Steady state error

Why would you expect the error to reach a steady state with a random
input.  One can do simulations to estimate the errors as a function of
time and random input but they will not be steady state


That wouldn't be random then, it would be a super-imposed input.  One
can combine the different functions and use that as a forcing
function.  Some engineering software packages can convert Laplace
transform to a response in the time domain but not if the forcing
function is too complicated.


You mean like this?
ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20T0C1-PID%20CTM%20NG.pdf
In this case I made simple step changes in the target position r but I
could have made the target position any profile I want.
If you look through this group you will find I have posted links to
many similar .pdf files.


It isn't clear what you mean.

Peter Nachtwey

I just like to give an input as a bell-shape function, instead of
step, ramp, and parabolic functions.
And then see what the steady state error is. I think the superposition
method is a good way to
evaluate the error function. EX: I can decompose the bell-shape
function as the combination
of step, ramp, or parabolic functions.

In fact, I want to see the transient error, which means t is not
infinity (steady-state error). The only
way is to integrate the error function, instead of using the final
value theorem. But it becomes
more complicated when I use integration to calculate the error.

Am I correct?


On 4$B7n(B9$BF|(B, $B2<8a(B8$B;~(B32$BJ,(B, pnachtwey

this?ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20T0C1-PID%20CTM%20N ...

On Thu, 10 Apr 2008 08:40:48 -0700, sofin wrote:
(top posting fixed)

I'm not sure what you're getting at.  If you really want to know the
error at infinity, and your input settles to some constant value, then
unless the error integrates you can use the final value theorem.


--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

On Wed, 09 Apr 2008 11:04:41 -0700, sofin wrote:


One can estimate the steady-state part of the error, but if you're
interested in how well the system tracks the input that's probably not
what you want.


It is possible, but won't result in a very good fit.  You _can_ estimate
the on-going error for a random input if you know the spectral
characteristics of the input.


It is correct to estimate errors by figuring out what is important to
you, and building your cost function accordingly.  The common measure
given an ever-changing input is the RMS error over time, probably because
it fits well with frequency-domain analysis.  If RMS error (or mean
squared error) is what's important to you, then spectral analysis will do
well.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

sofin wrote:

How would you define the steady-state error for a random trajectory?
Computing it comes later.


Can you do a simulation with a particular input? That would tell you
what the error is, but the sticking point is "steady state".

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

Transient, or start up, analysis pretty much requires a time domain
simulation.  For Steady-State analysis there are a couple of tools.

I do a frequency domain analysis that pretty simple in something like
MATLAB.  If you input can be represented with a PSD then you multiply
the PSD by the square of the magnitude of your system transfer
function.  (This is done at discrete frequencies for both magnitude
and transfer function.)  For a typical closed loop system the transfer
function of interest is the disturbance rejection.  You would multiply
the input PSD by the magnitude squared of the disturbance rejection.
The result is the PSD of your system residual error.

Caveats:
1)  This only account for linear behavior.  Start up, transient,
behavior is often nonlinear as filters, etc. are impulsed at system
start up and need time to settle before they behave linearly.  Systems
with nonlinearities need to be simulated in the time domain or certain
linearizing assumptions must be made and understood.
2)  The system must be stationary and ergodic.  (I will eventually add
a section on stochastics to my wiki - http://wikis.ControlTheoryPro.com .)
Typically if your system has an integrator applied to a random,
stationary process - such as random noise - then the output of the
integrator needs to be checked.  It is fairly likely that the output
of the integrator is not stationary.
3)  This can be a pretty miserable analysis to do without the aid of
something like MATLAB or MATHCAD.

Gabe
I've started a new wikipedia style site for control systems
  http://wikis.ControlTheoryPro.com
Forum for controls - without the spam
  http://forums.ControlTheoryPro.com