Controllability and the PBH Test [Control Bootcamp]

This video describes the PBH test for controllability and describes some of the implications for good choices of "B".

These lectures follow Chapter 8 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz

This video was produced at the University of Washington

[ahmedkamiss]

I had to watch it a few times to finally get the joke. Yay! Thank you professor!

[김기현-vy]

Thank you so much for your wonderful explanation, professor!

[DiggaDiggaDug]

Cheap vodka, I'm dying over here... XD prof. letting us know he had a totally normal undergrad experience.

[nikosips]

Thank you very much for your great and intuitive explanations!

[adeeljamal]

A not-so-straightforward topic tackled really well by you! Thumbs up! The importance of repeated eigenvalues, a concept I wasn't taught in my grad studies. I just wanted to clarify one little thing. At 12:04, you said that two eigenvectors that're really really close, what does it signify in the physical system?
Mathematically I understood PBH test but I was just thinking of a simplest system where A = Identity matrix, B=[1;1], then with a single control input variable u, I can influence system in both of the eigenvectors direction set by A. The two equations are essentially the same and decoupled with some input u available as well. Intuitively, I think it should be controllable but it isn't according to PBH criteria because the matrix of PBH test is rank-deficient. I could not fit the takeaways of this test here intuitively when my matrix A has repeated eigenvalues and B has components in all the eigenvector directions set by A, then essentially, all the eigenvector directions could be reached by my control input u or not? What am I missing here? Could you please explain a little? Thanks!

[roboquaternion]

Nice video, Professor Steve. Explanation is clear and easy to understand. You mentioned that you do not use PBH in practice though. Does that imply that you use something else? If so, what?

[kidcasco]

Thank you. Amazing work and incredibly helpful. Cheers!! Great Idea to record using a mirror, I think. The wedding ring gave it away else you'd be some kind of savant dyslexic.

Csn you suggest some linear algebra tutorial or book specifically for controls problem? Just to map linear algebra concepts to control problems.

[TJChang-kx]

A question about point 2: The purpose of adding B is to supplement the left null space of (A-\lambda I) so that the final matrix is full-rank. Then I don't quite understand the saying of point 2 that "B needs to have some components in each eigenvector direction" (Eigenvectors lie in the null space of (A-\lambda I), is there any relation between left null space and null space)?

[Kendric]

Great video, but not going to lie, that marker was killing me lol!

[animeshsinghal]

At 3:45, Are we assuming that the rank of A is n??

[kshitijbithel]

Hi Steve, in the 1st point of test, we need to test only for A-(lambda)I if lambda is not its eigenvalue, but if in this case B is pointing in a direction different than A-(lambda)I, rank may be greater than n, we didn't account for that case . Please correct me if I am wrong.

[AdityaMishra-ppgw]

Can't we say that if the determinant is 0 then the equation dimension is reduced .That means eigen values are reducing the dimension of equation(A- lamda*I) corresponding to their eigen vector directions ?

[bierundkippen]

Point 3 is false if A has eigenvalues with multiplicity larger than one! Example: A = 0 in IR^(2x2).

[NicoCatalano_plus]

Thank you so much for these videos, they are very useful.

I have been asked to show that controllability and observability are not affected by replacing A with (A+αI ).
How can be proved such statement? replacing A with A+αI would change its eigenvalues, so the PBH test can't be exploited.

And I can't think on how to prove that the rank of the Controllability Matrix will not change either.

Do you have any suggestion?

[marshallmykietyshyn]

Hi Steve. If you maximize the projection of B along the least stable directions of the state space, does that result in optimal control (assuming all eigenvalues of A have multiplicity 1 for now)? Or is it possible to set this up as a form of control optimization?
Edit: By optimization here I think I mean minimizing control input energy, but I'm not sure if it's necessary to specify what type of optimization I mean. It seems like it could be generally true.

[kubigaming]

will remember the 'cheap vodka' test lmao

[ekanshsaraf]

Amazing Video! But just one question. Shouldn't we look at left eigenvectors of A?

[eduardodossantos]

Hi Steve it is me here again. My state matrix (A) is 10x10 and my control matrix (B) is 10x6. So when I compute the controllability matrix it has rank 10 . So I belive that my system is controllable, but when I do a numeric simulation in MatLab I got two state (position and velocity) that I can not control. That means, my system is stable but not controllable ? . if I try to increase the number of columns in matrix B It could help my in someway ?

Thanks for attention !

[hbasu]

Regarding the PBH test, I have a bit of confusion over repeated eigenvalues. Lets say A matrix is an identity matrix (order 2). Hence the eigenvalues are repeated. The eigenspace is whole of R^2. I believe if I have an actuation matrix B with only one column (lets say [1, 3]^T), I can still control the system. In that case, the algebraic multiplicty of the repeated eigenvalues are not the same as the no. of columns of B. Can you please clarify this case in comparison to your statement at 11.51? Thanks so much.