arxiv: 2603.27374 · v2 · submitted 2026-03-28 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Safe Adaptive-Sampling Control via Robust M-Step Hold Model Predictive Control

Spencer Schutz , Charlott Vallon , Francesco Borrelli

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:47 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords model predictive controlrobust controladaptive samplinginvariant setsconstraint satisfactionmulti-step holdcruise control

0 comments

The pith

Robust M-step hold MPC allows online adjustment of input hold duration while guaranteeing constraint satisfaction in uncertain discrete-time systems.

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

The paper introduces a robust formulation of model predictive control that incorporates an adaptable multi-step input hold, called M-step hold, to support adaptive sampling rates during operation. It extends robust invariant sets to these variable holds so that the controller maintains recursive feasibility and robustly satisfies state and input constraints even when the underlying system model is uncertain. A sympathetic reader would care because real-time systems often need to vary control frequency to save computation or respond to changing conditions, yet doing so without a systematic guarantee risks violating safety limits. The formulation is demonstrated on a cruise control task where M can be chosen online without loss of safety.

Core claim

The central claim is that robust M-step hold MPC provides robust constraint satisfaction for an uncertain discrete-time system model with a fixed sampling time subject to an adaptable multi-step input hold. Recursive feasibility of the MPC is ensured by M-step hold extensions of robust invariant sets, and safe adaptive-sampling control is enabled by the online selection of M.

What carries the argument

The robust M-step hold MPC formulation, which extends robust invariant sets to variable-length input holds so that feasibility and constraint satisfaction are preserved when M is changed online.

If this is right

Recursive feasibility of the MPC is maintained across arbitrary online selections of M.
Robust satisfaction of state and input constraints holds for any admissible M.
Safe adaptive-sampling control is achieved in uncertain systems by treating M as a decision variable at each step.
The approach applies directly to tasks such as cruise control where sampling rate can be adjusted without jeopardizing safety.

Where Pith is reading between the lines

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

The same extension technique could be applied to other predictive controllers that vary their prediction horizon length.
Integration with online uncertainty estimation would allow M to be chosen based on current model confidence rather than worst-case bounds.
The formulation may reduce average computational cost in embedded systems by permitting larger M when the state is far from constraint boundaries.

Load-bearing premise

Robust invariant sets computed for the uncertain system can be extended to M-step input holds while keeping the invariance and feasibility properties intact.

What would settle it

A simulation or experiment in which an online change of M produces a state constraint violation under the stated model uncertainty bounds.

read the original abstract

In adaptive-sampling control, the control frequency can be adjusted during task execution. Ensuring that these changes do not jeopardize the safety of the system being controlled requires attention. We introduce robust M-step hold model predictive control (MPC) to address this. Our formulation provides robust constraint satisfaction for an uncertain discrete-time system model with a fixed sampling time subject to an adaptable multi-step input hold (referred to as M-step hold). We show how to ensure recursive feasibility of the MPC utilizing M-step hold extensions of robust invariant sets, and demonstrate how to enable safe adaptive-sampling control via the online selection of M. We evaluate the utility of the robust M-step hold MPC formulation in a cruise control example.

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

Robust M-step hold MPC for safe adaptive sampling, with a question mark on switching feasibility.

read the letter

The main thing to know is that this paper develops a robust MPC scheme with variable M-step input holds to support safe online adaptive sampling in uncertain discrete-time systems, using extended robust invariant sets for recursive feasibility. It does a solid job laying out the problem and proposing a structured way to adjust the hold length without losing safety. The cruise control example helps show potential use in automotive settings where you might reduce control effort when the system is stable. The formulation appears new in combining the M-step hold with online selection for adaptive sampling under uncertainty. Credit to the authors for focusing on a practical extension rather than a completely new framework. The soft spot is the recursive feasibility under switches. The invariant sets are built for specific M, but when M changes, the successor state after an M-hold must land in a set for the new M'. Because the multi-step reachable sets differ, this compatibility isn't automatic. The paper claims it works via the extensions, but the example likely doesn't probe frequent or arbitrary switches, so the online selection rule needs to be robust to that. If the full proofs address the transition conditions explicitly, it would strengthen the result. The abstract mentions demonstrating the approach in the cruise control example, but without quantitative results or details on how M is selected online, it's hard to gauge the practical performance. This paper is for control engineers and researchers working on robust MPC for resource-aware applications. A reader familiar with invariant sets in MPC would get the most out of it and could see how to adapt it to their problems. I recommend putting it through peer review to check the set constructions and the switching guarantees in detail.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a robust M-step hold model predictive control (MPC) formulation for uncertain discrete-time systems with fixed sampling time but adaptable multi-step input holds. It claims to achieve robust constraint satisfaction and recursive feasibility by extending robust invariant sets to these M-step holds, thereby enabling safe adaptive-sampling control through online selection of the hold parameter M, with utility shown in a cruise control example.

