In this work, we demonstrate how constrained Moving Horizon Estimation (MHE) and Model Predictive Control (MPC) can be simultaneously addressed via multi-parametric programming. First, we present a method for obtaining the error dynamics of constrained MHE for linear, time-invariant systems by solving the constrained optimization problem of the MHE by multi-parametric programming methods. Set-theoretical methods are then used to derive bounds on the estimation error - it is shown that the estimation error is bounded in an invariant set for the error dynamics described by a set of linear inequalities. The error dynamics and the error bounds can then be used for the design of a robust output feedback explicit/multi-parametric MPC yielding a simultaneous estimation and control design which is illustrated with an example of robust tube-based MPC.