Distributed stochastic model predictive control for systems with stochastic multiplicative uncertainty and chance constraints

The rapid development of technology and economy has led to the development of chemical processes, large-scale manufacturing equipment, and transportation networks, with their increasing complexity. These large systems are usually composed of many interacting and coupling subsystems. Moreover, the propagation and perturbation of uncertainty make the control design of such systems to be a thorny problem. In this study, for a complex system composed of multiple subsystems suffering from multiplicative uncertainty, not only the individual constraints of each subsystem but also the coupling constraints among them are considered. All the constraints with the probabilistic form are used to characterize the stochastic natures of uncertainty. This paper first establishes a centralized model predictive control scheme by integrating overall system dynamics and chance constraints as a whole. To deal with the chance constraint, based on the concept of multi-step probabilistic invariant set, a condition formulated by a series of linear matrix inequality is designed to guarantee the chance constraint. Stochastic stability can also be guaranteed by the virtue of nonnegative supermartingale property. In this way, instead of solving a non-convex and intractable chance-constrained optimization problem at each moment, a semidefinite programming problem is established so as to be realized online in a rolling manner. Furthermore, to reduce the computational burdens and amount of communication under the centralized framework, a distributed stochastic model predictive control based on a sequential update scheme is designed, where only one subsystem is required to update its plan by executing optimization problem at each time instant. The closed-loop stability in stochastic sense and recursive feasibility are ensured. A numerical example is employed to illustrate the efficacy and validity of the presented algorithm in this study.