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1、中文4350字外文原文Hindawi Publishing CorporationInternational Journal of Navigation and ObservationVolume 2021, Article ID 261384,8 pagesdoi:10.1155/2021/261384Research ArticleGPS Composite Clock AnalysisJames R. WrightAnalytical Graphics, In c., 220 Valle y Creek Blvd, E x ton, PA 19341, USA Correspondenc
2、e should be addressed to James R. Wright, jwrightagi Received 30 June 2007; Accepted 6 November 2007Recommended by Demetrios MatsakisCopyright 2021 James R. Wright. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution,
3、 and reproduction in any medium, provided the original work is properly cited.Abstract The GPS composite clock defines GPS time, the timescale used today in GPS operations. GPS time is illuminated by examination of its role in the complete estimation and control problem relative to UTC/TAI. The phas
4、e of each GPS clock is unobservable from GPS pseudorange measurements, and the mean phase of the GPS clock ensemble (GPS time) is unobservable. A new and useful obs e r vabilit y definition is presented, together with new observabilit y theorems, to demonstrate explicitly that GPS time is unobservab
5、le. Simulated GPS clock phase and frequency deviations, and simulated GPS pseudorange measurements, are used to understand GPS time in terms of Kalman filter estimation errors.1. INTRODUCTIONGPS time is created by processing GPS pseudorange measurements with the operational GPS Kalman filter. Brown
6、2refers to the object created by the Kalman filter as the GPS composite clock, and to GPS time as the implicit ensemble mean phase of the GPS composite clock. The fundamental goal by the USAF and the USNO is to control GPS time to within a specified bound of UTC/TAI. (I refer to TAI/UTC understandin
7、g that UTC has an accumulated discontinuity (a sum of leap seconds) when compared to TAI. But unique two-way transformations between TAI and UTC have been in successful operational use since 1972. I have no need hereinto further distinguish between TAI and UTC.) I present here a quantitative analysi
8、s of the GPS composite clock, derived from detailed simulations and associated graphics. GPS clock diffusion coefficient values used here were derived from Allan deviation graphs presented by Oaks et al. 12 in 1998. I refer to them as “realistic, and in the sequel I claim “realistic results from the
9、ir use. Figure 1 presents their diffusion coefficient values and my derivation of associated Allan deviation lines.My interest in the GPS composite clock derives from my interest in performing real-time orbit determination for GPS NAVSTAR spacecraft from ground receiver pseudorange measurements. (Ja
10、mes R Wright is the architect of ODTK (Orbit Determination Tool Kit), a commercial soft-ware product offered by Analytical Graphics, Inc. (AGI).)The estimation of NAVSTAR orbits would be in complete without the simultaneous estimation of GPS clock parameters. I use simulated GPS clock phase and freq
11、uency deviations, and simulated GPS pseudorange measurements, to study Kalman filter estimation errors.This paper was first prepared for TimeNav07 20 . I am indebted to Charles Greenhall (JPL) for encouragement and help in this work.2. THE COMPLETE ESTIMATION AND CONTROL P ROBLEMThe USNO operates tw
12、o UTC/TAI master clocks, each of which provides access to an estimate of UTC/TAI in real time(1 pps). One of these clocks is maintained at the USNO, and the other is maintained at Schriever Air Force Base in Colorado Springs. This enables the USNO to compare UTC/TAI to the phase of each GPS orbital
13、NAVSTAR clock via GPS pseudorange measurements, by using a UTC/TAI master clock in a USNO GPS ground receiver. Each GPS clock is a member of (internal to) the GPS ensemble of clocks, but the USNO master clock is external to the GPS ensemble of clocks. Because of this, the difference between UTC/TAI
14、and the phase of each NAVSTAR GPS clock is observable. This difference can be (and is) estimated and quantified. The root mean square (RMS) on these differences quantifies the difference between UTC/TAI and GPS time. Inspection of the differences between UTC/TAI and the phase of each NAVSTAR GPS clo
15、ck enables the USNO to identify GPS clocks that require particular frequency-rate control corrections. Use of this knowledge enables the USAF to adjust frequency rates of selected GPS clocks. Currently, the USAF uses an automated bang-bang controller on frequency-rate. (According to Bill Feess, an i
16、mprovement in control can be achieved by replacing the existing “bang-bang controller with a “proportional controller.)3. STOCHASTIC CLOCK PHYSICSThe most significant stochastic clock physics are understood in terms of Wiener processes and their integrals .Clock physics are characterized by particul
