Re: PID controlle design with OCST

[A S Hodel]

On Friday, April 9, 2004, at 06:16 AM, [name deleted] wrote:

> Hi. I am a uni student studying Comp. Sys. Eng. And I am
> currently doing a Discrete Systems control subject.
>
> I have an assignment to design a PID controller and optimise it with
> ITAE Steepest Descent minimisation. I am trying to use GNU octave for
> most of my subjects as I can't afford MATLAB (who CAN?) and I really
> like GNU Octave's OCST.
>
> I am having trouble thinking of how to develop a PID controller as
> Octave doesn't seem to support derivatives. I noticed you have some
> understanding (I saw this from searching the Octave forums) about this
> particular problem. I did not quite understand the solution you
> mentioned:
>
> "It's sloppy, but the only way I know to do PID design in the OCST is
> to have both y and ydot (output and its derivative) available in the
> plant model."
>
> I have to admit, I have no idea what you mean by y and ydot. Could you
> spare a few minutes and elaborate a little further on it?

The OCST data structure can only handle causal transfer functions, i.e.,
systems that can be written in the form
        dx/dt = A x + B u
        y = C x + D u
which have corresponding transfer function
        P(s) = C*inv(s*I - A)*B + D
A PID control law does not fit this form.  In point of fact, a "true" 
PID control
law is non-causal - it is impossible to implement a derivative in 
physical
hardware.  One reader once responded to me saying, "But wait, there's
that op-amp circuit ..."  The op-amp circuit is the answer to your 
question:
op-amps have very fast internal dynamics that we usually ignore.  
However,
they're still there.  So, to get a derivative, you will need to 
approximate
K_D s (derivative term) by K_D s/( tau s + 1 ).

For low frequencies (s = j omega, omega small) this will closely 
approximate
a derviative.  For higher frequencies, you get the same problem the 
op-amp
has: poor approximation of a derivative.

A good project for someone to do in the future is to extend the tf and 
zp
data structure forms in the OCST so that they will admit non-proper
transfer functions. (functions that do not admit state space 
realizations).
This change will percolate through nearly the entire OCST, so it's no 
small
task.


A. S. Hodel Dept. ECE, 200 Broun Hall, Auburn University AL 36849-5201
(334) 844-1854/fax(334) 844-1809, 
http://www.eng.auburn.edu/users/hodelas , address@hidden



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