Distributionally Robust Stochastic MPC under Disturbance-Affine Feedback Policies

Abstract: This study addresses the stochastic Model Predictive Control (MPC) problem for linear time-invariant systems subjected to unknown disturbance distributions. By leveraging the most recent disturbance data, we construct a set of distributions with similar statistical properties contained within a Wasserstein ball, thereby accounting for the worst-case impacts on constraint satisfaction. Numerous MPC strategies, particularly tube-based approaches, have been extensively studied under the Wasserstein ambiguity set, but these methods often introduce conservatism and can limit control performance. Unlike tube-based approaches, we adopt a disturbance-affine control strategy, which introduces additional control degrees of freedom. We begin by developing the Disturbance-Affine Distributionally Robust (DA-DR) MPC framework, subsequently reformulating the control problem into a tractable quadratic programming formulation. Furthermore, we establish the recursive feasibility and stability of the proposed MPC scheme. Finally, we present comprehensive theoretical analysis and simulation results, demonstrating the superiority of the DA-DR MPC over tube-based MPC in initial feasible sets, average performance, and state variance control.

• The paper introduces a novel distributionally robust stochastic MPC framework using disturbance-affine feedback to reduce conservatism and enhance performance.
• It reformulates chance constraints with convex CVaR approximations, resulting in a tractable quadratic program that guarantees recursive feasibility.
• Empirical validation on a GCAI system shows lower state variance and improved tracking compared to traditional tube-based MPC methods.

Distributionally Robust Stochastic MPC with Disturbance-Affine Feedback: Theory, Tractability, and Empirical Assessment

Introduction

This paper develops a novel Distributionally Robust Stochastic Model Predictive Control (MPC) framework for linear time-invariant (LTI) systems with uncertainty characterized by unknown, possibly time-varying, disturbance distributions. The key innovation lies in parameterizing the feedback policy as an affine function of past disturbances (disturbance-affine), rather than employing a tube-based approach. The disturbance-induced ambiguity set is constructed as a Wasserstein ball around the empirical distribution of recent disturbance data, directly leveraging observed system behavior while ensuring robustness to model misspecification and data limitations. The framework achieves tractability by an explicit reformulation into a quadratic program (QP) via convex approximations based on Conditional Value-at-Risk (CVaR) for chance constraints. Theoretical guarantees—including recursive feasibility and closed-loop stability—are rigorously established. The benefits and practical relevance are demonstrated in simulation for the stochastic control of a Gasoline Controlled Auto-Ignition (GCAI) system.

Distributional Robustness via Wasserstein Ambiguity Sets

The distributional ambiguity set is built as a Wasserstein ball of radius $\epsilon$ around the nominal empirical disturbance distribution, accommodating both sampling error and non-stationary/unmodeled effects. Given finite recent disturbance samples, the true underlying distribution is unlikely to be identified exactly; by accounting for the set of distributions close in Wasserstein distance, the controller achieves robustness without undue conservatism typical in worst-case or moment-based robust MPC.

Key points:

• Wasserstein distance is chosen for its strong performance guarantees in out-of-sample settings and its flexibility in accommodating continuous and discrete distributions.
• The ambiguity set contains all Borel probability measures with bounded $p$-moment within $\epsilon$ of the empirical distribution; the choice of $\epsilon$ balances robustness and conservatism.

This setup is consistent with recent advances on data-driven DRO and has proven theoretical benefits regarding finite-sample guarantees and small-sample behavior.

Disturbance-Affine Control Parameterization

Distinct from tube-based approaches, the framework models the control action at prediction step $k+i$ as

$$u_{k+i} = c_{i|k} + \sum_{j=0}^{i-1} K_{j,i|k} w_{k+j}$$

where the $K_{j,i|k}$ are decision variables optimizing the influence of previous disturbances across the prediction horizon. This direct disturbance-affine feedback structure grants greater degrees of freedom than static tube-based gains, which are strongly coupled to constraint tightening and typically lead to excessively conservative policies.

