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arxiv: 2604.11588 · v1 · submitted 2026-04-13 · 📡 eess.SY · cs.SY

Distributed State Estimation for Discrete-Time Systems With Unknown Inputs: An Optimization Approach

Ruixuan Zhao , Guitao Yang , Nicola Bastianello , Boli Chen This is my paper

Pith reviewed 2026-05-10 15:15 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords distributed state estimationunknown input observerdistributed optimizationdiscrete-time systemssensor networkserror boundsstrong convexity
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The pith

A distributed framework lets sensor nodes estimate the full state of systems with unknown inputs by locally reconstructing reachable parts and optimizing their fusion.

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

The paper establishes a Distributed Unknown Input Observer that has each node first compute the largest state portion it can reconstruct from its own measurements and local inputs. Nodes then run a distributed optimization routine to combine those partial estimates into a global state. The analysis proves the resulting estimation error stays bounded, with the bound explicitly tied to how many communication rounds occur per time step and to the strong-convexity constant fixed by the system matrices. A normalization step is added inside the optimizer to improve convergence when the system geometry is poorly conditioned. This matters for large-scale plants where sensors and actuators are spread out and full centralization is impractical.

Core claim

The DUIO framework achieves bounded estimation error by having each node compute its maximal locally reconstructible state from local inputs and outputs, then fusing the partial estimates through a distributed optimization algorithm whose error bound depends on the number of communication iterations per time step and the strong-convexity constant determined by the system parameters; a normalization step mitigates curvature anisotropy from ill-conditioned geometry.

What carries the argument

The Distributed Unknown Input Observer (DUIO) that combines local maximal-state reconstruction with a distributed optimization step that fuses the partial estimates while preserving strong convexity.

If this is right

  • The explicit error bound decreases monotonically with more communication iterations per time step.
  • The normalization step inside the optimizer reduces the effect of poor system conditioning on convergence rate.
  • The framework applies directly to large-scale discrete-time networks whose sensors and unknown inputs are spatially dispersed.

Where Pith is reading between the lines

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

  • The same local-reconstruction-plus-optimization pattern could be tested on continuous-time or nonlinear systems without changing the core convexity argument.
  • If the strong-convexity constant can be computed from local data alone, the method could run with minimal global knowledge.
  • The bounded-error guarantee supplies a concrete starting point for designing distributed controllers that use the produced state estimates.

Load-bearing premise

Each node can independently compute the largest state segment reconstructible from its own local inputs and outputs, and the global fusion problem remains strongly convex with parameters set by the system.

What would settle it

Run the algorithm on a system where the local reconstructibility condition fails for at least one node and observe whether the global estimation error grows unbounded even as communication iterations increase.

Figures

Figures reproduced from arXiv: 2604.11588 by Boli Chen, Guitao Yang, Nicola Bastianello, Ruixuan Zhao.

Figure 1
Figure 1. Figure 1: Time step of system dynamics and communication [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Commutative diagram of W ∗ g,i decomposition at Node i. In this subsection, we design for each node i a local state estimation that filters out the effect of the local unknown inputs while preserving the information that is still recon￾structible from the local measurements. The construction follows the geometric approach in Zhao et al. (2025b,a) and is briefly recalled here for completeness [PITH_FULL_IM… view at source ↗
Figure 3
Figure 3. Figure 3: Communication network for numerical example. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Estimation errors using initial Algorithm 1 with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimation errors using normalized Algorithm 1 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Estimation errors using normalized Algorithm 1 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

This paper proposes a novel Distributed Unknown Input Observer (DUIO) framework for state estimation in large-scale systems subject to local unknown inputs. We consider systems where outputs are measured by a network of spatially distributed sensors and inputs are introduced through multiple dispersed channels. In this framework, each local node utilizes only its local input and output measurements to estimate the maximal locally reconstructible state. Subsequently, nodes collaboratively reconstruct the whole system state via a distributed optimization algorithm that fuses these partial estimates. We provide a rigorous analysis showing that the estimation error is bounded, with the error bound explicitly dependent on the number of communication iterations per time step and strongly convexity constant determined by the system parameters. Furthermore, to counteract curvature anisotropy induced by poor conditioned system geometry, we embed a normalization step into the distributed optimization procedure. Simulation results demonstrate the effectiveness of the proposed framework and the performance improvements yielded by the normalization procedure.

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.

