【正文】 U. Kalman’s MU derives from Sherman’s theorem, Sherman’s theorem derives from Anderson’s theorem [1], and Anderson’s theorem derives from the BrunnMinkowki inequality theorem . The theoretical foundation for my linearized MU derives from these theorems. . Initial conditions Initialization of all sequential estimators requires the use of an initial state estimate column matrix 0|0?? and an intial state estimate error covariance matrix 0|0P for time t0 . . Linear TU and nonlinear MU The 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 parameter estimation, as demonstrated in Section 6 . Let ij|?? denote an n 1 column matrix of state estimate ponents, where the left subscript j denotes state epoch tj and the right subscript i denotes timetag ti for the last observation processed, where i, j ∈ {0, 1, 2, ...}.Let ijP| denote an associated n n square symmetric state estimate error covariance matrix (positive eigenvalues). . Linear TU For k∈ {0, 1, 2, 3,..., M }, the propagation of the true unknown n 1 matrix state KX is given by KKKKKK JXX ,1,11 ??? ?? ? （ 1） Where KKJ ,1? is called the process noise matrix. Propagation of the known n 1 matrix state estimate KKX|? is given by KKKKKK XX |,1|1 ??? ? ? （ 2） because the conditional mean of KKJ ,1? is zero. Propagation of the known n n matrix state estimate error covariance matrix KKP| is given by KKT KKKKKKKK QPP ,1,1|,1|1 ???? ?? ?? （ 3） where the n n matrix KKQ ,1? is called the process noise covariance matrix. . Nonlinear MU Calculate the n1matrix filter gain: 111|111|11 ][ ???????? ?? KTKKKKTKKKK RHPHHPK （ 4） The filter measurement update state estimate n 1matrix 1|1 ??? KKX , due to the observation yK+1, is calculated with )]([ |111|11|1 KKKKKKKK XyyKXX ????????? ??? （ 5） 5. UNOBSERVABLE GPS CLOCK STATES GPS 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 unobservable 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 partitioned in this manner into a subset of unobservable clock parameters and a subset of observable orbit parameters. This partition is performed by application of Sherman’s theorem in the MU. . Partition of KF1 estimation errors Subtract estimated clock deviations from simulated (true) clock deviations to define and quantify Kalman filter (KF1) estimation errors. Adopt Brown’s additive partition of KF1 estimation errors into two ponents. I refer to the first ponent as the unobservable error mon to each clock (UECC), and to the second ponent as the observable error independent for each clock (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 variances 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 ponents of KF1 estimation errors. Since there is no physical process noise on the UECC, an estimate of the UECC can also be achieved using a batch least squares estimation algorithm on the phase ponents of KF1 estimation errors—demonstrated previously by Greenhall [7]. (I apply sufficient process noise covariance for KF2 to mask the effects of doubleprecision puter word truncation. Without this, KF2 does diverge.) . Unobservable error mon to each clock There 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 sample variance about the mean。 中文 4350 字 外文原文 Hindawi Publishing Corporation International Journal of Navigation and Observation Volume 2021, Article ID 261384,8 pages doi: Research Article GPS Composite Clock Analysis James R. Wright Analytical Graphics, In c., 220 Valle y Creek Blvd, E x ton, PA 19341, USA Correspondence should be addressed to James R. Wright, Received 30 June 2021。 Accepted 6 November 2021 Remended by Demetrios Matsakis Copyright 169。 this would yield a sequential sampling procedure, but where each mean and variance is sequentially unconnected. Second, one can employ Ken Brown’s implicit ensemble mean (IEM) and covariance。 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 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 UECC. I have been successful with this approach. Figure 3 presents an ensemble of ―r