Smoothness properties of the MPC value function with respect to sampling time and prediction horizon

Sampling time and the prediction horizon length are two underlying design choices for the implementation of model predictive control (MPC). Smoothness properties of the open- and closed-loop value function with respect to the two parameters are essential to characterise for the purpose of providing knowledge that is useful in the context of MPC design optimisation to maximise system performance. Specifically, these properties are continuity, differentiability and monotonicity. This paper presents both numerical and analytical results to reveal the smoothness properties of the value function. Increasing sampling rate and/or prediction horizon does not necessarily improve closed-loop performance. Furthermore, the value function in open-loop under input constraints is differentiable, which may contradict the traditional thinking and expectations.