## 4 pLEO Time Scale Realizations
### Onboard Steering OCXO to IEM  
The realization of the IEM time scale generated by the clock-ensemble Kalman filter involves locking a physical oscillator to the IEM. Consequently, the MPS output from an OCXO onboard the pLEO platform \(i\) (for \(i=1,\ldots,N\)) adheres to the IEM and thus serves as a realization of it. The control value applied by the MPS is derived using various steering techniques [24].  
In real measurements, only the clock-to-clock difference measurement is directly available. Therefore, it is possible to simulate this situation by propagating and estimating the difference between the OCXO and IEM clocks for the \(i\)-th onboard platform (where \(i=1,\ldots,N\)). This is achieved using the two-state clock model (46), which describes the behavior of the difference between two participating clocks  
\[s_{i}(t_{k}) =x_{i,OCXO}(t_{k})-x_{0}(t_{k}) \tag{74}\] \[s_{i}(t_{k+1}) =A_{i}(h)s_{i}(t_{k})+v_{i}(t_{k})\,, \tag{75}\]  
Figure 9: Implicit ensemble mean realizations aboard pLEO platforms.  
where the process noise, \(\
u_{i}(t_{k})\) is a zero-mean Gaussian with its covariance, \(Q_{vi}(h)\) taken into account of the behaviors of the OCXO and IEM, i.e., \(Q_{vi_{i}}=Q_{OCXO}(h)+Q_{IEM}(h)\). The transition matrix on the differential clock state, \(s_{i}(t_{k})\), is described by  
\[A_{i}(h)=\left[\begin{array}{cc}1&h\\ 0&1\end{array}\right] \tag{76}\]  
and \(t_{k+1}-t_{k}=h\) is the control interval which is one of the few parameters used to tune the steered signal response corresponding to the target state errors. Intuitively, the impact of the control rates will influence the stability of the OCXO reference clock on each orbital platform.  
If repeated adjustment of oscillator frequency, called clock steering, is applied in the realization of IEM to take advantage of the best clock stability across different averaging intervals, the differential clock model (75) onboard each platform \(i\) and \(i=1,\ldots,N\) will then need to be extended by the control term \(B_{i}(h)u_{i}(t_{k})\) as follows:  
\[s_{i}(t_{k+1})=A_{i}(h)s_{i}(t_{k})+B_{i}(h)u_{i}(t_{k})+\
u_{i}(t_{k}) \tag{77}\]  
with  
\[B_{i}(h)=\left[\begin{array}{c}h\\ 1\end{array}\right]. \tag{78}\]  
Notably, the controllability matrix, \(C_{i}(h)\), associated with the differential clock model (77) onboard the \(i\)-th platform is given by  
\[C_{i}(h)=\left[\begin{array}{cc}B_{i}(h)&A_{i}(h)B_{i}(h)\end{array}\right] =\left[\begin{array}{cc}h&2h\\ 1&1\end{array}\right]. \tag{79}\]  
And its determinant satisfies \(det\{C_{i}(h)\}=-h\
eq 0\). Thus, \(C_{i}(h)\) has full rank for any \(h\in\Re^{+}\).  
It is also important to note that the independence of the selected control interval \(h\) on each orbital platform \(i\) means that the clock estimates from the Kalman-filter clock ensemble running on the anchor orbital platform \(j\) (where \(j\
eq i\)) are distributed to all participating orbital platforms in the network. This distribution occurs at a frequency that differs from the control values \(u_{i}(t_{k})\), which are applied to the MPS of the OCXO oscillator onboard orbital platform \(i\). As a result, the control gains for \(u_{i}(t_{k})\) need to be adjusted to reflect the difference between the control interval, or the communication period \(h\) between the anchor orbital platform and the participating orbital platforms, and the filter iteration \(\delta\), as discussed in Section 2.