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arxiv: 2510.04814 · v3 · submitted 2025-10-06 · 📡 eess.SY · cs.SY

Robust stability of event-triggered nonlinear moving horizon estimation

Isabelle Krauss , Victor G. Lopez , Matthias A. M\"uller This is my paper

Pith reviewed 2026-05-18 09:29 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords event-triggered estimationmoving horizon estimationnonlinear systemsrobust stabilityglobal exponential stabilitydetectabilityremote state estimation
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The pith

An event-triggered moving horizon estimator for nonlinear systems achieves robust global exponential stability under a detectability condition.

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

The paper introduces an event-triggered moving horizon estimation scheme that transmits a measurement and solves the optimization only when an event occurs, otherwise updating the estimate with open-loop prediction from the system dynamics. A new triggering rule is proposed to guarantee that the estimation error remains robustly globally exponentially stable whenever the nonlinear system satisfies a suitable detectability condition. The work further shows that allowing the horizon length to vary produces a stricter upper bound on the estimation error than a fixed horizon. These properties are established through stability analysis and illustrated on two example systems.

Core claim

For general nonlinear systems, the ET-MHE scheme with the introduced event-triggering rule guarantees robust global exponential stability of the state estimation error, provided a suitable detectability condition holds, while a time-varying horizon length yields a tighter bound on that error.

What carries the argument

The novel event-triggering rule that compares predicted and measured behavior to decide transmissions, combined with the moving-horizon optimization solved only at trigger instants and open-loop prediction in between.

If this is right

  • Communication between sensor and estimator is reduced without sacrificing global exponential stability of the error.
  • The estimation error converges exponentially even when disturbances are present, as long as the detectability condition holds.
  • Switching to a varying horizon length produces a strictly smaller guaranteed bound on the steady-state estimation error.
  • Between events the estimator can safely rely on open-loop prediction while preserving the overall stability property.

Where Pith is reading between the lines

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

  • The scheme could be combined with event-triggered control loops to further reduce network load in distributed systems.
  • Similar triggering logic might be applied to other observers such as extended Kalman filters for nonlinear plants.
  • Practical tests on systems with sensor noise or quantization would clarify how close real performance comes to the theoretical bound.

Load-bearing premise

The nonlinear system satisfies a suitable detectability condition that permits reconstruction of the state from available measurements.

What would settle it

A concrete nonlinear system violating the detectability condition for which the estimation error fails to converge exponentially under the proposed event-triggered scheme.

Figures

Figures reproduced from arXiv: 2510.04814 by Isabelle Krauss, Matthias A. M\"uller, Victor G. Lopez.

Figure 1
Figure 1. Figure 1: Diagram of the overall ET-MHE framework. The dashed arrows represent the communication that is only required in case of an event. J. The optimal estimate at the current time t is defined as xˆt := ˆx ∗ t|t . Moreover, the notation eˆt := xt − xˆt is used to describe the estimation error at time t. For t ≥ τ˜, the MHE’s NLP is given by (5) and the following additional constraint X j∈I[t−Mt,µt−1]\Ks η µt−j−1… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of ET-MHE results for α = 5, MHE estimates (without event-triggering, i.e., α = 0) and real system states. 0 10 20 30 40 50 60 0 1 t γ [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: the corresponding values of γt are plotted, i.e, the time instances when an event was scheduled are displayed. Next, we consider different values of α between 1 and 14. For each value of α we performed 200 simulations with different disturbance sequences. As expected, larger values of α result in fewer events being triggered. This correlation is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Average number of events for different choices of α for Exam￾ple 1 (batch reactor), simulated over 60 time steps (i.e., 6 seconds). B. Example 2: Robot Arm As a second example we consider a two-link robot arm moving in a 2D-plane as depicted in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two-link robot arm moving in a 2D-plane. 0 200 400 600 800 1000 0 1 2 3 t ||e|| α=30 α=5 α=0 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average number of events for different choices of α for Example 2 (robot arm), simulated over 1000 time steps (i.e., 5 seconds). C. Additional constraint of MHE’s NLP After having tested the functionality of the algorithm with these two examples, we now examine the impact of the additional constraint (6) on the simulation results. The inclu￾sion of this constraint generally had no significant effect on the… view at source ↗
Figure 9
Figure 9. Figure 9: Event scheduling variable γ for the experiment in [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

