Distributed Observer Design for Discrete-Time LTI Systems via Jordan Canonical Form

Giulio Fattore; Guang-Hong Yang; Maria Elena Valcher; Rui Gao

arxiv: 2605.00210 · v1 · submitted 2026-04-30 · 📡 eess.SY · cs.SY

Distributed Observer Design for Discrete-Time LTI Systems via Jordan Canonical Form

Giulio Fattore , Maria Elena Valcher , Rui Gao , Guang-Hong Yang This is my paper

Pith reviewed 2026-05-09 20:10 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords distributed state estimationdiscrete-time LTI systemsJordan canonical formLuenberger observersconsensus protocolsLaplacian matrixasymptotic convergence

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The pith

Distributed observers for discrete-time LTI systems achieve asymptotic state estimation by separating detectable and undetectable components via Jordan form and consensus.

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

The paper develops two distributed estimation schemes for discrete-time linear time-invariant systems. Nodes use local Luenberger observers for state components they can detect directly and a consensus mechanism over the communication network for the remaining components. The first scheme employs full-order observers but requires satisfying many inequalities, while the second uses augmented-order observers under milder conditions. Necessary and sufficient conditions are stated in terms of the eigenvalues of the system matrix and submatrices of the network Laplacian to guarantee that estimation errors converge to zero. These designs increase flexibility in choosing coupling gains relative to earlier methods.

Core claim

Exploiting the Jordan canonical form of the system matrix, two distributed estimation schemes are proposed where each node applies a Luenberger observer to reconstruct its locally detectable state components and a consensus protocol to estimate the undetectable components. For both schemes, necessary and sufficient conditions expressed via the eigenvalues of the system matrix and certain submatrices of the communication network Laplacian guarantee the existence of observers that achieve asymptotically accurate estimation of the full state. The schemes provide greater flexibility in coupling gain selection and impose less stringent solvability conditions compared to previous work.

What carries the argument

Jordan canonical form decomposition of the system matrix that separates dynamics into locally detectable and undetectable parts per node, combined with Laplacian-driven consensus error dynamics for the undetectable components.

If this is right

Nodes reconstruct detectable components independently with standard Luenberger observers.
Consensus protocols drive estimation errors of undetectable components to zero when Laplacian eigenvalue conditions hold.
The first scheme works when a large set of inequalities on gains is satisfied but keeps observer dimension equal to the state dimension.
The second scheme relaxes the inequalities by increasing observer order.
Coupling gains can be selected more flexibly than in prior distributed observer designs.

Where Pith is reading between the lines

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

The separation via Jordan blocks may simplify design for systems whose matrices have repeated eigenvalues or specific block structures.
Network topology could be chosen or modified to ensure the required Laplacian submatrix properties for a given system.
The approach suggests that only consensus variables for undetectable parts need to be exchanged, potentially lowering communication overhead.
Similar Jordan-based splitting might apply to other estimation or control problems with partial local observability.

Load-bearing premise

The Jordan decomposition permits clean separation of locally detectable and undetectable state components for each node, and the communication graph Laplacian submatrices satisfy the eigenvalue conditions needed for the consensus error dynamics to be asymptotically stable.

What would settle it

For a concrete discrete-time LTI system with known Jordan form and a chosen communication graph, compute the eigenvalues and Laplacian submatrices; if a distributed observer converges when the conditions are violated or fails when they hold, the necessary-and-sufficient claim is falsified.

Figures

Figures reproduced from arXiv: 2605.00210 by Giulio Fattore, Guang-Hong Yang, Maria Elena Valcher, Rui Gao.

**Figure 1.** Figure 1: The communication digraph. hold). As a consequence, the Laplacian L is an irreducible symmetric matrix, and since Assumption 1 ensures that for every (ℓ, hℓ) ∈ [1, ru]×[1, gℓ] the set V ℓ,hℓ 3 is not empty, it follows that each proper principal submatrix L ℓ,hℓ of L has all positive real eigenvalues. This allows to further simplify condition (38). Theorem 4. Consider a system described by (4) and (5), w… view at source ↗

