Hello all,

I am wanting to model a counter in a control loop with a transfer function so that I can apply control therory to examine stability. I am not quite sure what to use for a transfer function. The system is discrete time and the counter counts the input pules (random) before dumping them to its output. I have considered two possibilities so far:
1) the counter acts as an integrator and thus I could use 1/s or 1/(z-1) as TF.
2) the counter is a decimator and has the TF (1-z^-N)/(1-z^-1)

I am not quite happy with version 1) as 1/(z-1) does not look at what happens in between sampling instances. Remember that the input pulses occure randomly without a time reference. The TF 1/s does not apply since the overall system is sampled.
Version 2) has I guess the same issue and since there is no set decimation factor N, I don't think this equation applies.

Does anyone have any suggestions, hints, comments...

Thanks a lot,
Sven

Jess Chen,

Thanks for your feedback. Indeed the pulses enter the counter without any constant rate. The counter output, however, occurs at a constant rate Ts which is the rate the overall system is clocked.
To model the counter as a simple gain block does not suffice as it does not give insight into stability. An analogy would be a quantizer. A quantizer is sometimes modeled with a variable gain and a phase shift to obtain the systems overall stability (i.e. sigma delta modulators). A constant gain does not provide such insight.

So I am looking to find a TF for a counter wich counts input  pulses and the dumps them after a constant rate to the output.

Thanks a lot for any help.
Sven

I would like to obtain a quantized representation of the input signal which is a frequency modulated signal. That is why I choose a counter. I am sort of familiar with the describing function method. As mentioned above the quantizer can then be modelled with variable gains and with phase uncertainty. But I could not use the same methods for the counter as it has essentially a pulse train with variable frequency at its input (information is in frequency), and a quantized output at a constant frequency where the information is now in amplitude. A variable gain does not apply here even if we assume the output amplitude to be constant, i.e. just a one bit counter.

I will have a look in to ensemble averaging. Do you have any good references on ensemble averaging?

Thanks for your help Jess Chen.

Just in case you are not aware of it: This article can be downloaded from Ken's personal website at http://www.thekunderts.net/ken/docs/jssc04-09.pdf. You can find a list of his publications at http://www.thekunderts.net/ken/pubs.html, most of them are available for download.

Laplace domain transfer functions are usually applied only to linear time invariant systems. Are you sure that's what you have?

-Jess

I found the answer to the Clegg integrator.
When excited with a Asin(wt) input, the output of a Clegg integrator will be A/w*(1-cos(wt)). Using the describing function method, we can find the fourier coefficients. They equate to be a1=-A/w and b1=4A/(pi*w). Thus the transfer function is given as b1+j*a1 which will have a phase of 38.1 (pi/4) degrees.

Sven,

I agree that the VCO resets but I would point out two observations:

1. Since the argument of the VCO can be thought of as a trig function, the "reset" events are transparent to the VCO output. It is only at the phase detector that such events become important.

2. In a phase domain model of the VCO, where the output is a sawtooth waveform, the output is not a true steady state quantity. I usually pull the VCO integrator, and the reference integrator, into the PFD model. This has couple of advantages. First, small signal analysis is performed about a true DC quantity (i.e. frequency). And since the DC analysis is valid, the models of the various components can be written to map out lock ranges with just the DC analysis. Second, by placing the integrator (a resettable integrator to be more precise) inside the PFD, the PFD accurately models the hysteresis observed in the relationship between average output and input phase error. Furthermore, the reset value of the integrator can be adjusted to accurately simulate the frequency slewing properties of the PFD. Where was I going with this? In a phase domain model, I think the resettable integrator belongs in the PFD, not the VCO. Furthermore, for small signal analysis, the integrator indeed introduces 90 degrees of phase lag. For large signal analysis, the behavior is highly nonlinear, probably to the point that the usual nonlinear methods, such as describing functions, bear little fruit.

Sven,
It is too bad we could not meet face to face.  It seems we are both having trouble communicating. Im the California and based on when you respond, I suspect you are in Europe.

Before I explain my response, let me ask a couple of questions.

1. Are you working with a PLL?

2. When you replace the ideal integrator with a VCO, I assume you are also switching between phase domain and voltage domain models. Right?

3. Exactly how are you distinguishing between stability and instability?

-Jess