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Linear Control Theory: How to solve this matrix

  1. Oct 20, 2011 #1
    1. The problem statement, all variables and given/known data
    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


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


    3. The attempt at a solution

    I initially thought to diagonlize Phi(t) but didnt really get anywhere.
     
    Last edited: Oct 20, 2011
  2. jcsd
  3. Oct 22, 2011 #2

    Mark44

    Staff: Mentor

    If Phi(t) = eAt, then ln(Phi(t)) = At.

    See this wikipedia article on the logarithm of a matrix.
     
  4. Oct 22, 2011 #3
    Thanks for the reply Mark44. How do we deal with the ln(0) terms?
     
  5. Oct 22, 2011 #4

    Mark44

    Staff: Mentor

    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.
     
  6. Oct 23, 2011 #5
    Yes thats 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 dont know how to proceed from here.
     
    Last edited: Oct 23, 2011
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