This paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates. To this aim, the recently developed explicit model predictive control (eMPC) is reformulated and characterized using the delta-operator to enjoy its promising advantages compared to the time-shift operator. Using the proposed delta-model eMPC formulation, a systematic method is proposed for first designing a low-complexity approximate eMPC solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity. It is shown that the stability and feasibility of the proposed sub-optimal solution is guaranteed.