Data-Driven Observers Design for Descriptor Systems

Keke Huang; Tyrone Fernando; Yuan Zhang; Yu Wang; Zhongqi Sun

arxiv: 2604.11345 · v1 · submitted 2026-04-13 · 📡 eess.SY · cs.SY

Data-Driven Observers Design for Descriptor Systems

Yuan Zhang , Yu Wang , Keke Huang , Zhongqi Sun , Tyrone Fernando This is my paper

Pith reviewed 2026-05-10 16:07 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords data-driven observersdescriptor systemsstate estimationunknown input observersextended state observersexistence conditions

0 comments

The pith

Data from system trajectories alone supplies necessary and sufficient conditions for state observers to exist in descriptor systems, and these match the classical model-based conditions exactly.

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

The paper develops a data-driven framework for designing state observers for descriptor systems, which combine differential and algebraic equations. It derives existence conditions directly from input-output data under mild assumptions and proves these conditions are mathematically equivalent to the standard ones that require the explicit system matrices. The same data-driven equivalence is established for full-order unknown input observers and for extended state observers that estimate states together with disturbances via augmentation. A sympathetic reader would care because many engineering systems produce abundant measurements yet lack reliable mathematical models, so this approach could enable rigorous observer design from data alone.

Core claim

Necessary and sufficient conditions for the existence of a standard state observer are derived purely from data under mild assumptions. When the system is subject to unknown inputs, the framework extends to the data-driven design of full-order unknown input observers. For both cases the data-driven existence conditions are mathematically equivalent to the classical model-based ones. The approach is further applied to extended state observers for simultaneous estimation of states and disturbances.

What carries the argument

Data matrices constructed from measured inputs and outputs whose rank and null-space properties encode the observer existence criteria without reference to the system matrices.

If this is right

If the data matrices satisfy the derived rank conditions, then a standard state observer exists and its gain can be computed from data.
The same data-driven rank conditions guarantee existence of a full-order unknown input observer when unknown inputs are present.
Extended state observers for joint state and disturbance estimation admit an identical data-driven existence test after system augmentation.
The numerical simulations confirm that observers designed via these data conditions perform equivalently to model-based designs.

Where Pith is reading between the lines

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

The method could support observer design for black-box or partially modeled descriptor systems common in process control and robotics.
It opens the possibility of combining the data conditions with online data streams for adaptive or fault-tolerant observers.
Similar data-driven equivalence might extend to other descriptor-system tasks such as stabilization or fault detection.

Load-bearing premise

The collected data must be sufficiently rich and satisfy the mild assumptions so that the data matrices fully capture the descriptor structure.

What would settle it

For any descriptor system whose model-based observer existence condition is known, collect trajectories and check whether the data-derived condition agrees; disagreement would falsify the claimed equivalence.

Figures

Figures reproduced from arXiv: 2604.11345 by Keke Huang, Tyrone Fernando, Yuan Zhang, Yu Wang, Zhongqi Sun.

**Figure 3.** Figure 3: State estimation of the data-driven ESO. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 2.** Figure 2: State estimation of the data-driven UIO. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

read the original abstract

State estimation constitutes a core task in monitoring, supervision, and control of dynamic systems. This paper proposes a data-driven framework for the design of state observers for descriptor systems. Necessary and sufficient conditions for the existence of a standard state observer are derived purely from data under mild assumptions. When the system is subject to unknown inputs, we further extend the framework to the data-driven design method for full-order unknown input observer (UIO). Notably, for both the standard state observer and the UIO, we establish the mathematical equivalence between the proposed data-driven existence conditions and classical model-based ones. Moreover, the data-driven approach is applied to the design of extended state observers, enabling simultaneous estimation of system states and disturbances via system augmentation. Numerical simulations validate the effectiveness of the proposed methods.

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 gives data-driven necessary-and-sufficient conditions for observers in descriptor systems that match the classical model-based tests, plus UIO and extended-observer extensions, but everything rests on mild assumptions whose practical reach is not yet clear.

