Reply by robert bristow-johnson●April 25, 20112011-04-25
On Apr 21, 2:56�pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On 04/21/2011 11:35 AM, dhaval_shah wrote:
>
> > What is the role of "zeros" in deciding the stability of the system?
> > why only "poles" are being used to decide the stability of the system?
>
> See http://www.wescottdesign.com/articles/zTransform/z-transforms.html .
>
just to add to Tim's response.
The main mathematical reason that the pole loci exclusively determine
the stability of the system is because of partial fraction expansion.
Note Eqs (17) and (20) in Tim's document. Even though it's X(z), not
H(z) (the transfer function), the principle is the same. The factors
in the denominator of X(z) in Eq (17) are the same as the little
denominators in the partial fraction expanded form of Eq (20). There
is a double term (a double pole if this were H(z)), so maybe it can be
more simply explained without it.
Consider a system with transfer function (i suggest to view using
Google Groups with a fixed-width font):
(z-q1)(z-q2) ... (z-qM)
H(z) = -------------------------
(z-p1)(z-p2) ... (z-pN)
q1, q2, ... qN are the M zeros. p1, p2, ... pN are the N poles.
Usually we assume that the number of poles is at least as many as the
number of zeros: M <= N, otherwize we can perform a form of long
division and get terms like z and z^2 outside of this big fraction.
the zeros q1, q2, ... qN and poles p1, p2, ... pN should be considered
possibly complex. Now let's also assume (for simplicity) that none of
the poles are identical to any other poles. The above transfer
function can be represented as a sum of partial fractions (instead of
as a product of factors with zeros divided by a product of factors
with poles):
A_1 A_2 A_N
H(z) = -------- + -------- + ... + --------
z - p1 z - p2 z - pN
The denominators of each partial fraction are the same factors as
those in the denominator of the form of H(z) we started with. The
poles are the same in both cases. That makes sense because, if the
two forms are equivalent, when H(z) is evaluated at any pole location,
|H(z)| will blow up. In the first form, it blows up because the
denominator of the whole transfer function goes to zero. in the
second form (with partial fractions), it blows up because one of the
partial fractions blow up (because its denominator goes to zero).
The symbols A_1, A_2, ... A_N are *constants* that have to be just
right so that when all of these partial fractions are summed (and
placed over a common denominator), it adds up to the same H(z) as
before. Now the zeros have some effect in the latter transfer
function (they have to), but they do not affect any partial fractions
denominator, only the numerators.
Then when we perform the inverse Z transform of this transfer
function, H(z), we get the impulse response that looks like
h[n] = u[n]*( A_1*p1^n + A_2*p2^n + ... + A_N*pN^n )
where u[n] is the unit step function. p1^n is an exponential
function. in order for the impulse response to be stable and not blow
up when a unit impulse is applied to the input, then all of the pole
values must have magnitude of less than 1.
|p1| < 1, |p2| < 1, ... |pN| < 1 .
if any of those get as big as 1, the impulse response will never
settle down and decay to zero. if any exceed 1, the exponential
functions will blow up.
that is the fundamental answer to your question. if you're an EE
student, it's something you should learn by your 3rd year and
something similar for Laplace transforms, H(s).
r b-j
Reply by Tim Wescott●April 25, 20112011-04-25
On 04/25/2011 03:27 AM, dhaval_shah wrote:
> I got some idea, poles are considered bcoz real part of the poses are
> placed as a damping factor when u perform inverse Z-Transform.
You're confusing the z transform and the Laplace transform. In the
_Laplace_ transform the real parts of the poles determine how rapidly
they damp out (but are different from the damping factor). In the z
domain it's the magnitude of the pole -- a pole with magnitude greater
than 1 is unstable, one with magnitude equal to one is metastable, and
one with magnitude less than one is stable. The lower the pole
amplitude, the more rapidly it damps out.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by dhaval_shah●April 25, 20112011-04-25
I got some idea, poles are considered bcoz real part of the poses are
placed as a damping factor when u perform inverse Z-Transform.
Is this correct?
