anti aliasing filters for digital control loops

Steve Minshull wrote:

The filter design should not be a problem in a well conceived system.
For control loops, the sample rate ought to be 5 times the highest
significant frequency, and 10 is better. Part of the reason is that it
largely solves the aliasing problem. It really doesn't matter if
something at 12*Fmax aliases down to 8*Fmax, and there's not much phase
penalty in bringing the alias at 2*Fmax down enough so it won't matter.

A less obvious point is processing delay. With critical sampling where
Fs is a hair over 2*Fmax, a one-sample delay is a 180-degree phase
shift. The way most systems are implemented, you can get a one-sample
delay in the A/D and another in the D/A. Filters can add more.

Symmetric FIR filters' delays are half their number of taps. They aren't
much use in servos. Servos need filters whose impulse responses are
weighted toward the front. They show dispersion -- delay varies with
frequency -- but decent phase margin is about keeping the delays small,
not about keeping them equal. I call such filters "prompt". The
technical term "minimum phase" may be too restrictive. Almost minimum is
usually good enough. A quote from Chris Bore (http://www.bores.com/ ):
"... the worst possible design that just meets the specification -
almost a definition of practical engineering"

With a high sample rate, the critical frequencies most filters are low
compared to it, and IIR filters excel at that. Even impulse-invariant
designs work well and may be simpler than the more versatile biquads.

I'm cross posting to comp.dsp so others can critique me and pick up
where I left off. I'll be away for a while. Good luck!

Jerry
--
Engineering is the art of making what you want from things you can get.
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Jerry Avins wrote:


Yes to everything here, with the added comment that I have found a
single-stage comb filter to be very useful in anti-aliasing.  The idea
is to sample at 4, 8 or 16 times your desired control rate, add up all
the samples and shift to right to normalize, then go ahead and control.
  This is an FIR filter, and it does add delay, so you have to
compensate by keeping your control sampling rate up.

The really nice thing about this scheme is that the comb filter has big
wide notches at the controller's sampling rate and all it's harmonics.
You may have aliases that come from noise at (n + 1/2)Fs, but when it
gets applied to the controller it's at Fs/2, where either your
controller or your plant isn't going to pass much energy.  What you
_won't_ have is aliases that come out of the filter very close to DC,
which is usually where you can least afford aliasing problems.

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

How can you use a digital filter for anti-aliasing?

Rimmer

Rimmer wrote:


The same way you use an analog one -- filter out unwanted spectrum at
the alias frequencies before sampling (or in this case before downsampling).

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

To clear up confusion, this addresses aliasing that might be introduced by
decimation in time.  The assumption is that your original sampling is fast
enough so that you can ignore any aliasing effects at that stage.

Scott

Scott Seidman wrote:

Not exactly, even though that would be enough. (One likes to sample 5 to
  10 times the Nyquist rate in a servo system. A bit of garbage above
the unity-gain frequency can be tolerated. Etc.) The scheme also
guarantees null at those frequencies where the strongest and most
troublesome aliases would occur. It isn't free: to end up with 10x
samples and also use 16x anti-alias "filtering", the original sample
rate has to be at least 250 times the servo's unity gain. A
successive-approximation converter is needed to keep the latency low.

At least that's how I see it. I had not thought of it until Wescott
described it yesterday.

--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Thanks for all the help.  My problem was having a too low sampling
frequency.  The end application is a digitally controlled buck converter for
charging batteries where i'll be sensing current and voltage and then i can
vary the charging sequence in software (either in voltage control or current
control).  This probably isn't the best way of going about business but it's
purely an educational project in my summer holidays.

I was planning on using a 18 series PIC microcontroller (as i have the
development hardware+software) which can output a PWM of upto 40kHz at a 10
bit resolution.  Now presumably thats the limit on my sampling frequency as
I don't see any point in sampling more often than I can update the
output.....am I right?  So with a 40kHz sampling frequency do I need to
design my buck converter with a open loop bandwidth of ~4kHz?

