How Do You Determine Matrix A from Phi(t) in Linear Control Theory?

[aznkid310]

Homework Statement

So the first part of the problem asks us to solve for the state transition matrix, which I found to be:

Phi(t) = [2 0 0; (2-e^-t) e^-t 0; 0 0 e^-2t];
I need to find the matrix A, which I assume can be done with the following relation:

Phit(t) = e^(At)

where A is also a matrix

Homework Equations

Need to solve for A. Not sure how to solve this and what properties hold when matrices are concerned.

The Attempt at a Solution

I initially thought to diagonlize Phi(t) but didnt really get anywhere.

 

[Mark44]

Homework Statement

So the first part of the problem asks us to solve for the state transition matrix, which I found to be:

Phi(t) = [2 0 0; (2-e^-t) e^-t 0; 0 0 e^-2t];
I need to find the matrix A, which I assume can be done with the following relation:

Phit(t) = e^(At)

where A is also a matrix

Homework Equations

Need to solve for A. Not sure how to solve this and what properties hold when matrices are concerned.

The Attempt at a Solution

I initially thought to diagonlize Phi(t) but didnt really get anywhere.

If Phi(t) = e^At, then ln(Phi(t)) = At.

See this wikipedia article on the logarithm of a matrix.

 

[aznkid310]

Thanks for the reply Mark44. How do we deal with the ln(0) terms?

 

[Mark44]

The log of a matrix is not just the log of the entries in the matrix. Is that what you're trying to do?

The link I provided shows some examples of finding the log of a matrix.

 

[aznkid310]

Yes that's what I thought. Reading a little furthur, it seems that I should find the eigenvectors of phi(t), then do: ln(phi(t)) = V*ln(inv(V)*phi(t)*V)*inv(V), where V is the matrix of eigenvectors. But phi(t) is not diagonalizable, so I don't know how to proceed from here.