This is more of an extended comment than an answer.
The system may be inherently discrete-time. It may not make sense to find the continuous time plant model, as it may not exist. I am not familiar with atomic clock plant modeling, but the following points in the references indicate that the system is inherently discrete-time and that the input to the synthesizer is the incremental frequency steps.
Page 11. The word synthesizer and the word step.
What is the smallest step you can use on your synthesizer to correct the frequency?
Page 2. The input seems to be the frequency step by which the synthesizer needs to be adjusted. So with each input pulse, the frequency seems to step by a fixed amount. i.e., \$f_{k+1} = f_k + u_k\$.
The control vector ... corresponding to the fractional frequency change of the synthesizer ...
same page. The system seems to be inherently discrete-time.
is the time interval between measurements
However, ref 4 of the above paper provides a continuous time plant model.
page 2. confirms the above.
assumes frequency steps are used to implement the control
page 2
we consider only those in which the clock is controlled by shifting its frequency as fractions (gains) of its phase and frequency deviations from the reference standard
The above 2 references indicate that you have to just accept that this is the model.
Perhaps, References 3,4 to this paper may show the model derivation. I dont have access to them to be sure.
Best Answer
A discrete signal does not exist between the sampling increments, so it doesn't have an area under the signal. If you wanted to create a continuous signal from a discrete one you might use a zero order hold which gives it area by extending each sample into a pulse of width \$\Delta t\$, or \$dt\$ in the limit.