# Newbie question: MIMO transfer functions

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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z wrote:

I cannot do the answer justice in a single post. Try:
"Linear Systems", Thomas Kailath, Prentice-Hall, 1980.
--

Tim Wescott
Wescott Design Services
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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.
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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.
RRogers wrote:

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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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