Explicit Ensemble Mean Clock Synchronization for Optimal Atomic Time Scale Generation

TL;DR

This work addresses robust atomic time scale generation by unifying clock synchronization and oscillator control within a control-theoretic framework. It introduces explicit ensemble mean synchronization based on an observable canonical decomposition, showing that standard Kalman filtering is a special case and proposing a determinate, stationary Kalman filter to avoid covariance indeterminacy. The key contributions include a synchronization algorithm that explicitly targets the ensemble mean, a method to optimize Allan variance via optimal weighting, and intermittent unobservable-state feedback to balance short- and long-term stability. The approach offers a principled path to resilient timekeeping with improved long-term stability and practical clock-steering capabilities for large ensembles.

Abstract

This paper presents a novel theoretical framework for atomic time scale generation, called explicit ensemble mean synchronization, which unifies clock synchronization and time scale generation within a control-theoretic paradigm. By exploiting an observable canonical decomposition of a standard atomic clock ensemble model, the system is decomposed into two complementary components: the observable part, which represents the synchronization deviation, and the unobservable part, which captures the synchronization destination. Within this structure, we mathematically prove that standard Kalman filtering, widely used in current time scale generation, can be interpreted as a special case of the proposed framework that optimizes long-term frequency stability in terms of the Allan variance. Furthermore, by applying appropriate state feedback control to each component based on the Kalman filtering, both clock synchronization and optimal time scale generation are achieved within a unified framework. This framework provides a principled basis for robust timekeeping systems that goes beyond conventional approaches in both scope and performance.

Figures (7)

• Figure 1: The Allan variances. The black lines are the analytical Allan variance of the free-running clocks. The blue line is the statistical Allan variance of the reference time scale generated by the optimal Kalman filtering. The red line is the the statistical Allan variance of the reference time scale generated by the suboptimal Kalman filtering.
• Figure 2: Increments of the Kalman gain and error covariance. The upper subfigure shows the Frobenius norm of the increment of the Kalman gain. The lower subfigure shows that of the error covariance.
• Figure 3: The Allan variances. The black lines are the analytical Allan variances of the free-running clocks. The blue line is the analytical Allan variances of the free-running reference clock. The red line is the analytical Allan variance of the average of all free-running clocks. The purple lines are the statistical Allan variances of the controlled clocks.
• Figure 4: The Allan variances. The black lines are the analytical Allan variance of the free-running clocks. The blue line is the analytical Allan variance of the best reference time scale in short term. The red line is that in long term. The purple lines are the statistical Allan variance of the controlled clocks.
• Figure 5: The Allan variances. The black lines are the analytical Allan variance of the free-running clocks. The blue line is the analytical Allan variance of the best reference time scale in short term. The red line is the analytical Allan variance of the best reference time scale in long term. The purple lines are the statistical of the controlled clocks.

Theorems & Definitions (6)

• Lemma 1
• Theorem 1
• Theorem 2
• Lemma 2
• Theorem 3
• Theorem 4