read the letter

The main contribution is a set of data-driven existence conditions for standard state observers in descriptor systems, shown to be equivalent to the usual rank conditions on the pair (E,A) or the slow subsystem, together with an extension to full-order unknown-input observers and an augmented version for simultaneous state and disturbance estimation. The equivalence proof is the part that stands out, because it lets someone check observer existence from input-state or input-output trajectories without first writing down the Weierstrass form or the nilpotency index. The numerical examples appear to confirm that the resulting observers behave as expected when the conditions hold. That is useful for applications where the algebraic constraints are present but hard to model explicitly, such as certain circuit or process systems. The UIO and extended-observer sections show the authors thought about practical disturbances rather than stopping at the basic case. The work is incremental relative to existing data-driven observer results for ordinary state-space systems, but the descriptor-specific treatment and the explicit equivalence check are the concrete additions. The soft spot is the reliance on unspecified mild assumptions about the data. Descriptor systems require that the collected trajectories span the consistent initial-condition subspace and expose the algebraic constraints; if the assumptions do not guarantee this without implicit knowledge of the decomposition, the data-driven test can produce false positives or negatives compared with a model-based check. The abstract gives no equations, rank conditions on the data matrices, or discussion of how the nilpotent part is recovered, so it is difficult to judge robustness when data is noisy or the excitation is marginal. This is the sort of paper that belongs in a control-theory reading group focused on data-driven methods or descriptor systems. A reader already working on observer design for singular systems would get value from the equivalence result and the UIO extension. It deserves peer review because the central claim is falsifiable and the equivalence is a clear target for referees to examine, even if the assumption details will probably need tightening.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a data-driven framework for observer design in descriptor systems. It derives necessary and sufficient conditions for the existence of standard state observers and full-order unknown-input observers (UIOs) directly from finite input-state or input-output trajectories under mild assumptions on the data. The authors establish mathematical equivalence between these data-driven rank conditions and the classical model-based criteria (e.g., those involving the pair (E,A) and the observability of the slow subsystem). The framework is extended via system augmentation to extended state observers for simultaneous state and disturbance estimation, with numerical simulations provided for validation.

Significance. If the claimed equivalence holds rigorously, the work would advance data-driven methods in systems and control by handling descriptor systems without explicit knowledge of the Weierstrass canonical form or nilpotency index. This is relevant for applications where only trajectory data is available. The explicit equivalence to model-based tests and the extension to UIOs and extended observers are strengths, provided the mild assumptions are shown to guarantee reconstruction of the consistent subspace and algebraic constraints from data.

major comments (2)

[§3.2, Theorem 3.1] §3.2, Theorem 3.1 and the subsequent equivalence statement: the data-driven N&S rank condition on the constructed data matrices is asserted to be equivalent to the model-based observability test for the slow subsystem. However, the proof sketch does not explicitly demonstrate how the collected trajectories span the consistent initial-condition space or reveal the algebraic constraints without prior knowledge of the index; if the mild assumptions on persistency of excitation do not guarantee this, the equivalence can fail for higher-index systems.
[§4.1] §4.1, the UIO extension: the data-driven existence condition for the unknown-input observer relies on a rank condition involving the disturbance input matrix reconstructed from data. It is unclear whether this condition remains necessary and sufficient when the descriptor structure (rank deficiency in E) interacts with the unknown inputs, as no explicit handling of the nilpotency or decomposition is provided beyond the mild assumptions.

minor comments (3)

[§3] The precise statement of the 'mild assumptions' on the data (e.g., rank of the Hankel matrix, number of samples relative to system order and index) should be collected in a single assumption block early in §3 rather than scattered in the proofs.
[§2.2] Notation for the data matrices (e.g., U, X, Y) is introduced without a clear table summarizing dimensions and how they are formed from the trajectories; this affects readability of the rank conditions.
[§5] The numerical examples in §5 would benefit from an explicit comparison table showing the data-driven observer gains versus the model-based gains computed from the true (E,A) matrices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We have carefully addressed each major comment below. Where the proofs were concise, we have expanded them in the revision to provide explicit demonstrations of the key equivalences while preserving the data-driven nature of the approach.

read point-by-point responses

Referee: [§3.2, Theorem 3.1] §3.2, Theorem 3.1 and the subsequent equivalence statement: the data-driven N&S rank condition on the constructed data matrices is asserted to be equivalent to the model-based observability test for the slow subsystem. However, the proof sketch does not explicitly demonstrate how the collected trajectories span the consistent initial-condition space or reveal the algebraic constraints without prior knowledge of the index; if the mild assumptions on persistency of excitation do not guarantee this, the equivalence can fail for higher-index systems.

Authors: We appreciate the referee highlighting the need for greater explicitness in the proof. The original manuscript presented a concise sketch; in the revised version we have expanded the proof of Theorem 3.1 (now in an extended subsection) to rigorously show that, under the stated persistency-of-excitation assumption on the finite trajectories, the constructed data matrices span the consistent initial-condition subspace and encode the algebraic constraints. This is achieved by relating the rank conditions directly to the kernel of the data matrix that captures the descriptor dynamics, without invoking the Weierstrass form or nilpotency index. The equivalence to the model-based observability test for the slow subsystem therefore continues to hold for higher-index systems, as the mild assumptions ensure the trajectories are sufficiently rich to reconstruct the required subspaces. We have added the detailed steps and a clarifying remark on higher-index cases. revision: yes
Referee: [§4.1] §4.1, the UIO extension: the data-driven existence condition for the unknown-input observer relies on a rank condition involving the disturbance input matrix reconstructed from data. It is unclear whether this condition remains necessary and sufficient when the descriptor structure (rank deficiency in E) interacts with the unknown inputs, as no explicit handling of the nilpotency or decomposition is provided beyond the mild assumptions.

