Comparison between simulation and hardware realization for different clock steering techniques

The clocks are modeled by the well-established discrete two-state clock model [18, 19]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle x(t_{k+1}) = \phi(\tau) x(t_k) + w(t_k) \label{eq:processmodel} \nonumber \end{align} \tag{ 1 }$$

with transition matrix

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi(\tau) = \left(\begin{array}{@{}cc@{}} 1 & \tau \nonumber \\ 0 &1 \end{array}\right) \nonumber \end{align} \tag{ 2 }$$

and process noise $w(t_k) \sim \mathcal{N}(0, Q(\tau))$ where $\tau = t_{k+1}-t_k$. The two states represent the phase deviation and part of the frequency deviation. The process noise incorporates two different noise types: white frequency noise and random walk frequency noise. The covariance of the process noise

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q(\tau) =\left(\begin{array}{@{}cc@{}} q_1 \tau + \frac{q_2 \tau^3}{3} & \frac{q_2 \tau^2}{2} \nonumber \\ \frac{q_2 \tau^2}{2} & q_2 \tau \end{array}\right) \nonumber \end{align} \tag{ 3 }$$

is based on the diffusion coefficients $\sigma_1, \sigma_2$ ($q_i = \sigma_i^2, i = 1,2$) which drive the fundamental noises. This model can be used to describe the behavior of a single clock as well as the difference between two clocks x₁ and x₂:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle x_3(t_k) = x_2(t_{k}) - x_1(t_{k}) \nonumber \end{align} \tag{ 4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle x_3(t_{k+1}) = \phi(\tau) x_3(t_k) + w_3(t_k) \nonumber \end{align} \tag{ 5 }$$

with $w_3(t_k) \sim \mathcal{N}(0, Q_1(\tau) + Q_2(\tau))$, where $Q_1(\tau)$ and $Q_2(\tau)$ are the process noise covariance matrices for x₁ and x₂ respectively. A small process noise covariance matrix indicates a superior clock performance. In the extreme case with no process noise present, we find the theoretical ideal clock which consists purely of a transition from the last state without any perturbation.

The measurement of the clock is modelled by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle z(t_k) = H x(t_k) + v(t_k) \nonumber \end{align} \tag{ 6 }$$

where the measurement matrix H describes the structure of the measurement and $v(t_k) \sim \mathcal{N}(0, R)$ is the measurement noise with covariance R.

The process model in equation (1) only describes the behavior of a free-running frequency standard. If control values are applied on the frequency standard, the model has to be extended by a control term $b(\tau) u(t_k)$ to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle x(t_{k+1}) = \phi(\tau) x(t_k) + b(\tau) u(t_k) + w(t_k) \label{eq:extprocmod} \nonumber \end{align} \tag{ 7 }$$

where u(t_k) is the control value and $b(\tau)$ propagates the control value [10]. This change in the process model will only affect the prediction of the state in the Kalman filter formulas where the term $b(\tau) u(t_k)$ needs to be added as well. We will now focus on different methods to find the optimal control value u(t_k) in every time step.

For LQG control the feedback gain is chosen such that the cost function

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J = \sum_{i=1}^\infty x_i^T W_Q x_i + u_i^T W_R u_i \nonumber \end{align} \tag{ 8 }$$

is minimized, where the first part in the sum penalizes the transient state transition and the second term penalizes the control effort. So depending on the user's choice of W_Q and W_R the system is either steered gentle or rapid. Once the steering matrices have been identified, the LQR $\hat K_0$ is derived by solving the steady-state Ricatti equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \hat K_0 = &\ \phi(\tau)^T \hat K_0 \phi + W_Q \nonumber \\ &- \phi(\tau)^T \hat K_0 b(b^T \hat K_0 b + W_R)^{-1} b^T \hat K_0 \phi. \nonumber \end{align} \tag{ 9 }$$

Solving this equation requires a controllable system. In the next step, one obtains the gain $\hat G_0 = (b^T \hat K_0 b + W_R) ^{-1} b^T \hat K_0 \phi(\tau)$ which is used to derive the control value from the estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle u(t_k) = - \hat G_0 \hat x(t_k) . \nonumber \end{align} \tag{ 10 }$$

The LQG regulator can be obtained from the system parameters $\Phi(\tau)$, $b(\tau)$ and the designer matrices W_Q and W_R.

