Soft‐constrained model predictive control based on data‐driven distributionally robust optimization

Abstract

This article proposes a novel distributionally robust optimization (DRO)‐based soft‐constrained model predictive control (MPC) framework to explicitly hedge against unknown external input terms in a linear state‐space system. Without a priori knowledge of the exact uncertainty distribution, this framework works with a lifted ambiguity set constructed using machine learning to incorporate the first‐order moment information. By adopting a linear performance measure and considering input and state constraints robustly with respect to a lifted support set, the DRO‐based MPC is reformulated as a robust optimization problem. The constraints are softened to ensure recursive feasibility. Theoretical results on optimality, feasibility, and stability are further discussed. Performance and computational efficiency of the proposed method are illustrated through motion control and building energy control systems, showing 18.3% less cost and 78.8% less constraint violations, respectively, while requiring one third of the CPU time compared to multi‐stage scenario based stochastic MPC.

Data-driven distributionally robust MPC for constrained stochastic systems

“…Similar results were achieved in [10] for a tube-based approach with Wasserstein ambiguity, the radius of which is not data-driven; in [11] for relaxed, robust constraints; in [4] for moment-based ambiguity which is not data-driven and does not guarantee recursive feasibility; and [12] for discrete distributions. The advantage of our framework is that it allows for truly data-driven ambiguity sets, risk constraints and supports online learning of the ambiguity, while retaining recursive feasibility guarantees.…”

Uncertainty-Aware Model Predictive Control for Residential Buildings Participating in Intraday Markets

“…In distributionally robust MPC, ambiguity sets are used to define possible deviations from a given probability distribution functions [19]. A case study for using distributionally robust MPC for thermal control in buildings is presented in [20]. However, implementing such optimization need more advanced statistical approaches to determine the required ambiguity sets from historical data.…”

“…Stochastic Optimization [17], [18] Uses probability distribution functions of forecasted variables or scenario trees Allows optimization across scenarios with the underlying probability distribution function Does not provide a guarantee in finding the global optimum solution in polynomial time May be slow depending on the implementations (e.g. using monte-carlo methods) The scenario trees may not be accurate depending on the assumptions on the underlying probability distribution functions Distributionally Robust Optimization [20] Uses ambiguity sets to define possible deviations in the probability distribution functions describing the uncertainties in forecasted quantities Statistical approaches can be used to define ambiguities in the probability distribution function Does not provide a guarantee in finding global optimum solution in polynomial time, and approximations may be required Robust Optimization [22]- [25] Uses uncertainty sets and optimize under the worstcase condition Does not need to assume a probability distribution function random variables Pessimistic and may lead to higher costs.…”

Data-driven distributionally robust MPC using the Wasserstein metric

“…Distributionally robust optimization in control problems has been studied with regard to different formulations of objective functions Lu, Lee and You (2020). In the setting of multi-stage stochastic optimal power flow problem, a framework is proposed to solve multi-stage feedback control policies with respect a Conditional Value at Risk (CVaR) objective Guo, Baker, Dall'Anese, Hu and Summers (2018).…”