Exercise 7.2#
Control design ∙ State feedback
Consider the system:
\[\begin{split}
\begin{dcases}
\dot{x} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u \\
y = \begin{bmatrix} 1 & 0 \end{bmatrix} x
\end{dcases}
\end{split}\]
Construct a state observer whose state estimation error decays as \(\varepsilon(t) \propto e^{-10t}\).
Suppose that \(u(t) = 0\), and let \(\hat{y} = C \hat{x}\). Compute the transfer function between \(y\) and \(\hat{y}\).
Estimation error decay rate
Recall that the state estimation error evolves according to \(\dot{\varepsilon} = (A-KC) \varepsilon\).
Then, \(\varepsilon\) decays proportionally to \(e^{-10t}\) if all eigenvalues of \(A-KC\) have \(\Re (\lambda) \leq -10\).