17、ar values of clock-dependent diffusion coefficients, and are conveniently studied with aid of a relevant clock model that relates diffusioncoefficient values to their underlying Wiener processes. For my presentation here I have selected “The clock model and its relationship with the Allan and relate
18、d variances presented as an IEEE paper by Zucca and Tavella 19 in 2005.Except for FM flicker noise, this model captures the most significant physics for all GPS clocks. I simulate and validate GPS pseudorange measurements using simulated phase deviations and simulated frequency deviations, according
19、 toZucca and Tavella.4. KALMAN FILTERSI present my approach for the optimal sequential estimation of clock deviation states and their error covariance functions. Sequential state estimates are generated recursively from two multidimensional stochastic update functions, the time update (TU) and the m
20、easurement update(MU). The TU moves the state estimate and covariance forward with time, accumulating integrals of random clock deviation process noise in the covariance. The MU is performed at a fixed measurement time where the state estimate and covariance are corrected with new observation inform
21、ation.The sequential estimation of GPS clock deviations re-quires the development of a linear TU and nonlinear MU. The nonlinear MU must be linearized locally to enable application of the linear Kalman MU. Kalmans MU derives from Shermans theorem, Shermans theorem derives from Andersons theorem 1, a
22、nd Andersons theorem derives from the Brunn-Minkowki inequality theorem . The theoretical foundation for my linearized MU derives from these theorems.4.1. Initial conditionsInitialization of all sequential estimators requires the use of an initial state estimate column matrix and an intial state est
23、imate error covariance matrix for time t0.4.2. Linear TU and nonlinear MUThe simultaneous sequential estimation of GPS clock phase and frequency deviation parameters can be studied with the development of a linear TU and nonlinear MU for the clock state estimate subset. This is useful to study clock
24、 parameter estimation, as demonstrated in Section 6 .Let denote an n 1 column matrix of state estimate components, where the left subscript j denotes state epoch tj and the right subscript i denotes time-tag ti for the last observation processed, where i, j 0, 1, 2, .Let denote an associated n n squ
25、are symmetric state estimate error co-variance matrix (positive eigenvalues).4.2.1. Linear TUFor k0, 1, 2, 3,., M , the propagation of the true un-known n1 matrix state is given by 1Whereis called the process noise matrix. Propagation of the known n 1 matrix state estimate is given by 2because the c
26、onditional mean of is zero. Propagation of the known n n matrix state estimate error covariance matrixis given by 3where the n n matrixis called the process noise co-variance matrix.4.2.2. Nonlinear MUCalculate the n1matrix filter gain: 4The filter measurement update state estimate n 1matrix, due to
27、 the observation yK+1, is calculated with 55. UNOBSERVABLE GPS CLOCK STATESGPS time is created by the operational USAF Kalman filter by processing GPS pseudorange observations. GPS time is the mean phase of an ensemble of many GPS clocks, and yet the clock phase of every operational GPS clock is uno
28、bservable from GPS pseudorange observations, as demonstrated below. GPS NAVSTAR orbit parameters are observable from GPS pseudorange observations. The USAF Kalman filter simultaneously estimates orbit parameters and clock parameters from GPS pseudorange observations, so the state estimate is partiti
29、oned in this manner into a subset of unobservable clock parameters and a subset of observable orbit parameters. This partition is performed by application of Shermans theorem in the MU.5.1. Partition of KF1 estimation errorsSubtract estimated clock deviations from simulated (true) clock deviations t
30、o define and quantify Kalman filter (KF1) estimation errors. Adopt Browns additive partition of KF1 estimation errors into two components. I refer to the first component as the unobservable error common to each clock (UECC), and to the second component as the observable error independent for each cl
31、ock (OEIC). (Observability is meaningful here only when processing simulated GPS pseudorange data.) On processing the first GPS pseudorange measurements with KF1 the variances on both fall quickly. But with continued measurement processing the variances on the UECC increase without bound while the v
32、ariances on the OEIC approach zero asymptotically.For simulated GPS pseudorange data I create an optimal sequential estimate of the UECC by application of a second Kalman filter KF2 to pseudo measurements defined by the phase components of KF1 estimation errors.Since there is no physical process noi
