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Hierarchical parameter estimation for distributed networked systems: a dynamic consensus approach

Ariana R. Mendez-Castillo, Rodrigo Aldana-Lopez, Antonio Ramirez-Trevino, Rosario Aragues, David Gomez-Gutierrez

TL;DR

A novel two-stage distributed framework to globally estimate constant parameters in a networked system, separating shared information from local estimation, and supports relaxed excitation requirements is introduced.

Abstract

This work introduces a novel two-stage distributed framework to globally estimate constant parameters in a networked system, separating shared information from local estimation. The first stage uses dynamic average consensus to aggregate agents' measurements into surrogates of centralized data. Using these surrogates, the second stage implements a local estimator to determine the parameters. By designing an appropriate consensus gain, the persistence of excitation of the regressor matrix is achieved, and thus, exponential convergence of a local Gradient Estimator (GE) is guaranteed. The framework facilitates its extension to switched network topologies, quantization, and the heterogeneous substitution of the GE with a Dynamic Regressor Extension and Mixing (DREM) estimator, which supports relaxed excitation requirements.

Hierarchical parameter estimation for distributed networked systems: a dynamic consensus approach

TL;DR

A novel two-stage distributed framework to globally estimate constant parameters in a networked system, separating shared information from local estimation, and supports relaxed excitation requirements is introduced.

Abstract

This work introduces a novel two-stage distributed framework to globally estimate constant parameters in a networked system, separating shared information from local estimation. The first stage uses dynamic average consensus to aggregate agents' measurements into surrogates of centralized data. Using these surrogates, the second stage implements a local estimator to determine the parameters. By designing an appropriate consensus gain, the persistence of excitation of the regressor matrix is achieved, and thus, exponential convergence of a local Gradient Estimator (GE) is guaranteed. The framework facilitates its extension to switched network topologies, quantization, and the heterogeneous substitution of the GE with a Dynamic Regressor Extension and Mixing (DREM) estimator, which supports relaxed excitation requirements.
Paper Structure (16 sections, 60 equations, 6 figures)

This paper contains 16 sections, 60 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Centralized estimation
  4. Hierarchical Loosely Coupled Estimation
  5. Distributed DREM
  6. Discussion
  7. Numerical Examples
  8. Conclusion
  9. Proofs
  10. Proof of Proposition \ref{['prop:prime_equivalence']}
  11. Auxiliary lemmas for Theorem \ref{['the:gain_PE']}
  12. Proof of Theorem \ref{['the:gain_PE']}
  13. Proof of Corollary \ref{['cor:quantization']}
  14. Proof of Corollary \ref{['cor:switched']}
  15. Proof of Corollary \ref{['cor:DREM_L2']}
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Distributed parameter estimator framework. (Top) Scheme of the consensus+innovations algorithm, implemented within the same block. (Bottom) Scheme of the proposed approach, where consensus and parameter estimation are decoupled.
  • Figure 2: Collection of communication topologies $\mathcal{G}_\sigma$ used for the switched dynamic network in the numerical example.
  • Figure 3: Simulation results. Colors are related to the parameters $\theta_1$ (blue), $\theta_2$ (red), and $\theta_3$ (green). (Top) Parameters estimation $\hat{\theta}_{i,\mu},$ for each agent for GE (green) and DREM (blue); dashed lines show $\bm{\theta}$. (Top-middle) Dashed line represents the average value ($\overline{\mathbf{y}}(t)$), while solid lines represent the individual consensus outputs $\hat{\mathbf{y}}_i(t)$. (Low-middle) Magnitude of the estimation error $\mathbf{e}_i$, dashed for GE and solid for DREM. (Bottom) The switching signal $\sigma(t)$.
  • Figure 4: Simulation results using quantization. Colors are related to the parameters $\theta_1$ (blue), $\theta_2$ (red) $\theta_3$ (green). (Top) Parameter estimation for each agent for GE (green) and DREM (blue); dashed lines show $\bm{\theta}$. (Bottom) Magnitude of estimation error $\mathbf{e}_i$, dashed lines for GE and solid for DREM.
  • Figure 5: Simulation results with additive Gaussian sensor noise modeled as $\mathbf{y}_i(t)=\mathbf{C}_i(t)\bm{\theta}+\bm{\eta}_i(t)$, where $\bm{\eta}_i(t)$ is Gaussian noise with mean $0$ and standard deviation equal to $0.2$. Estimation error magnitudes $\mathbf{e}_1$ (blue), $\mathbf{e}_2$ (red), and $\mathbf{e}_3$ (green) are shown; dotted lines indicate GE and solid lines DREM.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Definition 1