GATE 2021 EC – Question 43
The electrical system shown in the figure converts input source current $i_s(t)$ to output voltage $v_o(t)$.
Current $i_L(t)$ in the inductor and voltage $v_C(t)$ across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., $i_L(0)=0$ and $v_C(0)=0$. The system is
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Correct answer: (D) neither state controllable nor observable
Explanation
The $1\ \text{H}$ inductor is in parallel with a $1\ \Omega$ resistor, so $\dot i_L=i_s-i_L$. The capacitor node gives $\dot v_C=i_s-v_C$ (using $C=1$ F and the $1\ \Omega$ load). In state form $A=-I$, $B=\begin{bmatrix}1\\1\end{bmatrix}$ and $C=\begin{bmatrix}0&1\end{bmatrix}$ (the output is $v_o=v_C$). The controllability matrix $[B\ AB]=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}$ has rank 1, and the observability matrix $\begin{bmatrix}C\\CA\end{bmatrix}=\begin{bmatrix}0&1\\0&-1\end{bmatrix}$ has rank 1. The system is neither controllable nor observable.