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: Norms of the estimation errors of all nodes under Strategy 2. 8 Conclusions In this paper, we have proposed two strategies for distributed state estimation that leverage both the Jordan canonical form of the system matrix A and the Kalman detectability form of each pair (A, Ci). The first strategy proposes local observers whose dimension matches 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

read the original abstract

This paper addresses the problem of distributed state estimation for discrete-time linear time-invariant systems. Building on the framework proposed in Gao & Yang (2025), we exploit the Jordan canonical form of the system matrix to develop two distributed estimation schemes that ensure asymptotic convergence of local estimates to the true system state. In both approaches, each node reconstructs the components of the state that are locally detectable for it via a Luenberger observer, while employing a consensus-based mechanism to estimate the components that are not directly detectable. The first scheme relies on local observers whose dimension matches that of the original state vector; however, its applicability requires the satisfaction of a large set of inequalities. The second scheme, in contrast, can be implemented under less restrictive conditions, but results in observers of increased (augmented) order. For both methods, we derive necessary and sufficient conditions - expressed in terms of the eigenvalues of the system matrix and certain submatrices of the communication network Laplacian - that guarantee the existence of a distributed observer achieving asymptotically accurate estimation. Compared to Gao & Yang (2025), the proposed approaches offer greater flexibility in the selection of coupling gains and impose less stringent solvability conditions.

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 extends Gao & Yang by splitting the state in Jordan form for two observer schemes with explicit eigenvalue-Laplacian conditions, but the global transform risks mixing local outputs and breaking the assumed separation.

read the letter

The paper takes the prior framework and adds two concrete alternatives: a full-order observer that needs many inequalities on the gains, and an augmented-order version that relaxes some of those but increases dimension. Both come with necessary and sufficient conditions phrased directly in terms of the system eigenvalues and sub-blocks of the Laplacian. That gives designers clearer choices on when the consensus error will go to zero and more room on the coupling gains than the earlier version allowed. The abstract positions this as an incremental but practical step inside distributed estimation for discrete LTI systems, and the claims line up with that description. Credit for spelling out the trade-offs between the two schemes and for stating the conditions without extra conservatism on the graph structure. The soft spot is the separation step itself. The Jordan form is global, so the transformed local outputs C_i T can easily couple different Jordan chains or sub-blocks. If a node only sees part of a chain, the local Luenberger piece no longer decouples cleanly from the consensus variables, and the block-triangular error system the conditions rely on may not hold. The abstract does not show how the proofs rule this out or whether the conditions are derived after accounting for the mixing. Without the full derivations it is hard to judge whether the necessity and sufficiency survive. The work is aimed at control engineers who already work with networked observers and want explicit design rules rather than existence results. A serious referee should see it because the conditions are new, the two-scheme comparison is useful, and the potential coupling issue is checkable in the proofs. I would send it for review.

Referee Report

3 major / 3 minor

Summary. The manuscript develops two distributed observer schemes for discrete-time LTI systems by transforming the dynamics into Jordan canonical form. Each node applies a local Luenberger observer to reconstruct the state components it can detect directly and uses a consensus protocol over the communication graph to recover the remaining components. Necessary and sufficient conditions for asymptotic convergence of the estimates are stated in terms of the eigenvalues of the system matrix and selected submatrices of the Laplacian; the first scheme uses observers whose dimension matches the original state but requires a large set of inequalities, while the second employs augmented-order observers under milder conditions. The approaches are presented as extensions of Gao & Yang (2025) that afford greater flexibility in coupling-gain selection.

Significance. If the central derivations hold, the work supplies explicit necessary-and-sufficient existence conditions for distributed observers, a useful contribution in the discrete-time setting. The Jordan-form separation into detectable and consensus-driven modes offers a structured design route, and the two schemes provide a concrete trade-off between observer order and solvability restrictions. The explicit dependence on eigenvalues and Laplacian sub-blocks is a clear strength that could aid verification on concrete graphs.

major comments (3)

[§3–4 (error-system construction for both schemes)] §3–4 (error-system construction for both schemes): the global similarity T that places A in Jordan form yields transformed outputs C_i T whose blocks generally mix distinct Jordan chains. Consequently the local Luenberger observer cannot be designed independently on the “detectable” sub-blocks; the resulting error dynamics lose the block-triangular structure implicitly required for the claimed eigenvalue conditions on the Laplacian submatrices to be necessary and sufficient. This coupling is not ruled out by the stated assumptions and directly affects the central existence claims.
[§4.1 (dimension-matched scheme)] §4.1 (dimension-matched scheme): the necessity part of the eigenvalue conditions on the Laplacian submatrices presupposes that the consensus error subsystem is decoupled from the local-observer error once the detectable components are removed. Because C_i T can introduce cross terms between chains, the closed-loop matrix is no longer block-triangular, so the necessity argument does not go through without additional restrictions on the output matrices or the graph.
[§4.2 (augmented-order scheme)] §4.2 (augmented-order scheme): while the augmentation is said to relax the inequalities, the same global-T coupling persists in the augmented error system; the paper must show that the extra states eliminate the cross-chain terms or that the eigenvalue conditions remain valid despite them. Absent such a demonstration the sufficiency claim for the milder conditions is not yet established.

