Consider the single input single output linear system

x$\_=$ Ax $+$ bu where A $=$

$0$

@

$210$

$110$

$101$

$1$

A; b $=$

$0$

@

$1$

$1$

$0$

$1$

A; c $=(0$; $0$; $1)$

$($a$)$ Find the characteristic polynomial of A$.$

$($b$)$ Write down the matrices A$0$ and b$0$ in controllable canonical form.

$($c$)$ Find the state feedback gain K$0=($k$01$

; k$02$; k$03$

$)$ which places the poles of the system at $1$;$1$;$1.$

$($Note that K$0$ is in the controllable canonical form coordinates.$)$

$($d$)$ Find the basis fe$1$; e$2$; e$3$g of the controllable canonical form in the undashed coordinates.

$($e$)$ Let the matrix P have columns e$1,$ e$2,$ e$3.$ Write down a formula for the feedback K in the undashed

coordinates in terms of K$0$ and P$.$

If we change the state to dashed coordinates x$0$ with x $=$ Tx$0,$ the system in the dashed coordinates is

then de$$ned by matrices A$0,$ B$0$ C$0$ so that x$\_0=$ A$0$x$0+$ B$0$u and y $=$ C$0$x$0.$

Express A$0,$ B$0,$ C$0$ in terms of A$,$ B$,$ C$,$ and T$.$