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GATE 2026 EE – Question 52

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

A system is characterized by the following state equation and output equation ($u$: input, $\mathbf x$: state vector, $y$: output)

$$\dot{\mathbf x}=\begin{bmatrix}a&b\\-a&0\end{bmatrix}\mathbf x+\begin{bmatrix}1\\0\end{bmatrix}u,\qquad y=\begin{bmatrix}1&2\end{bmatrix}\mathbf x$$

What are the values of $a$ and $b$ for which the poles of the transfer function are at $-2+j3$ and $-2-j3$?

  1. $a=4,\ b=3.25$
  2. $a=-4,\ b=3.25$
  3. $a=4,\ b=-3.25$
  4. $a=-4,\ b=-3.25$

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Correct answer: (D) $a=-4,\ b=-3.25$

Explanation

The poles are the eigenvalues of the matrix: $\det(sI-A)=s(s-a)+ab=s^2-as+ab$. The required polynomial is $(s+2)^2+9=s^2+4s+13$. So $-a=4$, i.e. $a=-4$, and $ab=13$, i.e. $b=13/(-4)=-3.25$.