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Clock error prediction is important for satellites while their clocks could not transfer time message with the stations in earth. It puts forth a novel short-medium term clock error prediction algorithm based on modified differential exponential smoothing (ES). Firstly, it introduces the basic double ES (DES) and triple ES (TES). As the weighted parameter in ES is fixed, leading to growing predicted errors, a dynamic weighted parameter based on a sliding window (SW) is put forward. And in order to improve the predicted precision, it brings in grey mode (GM) to learn the predicted errors of DES (TES) and combines the DES (TES) predicted results with the results of GM prediction from error learning. From examples' analysis, it could conclude that the short term predicted precisions of algorithms based on ES with GM error learning are less than 0.4ns, where GM error learning could better the performances slightly. And for the medium term, it could conclude that the fusion algorithm in DES (TES) with error learning in GM based on SW could reduce the predicted errors in 35.37% (66.34%) compared with DES (TES) alone. In medium term clock error prediction, the predicted precision of TES is worse than DES, which is roughly in the same level of GM.

High precision time synchronization is not only of vital importance for the operation of satellites, which will influence the precision of satellites’ navigation, locating, and timing, but also for distributed weapon systems, like the distributed netted radar system or the multistation radar system, etc., which will determine the precision of tracking, guiding, and locating directly. In order to keep high precision time synchronization, satellites usually take two way time transfer (TWTT) with the stations in the earth during the period of time when satellites are in the visual angle of earth stations. While satellites are out of the visual angle, the clocks in satellites have to operate by themselves, which have to predict the clock errors between satellites and earth stations. In clock error prediction, the predicted time length is always not long. When satellites fly into the visual angle of earth stations, the transfer could be reestablished. The break of transfer resulted from comparative position is usually in several hours. Also there are other factors resulting in transfer break, like interference, clock trouble, satellite fault, etc., which have to take medium term clock error prediction. The long term clock error prediction is few [

Aiming at the short-medium term clock error prediction, there are many studies which had been done, like grey model, quadratic polynomial model, LS-SVM algorithm, ARMA algorithm, functional network, etc. [

In single ES (SES), the weighted parameter (WP)

The rest of the paper is organized as follows. In Section

As some satellite clock error series have big absolute value, which will bring extra computational complexity, especially in exponential calculation, so we make the original clock error series differential operation. If the clock time series are

SES model could be presented as follows.

The differential clock time series are

From (

The predicted value

Then we could get the double ES (DES) predicted value

As ES has the obvious defect of fixed weighted parameter

Sketch map of ES with sliding window.

We choose root mean square error (RMSE) to evaluate the prediction performance.

As

As the ES algorithm has clear defect of error accumulation, we bring in SW to update the learning swatches to reduce the effect, which refreshes the learning series by new predicted series. In part 2, we choose the predicted series as the learning series. Then we repeat the steps in part 1 to search the best WP

Make difference of the original clock series.

The parameters related to the algorithm should be initialized firstly, including those in ES and SW.

In the learning window, we search the minimal

Based on the best WP, we use ES to predict the following series.

We take part 1 for example. While the predicted length reaches

In order to improve the precision of the prediction, we bring in the GM to predict the ES predicted errors based on foregoing algorithm, which is showed in Figure

Sketch map of ES+GM with sliding window.

Step 1 to Step 3 are the same as the steps in ESSW

Based on the best WP, we take ES to predict the following series. We call the step as ESP.

In the ES learning window, based on the best

Based on Step 5, we take GM to predict the ES prediction errors.

We combine the ESP and GEP as FEP.

Step 8 is the same as Step 5 in ESSW, while the difference is that the updating swathes are the FEP series.

The GM for ES predicted errors could be interpreted as

After we get the complete predicted series, we take inverse differential operation for the predicted clock error series. We will get the predicted errors compared with the original clock series.

In order to study the performance of the algorithm above, we take the 1980th and 1981th GPS week clock error data (2017.12.17-2017.12.30), for example. We choose clocks from Rb clock and Cs clock in different types stochastically, which is showed in Table

The chosen clocks.

