Sampled-Data Observers for Delay Systems and Hyperbolic PDE-ODE Loops

Fouad Giri; Iasson Karafyllis; Tarek Ahmed-Ali

arxiv: 1907.06691 · v1 · pith:JQA77EBWnew · submitted 2019-07-15 · 🧮 math.OC · cs.SY· eess.SY

Sampled-Data Observers for Delay Systems and Hyperbolic PDE-ODE Loops

Tarek Ahmed-Ali , Iasson Karafyllis , Fouad Giri This is my paper

Pith reviewed 2026-05-24 21:11 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords sampled-data observerstime-delay systemshyperbolic PDEsexponential stabilitysmall-gain analysisinter-sample predictoroutput feedbackPDE-ODE loops

0 comments

The pith

Modifying continuous-time observers with an inter-sample output predictor preserves robust exponential error stability under sampled measurements if the sampling period is small enough.

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

The paper develops sampled-data observers for systems that include both discrete and distributed time delays by starting from an existing continuous-time observer that already guarantees robust exponential convergence of the estimation error. It adds a predictor that estimates the output between sampling instants and applies Lyapunov and small-gain arguments to prove that the same exponential stability property carries over to the sampled-data case. The approach covers strict-feedback delay systems, transport PDEs with nonlocal terms, and PDE-ODE feedback loops, and is illustrated on chemical-reactor examples that include both state observation and output-feedback stabilization.

Core claim

If a continuous-time observer achieves robust exponential error convergence for continuous output measurements, then the same observer modified by an inter-sample output predictor achieves the same robust exponential convergence for sampled measurements, provided the sampling period is not too large; this is shown via Lyapunov stability tools and small-gain analysis for the listed classes of delay systems and hyperbolic PDE-ODE loops.

What carries the argument

The inter-sample output predictor, which compensates for the absence of continuous output data and is combined with small-gain analysis to bound the sampling-induced perturbation.

If this is right

The same construction yields sampled-data output-feedback stabilizers for the considered system classes.
The result applies directly to state observation and stabilization of a chemical reactor.
Robustness margins established for the continuous observer carry over to the sampled-data version.
The method covers both finite-dimensional delay systems and infinite-dimensional transport PDEs.

Where Pith is reading between the lines

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

Digital implementations of observers for infinite-dimensional systems become feasible without a complete redesign from continuous-time theory.
The sampling-period bound could be tightened or relaxed by replacing the basic predictor with a more accurate inter-sample model.
Similar predictor-based extensions might apply to other classes of distributed-parameter systems once a continuous-time observer is known.

Load-bearing premise

An existing continuous-time observer already delivers robust exponential error convergence when output measurements are available at every instant.

What would settle it

A concrete delay system or PDE-ODE loop for which the modified observer loses exponential stability even when the sampling period is made arbitrarily small.

read the original abstract

This paper studies the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results can be applied to time delay systems of strict-feedback structure, transport Partial Differential Equations (PDEs) with nonlocal terms, and feedback interconnections of Ordinary Differential Equations with a transport PDE. The proposed design approach consists in exploiting an existing observer, which features robust exponential convergence of the error when continuous-time output measurements are available. The observer is then modified, mainly by adding an inter-sample output predictor, to compensate for the effect of data-sampling. Using Lyapunov stability tools and small-gain analysis, we show that robust exponential stability of the error is preserved, provided the sampling period is not too large. The general result is illustrated with different examples including state observation and output-feedback stabilization of a chemical reactor.

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 shows how to turn a working continuous-time observer into a sampled-data one for delay systems and PDE-ODE loops by adding an inter-sample predictor, with stability recovered for small enough sampling periods via small-gain.

read the letter

The main thing here is a modular conversion: start with an existing continuous observer that already gives robust exponential error decay under full measurements, then insert an inter-sample output predictor to handle sampling and use Lyapunov plus small-gain to show the exponential property survives as long as the sampling period is below some threshold. This is applied to strict-feedback delay systems, transport PDEs with nonlocal terms, and ODE-transport PDE loops, with an example on a chemical reactor for both observation and output-feedback stabilization. The construction is straightforward and reuses prior continuous-time results without redesigning the observer core. That reuse is the practical strength; it gives a clear recipe rather than a wholly new framework. The soft spots are minor but real. Everything rests on the continuous observer already delivering quantifiable exponential or ISS estimates, so the paper does not help when that starting point is missing or weak. The abstract claims explicit conditions exist, but without seeing the precise gain calculations or how conservative the resulting sampling bound turns out to be, it is hard to judge how useful the threshold is in applications. The examples illustrate the method but do not appear to include numerical checks on the actual sampling limit. This is for people already working on sampled-data control of infinite-dimensional or delay systems who have a continuous observer in hand. A reader outside that niche will find the setup too specialized. The argument looks internally consistent and the extension is concrete enough that it deserves a serious referee rather than a desk reject.

