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The state variable description of an LTI system is given by

$$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix}=\begin{pmatrix} 0 & a_1 & 0\\ 0 & 0 & a_2\\a_3 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}+\begin{pmatrix} 0\\ 0\\1 \end{pmatrix}u$$

$$y=\begin{pmatrix} 1 &0&0\end{pmatrix}\begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}$$

where $y$ is the output and $u$ is the input. The system is controllable for

  1. $a_1\neq 0,a_2=0,a_3\neq 0$
  2. $a_1=0,a_2\neq0,a_3\neq 0$
  3. $a_1=0,a_2\neq0,a_3=0$
  4. $a_1\neq 0,a_2\neq0,a_3=0$
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