Steady state error

For unit-step, ramp, parabolic inputs, we can estimate the steady- state errors. But how to estimate the steady-state errors, if I give an input is not
either of them (a random trajectory)? Or is it possible to treat the input as the combination of unit step, ramp, and parabolic inputs and then to analyze the steady-state error?
Actually, the input trajectory is a real time command, and I want to estimate the output response error. It's no sense to do it by steady-state (time -> infinity). Is it correct to estimate errors by integrating the error function with respect to time?
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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
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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

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

20N...
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.
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Tim Wescott
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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.
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Tim Wescott
Control systems and communications consulting
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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
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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
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