A distributionally robust optimization approach to two-sided chance constrained stochastic model predictive control with unknown noise distribution

In this work, we propose a distributionally robust stochastic model predictive control (DR-SMPC) algorithm to address the problem of two-sided chance constrained discrete-time linear system corrupted by additive noise. The prevalent mechanism to cope with two-sided chance constraints is the so-called risk allocation approach, which conservatively approximates the two-sided chance constraints with two single chance constraints by applying the Boole's inequality. In this proposed DR-SMPC framework, an exact tractable second-order cone (SOC) approach is adopted to abstract the two-sided chance constraints by considering the first and second moments of the noise. The proposed DR-SMPC algorithm is able to guarantee that the worst-case probability of violating both the upper and lower limits of safety constraints is within the pre-specified maximum probability (PsMP). By flexibly adjusting this PsMP, the feasible region of the initial states can be increased for the SMPC problem. The recursive feasibility and convergence of the proposed DR-SMPC are established rigorously by introducing binary initialization strategy of nominal state. Simulation studies of two practical cases are conducted to demonstrate the effectiveness of the proposed DR-SMPC algorithm.