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arxiv: 2604.15252 · v1 · submitted 2026-04-16 · 📡 eess.SY · cs.SY· math.OC

Tube-Based Robust Data-Driven Predictive Control

Chi Wang , David Angeli This is my paper

Pith reviewed 2026-05-10 10:21 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords tube-based controldata-driven predictive controlrobust controlHankel matrixinput-to-state stabilityrobust positively invariant setslinear time-invariant systems
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The pith

A tube-based robust data-driven predictive controller for unknown discrete-time LTI systems can be synthesized from one finite noisy input-state trajectory and guarantees recursive feasibility along with practical input-to-state stability.

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

The paper constructs a tractable robust predictive control law that operates directly on a single noisy data record of an unknown linear system. A simplex constraint is placed on the vector of Hankel coefficients to produce explicit polyhedral bounds on the prediction error caused by bounded measurement noise. These bounds are then used to tighten the state and input constraints inside a tube-based formulation that employs certified initial and terminal robust positively invariant sets. The resulting online problem is a strictly convex quadratic program whose solution delivers recursive feasibility, robust constraint satisfaction, and practical input-to-state stability of the closed loop with respect to the noise.

Core claim

The central claim is that, for an unknown discrete-time LTI system, a single finite noisy input-state trajectory suffices to build a tube-based robust data-driven predictive controller: a simplex constraint on the Hankel coefficient vector yields polyhedral bounds on the noise-induced prediction mismatch; certified initial and terminal robust positively invariant sets then produce a tightened constraint set whose online optimization is a strictly convex quadratic program; the closed-loop system under this controller satisfies recursive feasibility, robust input and state constraint satisfaction, and practical input-to-state stability with respect to bounded measurement noise.

What carries the argument

The simplex constraint imposed on the Hankel coefficient vector, which supplies explicit polyhedral bounds on the prediction mismatch induced by bounded measurement noise, together with certified initial and terminal robust positively invariant sets for the tightened system.

If this is right

  • The online optimization reduces to a strictly convex quadratic program that can be solved reliably in real time.
  • Recursive feasibility holds for all future time steps once the initial problem is feasible.
  • Input and state constraints are satisfied robustly despite the bounded measurement noise.
  • The closed-loop trajectory remains practically input-to-state stable, converging to a neighborhood whose radius scales with the noise bound.

Where Pith is reading between the lines

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

  • The method's reliance on a single trajectory suggests it may be useful in settings where repeated experiments are costly or unsafe.
  • If similar polyhedral bounds can be derived for other uncertainty classes, the same tube-tightening strategy could be applied to nonlinear or switched systems.
  • The explicit polyhedral nature of the noise bounds opens the possibility of trading off online computation against offline set computation by adjusting the simplex dimension.

Load-bearing premise

The plant must be discrete-time linear time-invariant, the measurement noise must be bounded, and certified robust positively invariant sets for the tightened system must be computable in advance.

What would settle it

An explicit numerical counterexample consisting of a discrete-time LTI system, a bounded-noise trajectory, and a computed controller for which the online quadratic program either loses feasibility at some step or violates a state or input constraint for some admissible noise realization.

Figures

Figures reproduced from arXiv: 2604.15252 by Chi Wang, David Angeli.

Figure 1
Figure 1. Figure 1: Closed-loop state trajectories for the flight-vehicle example under the robust tube-based MPC of [13] and the proposed TRDDPC scheme. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: First-step prediction/rollout and the RPI polytope E (zoomed in (b)). chosen as L = 6. The stabilizing feedback gain K is computed according to Proposition 2, which simultaneously provides the design of the terminal weight PL. For comparison, we apply the same settings (Q, R, L, K) and the same noise level to the robust MPC scheme in [13]. The resulting closed-loop state trajectories for both controllers a… view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop state and input trajectories for the scalar system under the DDMMMPC of [31], the robust DeePC scheme of [28], and the proposed TRDDPC scheme. TABLE I COMPARISON OF DATA-DRIVEN ROBUST MPC SCHEMES Sum of stage costs J Avg. comp. time t¯ [s] Max. admissible ∥w∥∞ Design params. from noisy data? DDMMMPC 1.395332 5.426 × 10−2 2.06 × 10−1 partially1 Robust DeePC 1.376511 2.053 × 10−2 1.29 × 10−4 no2 … view at source ↗
read the original abstract

This paper presents a tractable tube-based robust data-driven predictive control scheme that uses only a single finite noisy input-state trajectory of an unknown discrete-time linear time-invariant (LTI) system. A simplex constraint is imposed on the Hankel coefficient vector, yielding explicit polyhedral bounds on the prediction mismatch induced by bounded measurement noise. Using certified initial and terminal robust positively invariant (RPI) sets, we derive a tube-tightened formulation whose online optimization problem is a strictly convex quadratic program (QP). The resulting controller guarantees recursive feasibility, robust satisfaction of input and state constraints, and practical input-to-state stability of the closed loop with respect to measurement noise. Numerical examples illustrate the effectiveness, robustness, and closed-loop performance of the proposed method.

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.

