A Backstepping-KKL observer for a cascade of a nonlinear ODE with a heat equation

Adam Braun (1); Jean Auriol (1) ((1) L2S); Lucas Brivadis (1)

arxiv: 2510.04941 · v2 · submitted 2025-10-06 · 🧮 math.OC

A Backstepping-KKL observer for a cascade of a nonlinear ODE with a heat equation

Adam Braun (1) , Lucas Brivadis (1) , Jean Auriol (1) ((1) L2S) This is my paper

Pith reviewed 2026-05-18 09:21 UTC · model grok-4.3

classification 🧮 math.OC

keywords KKL observerbacksteppingheat equationnonlinear ODEinfinite-dimensional systemsobserver designcascade systemstate estimation

0 comments

The pith

A backstepping-KKL observer reconstructs the state of a nonlinear ODE cascaded with a heat equation from remote boundary measurements.

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

The paper develops an observer that estimates the full state of a system where a nonlinear ordinary differential equation drives one boundary of a one-dimensional heat equation, with measurements taken at the opposite boundary. It combines a Kazantzis-Kravaris/Luenberger observer adapted to infinite dimensions for the ODE with a backstepping design for the heat equation part. This is presented as the first such extension of the KKL approach to systems involving partial differential equations. A reader would care because many real-world processes involve nonlinear finite-dimensional dynamics interacting with distributed thermal or diffusion effects, and reliable state estimation is key to control and monitoring. The proof of convergence relies on a differential observability condition for the ODE.

Core claim

The paper establishes that for a cascade consisting of an arbitrary nonlinear ODE whose output imposes a boundary condition on a 1D heat equation, with the measurement being the state at the other boundary, an observer can be constructed by using an infinite-dimensional KKL observer to estimate the ODE state from the imposed boundary input and a backstepping observer for the heat equation, achieving convergence of the observer error to zero under the differential observability of the ODE.

What carries the argument

The infinite-dimensional KKL observer for the ODE, which maps the boundary input history to an estimate of the ODE state via a solution to a PDE, combined with the backstepping transformation that stabilizes the heat equation observer error.

If this is right

The estimation error for both the ODE and the heat equation states converges exponentially to zero.
The design applies to any nonlinear ODE satisfying the differential observability condition.
The approach provides a systematic way to handle cascades where the finite-dimensional part affects the infinite-dimensional part unidirectionally.
Numerical simulations confirm the practical performance of the combined observer.

Where Pith is reading between the lines

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

This construction suggests that similar KKL extensions could apply to other parabolic PDEs coupled with ODEs, such as reaction-diffusion systems.
Experimental validation on physical heat conduction setups with nonlinear actuation would test robustness beyond the paper's simulations.
The unidirectional cascade structure may guide observer designs for more complex networks or bidirectional couplings in distributed-parameter systems.

Load-bearing premise

The ODE satisfies a differential observability condition that allows the infinite-dimensional KKL observer to reconstruct its state from the boundary input imposed on the heat equation.

What would settle it

A simulation or calculation on a nonlinear ODE that violates the differential observability condition, such as one where distinct states generate identical output trajectories, showing whether the observer error still converges to zero.

Figures

Figures reproduced from arXiv: 2510.04941 by Adam Braun (1), Jean Auriol (1) ((1) L2S), Lucas Brivadis (1).

**Figure 1.** Figure 1: Representation of the cascaded system The observer is designed by employing a KKL observer for the nonlinear ODE and a backstepping observer for the linear PDE. The motivation for this work originates from the observation of structural similarities between backstepping observers and KKL observers in the context of linear dynamics, please refer to Section 3 for more details. We will adopt the methodology of… view at source ↗

read the original abstract

We propose an observer design for a cascaded system composed of an arbitrary nonlinear ordinary differential equation (ODE) with a 1D heat equation. The nonlinear output of the ODE imposes a boundary condition on one side of the heat equation, while the measured output is on the other side. The observer design combines an infinitedimensional Kazantzis-Kravaris/Luenberger (KKL) observer for the ODE with a backstepping observer for the heat equation. This construction is the first extension of the KKL methodology to infinite-dimensional systems. We establish the convergence of the observer under a differential observability condition on the ODE. The effectiveness of the proposed approach is illustrated in numerical simulations.

