FORUM ON BEER, HOMEBREWING, AND RELATED ISSUES
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Contents:
PID Algorithm ("A. J. deLange")
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Date: Mon, 12 Mar 2012 04:13:20 -0400
From: "A. J. deLange" <ajdel at cox.net>
Subject: PID Algorithm
"Error is not the proportional term."
Jeff is apparently still confused on how PID controllers work. They start by
computing an error, the difference between the process variable and the set
point. For a heating load
E = SP - PV
and the set point is the upper edge of the proportional band. The lower edge
is at SP - Band where 'Band' is the band width in PV units (i.e. degrees).
The proportional term from the controller is G*(SP - PV)/Band where G limits
the output to 0 - 1 (0 to 100%). Thus if PV = SP - Band the output of the
proportional term is 0. If PV< SP - Band G sets the output to 0. If PV> SP
the output is 0 (unless the controller is set up to simultaneously control
heating and cooling loads). If PV = SP-Band then the proportional output is 1
(100%). If PV< SP - Band then the output is still 100%.
Thus, clearly the error scaled by gain = 1/Band is indeed the proportional
term.
" With only proportional gain the error will oscillate around the setpoint,
but never
eliminate/minimize the error."
If a controller is run with only the proportional term it cannot (unless the
load is 0) ever bring the PV to SP so the second part is true. There will be
some finite error necessary to establish the output required to supply the
load (heat losses). But a P-only controller does not oscillate at all unless
the proportional gain is set absurdly high (band to a fraction of a degree)
in which case oscillation does occur but not around the set point. In the
usual case the PV simply assymptotically approaches some value less than the
SP.
"It's take the integral term which integrates
its gain value as a function of time to reduce the error beyond the ability
of proportional gain."
It is not the gain which is integrated but rather the error. The integrated
error is then scaled by a gain and that plus the proportional error is the
output. The constant error from P integrates up calling for more output until
such time as the total error goes to 0. The integrator holds its value for
ever - it computes the integral from startup to the current time. E is also
differentiated. The differential is also scaled by a gain and added to the
output.
Output = G*[E/Band + g2*Integral_-infinity_to_t(E) + g3*dE/dt]
The differential term responds to sudden changes in PV (SP) such as throwing
a block of ice into the mash tun and also helps damp overshoot after an SP
step. Proper tuning (correct choice of Band, g2 and g3 are required for
optimum performance of a PID controller. These differ as thermal mass and
rate of heat loss from the thing being heated vary.
"Lacking a method to infinitely vary the output in
response to error makes the ability of the integral term to extinguish the
error not managed by proportional gain a pretty tough putt in a closed loop
system."
Not so. This can easily be verified by writing a simple simulation in which
you coarsely quantize the output. But more to the point, as I explained in
the last post controllers that use Output, as computed above, to control
relative on and off times, be it within a cycle of the 60 Hz line or within a
30 second or longer cycle time do provide infinite control (subject to the
quantization of Output i.e. whether the controller resolves to 1% or 0.1%).
"My best guess is with an on/off output, any integral gain value
applied will wind-up in value and put the system back in oscillation."
The arguments I gave above, the simulation I ran and most important, the
fact that I have three channels of PID control in my brewery that use on/off
output with a PID algorithm make it clear that this guess is not a good one
at all. We might also ask ourselves why, if this guess had any validity,
manufacturers would offer this output mode in a PID controller.
"Most
pid controllers also have a filter on the PV signal before it is passed to
the pid algorithm. If this value is increased to the point that the pv
signal is effectively dampened out of it's normal responsive range the loop
can also appear to be in a steady state."
Again simple simulation shows what happens if the input is excessively low
pass filtered. The loop will come to equilibrium at the set point eventually
but the overshoot/undershoot after a SP step can get kind of wild. Increasing
the differential gain will take a lot of that out and speed up convergence
but obviously if the input is low pass filtered it is going to take longer to
respond to a step in SP or load.
Details of how PID works must be in dozens of textbooks. Suggest Jeff look
at one of those. Playing around with simulations is also most instructive.
Most convincing of all, I suppose, would be to set up a PID controller with
on/off output and watch it play.
A.J.
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End of HOMEBREW Digest #5914, 03/12/12
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