Dhaval
Reply by HardySpicer●April 23, 20112011-04-23
On Apr 23, 5:50�pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On 04/22/2011 08:36 PM, HardySpicer wrote:
>
>
>
> > On Apr 22, 9:22 am, Tim Wescott<t...@seemywebsite.com> �wrote:
> >> On 04/21/2011 01:26 PM, HardySpicer wrote:
>
> >>> On Apr 22, 6:56 am, Tim Wescott<t...@seemywebsite.com> � �wrote:
> >>>> On 04/21/2011 11:35 AM, dhaval_shah wrote:
>
> >>>>> What is the role of "zeros" in deciding the stability of the system?
> >>>>> why only "poles" are being used to decide the stability of the system?
>
> >>>> Seehttp://www.wescottdesign.com/articles/zTransform/z-transforms.html.
>
> >>>> --
>
> >>>> Tim Wescott
> >>>> Wescott Design Serviceshttp://www.wescottdesign.com
>
> >>>> Do you need to implement control loops in software?
> >>>> "Applied Control Theory for Embedded Systems" was written for you.
> >>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> >>> That part on PID in your article uses "pure" differentiation. Not so
> >>> good if you have a structural resonance
> >>> to deal with (which you always do). You don't discuss filtering this
> >>> with a pole?
>
> >> I originally wrote that intending to submit it to Embedded Systems
> >> Programming magazine, and was adhering to a 5000 word limit.
>
> >> Were I to re-write it (which it needs), I would include that tidbit --
> >> particularly given that frequency domain design really makes it clear
> >> why you want to band limit your differentiator.
>
> >> --
>
> >> Tim Wescott
> >> Wescott Design Serviceshttp://www.wescottdesign.com
>
> >> Do you need to implement control loops in software?
> >> "Applied Control Theory for Embedded Systems" was written for you.
> >> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> > I woudn't differentiate from dc either as this knocks out an
> > integrator (or part of it).
>
> Only if you cascade the differentiator with the integrator. �If you run
> it in parallel with the integrator and sum the results then that does
> not happen.
>
> > Better to have a differentiator from freuqency f1 to f2 spanning unity
> > gain crossover.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" was written for you.
> See details athttp://www.wescottdesign.com/actfes/actfes.html
Yes of course, that's how I do it. The PID stuff tends to come from
process control whereas the lag-lead compensation
is your traditional motor servo stuff - historically. I am guessing
that in proces control it is more guesswork as you don't have a good
model of the plant. Hence the fiddling with three-term controllers.
Whereas for motors, PLLs etc we can draw a fairly accurate Bode Plot.
Hardy
Reply by Tim Wescott●April 23, 20112011-04-23
On 04/22/2011 08:36 PM, HardySpicer wrote:
> On Apr 22, 9:22 am, Tim Wescott<t...@seemywebsite.com> wrote:
>> On 04/21/2011 01:26 PM, HardySpicer wrote:
>>
>>
>>
>>> On Apr 22, 6:56 am, Tim Wescott<t...@seemywebsite.com> wrote:
>>>> On 04/21/2011 11:35 AM, dhaval_shah wrote:
>>
>>>>> What is the role of "zeros" in deciding the stability of the system?
>>>>> why only "poles" are being used to decide the stability of the system?
>>
>>>> Seehttp://www.wescottdesign.com/articles/zTransform/z-transforms.html.
>>
>>>> --
>>
>>>> Tim Wescott
>>>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>>
>>> That part on PID in your article uses "pure" differentiation. Not so
>>> good if you have a structural resonance
>>> to deal with (which you always do). You don't discuss filtering this
>>> with a pole?
>>
>> I originally wrote that intending to submit it to Embedded Systems
>> Programming magazine, and was adhering to a 5000 word limit.
>>
>> Were I to re-write it (which it needs), I would include that tidbit --
>> particularly given that frequency domain design really makes it clear
>> why you want to band limit your differentiator.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> I woudn't differentiate from dc either as this knocks out an
> integrator (or part of it).
