Electronic – State Space Clock Model & Control

I was hoping to get some help understanding a state space model that I have found in time and frequency literature. The literature pertains to a discrete-time model of a steered clock. A phase measurement is available, and the control input is via a frequency control element (e.g. a VCO).

Here is the two-state clock model:
$$x_{k+1} = \Phi x_k + w_{k+1}$$
$$\vec{x} =\begin{bmatrix} x_1 \\ x_2\end{bmatrix}$$, where x1 refers to a phase offset and x2 refers to a frequency offset between two oscillators. The state transition matrix is:

$$\Phi = \begin{bmatrix} 1 & T_s \\ 0 & 1 \end{bmatrix} $$

Which I believe originates from a continuous state matrix $$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$$. Phase is the integral of frequency, so this makes sense.

The output equation for the phase measurement is: $$z_{k+1} = Hx_k + v_k$$ where $$H=[1,0]$$.
My confusion is related to the model of the input/control matrix B whether in discrete or continuous time.

In core literature, the ideal discrete matrix for a frequency-steering element is described as

$$B = \begin{bmatrix} T_s \\ 1 \end{bmatrix}$$
This comes from: $$B=\Phi \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ which is described as the "propagation forward of the steer input."

Just looking at the equation, the form of discrete B proposed as such makes intuitive sense– if we can apply a delta to the frequency of an oscillator, that will be integrated into the phase variable and so on. Also I like this model because it is controllable.

I would be fine with working in the discrete domain, but I am quite confused how this model was derived from a continuous model. What would the continuous version of B be, and how is this justified?

The paper uses the phrase "fractional frequency" to describe the frequency steering mechanism. Perhaps this suggests that $$\ddot \phi = k u(t)$$.

then $$x_1 = \phi, x_2 = \dot\phi, \dot x_1 = x_2, \dot x_2 = \ddot\phi = k u(t)$$.
Then,
$$\begin{bmatrix} \dot{x_1} \\ \dot x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\begin{bmatrix}{x_1} \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ k \end{bmatrix}u(t).$$

This seems to match what is in the paper, because $$B_{discrete} = \Phi B_{continuous} = \begin{bmatrix} Ts \\ 1 \end{bmatrix}$$, which is the computation the authors of [2] use for deriving B in the paper. However, I thought the expression for computing discrete control was:

$$B_{discrete}= \int_0^{T}e^{A\tau}\vec{u}(\tau)d\tau B$$

When I do this integration for constant u, I don't get the same answer for discrete B, but I am probably doing it wrong (u isn't constant, error in doing the integration, etc). In any case I'm not sure how to fix it. But, nevertheless, I feel like I am on the right track for understanding some of the author's assumptions. Any pointers or insights would be appreciated.

By contrast, an alternate form of modeling a frequency steer assuming a voltage-controlled frequency synthesizer obey the following relationship (the assumption that the transfer function of the steering mechanism is linear when the system is operating in closed loop).

$$\dot \phi = f(t) + k u(t)$$
$$\implies $$
$$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} k \\ 0 \end{bmatrix}$$

This model is not controllable (though it is potentially stabilizable).

Key references:
Reference 1: Koppang. State space control of frequency standards https://iopscience.iop.org/article/10.1088/0026-1394/53/3/R60/pdf
Reference 2: Matsakis 2019 The effects of proportional steering strategies onthe behavior of controlled cloc https://iopscience.iop.org/article/10.1088/1681-7575/ab0614/pdf

Older refs:
Reference 3: Koppang & Matsakis, New Steering Strategies for the USNO master clocks https://apps.dtic.mil/dtic/tr/fulltext/u2/a496156.pdf
Reference 4: Koppang and Leland, Steering of frequency standards by the use of linear quadratic gaussian control theory https://ntrs.nasa.gov/api/citations/19960042633/downloads/19960042633.pdf

$$\Phi = exp(A \tau)$$