Newbie question: MIMO transfer functions

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z's Avatar (by Gravatar)z posted on
August 1, 2006, 6:21 pm
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Hi,

I have a transfer function for a 2x2 MIMO problem, of the form G=[G11
G12; G21 G22], consisting of first-order lags with time delay.  I also
have two PI controllers that are supposed to control this system.  I
would like to plot the performance of these controllers, but I only
know how to work with ODEs in the time domain.

I know how to convert a SISO transfer function into an ODE, but after
much frustration I'm stuck on the MIMO case.  I am aware of this
equation:

   Y1(s) = G11(s)R1(s) + G12(s)R2(s)
   Y2(s) = G21(s)R1(s) + G22(s)R2(s)

How do I convert that into:

  dy1/dt = f(u1,u2)
  dy2/dt = f(u1,u2)

where u1 and u2 are the outputs of the PI controllers?

Or is there some other way of plotting y1 and y2 as a function of time?

Many heart-felt thanks.


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Tim Wescott's Avatar (by Gravatar)Tim Wescott posted on
August 1, 2006, 8:41 pm
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Re: Newbie question: MIMO transfer functions



z wrote:

I cannot do the answer justice in a single post.  Try:

"Linear Systems", Thomas Kailath, Prentice-Hall, 1980.

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Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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z's Avatar (by Gravatar)z posted on
August 4, 2006, 12:09 am
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Re: Newbie question: MIMO transfer functions



I think I asked the question in a way that was more complicated than
necessary.  Let me start over:

I have transfer functions (first-order lags with time delay) for a 2x2
MIMO system.  I also have the parameters for 2 PI controllers, and I
know the right pairing.  How do I get from there to being able to
generate one of those plots that show how the controllers respond to
unit step changes in setpoints?  

Many thanks.


z's Avatar (by Gravatar)z posted on
August 4, 2006, 1:35 pm
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Re: Newbie question: MIMO transfer functions



Believe me I've been hitting the books.  They explain everything well
for SISO systems, and I've got that all figured out, but then at MIMO
they all leave me hanging.   Can you please verify one thing for me?
Again, a 2x2 MIMO system with first order lags [Gij = Kpij /
(Tpij*s+1)], is the following equation accurate:

     y1dot  = (1/Tp11)*(Kp11*u1 - y1) + (1/Tp12)*(Kp12*u2 - y1)
     y2dot  = (1/Tp21)*(Kp21*u1 - y2) + (1/Tp22)*(Kp22*u2 - y2)

If true, then I'm all set. If not, then I'm about ready to give up.

Thank you for your help.


RRogers wrote:


NASSER ALHATMI's Avatar (by Gravatar)NASSER ALHATMI posted on
August 4, 2006, 7:05 pm
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Re: Newbie question: MIMO transfer functions



Hi Mate:
If you know a program called MATLAB/SIMULINK then I will be able to help
you.
All the best
ASSad


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RRogers's Avatar (by Gravatar)RRogers posted on
August 7, 2006, 9:26 am
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Re: Newbie question: MIMO transfer functions



Yes your answer is right.  Just take the Laplace xform of your y_dot
equations and you will have the Gij equations; with the exception of
the starting transient terms.  The transients die away for any stable
systems, and so are often discarded.
You do know that you xform the MIMO system to a sum of MISO terms using
the characteristic equation  ( I am in a hurry and can't check if that
is the right term)?




z wrote:


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