The GATE Grind

GATE 2019 EC – Question 41

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

Consider a causal second-order system with the transfer function

$$G(s)=\frac{1}{1+2s+s^2}$$

with a unit-step $R(s)=\frac1s$ as an input. Let $C(s)$ be the corresponding output. The time taken by the system output $c(t)$ to reach 94% of its steady-state value $\lim_{t\to\infty}c(t)$, rounded off to two decimal places, is

  1. 5.25
  2. 4.50
  3. 3.89
  4. 2.81

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) 4.50

Explanation

$G(s)=\frac{1}{(s+1)^2}$ is critically damped, so $c(t)=1-(1+t)e^{-t}$. Setting $c(t)=0.94$ gives $(1+t)e^{-t}=0.06$, which is satisfied at $t\approx4.5$ s ($5.5\times e^{-4.5}=0.061$).