# How do I check controllability and observability using Gramian matrixes?

I have a pending exam and this is one of the must-know questions. It will be about checking controllability/observability of a simple MIMO linear system, using Controlability Gramian/Observability Gramian.

More precisely, the system will be given in state-space representation, with the system matrix A of size 3x3.

Searching around, both Gramian matrixes are well defined. However, I am unable to solve the integral equation.

• Do I need to solve the integral equation from the definition?

EDIT:

Here is a very nice tutorial with an example. I still don't understand the final substitution for t in the first example.

I understand that a gramian matrix of full rank means the rows or columns of the matrix are linearly independent. But what has that to do with controllability? * If so, how do I compute exp(A) on paper for A being a 3x3 matrix? * From this example, I conclude that the definite integral of a matrix is computed element-wise. Is that so?

I know those are a bunch of silly question. What I am looking for is just a tutorial or a sample calculation

• By the evening i will have :( failed the exam. As the exam varians are rather consistent, I will post the exact problem, that was given to me. Hopefully then it will be easier to answer. – Vorac Feb 8 '13 at 8:38
• I need the answer to this also, namely, multiple input single output. how is the test for controllability changed? it seems checking the rank of [B AB A^B ...] wont work anymore because the resulting matrix is not square. – user7044 Nov 22 '13 at 6:19