## 3 pLEO Time Scale Development
### Kalman-Filter Based Clock Ensemble Solution  
In linear and Gaussian estimation environments, the estimates and covariances produced by the Kalman filter at time \(t_{k}\) serve as sufficient statistics for the clock system and consider all clock-to-clock measurements up to that point. Figure 8 illustrates an approach for designing a Kalman filter-based clock ensemble.  
Figure 8: Interconnections of clock-to-clock difference measurements and Kalman filters onboard anchor platform.  
It is necessary to define a state vector that describes the time-difference states among these onboard timing systems  
\[x_{ij}(t_{k})\triangleq\left[\begin{array}{c}p_{ij}(t_{k})\\ f_{ij}(t_{k})\\ r_{ij}(t_{k})\end{array}\right]. \tag{43}\]  
We also consider the noise vector, \(w_{ij}(t_{k})\), that causes the clock states to evolve  
\[w_{ij}(t_{k})\triangleq\left[\begin{array}{c}\epsilon_{ij}(t_{k})\\ \eta_{ij}(t_{k})\\ \alpha_{ij}(t_{k})\end{array}\right], \tag{44}\]  
and the state transition matrix that depends on the uniform sampling rate, \(\delta\), and contains the clock dynamics  
\[\Phi_{\delta}\triangleq\left[\begin{array}{ccc}1&\delta&\frac{\delta^{2}}{ 2}\\ 0&1&\delta\\ 0&0&1\end{array}\right]. \tag{45}\]  
Thus, the basic dynamical system (40)-(42) is now rewritten as  
\[x_{ij}(t_{k}+1)=\Phi_{\delta}x_{ij}(t_{k})+w_{ij}(t_{k})\,. \tag{46}\]  
Notably, the process noise, \(w_{ij}(\cdot)\), is Gaussian with zero mean and covariance matrix, \(Q_{ij}\)  
\[Q_{ij}=E\{w_{ij}w_{ij}^{\mathrm{T}}\}\,, \tag{47}\]  
which is the sum of \(Q_{i}(\delta)\) and \(Q_{j}(\delta)\), denoted by the process noise covariances for each ensemble clock. The method for computing these process noise covariances from the Allan deviation values associated with an oscillator is detailed in [23].  
Furthermore, the error covariance matrix associated with the filtered estimate of \(x_{ij}(t_{k})\), \(\hat{x}_{ij}(t_{k})\) is given by  
\[P_{ij}(t_{k})=E\{[x_{ij}(t_{k})-\hat{x}_{ij}(t_{k})][x_{ij}(t_{k})-\hat{x}_{ij }(t_{k})]^{T}\}\,, \tag{48}\]  
which can be computed recursively; just before a measurement and just after a measurement, e.g., \(P_{ij}(t_{k}^{-})\) and \(P_{ij}(t_{k}^{+})\).  
Interestingly enough, the filtered estimate, \(\hat{x}_{ij}(t_{k})\) uses information from the measurement equation, e.g.,  
\[z_{ij}(t_{k})=H_{ij}x_{ij}(t_{k})+\
u_{ij}(t_{k})\,, \tag{49}\]  
where \(H_{ij}\) is the observation matrix of \(1\) and \(\
u_{ij}(t_{k})\) is a Gaussian noise with zero mean and covariance of \(R_{ij}\).  
Finally, the mean-squared filtered estimator of \(x_{ij}(t_{k+1})\), \(\hat{x}_{ij}(t_{k+1})\), written in the predictor-corrector format, is  
\[\hat{x}_{ij}(t_{k+1})=\Phi_{\delta}\hat{x}_{ij}(t_{k})+K_{ij}(t_{k+1})[z_{ij}( t_{k+1})-H_{ij}\Phi_{\delta}\hat{x}_{ij}(t_{k})] \tag{50}\]and the Kalman gain, \(K_{ij}(t_{k+1})\) is specified by the following set of relations  
\[K_{ij}(t_{k+1}) =P_{ij}(t_{k+1}^{-})H_{ij}^{T}[H_{ij}P_{ij}(t_{k+1}^{-})H_{ij}^{T}+ R_{ij}]^{-1} \tag{51}\] \[P_{ij}(t_{k+1}^{+}) =K_{ij}(t_{k+1})R_{ij}K_{ij}^{T}(t_{k+1})+[I-K_{ij}(t_{k+1})H_{ij}] P_{ij}(t_{k+1}^{-})[I-K_{ij}(t_{k+1})H_{ij}]^{T}\] (52) \[P_{ij}(t_{k+1}^{-}) =\Phi_{\delta}P_{ij}(t_{k}^{+})\Phi_{\delta}^{T}+Q_{ij}. \tag{53}\]  
Substituting (46) into (50), the estimate of random shocks is  
\[\hat{w}_{ij}(t_{k}) =\hat{x}_{ij}(t_{k+1})-\Phi_{\delta}\hat{x}_{ij}(t_{k}) \tag{54}\] \[=K_{ij}(t_{k+1})[z_{ij}(t_{k+1})-H_{ij}\Phi_{\delta}\hat{x}_{ij}( t_{k})]. \tag{55}\]