On the centres and scalings of robust and stochastic tubes

Tubes have recently emerged as a convenient means of devising robust and stochastic Model Predictive Control (MPC) algorithms. Despite receiving a considerable amount of attention in the MPC literature, this framework retains some aspects that not only are unresolved but appear to be contradictory. One such issue is whether it is beneficial to use the centre of the initial cross-section of a robust state tube as a degree of freedom in the online MPC optimization, and it is one of the aims of the current paper to explore this issue. A further aim of the paper is to introduce, at minimal computational load, techniques to allow constraint relaxation through the use of information on predicted uncertainty, which, although not available at current time, will be known at the time that a future control action is applied. The results of the paper are illustrated by numerical examples.