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Regret and Conservatism of Distributionally Robust Constrained Stochastic Model Predictive Control

Maik Pfefferkorn, Venkatraman Renganathan, Rolf Findeisen

TL;DR

This work analyzes conservatism and regret in distributionally robust stochastic MPC using moment-based ambiguity sets for unknown disturbances. By decomposing the DR joint risk into per-constraint tightens and reformulating the problem as a finite-dimensional QP via a surrogate tightening, the authors derive analytic expressions for regret and conservatism and establish probabilistic recurrence results for the closed-loop system. Numerical results on a discretized double integrator demonstrate how conservatism and regret evolve over time and converge once the system enters a probabilistic invariant set. The framework guides adaptive risk allocation and robust MPC design under distributional uncertainty, with potential extensions to online estimation and nonuniform risk budgets.

Abstract

We analyse the conservatism and regret of distributionally robust (DR) stochastic model predictive control (SMPC) when using moment-based ambiguity sets for modeling unknown uncertainties. To quantify the conservatism, we compare the deterministic constraint tightening while taking a DR approach against the optimal tightening when the exact distributions of the stochastic uncertainties are known. Furthermore, we quantify the regret by comparing the performance when the distributions of the stochastic uncertainties are known and unknown. Analysing the accumulated sub-optimality of SMPC due to the lack of knowledge about the true distributions of the uncertainties marks the novel contribution of this work.

Regret and Conservatism of Distributionally Robust Constrained Stochastic Model Predictive Control

TL;DR

This work analyzes conservatism and regret in distributionally robust stochastic MPC using moment-based ambiguity sets for unknown disturbances. By decomposing the DR joint risk into per-constraint tightens and reformulating the problem as a finite-dimensional QP via a surrogate tightening, the authors derive analytic expressions for regret and conservatism and establish probabilistic recurrence results for the closed-loop system. Numerical results on a discretized double integrator demonstrate how conservatism and regret evolve over time and converge once the system enters a probabilistic invariant set. The framework guides adaptive risk allocation and robust MPC design under distributional uncertainty, with potential extensions to online estimation and nonuniform risk budgets.

Abstract

We analyse the conservatism and regret of distributionally robust (DR) stochastic model predictive control (SMPC) when using moment-based ambiguity sets for modeling unknown uncertainties. To quantify the conservatism, we compare the deterministic constraint tightening while taking a DR approach against the optimal tightening when the exact distributions of the stochastic uncertainties are known. Furthermore, we quantify the regret by comparing the performance when the distributions of the stochastic uncertainties are known and unknown. Analysing the accumulated sub-optimality of SMPC due to the lack of knowledge about the true distributions of the uncertainties marks the novel contribution of this work.
Paper Structure (9 sections, 9 theorems, 56 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. DR SMPC Problem Formulation
  3. System Dynamics & Constraints
  4. The DR SMPC Problem
  5. Conservatism & Regret Analyses
  6. Conservatism of DR SMPC
  7. Regret of DR SMPC
  8. Numerical Simulation
  9. Conclusion

Key Result

Lemma 1

(From ono2008iterative and ElGhaoui_DR_LP) Let $\mathbf{x} \in \mathbb{R}^{n}$ be a random vector with known mean $\bar{\mathbf{x}}$ and covariance $\mathbf{\Sigma}_{\mathbf{x}}$. Then, $\forall \delta_{i} \! \in \! (0, 0.5], i \in \llbracket n_x \rrbracket$, we have where the tightening constant $\psi_{i}, \forall i \in \llbracket n_{x} \rrbracket$ is given by and $\Phi_{\mathbb{P}^{\mathbf{x}}

Figures (2)

  • Figure 1: Simulation results for the DR and the fully informed case.
  • Figure 2: Evolution of closed-loop regret $\mathfrak{R}_k$ and suboptimality gap $\mathfrak{G}_k$ with respect to time.

Theorems & Definitions (26)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Corollary 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof
  • ...and 16 more