minor comments (3)

[Abstract] The abstract refers to “a large set of inequalities” for the first scheme; an explicit listing or a compact matrix inequality equivalent would improve readability.
[§2–3] Notation for the transformed output matrices (C_i T) and the partitioned Jordan blocks should be introduced once and used consistently; occasional reuse of the original C_i symbols creates ambiguity.
[Introduction] The comparison with Gao & Yang (2025) would benefit from a short table contrasting the solvability conditions and gain-selection freedom of the three approaches.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The concerns about the impact of the global similarity transformation on the error dynamics and the validity of the necessary-and-sufficient conditions are substantive. We will revise the manuscript to provide explicit derivations of the closed-loop error system that account for possible cross terms and to strengthen or qualify the existence claims accordingly.

read point-by-point responses

Referee: §3–4 (error-system construction for both schemes): the global similarity T that places A in Jordan form yields transformed outputs C_i T whose blocks generally mix distinct Jordan chains. Consequently the local Luenberger observer cannot be designed independently on the “detectable” sub-blocks; the resulting error dynamics lose the block-triangular structure implicitly required for the claimed eigenvalue conditions on the Laplacian submatrices to be necessary and sufficient. This coupling is not ruled out by the stated assumptions and directly affects the central existence claims.

Authors: We agree that the global transformation T produces C_i T matrices whose blocks can mix distinct Jordan chains, preventing fully independent local design on detectable sub-blocks and potentially destroying the block-triangular error structure. In the revision we will derive the full (non-block-triangular) closed-loop error dynamics explicitly, incorporating the cross terms, and prove that the stated eigenvalue-Laplacian conditions still guarantee asymptotic convergence of the overall error. Where the mixing cannot be ruled out, we will add a mild structural assumption on the local C_i (that each node’s detectable subspace aligns with complete Jordan chains) and show that the original claims hold under this assumption. revision: yes
Referee: §4.1 (dimension-matched scheme): the necessity part of the eigenvalue conditions on the Laplacian submatrices presupposes that the consensus error subsystem is decoupled from the local-observer error once the detectable components are removed. Because C_i T can introduce cross terms between chains, the closed-loop matrix is no longer block-triangular, so the necessity argument does not go through without additional restrictions on the output matrices or the graph.

Authors: The referee correctly identifies that the necessity argument relies on decoupling after removal of detectable components. With possible cross terms the matrix is not block-triangular, so the necessity proof must be revisited. We will revise §4.1 to either (i) restrict the admissible output matrices C_i so that cross-chain mixing does not occur for detectable modes, or (ii) analyze the spectrum of the full closed-loop matrix directly and prove that the eigenvalue conditions on the Laplacian submatrices remain necessary for stability. The revised necessity statement will be stated with the appropriate qualification. revision: yes
Referee: §4.2 (augmented-order scheme): while the augmentation is said to relax the inequalities, the same global-T coupling persists in the augmented error system; the paper must show that the extra states eliminate the cross-chain terms or that the eigenvalue conditions remain valid despite them. Absent such a demonstration the sufficiency claim for the milder conditions is not yet established.

Authors: We accept that the augmentation does not automatically cancel the cross terms induced by the global T. In the revised §4.2 we will augment the stability analysis with an explicit Lyapunov or eigenvalue argument showing that the additional observer states can be chosen so that the milder Laplacian-submatrix conditions still dominate the cross terms and guarantee asymptotic convergence. The sufficiency proof will be supplied in full, thereby establishing the claimed relaxation of the inequalities. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations presented as independent extensions

full rationale

The paper explicitly builds on Gao & Yang (2025) but states that it develops two new distributed estimation schemes and derives fresh necessary and sufficient conditions in terms of eigenvalues and Laplacian submatrices. No quoted step shows a result reducing by construction to the cited framework, a fitted parameter, or a self-referential definition; the central claims are framed as new derivations rather than inherited without re-derivation. The self-citation is acknowledged but not load-bearing for the novel contributions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear-algebra facts about Jordan forms, standard control-theoretic notions of detectability, and the prior distributed-observer framework of Gao & Yang (2025). No new physical entities are introduced; the coupling gains are free parameters chosen to satisfy the derived inequalities.

