[pulseaudio-discuss] [PATCH 04/13] loopback: Adjust rates based on latency difference

[Georg Chini]

On 11.11.2015 20:30, Alexander E. Patrakov wrote:
> 11.11.2015 23:36, Tanu Kaskinen wrote:
>> Sorry for being obtuse, but I don't follow what this simple bit of code
>> is doing. You mentioned "P-controller" and the "Ziegler-Nichols
>> method". I followed the Wikipedia link, and found that a P-controller
>> is a very simple thing:
>
> Actually it was me when splitting the patch :)
>
>>
>> u(t) = Kp * e(t)
>>
>> where
>>
>> u(t): the new control variable value (the new sink input rate)
>
> No. The control variable is new_rate - base_rate.
>
>>
>> Kp: a tunable parameter (a magic number)
>>
>> e(t): the error value, i.e. the difference between the current process
>> variable value and the target value (current latency minus configured
>> latency)
>
> Correct.
>
>> The Ziegler-Nichols method can be used to choose Kp. For a P-controller
>> Kp is defined as
>>
>> Kp = 0.5 * Ku
>>
>> where
>>
>> Ku: a number that, when used in place of Kp, makes u(t) oscillate in a
>> stable manner
>
> See below, I'll comment on that.
>
>>
>> (A sidenote: I probably have understood something wrong, because Kp is
>> a plain number, and u(t) and e(t) have different units, so there
>> appears to be a unit mismatch. u(t) is a frequency and e(t) is a time
>> amount.)
>
> Kp is not a plain number. It has the unit necessary to convert from 
> the unit of error value to the unit of the control variable.
>
>>
>> Figuring out Ku seems to require having an initial calibration phase
>> where various Ku values are tried and the oscillation of u(t) is
>> measured. The code doesn't seem to do this. Could you explain how you
>> have derived the formula in rate_controller()?
>>
>
> The formula is indeed not the most obvious one. We have exchanged some 
> emails with Georg. If he permits, I can forward his email with the 
> derivation of the non-linear part written on paper and scanned. But, 
> to answer the question about the "optimal tuning" in the sense of 
> Ziegler-Nichols method, we only need to talk about the linear 
> approximation in latency_difference_usec, that is, put min_cycles to 1.0.
>
> So:
>
> new_rate = base_rate * (1.0 + latency_difference_usec / adjust_time)
>
> I.e. here Kp = 1 / adjust_time, that's all.
>
> Assuming that the correct rate is the nominal one (i.e. base_rate), 
> which is a crude approximation but good enough for evaluating 
> stability, the latency difference accumulates with the speed which is 
> exactly (base_rate - new_rate) / base_rate. Indeed, in one second 
> according to the input, base_rate samples will be pushed, but only 
> new_rate samples will be pulled from the queue. So, each second, the 
> queue grows by base_rate - new_rate samples. According to base_rate, 
> it's (base_rate - new_rate) / base_rate seconds per second.
>
> Now note that the new latency difference will be evaluated again in 
> adjust_time. So, if we put Kp = 2 / adjust_time instead of what we 
> did, then see what happens: by the time we look again, the latency 
> difference will be overcorrected by a factor of 2. I.e. changes the 
> sign. Then the rate controller will try to correct that again, and 
> will again overshoot by a factor of 2, i.e. it will return to the 
> original value. I.e. it will exhibit oscillations with constant 
> amplitude - exactly what Ziegler-Nichols method calls for, when 
> calibrating. We actually use Kp = 1 / adjust_time, i.e. half of the 
> critical value, which is exactly what Ziegler-Nichols method prescribes.
>
> Note: I did not say that following this method is good for our 
> purposes. The PID controller recommended in these papers (and used in 
> Jack) is not optimal in the sense of Ziegler-Nichols method:
>
> http://kokkinizita.linuxaudio.org/papers/usingdll.pdf
> http://kokkinizita.linuxaudio.org/papers/adapt-resamp.pdf
>
Hi Alexander,

regarding your note: I was under the impression that we agreed that we 
could both not
come up with a better way of controlling the latency in this context. 
And - as another side
note - my controller is only optimal in the sense of Ziegler-Nichols for 
small latency differences.

Regards
            Georg