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arxiv: 2509.16836 · v1 · submitted 2025-09-20 · 📡 eess.SY · cs.SY

Prescribed-Time Observer Is Naturally Robust Against Disturbances and Uncertainties

Abedou Abdelhadi , Mameche Omar This is my paper

Pith reviewed 2026-05-18 15:16 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords prescribed-time observerrobustnessdisturbancesunmodeled dynamicsnonlinear systemsstate estimationhigh-gain observerpeaking phenomenon
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The pith

A prescribed-time observer for nonlinear systems completely rejects the effects of arbitrarily large bounded disturbances and unmodeled dynamics.

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

This paper proves that a prescribed-time observer for a class of nonlinear systems is inherently robust to disturbances and uncertainties. It shows the observer can estimate both the system states and the disturbances accurately regardless of how large the bounded disturbances become. Simulations confirm the complete rejection of these effects and compare the approach to standard high-gain observers, noting reduced peaking and higher accuracy. Readers would care because many practical systems include unknown forces or modeling errors that typically degrade estimator performance.

Core claim

The paper establishes that the prescribed-time observer completely rejects the effects of arbitrarily large bounded disturbances and unmodeled dynamics. This enables accurate estimation of both the states and the disturbances for the class of nonlinear systems considered. The result is proven analytically and verified in simulations, with an explicit comparison showing advantages over the standard high-gain observer in reducing the peaking phenomenon and improving overall estimation accuracy.

What carries the argument

The prescribed-time observer, whose convergence time is fixed in advance by the designer and which carries an internal structure that cancels the impact of bounded additive disturbances and uncertainties.

If this is right

  • Accurate state estimates are obtained without explicit knowledge or compensation of the disturbances.
  • Disturbance signals themselves are estimated as a byproduct for potential use in control laws.
  • The peaking effect during transient response is smaller than with high-gain observers.
  • Estimation accuracy holds even when unmodeled dynamics are present and large.

Where Pith is reading between the lines

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

  • The built-in rejection could allow the observer to be inserted directly into feedback loops on uncertain plants without extra disturbance-rejection layers.
  • The same structure might be tested on mechanical systems such as robotic arms or vehicles where disturbances arise from friction or wind.
  • If the boundedness condition is relaxed to slowly growing signals, the observer might still deliver useful estimates for a limited interval.

Load-bearing premise

The disturbances and unmodeled dynamics remain bounded for all time and the nonlinear system belongs to the specific class for which the prescribed-time observer is constructed.

What would settle it

If simulations or experiments apply a very large but bounded disturbance to a qualifying nonlinear system and the state or disturbance estimates fail to converge to their true values within the prescribed time, the central claim would be disproven.

Figures

Figures reproduced from arXiv: 2509.16836 by Abedou Abdelhadi, Mameche Omar.

Figure 2
Figure 2. Figure 2: System state x2 and its estimate xˆ2 using the prescribed-time observer [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Norm of the estimation error ∥x − xˆ∥ using the prescribed-time observer [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: System state x1 and its estimate xˆ1 comparison. state while preserving its triangular structure. The considered nonlinear system is the same as in Example [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: System state x2 and its estimate xˆ2 comparison [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: x2 and its estimate xˆ2 using the extended prescribed-time observer [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The disturbance d and its estimate dˆ using the extended prescribed￾time observer. V. CONCLUSION In this work, the robustness of a prescribed-time observer for a class of nonlinear systems has been rigorously analyzed [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
read the original abstract

This paper addresses the robustness of a prescribed-time observer for a class of nonlinear systems in the presence of disturbances and unmodeled dynamics. It is proven and demonstrated through simulations that the proposed observer completely rejects the effects of arbitrarily large bounded disturbances and unmodeled dynamics, enabling accurate estimation of both the states and the disturbances. Furthermore, a comparison with the standard high-gain observer is provided to highlight the superiority of the prescribed-time observer in reducing the peaking phenomenon and improving estimation accuracy.

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 claims to prove that a prescribed-time observer for a class of nonlinear systems is naturally robust to arbitrarily large bounded disturbances and unmodeled dynamics. It asserts that the observer achieves exact convergence of state and disturbance estimates at a user-specified time T independent of disturbance magnitude, supported by Lyapunov analysis and numerical simulations that also show reduced peaking relative to high-gain observers.

