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arxiv: 2604.21608 · v1 · submitted 2026-04-23 · 📡 eess.SY · cs.SY

ADMM-Based Distributed Kalman-like Observer with Applications to Cooperative Localization

Nicola De Carli , Nicola Bastianello , Dimos V. Dimarogonas This is my paper

Pith reviewed 2026-05-09 20:34 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords distributed state estimationADMMKalman-like observermulti-agent systemscooperative localizationinformation filtertwo-time-scale stability
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The pith

A distributed ADMM-based Kalman-like observer maintains uniform exponential stability for multi-agent state estimation using only local communications.

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

The paper develops a Kalman-like observer in information form for multi-agent systems that rely on local and relative measurements. An exponential forgetting factor keeps the prediction step sparse and avoids the dense Riccati updates of a standard information filter. The correction step is cast as a strongly convex quadratic program whose structure follows the sensing graph, allowing solution by ADMM so that each agent updates only local copies of its own and its neighbors' variables. A two-time-scale argument then proves uniform exponential stability by showing the reduced estimation-error system is uniformly exponentially stable while the ADMM iterations form an exponentially stable fast subsystem. This setup supports cooperative localization in networks whose global state dimension grows with the number of agents.

Core claim

The authors establish that the interconnected observer formed by the sparsity-preserving Kalman-like update and the ADMM solution of the graph-structured correction quadratic program is uniformly exponentially stable. The two-time-scale analysis combines uniform exponential stability of the reduced error dynamics with exponential stability of the fast ADMM dynamics to reach this conclusion for the overall distributed estimator.

What carries the argument

The two-time-scale stability analysis applied to the interconnected observer, separating the reduced estimation-error dynamics from the ADMM iteration dynamics.

If this is right

  • The estimation error of the global state converges exponentially to zero under the stated conditions.
  • Agents communicate only with neighbors and maintain local copies of correction variables without ever inverting a centralized matrix.
  • The prediction step remains sparse, so storage and computation scale better with network size than a standard information filter.
  • The scheme applies directly to cooperative localization problems where the global state vector length equals the number of agents.
  • No consensus step over the full global state is required at any iteration.

Where Pith is reading between the lines

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

  • The same two-time-scale separation could be reused to analyze distributed observers that incorporate communication delays or switching graphs.
  • Reducing the number of ADMM iterations per step might trade a small amount of estimation accuracy for lower per-agent communication load in very large networks.
  • Physical experiments on mobile robots would test whether the exponential stability margins survive model mismatch and sensor noise not captured in the linear analysis.
  • The quadratic-program structure might admit faster solvers than ADMM for particular graph topologies, potentially tightening the separation between time scales.

Load-bearing premise

The correction step admits a strongly convex quadratic-program formulation whose structure is induced by the sensing graph and therefore permits a distributed ADMM solution.

What would settle it

A cooperative localization simulation in which the number of ADMM iterations per time step is deliberately lowered while the network size is increased, to check whether the combined estimation error remains bounded or begins to diverge.

read the original abstract

This paper addresses distributed state estimation for multi-agent systems with local and relative measurements, motivated by cooperative localization problems in which the global state dimension scales with the size of the network. We consider a Kalman-like observer in information form and introduce a sparsity-preserving prediction step based on an exponential forgetting factor, thereby avoiding the dense Riccati recursion of the standard information filter. The correction step is recast as a strongly convex quadratic program with structure induced by the sensing graph, which enables a distributed solution based on the alternating direction method of multipliers (ADMM). In the resulting scheme, each agent updates local copies of its own correction variable and those of its neighbors using only local communication, thus avoiding centralized matrix inversion and consensus over full global-state quantities. A two-time-scale stability analysis is developed for the interconnected observer: the reduced estimation-error dynamics are shown to be uniformly exponentially stable, the ADMM dynamics define an exponentially stable fast subsystem, and these properties are combined to establish uniform exponential stability of the overall distributed observer. Numerical simulations in a multi-agent cooperative localization scenario illustrate the performance of the proposed distributed observer.

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

1 major / 2 minor

Summary. This paper proposes an ADMM-based distributed Kalman-like observer for multi-agent systems with local and relative measurements, aimed at cooperative localization. It introduces a sparsity-preserving prediction step using an exponential forgetting factor and recasts the correction step as a strongly convex quadratic program solved distributively via ADMM. The key theoretical result is a two-time-scale analysis establishing uniform exponential stability of the distributed observer by showing UES of the reduced estimation-error dynamics and exponential stability of the ADMM fast subsystem.

