Conformal Prediction-Based MPC for Stochastic Linear Systems

Andrea Carron; Dimos V. Dimarogonas; Eleftherios E. Vlahakis; Lukas Vogel

arxiv: 2512.10738 · v2 · submitted 2025-12-11 · 📡 eess.SY · cs.RO· cs.SY

Conformal Prediction-Based MPC for Stochastic Linear Systems

Lukas Vogel , Andrea Carron , Eleftherios E. Vlahakis , Dimos V. Dimarogonas This is my paper

Pith reviewed 2026-05-16 23:20 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY

keywords conformal predictionmodel predictive controlstochastic linear systemschance constraintsoutput feedbackrecursive feasibilityfinite-sample guarantees

0 comments

The pith

Conformal prediction builds finite-sample sets that convert joint chance constraints into a deterministic MPC problem for linear systems with unknown disturbances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a stochastic model predictive control approach for linear systems where disturbances follow an unknown distribution. It applies conformal prediction to a collection of disturbance samples to produce confidence regions around the resulting error trajectories. These regions allow the joint-in-time probabilistic constraints to be replaced by a deterministic optimization that uses indirect feedback. The resulting controller is recursively feasible and satisfies the original chance constraints at the prescribed level. The same construction extends to the case of output feedback when only noise samples are available.

Core claim

Conformal prediction applied to disturbance samples yields probabilistic sets for the closed-loop error trajectories. These sets enable a deterministic closed-loop MPC formulation based on indirect feedback that inherits recursive feasibility and meets the joint-in-time chance constraints without requiring parametric assumptions on the disturbance distribution.

What carries the argument

Conformal prediction sets for error trajectories, relaxed via an indirect-feedback deterministic MPC formulation.

If this is right

The MPC optimization remains recursively feasible at every time step.
Joint chance constraints are satisfied with the user-specified probability in closed loop.
The framework extends to output-feedback control using only noise samples from the measurement channel.
No expensive offline scenario generation or distribution estimation is required beyond collecting a modest number of disturbance samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sample-based construction could be combined with online adaptation of the conformal sets to reduce conservatism as more data arrives.
Similar error-set techniques might extend the guarantees to mildly nonlinear systems if suitable linearization or bounding errors are available.
In networked settings the method could share disturbance samples across agents to tighten collective constraints.

Load-bearing premise

Conformal sets computed from open-loop disturbance samples continue to provide valid coverage for the actual closed-loop error trajectories generated by the controller.

What would settle it

Closed-loop simulations or hardware experiments in which the empirical frequency of joint constraint violations exceeds the target probability level when the conformal sets are used.

Figures

Figures reproduced from arXiv: 2512.10738 by Andrea Carron, Dimos V. Dimarogonas, Eleftherios E. Vlahakis, Lukas Vogel.

read the original abstract

We propose a stochastic model predictive control (MPC) framework for linear systems subject to joint-in-time chance constraints under unknown disturbance distributions. Unlike existing approaches that rely on parametric or Gaussian assumptions, or require expensive offline computation, the method uses conformal prediction to construct finite-sample confidence regions for the system's error trajectories with minimal computational effort. These probabilistic sets enable relaxation of the joint-in-time chance constraints into a deterministic closed-loop formulation based on indirect feedback, ensuring recursive feasibility and chance constraint satisfaction. Further, we extend to the output feedback setting and establish analogous guarantees from output measurements alone, given access to noise samples. Numerical examples demonstrate the effectiveness and advantages compared to existing approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This conformal MPC paper has a promising distribution-free angle for chance constraints, but the closed-loop coverage needs proof.

read the letter

The main thing to know about this paper is that it uses conformal prediction to create finite-sample confidence sets for error trajectories in stochastic linear MPC, allowing joint chance constraints to be handled deterministically through an indirect feedback formulation that preserves recursive feasibility. They also extend the idea to output feedback using noise samples. This combination is new enough: most conformal work in control is either open-loop or doesn't tie directly into the closed-loop MPC with feasibility guarantees. The indirect feedback seems to be the key to making the coverage transfer without blowing up the conservatism. What it does well is avoid both parametric assumptions on the disturbances and the computational cost of offline scenario generation or robust optimization. For applications in robotics where you have access to disturbance samples but not the full distribution, this could be practical. The soft spots are in the verification. The abstract claims the guarantees but doesn't sketch the proof that the conformal coverage on open-loop samples carries over to the closed-loop error trajectories under the state-dependent policy. If the feedback alters the joint distribution in a way that breaks the exchangeability or whatever conformal relies on, then the chance constraint satisfaction could be compromised. The stress test concern is real here, and without seeing the derivation or error bars on the numerical results, it's hard to gauge how tight it is. The examples are said to demonstrate advantages, but lacking quantitative data in the summary makes it difficult to assess the practical gains. This paper is for control engineers and researchers focused on stochastic MPC who want non-parametric, sample-based methods. It deserves a serious referee to go through the proofs and the simulation setup. I'd recommend sending it to peer review so the details can be checked properly.

