Engineering domain: Control Design

## Task Description
You are given an aircraft pitch-axis plant model and asked to design a Robust-Servo LQR (RS-LQR) controller. Consider the continuous‐time state–space model
\[
\dot{\bm x}_p = A_p\,\bm x_p + B_p\,u,\quad
y = C_p\,\bm x_p + D_p\,u,
\]
where
\[
\bm x_p = [\alpha,\; q,\; \delta_e,\;\dot\delta_e]^T,\quad
u = \delta_{\mathrm{cmd}},\quad
y = [A_z,\;\alpha,\;q,\;\delta_e,\;\dot\delta_e]^T.
\]
The plant matrices are
\[
A_p = \begin{bmatrix}
-1.3046 & 1.0     & -0.21420 &    0     \\
 47.71  & 0       & -104.83  &    0     \\
 0      & 0       & 0        &  1.0     \\
 0      & 0       & -12769.0 & -135.6
\end{bmatrix},\quad
B_p = [0,\;0,\;0,\;12769.0]^T,
\]
\[
C_p = \begin{bmatrix}
-1156.9 &  0  & -189.95 & 0 \\
     1  &  0  &     0   & 0 \\
     0  &  1  &     0   & 0 \\
     0  &  0  &     1   & 0 \\
     0  &  0  &     0   & 1
\end{bmatrix},\quad
D_p = \mathbf{0}_{5\times1}.
\]
Design a dynamic state–feedback controller
\[
\dot{\bm x}_c = A_c\,\bm x_c + B_{c1}\,y + B_{c2}\,r,\quad
u = C_c\,\bm x_c + D_{c1}\,y + D_{c2}\,r,
\]
such that:
\begin{itemize}
  \item The closed‐loop rise time of \(A_z\) is \(<0.2\) s.
  \item The system is stable.
  \item The gain margin is \(>3\) dB.
  \item The phase margin is \(>30\) degrees.
\end{itemize}

Please output all six controller matrices Ac, Bc1, Bc2, Cc, Dc1, Dc2 that satisfies the above requirements.