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GATE 2024 IN – Question 17

Control Systems · State-space analysis · 1 mark · Multiple choice

Consider $\dot x=Ax+Bu$, where $x$ is the state vector, $u$ the input, $A$ the system matrix, and $B$ the input matrix. Which matrix $A$ gives a system pole at the origin?

  1. $\begin{bmatrix}0&1\\-2&-3\end{bmatrix}$
  2. $\begin{bmatrix}1&-1.5\\-2&3\end{bmatrix}$
  3. $\begin{bmatrix}1&1.5\\2&-3\end{bmatrix}$
  4. $\begin{bmatrix}0&1\\-2&3\end{bmatrix}$

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Correct answer: (B) $\begin{bmatrix}1&-1.5\\-2&3\end{bmatrix}$

Explanation

A pole at the origin is a zero eigenvalue, requiring $\det A=0$. For option B, $\det A=1(3)-(-1.5)(-2)=0$. Its trace is 4, so its eigenvalues are 0 and 4. The other determinants are nonzero.

Answer: **B**.