Clock types | Numbers |
---|---|

IIR Rb | PG13(No. 1) PG23(No. 2) |

IIR-M Rb | PG5(No. 3) PG17(No. 4) |

IIF Cs | PG8(No. 5) PG24(No. 6) |

We analyze the algorithm in short and medium term. We take the schemes as follows.

We take one day long prediction, for example, to analyze the short term predicted performances. As the predicted length is not long, we do not bring in sliding window in the algorithm. Besides, we choose GM as a contradistinctive algorithm to analyze the algorithm the paper put forward.

We choose one day clock error series for leaning, then we get the statistical predicted results of the chosen clocks, which are showed in Figure

The statistical chart of above schemes (unit in ns.).

Scheme | No. | 1 | 2 | 3 | 4 | 5 | 6 | Avg | Std | |
---|---|---|---|---|---|---|---|---|---|---|

S1 | DES | RMSE | 0.3363 | 0.2405 | 0.3225 | 0.2570 | 0.5088 | 0.8182 | 0.4139 | 0.2198 |

Max | 1.2321 | 0.1874 | 0.1672 | 0.5926 | 1.2149 | 2.1844 | ||||

Min | -1.4544 | -0.5476 | -0.7088 | -0.5061 | -0.7692 | -1.5594 | ||||

TES | RMSE | 0.4691 | 0.5258 | 0.8922 | 0.2881 | 0.5435 | 0.8214 | 0.5900 | 0.2267 | |

Max | 1.1801 | 0.9491 | 1.7628 | 0.5479 | 1.0825 | 1.9872 | ||||

Min | -1.5104 | -0.0705 | 0.0125 | -0.5682 | -0.9944 | -1.7465 | ||||

GM | RMSE | 0.3298 | 0.1253 | 0.2004 | 0.2509 | 0.4298 | 0.7968 | 0.3555 | 0.2404 | |

Max | 1.5075 | 0.2875 | 0.4857 | 0.5229 | 1.0631 | 1.6884 | ||||

Min | -1.1804 | -0.3155 | -0.4108 | -0.5805 | -1.1460 | -2.0537 | ||||

S2 | DES+GM | RMSE | 0.3321 | 0.2401 | 0.3214 | 0.2562 | 0.5037 | 0.7996 | 0.4089 | 0.2131 |

Max | 1.2165 | 0.1887 | 0.1696 | 0.5897 | 0.7448 | 2.0719 | ||||

Min | -1.4703 | -0.5475 | -0.7061 | -0.5134 | -1.3902 | -1.6737 | ||||

TES+GM | RMSE | 0.4367 | 0.3113 | 0.7305 | 0.2843 | 0.4335 | 0.8007 | 0.4995 | 0.2164 | |

Max | 1.0742 | 0.6572 | 1.5336 | 0.5605 | 0.7266 | 1.9872 | ||||

Min | -1.6213 | -0.2455 | -0.1339 | -0.5561 | -1.4685 | -1.7465 | ||||

S3 | DES | RMSE | 0.4001 | 1.0364 | 1.2312 | 0.3363 | 1.5164 | 0.7425 | 0.8772 | 0.4685 |

Max | 1.2322 | 0.1874 | 0.1963 | 1.1834 | 1.4221 | 2.1844 | ||||

Min | -1.4544 | -2.0241 | -2.4174 | -0.0686 | -2.1932 | -2.4237 | ||||

DES+SW | RMSE | 0.3171 | 0.5586 | 1.2093 | 0.2891 | 1.1880 | 0.7354 | 0.7163 | 0.4081 | |

Max | 1.2328 | 0.1849 | 2.5172 | 1.0120 | 1.4221 | 2.0767 | ||||

Min | -1.4537 | -1.1640 | -1.4977 | -0.6863 | -2.9796 | -2.4733 | ||||

DES+GM+SW | RMSE | 0.2793 | 0.3996 | 0.7033 | 0.2548 | 1.0269 | 0.7376 | 0.5669 | 0.3058 | |

Max | 1.1912 | 0.5105 | 1.7261 | 0.8462 | 0.7881 | 2.0767 | ||||

Min | -1.4963 | -1.2359 | -0.9158 | -0.8081 | -2.6918 | -1.6754 | ||||

TES | RMSE | 2.6601 | 7.2823 | 15.7942 | 3.3886 | 1.3800 | 0.7894 | 5.2158 | 5.6630 | |

Max | 1.1801 | 15.6289 | 34.2391 | 0.0548 | 1.1644 | 2.2027 | ||||

Min | -6.0656 | -0.0705 | 0.0125 | -7.5281 | -3.5282 | -2.4249 | ||||

TES+SW | RMSE | 1.4326 | 4.5575 | 6.4957 | 2.6798 | 1.2568 | 0.7520 | 2.8624 | 2.2443 | |

