  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.
