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?
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