Stochastic Model Predictive Control for Linear Systems with Unbounded Additive Uncertainties

TL;DR

This paper introduces two novel stochastic model predictive control methods for linear systems with unbounded uncertainties, using probabilistic reachable sets to handle chance constraints and ensure stability.

Contribution

The paper develops two new stochastic MPC algorithms employing probabilistic reachable sets and invariant sets, improving feasibility and stability for systems with unbounded uncertainties.

Findings

Algorithms are feasible and stable under specified conditions.

Numerical simulations demonstrate effectiveness of the proposed methods.

Enhanced feasibility through soft constraints on initial states.

Abstract

This paper presents two stochastic model predictive control methods for linear time-invariant systems subject to unbounded additive uncertainties. The new methods are developed by formulating the chance constraints into deterministic form, which are treated in analogy with robust constraints, by using the probabilistic reachable set. Firstly, the probabilistically resolvable time-varying tube-based stochastic model predictive control algorithm is designed by employing the time-varying probabilistic reachable sets as the tubes. Secondly, by utilizing the probabilistic positively invariant set, the probabilistically resolvable constant tube-based stochastic model predictive control algorithm is developed by employing the constantly tightened constraints in the entire prediction horizons. In addition, to enhance the feasibility of the algorithms, the soft constraints are imposed to the state initializations. The algorithm feasibility and closed-loop stability results are provided. The efficacy of the approaches are demonstrated by means of numerical simulations.