Linear stochastic MPC under finitely supported multiplicative uncertainty

Stochastic predictive control in the presence of uncertainty and constraints is an active area of research, but most results available to date concern the case of additive uncertainty or apply constraints in expected value only. In addition, the conventional assumption that model uncertainty is normally distributed prevents the development of suitable guarantees of feasibility and therefore closed loop stability. Some recent work considered the case of multiplicative uncertainty with bounded support and used multilayer tubes in conjunction with Markov chain model to provide feasibility results, but prohibitive computation implied the need to restrict the number of layers with the consequence of that the derived results were conservative. This is overcome in the current paper through the combined use of sampling and mixed integer programming. The novel contribution concerns the construction of terminal sets, the relaxation of constraints through the use of error feedback, the definition of the predicted cost as a quadratic function of the degrees of freedom, and the handling of constraints through sampling and mixed integer programming. The results of the paper are illustrated by means of a numerical example.