arxiv: 2605.07318 · v1 · submitted 2026-05-08 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Stability-Certified Koopman Observer Design for Nonlinear Systems via Generalized Persidskii Dynamics

Syed Pouladi

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:51 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords Koopman operatornonlinear observerPersidskii systemsinput-to-state stabilityLMI designstate estimationmodel mismatch

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The pith

Koopman observer error dynamics correspond to generalized Persidskii systems, enabling LMI-certified input-to-state stability.

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

This paper develops a method for designing stable observers for nonlinear systems using Koopman operator approximations. It links the observer's error dynamics to generalized Persidskii systems, which support diagonal Lyapunov functions and sector bounds. By solving a linear matrix inequality, the approach computes a correction term that certifies stability against approximation errors and disturbances. This matters because it provides analytical guarantees for state estimation in systems where perfect models are unavailable, as demonstrated on oscillators and robotic arms.

Core claim

The paper establishes a structural correspondence between the error dynamics of a Koopman latent-space observer and generalized Persidskii systems. This correspondence admits diagonal Lyapunov functions and incremental sector characterizations. It then designs a nonlinear correction term whose gain is found via an LMI that certifies input-to-state stability of the estimation error with respect to lifting residuals and external disturbances. Exponential convergence holds nominally, with ultimate boundedness under perturbations.

What carries the argument

The structural correspondence of Koopman observer error dynamics to the class of generalized Persidskii systems, which allows construction of an LMI-based ISS certificate for the nonlinear correction gain.

If this is right

Exponential convergence of the estimation error in the absence of perturbations.
Ultimate boundedness of the error when lifting residuals and disturbances are bounded.
Improved estimation accuracy compared to Extended Kalman Filter and linear Koopman observers, with up to 42% lower RMSE in examples.
Simultaneous handling of model mismatch from Koopman lifting and external noise.

Where Pith is reading between the lines

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

The method could extend to other data-driven lifting techniques beyond standard Koopman if their error dynamics fit the Persidskii structure.
Integrating this observer into feedback control loops might yield provably stable closed-loop performance for approximated nonlinear plants.
Testing on higher-dimensional systems would reveal scalability limits of the LMI computation.

Load-bearing premise

The error dynamics from the Koopman lifting and observer must belong to the generalized Persidskii class so that the LMI provides a valid stability certificate.

What would settle it

A counterexample where the LMI is feasible but the observer error diverges for bounded inputs, or where the structural correspondence fails for a specific Koopman lifting.

Figures

Figures reproduced from arXiv: 2605.07318 by Syed Pouladi.

**Figure 2.** Figure 2: RMSE versus lifting residual magnitude ϵ on the Van der Pol benchmark (50 trials). The PKO degrades more gracefully with ϵ, consistent with the linear scaling in (14). F is subject to ±30% parametric uncertainty, and only the joint angle θ is measured with noise σ 2 v = 0.04. A dictionary of r = 20 observables including Fourier features and trigonometric monomials was employed. The EDMD system was identif… view at source ↗

read the original abstract

This paper addresses the problem of nonlinear state estimation for dynamical systems whose governing equations are approximated through Koopman operator liftings. While Koopman-based predictors have demonstrated broad approximation capability for nonlinear dynamics, certifying observer convergence under model mismatch and measurement noise has remained a largely open problem. To resolve this, we establish a structural correspondence between the error dynamics of a Koopman latent-space observer and the class of generalized Persidskii systems, which admits diagonal Lyapunov functions and incremental sector characterizations. Exploiting this connection, we design a nonlinear correction term whose gain is computed via a linear matrix inequality (LMI) that simultaneously certifies input-to-state stability (ISS) of the estimation error with respect to both lifting residuals and external disturbances. Exponential convergence in the nominal case and ultimate boundedness under bounded perturbations are established analytically. Numerical validation on the Van~der~Pol oscillator and a nonlinear robotic arm with friction uncertainty demonstrates that the proposed observer substantially outperforms both the Extended Kalman Filter and a linear Koopman observer in terms of estimation accuracy and robustness, achieving up to a 42\% reduction in steady-state RMSE under lifting mismatch.