Significance. If the central claim of disturbance rejection at fixed prescribed time holds without retuning or dependence on disturbance bounds, the result would be significant for nonlinear observer design in uncertain systems. It would provide a strong robustness property that avoids the usual trade-offs between convergence speed and disturbance size, with practical value in applications requiring reliable estimation under severe uncertainties.

major comments (2)
  1. [§3] §3 (Error-system analysis) and the subsequent Lyapunov argument: the differential inequality satisfied by V includes an additive term bounded by D (the disturbance magnitude). Application of the comparison lemma then produces an explicit upper bound on the settling time that grows with both the initial error and D. This directly contradicts the headline claim that a single fixed-T observer completely rejects disturbances of arbitrary finite magnitude.
  2. [Theorem 1] Theorem 1 (or equivalent main result): the proof must explicitly demonstrate that the prescribed-time convergence property is preserved for any finite D without requiring T to increase or gains to be retuned. The current presentation treats D as an arbitrary constant but does not resolve the dependence of the comparison-system solution on D.
minor comments (2)
  1. [Abstract] The abstract states that the observer 'completely rejects' disturbances but does not specify the precise class of nonlinear systems or the boundedness assumption on disturbances; this should be stated explicitly.
  2. [Simulations] Simulation section: the disturbance signals and their magnitudes should be plotted alongside the estimation errors to make the rejection property visually clear; current figures focus mainly on state estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to strengthen the explicit demonstration of disturbance-independent prescribed-time convergence.

read point-by-point responses

  1. Referee: [§3] §3 (Error-system analysis) and the subsequent Lyapunov argument: the differential inequality satisfied by V includes an additive term bounded by D (the disturbance magnitude). Application of the comparison lemma then produces an explicit upper bound on the settling time that grows with both the initial error and D. This directly contradicts the headline claim that a single fixed-T observer completely rejects disturbances of arbitrary finite magnitude.

    Authors: We appreciate the referee highlighting the need for clarity in the comparison argument. The differential inequality is of the form Ḋ ≤ −α(t)V + D, where the time-varying coefficient α(t) diverges to +∞ as t → T. The explicit solution of the associated comparison system is V(t) ≤ ϕ(t) [V(0) + ∫₀ᵗ (D/ϕ(s)) ds], with ϕ(t) = exp(−∫ α(τ) dτ). Because ϕ(t) → 0 as t → T at a rate that dominates the accumulated integral term for any finite D, the upper bound on V(t) vanishes exactly at the prescribed time T. We have added this closed-form derivation to the revised §3 to eliminate any ambiguity regarding dependence on D. revision: yes

  2. Referee: [Theorem 1] Theorem 1 (or equivalent main result): the proof must explicitly demonstrate that the prescribed-time convergence property is preserved for any finite D without requiring T to increase or gains to be retuned. The current presentation treats D as an arbitrary constant but does not resolve the dependence of the comparison-system solution on D.

    Authors: We agree that the main result benefits from an explicit resolution of the D-dependence. In the revised manuscript, the proof of Theorem 1 now includes the closed-form solution of the comparison system and directly verifies that the estimation-error bound reaches zero at the user-specified T for any finite D, without any retuning of T or the observer parameters. This confirms that the prescribed-time property is retained under arbitrarily large but bounded disturbances and unmodeled dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity; robustness claim derived directly from observer structure and comparison lemma

full rationale

The derivation proceeds from the prescribed-time observer gains (which diverge at the fixed T) to an error-system Lyapunov inequality that absorbs an arbitrary bounded disturbance D via the comparison lemma, yielding exact convergence at T independent of D's magnitude. No step reduces the result to a fitted parameter, self-definition, or load-bearing self-citation; the central theorem is self-contained under the stated boundedness assumption and does not rename or smuggle prior results. External benchmarks (standard high-gain comparison) are used only for illustration, not as justification for the main claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard boundedness assumptions for disturbances and on the system satisfying the structural conditions needed for the prescribed-time design; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Disturbances and unmodeled dynamics are bounded.
    Boundedness is invoked to guarantee complete rejection of their effects.
  • domain assumption The plant belongs to the class of nonlinear systems admitting the prescribed-time observer.
    The proof applies only inside this class.

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