Distributed model predictive control of linear systems with stochastic parametric uncertainties and coupled probabilistic constraints

This paper investigates the distributed stochastic model predictive control (DSMPC) for multiple constrained dynamically decoupled subsystems subject to stochastic uncertainties in the parameters. To obtain a computationally tractable formulation for real control applications, a spectral method called generalized polynomial chaos expansions (gPCEs) is utilized to propagate the stochastic parametric uncertainties through the system model. By using gPCEs combined with the probabilistic information on the uncertainties, both local probabilistic constraints and coupled probabilistic constraints are converted explicitly into deterministic convex second-order cone constraints. The constraints can achieve satisfaction of coupled probabilistic constraints in a distributed way by permitting a single subsystem to optimize a local cost function at each time step, while "freezing" the plans of others. The proposed gPCEs-based DSMPC algorithm guarantees recursive feasibility with respect to both local and coupled probabilistic constraints and ensures asymptotic stability in all the moments for any choice of update sequence. A numerical example is used to illustrate the effectiveness of the proposed algorithm.