It would also be nice to get a simulation of the system running in Simulink
(to aid my understanding at make the right design decisions).  I realise
that you can model the a/d&d/a action as a zero order hold function but in
this sytem the input into the plant's TF is a PWM signal.  Any ideas on what
i should do here?

Thanks Again

Steve Minshull

Steve Minshull wrote:

You As you can see from Tim Wescott's suggestion, you might want to take
several samples for every one you end up using. Still, 40 KHz may be
higher than you need.


That's a good idea. A lower frequency is probably adequate, and would
mean more time for the computations.

Not from me! With luck, from others.

   ...

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

No.  You don't need to match the rate of the control loop to your PWM
signal, and life will be incredibly difficult if you try.  The PWM of 40kHz
can be COMPLETELY independent of the rate of the control loop!!

Set up the PWM frequency at 40 kHz according to the PIC.  This is all
handled by the PIC hardware (assuming that the 18 series you've chosen has
hardware PWM).  Then, use a timer to control the rate of the control
algorithm.  The best way is to set a flag with the timer ISR, and do all
your calculations outside of the ISR when the flag goes high (then reset
the flag of course).   Each pass through the control algorithm, adjust the
duty cycle of the PWM.  

For example, assume that 1kHz is a sufficient rate for your buck converter
(and that's probably 2 orders of magnitude faster than you need).  Now
let's say your control algorithm calls for a 35% duty cycle.  Set the duty
cycle at 35.  You'll then get 40 pulses of 35% duty with a PWM set to tick
at 40kHz.  The next algorithm check, you get 26% spit back.  Set the width,
and repeat every msec.

If we had to run our control loops at the PWM rates we sometimes need,
we'll all go crazy.

Scott

Steve Minshull wrote:


Without writing custom blocks or having subsystems sampled at very high
rates your best bet is probably to just output a continous value and
don't try to PWM it.  Unless you pour an incredible amount of energy
into it you can't get a simulation to match reality all that exactly --
at some point it's cheaper to just build a board and try it out.

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

@corp.supernews.com:


Tim, the 18-series has hardware PWM.  You set it up, and it goes at
whatever frequency you tell it to.  The control rate can be orders of
magnitude slower.

Scott

Scott Seidman wrote:


Sorry, I wasn't clear -- I was only addressing how to simulate it, not
how to make it work in practice.  In practice you use the PWM, most
certainly.  Given the amount of ripple the PWM will induce, it's
probably a good idea to try to sample synchronously with the PWM, if
possible, however.

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

@corp.supernews.com:


--

Keep in mind that the OP is trying to charge a battery!!  I suspect a
10Hz control loop, with a 1kHz simple 1-pole RC filter on the input will
do the job nicely.  The OP can PWM at whatever frequency he wants to, and
never worry about that rate again.

Also, the PIC series are not without their limitations--the relevant one
here being that none of them (short of the new dsPIC line-- real DSP's
for less than $30!!) have any analog output capability.  The PIC guys
often kluge one up by low pass filtering a PWM signal.  After that,
you're talking about adding a DAC.

You're perfectly right, though.  There's no way the OP should attempt to
bit-bang a PWM controller at 40kHz.

Scott

Hi.

Again thanks for the help.

With regard to the control loop bandwith.  The transfer function (output
volts vs duty) i have got from state space averaging is: Vo/d = Vi / ( LC
(s^2 + (s/RC) + (1/LC))).  So the undamped natural frequency fn = 1/(2pi *
root(LC)).  Am I right in thinking that I size L and C to give me the
required controll loop bandwidth?  If so then the inductor/capacitor size
seems rather large for a 10Hz bandwidth.  The biggest inductor i can find
(RS/Farnell) for the max current of 2A is 680uH so I'm a bit limited here
without designing my own inductor.

Steve Minshull

You tune things to the PWM frequency, not the control loop frequency.