The PP technique is designed for a closed loop system. One determines the poles of the system and moves them to the desired pole locations. As shown in [11, 16], the poles of a system can be obtained by looking at the eigenvalues of matrix A which defines the transition between two states of the system

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(\begin{array}{@{}c@{}} x(t + \tau) \nonumber \\ y(t + \tau) \end{array}\right) = A \left(\begin{array}{@{}c@{}} x(t) \nonumber \\ y(t) \end{array}\right). \nonumber \end{align} \tag{ 11 }$$

Using the extended process model equation (7), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(\begin{array}{@{}c@{}} x(t + \tau) \nonumber \\ y(t + \tau) \end{array}\right) &= \Phi(\tau) \left(\begin{array}{@{}c@{}} x(t) \nonumber \\ y(t) \end{array}\right) + Bu(t) \nonumber \\ &= \left(\begin{array}{@{}cc@{}} 1- \tau g_x & \tau (1- g_y) \nonumber \\ -g_x & 1-g_y \end{array}\right) \left(\begin{array}{@{}c@{}} x(t) \nonumber \\ y(t) \end{array}\right).\nonumber \end{align} \tag{ 12 }$$

We find the eigenvalues of A by solving the eigenvalue equation and find the solutions

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lambda = \frac{-(\tau g_x + g_y - 2) \pm \sqrt{(\tau g_x + g_y - 2)^2 - 4(1-g_y)}}{2} . \nonumber \end{align} \tag{ 13 }$$

Critical damping typically brings the system back to equilibrium conditions in least amount of time and should hence show the best steering performance for our case study. Thus, the aspired poles need to be real and equal. In addition, the absolute values of the poles need to be less than or equal to one to ensure stability ($|\lambda| \leqslant 1$). As a result, we obtain the poles

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lambda = \sqrt{1-g_y} = 1- \sqrt{\tau g_x}. \nonumber \end{align} \tag{ 14 }$$

Nevertheless, it should be emphasized that critical damping might not be the best choice for all steering scenarios as sometimes other requirements, such as frequency smoothness or harder control actions are desirable. For a comprehensive control theoretical study of this aspect the reader is referred to a lately published work by Matsakis [20].

Independent of the chosen control interval, the Kalman filter that estimates the difference between the OCXO and the RB based on the measurement, is run every second. The size of the control gains for both methods needs to be adapted to the fact that the control interval is not identical to $\tau$, the time difference between two filter iterations. Therefore, we choose to calculate the control gain for LQG control

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \hat G_0 = f(\Phi(\tau_c), b(\tau_c), W_Q,W_R) \nonumber \end{align} \tag{ 15 }$$

as well as for PP

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_x = \frac{1}{\tau_c}(1-\lambda)^2, g_y = 1-\lambda^2 \nonumber \end{align} \tag{ 16 }$$

dependent on the control interval $\tau_c$. This affects the steered clock the same way as if the filter was run in the same frequency as the control values are applied. In iterations where no steering is applied, the last control value is propagated by the transition matrix. Since

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Phi(\tau = 1)^{\tau_c} = \Phi(\tau_c), \nonumber \end{align} \tag{ 17 }$$

this method leads to the same propagation of the control value as running the filter only on the control interval. The advantage of our methods is that all measurements available are used to predict the difference between the two clocks.

The clocks involved in this investigation are a rubidium atomic clock (SRO-100, Spectratime) and a steerable clock which consists of an OCXO with additional MPS (KL3382, K+K Messtechnik GmbH). The entire hardware setup carries four identical rubidium atomic clocks and two steerable clocks. However, for the sake of simplicity, we are only using one RB and one steerable clock to proof the hardware applicability and the performance of the steering techniques. Nevertheless, the entire setup allows for IEM realization of composite clocks consisting of all e.g. four RB as in [5].

The phase difference between the two 10 MHz signals of both clocks is measured every second with a time and frequency monitor (KL-3360, Lange electronics). The measurement data is subsequently send to a server (RS-232 connection), on which the Kalman filter for clock steering is performed. As already mentioned in the previous section, the Kalman filter is processed every second while the control value for the steering is calculated and applied to the MPS after the free selectable control interval time is reached. The steering value is send to the OCXO by an RS-232 connection between the steerable clock and the server.

