Re: RGA og RHP

Dear Alfonso, Sigurd,

I have to admit that we have been a little unclear here. However, if you
consider the expression for the RGA of an individual element in Eq. 4 of
the paper, things should be a little more clear. Since the RGA at infinite
frequency is finite and different from 0, the numerator and denominator in
Eq. 4 must have the same order. Then, any phase change from 0 to infinity
must be due to different "number of factors" in the RHP for the numerator
and denominator. The plant is assumed to be stable, so only zeros can be
in the RHP. Thus, the theorem applies to:

zeros of g_ij (which is the same as transmission zeros, since this is a scalar)
transmission zeros of G
transmission zeros of G^ij

That is, individual elements of G, with the sole exception of g_ij, can
have zeros in the RHP. Then, the sign of the RGA will be the same at
fequencies 0 and infinity provided G and G^ij have the same number of RHP
zeros (ideally both have no transmission zeros in the RHP). Transmission
zeros of G and G^ij may be located at the same place in the complex plane
as the zeros of the individual elements, but they may just as well be
located elswhere.

I hope this is sufficient clarification

Sincerely,

Morten Hovd

>>From Mchpias2@fs4.umist.ac.uk Tue Sep 12 18:05:07 2000
>From: "Alfonso Serrano" <Alfonso.Serrano-Duque@stud.umist.ac.uk>
>Organization: UMIST
>To: skoge@chembio.ntnu.no
>Date: Tue, 12 Sep 2000 17:04:29 GMT
>
>Dear Professor Skogestad:
>
>I am completing my MSc dissertation in the area of RGA and I
>would be grateful if you could give me an acclaration about the
>theorem wich relates RGA and RHP zeros, which I took from your
>paper in Automatica "Simple frequency-dependent tools for control
>system analysis, structure and design".
>
>According to the paper, this theorem applies to "transfer matrix
>with stable elements and no zeros or poles at s=0".
>
>Are these zeros transfer zeros, zeros of individual transfer functions
>or both?. In other words, will be the theorem applicable for a
>system with zeros at the origin for some individual transfer
>functions, but no transmission zeros in the origin?
>
>Thank you very much in advance.
>
>---------------------------------------------------
>Alfonso Serrano Duque.
>term address: Lambert and Fairfield Hall. Room A6-3.
> Granby Row. M60 7LH. Manchester. UK
> 00 44 1619333492
>home: Reyes Católicos 14-6B. 09005 Burgos SPAIN
> 00 34 947217483
> serranoduque@hotmail.com

Morten Hovd Phone: +4773591426
Professor of Process Control Fax: +4773594399
Department of Engineering Cybernetics
Norwegian University of Technology and Science
N-7030 Trondheim, Norway e-mail: morten.hovd@itk.ntnu.no
Received on Wed Sep 13 09:01:36 2000