Look at some pseudo code

main(){
set PWM frequency to 40K
set PWM duty cycle to 50%

while (1){

            if timerflag=1 {
                sample the voltage and/or current
                calculate next PWM duty cycle
                set PWM duty cycle
                timerflag=0
            }

}

}
timer_isr(){
timerflag=1
timercount=timercount+offset  (to give you 1kHz control loop)
}


Whenever the timer rolls over (goes from ffff to 0) the interrupt service
routine gets triggered, setting the timer flag, and then adding any
offset to the count register that you need to give you the control rate
you need.  Thus, your main loop sits there doing nothing until the timer
rolls over, but your hardware PWM is ticking away at 40K, at whatever
duty cycle it was last set to.

The only waveform your buck converter sees is the PWM switching at 40K.  
This is the frequency that you select all your components for.  The
control loop is merely how often you change the duty cycle.


Scott

Hi

Thanks, I understand the difference between the PWM freq and the control
loop sampling frequency.  I understand you can choose the inductor size to
get a certain current ripple at the PWM switching frequency.  I was
referring to choosing the inductor size to get the converter bandwidth down
as low as 10Hz.  I think I may be getting confused in how to work out the
loop bandwidth as the sampling frequency should be ~10x this loop bandwitdh.
I was assumning the loop bandwidth was determined by the frequency response
(bandwidth) of the plant (buck converter) and such i would need a large
inductor/capacitor in order to limit the controller bandwidth to 10Hz.
Presumably i'm going wrong somewhere?

Steve

Sample at a rate reasonable to your platform, like 1K.  if you need to slow
down the loop even more, you can use digital filtering and decimation.

Scott

Steve Minshull wrote:

 > Hi.
 > Again thanks for the help.
 >
 > With regard to the control loop bandwith.  The transfer function
 > (output volts vs duty) i have got from state space averaging is: Vo/d
 > = Vi / ( LC (s^2 + (s/RC) + (1/LC))).  So the undamped natural
 > frequency fn = 1/(2pi * root(LC)).  Am I right in thinking that I
 > size L and C to give me the required controll loop bandwidth?  If
 > so then the inductor/capacitor size seems rather large for a 10Hz
 > bandwidth.  The biggest inductor i can find (RS/Farnell) for the max
 > current of 2A is 680uH so I'm a bit limited here without designing my
 > own inductor.
 > Steve Minshull
 >
No!  You size L and C to get the desired ripple and/or transient
response.  The only thing the filter has to do with the control
bandwidth in this case is that you won't have good transient response --
but for charging batteries that's not going to be a big issue.

You'll be limited by the PIC's ability to sample, which will limit your
bandwidth.  I'd choose a sampling rate, then see what I could tune the
PIC up to do.  You will, of course, want to have a PID controller on the
PIC to get good DC accuracy (depending on your exact driver topology you
may only need an integrator, in this case).

See http://www.wescottdesign.com/articles/pidwophd.html  for guidance on
writing the software, if you need it.

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Steve Minshull wrote:

If you want to respond quickly to things like an output short circuit,
you need a fast current control loop, though it can be crude and
sloppy.  The fast possible way to control current is cycle at a time.
   This might be as simple as a proportional only feedback loop that
samples the current once per PWM cycle (at the time you would expect
the current to peak, just as the PWM output is turning off) and alters
the PWM duty cycle once per PWM cycle, to correct the current on the
next cycle toward the current setpoint.  This gives you the ability to
react to over current conditions before any damage is done, even on a
short.  The output filter inductor is the only filtering component
involved in this process.

   The setpoint for this simple, high speed loop would be the output
of a much slower voltage control loop that probably incorporates both
proportional and integral effects, but in the case of a battery
charger may have a droop effect that allows the integrator to be
satisfied with a voltage somewhat less than the voltage setpoint, and
the amount of this "somewhat" is proportional to the output current.
The tuning of the voltage loop gain and integral time sets its
response to a voltage error, since the battery is the main filter
component that smooths the voltage.  The voltage measurement should be
low pass filtered enough that whenever it is sampled, the current
ripple from the PWM filter inductor is averaged out over at least a
few ripple cycles.  I doubt you need an output filter capacitor,
unless you want the regulator to be stable with no battery connected.