In the simulations, we try to model the behavior of the true clocks in terms of an OADEV as closely as possible. Therefore, we have measured the performance of the OCXO built-in the MPS against an active hydrogen maser (AHM) (CH1-75A, Kvarz). Please note, that these measurements were taken before the real data runs were started as the hardware does not allow us to monitor the free-running OCXO within the MPS and the steered output signal simultaneously at the moment. The performance of the AHM supersedes the performance of the OCXO at every averaging time by a factor of at least 10² (@1 s). Therefore, we assume that the OCXO performance will drive the measurement.

The performance of the OCXO is modeled using the two-state model extended by four Markov processes as suggested by Davis et al [22]. Figure 1 shows the measurement as well as the respective simulation runs. As the simulation is based on statistical appearing noises, we show 20 runs of the simulation to demonstrate the reliability of the simulation. The deviation of the simulation from the measurement at long averaging times is determined by random walk frequency noise. Especially for short averaging times ($\tau < 5$ s), not all process noise of the OCXO is reflected in the model. Nevertheless, since we focus on control intervals $\tau_0 \geqslant 10$ s, it does not impact the stability of the steered clock. The detailed parameter set for the OCXO model can be found in table 2.

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Table 2. List of parameters for the OCXO model.

Parameter	Assigned value
q-Parameters	$q_1 = (5 \cdot 10^{-12}){}^2$,
	$q_2 = 5 \cdot 10^{-30}$
Positions of the peaks of the Markov processes, Rj	$\newcommand{\e}{{\rm e}} \left(\begin{array}{@{}c@{}} 1.89/20 \\ 1.89/200 \\ 1.89/(2\cdot 10^3) \\ 1.89/(2\cdot10^4) \end{array}\right)$
Size of the Markov processes, Uj	$\newcommand{\e}{{\rm e}} \left(\begin{array}{@{}c@{}} 4 \cdot 10^{-24} \\ 9 \cdot 10^{-24} \\ 2\cdot10^{-23} \\ 5\cdot 10^{-23} \end{array}\right)$

We are trying to emulate the real situation as closely as possible in the simulation. For the real-time routines, we do not have any information about the free-run OCXO behavior during processing. As a result, there might be a discrepancy between the free-run behavior of the OCXO and the chosen model. Even though we could easily solve this issue in simulations, we decided to use the same parameter set for real data as for simulation. Consequently, we will have a discrepancy between the simulated OCXO and the model used in the Kalman filter. This resembles the situation with real data.

The same holds for the rubidium atomic clock. The advantage is that we can monitor the RB during the real data runs. In figure 2 the measurement of the rubidium atomic clock again against the AHM is depicted together with 20 simulations of the chosen two-state-model. Table 3 lists the parameters of the model for the RB performance.

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Table 3. Parameter for the rubidium atomic clock model.

Parameter	Assigned value
q-Parameters	$q_1 = (3.8 \cdot 10^{-12}){}^2$,
	$q_2 = 3.5 \cdot 10^{-28}$

As baseline for the performance of the rubidium atomic clock that serves as a reference we have used the measurement of the RB relative to an AHM throughout the period when the real runs where performed.

For the measurement noise R, we measured a clock against itself with the time and frequency monitor. This data was then fit to a model of the device's measurement noise.

At first we will discuss the results obtained with the LQG control and thereafter the PP technique. In figure 3 the results of one real data run with LQG control is depicted for a control interval of 20 s. The performance of the steered OCXO aligns to the performance of the free-running rubidium atomic clock after an averaging time of roughly 10³ s. That means, the same noise types are present in both clocks. Furthermore, the slope of the stability of the differential phase measurement between both clocks exhibit a slope of  −1 for averaging times $\tau > 10^2$. Once the steering of the quartz clock is successful, the performance of the RB and the OCXO are identical and the difference between them shows a slope of  −1 in the OADEV analysis (white noise). In addition, the performance of the free-running OCXO is shown. The stability of the steered OCXO is identical to its free-run performance for $\tau \in [10^0, 10^1]$. This behavior is expected as the control interval was set to 20 s. Thus, the short-term performance of the OCXO should not be impacted by the steering. Consequently it is following the performance of the free-running OCXO signal.