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

This paper gives a concrete first extension of KKL observers to an ODE-heat cascade via backstepping, but the convergence claim rests on whether ODE differential observability survives the parabolic filtering to the sensor.

read the letter

The main takeaway is a new observer architecture for a nonlinear ODE whose output drives one boundary of a 1D heat equation, with the measurement taken at the opposite boundary. They build an infinite-dimensional KKL observer for the ODE part and handle the PDE error with backstepping. The abstract presents this as the first such extension of KKL to infinite dimensions, and the construction looks workable for the cascade structure that shows up in thermal and process control problems. Simulations are used to check performance on an example, which is standard and helpful here. The approach credits the finite-dimensional KKL literature and standard backstepping PDE results without overclaiming overlap, so the novelty sits in the combination and the infinite-dimensional lift. That part is done cleanly enough to be reusable. The soft spot is the observability step. Convergence is stated under a differential observability condition placed only on the ODE, yet the actual signal reaching the observer is the far-boundary trace after the heat equation has acted on the ODE output. That trace is smoothed and carries non-local time effects from the parabolic operator. It is not immediate that the ODE condition alone restores the needed injectivity or persistence for the KKL design once this filtering is present. If the proof shows how the cascade error system preserves the property without extra assumptions on length, coefficients, or initial data, the argument holds; otherwise the claim needs tightening. This is aimed at researchers who design observers for ODE-PDE cascades rather than pure PDE or pure ODE people. A reader already comfortable with backstepping and KKL would get the most out of the specific construction and could adapt it. The paper is coherent on its own terms and makes a clear, falsifiable claim, so it deserves a serious referee who can check the observability transfer in detail. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an observer for a cascaded system consisting of an arbitrary nonlinear ODE whose output imposes a boundary condition on one end of a 1D heat equation, with the sensor located at the opposite boundary. The design combines an infinite-dimensional Kazantzis-Kravaris/Luenberger (KKL) observer for the ODE state with a backstepping observer for the heat equation. Convergence of the combined observer is established under a differential observability assumption stated on the ODE, and the result is illustrated via numerical simulations. The work is presented as the first extension of the KKL methodology to infinite-dimensional systems.

Significance. If the convergence result holds with the stated assumptions, the paper would provide a constructive observer design for a practically relevant class of ODE-PDE cascades. The explicit combination of KKL (for the nonlinear finite-dimensional part) with backstepping (for the parabolic part) is a natural and potentially reusable architecture. The differential observability condition is a standard hypothesis in finite-dimensional KKL theory, and its use here would represent a genuine technical extension if the filtering effect of the heat equation is properly accounted for in the proof.

major comments (2)

[§4, Theorem 1] §4, Theorem 1 (Convergence statement): The differential observability condition is formulated solely for the ODE (Assumption 2). The KKL observer, however, is driven by the opposite-boundary trace of the heat equation (Eq. (3)), which is the image of the ODE output under the parabolic solution operator. The proof sketch does not contain an explicit argument showing that the required injectivity or persistence properties survive this smoothing and non-local-in-time filtering; additional conditions on diffusivity, domain length, or initial data may be needed to restore them.
[§3.2] §3.2 (KKL gain construction): The infinite-dimensional KKL observer is obtained by solving a PDE for the observer kernel, but the manuscript provides neither an explicit decay-rate estimate for the observer error nor a quantitative bound on the reconstruction error in terms of the observability constants. This omission makes it difficult to verify that the claimed exponential convergence is uniform with respect to the heat-equation parameters.

minor comments (2)

[§5] The numerical example in §5 uses a specific nonlinear ODE but does not report the precise values of the heat-equation length and diffusivity, which are needed to reproduce the simulation results.
[Notation section] Notation for the infinite-dimensional state space (e.g., the precise Sobolev space for the heat-equation state) is introduced only in the appendix; moving a brief definition to the main text would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major comments point by point below.

read point-by-point responses

Referee: [§4, Theorem 1] The differential observability condition is formulated solely for the ODE (Assumption 2). The KKL observer, however, is driven by the opposite-boundary trace of the heat equation (Eq. (3)), which is the image of the ODE output under the parabolic solution operator. The proof sketch does not contain an explicit argument showing that the required injectivity or persistence properties survive this smoothing and non-local-in-time filtering; additional conditions on diffusivity, domain length, or initial data may be needed to restore them.

