Multivariable feedback control - coprime factorizations

Dear professor Postlethwaite,

I am currently preparing to lecture a course based on the book by yourself
and Skogestad.

I have a problem with the coprime factorization in section 4.1.5:

First, the coprime factorization is defined such that G=N_rM_r^{-1} and

U_rN_r+V_rM_r = I

with U_r and V_r stable.

Then you define the normalized coprime factorization such that

M^*_rM_r+N^*_rN_r = I

where M^*_r = M^T(-s).

Would not V_r=M^*_r =M^T_r(-s) mean that V_r is unstable for stable M_r?
For your Exercise 4.1, U_r and V_r are unstable.

What is it that I am missing? Is the requirement for stable U_r and V_r
unnecessary? Is the parametrization of all stabilizing controllers
(Eq. 4.91) valid also for unstable U_r and V_r?

Regards,

Morten Hovd

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 Nov 1 14:04:28 2000