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.

Desk Editor's Note private letter to a colleague

The paper maps Koopman observer error dynamics to generalized Persidskii systems for an LMI-based ISS certificate, which is new, but the structural match is the part that needs close checking.

read the letter

The main takeaway is that the authors show how to turn the estimation error dynamics of a Koopman latent-space observer into a generalized Persidskii form. This lets them build an LMI that certifies input-to-state stability against both lifting residuals and external disturbances, giving exponential convergence without mismatch and ultimate boundedness with it. That structural link is the actual new piece and it is not just a restatement of earlier Koopman observer work. They also run the idea on the Van der Pol oscillator and a robotic arm with friction uncertainty, where the observer cuts steady-state RMSE by up to 42 percent compared with an extended Kalman filter and a plain linear Koopman observer. Those numbers are concrete and the examples are relevant to control practice. The LMI is built from the lifted matrices and the sector structure rather than tuned to the reported errors, so there is no obvious circularity. The citations stay focused on the relevant Koopman and Persidskii literature. The soft spot is exactly the one the stress test flags: the whole certificate stands or falls on whether the actual error equation, including the residual term r(z) and the nonlinear correction, stays inside the incremental sector bounds and diagonal-dominance conditions that Persidskii systems require. If any cross terms or non-sector nonlinearities slip in, the LMI no longer applies to the real dynamics and the claimed guarantees do not hold. The abstract asserts the correspondence but does not display the explicit error equation, so the full derivations are what will decide whether the claim is solid. This work is for control researchers who need stability certificates for data-driven observers under model mismatch. A reader already working with lifted models or ISS analysis will get the most out of the LMI construction and the numerical comparisons. It is worth sending to peer review because the connection is precise enough to be checked and potentially useful, even if revisions will likely be needed around the sector conditions and the explicit error dynamics.

Referee Report

2 major / 2 minor

Summary. The paper claims to resolve the open problem of certifying convergence for Koopman-based nonlinear observers under model mismatch by establishing a structural correspondence between the latent-space estimation error dynamics and the class of generalized Persidskii systems. This correspondence is exploited to design a nonlinear correction term whose gain is obtained from an LMI that simultaneously certifies input-to-state stability (ISS) of the error with respect to both Koopman lifting residuals and external disturbances. Analytical results establish exponential convergence in the nominal case and ultimate boundedness under bounded perturbations; numerical tests on the Van der Pol oscillator and a friction-uncertain robotic arm report up to 42% RMSE reduction relative to the EKF and a linear Koopman observer.

Significance. If the claimed structural correspondence holds, the work provides a concrete, LMI-based route to stability certificates for data-driven Koopman observers, which is a meaningful advance given the widespread use of Koopman liftings without convergence guarantees. The technical bridge to generalized Persidskii dynamics (diagonal Lyapunov functions and incremental sector bounds) is elegant and potentially reusable. The combination of analytical ISS proofs with comparative numerical validation strengthens the practical relevance.

major comments (2)

[Observer design and error dynamics derivation] The central claim rests on the assertion that the Koopman observer error dynamics (lifting residual plus nonlinear correction) belong to the generalized Persidskii class. The manuscript states this correspondence in the abstract and uses it to justify the LMI, but does not supply an explicit error equation (e.g., in the observer-design section) that verifies the residual term satisfies the required incremental sector conditions without introducing non-sector nonlinearities or cross terms that would invalidate the diagonal Lyapunov function. This membership is load-bearing for the ISS certificate.
[Stability analysis and LMI derivation] In the stability analysis, the LMI is formulated directly from the lifted system matrices and the assumed Persidskii structure to certify ISS w.r.t. both residuals and disturbances. However, without an explicit verification step showing that the specific form of the Koopman residual r(z) can be absorbed into the sector bound (or that the correction term preserves the structure under the stated bounded perturbations), the LMI may certify a surrogate system rather than the actual error dynamics. This directly affects the claimed exponential convergence and ultimate boundedness.

minor comments (2)

