Engineering domain: Control Design

## Task Description
Consider the following continuous‑time uncertain system:
\[
\dot{x}(t) = \bigl(A_0 + \Delta A\bigr)\,x(t) + B_1\,w(t) + B_2\,u(t),
\]
\[
z(t) = C_1\,x(t) + D_{11}\,w(t) + D_{12}\,u(t),
\]
\[
y(t) = C_2\,x(t) + D_{21}\,w(t) + D_{22}\,u(t),
\]
where \(x\in\mathbb{R}^4\), \(u\in\mathbb{R}\), \(w\in\mathbb{R}\), and \(z,y\in\mathbb{R}\). The nominal matrices are
\[
A_0 = \begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
-5 & -4 & -3 & -2
\end{bmatrix},\quad
B_1 = \begin{bmatrix}0\\0\\0\\1\end{bmatrix},\quad
B_2 = \begin{bmatrix}0\\0\\1\\0\end{bmatrix},
\]
\[
C_1 = \begin{bmatrix}1 & 0 & 0 & 0\end{bmatrix},\quad
D_{11} = 0,\quad
D_{12} = 0,
\]
\[
C_2 = \begin{bmatrix}0 & 0 & 1 & 0\end{bmatrix},\quad
D_{21} = 0,\quad
D_{22} = 0.
\]

The uncertainties are
\[
\Delta A \in \mathrm{Co}\{\,0.2\,I_4,\;-0.2\,I_4\}, 
\qquad
w(s) = \Delta_{\mathrm{dyn}}(s)\,z(s),\;\|\Delta_{\mathrm{dyn}}\|_\infty \le 1.
\]

### Task 
By solving a SDP to meet the H-infinity performance, design a static state‑feedback gain \(K\in\mathbb{R}^{1\times 4}\) such that the closed‑loop system satisfies:

1. Robust stability: for all \(\Delta A \in \mathrm{Co}\{\,\pm0.2\,I_4\}\) and any \(\Delta_{\mathrm{dyn}}\) with \(\|\Delta_{\mathrm{dyn}}\|_\infty \le 1\), the closed‑loop matrix is Hurwitz.

2. \(\mathcal{H}_\infty\) performance: the transfer function \(T_{zw}(s)\) satisfies \(\|T_{zw}\|_\infty < 0.3\).