Significance. If the stability claims hold, this work provides a scalable, distributed method for state estimation in large networks without requiring centralized computations or full state consensus. It leverages standard tools from convex optimization and singular perturbation theory, offering both practical implementation benefits and rigorous stability guarantees. The simulations in cooperative localization scenarios add empirical support.

major comments (1)
  1. [Two-time-scale stability analysis] The combination of the reduced slow system UES and fast ADMM subsystem stability to conclude overall UES relies on the fast subsystem being exponentially stable uniformly with respect to the slow state (estimation error). The ADMM convergence rate depends on the strong convexity modulus of the quadratic program, which is determined by the Hessian involving the predicted information matrix and graph Laplacian. The manuscript should provide explicit arguments (e.g., via boundedness of the information matrix under the forgetting factor or Lipschitz continuity) ensuring the decay rate is bounded away from zero independently of the error; without this, the application of standard two-time-scale theorems (such as those in Khalil) may not be justified. This is load-bearing for the central stability claim.
minor comments (2)
  1. [Abstract] The abstract is clear but could briefly mention the specific assumptions on the sensing graph or measurement models for completeness.
  2. [Simulations] The numerical results illustrate performance but would benefit from comparison to a centralized information filter or other distributed methods to quantify the trade-offs in estimation accuracy and communication.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and constructive feedback on our manuscript. The major comment raises an important point about ensuring uniformity in the two-time-scale analysis, which we address below by clarifying the role of the forgetting factor and committing to explicit additions in the revision.

read point-by-point responses

  1. Referee: [Two-time-scale stability analysis] The combination of the reduced slow system UES and fast ADMM subsystem stability to conclude overall UES relies on the fast subsystem being exponentially stable uniformly with respect to the slow state (estimation error). The ADMM convergence rate depends on the strong convexity modulus of the quadratic program, which is determined by the Hessian involving the predicted information matrix and graph Laplacian. The manuscript should provide explicit arguments (e.g., via boundedness of the information matrix under the forgetting factor or Lipschitz continuity) ensuring the decay rate is bounded away from zero independently of the error; without this, the application of standard two-time-scale theorems (such as those in Khalil) may not be justified. This is load-bearing for the central stability claim.

    Authors: We appreciate the referee's careful identification of the uniformity requirement for applying two-time-scale results. The exponential forgetting factor in the sparsity-preserving prediction step is introduced specifically to ensure that the predicted information matrix remains bounded with eigenvalues uniformly bounded away from zero, independent of the estimation error. This prevents the accumulation of dense information and maintains a uniform strong-convexity modulus for the quadratic program (combined with the fixed graph Laplacian). Under the paper's observability assumptions on the multi-agent system, this yields an ADMM convergence rate bounded away from zero uniformly in the slow state. In the revised manuscript we will add an explicit lemma establishing the uniform eigenvalue bound on the predicted information matrix and will verify the conditions of the relevant singular-perturbation theorem (e.g., Khalil) with this uniformity made precise. This strengthens the central stability claim without altering the existing proof structure. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper recasts the correction step as a strongly convex QP solved via ADMM and combines UES of the reduced error dynamics with exponential stability of the ADMM fast subsystem via a two-time-scale theorem. These steps rely on standard results from convex optimization and singular perturbation theory (e.g., uniform bounds on the strong-convexity modulus and Lyapunov functions) rather than any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. No quoted equation reduces the central stability claim to its own inputs by construction; the analysis is self-contained against external benchmarks in control theory.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about the sensing graph structure and strong convexity of the quadratic program, plus standard results from ADMM convergence and stability theory; the forgetting factor is a design parameter.

free parameters (1)
  • exponential forgetting factor
    Introduced to preserve sparsity in the prediction step; its specific value is a design choice not derived from first principles.
axioms (2)
  • domain assumption The quadratic program for the correction step is strongly convex
    Invoked to guarantee a unique solution and convergence of the ADMM iterations.
  • domain assumption The sensing graph induces a structure that permits fully distributed ADMM updates using only local communication
    Required to avoid centralized matrix inversion and global consensus.

pith-pipeline@v0.9.0 · 5499 in / 1434 out tokens · 39223 ms · 2026-05-09T20:34:35.668843+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    The formulation of a discrete-time Kalman-like observer in information form that preserves the sparsity of the sensing graph during prediction, bypassing the dense Riccati recursions typical of standard information filters

    work page

  2. [2]

