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Distributed Adaptive Estimation over Sensor Networks with Partially Unknown Source Dynamics

Moh Kamalul Wafi, Hamidreza Montazeri Hedesh, Milad Siami

TL;DR

This work addresses distributed state estimation over sensor networks when the source dynamics are only partially known. It develops parallel continuous-time and discrete-time adaptive observers that rely on a directed graph topology and Kronecker-structured couplings to decouple local observer design from network connectivity. Key contributions include stability conditions for the coupling operators ($\mathcal{H}$ in continuous time and $\mathcal{S}$ in discrete time), ISS bounds under bounded disturbances, and, for the discrete-time case, a normalized gradient adaptation with per-step regressor normalization that guarantees bounded parameter updates. Numerical results on star, cyclic, and path networks confirm accurate tracking, robustness to disturbances, and scalability to large node counts. The framework provides a practical, scalable approach for robust distributed estimation in directed sensor networks with uncertain source dynamics, and sets the stage for integration with distributed control and learning-driven enhancements.

Abstract

This paper studies distributed adaptive estimation over sensor networks with partially known source dynamics. We present parallel continuous-time and discrete-time designs in which each node runs a local adaptive observer and exchanges information over a directed graph. For both time scales, we establish stability of the network coupling operators, prove boundedness of all internal signals, and show convergence of each node estimate to the source despite model uncertainty and disturbances. We further derive input-to-state stability (ISS) bounds that quantify robustness to bounded process noise. A key distinction is that the discrete-time design uses constant adaptive gains and per-step regressor normalization to handle sampling effects, whereas the continuous-time design does not. A unified Lyapunov framework links local observer dynamics with graph topology. Simulations on star, cyclic, and path networks corroborate the analysis, demonstrating accurate tracking, robustness, and scalability with the number of sensing nodes.

Distributed Adaptive Estimation over Sensor Networks with Partially Unknown Source Dynamics

TL;DR

This work addresses distributed state estimation over sensor networks when the source dynamics are only partially known. It develops parallel continuous-time and discrete-time adaptive observers that rely on a directed graph topology and Kronecker-structured couplings to decouple local observer design from network connectivity. Key contributions include stability conditions for the coupling operators ( in continuous time and in discrete time), ISS bounds under bounded disturbances, and, for the discrete-time case, a normalized gradient adaptation with per-step regressor normalization that guarantees bounded parameter updates. Numerical results on star, cyclic, and path networks confirm accurate tracking, robustness to disturbances, and scalability to large node counts. The framework provides a practical, scalable approach for robust distributed estimation in directed sensor networks with uncertain source dynamics, and sets the stage for integration with distributed control and learning-driven enhancements.

Abstract

This paper studies distributed adaptive estimation over sensor networks with partially known source dynamics. We present parallel continuous-time and discrete-time designs in which each node runs a local adaptive observer and exchanges information over a directed graph. For both time scales, we establish stability of the network coupling operators, prove boundedness of all internal signals, and show convergence of each node estimate to the source despite model uncertainty and disturbances. We further derive input-to-state stability (ISS) bounds that quantify robustness to bounded process noise. A key distinction is that the discrete-time design uses constant adaptive gains and per-step regressor normalization to handle sampling effects, whereas the continuous-time design does not. A unified Lyapunov framework links local observer dynamics with graph topology. Simulations on star, cyclic, and path networks corroborate the analysis, demonstrating accurate tracking, robustness, and scalability with the number of sensing nodes.

Paper Structure

This paper contains 17 sections, 11 theorems, 80 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Communication Network
  3. Problem Formulation
  4. Problem 1: Unknown Continuous-time Dynamics
  5. Problem 2: Uncertain Discrete-time Dynamics
  6. Unified Notation
  7. Adaptive Estimation for Problem 1
  8. Error Dynamics in Networked Systems
  9. Stability of the Coupling Operator $\mathcal{H}$
  10. Adaptive Laws and Stability
  11. Adaptive Estimation for Problem 2
  12. Error Dynamics in Networked Systems
  13. Stability of the Coupling Operator $\mathcal{S}$
  14. Normalized Adaptive Laws and Stability
  15. Numerical Results
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\mathcal{H} \coloneqq (\mathbb{L}\otimes I_n)\mathbf{H} + (\mathbb{A}_m\otimes I_n)\mathbf{F}_0$ where $\mathbb{L}\succ 0$ is the (positive-stable) augmented Laplacian under the balanced and source-reachability assumptions (cf. eq:ComNet:balance and Remark rem:reachability), $\mathbf{H}=\mathop If then $\mathcal{H}$ is Hurwitz.

Figures (3)

  • Figure 1: Example of a graph $\mathcal{G}$ with $m=5$, sensing-to-sensing subgraph $\mathcal{G}_m$, and source-to-sensing subgraph $\mathcal{G}_0$, showing the decoupling and assignment of $w_{ij}$.
  • Figure 2: Three network topologies $(\mathcal{G}_{\mathrm{s}}, \mathcal{G}_{\mathrm{c}}, \mathcal{G}_{\mathrm{p}})$ with weights used in the simulations.
  • Figure 3: Trajectories of sensing nodes for continuous-time cases (a)--(c) and discrete-time cases (e)--(g), along with corresponding error norms in (d) and (h).

Theorems & Definitions (14)

  • Remark 1: Source Reachability
  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Remark 2
  • Corollary 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Proposition 2
  • ...and 4 more