On Thu, 2015-11-12 at 00:30 +0500, 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. Thanks a lot for explaining! I think I'll put the rate controller details in a wiki page during the weekend. I first thought of putting it in code comments, but that's probably not the best medium in this case. --Â Tanu