    A distributed correction architecture based on a partition- based ADMM. This scheme asymptotically recovers the centralized estimate while requiring only local communi- cation, avoids consensus over the full global state, and allows for the direct, closed-form integration of local sensing

    work page

  3. [3]

    ∆Sr ij,ii,k + ∆Sr ji,ii,k ∆Sr ij,ij,k + ∆Sr ji,ij,k ∆Sij ji,k + ∆Sji ji,k ∆Sij jj,k + ∆Sji jj,k # be ij,k =δ r ij

    A rigorous stability analysis of the resulting intercon- nected observer. Using a two-time-scale framework and small gain arguments, we establish the uniform global ex- ponential stability (UGES) of the full system by showing that the fast ADMM dynamics track the slow estimation- error dynamics. Notation:Scalars, vectors, matrices, and sets are denoted by...

    work page

  4. [4]

    Local primal update.Each agent solves the quadratic subproblem ˆξNi,k,h = argmin ξNi   Ji(ξNi)− X j∈Ni q⊤ ij,i,k,hξ(i) i +q ⊤ ij,j,k,hξ(i) j + ρ 2  |Ni| ∥ξ(i) i ∥2 + X j∈Ni ∥ξ(i) j ∥2      . (29)

    work page

  5. [5]

    In particular, at each observer step we allow for the execution of multiple ADMM iterations, denoted byH≥1

    Dual update.For every{i, j} ∈ E c, the dual variables are updated as qij,i,k,h+1 = (1−α)q ij,i,k,h +α −q ji,i,k,h + 2ρ ˆξ(j) i,k,h , qij,j,k,h+1 = (1−α)q ij,j,k,h +α −q ji,j,k,h + 2ρ ˆξ(j) j,k,h , (30) whereρ >0is the ADMM penalty parameter,α∈(0,1)is a relaxation factor, andhdenotes the ADMM iteration index. In particular, at each observer step we allow f...

    work page

  6. [6]

    after Line 4 of the algorithm, each agent first computes the local correction ξℓ i,k = (Sℓ i,k|k)−1bℓ i,k

    work page

  7. [7]

    •the primal step becomes: ˆηNi,k,h =H −1 ρ,i,k ¯bη i,k +A ⊤ qiqi,k,h ,(40) where ˆηNi,k,h and ¯bη i,k are defined analogously to ˆξNi,k,h and ¯bi,k

    the distributed ADMM layer is then applied to the resid- ual problem (37), usingη k as optimization variable, i.e. •the primal step becomes: ˆηNi,k,h =H −1 ρ,i,k ¯bη i,k +A ⊤ qiqi,k,h ,(40) where ˆηNi,k,h and ¯bη i,k are defined analogously to ˆξNi,k,h and ¯bi,k. •the dual step is as in (30) replacing ˆξ(j) i,k,h and ˆξ(j) j,k,h with ˆη(j) i,k,h and ˆη(j) j,k,h

    work page

  8. [8]

    All subsequent observer and prediction updates are then per- formed using the correction ˆξi,k

    the full correction is reconstructed as ˆξi,k =ξ ℓ i,k + ˆη(i) i,k,H . All subsequent observer and prediction updates are then per- formed using the correction ˆξi,k. For simplicity, in the following we focus on the algorithm introduced in the previous section. Nevertheless, the same analysis extends directly to the present decomposition. IV. STABILITYANA...

    work page

  9. [9]

    Recursive decentralized localization for multi-robot systems with asynchronous pairwise communication,

    L. Luft, T. Schubert, S. I. Roumeliotis, and W. Burgard, “Recursive decentralized localization for multi-robot systems with asynchronous pairwise communication,”The International Journal of Robotics Re- search, vol. 37, no. 10, pp. 1152–1167, 2018

    work page 2018

  10. [10]

    Cooperative localization for mobile agents: A recursive decentralized algorithm based on kalman- filter decoupling,

    S. S. Kia, S. Rounds, and S. Martinez, “Cooperative localization for mobile agents: A recursive decentralized algorithm based on kalman- filter decoupling,”IEEE Control Systems Magazine, vol. 36, no. 2, pp. 86–101, 2016

    work page 2016

  11. [11]

