Question: Consider the discrete - time state - space equation: x [ k + 1 ] = [ 1 1 2 0 1 1 0 0

Consider the discrete-time state-space equation:
x[k+1]=[112011001]x[k]+[101]u[k],x[k+1]=\begin{bmatrix}1 & 1 & -2\\0 & 1 & 1\\0 & 0 & 1\end{bmatrix} x[k]+\begin{bmatrix}1\\0\\1\end{bmatrix} u[k],x[k+1]=100110211x[k]+101u[k],y[k]=[200]x[k].y[k]=\begin{bmatrix}2 & 0 & 0\end{bmatrix} x[k].y[k]=[200]x[k].
Find the state feedback gain so that the resulting system has all eigenvalues at z=0z =0z=0. Show that for any initial state, the zero-input response of the feedback system becomes identically zero for k3k \geq 3k3.

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