Key contributions:

• The affine feedback structure enables decoupling safety (via constraint satisfaction) from nominal performance more flexibly.
• The induced optimization remains convex after transformation and allows larger feasible sets—formally proved in the analysis.

Reformulation of Chance Constraints and Tractability

Distributionally robust chance constraints over a Wasserstein ambiguity set are generally NP-hard. The paper leverages a convex CVaR (risk envelope) approximation, using strong duality to derive explicit tractable reformulations as linear inequalities, indexed over the finite sample set. The resulting MPC problem is cast as a QP with all required constraints represented as convex, efficiently-testable conditions.

Technical merits:

• The crucial min-max order interchange (via Sion’s theorem and regularity conditions) ensures the equivalence between the original intractable program and the dual-based convex surrogate.
• The analysis provides explicit conditions for recursive feasibility, including a terminal invariant set with a stabilizing linear feedback, and a proof that recursive constraint enforcement remains possible by imposing $t$-step-ahead constraints.
• Sample average approximations replace potentially intractable expectations, yielding a scalable implementation compatible with standard optimization toolchains.

Closed-Loop Properties: Recursive Feasibility and Stability

Recursive feasibility is proven under mild regularity assumptions and the construction of a robustly positively invariant terminal set. The theoretical machinery ensures, for the adopted affine feedback and CVaR-approximated chance constraints, that the closed-loop system always admits a feasible control action at future steps if initially feasible.

Stability is addressed in the mean-square sense: it is established that the quadratic performance measure (weighted sum of state and input norms) admits a computable, explicit upper bound proportional to the disturbance covariance and the solution of the terminal Lyapunov equation.

Comparative Analysis: DA-DR MPC versus Tube-Based MPC

A formal dominance argument is presented regarding the initial feasible set: any feasible solution for a tube-based (static feedback) MPC maps to a feasible disturbance-affine solution, but not vice versa in general, except in the degenerate case ($N_h=1$). Thus, disturbance-affine feedback enlarges the feasible region and leads to controllers that are typically less conservative.

Empirical Validation: GCAI Application

The methodology is evaluated in simulation on a GCAI plant with data-derived disturbance generation (estimating the admissible uncertainty set via principal component analysis on engine measurements). CA50 (center of combustion) and IMEP (indicated mean effective pressure) are the principal states/outputs of interest. Both DA-DR MPC and tube-based distributionally robust MPC are evaluated with identical (recent) data and ambiguity set radii.

Key empirical results:

• Variance Reduction and Constraint Violation: DA-DR MPC achieves lower state variance (CA50) and reduced upper-bound violation frequency compared to the tube-based baseline.
• Tracking: Superior mean tracking of the reference, especially for time-varying IMEP, and better system responsiveness to reference changes.
• Statistical Significance: The improvement in closed-loop objective function (Wilcoxon test $p \approx 0.0016$) is statistically significant.

The disturbance-affine controller’s reduced conservatism directly translates into increased performance, more agile response, and better satisfaction of soft constraints under realistic measurement uncertainty.

Theoretical and Practical Implications

This framework exemplifies a paradigm shift in stochastic MPC—moving away from deterministic tube-based constraint tightening and static gains, towards data-driven, dynamically optimized feedback policies. The use of Wasserstein balls aligns with modern DRO theory, providing finite-sample robustness, theoretically justified tractability, and performance improvements. The results support deploying disturbance-affine, distributionally robust controllers in domains where sample efficiency, constraint reliability, and real-time computational feasibility are all critical.

The primary practical limitation is the growth in computational burden from multi-step-ahead feasibility constraints and the size of the disturbance-affine gain matrices, especially for long horizons. Future work is appropriately directed at computational efficiency and structure-exploiting solvers.

Conclusion

This work provides a tractable, theoretically-sound, and empirically-validated framework for distributionally robust stochastic MPC using disturbance-affine feedback parameterizations. It advances both the theory and practice of robust data-driven predictive control, establishes formal dominance over tube-based methods in feasible set size and average performance, and highlights directions for further research in scalable, robust, and adaptive MPC synthesis.