33、se on the UECC, an estimate of the UECC can also be achieved using a batch least squares estimation algorithm on the phase components of KF1 estimation errorsdemonstrated previously by Green-hall 7. (I apply sufficient process noise covariance for KF2 to mask the effects of double-precision computer
34、 word truncation. Without this, KF2 does diverge.)5.2. Unobservable error common to each clockThere are at least four techniques to estimate the UECC when simulating GPS pseudorange data. First, one could take the sample mean of KF1 estimation errors across the clock ensemble at each time and form a
35、 sample variance about the mean; this would yield a sequential sampling procedure, but where each mean and variance is sequentially unconnected. Second, one can employ Ken Browns implicit ensemble mean (IEM) and covariance; this is a batch procedure requiring an inversion of the KF1 covariance matri
36、x followed by a second matrix inversion of the modified covariance matrix inverse; this is not a sequential procedure. Third, one can adopt the new procedure by Greenhall 7 wherein KF1 phase estimation errors are treated as pseudo measurements, and are processed by a batch least squares estimator to
37、 obtain optimal batch estimates and covariance matrices for the UECC. Fourth, one can treat the KF1 phase estimation errors as pseudo measurements, invoke a second Kalman filter (KF2), and process these phase pseudo measurements with KF2 to obtain optimal sequential estimates and variances for the U
38、ECC. I have been successful with this approach. Figure 3 presents an ensemble of “realistic KF1 phase estimation errors, overlaid with “realistic KF2 sequential estimates of UECC in phase. (By “realistic I refer to realistic clock diffusion coefficient values.)5.3. Observable error independent for e
39、ach clockAt each applicable time subtract the estimate of the UECC from the KF1 phase deviation estimate, for each particular GPS clock, to estimate the OEIC in phase for that clock. During measurement processing, the OEIC is contained within an envelope of a few parts of a nanosecond (see Figure 4)
40、.Figure 4 presents a graph of two cases of the OEIC for ground station clock S1. For the blue line of intervals of link visibility and KF1 range measurement processing are clearly distinguished from propagation intervals with no measurements. During measurement processing, the observable component o
41、f KF1 estimation error is contained within an envelope of a few parts of a nanosecond. Calculation of the sequential covariance for the OEIC requires a matrix value for the cross-covariance between the KF1 phase deviation estimation error and the UECC estimation error at each time. I have not yet be
42、en able to calculate this cross-covariance.6. KALMAN FILTERS KF1 AND KF2I have simulated GPS pseudorange measurements for two GPS ground station clocks S1 and S2, and for two GPS NAVSTAR clocks N1 and N2. Here I set simulated measurement time granularity to 30s for the set of all visible link interv
43、als. Visible and nonvisible intervals are clearly evident in the blue line of Figure 4. I set the scalar root-variance for both measurement simulations and Kalman filter KF1 to = 1 cm. Typically 1 m for GPS pseudorange, but when carrier phase measurements are processed simultaneously with pseudorang
44、e, the root-variance is reduced by two orders of magnitude. So the use of = 1cm enables me to quantify lower performance bounds for the simultaneous processing of both measurement types.6.1. Create GPS clock ensembleTypically, one processes measurements with a Kalman filter to derive sequential esti
45、mates of a multidimensional observable state. Instead, here I imitate the GPS operational procedure and process simulated GPS pseudorange measurements with KF1 to create a sequence of unobservable multidimensional clock state estimates. Clock state components are unobservable from GPS pseudorange me
46、asurements. See Figure 2 for an example of an ensemble of estimated unobservable clock phase deviation state components created by KF1.6.1.1. Shermans theoremGPS time, the unobservable GPS clock ensemble mean phase, is created by the use of Shermans theorem 11、18 in the USAF Kalman filter measuremen
47、t update algorithm on GPS range measurements. Satisfaction of Shermans Theorem guarantees that the mean-squared state estimate error on each observable state estimate component is minimized. But the mean-squared state estimate error on each unobservable state estimate component is not reduced. Thus
48、the unobservable clock phase deviation state estimate component common to every GPS clock is isolated by application of Shermans theorem. An ensemble of unobservable state estimate components is thus created by Shermans theoremsee Figure 3for an example.6.2. Initial condition errorsA significant result emerges due to the modeling of Kalman filter (KF1) initial condition errors in phase and frequency. Initial estimated clock phase deviations are significantly displaced by the KF1 initial condition erro