    Distributed multirobot localization,

    S. I. Roumeliotis and G. A. Bekey, “Distributed multirobot localization,” IEEE transactions on robotics and automation, vol. 18, no. 5, pp. 781– 795, 2002

    work page 2002

  12. [12]

    Resilient and consistent mul- tirobot cooperative localization with covariance intersection,

    T.-K. Chang, K. Chen, and A. Mehta, “Resilient and consistent mul- tirobot cooperative localization with covariance intersection,”IEEE Transactions on Robotics, vol. 38, no. 1, pp. 197–208, 2021

    work page 2021

  13. [13]

    Adaptive observer from body-frame relative position measurements for cooperative localization,

    N. De Carli, E. Restrepo, and P. R. Giordano, “Adaptive observer from body-frame relative position measurements for cooperative localization,” IEEE Control Systems Letters, vol. 8, pp. 1337–1342, 2024

    work page 2024

  14. [14]

    A taxonomy of multi-area state estimation methods,

    A. G ´omez-Exp´osito, A. De La Villa Ja ´en, C. G ´omez-Quiles, P. Rousseaux, and T. Van Cutsem, “A taxonomy of multi-area state estimation methods,”Electric Power Systems Research, vol. 81, no. 4, pp. 1060–1069, 2011. 16

    work page 2011

  15. [15]

    Distributed kalman-like filtering and bad data detection in the large-scale power system,

    J. Yang, W.-A. Zhang, and F. Guo, “Distributed kalman-like filtering and bad data detection in the large-scale power system,”IEEE transactions on industrial informatics, vol. 18, no. 8, pp. 5096–5104, 2021

    work page 2021

  16. [16]

    Consensus-based linear and nonlinear filtering,

    G. Battistelli, L. Chisci, G. Mugnai, A. Farina, and A. Graziano, “Consensus-based linear and nonlinear filtering,”IEEE Transactions on Automatic Control, vol. 60, no. 5, pp. 1410–1415, 2014

    work page 2014

  17. [17]

    Information weighted consensus filters and their application in distributed camera networks,

    A. T. Kamal, J. A. Farrell, and A. K. Roy-Chowdhury, “Information weighted consensus filters and their application in distributed camera networks,”IEEE Transactions on Automatic Control, vol. 58, no. 12, pp. 3112–3125, 2013

    work page 2013

  18. [18]

    Eco-dkf: Event-triggered and certifiable optimal distributed kalman filter under unknown corre- lations,

    E. Sebasti ´an, E. Montijano, and C. Sag ¨u´es, “Eco-dkf: Event-triggered and certifiable optimal distributed kalman filter under unknown corre- lations,”IEEE Transactions on Automatic Control, vol. 69, no. 4, pp. 2613–2620, 2023

    work page 2023

  19. [19]

    Decentralized collaborative state estimation for aided inertial navigation,

    R. Jung, C. Brommer, and S. Weiss, “Decentralized collaborative state estimation for aided inertial navigation,” in2020 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2020, pp. 4673–4679

    work page 2020

  20. [20]

    Covariance intersection algorithm for distributed spacecraft state estimation,

    P. O. Arambel, C. Rago, and R. K. Mehra, “Covariance intersection algorithm for distributed spacecraft state estimation,” inProceedings of the 2001 American Control Conference.(Cat. No. 01CH37148), vol. 6. IEEE, 2001, pp. 4398–4403

    work page 2001

  21. [21]

    Decentralized multi-robot cooperative localization using covariance intersection,

    L. C. Carrillo-Arce, E. D. Nerurkar, J. L. Gordillo, and S. I. Roumeliotis, “Decentralized multi-robot cooperative localization using covariance intersection,” in2013 IEEE/RSJ international conference on intelligent robots and systems. IEEE, 2013, pp. 1412–1417

    work page 2013

  22. [22]

    Fault-tolerant covariance intersection for localizing robot swarms,

    J. Klingner, N. Ahmed, and N. Correll, “Fault-tolerant covariance intersection for localizing robot swarms,”Robotics and Autonomous Systems, vol. 122, p. 103306, 2019

    work page 2019

  23. [23]

    Online decentralized perception-aware path planning for multi-robot systems,

    N. De Carli, P. Salaris, and P. R. Giordano, “Online decentralized perception-aware path planning for multi-robot systems,” in2021 Inter- national Symposium on Multi-Robot and Multi-Agent Systems (MRS). IEEE, 2021, pp. 128–136

    work page 2021

  24. [24]