Significance. If the central claims hold, the work would provide a practical extension of robust MPC to variable input-hold lengths, allowing real-time adjustment of control frequency while preserving safety guarantees under uncertainty. This addresses a relevant problem in embedded and resource-constrained control applications. The approach builds directly on standard robust invariant-set techniques without introducing circular definitions, and the cruise-control demonstration offers a concrete testbed, though quantitative results are not detailed in the abstract.

major comments (1)

[Recursive feasibility via M-step invariant sets] The recursive feasibility argument via M-step hold extensions of robust invariant sets (as stated in the abstract and the section on recursive feasibility) requires that, after applying an M-hold, the successor state lies inside the terminal set for any subsequently chosen M'. Because the M-step reachable sets under uncertainty are different for each M, the corresponding robust invariant sets are generally incomparable. The manuscript must supply an explicit set construction, inclusion proof, or switching condition that guarantees this compatibility; without it the online M-selection claim is unsupported. The cruise-control example does not appear to exercise arbitrary online switches, leaving the property unverified.

minor comments (1)

[Abstract] The abstract asserts robust constraint satisfaction and recursive feasibility but supplies no explicit derivations, set constructions, or quantitative results from the cruise-control example, making evaluation of the central claims difficult.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. The major comment raises a valid point about ensuring explicit compatibility in the recursive feasibility argument for arbitrary online M-selection. We address this below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: The recursive feasibility argument via M-step hold extensions of robust invariant sets (as stated in the abstract and the section on recursive feasibility) requires that, after applying an M-hold, the successor state lies inside the terminal set for any subsequently chosen M'. Because the M-step reachable sets under uncertainty are different for each M, the corresponding robust invariant sets are generally incomparable. The manuscript must supply an explicit set construction, inclusion proof, or switching condition that guarantees this compatibility; without it the online M-selection claim is unsupported. The cruise-control example does not appear to exercise arbitrary online switches, leaving the property unverified.

Authors: We agree that the compatibility of terminal sets across different M values requires an explicit construction to rigorously support arbitrary online switches. In the revised manuscript, we will add a dedicated subsection detailing the M-step robust invariant set construction. We define the common terminal set as the intersection over all admissible M of the individual M-step robust invariant sets. We will prove that this intersection remains robustly positively invariant under any sequence of M-holds, because any M-step reachable set under uncertainty is contained in the union of the individual invariant sets, and the intersection ensures the successor state lies in the terminal set irrespective of the next chosen M'. This guarantees recursive feasibility for the online M-selection. We will also extend the cruise-control example with additional simulations that include arbitrary online switches between M values (e.g., cycling through M=1,2,3) to empirically confirm the property under uncertainty. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends standard robust invariant sets independently

full rationale

The paper constructs M-step hold extensions of robust invariant sets to guarantee recursive feasibility for uncertain systems under adaptable input holds, then uses online M selection for adaptive sampling. This chain relies on established robust MPC theory and set invariance properties without reducing any claimed prediction or feasibility result to a fitted parameter, self-definition, or self-citation load-bearing step. No equations redefine a quantity in terms of itself or rename an empirical pattern as a new unification. The cruise-control demonstration validates the construction rather than serving as its basis. The derivation remains self-contained against external benchmarks in robust control.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach relies on standard domain assumptions from robust MPC; no new free parameters or invented entities are identified in the abstract.

axioms (2)

domain assumption Existence of robust invariant sets for the uncertain discrete-time system
Invoked to guarantee recursive feasibility via M-step extensions
domain assumption The system admits a fixed sampling time with bounded uncertainty
Required for the robust constraint satisfaction formulation

pith-pipeline@v0.9.0 · 5412 in / 1220 out tokens · 79001 ms · 2026-05-14T21:47:24.774412+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.lean reality_from_one_distinction unclear

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

We show how to ensure recursive feasibility of the MPC utilizing M-step hold extensions of robust invariant sets
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

robust M-step hold control invariant set C^M_∞ ... maximal robust M-step hold positive invariant set O^M_∞

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.