Receding-horizon Stochastic Model Predictive Control with Hard Input Constraints and Joint State Chance Constraints

TL;DR

This paper develops a receding-horizon stochastic model predictive control method for linear systems with unbounded disturbances, hard input constraints, and joint chance state constraints, ensuring stability and feasibility.

Contribution

It introduces a convex SOCP formulation that guarantees hard input constraints and approximates joint chance state constraints for stochastic control.

Findings

Ensures feasibility via softening chance constraints with penalty functions.

Guarantees stochastic stability through a geometric drift condition.

Demonstrates effectiveness on a biofuel fermentation process.

Abstract

This article considers the stochastic optimal control of discrete-time linear systems subject to (possibly) unbounded stochastic disturbances, hard constraints on the manipulated variables, and joint chance constraints on the states. A tractable convex second-order cone program (SOCP) is derived for calculating the receding-horizon control law at each time step. Feedback is incorporated during prediction by parametrizing the control law as an affine function of the disturbances. Hard input constraints are guaranteed by saturating the disturbances that appear in the control law parametrization. The joint state chance constraints are conservatively approximated as a collection of individual chance constraints that are subsequently relaxed via the Cantelli-Chebyshev inequality. Feasibility of the SOCP is guaranteed by softening the approximated chance constraints using the exact penalty function method. Closed-loop stability in a stochastic sense is established by establishing that the states satisfy a geometric drift condition outside of a compact set such that their variance is bounded at all times. The SMPC approach is demonstrated using a continuous acetone-butanol-ethanol fermentation process, which is used for production of high-value-added drop-in biofuels.