Re: Question about frequency response techniques in MIMO systems

Dear Kerstin Lindén,

Thank you for your email with helpful comments.

At 11:35 21.08.2002 +0200, you wrote:
>Hej
>My name is Kerstin Lindén and I am a supervisor for a Diploma work
>concerning
>analysing MIMO-systems.
>
>I would really appreciate if your could answer my question about frequency
>response techniques in MIMO systems:
>
>We are planning to obtain the transfer function matrix for a system with 2
>inputs and 2 outputs, at each frequency,
>by using frequency response techniques. In this way, among other things, we
>can plot det(I+L(jw)).
>.
>I have studied your book "Multivariable feedback control" especially on
>page 68-69, and
>I get a bit confused. Here you talk about "...by simultaneously applying
>sinusoidal signals of
>the same frequency w to all input channels."
>
>Is this only for the analytical derivation of the expression (3.19)? Or
>should this be practically used
>when obtaining the different elements in the transfer function matrix?

This is for the interpretation of (3.19).
It should not be used for obtaining transfer function elements - for this
we normally obtain a state space model and then
obtain G(jw) = C (jwA-I)^-1 B + D (numerically, using a Matlab command,
e.g. frsp in the Mu toolbox).

>Before we saw this in your book, we were planning to obtain the different
>elements in the
>2x2 matrix by applying a sinusoidal signal to one input at a time, and
>measuring the frequency response at
>each output, with the different matrix elements defined as:
>gij(s)=yi(s)/uj(s) with all uk=0 for k not equal to j (here written in
>Laplace form).

Yes, this is correct and gives the same result (because of the
superposition principle), although I recommend computing it from the
A, B, C, D - matrices as mentioned above.

If you are talking about experimental data, then one often obtains data
from e.g. PRBS-inputs, and from
this obtains a model (A,B,C,D-matrices), and from this again obtains the
frequency response.
It is possible to obtain the frequency response directly as you outline -
this was common in the 60's - but few people do that today
because of the large effort involved.

>Is this a correct way of doing it, or should we disturb all inputs at the
>same time?!?
>
>Thank you for a well written book about MIMO-systems! To me the book has
>been a great introduktion to MIMO systems.
>I would also like to thank you for the summary about errors and misprints
>on you homepage. I think I also may have found
>some less serious misprints: p.23 Equation y=... where I think K should be
>replaced by Kr

I think this one is already on the list of misprints

>and figure 4.8 on page 148,
>where I think
>1+det(L(jw)) should be replaced by det(I+L(jw)).

Yes, this is an error.

Thank you.

-Sigurd Skogestad

>Best wishes,
>/Kerstin Lindén
>ABB University, Ludvika Sweden
>
>tel. +46 240 784205, fax. +46 240 148 14
>
>kerstin.linden@se.abb.com
>http://www.abb.se/tduniversity
>----------------------------------------------------

**********************************************************************************************
Sigurd Skogestad, Professor and Head of Department Phone: +47-7359-4154
Department of Chemical Engineering Home: +47-7393-6855
Norwegian Univ. of Science and Technology (NTNU) Fax: +47-7359-4080
N-7491 Trondheim email:
skoge@chemeng.ntnu.no
Norway
http://www.chemeng.ntnu.no/~skoge
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Received on Wed Aug 21 12:47:44 2002