Zero-Order Hold in control ystems

On Wed, 15 Feb 2017 05:39:48 -0800, robert bristow-johnson wrote:

> On Monday, February 13, 2017 at 7:41:57 PM UTC-5, gyans...@gmail.com
> wrote:
>> On Tuesday, February 14, 2017 at 8:29:43 AM UTC+13, robert
>> bristow-johnson wrote:
>> > On Monday, February 13, 2017 at 2:27:11 PM UTC-5, robert
>> > bristow-johnson wrote:
>> > > On Sunday, February 12, 2017 at 3:10:54 PM UTC-5,
>> > > gyans...@gmail.com wrote:
>> > > > When you have digital control with feedback we take account of
>> > > > the ZOH in the DAC. This is in all the textbooks and shows a
>> > > > pulse as the impulse response and you get the sinc for the
>> > > > magnitude of it and a linear phase characteristic. You then need
>> > > > to sample about 10 times the max freq of interest so as not to
>> > > > get too much phase lag from this.
>> > > > 
>> > > > However, many systems nowadays have H bridges and no ZOH as such.
>> > > > Of course there will always be a one-step delay to get round a
>> > > > loop so that too has a phase-lag. Are we to assume then that the
>> > > > ZOH is no longer a problem but we use a pure one step delay
>> > > > instead ie
>> > > > 
>> > > > z^-1 = exp(-sT)
>> > > > 
>> > > > where T is the sample interval in sec.
>> > > 
>> > > what'sa "H bridge"?
>> > > 
>> > > 
>> > okay, now i know what an H bridge is.  so is this question about
>> > replacing a DAC with PWM?  is that it?
>> > 
>> > 
>> Yes it is. What happens to the zero order hold equations. We think the
>> phase changes with amplitude and is linear for an H bridge
> 
> okay, so the question is how is a PWM output modeled rather than PAM
> (pulse amplitude modulation).  we know that PAM output with a regular
> DAC makes use of a ZOH.  even for sigma delta, but the upsampled sample
> rate is so high that the effect of ZOH is negligible for the baseband. 
> so what is the sample rate for the PWM and what is the baseband for the
> net control signal?

First, this is just my opinion, but I've been considering this question 
for ages.

If the PWM is feeding something like a coil, that has significant low-
pass effects, then its treatment is best split in two.  First make sure 
from a circuits point of view that your switching amplifier is behaving 
(i.e., the added triangular component to the motor current isn't too 
large, there's no noise or unexpected losses, and the transistors aren't 
going up in smoke).  Then treat the PWM as some mysterious and slight 
phase shift, possibly coupled with a mysterious and slight nonlinearity 
-- in other words, do what engineers do when confronted by a nonlinearity 
that can be swept under the rug.

If the PWM behavior _can't_ be swept under the rug, as, for instance, in 
a switching supply where you're trying to get the bandwidth close to 
1/10th the switching rate, then all bets are off, and you'll need to do a 
full nonlinear analysis of the circuit, or you'll have to simulate and 
hope.  There's usually ways to avoid this situation.

-- 
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work!  See my website if you're interested
http://www.wescottdesign.com

On Monday, February 13, 2017 at 9:40:08 PM UTC-5, Tim Wescott wrote:
> On Mon, 13 Feb 2017 11:29:40 -0800, robert bristow-johnson wrote:
> 
> > On Monday, February 13, 2017 at 2:27:11 PM UTC-5, robert bristow-johnson
> > wrote:
> >> On Sunday, February 12, 2017 at 3:10:54 PM UTC-5, gyans...@gmail.com
> >> wrote:
> >> > When you have digital control with feedback we take account of the
> >> > ZOH in the DAC. This is in all the textbooks and shows a pulse as the
> >> > impulse response and you get the sinc for the magnitude of it and a
> >> > linear phase characteristic. You then need to sample about 10 times
> >> > the max freq of interest so as not to get too much phase lag from
> >> > this.
> >> > 
> >> > However, many systems nowadays have H bridges and no ZOH as such. Of
> >> > course there will always be a one-step delay to get round a loop so
> >> > that too has a phase-lag. Are we to assume then that the ZOH is no
> >> > longer a problem but we use a pure one step delay instead ie
> >> > 
> >> > z^-1 = exp(-sT)
> >> > 
> >> > where T is the sample interval in sec.
> >> 
> >> what'sa "H bridge"?
> >> 
> > 
> > okay, now i know what an H bridge is.  so is this question about
> > replacing a DAC with PWM?  is that it?
> > 
> 
> More like replacing a DAC + power amp with an H-bridge, but yes.
 
