Suppose we have the following discrete time system in state space form:
\begin{align*} x[k+1] = \begin{bmatrix} 0 & a_k\\ 1 & 2 \end{bmatrix} x[k] + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u[k] \end{align*}
where $a_k = \begin{cases} -1,& \text{if } \text{mod(k,2) = 0}\\ 0, & \text{if } \text{mod(k,2) = 1} \end{cases} $
My aim is to design a feedback law in the form of \begin{equation} u[k] = -K[k]x[k] \end{equation} which makes the above system LTI and the system behavior dead-beat.
Here is my attempt: \begin{align*} x[k+1] &= \begin{bmatrix} 0 & a_k\\ 1 & 2 \end{bmatrix} x[k] - \begin{bmatrix} 1 \\ 0 \end{bmatrix} \begin{bmatrix} K_1[k] & K_2[k] \end{bmatrix} x[k] \\ &= \underbrace{\begin{bmatrix} -K_1[k] & a_k - K_2[k]\\ 1 & 2 \end{bmatrix}}_{F} x[k] \end{align*} To make the system LTI, we need to have constant coefficient matrix. Hence choose \begin{align*} \begin{bmatrix} K_1 \\ K_2 \end{bmatrix} = \begin{bmatrix} K_1 \\ a_k \end{bmatrix} \end{align*} Furthermore, for the dead-beat response, the eigenvalues should be at zero. So, \begin{align*} |zI - F| = \begin{bmatrix} z + K_1 & 0\\ -1 & z-2 \end{bmatrix} = z^2 \end{align*} which seems impossible. Any idea for such a design?
Since your system is controllable, you can place the eigenvalues wherever you want. The issue with your design is that you set $K_2(k)=a_k$, which leads to an infeasible problem. What about letting $K_1(k)=2$ and $K_2(k)=4+a_k$?