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GATE 2024 EE – Question 22

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 1 mark · Multiple choice

Consider the cascaded system as shown in the figure. Neglecting the faster component of the transient response, which one of the following options is a first-order pole-only approximation such that the steady-state values of the unit step responses of the original and the approximated systems are same?

Cascade: Input → $\dfrac{1}{s+1}$ → $\dfrac{s+40}{s+20}$ → Output

Diagram for GATE 2024 EE question 22
  1. $\dfrac1{s+1}$
  2. $\dfrac2{s+1}$
  3. $\dfrac1{s+20}$
  4. $\dfrac2{s+20}$

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Show answer and explanation

Correct answer: (B) $\dfrac2{s+1}$

Explanation

The overall transfer function is $\dfrac{s+40}{(s+1)(s+20)}$. The dominant (slow) pole is at $-1$; the pole at $-20$ is faster. Dropping the fast factor but keeping the DC gain, $\dfrac{s+40}{s+20}\to\dfrac{40}{20}=2$, which gives $\dfrac2{s+1}$. Its steady-state step-response value is 2, the same as the original ($\dfrac{40}{1\cdot20}=2$).