Probabilistically Recursively Feasible Stochastic MPC with Low Conservatism under Unbounded Disturbance

In this paper, a stochastic model predictive control (SMPC) method is proposed to handle unbounded additive Gaussian disturbance in linear time-invariant systems. This SMPC method is less conservative than robust model predictive control (RMPC) and other SMPC methods and guarantees recursive feasibility with a certain probability. Specifically, it calculates the inverse cumulative distribution of random noise, thereby enabling chance constraints to be transformed into deterministic constraints. The latter constraints are further tightened to determine sufficient conditions for recursive feasibility under bounded uncertainty. Subsequently, the terminal constraints in the finite-horizon problem are handled using a probabilistic invariant set. This handling ensures that there is a certain probability that a system implemented using an SMPC law is asymptotically stable and recursively feasible under unbounded normal distributions. The reduced conservatism of this SMPC method is subjected to quantitative analysis. Finally, this SMPC method is demonstrated in an illustrative example that shows that it has lower conservatism than RMPC and tube-based SMPC methods.