Help me solve a linear controls system task

[roofingboom]

Linaer Controls question

i am having trouble with a problem i am trying to work on from a linear controls system class.
Problem statement.
given system x'=Ax and A is a 2 by 2 matrix suppose that
x(0) =[0 1]' x(1)=[1 0]' x(2)=[0 -1]'

find e^A

i have been looking through my controls book and have not found anything yet. the problem looks so simple that i think i am overlooking the problem
if any one could point me in the right direction or give me and hint or answer that would be greatly apperciated!
thanks

 

[joi]

Re: Linaer Controls question

Following is the solution :

For, x' = Ax, x(t-T) = e^{A(t-T)}x(T), for all t>=T.
So,
x(1) = (e^A) x(0) and
x(2) = (e^A) x(1).

So, if (e^A) = M,

[1 0]' = M [0 1]' ----- (1)
and
[0 -1]' = M [1 0]' ----- (2)

By putting M = m[j], it is very easy to solve (1) & (2) above, to get

m[0][0] = 0, m[0][1] = 1, m[1][0] = -1 and m[1][1] = 0.

Thus, the answer is :

1st column of M = e^A = [0 -1]'
2nd column of M = e^A = [1 0]'