Krdytkyu

Which of the following discrete-time state-space models $(A,B,C,D)$ of the form

$x(t+1)=Ax(t)+Bu(t), quad y(t)=Cx(t)+Du(t), quad tin mathbb{N}$

with $A$ in jordan form has its impulse response given by

$h(t)=left{
begin{array}{ll}
delta(t-1)+2^{t-1} text{for} t = 1,2,3,...\
0 text{for} t = 0,\
end{array}
right.$
$quad$ where $delta = 0^t = left{
begin{array}{ll}
1 text{for} t = 0\
0 text{for} t = 1,2,3,...\
end{array}
right.$

Answer:

$A) quad left[ begin{array}{c|c} A&B \ hline C&D end{array} right] = left[ begin{array}{cc|c} 2&0 &1 \ 0&0&1 \ hline 1&1&0 end{array} right]$

$B) quad left[ begin{array}{c|c} A&B \ hline C&D end{array} right] = left[ begin{array}{cc|c} 1&1&1 \ 1&1&0 \ hline 2&0&0 end{array} right]$

$C) quad left[ begin{array}{c|c} A&B \ hline C&D end{array} right] = left[ begin{array}{c|c} 2&1 \ hline 1&0 end{array} right]$

$D) quad left[ begin{array}{c|c} A&B \ hline C&D end{array} right] = left[ begin{array}{cc|c} 2&0&0 \ 0&0&1 \ hline 1&0&0 end{array} right]$

$E) quad text{None of the above}$

I know that the DT impulse response output equals: $y(t) = CA^{t-1}B$

so $h(t) = CA^{t-1}B$.

Entering a few values of $t$ into $h(t)$
gives: $h(t) = 1 text{for} t = 1, $ $ h(t) = 2 text{for} t = 2, $ $h(t) = 4 text{for} t = 3, $ All answers fit this patern.

I know that $2^{t-1}$ means that the jordan form has at least a $2$ in it so $B$ is incorrect.

$A$ is the right answer, but I don't know how to get there.