    A distributed kalman-like observer with dynamic inversion-based correction for multi-agent estimation,

    N. De Carli and D. V . Dimarogonas, “A distributed kalman-like observer with dynamic inversion-based correction for multi-agent estimation,” IEEE Control Systems Letters, 2025

    work page 2025

  25. [25]

    On the semi-global stability of an ek-like filter,

    P. Bernard, N. Mimmo, and L. Marconi, “On the semi-global stability of an ek-like filter,”IEEE Control Systems Letters, vol. 5, no. 5, pp. 1771–1776, 2020

    work page 2020

  26. [26]

    Bertsekas and J

    D. Bertsekas and J. Tsitsiklis,Parallel and distributed computation: numerical methods. Athena Scientific, 2015

    work page 2015

  27. [27]

    Distributed control barrier functions for global connectivity maintenance,

    N. De Carli, P. Salaris, and P. R. Giordano, “Distributed control barrier functions for global connectivity maintenance,” in2024 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2024, pp. 12 048–12 054

    work page 2024

  28. [28]

    Exponential forgetting factor observer in discrete time,

    A. T ¸ iclea and G. Besanc ¸on, “Exponential forgetting factor observer in discrete time,”Systems & Control Letters, vol. 62, no. 9, pp. 756–763, 2013

    work page 2013

  29. [29]

    Distributed optimization and statistical learning via the alternating direction method of multipliers,

    P. Neal, C. Eric, P. Borja, and E. Jonathan, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine learning, vol. 3, no. 1, pp. 1–122, 2011

    work page 2011

  30. [30]

    A partition- based implementation of the relaxed admm for distributed convex opti- mization over lossy networks,

    N. Bastianello, R. Carli, L. Schenato, and M. Todescato, “A partition- based implementation of the relaxed admm for distributed convex opti- mization over lossy networks,” in2018 IEEE Conference on Decision and Control (CDC). IEEE, 2018, pp. 3379–3384

    work page 2018

  31. [31]

    Scalable distributed op- timization with separable variables in multi-agent networks,

    O. Shorinwa, T. Halsted, and M. Schwager, “Scalable distributed op- timization with separable variables in multi-agent networks,” in2020 American Control Conference (ACC). IEEE, 2020, pp. 3619–3626

    work page 2020

  32. [32]

    Admm- tracking gradient for distributed optimization over asynchronous and unreliable networks,

    G. Carnevale, N. Bastianello, G. Notarstefano, and R. Carli, “Admm- tracking gradient for distributed optimization over asynchronous and unreliable networks,”IEEE Transactions on Automatic Control, vol. 70, no. 8, pp. 5160–5175, 2025

    work page 2025

  33. [33]

    A unifying system theory framework for distributed optimization and games,

    G. Carnevale, N. Mimmo, and G. Notarstefano, “A unifying system theory framework for distributed optimization and games,”IEEE Trans- actions on Automatic Control, 2025

    work page 2025

  34. [34]

    Asynchronous distributed optimization over lossy networks via relaxed admm: Stability and linear convergence,

    N. Bastianello, R. Carli, L. Schenato, and M. Todescato, “Asynchronous distributed optimization over lossy networks via relaxed admm: Stability and linear convergence,”IEEE Transactions on Automatic Control, vol. 66, no. 6, pp. 2620–2635, 2020

    work page 2020

  35. [35]

    Proximal algorithms,

    N. Parikh and S. Boyd, “Proximal algorithms,”Foundations and Trends in optimization, vol. 1, no. 3, pp. 127–239, 2014

    work page 2014

  36. [36]

    Primer on monotone operator methods,

    E. K. Ryu and S. Boyd, “Primer on monotone operator methods,”Appl. comput. math, vol. 15, no. 1, pp. 3–43, 2016

    work page 2016

  37. [37]

    H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 1st ed., ser. CMS Books in Mathe- matics. New York, NY: Springer, 2011

    work page 2011

  38. [38]

    Ro- bust online learning over networks,

    N. Bastianello, D. Deplano, M. Franceschelli, and K. H. Johansson, “Ro- bust online learning over networks,”IEEE Transactions on Automatic Control, vol. 70, no. 2, pp. 933–946, 2024

    work page 2024

  39. [39]

    Berman and R

    A. Berman and R. J. Plemmons,Nonnegative matrices in the mathemat- ical sciences. SIAM, 1994

    work page 1994