For the linear system

\[\begin{gathered}\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B u} \\\mathbf{y}=\mathbf{C x}+\mathbf{D u}, \\\mathbf{A}=\left[\begin{array}{lll}0 & 0 & -1 \\1 & 0 & -3 \\0 & 1 & -3\end{array}ight], \quad \mathbf{B}=\left[\begin{array}{l}0 \\0 \\1\end{array}ight], \quad \mathbf{C}=\left[\begin{array}{lll}0 & 1 & 0 \\0 & 0 & 1\end{array}ight], \quad \mathbf{D}=[0] .\end{gathered}\]

(a) Design a full-order state estimator for the system.

(b) Find an optimal control that minimises the cost function

\[=\int_{0}^{\infty}\left(\mathbf{X}^{T} \mathbf{Q X}+\mathbf{u}^{T} \mathbf{R u}ight) d t\]

where

\[\mathbf{R}=\mathbf{I}, \mathbf{Q}=\left[\begin{array}{lll}3 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 1\end{array}ight]\]