Referee Report

0 major / 1 minor

Summary. The paper develops sampled-data observers for systems with discrete and distributed delays, including strict-feedback delay systems, transport PDEs with nonlocal terms, and PDE-ODE feedback loops. Starting from an existing continuous-time observer that achieves robust exponential error convergence under continuous measurements, the design adds an inter-sample output predictor to handle sampling. Lyapunov tools and small-gain analysis are invoked to conclude that robust exponential stability of the error is preserved provided the sampling period is sufficiently small. The general result is illustrated on state observation and output-feedback stabilization of a chemical reactor.

Significance. If the central claim holds, the work offers a modular extension of continuous-time observer designs to the sampled-data case for infinite-dimensional delay and PDE systems. The small-gain perturbation argument is a strength because it reuses existing exponential or ISS estimates without requiring a full redesign of the observer; this structure is internally consistent and directly addresses practical constraints where continuous output measurements are unavailable.

minor comments (1)

[Abstract] Abstract: the statement that stability holds 'provided the sampling period is not too large' is standard but would benefit from a brief indication of how the admissible upper bound depends on the continuous-time observer gains or the small-gain constants, even if only qualitatively.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no points to address point-by-point. We will incorporate any minor editorial or presentation improvements suggested during the revision process.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper takes as given an existing continuous-time observer with robust exponential error convergence and extends it to sampled data via an inter-sample predictor plus standard Lyapunov/small-gain perturbation analysis. No step reduces a claimed prediction or stability result to a fitted parameter, self-definition, or load-bearing self-citation chain; the modular premise is external and the extension is independent of the specific form of the delays or PDE-ODE loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a suitable continuous-time observer and on the small-gain/Lyapunov analysis yielding an explicit upper bound on the sampling period; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Existence of a continuous-time observer featuring robust exponential convergence under continuous output measurements
The sampled-data design explicitly starts from this pre-existing observer.

pith-pipeline@v0.9.0 · 5695 in / 1122 out tokens · 20957 ms · 2026-05-24T21:11:25.949350+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Sampled -Data Observers for 1 -D Parabolic PDEs with Non -Local Outputs

Karafyllis, I., T. Ahmed -Ali and F. Giri, “Sampled -Data Observers for 1 -D Parabolic PDEs with Non -Local Outputs ”, submitted to Systems & Control Lette rs (see also arXiv:1901.02434 [math.OC])

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Nonlinear Observer Design in the Presence of Delayed Output Measurements

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Multi-Rate Sampled-Data Observer Design for Nonlinear Systems with Asynchronous and Delayed Measurements

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Observer Design for Uniformly O bservable Systems with Sampled M easurements

Nadri, M., H. Hammouri and R. M. Grajales, “ Observer Design for Uniformly O bservable Systems with Sampled M easurements”, IEEE Transactions on Automatic Control , 58, 2012, 757-762

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Stabilization in the Sample -and-Hold Sense of Nonlinear R etarded Systems

Pepe, P., “Stabilization in the Sample -and-Hold Sense of Nonlinear R etarded Systems”, SIAM Journal on Control and Optimization, 52, 2014, 3053–3077

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On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems

Pepe, P. and E. Fridman, “On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems”, Automatica, 82, 2017, 295–300. 32

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Observer with Sample-and-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements

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Stability Analysis of Coupled Linear ODE -Hyperbolic PDE Systems with Two Time Scales

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[1] [1]

Cascade High Gain Predictors for a Class of Nonlinear Systems

Ahmed-Ali, T., E. Cherrier and F. Lamnabhi-Lagarrigue. “Cascade High Gain Predictors for a Class of Nonlinear Systems”, IEEE Transactions on Automatic Control, 57, 2012, 221–226

work page 2012

[2] [2]

Using Exponential Time-Varying Gains for Sampled-Data Stabilization and Estimation

Ahmed-Ali, T., E. Fridman, F. Giri, L. Burlion and F. Lamnabhi-Lagarrigue, “Using Exponential Time-Varying Gains for Sampled-Data Stabilization and Estimation”, Automatica, 67, 2016, 244–251

work page 2016

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PDE Based Observer Design for Nonlinear Systems with Large Output Delay

Ahmed-Ali, T., F. Giri, M. Krstic, M. Kahelras, “PDE Based Observer Design for Nonlinear Systems with Large Output Delay”, Systems & Control Letters, 113, 2018, 1–8

work page 2018

[4] [4]

Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements

Ahmed-Ali T., I. Karafyllis, and F. Lamnabhi-Lagarrigue, “Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements”, Systems & Control Letters , 62, 2013, 539-549

work page 2013

[5] [5]

Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems

Ahmed-Ali, T., I. Karafyllis, M. Krstic and F. Lamnabhi -Lagarrigue, “Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems”, in “ Recent Results on Nonlinear Time Delayed Systems”, Eds. M. Malisoff, F. Mazenc, P. Pepe and I. Karafylli s, Advances in Dela ys and Dynamics, Springer, 2016

work page 2016

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High Gain Observer Design for Some Networked Control Systems

Ahmed -Ali, T. and F. Lamnabhi-Lagarrigue, “ High Gain Observer Design for Some Networked Control Systems”, IFAC Proceedings Volumes, 43, 2010, 145-150

work page 2010

[7] [7]

Observer Design for Nonlinear System s with Output Distributed Delay

Ammeh, L. , F. G iri, T. Ahmed -Ali, E. Magarotto and H. El Fadil, “ Observer Design for Nonlinear System s with Output Distributed Delay”, submitted to the IEEE Conference on Decision and Control, Nice, France, 2019

work page 2019

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A Framework for Nonlinear Sampled -Data O bserver Design via Approximate Discrete-Time Models and Emulation

Arcak, M. and D. Nesic, “ A Framework for Nonlinear Sampled -Data O bserver Design via Approximate Discrete-Time Models and Emulation”, Automatica, 40, 2004, 1931–1938

work page 2004

[9] [9]

Stability of Linear Density -Flow Hyperbolic Systems Under PI Boundary Control

Bastin, G., J. -M. Coron and O. Tamasoiu, “Stability of Linear Density -Flow Hyperbolic Systems Under PI Boundary Control”, Automatica, 53, 2015, 37–42

work page 2015

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Lyapunov Stability of Linear Predictor Feedba ck for Distributed Input Delays

Bekiaris -Liberis, N. and M. Krstic. “Lyapunov Stability of Linear Predictor Feedba ck for Distributed Input Delays”, IEEE Transactions on Automatic Control, 56, 655-660, 2011

work page 2011

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Adaptive O utput-Feedback Stabilization of Non -Local Hyperbolic PDEs

Bernard, P. and M. Krstic, “ Adaptive O utput-Feedback Stabilization of Non -Local Hyperbolic PDEs”, Automatica, 50, 2014, 2692-2699

work page 2014

[12] [12]

Asymptotic State Prediction for Continuous - Time Systems with Delayed Input and Application to Control

Besancon, G., D. Georges, and Z. Benaya che. “Asymptotic State Prediction for Continuous - Time Systems with Delayed Input and Application to Control ”, Proceedings of the European Control Conference, Kos, Greece, 2007

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A Hybrid Redesign of Newton O bservers in the Absence of an Exact Discrete-Time Model

Biyik, E. and M. Arcak, “ A Hybrid Redesign of Newton O bservers in the Absence of an Exact Discrete-Time Model”, Systems & Control Letters, 55, 2006, 429–436

work page 2006

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An Observer for a Class of Nonlinear Systems with Time-Varying Measurement Delays

Cacace, F., A. Germani and C. Manes. “An Observer for a Class of Nonlinear Systems with Time-Varying Measurement Delays”, Systems & Control Letters, 59, 2010, 305-312. 31

work page 2010

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A Chain Observer for Nonlinear Systems with Multiple Time -Varying M easurement Delays

Cacace, F. , A. Germani, and C . Manes. “A Chain Observer for Nonlinear Systems with Multiple Time -Varying M easurement Delays ”, SIAM Journal on Control and Optimization , 52, 2014, 1862-1885

work page 2014

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Sampled -Data Emulation of Dynamic Output Feedback Controllers for Nonlinear Time-Delay Systems

Di Ferdinando, M. and P. Pepe, “Sampled -Data Emulation of Dynamic Output Feedback Controllers for Nonlinear Time-Delay Systems”, Automatica, 99, 2019, 120–131

work page 2019

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Boundary Control with Integral A ction for Hyperbolic Systems of Conservation Laws: Stability and Experiments

Dos Santo s, V., G. Bastin, J. -M. Coron and B. d’Andréa -Novel, “Boundary Control with Integral A ction for Hyperbolic Systems of Conservation Laws: Stability and Experiments”, Automatica, 44, 2008, 1310–1318

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Boundary Observer De sign for Coupled ODEs –Hyperbolic PDEs Systems

Ferrante, F. and A. Cristofaro, “Boundary Observer De sign for Coupled ODEs –Hyperbolic PDEs Systems”, Proceedings of the 18th European Control Conference , Naples, Italy, 2019, 2418-2423

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Fridman, E., Introduction to Time-Delay Systems: Analysis and Control, Birkhäuser, 2014

