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A robust consensus + innovations-based distributed parameter estimator

Nicolai Lorenz-Meyer, Juan G. Rueda-Escobedo, Jaime A. Moreno, Johannes Schiffer

TL;DR

The error dynamics of a continuous-time version of the widely used consensus + innovations-based distributed parameter estimator is recast to reflect the error dynamics induced by the classical gradient descent algorithm, paving the way for the construction of a strong Lyapunov function.

Abstract

While distributed parameter estimation has been extensively studied in the literature, little has been achieved in terms of robust analysis and tuning methods in the presence of disturbances. However, disturbances such as measurement noise and model mismatches occur in any real-world setting. Therefore, providing tuning methods with specific robustness guarantees would greatly benefit the practical application. To address these issues, we recast the error dynamics of a continuous-time version of the widely used consensus + innovations-based distributed parameter estimator to reflect the error dynamics induced by the classical gradient descent algorithm. This paves the way for the construction of a strong Lyapunov function. Based on this result, we derive linear matrix inequality-based tools for tuning the algorithm gains such that a guaranteed upper bound on the L2-gain with respect to parameter variations, measurement noise, and disturbances in the communication channels is achieved. An application example illustrates the efficiency of the method.

A robust consensus + innovations-based distributed parameter estimator

TL;DR

The error dynamics of a continuous-time version of the widely used consensus + innovations-based distributed parameter estimator is recast to reflect the error dynamics induced by the classical gradient descent algorithm, paving the way for the construction of a strong Lyapunov function.

Abstract

While distributed parameter estimation has been extensively studied in the literature, little has been achieved in terms of robust analysis and tuning methods in the presence of disturbances. However, disturbances such as measurement noise and model mismatches occur in any real-world setting. Therefore, providing tuning methods with specific robustness guarantees would greatly benefit the practical application. To address these issues, we recast the error dynamics of a continuous-time version of the widely used consensus + innovations-based distributed parameter estimator to reflect the error dynamics induced by the classical gradient descent algorithm. This paves the way for the construction of a strong Lyapunov function. Based on this result, we derive linear matrix inequality-based tools for tuning the algorithm gains such that a guaranteed upper bound on the L2-gain with respect to parameter variations, measurement noise, and disturbances in the communication channels is achieved. An application example illustrates the efficiency of the method.
Paper Structure (19 sections, 9 theorems, 112 equations, 4 figures, 1 table)

This paper contains 19 sections, 9 theorems, 112 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and literature review
  3. Contributions
  4. C+I-based distributed parameter estimation
  5. Nominal system
  6. Main results
  7. Strong Lyapunov function for C+I-based distributed parameter estimation
  8. $\mathcal{L}_2$-gain tuning in the presence of disturbances
  9. Application example
  10. Conclusions
  11. Proof of the main results
  12. Proof of Lemma \ref{['lemma:obs_gram_OI']}
  13. Proof of Lemma \ref{['lemma:obs_gram']}
  14. Proof of Theorem \ref{['th:stong_lya']}
  15. Proof of Corollary \ref{['corollary:convergence_bounds']}
  16. ...and 4 more sections

Key Result

Lemma 1

Under Assumptions ass:2 and ass:1, cpe eq:cPE implies the uco of the system eq:err_dyn2_OI, with the bounds on $M^\mathrm{OI}$ in eq:gram_OI as follows: with and where $a(t) \coloneqq [a_1(t), ..., a_N(t)]^\top \in \mathbb{R}^N$ is a vector with components $a_i(t) \ge 0, \ \forall t \in \mathbb{R}, \ \forall i \in \mathcal{V}$ defined in eq:linear_combination_EV_L.

Figures (4)

  • Figure 1: Communication structure among the six agents.
  • Figure 2: Simulation results of the system \ref{['eq:sys_mass_spring']}.
  • Figure 3: Estimation errors of the six agents.
  • Figure 4: The calculated values of the metric $\sqrt{\gamma}$ as defined in \ref{['eq:metric']} for the five scenarios given in Table \ref{['tab:scenarios']} using different gains, as well as the average value of $\sqrt{\gamma}$ for each gain over the five scenarios.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • ...and 11 more