Only if you cascade the differentiator with the integrator. If you run
it in parallel with the integrator and sum the results then that does
not happen.
> Better to have a differentiator from freuqency f1 to f2 spanning unity
> gain crossover.
On Apr 22, 9:22�am, Tim Wescott <t...@seemywebsite.com> wrote:
> On 04/21/2011 01:26 PM, HardySpicer wrote:
>
>
>
> > On Apr 22, 6:56 am, Tim Wescott<t...@seemywebsite.com> �wrote:
> >> On 04/21/2011 11:35 AM, dhaval_shah wrote:
>
> >>> What is the role of "zeros" in deciding the stability of the system?
> >>> why only "poles" are being used to decide the stability of the system?
>
> >> Seehttp://www.wescottdesign.com/articles/zTransform/z-transforms.html.
>
> >> --
>
> >> Tim Wescott
> >> Wescott Design Serviceshttp://www.wescottdesign.com
>
> >> Do you need to implement control loops in software?
> >> "Applied Control Theory for Embedded Systems" was written for you.
> >> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> > That part on PID in your article uses "pure" differentiation. Not so
> > good if you have a structural resonance
> > to deal with (which you always do). You don't discuss filtering this
> > with a pole?
>
> I originally wrote that intending to submit it to Embedded Systems
> Programming magazine, and was adhering to a 5000 word limit.
>
> Were I to re-write it (which it needs), I would include that tidbit --
> particularly given that frequency domain design really makes it clear
> why you want to band limit your differentiator.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" was written for you.
> See details athttp://www.wescottdesign.com/actfes/actfes.html
I woudn't differentiate from dc either as this knocks out an
integrator (or part of it).
Better to have a differentiator from freuqency f1 to f2 spanning unity
gain crossover.
Hardy
Reply by Tim Wescott●April 21, 20112011-04-21
On 04/21/2011 01:26 PM, HardySpicer wrote:
> On Apr 22, 6:56 am, Tim Wescott<t...@seemywebsite.com> wrote:
>> On 04/21/2011 11:35 AM, dhaval_shah wrote:
>>
>>> What is the role of "zeros" in deciding the stability of the system?
>>> why only "poles" are being used to decide the stability of the system?
>>
>> Seehttp://www.wescottdesign.com/articles/zTransform/z-transforms.html.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> That part on PID in your article uses "pure" differentiation. Not so
> good if you have a structural resonance
> to deal with (which you always do). You don't discuss filtering this
> with a pole?
I originally wrote that intending to submit it to Embedded Systems
Programming magazine, and was adhering to a 5000 word limit.
Were I to re-write it (which it needs), I would include that tidbit --
particularly given that frequency domain design really makes it clear
why you want to band limit your differentiator.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by HardySpicer●April 21, 20112011-04-21
On Apr 22, 6:56�am, Tim Wescott <t...@seemywebsite.com> wrote:
That part on PID in your article uses "pure" differentiation. Not so
good if you have a structural resonance
to deal with (which you always do). You don't discuss filtering this
with a pole?
Hardy
Reply by HardySpicer●April 21, 20112011-04-21
On Apr 22, 6:35�am, "dhaval_shah"
<shah.dhavalshah@n_o_s_p_a_m.gmail.com> wrote:
> What is the role of "zeros" in deciding the stability of the system?
> why only "poles" are being used to decide the stability of the system?
>
> Dhaval
Nothing, except if you use negative feedback around your system.
Zeros act like a magnet for the locus eminating from the poles, so if
the zeros are outside the unit circle (open loop) then there is a very
good chance
that at some value of gain the closed-loop poles will find there way
there.
Hardy
Reply by Randy Yates●April 21, 20112011-04-21
On Apr 21, 2:35�pm, "dhaval_shah"
<shah.dhavalshah@n_o_s_p_a_m.gmail.com> wrote:
> What is the role of "zeros" in deciding the stability of the system?
> why only "poles" are being used to decide the stability of the system?
>
> Dhaval
Dhaval,
If N(z) goes to zero, can that ever cause N(z)/D(z) to "blow up"?
--Randy