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Gauthier, J. P. and I. Kupka, Deterministic Observation Theory and Applications, New York: Cambridge Univ. Press, 2001

work page 2001

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A New Approach to State Observation of Nonlinear Systems with Delayed Output

Germani, A., C. Manes and P. Pepe. “A New Approach to State Observation of Nonlinear Systems with Delayed Output”, IEEE Transactions on Automatic Control, 47, 2002, 96–101

work page 2002

[22] [22]

Boundary Observer D esign for Hyperbolic PDE– ODE Cascade Systems

Hasan, A., O. M. Aamo and M. Krstic, “Boundary Observer D esign for Hyperbolic PDE– ODE Cascade Systems”, Automatica, 68, 2016, 75–86

work page 2016

[23] [23]

From Continuous -Time Design to S ampled-Data Design of Observers

Karafyllis I. and C. Kravaris, “From Continuous -Time Design to S ampled-Data Design of Observers”, Transactions on Automatic Control, 54, 2009, 2169–2174

work page 2009

[24] [24]

Karafyllis, I. and Z. -P. Jiang, Stability and Stabilization of Nonlinear Systems , Springer - Verlag, London (Series: Communications and Control Engineering), 2011

work page 2011

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Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers

Karafyllis, I. and M. Krs tic. “Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers”, Automatica, 49, 3623–3631, 2013

work page 2013

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Karafyllis, I. , M. Malisoff, F. Mazenc and P. Pepe, Recent Results on Nonlinear Delay Control Systems, Springer, 2016

work page 2016

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Karafyllis, I. and M. Krstic, Predictor Feedback for Delay Systems : Implementations and Approximations, Birkhäuser, Boston (Series: Mathematics, Systems & Control: Foundations & Applications), 2017

work page 2017

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Karafyllis, I. and M. Krstic, Input-to-State Stability for PDEs , Springer-Verlag, London (Series: Communications and Control Engineering), 2019

work page 2019

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Sampled -Data Observers for 1 -D Parabolic PDEs with Non -Local Outputs

Karafyllis, I., T. Ahmed -Ali and F. Giri, “Sampled -Data Observers for 1 -D Parabolic PDEs with Non -Local Outputs ”, submitted to Systems & Control Lette rs (see also arXiv:1901.02434 [math.OC])

work page arXiv 1901

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Nonlinear Observer Design in the Presence of Delayed Output Measurements

Kazantzis, N. and R. A. Wright. “Nonlinear Observer Design in the Presence of Delayed Output Measurements”, Systems & Control Letters, 54, 2005, 877–886

work page 2005

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Krstic, M., Delay Compensation for Nonlinear, Adaptive, and PDE Systems , Birkhäuser Boston, 2009

work page 2009

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Multi-Rate Sampled-Data Observer Design for Nonlinear Systems with Asynchronous and Delayed Measurements

Ling, C. and C. Kravaris, “Multi-Rate Sampled-Data Observer Design for Nonlinear Systems with Asynchronous and Delayed Measurements”, Proceedings of the A merican Control Conference, 2019

work page 2019

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Advances in Design and Control

Michiels, W. and S. -I. Niculescu, Stability, Control and Computation for Time -Delay Systems. An Eigenvalue Based Approach , SIAM, Phil adelphia, Series: “Advances in Design and Control”, vol. DC 27, 2014

work page 2014

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Observer Design for Uniformly O bservable Systems with Sampled M easurements

Nadri, M., H. Hammouri and R. M. Grajales, “ Observer Design for Uniformly O bservable Systems with Sampled M easurements”, IEEE Transactions on Automatic Control , 58, 2012, 757-762

work page 2012

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Stabilization in the Sample -and-Hold Sense of Nonlinear R etarded Systems

Pepe, P., “Stabilization in the Sample -and-Hold Sense of Nonlinear R etarded Systems”, SIAM Journal on Control and Optimization, 52, 2014, 3053–3077

work page 2014

[36] [36]

On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems

Pepe, P. and E. Fridman, “On Global Exponential Stability Preservation Under Sampling for Globally Lipschitz Time-Delay Systems”, Automatica, 82, 2017, 295–300. 32

work page 2017

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Observer with Sample-and-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements

Raff T., M. Kögel and F. Allgöwer, “Observer with Sample-and-Hold Updating for Lipschitz Nonlinear Systems with Nonuniformly Sampled Measurements”, Proceedings of the American Control Conference, Washington, USA, 2008, 5254-5257

work page 2008

[38] [38]

Stability Analysis of Coupled Linear ODE -Hyperbolic PDE Systems with Two Time Scales

Tang, Y. and G. Mazanti, “Stability Analysis of Coupled Linear ODE -Hyperbolic PDE Systems with Two Time Scales”, Automatica, 85, 2017, 386-396

work page 2017