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GATE 2018 EC – Question 39

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

The state equation and the output equation of a control system are given below:

$$\dot{\mathbf x}=\begin{bmatrix}-4&-1.5\\4&0\end{bmatrix}\mathbf x+\begin{bmatrix}2\\0\end{bmatrix}u,\qquad y=[1.5\ \ 0.625]\,\mathbf x.$$

The transfer function representation of the system is

  1. $\dfrac{3s+5}{s^2+4s+6}$
  2. $\dfrac{3s-1.875}{s^2+4s+6}$
  3. $\dfrac{4s+1.5}{s^2+4s+6}$
  4. $\dfrac{6s+5}{s^2+4s+6}$

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Show answer and explanation

Correct answer: (A) $\dfrac{3s+5}{s^2+4s+6}$

Explanation

$(sI-A)=\begin{bmatrix}s+4&1.5\\-4&s\end{bmatrix}$ has determinant $s^2+4s+6$, and its inverse is $\frac{1}{s^2+4s+6}\begin{bmatrix}s&-1.5\\4&s+4\end{bmatrix}$. Multiplying by $B=[2\ 0]^T$ gives $\frac{1}{s^2+4s+6}[2s,\ 8]^T$, and then $C(sI-A)^{-1}B=\frac{1.5(2s)+0.625(8)}{s^2+4s+6}=\frac{3s+5}{s^2+4s+6}$.