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Optimal GNSS Time Tracking for Long-term Stable Time Realisation in Synchronised Atomic Clocks

Maitreyee Dutta, Jiayu Chen, Masakazu Koike, Yuichiro Yano, Yuko Hanado, Takayuki Ishizaki

Abstract

In this manuscript, we propose a novel optimal Global Navigation Satellite System (GNSS) time tracking algorithm to collectively steer an ensemble consisting of synchronising miniature atomic clocks towards standard GNSS time. The synchronising miniature atomic clocks generate a common synchronised time which has good short term performance but its accuracy and precision, which is measured by Allan variance, deteriorates in the long run. So, a supervisor designs and periodically broadcasts the proposed GNSS time tracking control to the ensemble miniature atomic clocks that steer the average of ensemble towards the average of GNSS receivers, which are receivers of GNSS time. The tracking control is constructed using a Kalman filter estimation process that estimates the difference in average of GNSS receivers and average of ensemble clocks by using relative clock readings between GNSS receivers and their adjacent ensemble clock. Under the influence of the periodically received tracking control, the stabilised ensemble clocks have better long term accuracy and precision over long averaging periods. Since the tracking control is designed to solely influence the average of the ensemble, the tracking process does not interfere with the synchronisation process and vice versa. The feedback matrix associated with the tracking control is obtained from an optimisation problem that minimises steady-state Allan variance. Numerical results are provided to show the efficacy of the proposed algorithm for enhancing long term performance.

Optimal GNSS Time Tracking for Long-term Stable Time Realisation in Synchronised Atomic Clocks

Abstract

In this manuscript, we propose a novel optimal Global Navigation Satellite System (GNSS) time tracking algorithm to collectively steer an ensemble consisting of synchronising miniature atomic clocks towards standard GNSS time. The synchronising miniature atomic clocks generate a common synchronised time which has good short term performance but its accuracy and precision, which is measured by Allan variance, deteriorates in the long run. So, a supervisor designs and periodically broadcasts the proposed GNSS time tracking control to the ensemble miniature atomic clocks that steer the average of ensemble towards the average of GNSS receivers, which are receivers of GNSS time. The tracking control is constructed using a Kalman filter estimation process that estimates the difference in average of GNSS receivers and average of ensemble clocks by using relative clock readings between GNSS receivers and their adjacent ensemble clock. Under the influence of the periodically received tracking control, the stabilised ensemble clocks have better long term accuracy and precision over long averaging periods. Since the tracking control is designed to solely influence the average of the ensemble, the tracking process does not interfere with the synchronisation process and vice versa. The feedback matrix associated with the tracking control is obtained from an optimisation problem that minimises steady-state Allan variance. Numerical results are provided to show the efficacy of the proposed algorithm for enhancing long term performance.

Paper Structure

This paper contains 16 sections, 1 theorem, 44 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Notations
  4. Second order model of an atomic clock
  5. Networked System Setup
  6. Compact deviation model for a set of atomic clocks
  7. Compact model for an ensemble of miniature atomic clocks
  8. Two state model for clocks in the GNSS receivers
  9. Frequency stability: Allan variance(AVAR)
  10. Distributed synchronisation among ensemble clocks
  11. Problem statement
  12. Optimal GNSS time tracking control
  13. Estimation of the tracking error
  14. Tracking control broadcasted by the supervisor
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Theorem III.1

Consider a set of $n$ MACs compactx communicating by an undirected connected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$ and, synchronising as per usyni based on local edge state estimates. Then, $\underset{k\to \infty}{\lim} \mathbb{E}[\tilde{z}[k]]=0, \underset{k\to \infty}{\lim} \mathbb{E}[\til

Figures (8)

  • Figure 1: System setup for an ensemble consisting of MACs, GNSS receivers and supervisor.
  • Figure 2: An example: Communication network of three MACs and two GACs
  • Figure 3: AVAR of three free running clocks and average of these three clocks.
  • Figure 4: Statistical AVARs of the controlled MACs and the analytical AVARs of the free running MACs.
  • Figure 5: Time sequence of clock reading deviation of the MACs with respect to GACs.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem III.1
  • proof
  • Remark 2
  • Remark 3