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Let the state-space representation of an LTI system be $x(t)=A x(t)+B u(t), y(t)=Cx(t)+du(t)$ where $A,B,C$ are matrices, $d$ is a scalar, $u(t)$ is the input to the system, and $y(t)$ is its output. Let $B=[0\quad0\quad1]^{T}$ and $d=0$ .Which one of the following options for $A$ and $C$ will ensure that the transfer function of this LTI system is

$$H(s)=\dfrac{1}{s^{3}+3s^{2}+2s+1}?$$

  1. $A=\begin{bmatrix} 0&1&0\\ 0&0&1\\-1&-2&-3 \\\end{bmatrix} \text{and} \quad C=\begin{bmatrix} 1&0&0 \end{bmatrix}$
  2. $A=\begin{bmatrix} 0&1&0\\ 0&0&1\\-3&-2&-1 \\\end{bmatrix} \text{and} \quad C=\begin{bmatrix} 1&0&0 \end{bmatrix}$
  3. $A=\begin{bmatrix} 0&1&0\\ 0&0&1\\-1&-2&-3 \\\end{bmatrix} \text{and} \quad C=\begin{bmatrix} 0&0&1 \end{bmatrix}$
  4. $A=\begin{bmatrix} 0&1&0\\ 0&0&1\\-3&-2&-1 \\\end{bmatrix} \text{and} \quad C=\begin{bmatrix} 0&0&1 \end{bmatrix}$
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