Consider the third-order continuous-time LTI system

\[
\begin{aligned}
\dot{\mathbf{x}} & =\mathbf{A x}+\mathbf{B} u \\
y & =\mathbf{C x}
\end{aligned}
\]

\[
\begin{array}{r}
\text { with } \mathbf{A}=\left[\begin{array}{rrr}
0 & 2 & 0 \\
0 & 0 & 3 \\
0 & -8 & -6
\end{array}\right], \mathbf{B}=\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] \text {, and } \mathbf{C}=\left[\begin{array}{lll}
1 & 0 & 0
\end{array}\right] . \text { Using } \\
\mathbf{Q}=\left[\begin{array}{lll}
8 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 4
\end{array}\right], \quad R=1.5
\end{array}
\]

(a) First design a LQ controller for this continuous time-system using the MATLAB function lqr. Let the optimal controller gain vector be \(\mathbf{K}\). Simulate the closed-loop system

\[
\dot{\mathbf{x}}=(\mathbf{A}-\mathbf{B K}) \mathbf{x}
\]

with \(\mathbf{X}(0)=\left[\begin{array}{lll}2 & 0 & -2\end{array}\right]\). Sample the closed-loop output response \(y(t)\) with sampling time \(T=0.2 \mathrm{~s}\).

(b) Next, discretize the continuous-time system directly with sampling time \(T=0.2 \mathrm{~s}\) and design a LQ controller for this discrete-time LTI system with the same \(\mathbf{Q}\) and \(R\) as in part (a). Compare the closed-loop output response with the sampled continuous-time output obtained in part (a).

(c) Increase \(\mathbf{Q}\) and \(R\) by a factor of 10 and repeat both the continuous-time and discrete-time designs. Describe how the respective output responses change.