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GATE 2017 EE – Question 43

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Multiple choice

The transfer function of the system $Y(s)/U(s)$ whose state-space equations are given below is:

$$\begin{bmatrix}\dot{x}_1(t) \\ \dot{x}_2(t)\end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix}\begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix} + \begin{bmatrix}1 \\ 2\end{bmatrix}u(t)$$

$$y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix}\begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}.$$

  1. $\frac{(s + 2)}{(s^2 - 2s - 2)}$
  2. $\frac{(s - 2)}{(s^2 + s - 4)}$
  3. $\frac{(s - 4)}{(s^2 + s - 4)}$
  4. $\frac{(s + 4)}{(s^2 - s - 4)}$

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Correct answer: (D) $\frac{(s + 4)}{(s^2 - s - 4)}$

Explanation

Use $G(s) = C(sI - A)^{-1}B$. Here $sI - A = \begin{bmatrix} s - 1 & -2 \\ -2 & s \end{bmatrix}$ with determinant $s(s - 1) - 4 = s^2 - s - 4$. The first row of the inverse is $\frac{1}{s^2 - s - 4}[s, 2]$, and multiplying by $B = [1, 2]^T$ gives $\frac{s + 4}{s^2 - s - 4}$.