[Paper Review] Data-driven Distributionally Robust MPC: An indirect feedback approach

Christoph Mark, Steven Liu|arXiv (Cornell University)|Sep 20, 2021

Advanced Control Systems Optimization15 references11 citations

TL;DR

This paper proposes a data-driven distributionally robust model predictive control (MPC) framework for linear systems with unbounded, correlated disturbances, using an indirect feedback tube-based approach to reduce online complexity. By approximating chance constraints via distributionally robust Conditional Value-at-Risk (CVaR) over a Wasserstein ambiguity set and softening constraints with slack variables, the method ensures recursive feasibility and probabilistic state constraint satisfaction with guaranteed robustness against distributional uncertainty.

ABSTRACT

This paper presents a distributionally robust stochastic model predictive control (SMPC) approach for linear discrete-time systems subject to unbounded and correlated additive disturbances. We consider hard input constraints and state chance constraints, which are approximated as distributionally robust (DR) Conditional Value-at-Risk (CVaR) constraints over a Wasserstein ambiguity set. The computational complexity is reduced by resorting to a tube-based MPC scheme with indirect feedback, such that the error scenarios can be sampled offline. Recursive feasibility is guaranteed by softening the CVaR constraint. The approach is demonstrated on a four-room temperature control example.

Motivation & Objective

• To address the challenge of controlling linear discrete-time systems under unbounded, correlated additive disturbances with hard input and state chance constraints.
• To reduce online computational complexity in stochastic MPC by employing a tube-based indirect feedback structure with offline-sampled error scenarios.
• To ensure recursive feasibility and probabilistic constraint satisfaction under distributional uncertainty by leveraging distributionally robust CVaR constraints within a Wasserstein ambiguity set.
• To provide a tractable, sample-based optimization framework that maintains robustness without requiring explicit knowledge of the disturbance distribution.

Proposed method

• The approach uses a tube-based MPC scheme with indirect feedback, decoupling the nominal trajectory from the error dynamics to reduce online complexity.
• It formulates state chance constraints as distributionally robust CVaR constraints over a Wasserstein ambiguity set to handle uncertainty in the disturbance distribution.
• The optimization problem is relaxed using slack variables to soften the CVaR constraints, ensuring recursive feasibility even under distributional shifts.
• A scenario-based approximation is employed where error scenarios are sampled offline, enabling tractable online optimization via epigraph reformulation.
• The method leverages convex conjugate duality and norm relationships to reformulate the min-max CVaR problem into a solvable form involving dual norms and auxiliary variables.
• Terminal constraints and a terminal feedback law are incorporated to ensure stability and recursive feasibility, with the terminal set satisfying invariance and constraint conditions.

Experimental results

Research questions

• RQ1 How can the online computational complexity of scenario-based stochastic MPC be reduced while maintaining robustness to distributional uncertainty?
• RQ2 Can distributionally robust CVaR constraints over a Wasserstein ambiguity set effectively replace chance constraints in MPC while preserving recursive feasibility?
• RQ3 To what extent can an indirect feedback tube-based MPC scheme with offline-sampled error scenarios maintain closed-loop constraint satisfaction under unbounded, correlated disturbances?
• RQ4 How can slack variables be effectively used to soften distributionally robust CVaR constraints and guarantee recursive feasibility in the presence of uncertainty?
• RQ5 What is the probabilistic guarantee on state constraint satisfaction when using distributionally robust CVaR with a finite sample size of error scenarios?

Key findings

• The proposed method guarantees recursive feasibility by softening the distributionally robust CVaR constraints using slack variables, ensuring that a feasible solution exists at every time step.
• The probabilistic state constraint satisfaction is guaranteed with at least $1 - \beta$ probability under the $N_s$-fold conditional predictive error distribution, as established by Theorem 1.
• The use of a Wasserstein ambiguity set allows the method to be robust against distributional shifts in the disturbance, even when the true distribution is unknown.
• The tube-based indirect feedback structure enables the use of nonlinear controllers (e.g., saturated controllers) for handling hard input constraints, which is not easily achievable in standard SMPC.
• The reformulation of the distributionally robust CVaR problem into a finite-dimensional optimization problem with auxiliary variables $s_{i,j,t}$, $\tau_{i,t}$, and $\lambda_{i,t}$ ensures tractability and convergence to the optimal solution.
• The method achieves a closed-loop state constraint satisfaction guarantee that majorizes the VaR, meaning that both the frequency and severity of constraint violations are penalized, leading to improved safety performance.