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arxiv: 2606.29536 · v1 · pith:LTGGF36Wnew · submitted 2026-06-28 · 📡 eess.SY · cs.SY

High-Probability ISS Tubes for Continuous-Time State Estimation

Jerzy Baranowski This is my paper

Pith reviewed 2026-06-30 01:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords input-to-state stabilitystate estimationprobabilistic boundscontinuous-time systemsLyapunov inequalitiespositive systemserror tubesdisturbance envelope
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The pith

If the aggregated disturbance obeys a probabilistic envelope in the essential-supremum sense, deterministic ISS bounds immediately produce high-probability error tubes for continuous-time state estimation.

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

The paper shows that existing deterministic input-to-state stability bounds on estimation-error dynamics become high-probability guarantees once the disturbance satisfies a simple probabilistic envelope condition. This conversion requires only that the error system admit an ISS bound certifiable by quadratic Lyapunov inequalities. The construction is specialized to positive and cooperative systems and demonstrated on a compartment model, where the resulting tubes are compared against Kalman-Bucy and moving-horizon estimators. A reader would care because the approach supplies a computationally inexpensive probabilistic uncertainty baseline without requiring a full stochastic model of the disturbance.

Core claim

When the aggregated disturbance satisfies a probabilistic envelope in the essential-supremum sense, deterministic ISS bounds on the estimation-error dynamics immediately induce high-probability error tubes. Explicit sufficient conditions are obtained from quadratic Lyapunov inequalities, with specializations to positive and cooperative systems. The method is illustrated on a positive compartment model with aggregated measurements, where the ISS tubes serve as a conservative but light uncertainty baseline relative to Kalman-Bucy, Gaussian, and robust moving-horizon estimators.

What carries the argument

The probabilistic envelope condition on the aggregated disturbance (in the essential-supremum sense), which directly converts any deterministic ISS bound into a high-probability error tube.

If this is right

  • Any existing deterministic ISS bound certified by quadratic Lyapunov inequalities immediately yields a high-probability error tube once the disturbance envelope holds.
  • The construction applies without change to positive and cooperative systems.
  • ISS tubes supply a conservative but computationally light uncertainty baseline that can be computed faster than Kalman-Bucy or moving-horizon estimators.
  • Robust moving-horizon estimation remains less sensitive to outlier contamination than Gaussian-based estimators on the same data.

Where Pith is reading between the lines

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

  • The same envelope-to-tube conversion could be tested on discrete-time or sampled-data estimators if analogous envelope definitions are introduced.
  • The tubes could serve as cheap initial uncertainty sets that are later tightened by more expensive stochastic filters.
  • In safety-critical applications the method offers a quick way to obtain explicit probabilistic bounds before full stochastic analysis is performed.

Load-bearing premise

The estimation-error dynamics must admit an ISS bound that can be certified via quadratic Lyapunov inequalities, and the disturbance process must satisfy the stated probabilistic envelope in the ess-sup sense.

What would settle it

An explicit counterexample in which the disturbance meets the probabilistic envelope yet the estimation error exceeds the ISS-derived tube with probability strictly higher than the envelope allows.

Figures

Figures reproduced from arXiv: 2606.29536 by Jerzy Baranowski.

Figure 1
Figure 1. Figure 1: Representative trajectories and uncertainty descriptions for the most in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical coverage of nominal 95% state bands over time under regimes [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Negativity rate of state estimates for the positive-system example under [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

This paper studies a probabilistic interpretation of input-to-state stability (ISS) bounds for estimation-error dynamics in continuous-time systems. We show that, if the aggregated disturbance satisfies a probabilistic envelope in an essential-supremum sense, then deterministic ISS bounds immediately induce high-probability error tubes. To make this interpretation constructive, we also provide explicit sufficient conditions based on quadratic Lyapunov inequalities and specialize them to positive and cooperative systems. The approach is illustrated on a positive compartment model with aggregated measurements, where ISS tubes are compared with nominal uncertainty bands produced by a Kalman--Bucy filter and by Gaussian and robust moving-horizon estimators. The examples show that ISS tubes provide a conservative but computationally light uncertainty baseline, while robust MHE is less sensitive to outlier contamination than Gaussian-based

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 claims that if an aggregated disturbance process satisfies a probabilistic envelope condition in the essential-supremum sense, then any deterministic ISS bound on the continuous-time estimation-error dynamics immediately yields high-probability error tubes. It supplies explicit sufficient conditions for the ISS property via quadratic Lyapunov inequalities, specializes the conditions to positive and cooperative systems, and illustrates the resulting tubes on a positive compartment model with aggregated measurements, comparing them against Kalman-Bucy, Gaussian MHE, and robust MHE estimators.

Significance. If the central implication holds, the work supplies a lightweight, Lyapunov-based route to high-probability uncertainty quantification that re-uses standard deterministic ISS machinery rather than requiring a full stochastic analysis. The specialization to positive systems and the explicit quadratic conditions are constructive; the numerical example demonstrates that the tubes are conservative yet inexpensive to compute relative to robust MHE.

minor comments (3)
  1. [§3] §3, after Eq. (8): the transition from the deterministic ISS inequality to the high-probability tube is presented as immediate, but a short explicit statement of the measure-theoretic argument (using the definition of the ess-sup envelope) would improve readability.
  2. [§5] §5, Fig. 2: the caption does not state the numerical values of the probability level p and the envelope function eta used to generate the displayed ISS tubes, making direct reproduction difficult.
  3. [Abstract] The final sentence of the abstract is truncated; the published version should complete the comparison clause.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the recognition of the lightweight Lyapunov-based route to high-probability tubes, and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct logical implication from standard ISS and envelope definitions

full rationale

The paper's core result is that a deterministic ISS bound plus a probabilistic ess-sup envelope on the disturbance immediately yields high-probability tubes. This is a definitional implication, not a reduction to fitted quantities or self-citations. Quadratic Lyapunov conditions are supplied to certify the ISS bound (standard for linear/cooperative systems) and are independent of the probabilistic step. No equations rename a fit as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The derivation chain is self-contained against external benchmarks (ISS theory, Lyapunov inequalities) and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from ISS theory and Lyapunov analysis for continuous-time systems; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Estimation-error dynamics admit an ISS property with respect to aggregated disturbances
    Required to convert deterministic bounds into probabilistic tubes.
  • domain assumption Quadratic Lyapunov inequalities are sufficient to certify the ISS property
    Used to provide explicit sufficient conditions and specialization to positive systems.

pith-pipeline@v0.9.1-grok · 5647 in / 1320 out tokens · 40216 ms · 2026-06-30T01:58:57.681961+00:00 · methodology

discussion (0)

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Reference graph

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