Referee Report

2 major / 1 minor

Summary. The paper proposes a Distributed Unknown Input Observer (DUIO) framework for discrete-time large-scale systems with unknown inputs. Each local node uses only its local input and output measurements to compute the maximal locally reconstructible state subspace. Nodes then collaboratively reconstruct the full state by solving a distributed quadratic program that fuses the partial estimates. The manuscript claims a rigorous bounded-error analysis in which the estimation error is bounded by an expression that depends explicitly on the number of communication iterations per time step and on a strong-convexity constant μ determined solely by the system matrices (A,B,C,D). A normalization step is embedded in the distributed optimizer to mitigate curvature anisotropy, and simulation results are provided to illustrate effectiveness.

Significance. If the claimed error bound and its explicit dependence on iteration count and system-determined μ can be rigorously established, the work would supply a useful optimization-based route to distributed state estimation under unknown inputs, particularly for sensor networks where global observability is absent but local reconstructibility is exploitable. The normalization procedure for handling ill-conditioned geometry is a practical contribution. The manuscript does not supply machine-checked proofs or fully reproducible code, but the simulation validation and the attempt at an iteration-dependent bound are positive features.

major comments (2)
  1. [Abstract / error-bound derivation] Abstract and the section deriving the error bound: the headline claim requires that the fused quadratic program remain strongly convex with parameter μ > 0 fixed by the system matrices alone and independent of the communication graph. No rank, observability, or complementarity condition on the local reconstructible subspaces is stated that would guarantee the global Hessian is positive definite with such a μ. Without this, the derivation of the iteration-dependent error bound collapses.
  2. [Distributed optimization procedure] The normalization step (introduced to counteract curvature anisotropy): it is unclear whether this rescaling preserves the claimed strong-convexity constant μ or merely alters the effective conditioning while leaving the error-bound derivation intact. The manuscript must show that the normalized Hessian still satisfies the same μ lower bound used in the convergence analysis.
minor comments (1)
  1. [Abstract] Abstract: 'strongly convexity constant' is a grammatical error and should read 'strong-convexity constant'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will incorporate revisions to strengthen the theoretical claims.

read point-by-point responses

  1. Referee: [Abstract / error-bound derivation] Abstract and the section deriving the error bound: the headline claim requires that the fused quadratic program remain strongly convex with parameter μ > 0 fixed by the system matrices alone and independent of the communication graph. No rank, observability, or complementarity condition on the local reconstructible subspaces is stated that would guarantee the global Hessian is positive definite with such a μ. Without this, the derivation of the iteration-dependent error bound collapses.

    Authors: We agree that the conditions guaranteeing strong convexity of the fused quadratic program with μ determined solely by the system matrices (and independent of the communication graph) were not stated explicitly. In the revised manuscript we will introduce a new assumption on the complementarity of the local reconstructible subspaces (ensuring their direct sum spans the full state space) together with a supporting lemma that derives the explicit lower bound μ from the system matrices A, B, C, D alone. The iteration-dependent error bound derivation will then be restated under this assumption, restoring its validity. revision: yes

  2. Referee: [Distributed optimization procedure] The normalization step (introduced to counteract curvature anisotropy): it is unclear whether this rescaling preserves the claimed strong-convexity constant μ or merely alters the effective conditioning while leaving the error-bound derivation intact. The manuscript must show that the normalized Hessian still satisfies the same μ lower bound used in the convergence analysis.

    Authors: The normalization is a positive-definite diagonal scaling chosen to improve conditioning. We will add a short lemma in the revision proving that this scaling leaves the strong-convexity parameter μ unchanged: the minimal eigenvalue of the normalized Hessian is bounded below by the same μ derived from the original system matrices. Consequently the convergence analysis and the iteration-dependent error bound remain exactly as stated. revision: yes

Circularity Check

0 steps flagged

No circularity: error bound derived from stated assumptions on local subspaces and strong convexity

full rationale

The abstract and derivation outline a sequential process: local nodes compute maximal reconstructible states from their own (u,y) data, then solve a distributed QP whose Hessian is asserted strongly convex with μ fixed by (A,B,C,D). The error bound is then obtained explicitly in terms of iteration count and this μ. No equation reduces the bound to a fitted parameter, self-defined quantity, or prior self-citation chain; the convexity parameter is an input assumption rather than an output of the target result. The normalization step is an algorithmic addition, not a redefinition of the bound. The chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are detailed. Relies on standard discrete-time linear system assumptions and convergence properties of distributed optimization under strong convexity.

axioms (2)
  • domain assumption Discrete-time linear system dynamics with unknown inputs and distributed sensor measurements
    Invoked as the setting for the DUIO framework
  • domain assumption Distributed optimization converges with error bound governed by iteration count and strong-convexity constant
    Central to the bounded-error claim

pith-pipeline@v0.9.0 · 5457 in / 1245 out tokens · 75152 ms · 2026-05-10T15:15:18.932359+00:00 · methodology

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Reference graph

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