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Distributionally Robust Infinite-horizon Control: from a pool of samples to the design of dependable controllers

Jean-Sébastien Brouillon, Andrea Martin, John Lygeros, Florian Dörfler, Giancarlo Ferrari Trecate

TL;DR

The paper tackles infinite-horizon control of constrained linear systems under distributional uncertainty by employing Wasserstein ambiguity sets around an empirical disturbance distribution. It develops a tight, convex SDP-based framework (DRInC) that combines system-level synthesis with a finite impulse response parameterization to guarantee stability and CVaR-based safety offline, avoiding online re-optimization. A novel strong duality result for DRO of quadratic objectives and a tight convex relaxation enable tractable optimization even when the disturbance distribution is only known through samples. Numerical experiments demonstrate the value of empirical centers and distributional robustness under distribution shifts, suggesting practical benefits for data-driven control design in uncertain environments.

Abstract

We study control of constrained linear systems with only partial statistical information about the uncertainty affecting the system dynamics and the sensor measurements. Specifically, given a finite collection of disturbance realizations drawn from a generic distribution, we consider the problem of designing a stabilizing control policy with provable safety and performance guarantees despite the mismatch between the empirical and true distributions. We capture this discrepancy using Wasserstein ambiguity sets, and we formulate a distributionally robust (DR) optimal control problem, which provides guarantees on the expected cost, safety, and stability of the system. To solve this problem, we first present new results for DR optimization of quadratic objectives using convex programming, showing that strong duality holds under mild conditions. Then, by combining our results with the system-level parametrization of linear feedback policies, we show that the design problem can be reduced to a semidefinite program. We present numerical simulations to validate the effectiveness of our approach and to highlight the value of empirical distributions for control design.

Distributionally Robust Infinite-horizon Control: from a pool of samples to the design of dependable controllers

TL;DR

The paper tackles infinite-horizon control of constrained linear systems under distributional uncertainty by employing Wasserstein ambiguity sets around an empirical disturbance distribution. It develops a tight, convex SDP-based framework (DRInC) that combines system-level synthesis with a finite impulse response parameterization to guarantee stability and CVaR-based safety offline, avoiding online re-optimization. A novel strong duality result for DRO of quadratic objectives and a tight convex relaxation enable tractable optimization even when the disturbance distribution is only known through samples. Numerical experiments demonstrate the value of empirical centers and distributional robustness under distribution shifts, suggesting practical benefits for data-driven control design in uncertain environments.

Abstract

We study control of constrained linear systems with only partial statistical information about the uncertainty affecting the system dynamics and the sensor measurements. Specifically, given a finite collection of disturbance realizations drawn from a generic distribution, we consider the problem of designing a stabilizing control policy with provable safety and performance guarantees despite the mismatch between the empirical and true distributions. We capture this discrepancy using Wasserstein ambiguity sets, and we formulate a distributionally robust (DR) optimal control problem, which provides guarantees on the expected cost, safety, and stability of the system. To solve this problem, we first present new results for DR optimization of quadratic objectives using convex programming, showing that strong duality holds under mild conditions. Then, by combining our results with the system-level parametrization of linear feedback policies, we show that the design problem can be reduced to a semidefinite program. We present numerical simulations to validate the effectiveness of our approach and to highlight the value of empirical distributions for control design.
Paper Structure (25 sections, 10 theorems, 85 equations, 4 figures)

This paper contains 25 sections, 10 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. System dynamics and uncertainty description
  4. Control objectives, policies, and uncertainty propagation
  5. Expressivity of the problem formulation and related work
  6. Background
  7. System-level synthesis
  8. A stationary control problem
  9. Strong duality for DRO of piecewise linear objectives
  10. Main Results
  11. Non-convexity challenges
  12. Tight convex relaxation for DRO of quadratic objectives
  13. Convex formulation of DRInC design
  14. Numerical results
  15. The value of the empirical distribution
  16. ...and 10 more sections

Key Result

Proposition 1

Let $a_j \in \mathbb{R}^d$ and $b_j \in \mathbb{R}$, for $j = 1, \dots, J$, characterize the piece-wise linear cost $\max_{j \in [J]} a_j^\top \bm\xi_T + b_j$, and let $H$ and $h$ characterize the support $\bm\Xi$ as $\{\bm{\xi} \in \mathbb{R}^{d} : {H} \bm{\xi} \leq {h}\}$. If Assumption ass:supp can be equivalently computed as: for all $i = 1, \dots, N$ and $j = 1, \dots, J$.

Figures (4)

  • Figure 1: Illustration of two worst-case distributions $\mathbb{Q} \in \mathbb{B}_\epsilon(\delta)$ and $\mathbb{Q}' \in \mathbb{B}_{\epsilon'}(\delta)$ in different Wasserstein balls around the Dirac delta distribution. The support $\bm\Xi$ is represented by the horizontal blue square, and the left-most Dirac distribution represents a local minima in $\mathbb{B}_{\epsilon'}(\delta)$ for the risk $\mathcal{R}_\epsilon(Q)$ in \ref{['eq:prop_dro_general_quad_risk_def']}.
  • Figure 2: Probability density function of the real distribution $\mathbb{P}$ in red and the histogram of the empirical training distribution $\widehat{\mathbb{P}}$ for our first set of experiments in blue, marginalized to any dimension.
  • Figure 3: Closed-loop performance comparison: average control cost and constraint violations when $\mathbb{P}$ is given by Figure \ref{['fig:training_samples']} and $\mathbb{Q}$ is randomly sampled from $\mathbb{B}_\epsilon(\widehat{\mathbb{P}})$ as per \ref{['eq:ambiguity_set_definition']}. For increasing values of $\epsilon$, we use boxes of different colors to display the difference between the first and third quartiles of the data, using a solid line to denote the median value. The whiskers extend from the box to the farthest data point lying within 1.5x the inter-quartile range from the box, whereas the diamonds represent flier points that lie past the end of the whiskers.
  • Figure 4: Closed-loop performance comparison: average control cost and constraint violations when $\widehat{\mathbb{P}}$ is constructed from samples from the training distributions in the top left boxes, and $\mathbb{Q}$ from samples from increasingly more different distributions in the same class (examples of such distributions are shown in the small panels on top of the main figure). We use markers of different colors to display the performance of each method, and we plot the trend as a third-order polynomial using solid lines. As both $\bm{\pi}_{\operatorname{drinc}}$ and $\bm{\pi}_{\operatorname{rob}}$ do not violate the safety constraints, the blue and green trend lines associated with these two policies on the bottom plot overlap.

Theorems & Definitions (21)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Lemma 5
  • ...and 11 more