In this work, we propose an event-triggered moving horizon estimation (ET-MHE) scheme for the remote state estimation of general nonlinear systems. In the presented method, whenever an event is triggered, a single measurement is transmitted and the nonlinear MHE optimization problem is subsequently solved. If no event is triggered, the current state estimate is updated using an open-loop prediction based on the system dynamics. Moreover, we introduce a novel event-triggering rule under which we demonstrate robust global exponential stability of the ET-MHE scheme, assuming a suitable detectability condition is met. In addition, we show that with the adoption of a varying horizon length, a tighter bound on the estimation error can be achieved. Finally, we validate the effectiveness of the proposed method through two illustrative examples.

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

1 major / 1 minor

Summary. The manuscript proposes an event-triggered moving horizon estimation (ET-MHE) scheme for remote state estimation of general nonlinear systems. A single measurement is transmitted upon event triggering, after which the nonlinear MHE optimization is solved; otherwise the estimate is updated via open-loop prediction using the system dynamics. A novel event-triggering rule is introduced under which robust global exponential stability (GES) of the ET-MHE scheme is claimed, assuming a suitable detectability condition holds. The authors further show that adopting a varying horizon length yields a tighter bound on the estimation error. Effectiveness is illustrated via two examples.

Significance. If the stability analysis is rigorous, the result would be a useful addition to the event-triggered estimation literature by extending MHE techniques to communication-constrained nonlinear settings with explicit robustness guarantees. The varying-horizon improvement for tighter error bounds is a constructive feature that could enhance practical applicability in networked control systems.

major comments (1)
  1. [Main stability theorem / abstract] The central robust GES claim (abstract and main stability result) is conditioned on an external 'suitable detectability condition' for the general nonlinear system. The manuscript invokes this assumption without providing verifiable sufficient conditions (e.g., incremental observability or Lyapunov-like inequalities that hold uniformly) or demonstrating that the proposed event-triggering rule preserves the condition along closed-loop trajectories. Because the stability guarantee reduces to this unverified external property, the result remains conditional and its applicability to arbitrary nonlinear plants is not established.
minor comments (1)
  1. [Abstract] The abstract states that 'two illustrative examples' are used for validation, but does not identify the specific systems or report quantitative metrics (e.g., error norms, triggering rates) that would allow readers to assess the tightness of the derived bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the insightful comments and the recommendation for major revision. The feedback helps us improve the clarity and applicability of our results. Below, we provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: [Main stability theorem / abstract] The central robust GES claim (abstract and main stability result) is conditioned on an external 'suitable detectability condition' for the general nonlinear system. The manuscript invokes this assumption without providing verifiable sufficient conditions (e.g., incremental observability or Lyapunov-like inequalities that hold uniformly) or demonstrating that the proposed event-triggering rule preserves the condition along closed-loop trajectories. Because the stability guarantee reduces to this unverified external property, the result remains conditional and its applicability to arbitrary nonlinear plants is not established.

    Authors: We thank the referee for highlighting this important point. The detectability condition is indeed an assumption that the stability result relies upon, similar to many results in nonlinear observer design where such conditions are postulated for general classes of systems. To address the concern about verifiability, we have included in the revised version a discussion (new Remark 3) that provides sufficient conditions in terms of incremental observability, which is a standard and verifiable property for many nonlinear systems (with references to relevant literature). Furthermore, we have clarified in the proof of the main theorem that the event-triggering rule ensures that the open-loop prediction phases are of finite duration and the error remains within bounds that allow the detectability to hold uniformly. We believe these changes enhance the manuscript without altering the core contribution. revision: yes

Circularity Check

0 steps flagged

No circularity: stability result is conditional on external detectability assumption

full rationale

The paper derives robust global exponential stability for the event-triggered moving horizon estimator under an explicit external assumption of a suitable detectability condition on the general nonlinear system. This assumption is stated as a prerequisite rather than being constructed from or equivalent to the stability claim itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The additional result on tighter bounds via varying horizon length is presented as a direct consequence of the scheme without reducing to the input data by construction. The overall argument remains self-contained against the stated external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The stability claim rests primarily on the external detectability assumption and the design of the new triggering rule; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption suitable detectability condition for the nonlinear system
    Invoked as the assumption under which robust global exponential stability is demonstrated.

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Reference graph

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