Referee Report

2 major / 2 minor

Summary. The paper proposes a stochastic MPC framework for linear systems with joint-in-time chance constraints under unknown disturbance distributions. It uses conformal prediction to build finite-sample confidence regions for error trajectories, relaxes the constraints into a deterministic closed-loop formulation via indirect feedback, and claims recursive feasibility plus chance-constraint satisfaction. The approach is extended to output feedback using noise samples alone.

Significance. If the coverage transfer holds, the work supplies a distribution-free, finite-sample method for joint chance-constrained stochastic MPC that avoids parametric assumptions and heavy offline computation. This integration of conformal prediction with indirect-feedback MPC could enable practical deployment in settings where disturbance laws are unknown, while preserving recursive feasibility.

major comments (2)

[Main results / indirect-feedback formulation] The central claim that conformal sets calibrated on open-loop disturbance samples retain their nominal coverage for the closed-loop error trajectories generated by the state-dependent MPC policy is load-bearing for both recursive feasibility and the joint chance-constraint guarantee. The indirect-feedback relaxation introduces policy-dependent dependence into the error sequence; no explicit argument or theorem shows that this dependence preserves the coverage probability without extra conservatism that would violate the original 1-ε level. This transfer step requires a dedicated proof or counter-example analysis.
[Numerical examples] The abstract and numerical-examples section assert effectiveness and advantages over existing methods, yet no quantitative coverage rates, violation frequencies, or ablation data (e.g., effect of calibration-set size on closed-loop performance) are reported. Without these metrics it is impossible to verify that the claimed finite-sample guarantees materialize in closed loop.

minor comments (2)

[Preliminaries] Notation for the conformal prediction sets and the indirect-feedback law should be introduced with explicit dependence on the calibration data to avoid ambiguity when discussing coverage preservation.
[Output-feedback extension] The output-feedback extension is stated to yield analogous guarantees, but the precise mapping from output-noise samples to state-error sets is not spelled out; a short clarifying paragraph or lemma would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the two major comments below. We will revise the manuscript to include a dedicated proof for the coverage transfer and additional quantitative results from the numerical examples.

read point-by-point responses

Referee: [Main results / indirect-feedback formulation] The central claim that conformal sets calibrated on open-loop disturbance samples retain their nominal coverage for the closed-loop error trajectories generated by the state-dependent MPC policy is load-bearing for both recursive feasibility and the joint chance-constraint guarantee. The indirect-feedback relaxation introduces policy-dependent dependence into the error sequence; no explicit argument or theorem shows that this dependence preserves the coverage probability without extra conservatism that would violate the original 1-ε level. This transfer step requires a dedicated proof or counter-example analysis.

Authors: We acknowledge the referee's concern regarding the explicit justification for coverage transfer from open-loop calibration to closed-loop operation. The manuscript relies on the fact that conformal prediction provides marginal coverage guarantees that hold under the assumption of exchangeable samples, and the indirect feedback formulation is designed such that the closed-loop error trajectories satisfy the same distributional properties as the open-loop ones for the purpose of coverage. However, to make this rigorous and address the potential policy-dependent dependence, we will add a new theorem in the revised version that proves the coverage probability is preserved at the nominal level without additional conservatism. This theorem will explicitly handle the dependence introduced by the state-dependent policy. revision: yes
Referee: [Numerical examples] The abstract and numerical-examples section assert effectiveness and advantages over existing methods, yet no quantitative coverage rates, violation frequencies, or ablation data (e.g., effect of calibration-set size on closed-loop performance) are reported. Without these metrics it is impossible to verify that the claimed finite-sample guarantees materialize in closed loop.

Authors: We agree that the numerical section would benefit from more detailed quantitative validation. In the revised manuscript, we will augment the numerical examples with tables showing empirical coverage rates over multiple simulations, frequencies of constraint violations, and ablation studies varying the size of the calibration dataset to illustrate its impact on closed-loop performance and computational effort. This will provide concrete evidence supporting the finite-sample guarantees. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external conformal prediction coverage

full rationale

The paper applies standard conformal prediction (an external statistical result with finite-sample coverage guarantees) to construct confidence regions for error trajectories, then uses these sets to relax joint chance constraints in an MPC formulation. No equations reduce the claimed recursive feasibility, closed-loop guarantees, or chance-constraint satisfaction back to parameters fitted from the same data or to self-citations whose validity depends on the present work. The central claims rest on the transfer of conformal coverage to closed-loop trajectories under the stated assumptions, which is an independent statistical property rather than a self-definitional or fitted-input reduction. This is the most common honest non-finding for papers that import an established method without re-deriving its core properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external validity of conformal prediction coverage for the specific closed-loop trajectories generated by the MPC law; no free parameters are introduced in the abstract, and no new entities are postulated.