free parameters (1)

coupling gains
Scalar or matrix gains that must be selected to satisfy the large set of inequalities (first scheme) or the relaxed conditions (second scheme) for error stability.

axioms (2)

standard math Every square matrix over the complex numbers possesses a Jordan canonical form
Invoked to decompose the system matrix so that detectable and undetectable modes can be treated separately by each node.
domain assumption Individual nodes possess local detectability of at least some state components
Required so that a Luenberger observer of appropriate dimension can be designed for the locally observable subsystem.

pith-pipeline@v0.9.0 · 5515 in / 1430 out tokens · 91887 ms · 2026-05-09T20:10:14.574898+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Disarò, G., Fattore, G., & Valcher, M. E. (2025). Dis- tributed state estimation for discrete-time lti systems 16 in the presence of unknown inputs. IEEE Transac- tions on Automatic Control , (pp. 1–15)

work page 2025

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E., Gao, R., & Yang, G.-H

Fattore, G., Valcher, M. E., Gao, R., & Yang, G.-H. (2026). Distributed state estimation of discrete-time lti systems via jordan canonical representation. To be presented at the 24th European Control Conference (ECC), July 7–10, 2026, Reykjavík, Iceland

work page 2026

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Fioravanti, C., Makridis, E., Oliva, G., Vrakopoulou, M., & Charalambous, T. (2024). Distributed estimation and control for lti systems under finite-time agree- ment. IEEE Transactions on Automatic Control , 69, 7909–7916

work page 2024

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Gao, R., & Yang, G.-H. (2025). Distributed state esti- mation for discrete-time linear systems: A canonical decomposition-based approach. IEEE Transactions on Automatic Control , 70, 4825–4832

work page 2025

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L., Wang, Z., & Shen, Y

Han, W., Trentelman, H. L., Wang, Z., & Shen, Y. (2019). A simple approach to distributed observer de- sign for linear systems. IEEE Transactions on Auto- matic Control, 64, 329–336

work page 2019

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A., & Johnson, C

Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis

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Huang, S., Li, Y., & Wu, J. (2022). Distributed state es- timation for linear time-invariant dynamical systems: A review of theories and algorithms. Chinese Journal of Aeronautics, 35, 1–17

work page 2022

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Hwang, S.-G. (2004). Cauchy’s interlace theorem for eigenvalues of hermitian matrices. The American Mathematical Monthly, 111, 157–159

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Kailath, T. (1980). Linear Systems. Prentice Hall, Inc

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Kamgarpour, M., & Tomlin, C. (2008). Convergence properties of a decentralized kalman filter. In Pro- ceedings of the 47th IEEE Conference on Decision and Control (pp. 3205–3210). Cancun, Mexico

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Mitra, A., & Sundaram, S. (2018). Distributed observers for lti systems. IEEE Transactions on Automatic Con- trol, 63, 3689–3704

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Mitra, A., & Sundaram, S. (2019). Byzantine-resilient distributed observers for lti systems. Automatica, 108, 108487

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Olfati-Saber, R. (2005). Distributed kalman filter with embedded consensus filters. In Proceedings of the 44th IEEE Conference on Decision and Control (pp. 8179– 8184)

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Olfati-Saber, R. (2007). Distributed kalman filtering for sensor networks. In Proceedings of the 46th IEEE Conference on Decision and Control (pp. 5492–5498). New Orleans, USA

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Park, S., & Martins, N. C. (2012a). An augmented ob- server for the distributed estimation problem for lti systems. In Proceedings of the 2012 American Control Conference (pp. 6775–6780). Montreal, Canada

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Park, S., & Martins, N. C. (2017). Design of distributed lti observers for state omniscience. IEEE Transactions on Automatic Control , 62, 561–576

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Rego, F. F. C., Pascoal, A. M., Aguiar, A. P., & Jones, C. N. (2019). Distributed state estimation for discrete- time linear time invariant systems: A survey. Annual Reviews in Control , 48, 36–56

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Wang, L., Liu, J., Anderson, B. D. O., & Morse, A. S. (2024). Split-spectrum based distributed state esti- mation for linear systems. Automatica, 161, 111421

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Yang, G., Barboni, A., Rezaee, H., & Parisini, T. (2022). State estimation using a network of dis- tributed observers with unknown inputs. Automatica, 146, 110631. 17

work page 2022