Linsys lab forberedelser dag 3

Useful links: Closed-loop Estimator - Luenberger Observer Observability

Problem 1

Considering a state-space formulation of the system

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Where $D=0$. We also have

$$\begin{array}{r}x=\left[\begin{array}{c}p\\ \dot{p}\\ e\\ \dot{e}\\ \dot{\lambda }\end{array}\right]\\ u=\left[\begin{array}{c}{\tilde{V}}_s\\ V_d\end{array}\right]\end{array}$$

Where $\ddot{p}=K_1{V}_d$, $\ddot{e}=K_2{\tilde{V}}_{vs}$, $\ddot{\lambda }=K_{3}p$

$$A=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\\ K_3& 0& 0& 0& 0\end{array}\right]$$

And

$$B=\left[\begin{array}{cc}0& 0\\ 0& K_1\\ 0& 0\\ K_2& 0\\ 0& 0\end{array}\right]$$

Problem 2

Since we want to observe $y=\left[\begin{array}{c}p\\ e\\ \dot{\lambda }\end{array}\right]$, C will be

$$C=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$

Which gives us an observability matrix with rank $5$, which makes the system observable.

Problem 3

Problem 4

Given a state estimator

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