axioms (1)

domain assumption Conformal prediction sets constructed from i.i.d. disturbance samples achieve the stated finite-sample coverage for the closed-loop error trajectories
Invoked when the abstract states that the sets enable relaxation while preserving chance-constraint satisfaction.

pith-pipeline@v0.9.0 · 5420 in / 1293 out tokens · 61355 ms · 2026-05-16T23:20:43.475836+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use CP to address a stochastic MPC problem under joint-in-time chance constraints under unknown probability distributions, exploiting its distribution-independence and low computational requirements.
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the set E={E|s(E)≤q̂} ... satisfies Pr(E(0)∈E)≥p

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Scenario Optimization for MPC,

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Scenario-Based Probabilistic Reach- able Sets for Recursively Feasible Stochastic Model Predictive Con- trol,

L. Hewing and M. N. Zeilinger, “Scenario-Based Probabilistic Reach- able Sets for Recursively Feasible Stochastic Model Predictive Con- trol,”IEEE Control Systems Letters, vol. 4, no. 2, pp. 450–455, Apr. 2020

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Data- Driven Distributed Stochastic Model Predictive Control with Closed- Loop Chance Constraint Satisfaction,

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V ovk, A

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A. N. Angelopoulos and S. Bates, “Conformal Prediction: A Gentle Introduction,”Foundations and Trends® in Machine Learning, vol. 16, no. 4, pp. 494–591, Mar. 2023

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Formal Verification and Control with Conformal Prediction,

L. Lindemann, Y . Zhao, X. Yu, G. J. Pappas, and J. V . Deshmukh, “Formal Verification and Control with Conformal Prediction,” Aug. 2024

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A randomized approach to Stochastic Model Predictive Control,

M. Prandini, S. Garatti, and J. Lygeros, “A randomized approach to Stochastic Model Predictive Control,” in2012 IEEE 51st IEEE Conference on Decision and Control (CDC), Dec. 2012, pp. 7315– 7320

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Recursive Feasibility of Stochastic Model Predictive Control with Mission-Wide Probabilistic Constraints,

K. Wang and S. Gros, “Recursive Feasibility of Stochastic Model Predictive Control with Mission-Wide Probabilistic Constraints,” in 2021 60th IEEE Conference on Decision and Control (CDC), Dec. 2021, pp. 2312–2317

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Recursively Feasible Shrinking-Horizon MPC in Dynamic Environments with Conformal Prediction Guarantees,

C. Stamouli, L. Lindemann, and G. J. Pappas, “Recursively Feasible Shrinking-Horizon MPC in Dynamic Environments with Conformal Prediction Guarantees,” May 2024

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Conformal Prediction for Distribution-Free Optimal Control of Linear Stochastic Systems,

E. E. Vlahakis, L. Lindemann, P. Sopasakis, and D. V . Dimarogonas, “Conformal Prediction for Distribution-Free Optimal Control of Linear Stochastic Systems,”IEEE Control Systems Letters, vol. 8, pp. 2835– 2840, 2024

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Conformal Prediction Under Covariate Shift,

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Conditional Validity of Inductive Conformal Predictors,

V . V ovk, “Conditional Validity of Inductive Conformal Predictors,” in Proceedings of the Asian Conference on Machine Learning. PMLR, Nov. 2012, pp. 475–490

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Conformal Prediction Regions for Time Series Using Linear Complementarity Programming,

M. Cleaveland, I. Lee, G. J. Pappas, and L. Lindemann, “Conformal Prediction Regions for Time Series Using Linear Complementarity Programming,”Proceedings of the AAAI Conference on Artificial Intelligence, vol. 38, no. 19, pp. 20 984–20 992, Mar. 2024

work page 2024

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Ro- bust output feedback model predictive control of constrained linear systems,

D. Q. Mayne, S. V . Rakovi ´c, R. Findeisen, and F. Allg ¨ower, “Ro- bust output feedback model predictive control of constrained linear systems,”Automatica, vol. 42, no. 7, pp. 1217–1222, Jul. 2006

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LQG for Constrained Linear Systems: Indirect Feedback Stochastic MPC with Kalman Filtering,

S. Muntwiler, K. P. Wabersich, R. Miklos, and M. N. Zeilinger, “LQG for Constrained Linear Systems: Indirect Feedback Stochastic MPC with Kalman Filtering,” in2023 European Control Conference (ECC), Jun. 2023, pp. 1–7

work page 2023

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AMPyC: Advanced Model Predictive Control in Python,

J. Sieber, A. Didier, R. Rickenbach, and M. Zeilinger, “AMPyC: Advanced Model Predictive Control in Python,” Jun. 2025

work page 2025