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

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

  • \$\begingroup\$ 1) Is \$H\$, [0,1] or [1,0] ? The first state is phase; right ? 2) The continuous time \$B=[0,1]^T\$ indicates that the frequency is the integral of the input. \$\endgroup\$
    – AJN
    Mar 29, 2021 at 12:19
  • \$\begingroup\$ 1.) Yes, H should be [1,0]. \$\endgroup\$
    – mica
    Mar 29, 2021 at 13:01

1 Answer 1


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.

  • \$\begingroup\$ Wow, thank you. Yes, the Zucca and Tavella paper you mentioned discusses a derivation of a continuous clock plant model. I think you're probably right, though, that since we have a frequency synthesizer as control, this is an inherently discrete system. Though, it could probably be approximated by a continuous one, there may not have been any utility in doing so. \$\endgroup\$
    – mica
    Mar 29, 2021 at 13:49

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