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arxiv: 2604.04758 · v1 · submitted 2026-04-06 · 📡 eess.SY · cs.SY

Data-Driven Reachability Analysis with Optimal Input Design

Peng Xie , Davide M. Raimondo , Rolf Findeisen , Amr Alanwar This is my paper

Pith reviewed 2026-05-10 19:18 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven reachability analysisconstrained matrix zonotopeoptimal input designright inversereachable setssafety verificationlinear systemspiecewise affine systems
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The pith

Replacing the pseudoinverse with a row-norm right inverse and applying A-optimal input design reduces conservatism in data-driven reachability analysis.

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

This paper sets out to demonstrate that reachable-set over-approximations computed from finite input-state trajectories become substantially tighter when two specific changes are made inside an existing set-membership framework. The first change swaps the conventional pseudoinverse for a right inverse whose rows have minimal Euclidean norm, obtained by solving a second-order cone program; the second change selects the next inputs online according to an A-optimal design criterion. A sympathetic reader would care because reachable-set methods are routinely used to certify safety of systems whose dynamics are only partially known, and excess conservatism frequently renders the certificates useless even when the underlying system is safe.

Core claim

Within the constrained matrix zonotope representation of uncertain linear dynamics recovered from noisy input-state data, the Moore-Penrose pseudoinverse is replaced by a row-norm-minimizing right inverse computed via second-order cone programming, and the data-collection process is augmented by an online A-optimal input design. Both modifications shrink the resulting model set, yielding zonotopes with smaller generators. The combined procedure extends to piecewise-affine systems by partitioning trajectories according to active modes and is shown on a five-dimensional stable LTI system and a two-dimensional piecewise-affine system to produce markedly tighter reachable sets than the baseline,

What carries the argument

The row-norm-minimizing right inverse (computed via second-order cone program) together with A-optimal online input design, inside the constrained matrix zonotope that encloses all matrices consistent with observed trajectories and bounded process noise.

If this is right

  • The volume of the reachable-set over-approximation decreases, so safety-verification problems that previously returned inconclusive results can now return positive certificates.
  • The same two modifications apply without alteration to piecewise-affine systems once data are partitioned by mode.
  • Because input design can be performed online, the quality of the model set can improve continuously during operation rather than requiring a separate offline experiment.
  • Smaller generators translate directly into less conservative bounds on future states, supporting tighter planning and control under uncertainty.

Where Pith is reading between the lines

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

  • The same levers—choice of matrix inverse and quality of excitation—could be examined in other set-valued identification tasks that rely on matrix factorizations from finite data.
  • Numerical comparison on unstable or higher-dimensional systems would show whether the observed reduction in conservatism persists when the underlying dynamics are harder to excite.
  • The approach suggests that future data-driven verification pipelines may gain more from deliberate input selection and inverse design than from simply collecting longer trajectories.

Load-bearing premise

The constrained matrix zonotope correctly contains every possible system matrix consistent with the finite data and bounded noise, and neither the row-norm objective nor the A-optimal design compromises the soundness of the resulting over-approximation.

What would settle it

On the five-dimensional stable LTI example, simulate the true (but hidden) dynamics with the same inputs used for data collection and check whether any realized state trajectory ever leaves the computed reachable set; if it does, the over-approximation claim is false.

Figures

Figures reproduced from arXiv: 2604.04758 by Amr Alanwar, Davide M. Raimondo, Peng Xie, Rolf Findeisen.

Figure 1
Figure 1. Figure 1: Reachable-set comparison on the five-dimensional LTI system, projected onto [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: PWA system reachable-set comparison over 10 steps. Black filled: ˆ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

This paper addresses the conservatism in data-driven reachability analysis for discrete-time linear systems subject to bounded process noise, where the system matrices are unknown and only input--state trajectory data are available. Building on the constrained matrix zonotope (CMZ) framework, two complementary strategies are proposed to reduce conservatism in reachable-set over-approximations. First, the standard Moore--Penrose pseudoinverse is replaced with a row-norm-minimizing right inverse computed via a second-order cone program (SOCP), which directly reduces the size of the resulting model set, yielding tighter generators and less conservative reachable sets. Second, an online A-optimal input design strategy is introduced to improve the informativeness of the collected data and the conditioning of the resulting model set, thereby reducing uncertainty. The proposed framework extends naturally to piecewise affine systems through mode-dependent data partitioning. Numerical results on a five-dimensional stable LTI system and a two-dimensional piecewise affine system demonstrate that combining designed inputs with the row-norm right inverse significantly reduces conservatism compared to a baseline using random inputs and the pseudoinverse, leading to tighter reachable sets for safety verification.

