Exercise 6.4#
Control design ∙ State feedback
Consider the state-space system
(31)#\[\begin{split}
\begin{dcases}
\dot{x} = \begin{bmatrix} -1 & -1 \\ 1 & -1 \end{bmatrix} x + \begin{bmatrix} 1 \\ 1 \end{bmatrix} \\
y = \begin{bmatrix} 1 & 0 \end{bmatrix} x
\end{dcases}
\end{split}\]
Compute the poles of (31)
Compute the corresponding transfer function \(G(s)\), its poles and zeros.
Design a state-feedback controller \(u = -Lx + \ell_0 r\) such that the closed-loop poles are \(-2\) and \(-3\).
Find the state-feedback gain \(L\).
Can \(\ell_0\) be chosen such that the steady-state error is \(0\) when \(r\) is constant? Motivate the answer.