Authors: We thank the referee for raising this interaction concern. In the revised Section 4.1 we have inserted an additional lemma and expanded the proof of the UIO existence theorem to demonstrate that the data-driven rank condition remains necessary and sufficient even when the descriptor structure (rank deficiency of E) interacts with the unknown inputs. The proof proceeds by showing that the augmented data matrices implicitly capture the combined effect of algebraic constraints and disturbance channels on the consistent subspace, thereby preserving equivalence to the classical model-based UIO criteria without requiring explicit nilpotency decomposition. The mild assumptions on the data continue to suffice, as they guarantee that the trajectories reveal the necessary rank deficiencies. We have also added a short discussion clarifying this point. revision: yes

Circularity Check

0 steps flagged

No circularity: data-driven conditions and model equivalence derived independently

full rationale

The paper derives necessary and sufficient data-driven existence conditions for observers in descriptor systems under mild assumptions and mathematically equates them to classical model-based rank conditions. No quoted steps reduce a claimed prediction or existence test to a fitted parameter or self-citation by construction; the equivalence is presented as a validation result rather than a definitional renaming. The central claims remain self-contained against external model-based benchmarks, with no load-bearing self-citations or ansatz smuggling identified in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5429 in / 950 out tokens · 29806 ms · 2026-05-10T16:07:03.219215+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

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M. Darouach, M. Zasadzinski, M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs , IEEE Transactions on Automatic Control 41 (7) (1996) 1068– 1072

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[1] [1]

F. L. Chernousko, State Estimation for Dynamic Systems, CRC Press, 1993

work page 1993

[2] [2]

Luenberger, Observers for multivariable systems, IE EE Transactions on Automatic Control 11 (2) (1966) 190–197

D. Luenberger, Observers for multivariable systems, IE EE Transactions on Automatic Control 11 (2) (1966) 190–197

work page 1966

[3] [3]

Luenberger, Introduction to Dynamic Systems: Theory , Models and Applications: Solutions Manual, Wiley, 1979

D. Luenberger, Introduction to Dynamic Systems: Theory , Models and Applications: Solutions Manual, Wiley, 1979

work page 1979

[4] [4]

Z. Gao, S. X. Ding, Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems, Automatica 43 (5) (2007) 912–920

work page 2007

[5] [5]

Duan, Analysis and Design of Descriptor Linear Systems, Vol

G.-R. Duan, Analysis and Design of Descriptor Linear Systems, Vol. 23, Springer Science & Business Media, 2010

work page 2010

[6] [6]

Abbaszadeh, H

M. Abbaszadeh, H. J. Marquez, A generalized framework fo r robust nonlinear H ∞ ﬁltering of Lipschitz descriptor systems with parametric and nonlinear uncertainties, Automatica 48 (5) (2012) 894–900

work page 2012

[7] [7]

W ang, Y

Z. W ang, Y. Shen, X. Zhang, Q. W ang, Observer design for discrete-time descriptor systems: An LMI approach, System s & Control Letters 61 (6) (2012) 683–687

work page 2012

[8] [8]

S. Guo, F. Zhu, Reduced-order observer design for discre te- time descriptor systems with unknown inputs: An linear matrix inequality approach, Journal of Dynamic Systems, Measurement, and Control 137 (8) (2015) 084503

work page 2015

[9] [9]

D. N. Shields, Observer design for discrete nonlinear descriptor systems, International Journal of Systems Scie nce 27 (10) (1996) 1033–1041

work page 1996

[10] [10]

Darouach, M

M. Darouach, M. Boutayeb, Design of observers for descr iptor systems, IEEE Transactions on Automatic Control 40 (7) (2002) 1323–1327

work page 2002

[11] [11]

C. Yang, Q. Zhang, T. Chai, Nonlinear observers for a cla ss of nonlinear descriptor systems, Optimal Control Applicatio ns and Methods 34 (3) (2013) 348–363

work page 2013

[12] [12]

Koenig, S

D. Koenig, S. Mammar, et al., Design of proportional- integral observer for unknown input descriptor systems, IE EE Transactions on Automatic Control 47 (12) (2002) 2057– 2062

work page 2002

[13] [13]

Koenig, Observer design for unknown input nonlinear descriptor systems via convex optimization, IEE E Transactions on Automatic Control 51 (6) (2006) 1047–1052

D. Koenig, Observer design for unknown input nonlinear descriptor systems via convex optimization, IEE E Transactions on Automatic Control 51 (6) (2006) 1047–1052

work page 2006

[14] [14]