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 / 3 minor

Summary. The paper addresses conservatism in data-driven reachability analysis for discrete-time linear systems with unknown matrices and bounded process noise using constrained matrix zonotopes (CMZ). It proposes replacing the Moore-Penrose pseudoinverse with a row-norm-minimizing right inverse via SOCP and an online A-optimal input design strategy. The framework is extended to piecewise affine systems, and numerical results on a 5D LTI and 2D PWA system show reduced conservatism and tighter reachable sets.

Significance. If the central claims hold, this work is significant for improving the practicality of data-driven reachability analysis in safety-critical applications. By optimizing the right inverse and input design, it reduces over-approximation conservatism without sacrificing soundness, as the underlying consistent model set is preserved while the zonotope generators are tightened. This could lead to less conservative safety verification in control systems.

major comments (2)
  1. [Abstract] Abstract: The claim that the row-norm right inverse 'directly reduces the size of the resulting model set' is imprecise. The set of consistent matrices is independent of the right inverse R (equal to (X^+ - W)R for bounded W); the benefit arises from a zonotope generator representation with smaller row sums that tightens subsequent reachability propagations.
  2. [Numerical results] Numerical results section: The reported improvements on the 5D LTI and 2D PWA examples lack quantitative metrics (e.g., reachable-set volumes or reduction percentages), error bars, multiple random seeds, or statistical significance tests, undermining assessment of the magnitude and reliability of the conservatism reduction.
minor comments (3)
  1. [Introduction] Define all acronyms at first use (CMZ, SOCP, LTI, PWA) and ensure consistent notation for the constrained matrix zonotope throughout.
  2. [Input design] Elaborate on the computational complexity and real-time feasibility of the online A-optimal input design procedure.
  3. [Numerical results] Include a brief comparison table summarizing reachable-set sizes or conservatism metrics across the four method combinations (random/pseudoinverse vs. designed/row-norm).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the positive assessment and constructive feedback on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the row-norm right inverse 'directly reduces the size of the resulting model set' is imprecise. The set of consistent matrices is independent of the right inverse R (equal to (X^+ - W)R for bounded W); the benefit arises from a zonotope generator representation with smaller row sums that tightens subsequent reachability propagations.

    Authors: We agree that the abstract wording is imprecise. The set of consistent matrices is indeed independent of the specific right inverse chosen. The row-norm-minimizing right inverse improves the zonotope generator representation by yielding smaller row sums, which tightens the subsequent reachability propagations without altering the underlying consistent model set. We will revise the abstract to clarify this distinction and accurately describe the mechanism. revision: yes

  2. Referee: [Numerical results] Numerical results section: The reported improvements on the 5D LTI and 2D PWA examples lack quantitative metrics (e.g., reachable-set volumes or reduction percentages), error bars, multiple random seeds, or statistical significance tests, undermining assessment of the magnitude and reliability of the conservatism reduction.

    Authors: We acknowledge that quantitative metrics would strengthen the numerical results section. Although the current figures illustrate qualitative tightening, we will add explicit computations of reachable-set volumes, percentage reductions relative to the baseline, results from multiple random seeds, and standard deviations in the revised manuscript to provide a more rigorous assessment of the improvements. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation uses the constrained matrix zonotope (CMZ) representation of consistent system matrices from finite trajectories, then applies an SOCP to select a row-norm-minimizing right inverse (which yields an equivalent set but with smaller generators for tighter zonotope propagation) and an online A-optimal design for input selection. Neither step defines the reachable set in terms of itself; both are independent optimization procedures whose outputs feed forward into standard reachability recursion. Numerical validation on LTI and PWA examples is external to the construction. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the existing constrained matrix zonotope representation and standard assumptions of bounded noise and linear (or piecewise-affine) dynamics; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption System is discrete-time linear (or piecewise affine) with bounded process noise and unknown matrices
    Explicitly stated as the problem setting in the abstract
  • domain assumption Only finite input-state trajectory data are available
    Core premise of the data-driven setting

pith-pipeline@v0.9.0 · 5500 in / 1468 out tokens · 50300 ms · 2026-05-10T19:18:18.649936+00:00 · methodology

discussion (0)

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Reference graph

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