisfactory performance of the proposed distributed MPC design compared to centralized MPC schemes. (C) 2018 Elsevier Ltd. All rights reserved.