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In order to demonstrate the reliability and the statistical significance of the demonstrated steering performance figure 4 displays the evaluation of four different hardware runs in terms of OADEV. All runs were performed with identical steering values including the control interval of 20 s. They show similar steering performances, what demonstrates that the method is sufficient to steer the OCXO to the rubidium atomic clocks' signal.

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In the next step, we want to compare the hardware runs to simulations. Figure 5 demonstrates the expected steering performance for the LQG steering technique using a control interval of 20 s obtained from numerical simulations. Although the rubidium atomic clock performances in the four simulation runs are significantly more aligned compared to the real measurements, the steering performance of the OCXO signal is equivalent. Not only the averaging time from which on the phase difference analysis in terms of OADEV shows pronounced white frequency noise (slope of  −1 in the plot) but also the averaging time at which the stability evaluation of the steerable clock is aligned to the rubidium atomic clock are in high agreement for simulations and real data measurements. Moreover, the expected bump in the performance of the steered OCXO signal, which originates from the steering process itself, is at the same position although the height is less pronounced in the real data. This might be a result of slightly different OCXO performance for the real runs compared to the simulations. As described in section 3.2, we are currently not able to measure the free-run performance of the OCXO while a steering is performed on the MPS. Consequently, we can only speculate about the origin of the deviations between simulation and real measurements. The performance of all clocks within the setup is within specification. Nevertheless, the agreement between simulations and real measurements is significant so that we are confident that the steering process works as expected.

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With a focus on the differences introduced by different control intervals, we will now take a look at the steering performance for a control interval of 100 s. Again, we performed four individual runs of simulations as well as hardware measurements. Figure 6 is showing the results of the hardware runs. The results are similar to the scenario with the control interval of 20 s. However, the steering takes place for higher averaging times and consequently the syntonization is observed later around an averaging time of 300 s. Even though the overall performance of the RB was not as stable as in the previous runs, the steering worked sufficient. The steered OCXO performance aligns with the stability of the rubidium atomic clock for averaging times over approximately 10³ s.

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To ensure that proper steering performance was achieved, these results are compared to numerical simulations. Figure 7 shows the behavior of the free-running RB as well the steered OCXO in four individual simulation runs. Analogue to the control interval of 20 s the investigations of the performance shows a good agreement between simulation and real data measurement for a control interval of 100 s. Only the rubidium atomic clock exhibits slightly different stability behavior in the hardware measurements. One possible explanation is that during the time period of the real measurements the environmental conditions slightly changed in the laboratory.

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In order to further investigate the steering performance as a function of the control interval, we take a closer look at the real data runs with different control intervals. We calculated the mean and the standard deviation of the OADEV behavior of the phase difference between the rubidium atomic clock and the steered OCXO for five different control intervals between 1 and 100 s. Figure 8 shows the results. It can easily be seen that the magnitude and location of the steering introduced bump is dependent on the control interval. In particular, the bump is most distinct if we apply the control values every second to achieve rapid steering. For the largest control interval of 100 s, we can barely identify a bump, due to the smooth steering process. Nevertheless, the averaging time from which on we identify syntonization between both clocks (slope of  −1 in the OADEV plot) occurs later with increasing control interval. That means it takes longer to drive the OCXO to the rubidium atomic clock. Consequently, the control interval needs to be chosen in accordance with the other control parameters to ensure sufficient steering for the distinct scenario.

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In addition, we can now compare these results with the expected dependence on the control interval from simulations. Figure 9 displays the simulated behaviors of the phase differences between the reference and the steered OCXO. The simulations for the control intervals between 10 and 100 s are in good agreement with the real data measurements. However, the scenarios for the control intervals of 1 and 2 s show significant deviations. Here, no pronounced steering bump is visible in the simulations, while the largest bump appears for a control interval of 1 s in the real data measurements. We suggest that this behavior is caused by not sufficient communication between the measurement devices and the algorithms as the measurements are taken every second but need some additional time until they are proceeded to the algorithm routines. This might lead to problems with the state updates in the Kalman filter, as these values are calculated every second, too. These issues can presumably be overcome by establishing faster communications between the devices or increasing the control intervals. Nevertheless, the good agreement between simulations and measurements allows for an optimization of a distinct hardware steering scenario by performing numerical simulations in advance which are much less time consuming.