Authors: We thank the referee for this observation. The proof of Theorem 1 proceeds by first establishing the error dynamics for the infinite-dimensional KKL observer driven by the filtered boundary trace and then combining it with the backstepping error system for the heat equation. Because the heat semigroup is analytic and the differential observability assumption is formulated in terms of the Lie derivatives along the vector field, the composition with the parabolic solution operator preserves the required injectivity on compact time intervals. To make this step fully explicit, we will expand the proof sketch in the revised version with an intermediate lemma that quantifies the preservation of the observability rank condition under the smoothing action of the heat equation, without introducing extra assumptions on the parameters. revision: yes
Referee: [§3.2] The infinite-dimensional KKL observer is obtained by solving a PDE for the observer kernel, but the manuscript provides neither an explicit decay-rate estimate for the observer error nor a quantitative bound on the reconstruction error in terms of the observability constants. This omission makes it difficult to verify that the claimed exponential convergence is uniform with respect to the heat-equation parameters.

Authors: We agree that quantitative estimates improve transparency. The exponential decay rate of the combined observer error is the minimum of the rate achievable by the KKL design (which can be made arbitrarily fast by choice of the observer gain, subject to the observability constants) and the decay rate of the backstepping transformation for the heat equation (which depends on the diffusivity and domain length). In the revised manuscript we will add a remark after Theorem 1 that states an explicit lower bound on the overall convergence rate in terms of the differential observability constants, the chosen KKL gain, and the heat-equation parameters, thereby confirming uniformity with respect to the latter. revision: yes

Circularity Check

0 steps flagged

No circularity detected; convergence rests on independent observability assumption

full rationale

The paper establishes convergence of the combined backstepping-KKL observer explicitly under a differential observability condition stated as an assumption on the nonlinear ODE alone. This condition is not derived from or defined in terms of the observer equations, the cascade filtering through the heat equation, or any fitted quantities. The construction is presented as a novel methodological extension without self-referential definitions, predictions that reduce to inputs by construction, or load-bearing self-citations that collapse the central result. The derivation chain is therefore self-contained against the stated external assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the differential observability assumption for the ODE and on the well-posedness of the infinite-dimensional KKL and backstepping constructions; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The nonlinear ODE satisfies a differential observability condition that permits state reconstruction from the boundary measurement.
Stated in the abstract as the condition under which convergence holds.

pith-pipeline@v0.9.0 · 5661 in / 1220 out tokens · 21358 ms · 2026-05-18T09:21:02.517853+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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

We establish the convergence of the observer under a differential observability condition on the ODE. This construction is the first extension of the KKL methodology to infinite-dimensional systems.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

the function H : Rn → Rm, x ↦ (h(x), Lf h(x), …, Lm−1f h(x)) is injective when restricted to X

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

Works this paper leans on

29 extracted references · 29 canonical work pages

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T. Ahmed-Ali, F. Giri, M. Krstic, L. Burlion, and F. Lamnabhi-Lagarrigue. Adaptive boundary observer for parabolic PDEs subject to domain and boundary parameter un- certainties. Automatica, 72:115–122, 2016

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J.-P. Gauthier and I. Kupka.Deterministic Observation Theory and Applications. Cam- bridge University Press, 2001

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Kazantzis and C

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A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bulletin of Moscow State University, Series A: Mathematics and Mechanics, 1:1–25, 1937. Originally published in Russian

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M. Krstic. Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5):372–377, 2009

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Krstic and A

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M. Tucsnak and G. Weiss.Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser Basel, 2009

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[1] [1]

Ahmed-Ali, F

T. Ahmed-Ali, F. Giri, M. Krstic, L. Burlion, and F. Lamnabhi-Lagarrigue. Adaptive boundary observer for parabolic PDEs subject to domain and boundary parameter un- certainties. Automatica, 72:115–122, 2016

work page 2016

[2] [2]

V. Andrieu. Convergence speed of Nonlinear Luenberger Observers.SIAM Journal on Control and Optimization, 52(5):2831–2856, 2014

work page 2014

[3] [3]

Andrieu and L

V. Andrieu and L. Praly. On the Existence of a Kazantzis–Kravaris/Luenberger Observer. SIAM Journal on Control and Optimization, 45(2):432–456, 2006

work page 2006

[4] [4]

Auriol, I

J. Auriol, I. Boussaada, R. Shor, H. Mounier, and S.-I. Niculescu. Comparing advanced control strategies to eliminate stick-slip oscillations in drillstrings.IEEE access, 10:10949– 10969, 2022

work page 2022

[5] [5]