Referee Report

2 major / 2 minor

Summary. The paper presents a tube-based robust data-driven predictive control scheme for unknown discrete-time LTI systems that uses only a single finite noisy input-state trajectory. A simplex constraint is imposed on the Hankel coefficient vector to obtain explicit polyhedral bounds on the prediction mismatch induced by bounded measurement noise. Certified initial and terminal robust positively invariant (RPI) sets are then used to derive a tube-tightened formulation whose online problem is a strictly convex QP. The resulting controller is claimed to guarantee recursive feasibility, robust satisfaction of input and state constraints, and practical input-to-state stability of the closed loop with respect to measurement noise. Numerical examples illustrate the method.

Significance. If the guarantees hold, the work advances data-driven robust control by enabling explicit robustness margins and constraint satisfaction from a single noisy trajectory without an identified model. The reduction to a convex QP and the polyhedral mismatch bounds derived from the simplex constraint are technically valuable strengths that support practical deployment.

major comments (2)
  1. [Abstract and tube-based formulation section] Abstract and the main theoretical development (tube construction and feasibility arguments): The recursive feasibility, robust constraint satisfaction, and practical ISS claims rest on the existence of certified initial and terminal RPI sets for the tube-tightened data-driven system. No constructive procedure is supplied for obtaining or verifying these sets from the single noisy trajectory alone; standard RPI methods (set iteration or LMIs) require system matrices that remain unknown, leaving the central guarantees dependent on an unaddressed certification step.
  2. [Closed-loop analysis section] The section deriving the closed-loop properties: While the simplex constraint yields polyhedral mismatch bounds, the paper does not quantify how the resulting tube radius scales with the noise bound or data length, which directly affects the size of the RPI sets and the ultimate bound in the practical ISS statement.
minor comments (2)
  1. [Preliminaries] The notation for the Hankel matrix, coefficient vector, and simplex constraint should be introduced with an explicit definition and dimension table in the preliminaries to improve readability.
  2. [Numerical examples] Numerical examples: The plots would benefit from explicit indication of the computed tube boundaries and the tightened constraint sets to allow direct visual assessment of robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of the paper's contributions. We address each major comment below with clarifications and proposed revisions.

read point-by-point responses

  1. Referee: [Abstract and tube-based formulation section] Abstract and the main theoretical development (tube construction and feasibility arguments): The recursive feasibility, robust constraint satisfaction, and practical ISS claims rest on the existence of certified initial and terminal RPI sets for the tube-tightened data-driven system. No constructive procedure is supplied for obtaining or verifying these sets from the single noisy trajectory alone; standard RPI methods (set iteration or LMIs) require system matrices that remain unknown, leaving the central guarantees dependent on an unaddressed certification step.

    Authors: We acknowledge that the manuscript presents the tube-based formulation and associated guarantees under the assumption of certified RPI sets without detailing an explicit offline construction from the data. However, the polyhedral mismatch bounds derived via the simplex constraint on the Hankel coefficients enable a data-driven certification procedure that operates solely on the known error polytope and does not require the true system matrices. In the revised manuscript we will add a dedicated subsection describing how initial and terminal RPI sets can be computed by solving a convex optimization problem (e.g., via robust positive invariance LMIs or iterative set propagation) that uses only the mismatch bound obtained from the single noisy trajectory. This procedure directly supports the recursive feasibility and ISS arguments while remaining fully consistent with the data-driven setting. revision: yes

  2. Referee: [Closed-loop analysis section] The section deriving the closed-loop properties: While the simplex constraint yields polyhedral mismatch bounds, the paper does not quantify how the resulting tube radius scales with the noise bound or data length, which directly affects the size of the RPI sets and the ultimate bound in the practical ISS statement.

    Authors: We agree that an explicit scaling result would strengthen the theoretical claims and clarify the dependence of conservatism on data quality. In the revised closed-loop analysis we will derive an upper bound on the tube radius that is linear in the noise bound and inversely related to the data length (under the maintained persistency-of-excitation assumption). The derivation follows directly from the geometry of the simplex-constrained Hankel coefficients and the resulting polyhedral error set; the resulting expression will be substituted into the RPI-set size and the practical ISS ultimate bound to make the dependence explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit bounds from imposed constraint and standard RPI arguments

full rationale

The paper starts from one finite noisy trajectory, imposes a simplex constraint on the Hankel coefficient vector to derive explicit polyhedral mismatch bounds, and then applies tube tightening with externally certified initial/terminal RPI sets. Recursive feasibility, robust constraint satisfaction, and practical ISS are obtained by standard robust-control arguments on the tightened system. No equation reduces a claimed prediction or guarantee to a fitted parameter by construction, nor does any load-bearing step rely on self-citation or self-definition. The RPI certification assumption is external and does not create a circular reduction inside the paper's own chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The scheme rests on standard assumptions of discrete-time LTI dynamics and bounded measurement noise; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The unknown system is discrete-time linear time-invariant.
    Stated in the abstract as the class of systems considered.
  • domain assumption Measurement noise is bounded.
    Required to obtain explicit polyhedral bounds on prediction mismatch.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Data-Driven Synthesis of Robust Positively Invariant Sets from Noisy Data

    eess.SY 2026-03 unverdicted novelty 6.0

    A data-driven procedure constructs robust positively invariant tube sets from noisy data of unknown LTI systems and certifies them for use in tube-based robust predictive control.

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