Directions in MIMO systems

Dear Mr. Skogestad,

my name is Matthias Heller and I'm research assistant and supervisor for
mechanics, flight dynamics and flight control at the Technical
University of Munich.

>From a collegue of Linkoepings University (Sweden) we got a reference on
your book 'Multivariable Feedback Control' and now we are using it
extensively to get an intruduction to robust control techniques. I
think it is an excellent book and I will recommend it to our students
for learning multivariable control.

Since as yet, we didn't perform lectures on robust/multivariable flight
control (mainly we teached the well-know SISO approachs, see McRuer,
...) at our university and in our research we started to introduce
robust techniques only for a few years.
So, we as aircraft and not mainly control engineers are a kind of
beginners in robust control (and primary users of methods) and hope to
find more basic understanding from your book.

Therefore, I would like to give you some comments/questions:

1.) On page 78, MIMO bandwidth, you call the direction associated with
sigma_max(S) the worst case or low gain direction. On page 74 you call
the same direction 'high gain direction'. Similary, on page 78 you call
the direction associated with sigma_min(S) the best case or high gain
direction, and on page 74 this direction is called low gain direction!
Is this a misunderstanding from me or ?

2.) Directions of MIMO-Systems

You introduce input and output directions in form of input/output
singular vectors
(Chapter 3) and in chapter 4 you introduce pole & zero directions.

We got a little bit confused because in several literature we found
several, non-explained definitions of directions!

For example, in Stevens/Lewis: Aircraft Control and Simulation is given:

 C*vi = Output direction where where vi = right eigenvector for
eigenvalue lambda_i, C= measurement matrix

and additionally, when the closed loop state space equations under
output feedback are

x' = (A-BKC)x + B*r, r= reference signal

they call z_i = KC*vi the Input directions where K=Controller Matrix and
vi =
right eigenvector for eigenvalue Lambda_i, C= measurement matrix.

The definition of output direction is the same as the pole output
direction in your book (I believe) but the input direction makes us a
little bit confused.

And otherwise in some texts

wi'*B = input direction where wi=left eigenvector, B= Input Matrix which
is in your book called: pole Input direction.

and last but not least: r_i = wi'*B*K = is called output direction in
some texts, where wi=left eigenvector for Lambda_i,
B = Input Matrix.

So we are confused about directions and would like to ask you:
Is there any literature where the definition of directions is detailed
explained? In your reference of Chapter 4: Kailath, we didn't found
information about!
Whats the importance of this several directions defined by several
texts!

We would be very pleased for a short answer and wish your and your staff

 a Merry Christmas and a Happy New Year!

best regards

Matthias Heller

and the rest of our institute!

-- 
DIPL.-ING. MATTHIAS HELLER
LEHRSTUHL FUER FLUGMECHANIK UND FLUGREGELUNG
DER TECHNISCHEN UNIVERSITAET MUENCHEN
85747 Garching, Germany | Email: hellpc@lfm.mw.tu-muenchen.de 
Phone: +49 89 289 16065