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GATE 2025 EC – Question 53

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

Consider a system where $x_1(t)$, $x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by

$$\frac{d}{dt}\begin{bmatrix}x_1(t)\\x_2(t)\\x_3(t)\end{bmatrix}=\begin{bmatrix}2&0&0\\0&-2&0\\0&0&0\end{bmatrix}\begin{bmatrix}x_1(t)\\x_2(t)\\x_3(t)\end{bmatrix}+\begin{bmatrix}1\\1\\1\end{bmatrix}u(t).$$

Which of the following statements is/are TRUE?

  1. The signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ are bounded for all bounded inputs
  2. There exists a bounded input such that at least one of the signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ is unbounded
  3. There exists a bounded input such that the signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ are all unbounded
  4. The signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ are unbounded for all bounded inputs

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Correct answer: (B) There exists a bounded input such that at least one of the signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ is unbounded

Explanation

The equations decouple: $\dot x_1=2x_1+u$ (unstable pole at +2), $\dot x_2=-2x_2+u$ (stable), $\dot x_3=u$ (integrator). (A) is false: $u=1$ makes $x_1$ and $x_3$ grow without bound. (B) is true for the same reason. (C) is false: $x_2$ is a stable first-order system, so it stays bounded for every bounded input. (D) is false: $u=0$ with zero initial states keeps all states at 0.