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GATE 2022 EC – Question 50

Control Systems · Transient and Steady-State Analysis of LTI Systems · 2 marks · Multiple select

Two linear time-invariant systems with transfer functions

$$G_1(s)=\frac{10}{s^2+s+1}\quad\text{and}\quad G_2(s)=\frac{10}{s^2+s\sqrt{10}+10}$$

have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?

  1. $y_1(t)$ and $y_2(t)$ have the same percentage peak overshoot.
  2. $y_1(t)$ and $y_2(t)$ have the same steady-state value.
  3. $y_1(t)$ and $y_2(t)$ have the same damped frequency of oscillation.
  4. $y_1(t)$ and $y_2(t)$ have the same 2% settling time.

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Correct answer: (A) $y_1(t)$ and $y_2(t)$ have the same percentage peak overshoot.

Explanation

For $G_1$: $\omega_n=1$ and $\zeta=\frac12$. For $G_2$: $\omega_n=\sqrt{10}$ and $\zeta=\frac{\sqrt{10}}{2\sqrt{10}}=\frac12$. The same $\zeta$ gives the same percentage overshoot (A is true). The DC gains are $10$ and $1$, so the steady-state values differ. The damped frequency $\omega_n\sqrt{1-\zeta^2}$ and the settling time $\frac{4}{\zeta\omega_n}$ both depend on $\omega_n$, which differs, so C and D are false.