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Math Help - control systems - ODE confusion

  1. #1
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    control systems - ODE confusion

    Hi I'm confused about this past exam paper question:

    Suppose that is the solution to the initial value problem




    where is the state transition matrix.

    Find for the following



    I've done:



    So






    But since the left side is with respect to x_1 and not x_2 I'm not sure I can do this, and not sure where to go from here?

    Thanks in advance
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  2. #2
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    You have x1'= -x1- 4x2 and x2'= -x1- x2. From the first equation, 4x2= -x1'- x1. If you differentiate the first equation again, you get x1"= -x1'- 4x2'. From the second equation, x2'= -x1- x2 so that is x1"= -x1'- 4(-x1- x2)= -x1'- 4x1+ 4x2= -x1'- 4x1+ (-x1'- x1)= -2x1'- 5x1.

    Can you solve the single equation x1"= -2x1'- 5x1 which is the same as x1"+ 2x1'+ 5x1= 0? If so then you can find x2 from 4x2= -x1'- 1.

    Another way to do this problem, more in the "matrix" spirit, would be to find the eigenvalues and eigenvectors of \begin{bmatrix}-1 & -4 \\ -1 & -1\end{bmatrix}
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  3. #3
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    Thank you so much that post was really helpful.

    I'm having trouble with the same question but for matrix:



    I've done:



    So







    which gives rise to:





    However I can't see anyway of finding out what x_3 is, am I missing something?!

    Thanks again.
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