Engineering domain: Control Design

## Task Description
Consider the discrete‐time Lurye feedback interconnection composed of the following blocks:

Nonlinear block:
  \[
    w_k = \phi(v_k), 
    \quad
    \phi:\mathbb R\to\mathbb R\text{ satisfies }
    0 \;\le\; \frac{\phi(v_1)-\phi(v_2)}{v_1-v_2}\;\le\;1,
    \;\forall v_1\neq v_2.
  \]
  
Linear time‐invariant block:
  \[
    \begin{cases}
      x_{k+1} = A\,x_k + \alpha\,B\,w_k,\\
      y_k      = C\,x_k + D\,w_k,
    \end{cases}
    \quad
    \alpha>0\text{ multiplies the input matrix }B,
  \]
  with numerical values
  \[
    A = \begin{bmatrix}0.5000 & 0 \\1.0000 & 0\end{bmatrix},
    B = \begin{bmatrix}1 \\0\end{bmatrix},\quad
    C = \begin{bmatrix}2.0000 & 0.9200\end{bmatrix},
    D = [\,0\,].
  \]
  
Interconnection：
  \[
    v_k = d_k - y_k,\qquad
    w_k = \phi(v_k),\qquad
    e_k = w_k.
  \]
  
### Task 
Calculate the $\alpha$ where
\[
    \alpha_{\max}
    = \max\bigl\{\alpha>0 \mid \text{the closed‐loop is absolutely stable for all }\phi\text{ in the sector}\bigr\}.
\]
Please note the properties of the nonlinear operator $\phi$, you are encouraged to use semidefinite programming and bisection.