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A Unified Framework for Innovation-based Stochastic and Deterministic Event Triggers

Eva Julia Schmitt, Benjamin Noack

TL;DR

This paper tackles resource constraints in wireless remote state estimation by developing a unified framework that links stochastic and deterministic event-based triggers through a generalized Gaussian weighting function $\phi_\beta(\underline{z}_k) = \exp\left(-\tfrac{1}{2}\left(\sqrt{\underline{z}_k^\top \mathbf{Z}^{-1} \underline{z}_k}\right)^{\beta}\right)$. It demonstrates how innovation-based triggers can be interpreted within this framework and introduces two estimators: a stochastic event-based Kalman filter (SEBKF) and a sampling-based estimator, with extensions to arbitrary weighting functions and linear/nonlinear models. Through simulations on a 2D nearly-constant-velocity system, it shows that $\beta=2$ offers optimal performance for SEBKF, while larger $\beta$ can degrade consistency at low event rates unless more samples are used in the sampling-based approach. The work provides design guidelines for balancing transmission rate, estimation accuracy, and consistency, and highlights future directions for formal consistency analysis and nonlinear/generalized triggering schemes.

Abstract

Resources such as bandwidth and energy are limited in many wireless communications use cases, especially when large numbers of sensors and fusion centers need to exchange information frequently. One opportunity to overcome resource constraints is the use of event-based transmissions and estimation to transmit only information that contributes significantly to the reconstruction of the system's state. The design of efficient triggering policies and estimators is crucial for successful event-based transmissions. While previously deterministic and stochastic event triggering policies have been treated separately, this paper unifies the two approaches and gives insights into the design of consistent trigger-matching estimators. Two different estimators are presented, and different pairs of triggers and estimators are evaluated through simulation studies.

A Unified Framework for Innovation-based Stochastic and Deterministic Event Triggers

TL;DR

This paper tackles resource constraints in wireless remote state estimation by developing a unified framework that links stochastic and deterministic event-based triggers through a generalized Gaussian weighting function . It demonstrates how innovation-based triggers can be interpreted within this framework and introduces two estimators: a stochastic event-based Kalman filter (SEBKF) and a sampling-based estimator, with extensions to arbitrary weighting functions and linear/nonlinear models. Through simulations on a 2D nearly-constant-velocity system, it shows that offers optimal performance for SEBKF, while larger can degrade consistency at low event rates unless more samples are used in the sampling-based approach. The work provides design guidelines for balancing transmission rate, estimation accuracy, and consistency, and highlights future directions for formal consistency analysis and nonlinear/generalized triggering schemes.

Abstract

Resources such as bandwidth and energy are limited in many wireless communications use cases, especially when large numbers of sensors and fusion centers need to exchange information frequently. One opportunity to overcome resource constraints is the use of event-based transmissions and estimation to transmit only information that contributes significantly to the reconstruction of the system's state. The design of efficient triggering policies and estimators is crucial for successful event-based transmissions. While previously deterministic and stochastic event triggering policies have been treated separately, this paper unifies the two approaches and gives insights into the design of consistent trigger-matching estimators. Two different estimators are presented, and different pairs of triggers and estimators are evaluated through simulation studies.

Paper Structure

This paper contains 22 sections, 19 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Event-based Triggering
  4. System Model
  5. Stochastic and Deterministic Event-triggering
  6. Generalized Gaussian Weighting Function
  7. Popular Choices of ${\underline{c}} \IfNoValueF{-NoValue-}{_{-NoValue-}} \IfNoValueF{-NoValue-}{^{-NoValue-}} _k$
  8. Send-on-Delta
  9. Send-on-Delta with Prediction
  10. Choice of $\beta$
  11. Event-based Procedure
  12. $\gamma_k = 1$
  13. $\gamma_k = 0$
  14. Event-based Estimation
  15. Stochastic Event-based Kalman Filter
  16. ...and 7 more sections

Figures (5)

  • Figure 1: System setup consisting of transmitter and receiver with event-based transmissions of ${{\boldsymbol{\mathrm{{\underline{y}}}}}} \IfNoValueF{k}{_{k}} \IfNoValueF{-NoValue-}{^{-NoValue-}}$. The transmitter includes a measurement unit, a triggering unit and an optional local estimator which estimates ${\hat{{{\boldsymbol{\mathrm{{\underline{\IfNoValueTF{-NoValue-}{x}{-NoValue-}}}}}}}} } \IfNoValueF{k}{_{k}} \IfNoValueF{-NoValue-}{^{-NoValue-}} ^\text{S}$ depending on the choice of ${\underline{c}} \IfNoValueF{k}{_{k}} \IfNoValueF{-NoValue-}{^{-NoValue-}}$.
  • Figure 2: Univariate shaping function with varied $\beta$.
  • Figure 3: Trigger performance with 2D constant velocity system model.
  • Figure 4: $\mathop{\mathrm{\mathbb{E}}}\limits\{{{\boldsymbol{\mathrm{{\underline{z}}}}}}_k^\mathrm{T}{{\boldsymbol{\mathrm{{\underline{z}}}}}}_k\,|\,\gamma_k=0\}$ over ${{\mathbf{Z}}} \IfNoValueF{-NoValue-}{_{-NoValue-}} \IfNoValueF{}{^{}}$ for a 2D nearly constant velocity system model.
  • Figure 5: System performance under the use of Gaussian stochastic triggers using different values of $\beta$ and different estimators for state estimation at the receiver.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2