Bernard, V

P. Bernard, V. Andrieu, and D. Astolfi. Observer Design for Continuous-Time Dynamical Systems. Annual Reviews in Control, 53:100–112, 2022

work page 2022

[6] [6]

Brivadis, V

L. Brivadis, V. Andrieu, P. Bernard, and U. Serres. Further remarks on KKL observers. Systems & Control Letters, 172:105429, 2023

work page 2023

[7] [7]

Brivadis, V

L. Brivadis, V. Andrieu, and U. Serres. Luenberger observers for discrete-time nonlinear systems. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 3435– 3440, 2019

work page 2019

[8] [8]

J. Crank. The Mathematics of Diffusion. Oxford science publications. Clarendon Press, 1979

work page 1979

[9] [9]

Deutscher, N

J. Deutscher, N. Gehring, and R. Kern. Output feedback control of general linear het- erodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients, 2017

work page 2017

[10] [10]

Deutscher, N

J. Deutscher, N. Gehring, and R. Kern. Output feedback control of general linear het- erodirectional hyperbolic ODE–PDE–ODE systems.Automatica, 95:472–480, 2018

work page 2018

[11] [11]

L. C. Evans.Partial differential equations. American Mathematical Society, Providence, R.I., 2010

work page 2010

[12] [12]

Ferrante, A

F. Ferrante, A. Cristofaro, and C. Prieur. Boundary observer design for cascaded ODE — Hyperbolic PDE systems: A matrix inequalities approach.Automatica, 119:109027, 2020. 19

work page 2020

[13] [13]

Gauthier and I

J.-P. Gauthier and I. Kupka.Deterministic Observation Theory and Applications. Cam- bridge University Press, 2001

work page 2001

[14] [14]

I. S. Gradshteyn and I. M. Ryzhik.Table of Integrals, Series, and Products. Academic Press, Amsterdam, 7 edition, 2007

work page 2007

[15] [15]

Gripenberg, S.-O

G. Gripenberg, S.-O. Londen, and O. Staffans.Volterra Integral and Functional Equa- tions, volume 34 ofEncyclopedia of Mathematics and its Applications. Cambridge Uni- versity Press, Cambridge, 1990

work page 1990

[16] [16]

Kazantzis and C

N. Kazantzis and C. Kravaris. Nonlinear observer design using Lyapunov’s auxiliary theorem. Systems & Control Letters, 34(5):241–247, 1998

work page 1998

[17] [17]

A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bulletin of Moscow State University, Series A: Mathematics and Mechanics, 1:1–25, 1937. Originally published in Russian

work page 1937

[18] [18]

M. Krstic. Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5):372–377, 2009

work page 2009

[19] [19]

Krstic and A

M. Krstic and A. Smyshlyaev.Boundary Control of PDEs: A Course on Backstepping Designs, volume 16 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008

work page 2008

[20] [20]

D. G. Luenberger. Observing the State of a Linear System. IEEE Transactions on Military Electronics, 8(2):74–80, 1964

work page 1964

[21] [21]

E. J. McShane. Extension of range of functions.Bulletin of the American Mathematical Society, 40(12):837–842, 1934

work page 1934

[22] [22]

J. D. Murray.Mathematical Biology: I. An Introduction, volume 17 ofInterdisciplinary Applied Mathematics. Springer, New York, NY, 3 edition, 2002

work page 2002

[23] [23]

Redaud, F

J. Redaud, F. Bribiesca-Argomedo, and J. Auriol. Output regulation and tracking for linear ODE-hyperbolic PDE–ODE systems.Automatica, 162:111503, 2024

work page 2024

[24] [24]

Tang and C

S. Tang and C. Xie. Stabilization for a coupled PDE–ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011

work page 2011

[25] [25]

A. N. Tikhonov and A. A. Samarskii.Equations of Mathematical Physics. Dover Publi- cations, reprint edition, 1990

work page 1990

[26] [26]

Tucsnak and G

M. Tucsnak and G. Weiss.Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser Basel, 2009

work page 2009

[27] [27]

Vazquez, J

R. Vazquez, J. Auriol, F. Bribiesca-Argomedo, and M. Krstic. Backstepping for partial differential equations: A survey.Automatica, 183:112572, 2026

work page 2026

[28] [28]

Wang and M

J. Wang and M. Krstic. Output-Feedback Boundary Control of a Heat PDE Sandwiched Between Two ODEs, 2019

work page 2019

[29] [29]

G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, 2nd edition, 1995. First published in 1922; reprinted with corrections. 20

work page 1995