well, i know where the ZOH goes with a DAC + power amp.  but my only understanding of an H-bridge is as a means of connecting up and even reversing polarity to the motor.  so, it seems to me that if a constant-voltage power source is used, the only way to do speed control is to apply some form of PWM.  but it can be something more sophisticated than simple rectangular PWM.  it could even be a form of sigma-delta PWM.

but there aren't just two states.  there is the open-circuit motor and the shorted braking motor (i doubt you will slam the thing into reverse directly).  figuring out a duty-cycle and knowing what two states to be switching between is a little more sophisticated than simple PWM.

so if you were switching between these two states: 1. powered in the direction the motor is already turning and 2. open circuit (so the motor coasts), how is that modeled mathematically?  i am not sure.

i think i would know how to model switching between powered and short (braking) but i don't think that would be the best thing for the motor nor the most efficient.

r b-j

On Thu, 16 Feb 2017 10:19:21 -0800, robert bristow-johnson wrote:

> On Monday, February 13, 2017 at 9:40:08 PM UTC-5, Tim Wescott wrote:
>> On Mon, 13 Feb 2017 11:29:40 -0800, robert bristow-johnson wrote:
>> 
>> > On Monday, February 13, 2017 at 2:27:11 PM UTC-5, robert
>> > bristow-johnson wrote:
>> >> On Sunday, February 12, 2017 at 3:10:54 PM UTC-5, gyans...@gmail.com
>> >> wrote:
>> >> > When you have digital control with feedback we take account of the
>> >> > ZOH in the DAC. This is in all the textbooks and shows a pulse as
>> >> > the impulse response and you get the sinc for the magnitude of it
>> >> > and a linear phase characteristic. You then need to sample about
>> >> > 10 times the max freq of interest so as not to get too much phase
>> >> > lag from this.
>> >> > 
>> >> > However, many systems nowadays have H bridges and no ZOH as such.
>> >> > Of course there will always be a one-step delay to get round a
>> >> > loop so that too has a phase-lag. Are we to assume then that the
>> >> > ZOH is no longer a problem but we use a pure one step delay
>> >> > instead ie
>> >> > 
>> >> > z^-1 = exp(-sT)
>> >> > 
>> >> > where T is the sample interval in sec.
>> >> 
>> >> what'sa "H bridge"?
>> >> 
>> >> 
>> > okay, now i know what an H bridge is.  so is this question about
>> > replacing a DAC with PWM?  is that it?
>> > 
>> > 
>> More like replacing a DAC + power amp with an H-bridge, but yes.
>  
> well, i know where the ZOH goes with a DAC + power amp.  but my only
> understanding of an H-bridge is as a means of connecting up and even
> reversing polarity to the motor.  so, it seems to me that if a
> constant-voltage power source is used, the only way to do speed control
> is to apply some form of PWM.  but it can be something more
> sophisticated than simple rectangular PWM.  it could even be a form of
> sigma-delta PWM.

Correct.  I should have said H-bridge + PWM

> but there aren't just two states.  there is the open-circuit motor and
> the shorted braking motor (i doubt you will slam the thing into reverse
> directly).  figuring out a duty-cycle and knowing what two states to be
> switching between is a little more sophisticated than simple PWM.
> 
> so if you were switching between these two states: 1. powered in the
> direction the motor is already turning and 2. open circuit (so the motor
> coasts), how is that modeled mathematically?  i am not sure.
> 
> i think i would know how to model switching between powered and short
> (braking) but i don't think that would be the best thing for the motor
> nor the most efficient.

You're thinking slow PWM.

The OP and I should have maybe said H-bridge + PWM + motor inductance.  
The basic idea is that you build a switching amplifier using the motor's 
armature coil winding as the inductance.  So you run the thing so that 
the switches are almost always running one way or another, with a PWM 
rate that's fast enough that the sawtooth at zero speed isn't too much of 
a power drain.  The average voltage on the motor coils is then (mostly) a 
function of duty cycle and amplifier terminal voltage.

I have, actually, run this sort of fast PWM for efficient motor drive 
coupled with slower pulses to deal with friction -- but that's another 
story.

-- 
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work!  See my website if you're interested
http://www.wescottdesign.com