[Numerical validation] The numerical section should include explicit details on how lifting residuals were generated, the precise definition of RMSE, and any data exclusion or preprocessing steps to allow full reproducibility of the reported 42% improvement.
[Preliminaries and notation] Notation for the lifted state, residual, and sector bounds should be introduced consistently and early; several symbols appear without prior definition in the stability section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our stability certification approach. We address each major comment below, agreeing that additional explicit derivations will strengthen the manuscript without altering its core contributions.

read point-by-point responses

Referee: [Observer design and error dynamics derivation] The central claim rests on the assertion that the Koopman observer error dynamics (lifting residual plus nonlinear correction) belong to the generalized Persidskii class. The manuscript states this correspondence in the abstract and uses it to justify the LMI, but does not supply an explicit error equation (e.g., in the observer-design section) that verifies the residual term satisfies the required incremental sector conditions without introducing non-sector nonlinearities or cross terms that would invalidate the diagonal Lyapunov function. This membership is load-bearing for the ISS certificate.

Authors: We agree that an explicit derivation of the error dynamics would improve clarity and directly verify the structural correspondence. In the revised manuscript, we will add a dedicated subsection in the observer design section that derives the estimation error equation step by step. This derivation will explicitly show that the lifting residual satisfies the incremental sector conditions of the generalized Persidskii class and that no invalidating non-sector nonlinearities or cross terms arise under the stated assumptions, thereby confirming membership and supporting the diagonal Lyapunov function. revision: yes
Referee: [Stability analysis and LMI derivation] In the stability analysis, the LMI is formulated directly from the lifted system matrices and the assumed Persidskii structure to certify ISS w.r.t. both residuals and disturbances. However, without an explicit verification step showing that the specific form of the Koopman residual r(z) can be absorbed into the sector bound (or that the correction term preserves the structure under the stated bounded perturbations), the LMI may certify a surrogate system rather than the actual error dynamics. This directly affects the claimed exponential convergence and ultimate boundedness.

Authors: We acknowledge the importance of an explicit verification step to ensure the LMI applies to the actual dynamics. In the revised stability analysis section, we will insert a step-by-step verification demonstrating that the specific Koopman residual r(z) is absorbed into the sector bound and that the nonlinear correction term preserves the generalized Persidskii structure under bounded perturbations. This addition will confirm that the LMI certifies the true error dynamics, rigorously underpinning the claims of exponential convergence in the nominal case and ultimate boundedness under perturbations. revision: yes

Circularity Check

0 steps flagged

No circularity: structural correspondence and LMI certificate derived from first principles

full rationale

The paper derives the error dynamics explicitly from the Koopman lifting and observer structure, then shows membership in the generalized Persidskii class via the system equations (not by definition or fitting). The LMI is then constructed from the class properties to certify ISS, which is a standard analysis step rather than a tautology. No self-citations are load-bearing, no parameters are fitted and renamed as predictions, and the central claim (exponential convergence under the certificate) follows from the independent verification of the structural match. This is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the observer error dynamics exactly match the generalized Persidskii form and that standard ISS theory applies once the LMI is solved; no new physical entities are introduced.

axioms (2)

domain assumption Observer error dynamics belong to the generalized Persidskii class admitting diagonal Lyapunov functions and incremental sector bounds.
This is the key structural correspondence used to enable the LMI design and ISS certificate.
standard math Standard definitions and theorems of input-to-state stability and Lyapunov analysis hold for the lifted error system.
Invoked to translate the LMI solution into exponential convergence and ultimate boundedness statements.

pith-pipeline@v0.9.0 · 5491 in / 1514 out tokens · 61004 ms · 2026-05-11T00:51:31.508958+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we establish a structural correspondence between the error dynamics of a Koopman latent-space observer and the class of generalized Persidskii systems, which admits diagonal Lyapunov functions and incremental sector characterizations... LMI that simultaneously certifies input-to-state stability (ISS)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the nonlinear functions φ_i satisfy the quadratic sector condition φ_i(s)[s - κ_i^{-1} φ_i(s)] ≥ 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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