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Conformal Prediction for Distribution-free Optimal Control of Linear Stochastic Systems

Eleftherios E. Vlahakis, Lars Lindemann, Pantelis Sopasakis, Dimos V. Dimarogonas

TL;DR

This work addresses distribution-free optimal control of discrete-time linear stochastic systems with unknown disturbance distributions under joint chance constraints. It adapts conformal prediction to construct prediction regions for the error dynamics and proposes two data-driven routes to synthesize a feedback policy: a direct method that optimizes trajectory-based nonconformity quantiles and calibrates PRs, and an indirect method that uses CP on disturbances combined with the S-procedure to obtain an invariant PR and a stabilizing gain. The approach provides marginal coverage guarantees that are independent of dataset size and enables a deterministic relaxation that yields a feasible policy with provable probabilistic satisfaction. A numerical double-integrator example illustrates the trade-offs between the direct and indirect methods and highlights substantial calibration efficiency compared to traditional scenario-based approaches.

Abstract

We address an optimal control problem for linear stochastic systems with unknown noise distributions and joint chance constraints using conformal prediction. Our approach involves designing a feedback controller to maintain an error system within a prediction region (PR). We define PRs as sublevel sets of a nonconformity score over error trajectories, enabling the handling of joint chance constraints. We propose two methods to design feedback control and PRs: one through direct optimization over error trajectory samples, and the other indirectly using the $S$-procedure with a disturbance ellipsoid obtained from data. By tightening constraints with PRs, we solve a relaxed problem to synthesize a feedback policy. Our method ensures reliable probabilistic guarantees based on marginal coverage, independent of data size.

Conformal Prediction for Distribution-free Optimal Control of Linear Stochastic Systems

TL;DR

This work addresses distribution-free optimal control of discrete-time linear stochastic systems with unknown disturbance distributions under joint chance constraints. It adapts conformal prediction to construct prediction regions for the error dynamics and proposes two data-driven routes to synthesize a feedback policy: a direct method that optimizes trajectory-based nonconformity quantiles and calibrates PRs, and an indirect method that uses CP on disturbances combined with the S-procedure to obtain an invariant PR and a stabilizing gain. The approach provides marginal coverage guarantees that are independent of dataset size and enables a deterministic relaxation that yields a feasible policy with provable probabilistic satisfaction. A numerical double-integrator example illustrates the trade-offs between the direct and indirect methods and highlights substantial calibration efficiency compared to traditional scenario-based approaches.

Abstract

We address an optimal control problem for linear stochastic systems with unknown noise distributions and joint chance constraints using conformal prediction. Our approach involves designing a feedback controller to maintain an error system within a prediction region (PR). We define PRs as sublevel sets of a nonconformity score over error trajectories, enabling the handling of joint chance constraints. We propose two methods to design feedback control and PRs: one through direct optimization over error trajectory samples, and the other indirectly using the -procedure with a disturbance ellipsoid obtained from data. By tightening constraints with PRs, we solve a relaxed problem to synthesize a feedback policy. Our method ensures reliable probabilistic guarantees based on marginal coverage, independent of data size.

Paper Structure

This paper contains 9 sections, 5 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and problem setup
  3. Main results
  4. Open-loop control
  5. Prediction regions and feedback control
  6. Direct method
  7. Indirect method
  8. Example
  9. Conclusion

Key Result

Lemma 1

TibshiraniNeurIPS2019 If $\mathcal{R}^{(0)},\ldots,\mathcal{R}^{(k)}$ are i.i.d. random variables, then for any $\theta\in(0,1)$, we have

Figures (1)

  • Figure 1: Direct method (left), Indirect method (right): Representation of the original state constraint $\mathcal{X}_t$ (solid black), the tighter state constraint $\mathcal{Z}_t=\mathcal{X}_t\ominus \mathscr{E}^{1:N}_{0.95}(e(t))$ (dashed black), the PR $\mathscr{E}^{1:N}_{0.95}(e(t))$ (dashed blue), 100 sample trajectories $\bm{x}^{(j)}(0:N)$ (light red), and the deterministic trajectory $\bm{z}(0:N)$ (red), with initial condition $x(0)$ (red cross).

Theorems & Definitions (11)

  • Lemma 1
  • Remark 1
  • Definition 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 1 more