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GATE 2026 EC – Question 41

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

The state and output equations for a control system are: $\dot x=\begin{bmatrix}-4&-1.5\\4&0\end{bmatrix}x+\begin{bmatrix}2\\0\end{bmatrix}u$, $y=[1.5\ \ 0.625]\,x$. Which of the following expressions correctly represents the transfer function $\frac{Y(s)}{U(s)}$ of the system with zero initial conditions?

  1. $\frac{3s}{s^2+4s-6}$
  2. $\frac{3s+5}{s^2+4s-6}$
  3. $\frac{3s+5}{s^2+4s+6}$
  4. $\frac{3s}{s^2+4s+6}$

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

Explanation

$\det(sI-A)=s(s+4)+6=s^2+4s+6$ and $(sI-A)^{-1}B=[2s,\ 8]^T/\det$. Then $C(sI-A)^{-1}B=(1.5\cdot2s+0.625\cdot8)/\det=(3s+5)/(s^2+4s+6)$.