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arxiv: 2511.18374 · v2 · submitted 2025-11-23 · 💻 cs.RO · cs.SY· eess.SY· math.DS

Explicit Bounds on the Hausdorff Distance for Truncated mRPI Sets via Norm-Dependent Contraction Rates

Jiaxun Sun , Hengyu Xue , Yuyang Zhao This is my paper

Pith reviewed 2026-05-17 06:27 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SYmath.DS
keywords Hausdorff distancemRPI setsrobust invariant setscontraction ratesnorm selectiontube MPCset approximationrobust stability
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The pith

A closed-form upper bound on the Hausdorff distance for any truncated minimal robust positively invariant set is given by a formula using only disturbance-set size and the induced-norm contraction factor of the system matrix.

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

The paper establishes that for a robustly stable linear system one can compute an explicit upper bound on how far a finite truncation of the minimal robust positively invariant set lies from the true infinite-horizon set. The bound is driven solely by a measure of the disturbance set and by the contraction rate induced by the system matrix under a chosen vector norm. Selecting the norm via diagonal weighting or Lyapunov-based methods can shrink the contraction factor and therefore tighten the resulting error certificate. This immediately produces an analytic rule for choosing the truncation horizon that meets any prescribed accuracy tolerance without performing repeated set operations. The approach directly supports more precise constraint tightening inside tube-based model predictive control.

Core claim

We derive a computable closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit. The bound depends only on a disturbance-set size measure and an induced-norm contraction factor of the system matrix, and it yields an explicit, fully analytic horizon-selection rule that guarantees a prescribed approximation tolerance without iterative set computations. The choice of vector norm enters as a design lever: norm shaping through diagonal or Lyapunov-based weighting tightens both the contraction factor and the resulting certificate.

What carries the argument

The induced-norm contraction factor of the system matrix under a suitably chosen vector norm, which multiplies the disturbance-set size to produce the explicit Hausdorff-distance upper bound.

If this is right

  • The required truncation length follows directly from the target error tolerance via a simple algebraic expression.
  • Norm shaping reduces the contraction factor and thereby produces a smaller bound for the same horizon.
  • Tube-based MPC can use the bound to tighten constraints with a known, non-conservative margin.
  • Horizon selection no longer requires any Minkowski-sum or set-iteration computations.

Where Pith is reading between the lines

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

  • The same contraction-rate technique could be applied to obtain error bounds for other families of invariant sets in switched or parameter-varying systems.
  • A numerical search over weighting matrices could be used to minimize the contraction factor for a given system and thereby obtain the tightest possible certificate.
  • The bound's scaling with state dimension could be characterized to predict computational effort in high-dimensional problems.
  • Similar explicit certificates might reduce conservatism when approximating reachable sets in robust reachability analysis.

Load-bearing premise

The system matrix admits at least one vector norm under which its induced norm is strictly less than one, and the disturbance set is compact.

What would settle it

For a low-dimensional stable linear system, iterate the set recursion to a large horizon to obtain a numerical Hausdorff distance for several truncation levels and verify whether the closed-form bound is ever exceeded.

Figures

Figures reproduced from arXiv: 2511.18374 by Hengyu Xue, Jiaxun Sun, Yuyang Zhao.

Figure 1
Figure 1. Figure 1: Six-dimensional example: numerical Hausdorff error and explicit [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experiment 2: theoretical bound under different induced norms. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experiment 3: numerical and theoretical errors for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment 4 (Part A): Enlarged nominal feasible set. Baseline tightening (green), based on the classical invariant radius rtube [13], [2], is significantly more conservative than our truncation-bound-based tightening (red). The resulting feasible set X ⊖ E is therefore much larger under our method. VII. CONCLUSION This paper established the first explicit and computable Hausdorff-distance bound between th… view at source ↗
read the original abstract

We derive a computable closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit. The bound depends only on a disturbance-set size measure and an induced-norm contraction factor of the system matrix, and it yields an explicit, fully analytic horizon-selection rule that guarantees a prescribed approximation tolerance without iterative set computations. The choice of vector norm enters as a design lever: norm shaping -- through diagonal or Lyapunov-based weighting -- tightens both the contraction factor and the resulting certificate, with direct consequences for robust invariant-set approximation and tube-based model predictive control (MPC) constraint tightening. Numerical examples illustrate the accuracy, scalability, and practical impact of the proposed bound.

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

0 major / 3 minor

Summary. The paper derives a computable closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit for linear systems subject to bounded disturbances. The bound is expressed solely in terms of a scalar measure of the disturbance set and the induced-norm contraction factor of the system matrix (under a chosen vector norm for which the operator norm of A is strictly less than one). This yields an explicit analytic rule for selecting the truncation horizon N to guarantee a prescribed approximation tolerance, without requiring iterative set computations. Norm shaping (diagonal or Lyapunov weighting) is presented as a design choice that simultaneously reduces the contraction factor and tightens the bound, with direct implications for robust invariant-set approximation and tube-based MPC constraint tightening. Numerical examples are used to illustrate accuracy and scalability.

Significance. If the central derivation holds, the result supplies a practical, non-iterative certificate for mRPI-set truncation that is directly usable in robust control synthesis. The explicit dependence on the contraction rate and disturbance measure, together with the parameter-free character of the final expression once the norm is fixed, constitutes a clear strength for both theoretical analysis and numerical implementation. The treatment of norm selection as an optimizable design lever is a useful contribution that can improve bound tightness in applications such as tube MPC.

minor comments (3)
  1. [§2] §2, Definition 1: the precise scalar measure used for the size of the disturbance set W (e.g., radius in the chosen norm) should be stated explicitly rather than left as a generic “size measure,” to make the bound fully reproducible from the text alone.
  2. [§4] §4, Algorithm 1: the pseudocode for horizon selection would benefit from an explicit statement of the arithmetic operations required to evaluate the closed-form expression, including how the chosen norm is incorporated.
  3. [Figure 3] Figure 3: the caption should indicate the specific system dimensions and the norm employed in each subplot so that the reader can directly relate the plotted error decay to the contraction factor ρ reported in the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contribution—an explicit, non-iterative bound on the Hausdorff distance for truncated mRPI sets that depends only on the disturbance-set measure and the induced-norm contraction factor—and its utility for horizon selection in robust control and tube MPC. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central bound is obtained by applying the contraction-mapping theorem to the set-valued recursion S_{k+1}=A S_k ⊕ W in the Hausdorff metric induced by a vector norm whose induced operator norm satisfies ||A||<1. The tail distance is then bounded by a standard geometric-series sum whose only inputs are the contraction factor ρ and a scalar size measure of the compact disturbance set W. This construction is parameter-free once the norm is chosen and does not reduce to any fitted quantity, self-referential definition, or load-bearing self-citation. Norm shaping is presented as an explicit design choice that improves ρ and the numerical value of the bound, without smuggling in prior ansatzes or renaming known results. The argument therefore stands on independent mathematical properties of contraction mappings and Minkowski sums.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard definitions of mRPI sets and induced norms from prior robust-control literature; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The discrete-time linear system is robustly stable, i.e., there exists a vector norm such that the induced norm of the system matrix is strictly less than one.
    Required for the infinite-horizon mRPI set to exist and for the contraction factor to be usable in the bound.
  • domain assumption The disturbance set is compact and bounded.
    Needed to define a finite size measure that enters the error bound.

pith-pipeline@v0.9.0 · 5437 in / 1403 out tokens · 64486 ms · 2026-05-17T06:27:32.109412+00:00 · methodology

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