Max | 4.2711 | 4.8356 | 10.1357 | 5.8127 | 1.4003 | 2.2027 | ||||

Min | -2.0166 | -12.4834 | -16.7888 | -2.4607 | -3.3671 | -1.5969 | ||||

TES+GM+SW | RMSE | 0.7098 | 3.0143 | 3.2632 | 1.7600 | 1.1082 | 0.6773 | 1.7555 | 1.4300 | |

Max | 1.8674 | 4.3660 | 9.8627 | 3.3411 | 0.9642 | 1.5817 | ||||

Min | -1.8219 | -7.9424 | -0.8799 | -2.4694 | -3.6636 | -2.8538 | ||||

GM | RMSE | 0.2429 | 0.1440 | 0.2624 | 0.2692 | 0.7284 | 0.7310 | 0.3963 | 0.2621 | |

Max | 1.5075 | 0.4878 | 0.4857 | 0.8203 | 2.2155 | 2.5908 | ||||

Min | -1.1804 | -0.4490 | -0.9562 | -0.8304 | -1.1460 | -2.0537 |

Short predicted performances in three methods.

From Figure

In this part, we take GM algorithm to learn the predicted errors of DES/TES and combine them for the fusion algorithm. The performances are showed in Figure

Short predicted performances in DES+GM and TES+GM.

We make analyses DES/TES+GM compared with DES/TES. From Figures

From above analyses, we could get that the short term predicted precisions of algorithms based ES are less than 0.4ns and GM error learning could better the performances slightly.

We take one week long prediction for example to analyze the medium term predicted performances.

In this part, we take DES (TES), DES (TES)+SW, DES (TES)+GM+SW, and GM algorithms to predict the medium term clock errors. We choose clock errors in the

Medium term clock errors in different methods.

From Figure

In this paper, we have mainly presented a novel fusion algorithm of DES (TES) in sliding window based on predicted error learning in GM for short-medium term clock error prediction. We firstly analyze the basic DES and TES algorithm. Then we present ES with sliding window and ES+GM with sliding window algorithms in detailed steps. Furthermore, we take two-GPS week clock error data to analysis the algorithms we put forward. From the calculation results, for short term clock error prediction, we could get that the predicted precisions of algorithms based ES are less than 0.4ns and GM error learning could better the ES predicted performances slightly. As for the medium term clock error prediction, we could conclude that error learning in GM could better the performances and fusion algorithm of DES (TES) with sliding window based on error learning in GM could reduce the predicted errors in 35.37% (66.34%) compared with DES (TES) alone. And the precision of TES is worse than DES in medium term clock error prediction, which is roughly in the same level of GM. The fusion algorithm could not only be applied in clock error prediction in satellites, but also in distributed netted systems, like netted radars systems, which is of vital importance for the performances of the systems’ efficiency.

The clock data used in the paper are from the web-page of

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The paper was supported by National Natural Science Foundation of China (NSFC) under Grant no. 61701525 and China Postdoctoral Science Foundation under Grant no. 2017M623351.