In one of my modern control's class, the prof said we should be able to
prove (as a possible midterm/final problem):
Phi = M L M^-1
Where Phi = state transition matrix
M = modal matrix
L = diagonal matrix of Eigen values
M^-1= inverse of M
I really have no idea where to start and am hoping someone here can help.
midterms are coming up soon and I want to be prepared.
Is this not in your book? It's in mine from nearly 20 years ago:
"Linear Systems" by Kailath. Look in your table of contents and your
index for "Modal Decomposition". If you don't find it, see if your
library has Kailath's book.
It may not be a bad idea to bring it up with your prof -- sensible
teachers know that if there's one person asking a question, there's
probably ten more who are too shy to speak up.
You really should have covered this already in a linear algebra class.
The chapter will be called eigensystems or some such. The idea
A*[v1 v2 v3]=[v1 v2 v3] *diag([lambda1,lambda2,lambda3])
What is this used for besides passing a test? Some how I missed all of this
when I was in college.
I thought you might know since you seem to be mathematically inclined.. I
remember you helped me show the IAE tuning coefficient were in need of
I would be an annoying student now. I would always be asking why bother?
Because it is on a test is not enough. The answer may help Bo too.
I had been thinking that if I ever get my seminar business off the ground
I would invite you to a local one just to hear you heckle.
Now I'm not so sure I wouldn't tie you up and leave you in a corner half
way through. :-).
The most important part of the modal decomposition for me was the insight
in how one _could_ if one wanted to, separate the system such that there
was one integrator per mode. This isn't something that's practically
useful, but much like knowing that I-84 will take you farther than Boise
is useful even if that's only as far as you ever go, it's nice to have
that modal decomposition model in the back of your head when you do work
in state space.
I also strongly suspect that if you ever get it piled higher and deeper
you'll find some branch of mathematics where the modal decomposition will
come in extremely handy. Fer instance, as I was writing the last sentence
I realized that you could probably use modal decomposition to do a
numerically robust "partial fraction" decomposition on a system expressed
in state-space using modal decomposition, avoiding the inevitable problems
with finding the roots of large polynomials in the process.
Linear systems covers a huge swath of the field of engineering--many
disciplines such as control systems, filter design, mechanical
vibrations, etc. The question implicitly dealt with how one studies
the inner works of such systems, by decomposing them into their
essential characteristics (i.e. the eigenvectors/eigenvalues of the
system). For example, in the field of vibrations, the eigenvectors
represent the mode shapes of the vibration and the eigenvalues
represent the resonant frequencies of those modes. To study these
fields of engineering you need a language to describe how systems
work, and that language is linear algebra.