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GATE 2026 IN – Question 44

Control Systems · State-space analysis · 2 marks · Multiple choice

A linear time-invariant (LTI) system is described by

$$\begin{bmatrix}\dot x_1\\\dot x_2\end{bmatrix}=\begin{bmatrix}0&1\\-10&-7\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}+\begin{bmatrix}1\\1\end{bmatrix}u,\qquad y=\begin{bmatrix}1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}.$$

The poles of the system are ________.

  1. −2, −5
  2. 2, 5
  3. −2, 5
  4. 2, −5

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Correct answer: (A) −2, −5

Explanation

The poles are the eigenvalues of the state matrix. Its characteristic polynomial is $\det(sI-A)=s(s+7)+10=s^2+7s+10$.

Factorization gives $(s+2)(s+5)=0$, so the poles are $-2$ and $-5$, **A**.