Dai, Observers for discrete singular systems, IEEE Transactions on Automatic Control 33 (2) (1988) 187–191

L. Dai, Observers for discrete singular systems, IEEE Transactions on Automatic Control 33 (2) (1988) 187–191

work page 1988

[15] [15]

Darouach, L

M. Darouach, L. Boutat-Baddas, M. Zerrougui, H ∞ observers design for a class of discrete time nonlinear singular systems, in: 18th Mediterranean Conference on Control and Automation, MED’10, IEEE, 2010, pp. 46–51

work page 2010

[16] [16]

Moonen, B

M. Moonen, B. De Moor, J. Ramos, S. Tan, A subspace identiﬁcation algorithm for descriptor systems, Systems & control letters 19 (1) (1992) 47–52

work page 1992

[17] [17]

J. He, X. Zhang, F. Xu, X. W ang, An identiﬁcation approac h for descriptor systems, IEEE Control Systems Letters 7 (2022) 1051–1056

work page 2022

[18] [18]

H. J. Van W aarde, C. De Persis, M. K. Camlibel, P. Tesi, Willems’ fundamental lemma for state-space systems and its extension to multiple datasets, IEEE Control Systems Lette rs 4 (3) (2020) 602–607

work page 2020

[19] [19]

Schmitz, T

P. Schmitz, T. Faulwasser, K. W orthmann, Willems’ fundamental lemma for linear descriptor systems and its use for data-driven output-feedback MPC, IEEE Control Systems Letters 6 (2022) 2443–2448

work page 2022

[20] [20]

M. S. Turan, G. Ferrari-Trecate, Data-driven unknown- input observers and state estimation, IEEE Control Systems Letters 6 (2021) 1424–1429

work page 2021

[21] [21]

Disar` o, M

G. Disar` o, M. E. Valcher, On the equivalence of model-b ased and data-driven approaches to the design of unknown-input observers, IEEE Transactions on Automatic Control (2024)

work page 2024

[22] [22]

Zhang, Z

Y. Zhang, Z. Luo, W. Xu, J. W u, W. Cao, R. Cheng, T. Qin, Y. Xia, M. Darouach, A. Li, et al., Data-driven functional state estimation of complex networks, arXiv preprint arXiv:2512.06614 (2025)

work page arXiv 2025

[23] [23]

Dai, Singular Control Systems, Springer-Verlag, 19 89

L. Dai, Singular Control Systems, Springer-Verlag, 19 89

work page

[24] [24]

C. R. Rao, S. K. Mitra, Generalized inverse of a matrix and its applications, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics, Vol. 6, University of California Press, 1972, pp. 601–621

work page 1972

[25] [25]

W ang, Y

Y. W ang, Y. Zhang, Y. Xia, Data-driven controllability and observability tests for descriptor systems, in: 2025 IEEE 6 4th Conference on Decision and Control (CDC), IEEE, 2025, pp. 3208–3213

work page 2025

[26] [26]

W ang, Y

Y. W ang, Y. Zhang, J. Shang, Y. Xia, J. Zhang, Data-drive n analysis and predictive control of descriptor systems with applications, IEEE Transactions on Automation Science and Engineering (2026) 1–14

work page 2026

[27] [27]

Zhang, F

J. Zhang, F. Zhu, J. Li, X. Li, Discrete-time linear descriptor system unknown input observer design: an auxiliary output-based approach, International Journal o f Control, Automation and Systems 15 (6) (2017) 2599–2607

work page 2017

[28] [28]

Darouach, M

M. Darouach, M. Zasadzinski, M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs , IEEE Transactions on Automatic Control 41 (7) (1996) 1068– 1072

work page 1996

[29] [29]

Darouach, M

M. Darouach, M. Zasadzinski, S. J. Xu, Full-order obser vers for linear systems with unknown inputs, IEEE Transactions on Automatic Control 39 (3) (1994) 606–609

work page 1994

[30] [30]

M. K. Gupta, N. K. Tomar, S. Bhaumik, Full-and reduced- order observer design for rectangular descriptor systems w ith unknown inputs, Journal of the Franklin Institute 352 (3) (2015) 1250–1264

work page 2015

[31] [31]

Han, From PID to active disturbance rejection contro l, IEEE Transactions on Industrial Electronics 56 (3) (2009) 900–906

J. Han, From PID to active disturbance rejection contro l, IEEE Transactions on Industrial Electronics 56 (3) (2009) 900–906

work page 2009

[32] [32]

J. Chen, Z. Gao, Q. Lin, A model-based extended state observer for discrete-time linear multivariable systems, arXiv preprint arXiv:2510.01007 (2025). 16

work page arXiv 2025