Comparison of Clock Models in View of Clock Composition, Clock Steering and Measurement Fitting

Christian Trainotti, Tobias D. Schmidt, Johann Furthner

Abstract: Global navigation satellite systems need a stable and robust system time in order to provide services with high precision. A common approach for its generation is the composite clock method, where an ensemble of clocks contributes to the definition of the system time as a weighted average of the single clocks. In this way the robustness of the so generated time scale is enhanced, since a failure in one clock can be compensated by the others. The core part of the composite clock method is a Kalman filter, which estimates the future states of each clock depending on the relative measurements between all clocks. The Kalman filter implicitly provides the system time of the ensemble, in terms of the implicit ensemble mean (IEM) – a so called paper clock. However, this quantity is not directly available, since it requires knowing the exact state of each clock at each time step which is not directly measurable. A solution consists in realizing the IEM by steering a clock signal towards the IEM. Different steering techniques can be used, for instance the pole placement method and the linear quadratic Gaussian regulator. Both the Kalman filter and the regulator are based on a clock model, which describes the clock as a linear dynamic system. Hence, the performance of the used clock has implications in every step of the ensembling and realization algorithms. This paper discusses the advantage and disadvantages of different clock models and gives a recommendation on which one is most promising for future investigations. The 3-state clock model and the 2-state clock model with and without drift are described and expanded with additional Gauss-Markov processes, to account for floors of flicker noise. These models are then compared by considering four properties, namely their precision, the number of states, their theoretical formulation and controllability. The precision refers to which noise characteristics a model is able to describe and how well each model is suited to fit a given clock behavior. The theoretical formulation is studied by means of the theoretical expression of the overlapping Allan deviation (OADEV), which is used to fit a model to real measurements. Finally, the controllability of the dynamic model is needed in order to use the aforementioned control techniques. In light of these criteria, the 2-state clock model with drift and Gauss-Markov processes has been chosen, since it is the one providing a good modeling precision and controllability while being simple and flexible. Furthermore, in order to more easily determine the model parameters, a semi-automatic fitting procedure has been developed. Previously, once a clock measurement was given in terms of OADEV, the parameters were found by trial and error, meaning that the model’s dynamics was simulated, its OADEV computed and compared with the measured one. This procedure was repeated until the simulation described the measurement in a reasonable way. However, it is possible to describe the OADEV with a function which can be directly used to fit the measurement data. Several techniques are here described and have been implemented into a graphical interface which eases the fitting procedure.
Published in: Proceedings of the 50th Annual Precise Time and Time Interval Systems and Applications Meeting
January 28 - 31, 2019
Hyatt Regency Reston
Reston, Virginia
Pages: 265 - 283
Cite this article: Trainotti, Christian, Schmidt, Tobias D., Furthner, Johann, "Comparison of Clock Models in View of Clock Composition, Clock Steering and Measurement Fitting," Proceedings of the 50th Annual Precise Time and Time Interval Systems and Applications Meeting, Reston, Virginia, January 2019, pp. 265-283.
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