i need some help with explanation on linear control theory the system is x (k+1)

Question

i need some help with explanation on linear control theory

the system is

x(k+1) = Ax(k) + Bu(k) x(0) = 0

what is x(1) in terms of u(0), x(2) in terms of u(0) and u(1), and what is x(k) in terms of u(0), u(1), and u(k-1)? please help and give a brief explanation. id really like to understand this concept.

also can hou write the answers in terms of a vector build from u(0), u(1), u(k-1) and the matrix build up asM = (B, AB, A2B,. . . , A^kB)? i have issued 1500 points for this so i need some explanation as to why

Explanation / Answer

x(1)=Ax(0)+Bu(0)=Bu(0)

x(2)=ABu(0)+Bu(1)

So we conjecture x(k)=A^(k-1)Bu(0)+A^(k-2)Bu(1)+...+ABu(k-2)+Bu(k-1)

The result is true for k=1 and 2 as we have just seen.

Suppose it is true for k, that is x(k)=A^(k-1)Bu(0)+A^(k-2)Bu(1)+...+ABu(k-2)+Bu(k-1)

Then : Ax(k)+Bu(k) = A^kBu(0)+A^(k-1)Bu(1)+...+ABu(k-1) + Bu(k) which is the formula for x(k+1) as expected. So it is true for k+1.

So by induction we see this is true.

Then M = ( B AB ... A^(k-1)) and u=( u(k-1) u(k-2) ... u(0))^T ( transpose )

Then x(k) = Mu

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