Introduction
In a previous post I described some of my experiments with building a GPS disciplined oscillator. In my experiments, I lock a voltage controlled oscillator to the PPS signal of a GPS receiver. The locking is achieved by a control loop, which observes the phase error between GPS and the VCO and produces a control signal to adjust the frequency of the VCO.
The GPS PPS signal is very noisy at higher frequencies (above say 1 milli-Hz), so the controller must have a low bandwidth to reject the higher frequency noise. Also, since I don't want to spend a lot of time tuning the controller, I want the bandwidth to be the only parameter.
VCO phase error model
The VCO has a control voltage input \(u\), which controls its frequency \(f_\text{vco}\). This relationship is well approximated as linear around an operating point. Thus near our target frequency \(f_\text{target}\) we can model the VCO frequency as \[f_\text{vco}(t) = f_\text{target} + g (u(t) - u_o),\] in which \(g\) is the VCO gain and \(u_o\) is the needed control value to attain the target frequency. The frequency error is thus given by \[ f(t) = f_\text{target} - f_\text{ocxo}(t) = -g (u(t) - u_o) \]
As the goal is to achieve phase lock, the quantity of interest is actually the phase error. We take the unit of phase to be a full cycle. This makes the phase error \(e\) simply the integral of the frequency error over time. \[e'(t) = - g (u(t) - u_o) \]
As the controller will be implemented in the digital domain, \(u\) will be piecewise constant. This allows us to discretize the model as the recurrence \[ e_{n+1} = e_n - \Delta t g (u_n - u_o),\] in which \(\Delta t\) is the time between the discrete changes of \(u\), \(u_n\) is the value of \(u\) at \(t \in [n \Delta t, (n+1) \Delta t) \) and \(e_n = e(n \Delta t)\).
Controller
The simplest form of controller which can control the phase error to zero is a PI controller. Here the control is determined as a linear combination of the phase error \(e\) and its integral \(i\) as \[ u_n = P e_n + I i_n, \] in which \(P\) and \(I\) are tuning parameters of the controller.
Unfortunately for us, our phase error measurement is very noisy at short time scales. With the simple PI controller the noise would couple straight to the control output through the P term. We thus need to consider something else. One could obviously use a longer interval between the measurements, but that introduces its own set of problems. Our solution is to instead apply a simple first order low-pass filter to the phase error measurement and then combine that with a PI controller.
Since the filter and the controller are implemented in the digital domain, let's transition now to an entirely discretized domain and ignore the continuous domain altogether. In this context, we take our low-pass filter as \[ \hat{e}_{n+1} = (1-\alpha) \hat{e}_n + \alpha e_n, \] in which \(\hat{e}\) is the filtered phase error and \(0 < \alpha < 1\) is a parameter defining the filter bandwidth.
The integral of the filtered phase error, similarly, is considered only in the discretized sense, and is given as \[ \hat{i}_{n+1} = \hat{i}_n + \hat{e}_n \]
The control law of the controller is then \[ u_n = P\hat{e}_n + I\hat{i}_n, \] in which \(P\) and \(I\) are tuning parameters to set the behavior of the controller.
Combining the VCO phase error model with the controller gives the model of the entire system as \[ \begin{equation} \begin{pmatrix} e_{n+1} \\ \hat{e}_{n+1} \\ \hat{i}_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & -\Delta t g P & -\Delta t g I \\ \alpha & 1-\alpha & 0 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} e_n \\ \hat{e}_n \\ \hat{i}_n \end{pmatrix} + \begin{pmatrix} \Delta t g u_o \\ 0 \\ 0 \end{pmatrix} \end{equation} \]
For the controller to work, we want the errors to converge. The dynamics of the convergence are determined by the eigenvalues of the matrix. For simplicity, we'll choose all three eigenvalues as \(r\), where \(0 < r < 1\).
The characteristic polynomial of the matrix is \[ \lambda^3 + (\alpha - 3) \lambda^2 + (\Delta t g P \alpha - 2\alpha + 3) \lambda + \Delta t g I \alpha - \Delta t g P \alpha + \alpha - 1. \] On the other hand, to have the desired eigenvalues, we want the characteristic polynomial to be \[ \lambda^3 - 3r\lambda^2 + 3r^2 \lambda - r^3. \] Matching the coefficients, we get the equations \[ \begin{align} \alpha - 3 &= -3r \\ \Delta t g P \alpha - 2\alpha + 3 &= 3r^2 \\ \Delta t g I \alpha - \Delta t g P \alpha + \alpha - 1 &= -r^3 \end{align} \]
Solving those equations yield \[ \begin{align} \alpha &= 3(1-r) \\ P &= \frac{1-r}{\Delta t g} \\ I &= \frac{(1-r)^2}{3 \Delta t g} \end{align} \]
The resulting matrix is unfortunately non-diagonalizable, making analyzing the resulting dynamics a bit tedious. The controller however appears to be well-behaved and produces convergence without too much overshoot.
The following figures show simulated responses from a controller with \( r = 0.001 \).
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| Impulse response of the controller against phase error change |
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| Impulse response of the controller against control offset change |
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