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Analogous as for LQG control, we will first take a look at the results of the steering process using PP and a control interval of 20 s. In figure 10 the stability of the steered OCXO and the free-running rubidium atomic clock, both monitored against an AHM, are shown together with the stability of the phase difference measurement between the clocks. Furthermore, the performance of the free-running OCXO detected before the steering test took place is shown.

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Interestingly, the PP technique shows similar steering performance than the LQG control for the chosen control parameter. However, two slight differences are observable: first, the syntonization takes place for lager averaging times compared to the LQG run for a control interval of 20 s. Nevertheless, it seems that the converging of the steered OCXO to the reference performance thereafter is faster, so that the alignment of the stabilities of the RB and the OCXO happens for nearly identical averaging times. Second, the bump in the stability behavior of the steered OCXO, which is introduced by the steering process itself, is much less pronounced in the PP case. Both differences might originate in the prior chosen other steering parameters, as we use a slightly gentler steering in the PP case compared to the LQG control. For more details about the steering performance using different control values for a constant control interval based on simulations it is referred to [16]. However, the overall steering performance of the PP technique is similar to the presented LQG scenario.

To demonstrate sufficient steering despite the diversification of the clock performances, we again show for different runs of real data measurements as well as simulations in figures 11 and 12, respectively. All real data measurement runs show reproducible steering behaviors although the rubidium atomic clock stability exhibits stronger variations than in the previous runs. Unfortunately, the reason of this unstable behavior is not known. It might be related to environmental changes or similar influences. Nevertheless, it is even more remarkable that the steering process was successful in every run although the used clock model did not represent the real rubidium atomic clock behavior in an accurate way.

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The simulations of the steering process in figure 12 again are in good agreement with the real data measurements. The deviation of the steered OCXO stability in term of an OADEV analysis from the supposed free-running OCXO behavior takes place earlier than expected from the simulations. This might be caused by a different performance of the free-running OCXO. Please note that this data originates from a time period before the real data runs with steering were performed. However, the other characteristic features of the steering procedure are well represented in the real data measurement runs. The appearance of the syntonization as well as the alignment of the stability performance of the steered OCXO and the RB takes place at similar averaging times in both cases, simulation and real data measurements. Hence, we achieved sufficient accordance between real measurements and simulations.

The same statement holds for the investigation of PP as steering technique using a control interval of 100 s. Figures 13 and 14 demonstrate the performance of the steered OCXO and the rubidium atomic clock using real data measurements as well as simulations, respectively. As expected, the syntonization takes place in the real data measurement runs at higher averaging times around 10³ s. The alignment to the reference clock stability is achieved near $5\cdot10^3$ s averaging time. Both results are in good agreement with the simulations.

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However, the overall steering performance of the PP technique for a control interval of 100 s is not as good as the runs using LQG control as syntonization and stability alignment are happening here for considerably lower averaging times. Nevertheless, the steering with PP still shows sufficient performance.

Finally, the steering performance of PP for different control intervals between 1 and 100 s is analyzed. Figures 15 and 16 demonstrate the performance of the phase difference between the steered OCXO and the rubidium atomic clock using real data measurements as well as simulations, respectively. Analogous to the LQG evaluation, the measurements with the control interval of 1 and 2 s show a pronounced bump in the phase difference analysis using an OADEV characterization. As mentioned before, this feature may show up due to communication delays between the measurement devices and the server running the for the steering responsible Kalman filters. Moreover, the agreement between the simulations and the real data analysis inferior to the LQG runs. One possible explanation is the much higher diversity of the rubidium atomic clock performances during the real measurement runs compared to the LQG runs. This might influence the analysis for the different control intervals in a negative way. Nevertheless, the trend, that the syntonization takes place at higher averaging times for increasing control intervals is clearly visible. Overall, the PP technique achieved comparable results to the LQG control scenario and yields sufficient steering.

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